An Ancient Greek Treatise on Magic Squares: 72 (Boethius) 3515128522, 9783515128520

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Jacques Sesiano

An Ancient Greek Treatise on Magic Squares

Wissenschaftsgeschichte Franz Steiner Verlag

Boethius | 72

Boethius Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaften Begründet von Joseph Ehrenfried Hofmann, Friedrich Klemm und Bernhard Sticker Weitergeführt von Menso Folkerts Herausgegeben von Richard L. Kremer und Friedrich Steinle Band 72

An Ancient Greek Treatise on Magic Squares Jacques Sesiano

Franz Steiner Verlag

Bibliografische Information der Deutschen Nationalbibliothek: Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar. Dieses Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist unzulässig und strafbar. © Franz Steiner Verlag, Stuttgart 2020 Druck: Memminger MedienCentrum, Memmingen Gedruckt auf säurefreiem, alterungsbeständigem Papier. Printed in Germany. ISBN 978-3-515-12852-0 (Print) ISBN 978-3-515-12856-8 (E-Book)

Carissimae

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Table of contents I. General notions on magic squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Note on the history of magic squares . . . . . . . . . . . . . . . . . . . . . . . 4 II. The ancient work preserved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A. Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Manuscript A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Manuscript D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

B. Brief survey of the treatise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 III. Text and translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Bordered odd-order squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Bordered odd-order squares with separation by parity . . . . . . . . 31 Placing the odd numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Placing the even numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Square of order 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Squares of orders n = 4t + 1, t > 1 . . . . . . . . . . . . . . . . . . . . . . 57 Squares of orders n = 4t + 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Bordered even-order squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Square of order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Bordered squares of orders n = 4k + 2 . . . . . . . . . . . . . . . . . . . 91 Bordered squares of orders n = 4k . . . . . . . . . . . . . . . . . . . . . . 95 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Composite even-order squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Division into equal subsquares . . . . . . . . . . . . . . . . . . . . . . . 125

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Table of contents

Division into unequal parts . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Cross in the middle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 IV. General commentary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Construction of odd-order bordered squares . . . . . . . . . . . . . . . . 165 Odd-order bordered squares with separation by parity . . . . . . . 174 A. Placing the odd numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B. Placing the even numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 183 C. Completing the squares for the three order types . . . . . . . 188 Square of order 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Squares of orders n = 4t + 1, t > 1 . . . . . . . . . . . . . . . . . . . 189 Squares of orders n = 4t + 3 . . . . . . . . . . . . . . . . . . . . . . . . 197 Construction of even-order bordered squares . . . . . . . . . . . . . . . 204 Square of order 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A. Bordered squares of evenly-odd orders (n = 4k + 2) . . . . 205 B. Bordered squares of evenly-even orders (n = 4k) . . . . . . . 209 Construction of even-order composite squares . . . . . . . . . . . . . . 213 A. Division into equal subsquares . . . . . . . . . . . . . . . . . . . . . . 214 B. Division into unequal parts . . . . . . . . . . . . . . . . . . . . . . . . . 215 C. Cross in the middle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Arabic glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Greek glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

I. General notions on magic squares A magic square is a square divided into a square number of cells in which natural numbers, all different, are arranged in such a way that the same sum is found in each horizontal row, each vertical row, and each of the two main diagonals. A square with n cells on each side, thus n2 cells altogether, is said to have the order n. The constant sum to be found in each row is called the magic sum of this square. Usually what is written in a square of order n, thus with n2 cells, are the first n2 natural numbers. Since the sum of all these numbers equals n2 (n2 + 1) , 2 the sum in each row, thus the magic sum for such a square, will be n(n2 + 1) . 2 I. A square displaying this magic sum in the 2n + 2 aforesaid rows is an ordinary magic square (Fig. 1). It meets the minimum number of required conditions, and such a square can be constructed for any given order n ≥ 3 (a magic square of order 2 is not possible with different numbers). Mn =

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II. A bordered magic square is one where removal of its successive borders 6 8 10 7 9 leaves each time a magic square (Fig. 2). With an odd-order square, 11 12each 14 15 13 (odd-order) after removing border in turn, we shall finally reach the smallest possible square, that of order 3. With an even-order square, 16 17 18 19 20 that will be one of order 4 (its border cannot be removed since there is no 21

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General notions on magic squares

magic square of order 2). For order n ≥ 5, bordered squares are always possible. 16

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As seen above, the magic sum for a square of order n filled with the n2 first natural numbers is n(n2 + 1) Mn = . 2 2

Clearly, the average sum in each case is n 2+1 . Accordingly, for m cells, the average sum should be m times that quantity; this will be called the sum due for m cells. Thus, the inner square of order m (m ≥ 5) within a bordered square of order n must contain in each row its sum due, namely m(n2 + 1) 2 if the main square is filled with the n2 first natural numbers. The sum in a row in one border will therefore differ from the next by n2 +1. From this it follows that pairs of elements of the same border which are horizontally and vertically (diagonally for the corner cells) opposite add up to n2 + 1. Such pairs are what we call complements: to each ‘small’ number a (less 2 than n 2+1 ) is associated the ‘large’ number (n2 + 1) − a. For example, in each border of the above figure the sum of opposite elements is 82, while 41 belongs to the centre and 1, . . . , 40 is the set of small numbers. We therefore infer that, for a bordered square of odd order filled with the first 2 natural numbers, the central element must be n 2+1 (the median). The case of even-order squares is similar, except that there are two median 2 2 numbers, n2 and n2 + 1, to be placed in the central 4 × 4 square, the 2 small numbers being those less than (or equal to) n2 . Mn(m) =

General notions on magic squares

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A more particular kind of magic square is that of composite squares: the main square comprises subsquares which, taken individually, are also magic (Fig. 3). The possibility of such an arrangement depends on the divisibility of the order of the main square.

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There exist ways of directly constructing a square of given order and given type, that is, belonging to one of the above two kinds: once the empty square is drawn, a few, easily remembered instructions will enable us to place the sequence of consecutive numbers without any computation or recourse to trial and error. These are called general construction methods. Now there exists no general method uniformly applicable to any magic square. Indeed, general methods are applicable to, at most, one of three categories of order, which are: — The squares of odd orders, also called odd squares, thus with n = 2k+1, of which the smallest is the square of order 3. — The squares of evenly-even orders, also called evenly-even squares, thus with n = 4k, of which the smallest is the square of order 4. — The squares of evenly-odd orders, also called evenly-odd squares, thus with n = 4k + 2, of which the smallest is the square of order 6. These methods are, by definition, applicable whatever the size of the order; but they may require some adapting for squares of lower orders, like 3 or 4, sometimes also 6 and 8. Remark. Speaking of different methods suggests that a square of given order may take different aspects. As a matter of fact, the number of possible configurations (excluding mere inversions and rotations of the square) rapidly increases with the size of the order. Whereas there is just one form of the magic square of order 3, there are already 880 for the order 4, as discovered in 1693.1

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By Frénicle de Bessy in his Table generale des quarrez de quatre.

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General notions on magic squares

Note on the history of magic squares The origin of the science of magic squares is unknown. It is commonly said, and generally accepted, that the earliest magic square appeared in China at the beginning of our era. Now, first, it is the square of order 3, and, second, higher-order squares involving general construction methods do not occur in China before the 12th century, and are clearly of Arabic or Persian origin. For there are from the tenth century onwards numerous texts in Arabic, later also in Persian, displaying a continuous development with earlier methods being extended or new ones invented. Now trying to fix the starting point of this development led to a major problem: two tenth-century Arabic texts are preserved, one by a competent mathematician, Ab¯ u’l-Waf¯a’ B¯ uzj¯an¯ı (940-997/8) and the other, with only a part on magic squares, by a less competent one, ‘Al¯ı ibn Ah.mad al-Ant.¯ ak¯ı (d. 987). Their edition, though, shows that their state of knowledge was just in reverse proportion: whereas B¯ uzj¯an¯ı seemed to write the first steps in the construction of magic squares, and had only one general method, namely for constructing bordered odd-order squares, the second had general methods for all three types of bordered squares and applied them to quite elaborate cases. As the present author wrote in the edition of these texts, the tenth century, for the (mediaeval) history of magic squares, gives the impression of being both a beginning and an end.2 The discovery of the text studied here, a translation of an anonymous Greek text, brought the solution: Ant.¯ ak¯ı merely reproduced it, and, since it does not, for the case of bordered squares, give the foundation of these general methods, B¯ uzj¯ an¯ı attempted to do so (succeeding only in the case of the odd orders). As to the question of the first studies on magic squares, it is still unresolved, for those have yet to come to light — and probably never will. Our text does show that already in antiquity elaborate methods were being invented. But it leaves us in ignorance of the earlier history. Besides being anonymous, it does not refer to any person or treatise. We are only told about the existence of ways to construct ordinary magic squares, moreover not considered by our text (see its § 2, or below, p. 165). In short, apart from this isolated treatise and this scarce information, we know nothing about the studies of magic squares in Greek antiquity. Returning to Arabic times, we do see continuous development in the 11th and early 12th centuries, with the discovery of various constructions for ordinary magic squares. B¯ uzj¯ an¯ı spares no effort in attempting to 2

Magic squares in the tenth century, p. 8.

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General notions on magic squares

construct a few ordinary squares of small orders by individual methods, but without finding any general method. Indeed, general methods only begin to appear in the 11th century, first for odd and evenly-even orders; here the contribution of Ibn al-Haytham (ca. 965–1041) has proved to be essential. He considered the properties of the natural squares, that is, of the squares filled with the natural numbers taken in succession, and found that their main diagonals give the magic sum for the magic square of the same order while the sums displayed in opposite (horizontal and vertical) rows differ from the magic sum by equal amounts, but with opposite sign. Thus, rather than considering the squares by individual order, he sought (and found) ways to compensate the differences with exchanges between opposite rows. The less simple case of evenly-odd orders was finally solved towards the end of the 11th century.3 Squares filled with non-consecutive numbers also began to appear at that time. The origin of that has to do with the association of Arabic letters with numerical values — an adaptation of the Greek numerical system (Fig. 4); this adaptation appeared in early Islamic times, before the adoption of Indian numerals, but remained in use later. Thus to the letters of a word or the words of a sentence can be associated a set of numbers. So, assuming that the word or sentence is written in one row of a square (of which it thus determines the order), the task was then to complete the square numerically so that it would display in each row the sum in question — a mathematically interesting problem since this is not always possible. g

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Fig. 5 About this development, in particular how general methods were discovered, and contributions of Ibn al-Haytham and his successors, see our Magic squares, their history and construction from ancient times to AD 1600 (hereafter simply ‘Magic squares’), pp. 25–29, 51–56, 88–93; or earlier editions: Les carrés magiques, pp. 25–28, 49–51, 85–89; , pp. 33–37, 58–61, 96–100. 3

N xki N N

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General notions on magic squares

Whether original or inspired by earlier texts, quite an elaborate study on the construction of such squares with a set of given numbers not in arithmetical progression appeared already in the early 11th century. We do not know the author’s name, but we do at least know his motivation: as he tells us in the introduction, he devoted himself to the study of magic squares in order to find relief from worries and preoccupations.4 This, apparently efficient, remedy ultimately resulted in a treatise in which, after teaching various methods for the construction of ordinary squares of odd and evenly-even orders and bordered squares of all three types, he constructs squares of orders 3 to 8 with a given sequence of numbers in a row, these numbers corresponding to sacred names or sentences. He makes no allusion to constructing amulets. But any reader might have thought about such a use. General studies did not stop after the 11th century, although they tended to consist in finding other, simpler or more elegant methods of constructing squares. The treatises also became more numerous. This meant, however, that they less demanded of readers, the main concern being to teach methods without going into the mathematical background. This reached such a point that, as the 13th-century Persian author ‘Abd al-Wahh¯ab Zanj¯ an¯ı tells us, he had to write, at the request of some friends, a treatise that was shorter than he would have wished, for they merely wanted to have practical rules for constructing magic squares of any order.5 And such indeed is the characteristic of later treatises: written in response to public demand, they were to teach methods aiming at results, without bothering the reader with questions of foundation or feasibility. Meanwhile, however, popular use of magic squares as amulets grew steadily. Authors then followed this trend: many late, shorter texts are just not interested in teaching the construction of magic squares — to say nothing of their mathematical foundation. They give the figures of a few magic squares, most commonly squares of the orders 3 to 9 associated with the seven then known planets (including Moon and Sun) of which they embodied the respective, good or evil, qualities — abundantly described and commented in these texts. The reader is taught on what material and when he is to draw each of such squares; for both the nature of the material and the astrologically predetermined time of drawing are presumed to increase the square’s efficacy. It merely remains to put this object in the vicinity of the chosen person, beneficiary or victim. This must have been of great use in solving personal or business problems. 4 5

See our Un traité médiéval, pp. 21 & (Arabic) 208. See our Herstellungsverfahren II, II 0 .

General notions on magic squares

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Of such kind were the first Arabic texts translated into Latin in 14thcentury Spain. A characteristic example of theory and application is the 5 × 5 square, attributed to Mars, which uses its (mostly unfavourable) properties.6 The figure of Mars when unfavourable means war and exactions. It is a square figure, five by five, with 65 on each side. If you wish to operate with it, take a copper plate in the day and hour of Mars when Mars is decreasing in size and brightness, or malefic and retrograding, or in any way unfavourable, and engrave the plate with this figure; and you will fumigate it with the excrement of mice or cats. If you place it in an unfinished building, it will never be completed. If you place it in the seat of a prelate, he will suffer daily harm and misfortune. If you place it in the shop of a merchant, it will be wholly destroyed. If you make this plate with the names of two merchants and bury it in the house of one of them, hatred and hostility will come between them. If you happen to fear the king or some powerful person, or enemies, or have to appear before a judge or a court of justice, engrave this figure as said above when Mars is favourable, in direct motion, increasing in size and brightness; fumigate it with one drachma (= 81 ounce) of carnelian stone. If you put this plate in a piece of red silk and carry it with you, you will win in court and against your enemies in war, for they will flee at the sight of you, fear you and treat you with deference. If you place it upon the leg of a woman,7 she will suffer from a continuous blood flow. If you write it on parchment on the day and the hour of Mars and fumigate it with birthwort and place it in a hive, the bees will all fly away. It was thus the arrival of such texts in late mediaeval Europe which first aroused interest in, and later led to the study of, such squares there. This incidentally explains the use of the term ‘magic’ — formerly also ‘planetary’, which we find still employed by Fermat.8 One of the two 4×4 squares thus transmitted is that used by Dürer in his Melencolia in order to date it (Fig. 5; 15 14 occupy the two median cells on the bottom). As to the mediaeval Arabic (perhaps originally Greek) denomination, ‘Harmonious arrangement of numbers’ (wafq al-a‘d¯ ad), which had a more mathematical connotation, it remained unknown, as well as the various constructions described in Arabic and Persian manuscripts. 6

This and other examples in our Magic squares for daily life. The magic square in question is that of our figure 19, p. 174 below. 7 Reverting to evil uses. 8 Varia opera mathematica, p. 176; or Œuvres complètes, II, p. 194.

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General notions on magic squares

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The earlier transmission towards the East was more fruitful. India 36 examples 23 64 63 50 10 sometimes 9 77 37also conand China received many more of squares, struction methods. In the Byzantine Empire, a treatise was written at the 11 61 51 38 7 78 65 34 24 very beginning of the 14th century by Manuel Moschopoulos. From a set 40 from 3 some 80 67 30 26 13 57 manuscript, 53 of examples obviously received Persian or Arabic he reconstructed a few methods squares of odd 27 for 68 ordinary 28 54 14 55 81 41 and 1 evenlyeven orders. (He obviously does not know anything about the existence 56 52 15 2 79 42 29 25 69 of magic squares in ancient Greece.) 4 75 44 21 71 century 58 48 did 17 just the European scholars of the fifteenth and31sixteenth same with the squares received, so these initial studies met limited suc72 32 19 18 59 46 45 5 73 cess. A notable exception is Michael Stifel (1487–1567), who gives in9 he 47 16 60 74 43 20 70 33 does not structions for constructing bordered squares of all6 orders; mention any source, nor does he give himself any credit, and his methods are not exactly like the ones we know from Arabic sources.

European research had thus to start afresh. A return to studying their mathematical foundations was initiated by prominent mathematicians like Fermat and Euler, and this helped to dissociate these squares from their unfavourable reputation. Methods of construction were discovered, sometimes rediscovered. New categories were introduced later, such as the ‘bimagic’ or ‘trimagic’ square (one remaining magic when its numbers are replaced by their squares or cubes). Pandiagonal squares, with both main and broken diagonals magic, received greater attention than before.10 A recurrent problem was that of the number of given possibilities for a given order larger than 4 (above, p. 3, Remark), closely linked to that of classifiying magic squares by category — which is still the subject of contemporary research. 9 See his Arithmetica integra, fol. 24v – 30r ; or Magic squares, pp. 149–150 & 154, 161–163 & 166–167, 170–171 & 173 (Les carrés magiques, pp. 125–126 & 130, 137– 139 & 141–142, 146 & 148–149; , pp. 136 & 141, 148–150 & 153–154, 157–158 & 160–161). 10 One example below (p. 204). Other Arabic examples are given in our Magic N squares, see index there. xki N N k N

II. The ancient work preserved A. Manuscripts Manuscript A

The University of Ankara, or, more precisely, its Faculty of Language, History and Geography (Dil ve Tarih-Coˇgrafya Fakültesi), received a donation from a well-read book collector, Ismail Saib Sencer (d. 1940), of more than ten thousand manuscripts divided into two collections. Among the first, one finds the manuscript I, 5311, of 84 leaves, written on paper in the 13th century (7th century of the hegira). It contains ten treatises, all on mathematical subjects (geometry, algebra, magic squares), and copied by the same hand. The first of these (fol. 1r – 36r ) belongs to a commentary on Nicomachos’ Introduction to arithmetic by Ant.¯ak¯ı (above, p. 4). The text preserved, though, is not the whole of that commentary. For, as indicated by the title, that is only its third book, thus the third part of it, and also the last since the colophon clearly indicates that the work ends there. The writing (naskh¯ı) is quite legible, except in some worm-eaten places. There are between 18 and 24 lines on a page, mostly 20 or 21. The copyist sometimes used red ink, namely for some headings, numerical symbols and framing the figures; he did it later on, and that is why one correction is in red (l. 484 in this edition of the Arabic text). The copy is on the whole in good condition, and the copyist is responsible for only a few textual omissions which he partly or completely remedied upon rereading. We may, though, remark that there is some disorder in the Arabic text on the leaves 18 and 19, and that only half of fol. 18r contains text. It appears that there has been an error in an earlier copy, just reproduced by the present copyist, together with, in red ink, the explanation (fol. 18v , see p. 90, note to § 39 of the Arabic text): This part is left blank for the figure, where K is (written) in red on the left-hand page; now the place referred to is not found in the present copy. Nicomachos of Gerasa is a relatively late Greek mathematician (end of the 1st century). His Introduction to arithmetic (᾿Αριθμητικὴ εἰσαγωγή) is extant in Greek and was widely known during late antiquity; some Greek commentaries on it have also survived. The Introduction had a notable influence later as well. It was first translated into Syriac, then into Arabic in the 9th century by Th¯ abit ibn Qurra, and its modern edition has shown that the Arabic translation is quite faithful to the Greek original. The

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The ancient work preserved

rôle of the Introduction in mediaeval Europe is important as well. There it was, until the 12th century, the only surviving work of ancient Greek mathematics, through an adaptation by Boëtius (who, in the early 6th century partly summarized the original text and added remarks). In short, by means of Nicomachos’ book, ancient Greek, Byzantine, Arabic and mediaeval European readers could learn some basic number theory. Indeed, according to its Greek meaning, arithmetic has to do with what we would call today number theory, and not arithmetical calculations. This notoriety does not go, though, hand in hand with the mathematical level. It is in no way comparable to that of classical authors like Euclid (around −300) and Archimedes (around −250) — or also Diophantos (who, unlike the other two, lived after Nicomachos). With Nicomachos the science of mathematics seems to regress: an opinion becomes a theorem and a test serves as a demonstration. In short, Nicomachos’ Introduction, though undoubtedly of use at the time to anyone wishing to acquire elementary notions of number theory, must have disappointed others expecting to be trained in mathematical thinking. It embodies perfectly what would be called today ‘democratization of knowledge’. Book III of Ant.¯ ak¯ı’s Commentary consists of three distinct parts without any connection between them.11 The first part consists of an enumeration of definitions, statements of propositions and identities. The definitions and propositions are chiefly taken from Euclid’s Elements, but only those relating to elementary number theory. There is also a set of arithmetical identities, some of which are taken from the algebraic books of al-Khw¯arizm¯ı, c. 820, and Ab¯ u K¯amil, c. 880. This first part at times leaves us in doubt as to Ant.¯ak¯ı’s competence. Certain formulations are strange or nonsensical, incomplete or imprecise; definitions of terms do not appear together with the first use of these terms or are omitted; a demonstration is wrong, as well as two assertions. The second part is on the construction of magic squares. It is just copied from the translation of our ancient text, but without any mention of it, as if the author were Ant.¯ ak¯ı himself.

The third subject is that of hidden numbers, that is, of guessing a number thought of by someone else. The latter is asked to mentally perform various computations with that number and communicate the result. The asker will then be able to work out what that number was.

11 The reader interested in a detailed analysis is referred to its edition in Magic squares in the tenth century.

The ancient work preserved

11

Comparing now the content of Nicomachos’ work with that of Ant.¯ak¯ı’s Commentary, we make the surprising discovery that there is no connection between the two works — at least not in Book III of said Commentary since that is all we know. For the first part, Euclid and not Nicomachos would rightly be indicated as the source for most propositions. The second part, on magic squares, which reproduces our text, has no connection whatsoever with Nicomachos.12 Finally, although we guess from later sources that problems on hidden numbers existed in antiquity,13 they have nothing to do with Nicomachos’ Introduction. In short, Ant.¯ak¯ı must have taken the essence of these subjects from other sources, as, by the way, his often weak mathematical knowledge obliges him to, and probably without much change, as his literal copying of the part on magic squares suggests. Manuscript D

Manuscript D belongs to the Delhi collection preserved since 1876 in the India Office Library in London. The Delhi collection represents what remained in 1858 of the Mughal Imperial Library after parts had been given away, sold or seized during the 18th and 19th centuries. It consists of over 3500 volumes of manuscripts mainly in Arabic, Persian and Urdu. Of these, manuscript Delhi Arabic 110 contains two unrelated works, the v r first being the Nukhbat al-fikr by Ibn H . ajar al-‘Asqal¯an¯ı (fol. 2 – 27 ), a known expert on religious science (early 15th c.), and the other (fol. 28r – 119v ), with the title Kit¯ ab d¯ıw¯ an al-‘adad al-wafq, a compilation on the science of magic squares going back to the second half of the 12th century. This last copy (now made available on line by the Qatar National Library) is a beautiful item, carefully copied and with the use of several colors both for titles and subtitles and to distinguish the parts of the magic squares represented. We do not know, however, the author of the original text, for both the beginning and end leaves are missing. It is divided into eight ‘books’ (maq¯ al¯ at), covering the following subjects (fol. 28r – 29r contain part of the introduction and a — not very accurate — table of contents): Book Book Book Book Book 12

I (fol. 29v – 35v ): Preliminaries II (fol. 35v – 58v ): Ordinary magic squares III (fol. 58v – 80v ): Bordered magic squares IV (fol. 80v – 93v ): Curiosities V (fol. 93v – 98v ): Composite squares with equal subsquares

Although a 3×3 magic square is reported to have been in Nicomachos’ Introduction (see below, p. 163), that is not the case. 13 Magic squares in the tenth century, pp. 15 and 185 seqq.

12

The ancient work preserved

Book VI (fol. 99r – 109v ): Composite squares with unequal parts Book VII (fol. 109v – 116v ): Magic cubes14 Book VIII (fol. 116v – 119v ): Magic triangles. The borrowings are indeed from various early Islamic mathematicians, such as Ab¯ u’l-Waf¯ a’ B¯ uzj¯ an¯ı (940–997/8; above, p. 4), Ibn al-Haytham (ca. 965–1041; above, p. 5), Ab¯ u H atim Muz.affar Asfiz¯ar¯ı (d. before .¯ 1121/2), and his contemporaries ‘Abdarrah.m¯an al-Kh¯azin¯ı and ‘Umar Khayy¯am.15 Finally the author refers to a person variously named as alMufad.d.al ibn Th¯ abit al-H an¯ı (60v ; see below, p. 16n); al-Mufad.d.al ibn . arr¯ ar v Th¯abit ibn Qurra (28 , 80 ; p. 30n); al-Mufad.d.al ibn Th¯abit (94r , 95v , 102r , 107v ; pp. 124, 152n, 154n). From his name it appears that he was a Sabean, thus one of the group of persons educated in Greek culture and religion which produced numerous excellent translators of Greek works in the 9th century. He is doubtless related to the known translator Th¯abit ibn Qurra (836–901), perhaps his son. In the edition of Ant.¯ ak¯ı’s text, the present writer, considering the weakness of some of Ant.¯ ak¯ı’s work and the high level of the part on magic squares, observed that this latter part suggested a much earlier, possibly Greek time for the first discoveries.16 This is fully confirmed by MS. D, which reproduces al-Mufad.d.al ibn Th¯abit’s introduction, omitted in Ant.¯ak¯ı’s copy, which attests that we have here the translation of an ancient treatise. MS. D also reports examples of constructions of larger squares being alluded to in the translation but mostly omitted by Ant.¯ak¯ı. Thus, although lacking two main parts of our treatise (see below, pp. 13– 14), MS. D does complete substantially the text transmitted by Ant.¯ak¯ı. Since al-Mufad.d.al ibn Th¯ abit mentions in the introduction to his translation late ninth-century scholars as persons of former times, while his translation is copied by Ant.¯ ak¯ı in the middle of the tenth century and commented during its second half, that translation must go back to the early tenth century. Remark. Since a work on magic squares (Ris¯ ala fi’l-‘adad al-wafq) is attributed by Ibn al-Qift.¯ı and Ibn ab¯ı Us.aybi‘a (both 13th c.) to Th¯abit ibn Qurra,17 we first thought he might be the translator and so attributed it to him in our Magic squares. But it now seems evident 14

On this and the next particular subjects, see our Magic squares, pp. 271–273, 280. On their works on magic squares, see Magic squares, index. 16 Magic squares in the tenth centuries, p. 9. 17 See A. Müller’s edition of Ibn ab¯ı Us.aybi‘a’s work, I, p. 220, or J. Lippert’s edition of Ibn al-Qift.¯ı’s, p. 119. 15

The ancient work preserved

13

that he has not written such a work (our translator was not aware of any Arabic writing on this subject, see his introduction) and that the work referred to by Ibn al-Qift.¯ı and Ibn ab¯ı Us.aybi‘a must be our ancient text’s translation.

B. Brief survey of the treatise After the translator’s introduction (§ 0), we find a few generalities on magic squares (§§ 1–2). Then follow the four parts of our treatise. Part I (§§ 3–6). The first part is on the construction of bordered squares of odd orders. The author explains, but without justification, how to place each number in each border considered successively. A 12th-century reader who had read several treatises on magic squares, and knew Ant.¯ak¯ı’s treatise (but not its source), expresses his opinion as follows: As for other (authors), among whom al-Ant.¯ ak¯ı, I found that they (merely) said to put such and such a number in such and such a cell without explaining the reason for that, although this subject is difficult to understand and requires to be memorized.18 This allusion to the tedious construction of usual bordered squares indeed confirms the impression left by our text. Furthermore, not only does it list all the places of smaller numbers but also those of their complements, which is superfluous since their places are ipso facto known. To avoid such verbosity a later author, Kh¯azin¯ı (mentioned above, p. 12), represented a single odd-order square of seventh order with the places of the small numbers in its various borders (below, pp. 16n for the Arabic reference and 173 for the square). This being indeed sufficient, the author of the compilation preserved in MS. D decided to omit this part, presenting Kh¯ azin¯ı’s table instead. He did the same for Part III, on the construction of even-order squares, for the same reason. Part II (§§ 7–35). This is by far the longest and most difficult part of the work. The text of it is preserved in both manuscripts (for D, in Book IV), while for the examples of larger squares D completes A. We are to construct bordered squares of odd orders displaying separation by parity, with the odd numbers being grouped together in a rhomb (actually an oblique square) within the main square. This second part is doubtless one of the finest pieces of ancient mathematics. The construction presented is not only most subtle but is, unlike 18

See our Une compilation arabe, p. 163, lines 239–241 of the Arabic text.

14

The ancient work preserved

the previous part, justified: every step of the intricate placing of the surrounding even numbers is explained, and we are never left in doubt about the foundation of the procedure. Obviously, this was a topic for advanced readers, beyond the reach of most, and the previous part was perhaps just a preparation for it since these constructions are required for placing the odd numbers. Part III (§§ 36–43). Here we are told how to fill bordered squares of both even orders (divisible by 4 or by 2 only). This can be explained more succinctly than for odd-order squares, and the explanations are clear. This part is, again, omitted in MS. D since Kh¯ azin¯ı gave instead a table, including the two types of even orders (see p. 212). Part IV (§§ 44–54). Just as Part I was preparatory to Part II, so Part III paves the way to the last section. We learn how to construct composite squares for even orders, divided either into identical or different even-order subsquares, or also rectangular parts with both dimensions even (and not meeting the diagonals); for evenly-odd orders, a central cross (thus meeting the diagonal) separates the subsquares. Here too, the placing is clearly explained and justified. This may have been the second main purpose of our treatise, also intended for advanced readers. As for the manuscripts, the situation is the same as for the second part: the text is reproduced in both (for D, in Books V–VI), but A omits most of the figures, in this case a sizable set.

III. Text and translation Besides explanatory footnotes, the translation contains additional intermediate (incomplete) figures and tables to aid the reader. (A more complete analysis will be found in the General commentary, pp. 163–219 below). Round brackets mark our additions to the text, used mainly in the translation to clarify it, while square brackets, employed both in the Arabic text and the translation, enclose interpolations. As to the Arabic text, each section is located in the manuscripts (in addition, for A, in the edition) and supplemented with a critical apparatus.

16

Text and translation

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Text and translation

17

(§ 0. Translator’s introduction) My first acquaintance with the subject of magic squares was the figure of (order) three mentioned by Nicomachos in the Arithmetic.19 Then Ab¯ u al-Q¯asim al-H az¯ı came across a figure of four, (with the numbers) . ij¯ beginning with 1 and ending with 16 with constant difference 1, and was filled with admiration. Next I found a figure of six on the back of Euclid’s book (of the Elements) in the handwriting of Ish.¯aq ibn H . unayn. Afterwards I came across a book containing three or four figures, all less than (the order) ten. Then I found in the Library, among the books of the caliphs’ collection, two books, for the greater part damaged by termites, so that one could understand just little of them; the summary note about them was in al-M¯ ah¯ an¯ı’s handwriting and the first page for the most part in H us¯ a al-Nawbakht¯ı’s handwriting. Then I examined . usayn ibn M¯ them, found elucidating them very arduous, (but) it occurred to me that it might be possible to make sense of those parts which had been damaged in one by what had been preserved in the other and to restore the proper meaning by replacing a word by another until the account was correct. (§ 1. Definition of a magic square) The treatment for this kind consists in drawing square areas in a quantity (equal) to the product of an (integral) number by itself, in which one will put numbers from 1 to the quantity of these (small) squares so that hthe content ini [ the sum of these ] [ that is, the numbers ] each row of them is concordant with the content in (any) other row.20 (§ 2. Two ways of constructing magic squares) Some people begin by placing these numbers according to the succession of the natural order, from 1 to the number of squares in the figure where they wish to construct the magic square.21 Then they move its 19

About the authors mentioned here, see Commentary, pp. 163–164. All except Nicomachos are 9th-century scholars, well known from Ibn al-Nad¯ım’s Fihrist, a biobibliographical work written towards the end of the 10th century. 20 In an earlier copy the lacuna of the text (in angular brackets) has been filled with two glosses (in square brackets), which specify that the sum of the numbers is meant; the second gloss clarifies the first. Note that it is not explicitly stated that the constant sum will also be found in the two main diagonals. 21 The natural square is a square of the same order as that of the square to be constructed, filled with the natural numbers taken in order (above, p. 5).

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19

numbers — (which are) always in excess in some rows and in deficit in the rows opposite — and arrange all the rows in a certain manner. This is a method presenting difficulty for the beginner. Other people proceed with that in a different, and easier, way.22

(Bordered odd-order squares) For the figure of odd (order, they do it) as follows. (§ 3. Construction of the bordered square of order 3)23 144 Science of the magic square They put the median number of these numbers [ which the placing has reached ]24 in the centre of the square. (Next) they place, diagonally to its 25 above, to its left,213 the (above,) thepreceding number preceding it and, diagonally opposite left, number it and, diagonally opposite to it, the to it, the number following it. Then they put the small number preceding number following it. Then they put the number preceding that smaller that small number it, underneath it, and to it the large number number underneath and opposite to it opposite the number following that larger following that large number, (namely) above the number which follows number, (namely) above the number which follows that median number. that median number. Then they place the preceding Then they place the number preceding the small secondnumber (smaller number)the in second (small number) corner in the cell right-hand corner cell opposite it the (upper) right-hand and, opposite to itand, in the corner,tothe in the corner, thethe large number following theNext second large number. number following second larger number. they place in the Next (cell) they place in the (cell) following (that of) this large number small following (that of) this larger number the smaller number whichthe precedes number precedestothe third, and opposite it the large number the third,which and opposite it the number followingtothis third larger numfollowing this third large number. With this, the square of 3 by have 3 is ber. With this, the square of three by three is completed, and it will completed, and it will have the following aspect. the following aspect. 4

9

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Fig. n = a31 214 If the square is 5 by 5 (Fig. a 2), you put in the central cell 22 (A.II.4) By constructing successive borders, the description of which follows. ‘Some people’ (.the . . ) ‘other people’ seemsofto 25, be τινὲς μὲν (.13. . . )215 τινὲςThen δὲ. you place the number median number namely 23 Some words or expressions constantly used in constructing bordered squares must preceding 13, thus 12, in the cell (next) diagonally, where you had placed now be defined. If the numbers to be placed are 1, 2, . . . , n2 , with n odd, the median 4 in the square2 of 3; opposite to it diagonally, you place [ this figure ] the number will be n 2+1 , those preceding it are the small(er) numbers, the others the number following 13.216 Next you place the number preceding 12, thus large(r) ones. To each ‘small’ number a corresponds a ‘large’ number n2 + 1 − a, 11,complement. where you (Remember had placed that 3 in inthis figure,squares, and younumbers place opposite to it the its bordered complementing one large number 14, same thus border, 15, in the place horizontally of 7 in thisor, figure. another face eachfollowing other in the vertically, in theYou case ofplace cornerthe cells, diagonally.) In the11subsequent description, the numbers arecell placed number preceding in the (upper) right-hand corner of beginning with the median in the central cell, then — with the small numbers taken the inner square of 3, and you place opposite to it, in the cell where 8 had in descending sequence and thus their complements in ascending sequence — in the been put, the large number following 15, thus 16. Then you place in the successive borders. 24 following the (upper) right-hand corner [ thus 10 ] the large number cell This gloss alludes to the natural square (§ 2), although we have finished with it. 25 Actually, here and17, throughout, left of the following 16, thus then opposite to itreader. the number following 10 [ which 217 precedes it ] , thus 9. With this treatment, the square in the centre of 213

Here and throughout: to the left of the reader. Construction of the magic square of order 5, first the inner 3 × 3 square. 215 Whereas the previous figure was inserted at the appropriate place in the text, the

214

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Text and translation

(§ 4. Construction of the bordered square of order 5) If the square is five by five, you put in the central cell the median number of 25, namely 13.26 Then you place the number preceding 13, thus 12, in the cell (next) diagonally, where you had placed 4 in the square of three; opposite to it diagonally, you place [ this figure ] the number following 13.27 Next you place the number preceding 12, thus 11, where you had placed 3 [ in this figure ], and you place opposite to it the number following 14, thus 15, in the place of 7 [ in this figure ]. You place the number preceding 11 in the (upper) right-hand corner cell of the inner square of three, and you place opposite to it, in the cell where 8 had been put, the number following 15, thus 16. Then you place in the cell following the (upper) right-hand corner [ thus 10 ] the number Translation of Text A 10 [ which 145 following 16, thus 17, then opposite to it the number next to precedes it ]28 , thus 9. With this treatment, the square in the centre of the square of five five is and there and remain, the square, the by square ofcompleted, 5 by 5 is completed, therefrom remain, from the square, 16 16 (empty) cells and, from (available) numbers. numbers. (empty) cells the and,numbers, from the 16 numbers, 16 (available)

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Fig. a 2 n=5

8 23 24place 4 You then

the number preceding 9, thus 8, in the upper cell next You then put the left-hand number preceding 9, above thus 8,12, in the to to the leftto7 the corner cell, andupper 18 in cell the next cell next 12 17 10 19 the left-hand corner cell, above 12, and 18 in the cell next to the left-hand 218 hand corner in the lower row, underneath 16. You put 7 in the cell next 11row, 13 underneath 15 21 corner in the lower put 7 in it, thenext cellto next the to5 the left-hand corner 16. cell,You underneath 12, to and 19 opposite to left-hand corner cell, underneath it, next to 12, and 19 in the opposite cell it, row, next to 10. You put 6 in the upper left-hand 25 in16the9 right-hand 14 1 of the right-hand row,above next to put 6 in upper left-hand corner, corner, 7, 10. and,You opposite to the it, in the (other) corner of the diagonal, 22

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33

43

You put in the middle left-hand cell,place underneath 7, the and you put 21 Whereas the20. previous figure5 was inserted at the appropriate in the text, in the opposite the right-hand side, underneath others (all but one: § 7) cell appear togethertoatit, theon end of the main chapters. We shall put 19. You put some of them where add some upper intermediate results (incomplete 4 inthey the belong, cell ofand thealso right-hand corner, above 19, and opposite to figures), as well as tables when, later on, series of numerical quantities arecell, mentioned. it diagonally, 22. You put 3 in the lower middle underneath 9, and 27 8 12 words 10 45 48 6 The bracketed had,46presumably, once been written by a reader below the opposite to it in the upper row 23, above 17. You put 2 in the second figure for n = 3 and came to be incorporated in the text here. His purpose was 11 cell 18 219 20 of35the36lower 16 row, 39 underneath 14, and 24 opposite to it in the upper to specify to what the (also added) allusions ‘in this figure’ found twice below were referring. 9 row, 19 24next 29 to22 4. 31You 41 put the remaining (number) 1 in the second cell 28 following (verb tal¯ a )right-hand is indeed used both for the de-the left. The of the row, above 20,ascending and 25 sequence oppositeand to the it on 23 25by 27 33 43 whether it is smaller scending one,7 but17specified indicating or larger than the pre220 square of 5 by 5 is (now) completed. ceding one (we omit this in the translation since we use ‘preceding’ and ‘following’); 47 37 28 21 26 13 3 there was no specification here, whence the reader’s addition. 26

22

Text and translation

55 55 55 55

60 60 60 60

( 4 ) A: 10r ,5 - 10v ,11 (ed. ll . 453-477). 32

cod. || 32

]

( 4 ) A: 10 ,5 -]10 ,11 (ed.(pro ll . 453-477). ) r

32 || 36 cod. 65 65 65 65

cod.||||3936 cod.

cod. ] || 38

]

44 47

70 268 262

70 70

6070149

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

64

171

]

cod. || 33

cod. || 34 ] ]

]

cod. cod. |||| 33 37

cod.cod. || 38|| 34

]]

cod. || 37 ] ] cod. || 41

cod. ]|| 43 cod. || 38]

5) cod. || (33-4

]

cod. ||] 43

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cod. || 34

]]

(5)

]

cod. cod. |||| 33-4 38

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cod. || 39 ]

cod. ] ||

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cod. ||] 47 (sic) cod. ] cod. 49 ] ]cod. || 46 cod. || 49 ] add. supra ] || 46 cod. || 50 ] || 50|| r (47 4 ) A: 10 ,5cod. - 10v || ,1148(ed. ll . 453-477). ] ] (sic) cod. || 48 ] cod. ] cod. || 51 ] cod. || 51 ] cod. || 51 ] cod. || 52|| 32 ] cod. || 32 ] cod. || 33 ] cod. || ||33-4 18049 146 178 172 150 ] cod. || 53 cod. || 49 ] cod. 50 || 53 ] (sic) ] ] cod. || 53 ] || cod. ] cod. cod. ||50 158 169 164 159 154 ] (pro ) cod. || 34 ] cod. || 34 ] ] cod. ] || 51 cod. || 54 ] cod. || 51 || 55 ] cod. ] || 51 cod. ]|| 55-6 cod. || 52 54 ] cod.

56

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270

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(pro

|| 36-7 ] cod. cod. || 41

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] || 53 cod.||||||57 37 ] ] ] cod. || ||38 cod. ] cod. cod.|| ||5653 ] cod. 53 cod. || 56cod. ||] 36-7 ] ] cod. cod. cod. || cod. 38cod.|| ||5854 ] ] cod. ] ] cod.cod. ] ||3855|| 59 ||cod. 55-6|| 59 ]cod.|| cod. ] || 39 162153165176173 ] ] 194 26562 5525826965 5772 63255266263175152145168179155147 cod. || 39 ] cod. || 41 ] cod. || 41 ] cod. || 42 192 256 257 264 166 167 174 61 71 66 151 161 156 24 113 216 110 214 208 114 41 288 38 286 ] 280 ] 42cod.cod. || 56 ]] cod. ] ] cod.cod. || 57|| 60 ] cod. || || 59 cod. |||| 56 60 ] 253205702006712326011858279177501632771602721575117046148 130207267122 ] cod. || 42 ] cod. || 43 ] cod. || 43 ] cod. 55 57 || 61 ] cod. ] || 58 ] cod. || 59 ] cod. || 59|| cod. cod. 194116 26520455119 2691265720163209266441752761454717954147273153281176 44 ] cod. || 44 ] cod. || 44 ] corr. ex cod. || 45] 75 41 288 38 286 280 42 216202110203214120208210114 113125 cod. || 60 ] cod. || 60 30024115 43 53 274 275 ] v cod. ||r 59 ( 5 ) A: 4810282 ,11 - 11 ,6 (ed. ll . ]478-492). 75 75 75 207 205 200 279 277 272 118285 27150 52 49 27851 40 46 22 213 199122124 121 206123112 ] add. supra ] cod. || 47 ] cod. || cod. ] || 6155cod. || 46 ] cod. cod. || 46 ] || 63-4 ] cod. || 64 ] 204215119111 27628747 39 54 45273284281 12611720121220928344 3762 302 211116109 v r ( 5 ) A: ]10 ,11 cod.- 11 || ,6 48 (ed. ll . 478-492). ] (sic) cod. || 48 ] cod. || 47 78300 5115324125 2 202322203316120 6 210954323453 9227423227522648 96282 cod. || 64 ] cod. || 66 ] cod. || 66 ] cod. || 68 ] 22315213141993131243081211520610112225285104271223522184910527810040 62 ]cod. || 49cod. || 63-4 ] cod. || ]64 (sic) cod. ( ||6])50 49 ] cod. ||] 50 quasi ] cod. || 76 ] cod. 302 8 211312109112151811130911731721298283222 371012871083921945227284(sæpius) cod. || 76 (6) 60cod. || 64|| 51 ] cod. ||cod. 66 || 51] cod. ] cod. ] ] || 66 cod. ||] 51 cod.] || 68 cod. || 52] 24678 7 5 17324310 2 31132212316318 6 97951072342209222123210222622896 149161180166146167178156172174150 1921326159 71270256562572686626226460151

154 130 26726125368 702596725426069 5864177171163158160169157164170159148

cod. || 53 cod. ||] 76 cod. ]|| 53 cod. || ]76 cod. ||] 53 (sæpius)

132 59 270 56 268 262 60 149 180 146 178 172 150 76 32131530714 163131330831415 4 1023122521710410622310321822410594100

quasi ]

261 68 259 254 69 64 171 158 169 164 159 154 248 319 8 1 31232311 3 18 9 30932031722998 912222331019310899219230227 60

246 62 310 72 311 255 318 152 220 162 221 165 228 7 258 17 65 12 263 97 168 107 155 102 173

192 321 71 307 256 314 264 231 161 217 166 224 174 76 61 16 257 13 66 4 151 106 167 103 156 94 130 248 267 319 253 1

70 323 67 3

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58 320 177 229 163 233 157 230 91 160 93 170 99 148

194 265

269

63

266 175 145 179 147 153 176

55

57

ta/ta4.pdf

3 3

ta/ta4.pdf

cod.

]

(5)

(5)

cod. ||

22

18

3

2

20

Fig. a 2

Text preceding and translation 23 You then place the number 9, thus 8, in the upper cell next to the left-hand corner cell, above 12, and 18 in the cell next to the lefthand corner in the lower row, underneath 16.218 You put 7 in the cell next above and 20 opposite to it, in the (other) corner (cell) diagonal. to the7,left-hand corner cell, underneath it, next to 12, andof19the opposite to You put 5 in the middle left-hand cell, underneath 7, and you put 21 in it, in the right-hand row, next to 10. You put 6 in the upper left-hand the cell opposite to it, on the right-hand side, underneath 19. You put corner, above 7, and, opposite to it, in the (other) corner of the diagonal, 420. in the the upper corner, above 19,7,and to You cell put of 5 in theright-hand middle left-hand cell, underneath and opposite you put 21 itindiagonally, 22. You put 3 in the lower middle cell, underneath 9, and the cell opposite to it, on the right-hand side, underneath 19. You put opposite in the the right-hand upper row upper 23, above 17. above You put 2 in opposite the second 4 in the to cellit of corner, 19, and to 29 cell of the lower andmiddle 24 opposite to it in the9,upper it diagonally, 22. row, You underneath put 3 in the14, lower cell, underneath and row, next to to it4.inYou put therow remaining in 2the cell opposite the upper 23, above(number) 17. You 1put in second the second 219 ofcell the right-hand 20, and opposite to it to onitthe left.upper The of the lowerrow, row,above underneath 14,25and 24 opposite in the square of five (now) row, next to by 4. five Youis put thecompleted. remaining (number) 1 in the second cell of the right-hand row, above 20, and 25 opposite to it on the left. The (§ 5. Construction of bordered squares220 of larger odd orders) square of 5 by 5 is (now) completed. 8

12

10

45

46

48

6

11

18

20

35

36

16

39

9

19

24

29

22

31

41

7

17

23

25

27

33

43

47

37

28

21

26

13

3

49

34

30

15

14

32

1

44

38

40

5

4

2

42

Fig. n = a73 in numbers the text B,of s.aff refer to as well to ‘border’ (note 505). IfAs the themay square are‘row’ more thanas five by five, (you proceed) Counted from right to left, as it should be. thus. Put the median number in the central cell and the next ones where 220 We put the figures where they should be (see note 215). We shall even add some were put, in the squares of three by three and five by five, the correspond(marked with an asterisk). ing ones. When there remain the border surrounding this (latter) square and the still unplaced numbers, you deal with the remaining small and large numbers. You take the first two of them30 — which will always be [email protected] even if the sequence (of numbers) proceeds from 1 by successive increments of 1 — and you put the small one (12) in the cell next to the upper left-hand corner and, in the opposite cell, next to the lower left-hand corner, the larger even number (38). Take next the two numbers following these two numbers, which will always be odd;31 put the small one (11) 218

219

29

Counted from right to left, as in Arabic. On either side of the numbers already used, thus one ‘small’ and the corresponding ‘large’. For convenience, we shall add the numbers in round brackets. 31 This mention of parity serves no purpose other than, perhaps, to draw attention to the fact that the even numbers will mostly occupy the horizontal rows and the odd ones, the vertical rows. 30

cod. || 36

65 65 65 65 65 65

24 cod.

|| 39

] 44

]

cod. || 42

56

68

259 254

69

62

258

65

255 263

61

71

256 257

66

130 267 253

70

67

260

58

194 265

269

57

63

266 175 145 179 147 153 176

55

60

264

116 204 119 126 201 209 300 115 125 202 203 120 210 213 199 124 121 206 112

302 211 109 215 111 117 212 324

5

322 316

2

cod. ||

]

] corr. ex

cod. || 45

cod. ||

]

cod. || cod. 33-4 (sic)cod. cod.|||| ||33-4 50 || 34 ] || ]34 cod. ]|| 52

cod. 44 ] cod. 44 corr. ex60 cod. |||| 45 45] ]] cod. cod. ] ]] 42cod. ||cod. 59 |||| 44 cod. || 60 |||| 44 ] ]] corr. cod. ||ex 51 46 cod. |||| 46 46 add. supra supra 46 cod. |||| 47 47 cod. |||| cod.]] || 6155cod. ] ]] add. cod. |||| 46 ]] cod. ]] cod. 44 276 47 54 273 281 v r ( 5 ) A: 10 ,11 11 ,6 (ed. ll . 478-492). ] cod. || 48 ] (sic) cod. || 48 ] cod. 47 43 47 53 274 275 ]48 282cod. || 48 ] (sic) cod. || 48 ] cod. |||| 285 271 52 49 278 40 62 ]cod. |||| 49 cod. || ]]64 (sic) 49 49cod. || 63-4]] cod. ||||] 50 50 (sic) cod. cod. |||| ]50 50 49 ]] cod. cod. 283 37 287 39 45 284 ( 66 )) ( || 64 ||cod. 66 |||| 51 cod. 6 )52 ] cod.cod. cod. 51 ] ] cod. 51] cod. ||||] 51 51 cod.]] || 68 cod.(|||| 52] 95 234 92 ] 232 226 96 |||| 51 ]cod. ]] || 66 cod. cod.

75 75 75

75 279 207 122 205 200 123 75 118 75

78

cod. || 44

cod. || 42

]

cod. || 43

171 158 169 164 159 154

64

24 113 216 110 214 208 114 41

22

]

]

]

cod.cod. 38 || ]] ] cod. |||| 38 ] cod. || 38 ] cod. || 38 ] cod. || 39 ] 54 155 ]162 165 173cod.] || 38cod. || 54 ] ] cod. cod.|| ||3855 || 55-6 ] ] ] cod.cod. || 39 152 168 cod. 41 || 56 ]] cod. cod. 41 || 57 cod. 42 || 151 cod. 161 166 ||16739 ]156 174 cod. || ]]56 cod. ] ||||cod. ] ||||cod. ] |||| 42 cod. cod. || 39 41 41 ]] cod. 177 163 160 157 170 148 cod. ]|||| 42 42 cod. ]|| ] 58 cod. cod. |||| 43 43 ] cod. |||| 43 43 cod. cod. ]] || 59 cod. 57 ]] cod. ] ]] cod.cod. || 59||||

270

261

72

cod. || 43

cod. || 38

] cod. || 39

]

cod. Text || 41 and translation ] cod. || 41

cod. || 44

]

132 59

192

]

cod. || 37

] cod. || 38

]

] cod. || 46 ] add. supra || 46 ] cod. || 47 r v ( 4 ) A: 10 ,5 10 ,11 (ed. ll . 453-477). r v ] ,5cod. || 48(ed. ll . 453-477). ] (sic) cod. || 48 ] ( 47 4 ) A: 10 - 10 ,11 32 ] cod. || 32 ] cod. || 33 49 ] cod. || 49 ] cod. || 50 ] 32 ] cod. || 32 ] cod. || 33 ]] ] (pro ) cod. || 34 ] cod. ] cod. ]|| 51 51 || 34 ] cod. (pro] )cod. || cod. ] || 51 cod. 180 146 || 178 36 172 150 149 cod. ] cod. || 36-7 ] cod. || 37 ] cod. || 53 ] ] cod. || 53 cod. || ]36 cod. ] || 53 cod. || 36-7 37

70 70 70 70 70 70

268 262

cod. || 36-7

]

cod. || 38

]

6

44 288 44 50

38

286 280

277 272

cod. 53 cod. ||]] 76cod. cod. ]|||| 53 53 cod. || ]]76 cod. cod. ||||] 53 53 (sæpius) cod. |||| 53

270 268 262 180 178 172 132 59 56 60 149 146 150 315 270 313 268 308 262 225 180 223 178 218 172 14 56 15 60 10 149 104 146 105 150 100 132 59

quasi ]]

60

cod. |||| cod.

]]

cod.

261 68 259 254 171 169 164 69 64 158 159 154 312 259 309 64 317 171 222 169 219 154 227 11 254 18 69 98 158 101 164 108 159 2618 68 258 255 263 168 165 173 62 65 72 152 155 162 246 62 310 72 311 255 318 152 220 162 221 165 228 7 258 17 65 12 263 97 168 107 155 102 173 192 256 257 264 166 167 174 61 71 66 151 161 156 321 71 307 256 314 264 231 161 217 166 224 174 76 61 16 257 13 66 4 151 106 167 103 156 94 192

333

267 253 260 130 70 67 58 248 267 319 253 323 67 320 177 229 163 233 157 230 1 70 3 260 9 58 91 160 93 170 99 148 130 80 177 163 160 157 170 148 194 265 55 269 57 194 265 55 269 57

63 266 175 145 179 147 153 176 63 266 175 145 179 147 153 176

(5)

ta/ta4.pdf

24 113 216 110 214 208 114 41 288 38 286 280 42 24 113 216 110 214 208 114 41 288 38 286 280 42 207 122 205 200 123 118 279 50 277 272 51 46 207 122 205 200 123 118 279 50 277 272 5155 46

116 204 119 126 201 209 44 276 47 116 204 119 126 201 209 44 276 47 300 115 125 202 203 120 210 43 300 115 125 202 203 120 210 43

55

54 273 281 54 273 281 65

53 274 275 48 282 53 274 275 48 282

22 213 199 124 121 206 112 285 271 52 22 213 199 124 121 206 112 285 271 52

49 278 40 49 278 40

302 211 109 215 111 117 212 283 37 287 39 302 211 109 215 111 117 212 283 37 287 39 78 78

324 324

5 5

322 316 322 316

2 2

315 14 313 308 15 315 14 313 308 15

246 246

45 284 45 284

6 6

95 234 92 232 226 96 95 234 92 232 226 96

10

225 104 223 218 105 100

10 225 104 223 218 105 100 85

8 8

312 11 312 11

18 309 317 98 222 101 108 219 227 18 309 317 98 222 101 108 21960 227

7 7

17 310 311 12 318 97 107 220 221 102 228 17 310 311 12 318 97 107 220 221 102 228

60

76 321 307 16 76 321 307 16

2488 319 248 319

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70

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4 4

9 320 229 91 233 93 320 229 91 233 93

9 9

99 230 99 230

((55))

ta/ta4.pdf ta/ta4.pdf

22

2

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]

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( 6 ) A: 11r ,6 - 11v ,4 (ed. ll . 493-514).

60 149 180 146 178 172 150

78

64

] 70

70 171 158 169 164 159 154

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4 255 263 1529 168 155 162 165 173 9

72 4

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266 1753 145 179 147 153 176

61

71

55

7

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3

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288

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207 122 205 200 123 118 279

50

277 272

116 204 119 126 201 209

44

276

cod. 47 54

115 125 202 203 120 210

43

53

274 275

48

213 199 124 121 206 112 285 271

52

49

278

40

109 215 111 117 212 283

287

39

45

284

1 1

] (sic)

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216 110 214 208 114 41

5 5

cod. || 79

6 6

37

286 280 51

42

46

|| 81

]

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cod. || 79

cod.

|| 81 3 cod. || 82 ] 22 22 ] cod. || 94

]

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(6)

cod. || 95

]

] corr. ex

|| 79

(sæpius) cod.

] cod. || 91 ] cod. || 96

] cod. || 94 ]

282

(6)

|| 82

cod. ||

cod. || 97

75 273 281 75

23

]

] ]

Text and translation

25

in the left-hand row, in the cell below the left-hand corner, and the large one (39) opposite to it in the right-hand row. Take next the two numbers following them, which will (always) be even; put the small one (10) on the top, next to the cell where the previous even number had been put, and the large one (40) opposite to it on the bottom. Next take the two numbers following them, which will always be odd; you put the small one (9) on the left, next to the cell where the previous odd number had been put, and the large one (41) opposite on the right. (§ 6. More general description of the placing) Next you put, in like manner, the small odd numbers on the left and the large ones which are their complements opposite to them on the right, and the small even (numbers) on the top and the large ones which are their complements opposite to them on the bottom, until you reach on all (four) sides the middle cell.32 Once you have done that, take the two numbers following the last two of the numbers placed, which will always be even; put the small one in the upper left-hand corner cell and the large one opposite to it diagonally, in the right-hand (corner) cell. Take then the two subsequent numbers, which will always be odd; put the small one in the left-hand middle cell and the large one opposite to it, on the right. Take then the two subsequent numbers, which will (always) be even; put the small one in the upper right-hand corner cell and the large one opposite to it diagonally, in the lower left-hand (corner) cell. Take then the two subsequent numbers, which will always be odd; put the small one in the lower middle cell and the large one opposite to it, in the upper middle cell. Deal next with the remaining numbers:33 [ and after that ]34 place then the small even numbers in the lower row and each of their complements in the upper row, opposite to its associate, and place the small odd (numbers) in the right-hand row and each of their complements in the left-hand row, opposite to its associate, until you have finished with all the numbers. Then you will see the (previously) empty square filled with numbers, all meeting this (required) condition.35 By following the way of filling the squares of three and five and always placing the remaining (numbers) as for them, this method will lead you 32

Without filling it. We shall thus have reached 9 and 41 for order 7, 11 and 71 for order 9. The word we translate by ‘complements’, which in this case also means ‘facing them’ (m¯ a yuq¯ abilh¯ a ), might be ἀντικείμενοι (ἀλλήλοις). 33 Starting from the middle cells just filled. 34 Some early reader wanted to emphasize the steps. See also §§ 9, 33, 34. 35 See § 1. It is implicit that, by the construction, the squares are bordered ones, thus that each inner square is itself a magic square.

[[

26

90

90

]

]

Text and translation An ancient Greek treatise on magic squares

23

95

95

(7)

( 6 ) A: 11r ,6 - 11v ,4 (ed. ll . 493-514). 78

]

100

add. et del.

cod. || 79 || 81

cod.

]

] (sic) ( 6 ) A: 11r ,6 - 11v ,4 (ed. ll . 493-514).cod.

78

]

cod. || 94

]

cod. || 79 cod. || 94

60 149 180 146 178 172 150 100

132 59

270

56

141 136 261

68

259 254

69

183 191

62

258

65

255 263 152 168 155 162 165 173

138 192

61

71

256 257

66

188 130 267 253

70

67

260

58

194 265

269

57

63

266 175 145 179 147 153 176

55

268 262

92

64

171 158 169 164 159 154

]

cod. || 81 add. et del. 72

105

264 151 161 166 167 156 174

] (sic) (D, fol. -

177 163 160 157 170 v 148

; A, p. 309)

132 59 270 56 268 262 60 149 180 146 178 172 150 24 113 216 110 214 208 114 41 288 38 286 280 42

92

]

cod. || 94

141 136 261 68 259 254 69 64 171 158 169 164 159 154 33 28 207 122 205 200 123 118 279 50 277 272 51 46

cod. || 94

183 191 62 258 65 72 255 263 152 168 155 162 165 173 291 299 116 204 119 126 201 209 44 276 47 54 273 281 138 192 61 71 256 257 66 264 151 161 166 167 156 174 30 300 115 125 202 203 120 210 43 53 274 275 48 282

cod.

cod. || 79

]

]

188 130 267 253 70 67 260 105 58 177 163 160 157 170 148 296 22 213 199 124 121 206 112 285 271 52 49 278 40

]

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cod. || ]79

cod. || 82

|| 81 4

cod. || 94

]

cod. || 95

]

|| 79

cod. || 91 ]

] cod. || 94

] corr.] excod.

cod. || 96

|| 82

cod. ||

(7)

]

|| 97

|| ]79

]

(sæpius) cod.

]

cod. || 91

]

]

(sæpius) cod.

]

]

cod. || 94

cod.

]

|| 81

|| 82

cod. || 95

] corr. ex

] cod. || 96

] cod. || 94 ]

|| 82

cod. ||

cod. || 97

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28 82

291 299 116 204 119 126 201 209 237 245 8 312 11 18 309 317 30 84 296 242

300 115 125 202 203 120 210 246 7 17 310 311 12 318 22 76

5

324

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315

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312

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225 104 223 218 105 100 (D, fol. 84v ; 309 317 98 222 101 108 219 227 15

10

310 311

12

318

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314

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3

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18

4

276 47 54 273 281 222 101 108 219 227

213 199 124 121 206 112 285 271 52 49 278 321 307 16 13 314 4 231 217 106 103 224

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288 234

207 122 205 200 123 118 279 50 277 272 51 46 315 14 313 308 15 10 225 104 223 218 105 100

4

(3

tit. :

3) A : 23A,r (ed. p. 309)p. 309).

(D, fol. -v ; A, p. 309)

97 107 220 221 102 228 231 217 106 103 224

320 229

94

Eandem fig. præb. D, fol. 61r , cum tit. : 91 233 93 99 230 (5 tit. :

5) A : 23r (ed. p. 309).

Eandem fig. præb. D, fol. 61r , cum tit. : 8

4

9

(7 tit. :

2

7

3

8

4

9

5

1

6

2

7

3

Eandem fig. præb. D, fol. 61r , cum tit. : (9 tit. :

1

6

9) A : 23v (ed. p. 310).

(D, fol. 84v ; A, p. 309)

; add. etiam:

(= n2 + 1) et

(= M9 ).

Eandem fig. præb. D, fol. 61 , cum tit. : r

(11 5

7) A : 23r (ed. p. 309).

tit. :

11) A : 24r (ed. p. 311).

25 v

(319 ; D, fol. 84v ; A, p. 309)

] ]

Translation of Text A 145 to its left,213 (above,) the number preceding it and, diagonally opposite to it, the number following it. Then they put the small number preceding Text and translation 27 small it,and andthere opposite to it from the large number 16 thethat square of number 5 by 5 isunderneath completed, remain, the square, following that large number, (namely) above the number which follows (empty) cells and, from the numbers, 16 (available) numbers. that median number. Then they place the small number preceding the to the filling of all odd squares, whatever corner their (order), second (small number) in the cell and,provided opposite you to it fill 6 right-hand 8 23 24 4 (first) thecorner, centralthe (parts) and then deal with the outer as you in the large number following the second largeparts number. Nextdid 7 to 12 the 17 square 10 19 of three — as long as the for they the square five(cell) relative place inofthe following (that of) this large number the small number which precedes the5 third, and to itincrements the large number number sequence is the natural one with15opposite successive of 1.36 11 13 21 following this third large number. With this, the square of 3 by 3 is 25 16 9 14 1 completed, and it will have the following aspect. 22

18

3

2

4

9

2

20

Fig. a 2 3

5

7

You then place the number preceding 9, thus 8, in the upper cell next Translation A the cell next to the 145 8 12, 1 of 6 Text to the left-hand corner cell, above and 18 in left218 hand corner in the lower row, underneath You put 7 in the cell next Fig. a 1 16. n=3 to the the left-hand corner cell, underneath it,a 2next to 12, andthe 19central opposite 214 square of 5 by 5 is completed, there from square, 16 to (A.II.4) If the square is 5 by 5and (Fig. ),remain, you put in the cell it, (empty) in the right-hand row, next to 10. put 6you innumbers. the upper left-hand 215 cellsnumber and, from the numbers, 16You (available) the median of 25, namely 13. Then place the number corner, above 7, and, opposite to it, in the (other) corner of the diagonal, preceding 13, thus 12, in the cell (next) diagonally, where you had placed 6 8 23 24 4 20.4 You put 5 in the left-hand cell, underneath andfigure you ]put in the square of 3;middle opposite to it diagonally, you place 7, [ this the 21 216 7 12 17 10 19 in the cell opposite theyou right-hand underneath 19.12,You number following to 13.it, on Next place theside, number preceding thusput 4 in cellyou of had the placed right-hand upper corner, above 19, and opposite 11,the where 3 in this figure, and you place opposite to it the to 5 11 13 15 21 large number 14,3 thus 15,lower in themiddle place ofcell, 7 inunderneath this figure. 9, You it diagonally, 22.following You put in the and 25 16 9 14 1 place the number preceding 11 in the (upper) right-hand corner cell of opposite to it in the upper row 23, above 17. You put 2 in the second 219 inner 22 place 18 3opposite 2 20 to it, in the cell where 8 had 3, and you cellthe of thesquare lowerofrow, underneath 14, and 24 opposite to it in the upper been put, the large number following 15, thus 16. Thenin youthe place in thecell Fig. a 2 (number) row, next to 4. You put the remaining second n = 5corner [ thus 10 1] the cell following the (upper) right-hand large number then place the above number20, preceding thus 8, in upper next of theYou right-hand row, and 25 9,opposite tothe it on thecell left. The following 16, thus 17, then opposite to it the number following 10the [ which 220 to the left-hand corner cell, above 12, and 18 in the cell next to leftsquare of 5 by2175 is (now) completed. thus 9. With this treatment, square of precedes it ] in ,the hand corner lower row, underneath 16.218the You put 7in in the the centre cell next to213the left-hand corner cell, underneath it, next to 12, and 19 opposite to 8 12 10 45 46 48 6 Here and throughout: to the left of the reader. it,214inConstruction the right-hand row, next putthe6 inner in the left-hand of the magic squaretoof 10. orderYou 5, first 3 ×upper 3 square. 11 18 20 35 36 16 39 215 Whereas the7,previous figure wastoinserted at the appropriate place in the text, the corner, above and, opposite it, in the (other) corner of the diagonal, others (all but one: Fig. a6) appear at the end of the study of magic squares. 9 19 24 29 22 31 41 20. You put 5 in the middle left-hand cell, underneath 7, and you putThis 21 choice is the author’s: their presence there is announced. This is hardly pedagogical in the cell opposite to it, on the right-hand side, underneath 19. You put and makes the text of little 7use 17 to the23reader. 25 27 33 43 4 216 inThe thebracketed cell of the right-hand upper corner, 19, and to words had once been written by a above reader below Fig. opposite a1 and came 47 here. 28 21 26 was 13to make 3cell,clear to diagonally, be incorporated the text to what the it 22.inYou put 337in His thepurpose lower middle underneath 9,words and ‘this figure’, thereafter, were referring. opposite tooccurring it in thetwice upper row 23, above 17. You put 2 in the second 217 49 both 34 for30the 15 14 32sequence 1 following is indeed used ascending and the descending one 219 cell of the lower row, underneath 14, and 24 opposite to it in the upper (in the latter case we render it by preceding), sometimes specified by ‘before it’ or ‘after 44 the 38 remaining 40 5 4 (number) 2 42 1 in the second cell row, next to 4. You put it’; not here, whence the reader’s addition. of the right-hand row, above 20,Fig. anda25 opposite to it on the left. The n =220 7337 square of 5 by 5 is (now) completed.

218 As in the text B, s.aff 36 219 In fact, the placing of

may refer [email protected] to ‘row’ as well as to ‘border’ (note 505). terms be the same with numbers in any 8 as12it should 10 45 would 46 48 6 Counted from right tosuccessive left, be. 220 arithmetical progression. other We put the figures where they should be (see note 215). We shall even add some 37 20 with 35 Indo-Arabic 36 16 39 numerals. Since the squares In MS.with A the squares11are 18filled (marked an first asterisk). are represented together, first the odd-order 9 19 24 29 ones 22 then 31 the 41 even-order ones, those containing Indo-Arabic numerals are that for n = 3, the two for n = 5 and n = 7 (here 7 n 17 33 keeping 43 and pp. 76–77) and the first for = 9,23all 25 the 27 other the original alphabetical [email protected] numerals. Thus, obviously, a later change. 47 37 28 21 26 13 3 49

34

30

15

14

32

1

44

38

40

5

4

2

42

Fig. a 3

28

Text and translation

70

Translation (D, fol. 69v ; A, p. 310)

(3 ⇥ 3) A : 23r (ed. p. 309). tit. :

Eandem fig. præb. D, fol. 61r , cum tit. : (5 ⇥ 5) A : 23r (ed. p. 309). tit. :

Eandem fig. præb. D, fol. 61r , cum tit. : (7 ⇥ 7) A : 23r (ed. p. 309). tit. :

Eandem fig. præb. D, fol. 61r , cum tit. : (9 ⇥ 9) A : 23v (ed. p. 310).

(D, fol. 85r ;(= A, p. ; add. etiam: n2311) + 1) et

tit. :

(= M9 ).

Eandem fig. præb. D, fol. 61r , cum tit. : (11 ⇥ 11) A : 24r (ed. p. 311). tit. :

Eandem fig. præb. D, fol. 61v , cum tit. : (8, 10) ]

||

(11, 7) ]

||

(10, 10) ]

||

(2, 5) ]

||

(10, 9) ]

||

(11, 8) ]

(v. loculum supra) ||

||

(2, 3) ]

(7, 7) ]

||

||

(10, 7) ]

(4, 2) ] corr. ex

.

to the lower left-hand corner, the larger even number (38). Take next the two numbers following these two numbers, which will always be odd.224 Put the small one (11) Text in theand celltranslation below the left-hand corner in the left29 hand row and the large (39) opposite to it in the right-hand row. Take next the two subsequent numbers, which will (always) be even. Put the small one (10) on the top, next to the cell where the even number before it had been put, and the large one (40) opposite to it on the bottom. Next take the two subsequent numbers, which will always be odd. You put the small one (9) on the left, next to the cell where the odd number preceding it had been put, and the large one (41) opposite on the right. 10

16

14

12

75

76

78

80

8

15

24

28

26

61

62

64

22

67

13

27

34

36

51

52

32

55

69

11

25

35

40

45

38

47

57

71

9

23

33

39

41

43

49

59

73

77

63

53

44

37

42

29

19

5

79

65

50

46

31

30

48

17

3

81

60

54

56

21

20

18

58

1

74

66

68

70

7

6

4

2

72

Fig. a 4

n=9

221

Construction of bordered squares of larger odd orders. 222 In constructing the border of order 7 we shall indicate the numbers (in brackets). Afterwards (A.II.6), the author will switch from this particular to the general mode of 148filling. Science of the magic square 223 On either side of the numbers already used, thus one ‘small’ and the corresponding ‘large’. 224 This mention parity no purpose than,120 perhaps, to draw attention 12 of 20 18 serves 16 14 113 114 other 116 118 10 to the fact that the even numbers will mostly occupy the horizontal rows and the odd 30 36 34 32 95 96 98 100 28 103 ones, the vertical19rows. 17

35

44

48

46

84

42

87

105

15

33

47

54

56 71 72 52 [email protected]

75

89

107

13

31

45

55

60

65

58

67

77

91

109

11

29

43

53

59

61

63

69

79

93

111

115

97

83

73

64

57

62

49

39

25

7

117

99

85

70

66

51

50

68

37

23

5

119 101

80

74

76

41

40

38

78

21

3

121

86

88

90

27

26

24

22

92

1

112 102 104 106 108

9

8

6

4

2

110

94

81

82

n = 11

Fig. a 5

(A.II.7)231 You may construct these squares in another way, namely by putting all odd numbers in the central part of the successive squares and the even ones in the outer parts. The construction consists in this. You construct a square resulting from the multiplication of an odd number by itself, as you did before. Then you consider (first) the odd numbers from 1 to the last of those which will be in this square; you arrange them inside the square so as to give them the shape of a rhomb within the main square, leaving in the

95

30

Text and translation

(7) 100

105

4

(8) (319 ; D, fol. 84 ; A, p. 309) v

132 59

270

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

130 267 253

70

67

260

194 265

269

57

63

192

55

268 262

( 7 ) A: 11v ,4 - 11v ,15 (ed. ll . 515-526); D : 80v ,22 - 81r ,4.

72

60 149 180 146 178 172 v150 64

tit. D (80 ,15 - 80v ,22) :

264 151 161 166 167 156 174 58

177 163 160 157 170 148

288

38

286 280

42

207 122 205 200 123 118 279

50

277 272

51

116 204 119 126 201 209

44

276

47

273 281

300 115 125 202 203 120 210

43

53

274 275

48

52

49

278

40

37

287

39

45

284

234

92

232 226

96

213 199 124 121 206 112 285 271

302 211 109 215 111 117 212 283 78

246 76

5

324

315

14

313 308

15

8

312

11

309 317

7

17

310 311

12

318

16

13

314

4

323

3

9

321 307

248 319

1

322 316

2

18

(8)

Post haec hab.:

266 175 145 179 147 153 176

110

24 113 216 110 214 208 114 41

22

(. . .)

171 158 169 164 159 154

54

46

282

6

95

10

225 104 223 218 105 100

115

98

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

320 229

91

233

93

99

(319 ; D, fol. 84v ; A, p. 309)

94

(!)

(9)

230

(infra, § 8)

(§§ 12-35)

99

]

A || 109 120

5

8

4

9

A || 105 ]

(§§ 9-11)

]

A || 107 A || 109

A || 108

] ]

A.

]

117

99

85

119 101

80

121

94

86

70

66

51

50

68

37

23

5

74

76

41

40

38

78

21

3

Text and translation 88

90

27

26

24

22

92

1

112 102 104 106 108

9

8

6

4

2

110

31

Fig. a 5

(Bordered odd-order squares with separation by parity) 231 You may construct these squares in another way, namely by (A.II.7) (§ 7. Introduction) putting odd numbers in the central squares part of in the successive squares and Youall may construct these (odd-order) another way, namely the ones the outer parts. byeven putting all in odd numbers in the central part of the successive squares and theconstruction even ones in the outer parts. The consists in this. You construct a square resulting 8 56 55 54 53 13 14 7 from The the construction multiplication of an number itself, as you did consists in odd this. You drawby a square resulting frombefore. 59 6 the multiplication of an odd number by itself, as you did before. Then Then you consider (first) the odd numbers from 1 to the last of those 60 those which will you consider thesquare; odd numbers from 1 to the last of which will be (first) in this you5 arrange them inside the square so as to 61 in the central part of 4 the square so as be in this square; you arrange them give them the shape of a rhomb within the main square, leaving in the 38 This leaves to give them the shape of a rhomb3 within the main square. 62 main square (groups of empty) cells with the shape of triangles having in the main square (groups of empty) cells with the shape of triangles 2 63 each the same number of cells. Then you write there the then even you numbers having on each (of the four) sides the same number of cells; 64 1 from to the lasteven to be foundfrom in these squares, in suchin athese way that write2 there (the) numbers last52 to51and be57 (found) 58 92 to 10 the 11 12 the sums and be equal Thus doing, oddsquare) numbers will be in squares, in sucheverywhere. a way that the sums (in thethe main be equal the larger square within a rhombic figure and the even numbers everywhere. Thus doing, the odd numbers will be in the larger squarewill be within a rhombic and the even and numbers will sides be enclosed by this as in surrounded by thisfigure (rhombic figure) all four of the square, 12 132 131 130 129 17 18 19 125 124 22 11 39 232 (rhombic figure) and all four sides of the (square), as in this figure. this figure (Fig. a 6). 10

9

136

ev en

140

en ev

6

8

odd

7

138

ev en

137

en ev

135

139

Fig. a 6

5

141

4

Bordered odd-order squares of the numbers142by parity, general 3 with (Placing the separation odd numbers) description (without describing 2the structure of the rhomb and its largest inner square, 143 (§ 8. inFilling the largest square within the rhomb) unlike B.21i–ii). 144 1 232 This the to only otherthem figureis inserted in the texttake (see1note 215). It isterm understood Theisway place as follows. You and the last 134 13 14 15 16 128 127 126 20 21 123 133 that the squares will be bordered, as before. belonging to this square, namely its largest number, then 3 and the odd 231

number preceding this largest one, and so on always until you reach their middle.40 1 2

n

38

3 n −2 2

2

| . [email protected] . | n 2−7 | ... |

n2 +9 2

n2 −3 2 n2 +5 2

n2 +1 2

‘a rhomb’; in fact, an oblique square. This is the only other figure inserted in the text (see above, p. 21, n. 26). 40 Grouping the odd numbers by pairs of complements, as in our table. The same will be done later for even numbers before placing them (§ 18). 39

32

Text and translation

(8)

110

(319 ; D, fol. 84v ; A, p. 309)

115

120

(9) 125 56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

130 267 253

70

67

260

58

194 265

269

57

63

266 175 145 179 147 153 176

192

55

268 262

( 8 ) A: 11v ,16 - 12r ,4 (ed. ll . 527-536); D : 81r ,8 - 81r ,15.

270

132 59

72

60 149 180 146 178 172 150 64

tit. D :

171 158 169 164 159 154

110-3

288

38

286 280

42

207 122 205 200 123 118 279

50

44

A 277|| 27211251

46

116 204 119 126 201 209

276

47

300 115 125 202 203 120 210

43

53

274 275

48

282

|| 115 52 49

278

40

246 76

54

]

213 199 124 121 206 112 285 271

302 211 109 215 111 117 212 283 78

5

324

2

322 316

287

39

234

92

232 226

15

11

309 317

7

17

310 311

12

16

13

314

323

3

9

318 4

98

5

]

A 284|| 116 96

222 101 108 219 227

] om. D || 116

]

94

99

230

231 217 106 103 224 91

233

93

A, A || 122 || 122-3

]

(sc. D || 114

] A,

...

] om. D.

) D || 114 ]

D || 115

A || 116

]

add. A || 116-7

97 107 220 221 102 228

320 229

A || 114

(post.) ]

] om.

(saepissime) A || 112

] D || 114

]

45

225 104 223 218 105 100

313 308

1

37

A || 111

]

A || 113

D || 114

10

14 312

321 307

]

273 281

95

8

248 319

D || 110

6

315

18

]

177 163 160 157 170 148

24 113 216 110 214 208 114 41

22

...

264 151 161 166 167 156 174

A ] ] ]

D || 117-22

] add. D: (

... add. in marg.)

; v. enim D, fol. 58 v :

33

Text and translation

You put the median term in the central cell of the square. You put its two adjacent odd terms in the two cells where you had put, of the nine terms which were in the square of three by three, the two terms adjacent to the median.41 You do the same for the remaining numbers until you complete the square of three by three — if you are dealing with the square of five by five or the square of seven by seven.42 If you are dealing with (the squares of) nine and eleven, you do the same until you complete the square of five by five in the centre of these two squares. For the (squares of) thirteen and fifteen, you do the same until you complete the square of seven by seven. Likewise, for the (squares of) seventeen and nineteen, you place the (term)treatise and its odd terms 71in the square of An middle ancient Greek on adjacent magic squares nine by nine. You will always do this until you complete the whole square contained by the inner rhombic figure. n = 2k + 1

3

5

7

9

11

13

15

17

19

1

3

3

5

5

7

7

9

9

n2 ≠1 2

n2 ≠5 2

n2 +1 2

n=7 24 22 n = 11 60 58 n = 15 112 110 nAn = 19 178 ancient 180 Greek treatise on L ***

C ***

i

i

n2 +3 2

n2 +7 2

25 26 62 61 113 114 181 squares 182 magic

N2

***

N3

***

28 64 116 18471

N4

***

43 (§ 9. Filling the for6 the first n =remainder 9 2 of the ≠2 rhomb 2 10 three orders) n2 + 1 − (α + 4)

158

277

α

196 154

n2 + 1 − (α + 2) α + 6 194

271 253

α+2

184 150

n = 2k + 1 α+4

95

91

94

116 114 138 106

96

n =513 n = 17 4·1 51

31

3

93

111 119 115 185 187 191 107 197 265

45

89

117 131 135 165 167 127 173 201 245 269

41

57

85

39

55

63

α+2

81

79

142 118 108

25

17

213 215 219 223

2

83

n2 + 1 − (α + 2)

87

130

134

170 286

21

15

n2 + 1 − α

82

98

100

267 247 n2 237 + 1 − (α35 + 6) 7

182 172 144 256 263 243 231 229

84

11

4

120

=⇒ n2 + 1 − 2

α

113 133 143 141 151 157 177 205 233 249 273

27 2 4·4

n + 1153 − α 145 137 161 =⇒ n181 + 1 + 209 2 109 129 227 235 251 275 2

2

2 11 2 4·9

≠2 9 ≠2 4·4

L ****

C ****

6 13 10 15 6 10 4·9 4 · 16

****

****

****

17

19

4 · 16

4 · 25

N3 2 4 i Once you have done that,i takeNthe two oddNterms reached,44 and put 93 the 5left, 0 0 11 cell 4 13of the 8 15 border thensmall in the = 2k + one 1 n =on 7 0the9 middle 17 following 19 n = 13 0 0 0 4 8 inner square you have (just) filled, and put the large one opposite to 1 30 3 0 5 0 5 9 9 n = 17 4 7 8 7 it on the right. Then put the following small number on the bottom, in the middle cell of the border following the inner square, and opposite to it 281 257 241 217 189 169 139 149 147 121 101 285 261 221 193 163 155 125 123 159 n2 + 1 − (α + 2s)

α

289 225 183 171 175 105 103 36

148

211 195 199 203 2

α + 2s

34

152

n +1−α

27

47

23

156

160

59

77

61

75

97

73

49

33

69

29

5

=⇒ n2 + 1 − 2s

99

179

65

71

67

207 254

9

1

=⇒ n2 + 1 + 2s

239 259 287 288 146

43

53

255 283

19

37

279

174 176 140

190 192 136

13

206 208 132

n2 ≠1 2

41

n2 ≠5 2

n2 +1 2

n2 +3 2

n2 +7 2

In all subsequent inner squares of order 3 the arrangement is, relative to that seen 24 around 22 the descending 25 26 28 n= 7 inverted above (p. 19), diagonal. 42 n ≠5 n = 11 60 58 61 62 64 +3 +7 ≠1 ≠5 ≠1 +7 +3 Our gives the order of the largest square comprised by the rhomb if the main | n2.table | 42.≠ +. 314m 2n 2 n= 110 113 114show why 116 successive pairs of orders square is of 15 order n.112The subsequent illustrations n = 19 a same 180largest178 181 182 184 each comprise central square. C* 1 C* 43 n2 + 1 of the —i + “i small “i ≠odd 2, “numbers —i , —is 2 i i +common Thus n = 5, 7, i9. Placing to the 2 iremaining *** *** *** *** *** L C last twon orders since number of8 N cells to be filled is the same. Likewise for the =7 16 the 50 42 30, 32 10, 12 N3 N4 2 i i subsequent pairs of as the table n = 11 24orders,122 12 below 110 shows. 66, 68 42, 44 44 =9 2 placed. ≠2 Thus 2 3 and 6 10 the order 5. Reached nwithout 48 being 122 24 98 23 for 70, 72 26, 28 2

2

n = 13 n = 15 32 n = 17 64 96

2 ≠2 2 226 16 2 ≠2 2 226 32 226 48

L ****

C ****

i

i

N2

****

6 210 6 194 178 N3

****

10 118, 120 10 122, 124 126, 128 N4

****

90, 92 70, 72 50, 52

34

Text and translation

(9) 125

5

130

5

] [ 135

140

145 270

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

130 267 253

70

67

260

58

194 265

269

57

63

266 175 145 179 147 153 176

192

55

268 262

( 9 ) A: 12r ,4 - 12r ,16 (ed. ll . 537-555); D : 81r ,18 - 81v ,8.

132 59

72

60 149 180 146 178 172 150 64

tit. D :

124

hic et sæpissime D || 127

288

38

286 280 51

54

273 281

207 122 205 200 123 118 279

50

116 204 119 126 201 209

44

276

47

15043 300 115 125 202 203 120 210

53

274 275

48

213 199 124 121 206 112 285 271

302 211 109 215 111 117 212 283

246 76

5

324

2

322 316 15

11

309 317

7

17

310 311

12

16

13

314

323

3

9

321 307 1

52

49

278

40

37

287

39

]

45

284

234

92

232 226

96

318 4

98

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

320 229

91

233

93

99

94 230

D || 140

] om. A || 140

] om. (homoeotel.) A || 142

225 104 223 218 105 100

313 308

248 319

282

10

14 312

18

]

95

8

]

sæpius AD (‘alif otiosum’) || 138-40 46

6

315

pr. scr. et del. D || 130

]

D || 133

]

42

A || 145

] add. A.

( 10 )

(l. 130) tit.:

] om. (homoeotel.) A || 126

...

177 163 160 157 170 148

] 277 272

78

A || 125-33

]

264 151 161 166 167 156 174

24 113 216 110 214 208 114 41

22

; hab. ante

171 158 169 164 159 154

]

D || 145

D || 134

D || 131

]

D || 134

(post.) ]

] om. A || 138

...

D || 141-43 . . .

]

D || 143 ]

] om. D || 144

A || 145

]

]

13

31

45

55

60

65

58

67

77

91

109

11

29

43

53

59

61

63

69

79

93

111

115

97

83

73

64

57

62

49

39

25

7

117

99

85

70

66

51

50

68

37

23

119 101

80

74

76

41

40

38

78

21

121

86

88

90

27

26

24

22

92

94

35

5 Text and translation 3

1

on the top its complement. Once you have done that for the five by five (square), you will have completed (the placing of) all odd numbers. 112 102 104 106 108

9

8

6

4

2

110

25

150 150 3

11

9

19

21

13

5

Science Science of of the the magic magicsquare square 23

239 15 of the square the case case of of 7the the17 square of 9, 9, your your treatment treatment isis then then completed. completed.239

1

2

9

4

7

5

3

6

1

8

77 77 71 71

69 69

33

27 27

31 31

61 61

63 63

23 23

99

29 29

39 39

37 37

47 47

53 53

73 73

15 15

25 25

49 49

41 41

33 33

57 57

67 67

81 81

65 65

35 35

45 45

43 43

17 17

11

59 59

51 51

21 21

19 19

55 55

11 11

13 13

79 79

45 45

77

39 39

37 37

33

99

23 23

21 21

31 31

41 41

15 15

33 33

25 25

17 17

35 35

49 49

19 19

29 29

27 27

11

11 11

13 13

47 47

77 43 43

55

55

Fig. a 7*

Fig. a 8*

75 75

Fig. a 7* with the two squares of seven Fig. a 8*and nine, fill the If you are dealing inner square,240 andInput the odd numbers you haveof 13, reached assame youuntil did for (A.II.10) the the (A.II.10)240 In the case case of of the thesquare squareof of 11, 11, and andof 13, do do45 thesame until 241 the the square of situation five, until the same241situation is write attained. Put then the Then the number, isis attained. Then you you write the small small number, the above above situation attained. following the last of the small numbers already placed, in the first border small number you have reached [ after that ] (11) on the bottom, to the following the last of the small numbers already placed, in the first border the bottom, in the third cell from the middle (cell) on the left, and left on of the middle cell, and, opposite to it on the top, its complementary on the bottom, in the third cell from the middle (cell) on the left, and opposite to Then you opposite to itit its its complement. complement. Thenterm you put put the following small number term. Put then the following small (9)the onfollowing the left,small justnumber above the in the first border on the left, in the third cell above the middle in the on the left, theright thirdits cell above the middle (cell), middle cell,first andborder opposite to it oninthe complement. Then(cell), put the and its complement opposite to itit on the right. Then you put the subseand its complement opposite to on the right. Then you put the subsefollowing small number (7) in the middle cell of the second border on the quent the second quent small small term term in second border borderon on the the bottom, bottom, next nextto to the the middle middle left cell, and, opposite toinit the in the second border on the right, its complement. 242 242 and its complement opposite to it on the top. Then you cell, and its complement opposite to it on the top. Then you put put the the Then put the following smalltonumber (5) oninthe bottom, in the the middle subsequent subsequentsmall smallterm termnext next tothe themiddle middlecell cell inthe thesecond secondborder borderon on the cell left, of the border, opposite and opposite tothe it, right. on the topyou of the second and its to Then put left, andsecond its complement complement opposite to itit on on the right. Then you put the the border, its complement. Then put the following small term (3) in the small term reached in the middle cell of the third border on the left, and small term reached in the middle cell of the third border on the left, and opposite to it its complement. Then you put the subsequent small term first opposite border on theitstop, next to the middle cellthe onsubsequent the right, small and opposite to it complement. Then you put term in middle of border on bottom, and to inthe thethe middle cell ofthe thethird third border onthe the bottom, andopposite opposite toitput iton on the to it, in firstcell border on the bottom, its complement. Then the top its complement. Then you put the subsequent small term next to the topsmall its complement. you first put the subsequent small term nextto to the following number (1)Then in the border on the right, next the middle cell in the second border on the top, and opposite to itit on the the middle cell in the second border on the top, and opposite to on the middle cell, below it, and opposite to it on the left its complement. Once bottom its Then you put subsequent small number in bottom its complement. complement. Then you put the the subsequent small number in the you the have done that for these twonext squares, you willcell, have reached 1 to and second border on the right, to the middle and opposite the second border on the right, next to the middle cell, and opposite toitit last on (odd) left term and you will have you performed this (placing) in the desired on the the left its its complement. complement. Then Then you put putthe thesubsequent subsequentsmall small(number) (number) way. 239 239From this it may be inferred that the part of the rhomb outside the inner square From this it may be inferred that the part of the rhomb outside the inner square comprises the comprises thesame samenumber numberof ofcells cells for forthese thesetwo twoconsecutive consecutiveorders. orders. 45 240 Thus with the placing the next two pairsfor in both the cells of next border, 240Filling of rhomb the 13 Filling theremainder remainder ofthe the rhomb for bothmiddle the11 11× ×11 11and andthe 13× ×13 13squares. squares. namely 241 15, 35 241 &With 13, 37 7), or 15, 67placed & 13,last. 69 (order 9). 21 (Fig. and 101 Since With 21(order (Fig. a9*) a9*) and 101 placed last. Since for for orders orders 11 11 and and 13 13 there there isis the the same same quantity quantityof of cells cells to tobe befilled filled with with odd odd numbers numbersoutside outsidethe theinner innersquare, square, we wegive give just justone oneillustration. illustration. 242 242Only one adjacent cell still empty. Same in what follows. Only one adjacent cell still empty. Same in what follows.

140 140

145

36

Text and translation

( 10 ) 145 145 145

( 10 ) ( 10 )

150

150 150 150

155

155 155 155

6 ( 9 ) A: 12r ,4 - 12r ,16 (ed. ll . 537-555); D : 81r ,18 - 81v ,8. tit. D : 124

; hab. ante

A || 125-33

]

hic et sæpissime D || 127

]

sæpius AD (‘alif otiosum’) || 138-40

]

] om. (homoeotel.) A || 126

pr. scr. et del. D || 130

]

D || 133

] 160

6 6

...

(l. 130) tit.:

D || 134

D || 131

]

D || 134

(post.) ]

...

]

] om. A || 138

] D || 140 ] om. A || 140 ] D || 141-43 . . . ( 9 ) A: 12r ,4 - 12r ,16 (ed. ll . 537-555); D : 81r ,18 - 81v ,8. ] om. (homoeotel.) A || 142 ] D || 143 ] om. D || 144 tit. D : ; hab. ante (l. 130) tit.: ] A || 145 ] D || 145 ] A || 145 ] ]172r ,4150- 12 Ar ,16 || 125-33 . . . D : 81r ,18 - 81v ,8. ] om. (homoeotel.) A || 126 ] 270 56 268 262 60 149 (124 9180) 146 A:17812 (ed. ll .add. 537-555); A. 261 68 259 254 69 64 171 158 169 164 159 154 pr. scr. (l. et del. || 130 ] D || 131 tit. D hic : et sæpissime D || 127 ;] hab. ante 130)Dtit.: 62 258 65 72 255 263 152 168 155 162 165 173 ( 10 ) A: 12r, 16 - 12v ,17 (ed. ll . 556-576); D : 81v ,9 - 81v ,18. ] D || 133 ] D || 134 (post.) ] D 61 71 256 257 66 264 151 161 166 167 156 174 124 ] A || 125-33 ... ] om. (homoeotel.) A || 126 || 134] 165 tit.160D157: 170 148 267 253 70 67 260 58 177 163 ] hic etsæpius AD (‘alif otiosum’) ]|| 138-40 . . . D || 130] om. ] A || D 138|| 131 sæpissime D || 127 pr. scr. et del. 55 269 57 63 266 175 145 179 147 153 176 146 . . . ] (sic) A || 146 D || 140 D || ] om. D ||(post.) 141-43] . . . D ||] 134 216 110 214 208 114 41 288 38 286 280] 42 ] 133 A || ]140 D || ]134 207 122 205 200 123 118 279 50 277 272 46 D 51|| 147 ] DA|| ||147 ] AD ||D148 ] om. D || 144 A, ] om. AD (homoeotel.) 142 ||. . 143 ] sæpius (‘alif otiosum’) || 138-40] . ] ]om. A || 138 116 204 119 126 201 209 44 276 47 54 273 281 ] || 149 A ]|| 145 D || 149-50 ] D || 145 . . . ] ] om. A D D |||| 145 151 ]] D || 115 125 202 203 120 210 43 53 274 275 48] 282 D || 140 ] om. A || 140 ] D || 141-43 . . . 270 56 268 262 60 149 180 146 178 172 150 ( 11 ) 213 199 124 121 206 112 285 271 15252 49 278] 40 || 153 add. A. ... om.D(homoeotel.) A] || 142 D || 154-60 ] D || 143 ] om. D || 144] om. 261 68 259 254 69 64 171 158 169 164 159 ]154 109 215 111 117 212 283 37 287 39 45 284 ] A] || 155 A ||] add. ] pr. scr. et corr. in marg. A || 161 168 155 [162 165 173 D || 154 62 258 65 72 255 263 152 (homoeotel.) 324 2 322 6 149 95 234 92] 232 96 270 56 268 316 262 60 180 146 178 226 172 150 A || 145 ] D || 145 145 ] 61 71 256 257 66 264 151 161 166 167 156 174 315 D || 161 ] om. D || 162 ] om. A || 163 ] om. D || 164 261 14 259 308 254 15 171 104 169 218 164 ]105 68 313 69 10 64 225 158 223 159 100 154 add. A. 267 253 70 67 260 170 58 177 163 160 157 170 148 8 312 98 222 258 11 255 317 263 152 168 101 165 227 173 62 65 18 72 309 155 108 162 219 ] om. A || 165 ] A || 165 ] 55 269 57 63 266 175 145 179 147 153 176 7 61

17 97 107 256 311 257 12 264 151 166 221 167 102 174 71 310 66 318 161 220 156 228

24 113 216 110 214 208 114 41 288 38 286 280 42 321 4 231 94 267 307 253 16 260 58 177 217 163 106 170 148 70 13 67 314 160 103 157 224

207 122 205 1 323 269 55 116 204 119 216 110 115 125 202 207 122 205 22 213 199 124 116 204 119 109 215 115 125 202 78 5 324 2

200 123 118 279 3 9 320 266 229 175 57 63 126 201 209 44 214 208 114 41 203 120 210 43 200 123 118 279 121 206 112 285 126 201 209 44 111 117 212 283 203 120 210 43 322 316 6 95

D || 166

50 277 91 233 179 145 276 47 288 38 53 274 50 277 271 52 276 47 37 287 53 274 234 92

272 51 93 153 99 147 54 273 286 280 275 48 272 51 49 278 54 273 39 45 275 48 232 226

46 230 176

]

A,

D || 166

]

A,

D || 167

]

D.

v 281 ( 11 ) A: 1242 ,17 - 13r ,20 (ed. ll . 577-599); D : 81v ,19 - 82r ,6.

tit. D : 168

282 46 40 281 284 282 96

]

A || 169

]

in the first border on the top, in the third cell from the middle (cell), and opposite to it on the bottom its complement. Then you put the subseText and translation 37 quent small number in the third cell from the middle (cell) in the first border on the right, and opposite to it on the left its complement. Once you have done that, you will have placed the odd numbers in these two (§ 10. Filling the remainder of the rhomb for the orders 11 and 13) squares,243 to the last, in the centre (of the squares).244 113

11

107

97

7

103

91

89

23

3

17

47

51

81

83

43

105

13

29

49

59

57

67

73

93

109

27

35

45

69

61

53

77

87

95

117 101

85

55

65

63

37

21

5

121

79

71

41

39

75

1

19

31

33

99

119

15

25

115

111

9

Fig. a 9*

If the treatment is with the squares of eleven and thirteen, do the same until the above situation is attained.46 Then you write the small (A.II.11)245 If you are dealing with the square of 15, or of 17, you do numberthe following the last of the small numbers already placed (thus 19) in same as you did for the square of 13 until you reach the third borthe firstder. border on the in the thirdsmall cell number from theinmiddle on 246 Then youbottom, put the subsequent the first(cell) border 47 Then you put the following the left,onand opposite to it its complement. the bottom, in the fourth cell from the middle one, and its complesmall number (17) in firstthe border on the left,number in theinthird cellborder above ment opposite to the it; then subsequent small the first on the left, in theitsfourth cell fromopposite the middle to it the middle (cell), and complement to one, it onand theopposite right. Then 247 then the subsequent number in the second border on complement; you putitsthe following small term (15) in the second border on the bottom, the 48 third the middleopposite one, and to opposite to ittop. its next tothe thebottom, middleincell, andcell itsfrom complement it on the 243 put the following small term (13) next to the middle cell in Then you Of orders 11 and 13. 244 ‘inborder the centre’: meaning, in A.II.7, in the central opposite (rhombic) part. the second on the left, asand its complement to itNote on the the tedious cell-by-cell approach where a general method could have been described (see right. Then you put the small term reached (11) in the middle cell of the Commentary). 245 third border on the theremainder left, andof opposite tothe it its Then you put Filling the rhomb in 15 ×complement. 15 and 17 × 17 squares. Generalization. small term (9) in the middle cell of the third border on the the following 246 We shall have placed the pairs up to and including 29, 197. In Fig. a10*, the bottom, and opposite tonow it on the itsborder complement. Then the numbers to be placed form thetop outer of the rhomb. Fromyou whatput follows could be inferred filling the lateral the rhomb. following small termthe (7)general nextway to ofthe middle celltriangles in the ofsecond border on 247 Hereopposite and below:to exact situation relative toits thecomplement. middle cell not specified. the top, and it on the bottom Then you put 46

With 21, 101 (order 11) and 21, 149 [email protected] (order 13) placed last. Since for orders 11 and 13 there is the same quantity of cells to be filled with the smaller odd numbers outside the inner square — namely nine on each side, see p. 33 — we give just one illustration. (The numbers of cells on each side are each time square numbers for they are the sum of the successive odd numbers.) 47 ‘in the third cell’, here and (mutatis mutandis) in what follows: counting the initial cell. 48 The situation relative to the middle cell being as in the previous cases.

66

160 160

(38 9 ) A: 12r ,4 - 12r ,16 (ed. ll . 537-555); : 81r ,18 - 81v ,8. TextDand translation

160 160 160

tit. D : ; hab. ante (l. 130) tit.: ( 9 ) A: 12r ,4 - 12r ,16 (ed. ll . 537-555); D : 81r ,18 - 81v ,8. ] rr,4 Arr,16 || 125-33 ... D ] om. (homoeotel.) A || 126 ] ((124 99)) A: (ed. A: 12 12 ,4--12 12 ,16 (ed. llll.. 537-555); 537-555); D:: 81 81rr,18 ,18 -- 81 81vv,8. ,8. tit. D : ; hab. ante (l. 130) tit.: pr. scr. (l. et || 130 ] D || 131 tit. :: et sæpissime D || 127 ;]; hab. 130) tit. D D hic hab. ante ante (l. del. 130)Dtit.: tit.: v,8. ] rr,4 Arr,16 || 125-33 ... D ] om. (homoeotel.) A || 126 ] ((124 99)) A: --12 (ed. --81 A: 12 12 ,4 12 ,16 (ed. llll.D . 537-555); 537-555); D:]: 81 81rr,18 ,18 81v134 ,8. ] || 133 D || (post.) ] D || 134] 124 ]] A ...... ]] om. 124 A |||| 125-33 125-33 om. (homoeotel.) (homoeotel.) A || 126 et sæpissime D || 127 ]; hab. pr. scr. (l. et del. Dtit.: || 130 ] D || 131 165 165 tit. :: sæpius tit.] D D hic hab. ante ante (l. AD (‘alif otiosum’) ]|| ;138-40 . . 130) .130) tit.: ] om. ] A || D 138|| 131 hic pr. hic et et sæpissime sæpissime D D |||| 127 127 ] pr. scr. scr. et et del. del. D D |||| 130 130 ] A || 125-33 D || 133 ] D || ]134 (post.) ] || 126 D || 134] 124 ]]] (homoeotel.) 124 A ... || 140 om. D (homoeotel.) D |||| 125-33 140 D || ] om. ...A ] ] om. ||(post.) 141-43 . A .A. || 126 ]] ]] D D || 134 165 D || 133 133 D |||| 134 134 (post.) ] 165 165 ] hic etsæpius AD (‘alif otiosum’) || 138-40 . . . ] om. A || 138 sæpissime D ]]D || 144 D hic et sæpissime D |||| 127 127 pr. scr. scr.Det et del. D |||| 130 130 D |||| 131 131 ] om. (homoeotel.) A || 142 ]] ] pr. ||. .del. 143 D om. A ]] sæpius ] ]om. || 138 sæpius AD AD (‘alif (‘alif otiosum’) otiosum’) |||| 138-40 138-40 . . .. ] ] D || 140 D || ] om. A || 140 ] D || 141-43 . . . D ] A || 145 D] || 133 133 D || ]145 ] D ||||] 134 134 A || (post.) (post.) ]] D |||| 134 134 ] 145 ] D ]] D ]] om. ]] D ... D |||| 140 140 om. A A |||| 140 140 ] D |||| 141-43 141-43 D || 143 ] om. D || 144 270 56 268 262 60 149 180 146 178 172 150] om. (homoeotel.) A || 142 ( 11 ]] sæpius otiosum’) |||| 138-40 ...... ]] om. sæpius AD AD (‘alif (‘alif otiosum’) 138-40 om. A A |||| 138 138 ) add. A. A D ] om. D || 144 ] om. om. (homoeotel.) (homoeotel.) A ||||] 142 142 D || ]145 ] D ]|||| 143 143 A || 145 261 68 259 254 69 64 171 158 169 164 159 ]154 ] A || 145 ] D ]] om. ]] D D |||| 140 140 om. A A |||| 140 140] D |||| 141-43 141-43 .. .. .. 62 258 65 72 255 263 152 168 155 [[162 165 ]]173 270 268 262 180 178 172 56 60 149 146 150 132 59 ]]146 ]] A ] 270270 268268 262262 180180 178178 172172 ((11 56 56 60 60 149149 146 150150 A A |||| 145 145 add.]] A. D D |||| 145 145 A |||| 145 11)) 61 71 256 257 66 264 151 161 166 167 156 174 ] om. (homoeotel.) A || 142 ] D || 143 ]] om. ] om. (homoeotel.) A || 142 ] D || 143 om. D D |||| 144 144 261 259 254 171 169 164 68 69 64 158 159 154 261261 259259 254254 171171 169169 164164 68 68 69 69 64 64 158158 159159 154154 add. add. A. A. 170 267 253 70 67 260 170 58 177 163 160 157 170 148 ]] [[162 258 255 263 168 173 62 65 72 152 155 162 [178165165172165 258258 255255 263263 168168 173173 62 62 65 65 72 72 152152 155155 132 ]162 A ]] D ]] A ]] 270 56 268 262 262 60 180 146 172 150 132 59 59 270 56 268 60 149 149 180 146 150 ] 178 A |||| 145 145 D |||| 145 145] A |||| 145 145 55 269 57 63 266 175 145 179 147 153 176

256 257 264 166 167 174 61 71 66 151 161 156 192 256256 257257 264264 166166 167167 174174 61 61 71 71 66 66 151151 161161 156156 261 261 68 259 254 254 69 171 158 169 164 164 159 68 259 69 64 64 171 158 169 159 154 154 216 110 214 208 114 41 288 38 286 280 42 170 267 253 260 177 163 170 130 58 70 67 58 160 157 148 267267 253253 260260 177177 163163 170170 70 70 67 67 58170 160160 157157 148148 170 62 258 65 255 263 263 152 168 155 165 173 173 62 258 65 72 72 255 152 168 155 162 162 165 207 122 205 200 123 118 279 50 277 272 51 46 194 265 269 266 175 179 176 55 57 63 145 147 153 269269 266266 175175 179179 176176 55 55 57 57 63 63 145145 147147 153153 192 192 61 256 257 257 66 264 151 166 167 167 156 174 61 71 71 256 66 264 151 161 161 166 156 174 116 204 119 126 201 209 44 276 47 54 273 281 24 113 216 214 208 288 286 280 110 114 41 38 42 216216 214214 208208 288288 286286 280280 110110 114114 41 41 38 38 42 42 130 267 253 253 70 260 58 177 163 163 160 170 148 130 267 70 67 67 260 58 177 160 157 157 170 148 115 125 202 203 120 210 43 53 274 275 48 282 207 205 200 279 277 272 122 123 118 50 51 46 207207 205205 200200 279279 277277 272272 122122 123123 118118 50 50 51 51 46 46 194 194 265 265 55 269 57 266 175 175 145 179 147 176 55 269 57 63 63 266 145 179 147 153 153 176 213 199 124 121 206 112 285 271 52 49 278 40 204 201 209 276 273 281 116 119 126 44 47 54 204204 201201 209209 276276 273273 281281 116116 119119 126126 44 44 47 47 54 54 24 216 110 214 208 208 114 288 38 286 280 280 42 24 113 113 216 110 214 114 41 41 288 38 286 42 109 215 111 117 212 283 37 287 39 45 284 300 202 203 210 274 275 282 115 125 120 43 53 48 202202 203203 210210 274274 275275 282282 115115 125125 120120 43 43 53 53 48 48 207 207 122 205 200 200 123 279 50 277 272 272 51 122 205 123 118 118 279 50 277 51 46 46 324 2 322 316 6 95 234 92 232 226 96 22 213 199 206 285 271 278 124 121 112 52 49 40 213213 199199 206206 285285 271271 278278 124124 121121 112112 52 52 49 49 40 40 116 204 119 201 209 209 44 276 47 273 281 281 116 204 119 126 126 201 44 276 47 54 54 273 315 14 313 308 15 10 225 104 223 218 105 100 302 211 215 212 283 287 284 109 111 117 37 39 45 215215 212212 283283 287287 284284 109109 111111 117117 37 37 39 39 45 45 175 175 300 210 300 115 202 203 203 120 210 43 274 275 275 48 282 115 125 125 202 120 43 53 53 274 48 282 8 312 11 18 309 317 98 222 101 108 219 227 78 324 5324324 2322322 316 234 232 226 22 322 92 96 31631666 695 234234 232232 226226 95 95 92 92 96 96 22 213 199 199 124 206 112 285 271 271 52 278 40 22 213 124 121 121 206 112 285 52 49 49 278 40 7 17 310 311 12 318 97 107 220 221 102 228 r, r, 315 313 308 225 223 218 315315 313313 308308 225225 223223 218218 14 15 10 104 105 100 14 14 15 15 10 10 104104 105105 100100 302 302 211 211 109 215 111 212 283 283 37 287 39 284 109 215 111 117 117 212 37 287 39 45 45 284 175 321 307 16 13 314 4 175 231 217 106 103 224 94 175 8312312 309 317 222 219 227 309309 317317 222222 219219 227227 88 312 11 18 98 101 108 11 11 18 18 98 98 101101 108108 78 324 22 322 322 316 316 66 95 234 92 232 226 226 96 78 55 324 95 234 92 232 96 1 323 3 9 320 229 91 233 93 99 230 246 310310 311311 318318 220220 221221 228228 310 311 318 220 221 228 17 17 12 12 97 97 107107 102102 77 717 12 97 107 102 315 315 14 313 308 308 15 225 104 223 218 218 105 14 313 15 10 10 225 104 223 105 100 100 76 321321 307307 31431444 4 231231 217217 224224 16 16 13 13 106106 103103 94 94 321 307 314 231 217 224 16 13 106 103 94 312 11 309 317 317 98 222 101 219 227 227 88 312 11 18 18 309 98 222 101 108 108 219 248 319 1323323 320320 229229 233233 230230 91 91 93 93 99 99 320 229 233 230 11 323 33 399 9 91 93 99 246 77 17 310 311 311 12 318 97 220 221 221 102 228 246 17 310 12 318 97 107 107 220 102 228

add. add. A. A.

10)) A: A: 12 12 16 16--12 12vv,17 ,17 (ed. (ed. llll.. 556-576); 556-576); D D:: 81 81vv,9 ,9--81 81vv,18. ,18. ((10 tit. D D :: tit. 146 146

......

D |||| 147 147 D

]]

321 307 307 16 314 44 231 231 217 217 106 224 94 76 321 16 13 13 314 106 103 103 224 94 76 248 319 319 11 323 323 33 248

132 59

192

8

270

56

om. A A |||| 165 165 tit. D :]] om. D |||| 166 166 ] A, D A, 146 . .] . 171 158 169 164 159 154

60 149 180 146 178 172 150

68 4259 254

69

62

258

65

255 263 152 168 155 162 165 173 vv

61

71

256 257

66

70

260

130 267 253

7

67

AD |||| 148 148 AD

]]

]]

]] A, A,

D |||| 149-50 149-50 om. D D |||| 151 151 D |||| ]] D ...... ]] om. ]] D D |||| 153 153 D |||| 154-60 154-60 om. 180 180 D ]] D ...... ]] om. (homoeotel.) D D |||| 154 154 pr. scr. scr. et et corr. corr. A A |||| 155 155 add. in in marg. marg. A A |||| 161 161 (homoeotel.) ]] pr. ]] add. 180 180 180 r, v || 161 v v om. A || 163 ] D ] om. D || 162 ] ] om. D || 164 ( 10 ) A: ]12 16 - 12 (ed. ll . 556-576); 81 ,9 - 81] ,18. D ,17 || 161 ] om. D D || :162 om. A || 163 ] om. D || 164 D |||| 149 149 D 152 152 ]]

261

44 4

D |||| 147 147 D

A |||| 146 146 A

(sic) (sic)

320 229 229 91 233 93 230 91 233 93 99 99 230 99 320

268 262

72

]]

64 9

]]

A |||| 165 165 A D |||| 166 166 ] A, D ](sic) A, ]

]]

D |||| 167 167 D

] D. A ||] 146 D. ]

|| 147 D llll||.. 147 ] D AD || 148 ] 11)D ) A: A: 12 ,17 ,17--13 13r]r,20 ,20 (ed. (ed. 577-599); D:: 81 81vv,19 ,19 82rr,6. ,6. ((11 12 577-599); --82 9 9 9 tit.160D D||157:149 : 170 148 ] D || 149-50 ... ] om. D || 151 163 D 58 177 tit.

A,

264 151 161 166 167 156 174

]

3

D ||

168 A |||| 169 169. . . 152 ] D || 153 ] ] om. 168 ]] D || 154-60 A ]] 288 38 286 280 42 7 (homoeotel.) D || 154 ] pr. scr. et corr. A || 155 ] add. in marg. A || 161 A |||| 170 170 om. A A |||| 170 170 om. D D |||| 171 171 A |||| 171 171 46 A ]] om. ]] om. ]] A ]] 28 116 204 119 126 201 209 44 276 47 ]54 273 281 D || 161 ] om. D || 162 ] om. A || 163 ] om. D || 164 7 post præb. A, om. D || 171 ] D || 174-5 . . . ] om. 7 post præb. A, om. D || 171 ]7 D || 174-5 ... ] om. 300 22 115 125 202 77 203 120 21033 43 53 274 275 48 282 ] om. A || 165 ] A || 165 ] (homoeotel.) D || 175 ] (sc. ) add. A || 176 ] AD || 178-9 5 213 199 111241 121 206 112 271 52 49 278 40 D || 175 22 ] (sc. )28add. A || 176 ] AD || 178-9 66 6285 (homoeotel.) 270 56 268 262 60 149 180 146 178 172 150 302 211 109 215 111 117 212 283 D 37|| 287 16639 45 284 ] A, D || 166 ] A, D || 167 ] D. 28 28 ] 261 68 259 254 69 64 171 158 169 164 159 154 ] 194 265

55

269

57

63

266 175 145 179 147 153 176

88 113 216 110 44 214 208 11499 41 24 2

78

55

62

61 267 246 76

7 33 3279 50 277 272 51 207 122 77205 200 123 118 1 6

5

324

2

11

322 316

6

66

234

95

92

232 226

96

258 65 72 255 263 152 168 155 162 165 173 315 14 313 308 15 10 225 104 223 218 105v 100

A, ,17 - 13r ,20 (ed. ll . 577-599); D : 81v ,19 - 82r ,6. ( 11 ) A: A, 12 28 28 71 256 257 66 264 151 161 166 167 156 174 8 312 11 18 309 317 98 222 101 108 219 227 hab. et iter. iter. (....) .) D 253 70 67 260 58 177 tit. 163 postea 160 157: 170 148 A postea et (. 220 221hab. 7 17 310 311 12 318 97 107 102 228A

55 269 57 63 266 175 145 179 147 153 176 321 307 16 13 314 4 231 217 106 103 224 94

168

om. (homoeotel.) (homoeotel.) A A |||| 183-5 183-5] ]] om.

216 110 214 208 114 41 288 38 286 280 42 248 319 1 323 3 9 320 229 91 233 93 99 230 207 122 205 200 123 118 279 116 204 119 126 201 209

44

115 125 202 203 120 210

43

D |||| 180 180 D

181-2 ...... |||| 181-2 om. (homoeotel.) D D |||| 189 189] A]] om. || 169 (homoeotel.)

......

277A272|| 46 D || ]189 (pr.) om. D ||||] 190 190 D ||]] 171 D D |||| 190 190 170 om. A ||]] 170 D om. ] (pr.) ]] 51 om. D || 189 47 54 273 281 ]] A, om. D. post præb. A, om. D || 171 ] D || 174-5 A, om. D. 53 274 275 48 282 50

276

52 49 278 40 homoeotel.

213 199 124 121 206 112 285 271

175

sc.

add.

]]

176

AD |||| 190 190 A ] ]] || 171AD

...

] om.

178-9

Text and translation

39

the following small number (5) in the second border on the right, next to the middle cell, and opposite to it on the left its complement. Then you put the following small number (3) in the first border on the top, in the third cell from the middle (cell), and opposite to it on the bottom its complement. Then you put the following small number (1) in the third cell from the middle (cell) in the first border on the right, and opposite to it on the left its complement. Once you have done that, you will have put the odd numbers, to the last, in the central part of these two squares.49 (§ 11. Filling the remainder of the rhomb for the orders 15 and 17) If you are dealing with the squares of fifteen and seventeen, you do the same until you reach the third border [ as you did for the square of thirteen ].50 Then you put the following small number (27) in the first border on the bottom, in the fourth cell from the middle one, and its complement opposite to it; then the following small number (25) in the first border on the left, in the fourth cell from the middle one, and opposite to it its complement; then the following one (23) in the second border on the bottom, in the third cell from the middle one, and opposite to it its complement; then the following one (21) in the second border on the left, in the third cell from the middle one, and opposite to it its complement; then the following one (19) in the third border on the bottom, next to the middle cell, and opposite to it its complement; then the following one (17) in the third border on the left, next to the middle cell, and opposite to it its complement; then the following one (15) in the fourth border on the left, in the middle cell, and opposite to it its complement; then the following one (13) in the fourth border on the bottom, in the middle cell, and opposite to it its complement; then the following one (11) in the third border on the top, next to the middle cell, and opposite to it its complement; then the following one (9) in the third border on the right, next to the middle cell, and opposite to it its complement; then the following one (7) in the second border on the top, in the third (cell) from the middle one, and opposite to it its complement; then the following one (5) in the second border on the right, in the third (cell) from the middle one, and opposite to it its complement; then the following one (3) in the first border on the top, in the fourth (cell) from the middle one, and opposite to it its complement; then the following one (1) in the first border on the right, in the fourth (cell) from the middle one, and opposite 49

‘in the central part’: meaning, as in § 7 (p. 31), in the inner (rhombic) part. We shall have placed the pairs up to and including 29, 197 and 29, 261 (29 taking the place of 1 in the previous square). From what follows can be inferred the general way of filling the lateral triangles in the rhomb. 50

323 33 11 323

320 229 229 91 233 93 230 91 233 93 99 99 230 99 320

40

180 180

180

Text and translation

( 10 ) A: 12r, 16 - 12v ,17 (ed. ll . 556-576); D : 81v ,9 - 81v ,18. tit. D :

4

146

9

44

99

7

3

33

1

6

11

268 262

AD || 148

]

66

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173 v

61

71

256 257

66

267 253

70

67

260

58

55

269

57

63

266 175 145 179 147 153 176

60 149 171

( 11(homoeotel.) ) A: 12 ,17 D - 13||r ,20 D : 81 ,19 - 82 154 (ed. ll .] 577-599); pr. scr. et vcorr. Ar ,6. || 155 tit. D :

D || 161

]

177 163 160 157 170 148

190 190

168 288

207 122 205 200 123 118 279

50

116 204 119 126 201 209

44

276

115 125 202 203 120 210

43

38

] om. A || 165

286 280

47

54

] om. A || 163

] A || 165

]

42

DA272 ||||166 277 51 17046

] om. D || 162

273 281

109 215 111 117 212 283 324

234

]] 92 232 168

45

284

226

96

10

225 104 223 218 105 100

15

8

312

11

309 317

7

17

310 311

12

16

13

314

323

3

9

1

39

95

313 308

321 307

287

6

14

18

37

4

222 101 108 219 227

91

] om. 94(homoeotel.) (homoeotel.) D || 175 A ||] 183-5(sc.

233

93

]A,

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

130 267 253

70

67

260

194 265

269

57

63

192

55

268 262

72

58

]

177 163 160 157 170 148

266 175 145 179 147 153 176

tit.277 D272 : 51

50

44

276

300 115 125 202 203 120 210

43

53

191 274

275

48

282

49

278

40

212 283 37 287 39 2 302 211 109 7215 111 117 3

45

284

232 226

96

5 246 76

213 199 124 121 206 112 285 271

52

|| 192

324 14

8

312

7

17

321 307

248 319

1

2

322 316

309 317

18

234

205 205 6

310 311

12

318

16

13

314

4

323

3

9

98

92

D || 192

]

95

225 104 223 218 105 100

11

1

] om. A || 194

A || 193

AD || 190

]

]

231 217 106 103 224

infra post

91

[[

233

93

99

94

230

]]

add. D || 192-3

(l. 202) AD || 195

A || 195-6 ]

A || 196

[[

((14 14))

A || 193-4

]

] om. D || 195

A || 194

29

D

]

AD || 193

]

222 101 108 219 227

] om. D || 195

A || 191

]

]

]

97 107 220 221 102 228

320 229

D || 190

]

AD || 191

]

6

15

] om. (homoeotel.) D || 189

... D || 190

...

v

]

10

313 308

|| 181-2

46

54

5

((13 13)) ]

273 281

47

315

] om.

288

207 122 205 200 123 118 279

78

A, om. D.

[[

AD || 190

]

(. . .)

]

] ] ]

( 1238 ) 286A:2801342 ,20 - 13 ,3 (ed. ll . 600-603); D : 82r ,9 - 82r ,11. r

209 116 204 4119 126 201 9

22

D || 190

D || 180

] om. (homoeotel.) A || 183-5

(pr.) ] om. D || 189

AD || 178-9

]]

]

et iter.

264 151 161 166 167 156 174

200 200

]

((12 12)) ] om.

. ] om. (homoeotel.) D || || 189 ). .add. A || 176 ] AD 178-9

171 158 169 164 159 154

24 113 216 110 214 208 114 41

8

A, om. D.

60 149 180 146 178 172 150 64

D || 190

]

postea hab. A

270

132 59

[[

230

99

(pr.) ] om. D || 189

...

]

A || 169 D || ]180 ] om. D || 171 A || 171 ][[(. . .) D || ]]174-5 || 181-2 . . . . . .

231 217 106 103 224

320 229

]

A] || 171D.

]

]

A,|| 170 A ] om. A || 170 postea iter.D || 171 318 195 220 221hab. 97 107 102 228A post præb. A, etom. 195 98

] om. D || 164

]

A || 169

] ] om. A, A,171 D || 167 A || 170D || 166] om.] D || ]

post præb.v A, om. ] D r D || 171 v || 174-5 r 11275) A: 274 53 ( 48 28212 ,17 - 13 ,20 (ed. ll . 577-599); D : 81 ,19 - 82 ,6. 52 49 278 40 D || 175 (homoeotel.) ] (sc. ) add. A || 176 tit. D :

315

D ||

]

] add. in marg. A || 161

213 199 124 121 206 112 285 271

322 316

A,

]

264 151 161 166 167 156 174

216 110 214 208 114 41

2

]

152tit. D ]: D || 153 ] D || 154-60 ... ] om. 7 (homoeotel.) D || 154 ] pr. ] add. A in ||marg. 146 ... ] scr. et corr. (sic) A || 155 146 A || 161 ] ] D || 161 ] om. D || 28 1627 ] om. A || 163 ] om. D || 164 D || 147 ] D || 147 ] AD || 148 ] A, ] om. A || 165 ] A || 165 ] D || 149 ] D || 149-50 ... ] om. D || 151 ] D || 180 146 178 172 150 D || 166 ] A, D || 166 ] A, D || 167 ] D. 28 28 164 158 169 159 154 152 ] D || 153 ] D || 154-60 ... ] om.

56

64

D || 147

]

A || 146

(sic)

D )|| A: 14912r, 16 -]12v ,17D(ed. || 149-50 . .v ,9 - 81] vom. ( 10 ll . 556-576); D : .81 ,18. D || 151

270

72

]

D || 147

185 185

77

...

] inven.

... ]

]]

D || 196

]

AD.

( 13 ) A: 13v ,3 - 13v ,9 (ed. ll . 604-609); D : 82r ,12 - 82r ,16. tit. D :

197 210 210

]

AD || 197 A || 200-1

]

29 D || 198 ...

]

A || 200

] om. A || 201

] ]

D || 202

41

Text and translation

to it its complement. Once you have done that, you will have finished with the odd (numbers) for these two squares in the desired way. 213

15

207 189

11

203 183 173

35

7

199 179 167 165

51

31

3

153 155 159

75

25

79

87

83

21

45

85

99

103 133 135

17

41

57

81

101 111 109 119 125 145 169 185 209

39

55

63

77

97

95

201

141 181 205

121 113 105 129 149 163 171 187 211

217 193 177 157 137 107 117 115

89

69

49

33 5

221 197 161 131 123

93

91

127

65

29

225 151 139 143

73

71

67

147

1

27

47

59

61

175 195 223

23

43

53

191 219

19

37

215

9

13

If you wish (to fill squares) above these two, continue placing (the numbers) step by step in the same way.51

(Placing the even numbers) 12 17 10

52 (§ 12. Different treatments for placing 11 13 the 15 even numbers) At this point you will find that 16 9the14 squares are divided into classes requiring each, for the arrangement of the even numbers in the corner (triangles), a treatment distinct from that of the other.53 There are (the squares of) nine, thirteen, seventeen, and those larger numbers (resulting 6 8 23 24 4 from adding) successively 4 [ in class ]; and (the squares of) seven, eleven, 7 12 17 10 19 fifteen, nineteen, and those larger numbers (resulting from adding) suc5for11treating 13 15 the 21 cessively 4 [ in class ]. Only square of five one cannot   54 proceed as for the others of 25 this16section . 9 14 1 51

22 18 the 3 rhomb’s 2 20 sides filled previously. Surrounding with the next numbers Namely for n = 5 (isolated case), n = 4t + 1 (t > 1), n = 4t + 3. 53 Meaning: the treatment of one is different from that of its immediate neighbours (this will be explained now). An early reader pointed out what these classes are. 54 This last sentence is found between § 13 and § 14 in the two MSS. 52

] 195 195

42

[ [

[ ] Text and translation [

[ [

]

] ]

( 13 ) ( 13 )

200 200

( 14 ) ( 14 ) 205 205

[

[ [

]

] ]

([12 ) A: 13r ,20] - 13v ,3 (ed. ll . 600-603); D : 82r ,9 - 82r ,11. tit. D : 191 210 210

AD || 191

]

192

D || 192

]

A || 193

] ] om. A || 194 infra post

268 262

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

267 253

70

67

260

58

55

269

57

63

266 175 145 179 147 153 176

64

197

A,

280

42

51

46 v

116 204 119 126 201 209

44

276

47

54

273 281

115 125 202 203 120 210

43

53

206 274

275

76

282

49

278

40

109 215 111 117 212 283

37

287

39

45

234

92

232 226

8

]

D || 196

]

AD.

11

309 317

310 311

12

318

16

13

314

4

323

3

9

98

D || 209 ]

231 217 106 103 224

320 229

]

]

] om. A || 201

A || 211

91

233

93

99

94

D || 210 ]

230

] || 215

A || 212 ]

A, om. D || 217 ]

AD (

] om. D || 206

D || 202

]

pro

]

A || 211 ] A || 215 ]

A || 219

sæpissime) A,

] om. D || 209

A || 209

(sic) A || 210

A || 217

pr. scr. et del. A) || 219

]

]

]

A || 219

] om.

D || 209 (post.) ] ]

] A || 217

]

D || 210

]

] om. D || 211

A || 213

A || 206

]

] om. D || 208

D || 207

(

(pr.) ]

222 101 108 219 227

97 107 220 221 102 228

A,

]

A || 96208

95

17

A || 200

]

(sic) A.

]

A || 206

284

225 104 223 218 105 100

312

1

A || 196

...

D || 202

A || 207

6

15

321 307

]

10

313 308 18

] inven.

...

v

] corr. ex

48

52

7

A || 195-6

( 14 ) A: 13 ,9 - 13 ,21 (ed. ll . 610-623); D : 82r ,16 - 82v ,1.

213 199 124 121 206 112 285 271

322 316

8 8]

D || 198

]

A || 200-1

177 163 160 157 170 148

]286

2

] om. D || 195

A || 194

]

AD || 197

]

264 151 161 166 167 156 174

38

14

A || 193-4

tit. D :

277 272

324

]

171 158 169 164 159 154

50

5

D ||

60 149 180 146 178 172 150

288

315

]

( 13 ) A: 13v ,3 - 13v ,9 (ed. ll . 604-609); D : 82r ,12 - 82r ,16.

56

207 122 205 200 123 118 279

78

AD || 193

]

(l. 202) AD || 195

24 113 216 110 214 208 114 41

22

add. D || 192-3

] ]

] om. D || 195

270

72

A || 191

]

A || 209 ]

A || 211-2

A || 214

]

A

] A || 218

]

D || 219

(pr.) ]

] om. A || 219

A || 218-9

]

A || 219

]

43

Text and translation

(§ 13. Quantity of cells remaining empty in squares of orders n = 4t + 1) For the square of five and those of the same kind (the situation is the following).55

There remain as empty cells in the first border — which surrounds the inner square — the four corner cells and eight cells adjacent to them, two on each side;56 in the second border, there remain as empty cells those at the corners and twenty-four cells adjacent to them; in the third border, the empty cells are those at the corners and forty cells adjacent to them. It will 42 always be like that:Science each of border has, square excepting the corner cells, the magic sixteen more (empty cells) than the preceding border.57 (n = 4t + 1)

N1

N2

N3

N4

N5

4+8

4 + 24

4 + 40

4 + 56

4 + 72

(§ 14. Excesses and deficitsNin N2 row for N3the order N4 n = 4t N5 + 1) 1 each (n = 4t + 3) h For the square of five, 4after placing the in it, the upper 4+ 16 4 + 32 odd 4 +numbers 48 4 + 64 row is in excess over its sum due by 12, the lower one is in deficit by 12, L C L C L C L C L C the right-hand row is in over 1 excess 1 2 its 2sum due 3 by3 10, and 4 the4 left-hand 5 5 58 one is in deficit n = 5 by 10. 12 i 10For the other squares associated with it [ in class ] (the nsituation is the18following). =9 20 36 34After arranging the odd numbers in the above manner, the first upper of nine is in excess n = 13 28 26 52 row 50 of the 76 square 74 over its sum ] by 66 20, and one 130 is in deficit n =due 17 [ required 36 34 for it68 100 the98lower132 by 20;59 the first right-hand row is in excess over its sum due by 18, and202 n = 21 44 42 84 82 124 122 164 162 204 the left-hand one is in deficit by 18. The second upper (row) is in excess by 36, and the lower one isC in deficit by the LsameC (amount); the second L L C L C L C 1

1

2

2

3

3

4

4

5

5

55 n = of 7 order 45 belongs 2 18 class for determining the number of The square to20 the first n = in 11 the border, 4 2 but not28for 26 52 50 still empty cells filling them. Its filling will be explained separately inn§= 23.15 4 2 36 34 68 66 100 98 56 Meaning: on19each of4the four each side corner n= 2 sides 44of the 42 border 84 (not: 82 on 124 122 of a164 162 cell). 57 In our table, Ni is the whole quantity of cells remaining empty in the ith border (counted from the centraln2square already filled). L ≠1 n2 ≠5 58 2 appearing + ( L1 ≠ 4)in theC1two ≠ 2MSS. 4 + ( C1 ≠ 2) 1 ≠ 4 not We deemed it necessary 2 to add 2this sentence, 59 ‘requirednfor be an early ‘sum due’ will20 = 9it’ must 40 38 reader’s16addition: the 18concept of 16 be explained below.

n = 13 n = 17

84 144

n=9

82 142

24 32

n2 +1 2

2·

41

26 34

n2 +1 2

70

≠ 12

24 32

–, – + 2 34 36

28 36

44

Text and translation

215

215 215

220

( 15 )

]

220 220

]

[ ( 15 )

[

] 225

]

[ [

( 14 ) A: 13v ,9 - 13v ,21 (ed. ll . 610-623); D : 82r ,16 - 82v ,1. ] corr. ex

206

225 225

A || 207

]

] om. D || 206

A || 206

]

A,

D || 207

]

A || 228-9

A || 206

]

] om. D || 208

[

]

]

A || 208 ] ( pro sæpissime) A, D || 209 ] om. ( 14 ) A: 13v ,9 - 13v ,21 (ed. ll . 610-623); D : 82r ,16 - 82v ,1. D || 209 (pr.) ] A || 209 ] om. D || 209 (post.) ] A || 209 206 ] corr. ex A || 206 ] om. D || 206 ] A || 206 230 ] D || 210 ] (sic) A || 210 ] D || 210 [ ] ] ] A || 207 ] A, D || 207 ] om. D || 208 ] A || 211 ] A || 211 ] om. D || 211 ] A || 211-2 ( 16 ) A || 208 ] ( pro sæpissime) A, D || 209 ] om. ] A || 212 ] A || 213 ] A || 214 ] A 230 D || 209 (pr.) ] A || 209 ] om. D || 209 (post.) ] A || 209 132 59 270 56 268 262 230 60 149 180 146 178 172 150 || 215 ] A || 215 ] A || 217 ] 261 68 259 254 69 64 171 158 169 154 || 210 ] 164 159 D ] (sic) A || 210 ] D || 210 ] A, om. D || 217 ] A || 217 ] A || 218 ] A ||(218-9 62 258 65 72 255 263 152 168 155 162 165 173 16 )) A || 211 ] A || 211 ] om. D || 211 ] A || 211-2 ( 16 192 61 71 256 257 66 264 151 161 166 167 156 174 ] A || 219 ] A || 219 (pr.) ] D || 219 ] 130 267 253 70 67 260 58 177 163 160 157 148 || 212 ] 170 A ] A || 213 ] A || 214 ] A AD ( pr. scr. et del. A) || 219 ] om. A || 219 ] A || 219 194 265 270 5556 269 268 57 262 6360 266 180145 178147 172153 149175 146179 150176 ] A || 215 ] A || 217 ] 235 || 215 24 261 113 259 110 254 21469 20864 114 171 41 169 38164 286 68 216 158288 159 154 42 A || 220 (post.) ] 280 om. ] (sic) A || 221 ] om. A. 207 205 200 279 277 272 122 123 118 50 51 46 A, om. D || 217 ] A || 217 ] A || 218 ] A || 218-9 62 258 65 72 255 263 152 168 155 162 165 173 v r v v (16615 )54156 A:27317413 281 ,21 - 14 ,9 (ed. ll . 624-632); D : 82 ,1 - 82 ,6. 47167 256 119 257126 264 209 61 116 71 204 66 201 151 44 161276 ] A || 219 ] A || 219 (pr.) ] D || 219 ( 17 ] ) 300 275 48 282 267 115 253 125 260120 177 43 163 53 70 20267 203 58 210 160274 222157 170] 148 om. D || 222 ] AD || 223 ] om. D || 223 ] A, 235 285 271 52 AD153(278176 40 pr. scr. et del. A) || 219 ] om. A || 219 ] A || 219 22 21355 199 235 269 124 266112 175 145 179 147 49 57 121 63 206 D || 223 ] A || 223 ] A || 224 ] A, D || 302 211 216 109 214111 208117 288 3738 287 286 280 4542 284 110215 11421241 283 (post.) ] 39om. A || 220 ] (sic) A || 221 ] om. A. 78 92272 23251 22646 96 207 5122324 205 2200 322 277 123316 118 6279 9550 234 224 ] A || 225 ] D || 226 ] D || 226 ] D( 17 || 226 )

( 17 )

204 14 201 15 209 1044 225 276104 273105 281100 116315 119313 126308 47 22354 218

D || 227

]

202 11 203 18 210 31743 9853 222 274101 275108 282 227 115 8125312 120309 48 219 246 213 7199 17 206 12 285 97 271107 278102 124310 121311 112318 52 22049 221 40 228 76 248

]

321 215 16 212 4283 23137 217 287106 284 94 109307 111 13 117314 39 103 45 224

add. D. 319 324 1 2 323 322 3316 9 6 32095 229 234 9192 233 232 93226 9996 230

315

14

313 308

15

8

312

11

309 317

7

17

310 311

12

318

16

13

314

4

323

3

9

321 307 1

18

10

AD || 229

]

9

225 104 223 218 105 100 98

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

320 229

91

233

93

99

94 230

99

D || 230

] ]

AD || 229 A || 230

]

45

Text and translation

right-hand (row) is in excess by 34, and the left-hand one is in deficit by the same (amount). For the square of thirteen, the first upper (row) is in excess by 28, and the lower one is in deficit by the same (amount); the first right-hand (row) is in excess by 26, and the left-hand one is in deficit by the same (amount). The second upper (row) is in excess by 52, and the lower one is in deficit by the same (amount); the second right-hand (row) is in excess by 50, and the left-hand one is in deficit by the same (amount). The third upper (row) is in excess by 76, and the lower one is 42 Science of the magic square in deficit by the same (amount); the third right-hand (row) is in excess by 74, and the left-hand one is in deficit by the same (amount). Likewise for the others.60 L

C

1

n=5 n=9 n = 13 n = 17 n = 21

L

C

36 52 68 84

34 50 66 82

1

12 20 28 36 44

2

10 18 26 34 42

L

2

3

C

L

3

76 74 100 98 124 122

C

4

L

4

5

132 130 164 162

C

5

204 202

(In the square of) nine, the second (row) has 16 more than the (row) L C L C L C L C L C before. (In the square the3 first3 (row)4 has 48 more 1 1 of) thirteen, 2 2 5 than 5 the first n(of= the square) of nine, then each row has 24 more than the (row) 7 4 2 20 18 before. (In the square of) seventeen, the first (row) has 8 more than the n = 11 4 2 28 26 52 50 first nof=(the each66 row 32 15 square 4 2of) thirteen, 36 34then 68 100 more 98 than the (row) before. Then, in the same manner always, each row will 8 more n = 19 4 2 44 42 84 82 124 122have164 162 than 61 the one before. (§ 15. The concept n2 ≠1 of ‘sum n2 ≠5 due’)L ≠4 2

1

2

2+(

L

1

≠ 4)

C

1

≠2

4+(

C

1

≠ 2)

‘Excess its sum 38 due’ means The n = 9 over 40 16 (the following). 18 16 proper required 20 62 equals the quan(amount) for a cell [ of the totality of the numbers ] n = 13 84 82 24 26 24 28 tity of the median [ thus each cell of the square of five has a sum due n = 17 144 142 32 34 32 36 of 13 ]63 . Therefore you will add the odd numbers found in each row and 2 +1 n2 +1 divide the (result) by the number2 of concerned; · ncells –, –if+the 2 quotient is less 2 2 ≠ 12 than the median, the cells will contain less than their sum due by the ren=9 41 70 34, 36 sult of multiplying this deficit by the (number of) cells (filled); accordingly 60

n = 13 85 158 78, 80 L 145 n = 17 278 138, 140 See our table, with ∆i the excesses, or deficits, in the horizontal rows (‘Lines’) of

the ith border and ∆i those vertical rows (‘Columns’). 2 +1 n2in +1 the corresponding 2 · nthe ≠ 12 it will –, be – +successive 2 61 2 2 others, This is true for the first border. For multiples of 8. n=9 41 70 34,2 36 62 n +1 Early (now gloss ‘median’ (thus 78, for a square of order 280 n misplaced) = 13 85 explaining158 n); this term needs no explanation since it has been encountered regularly from § 3 on. n = 17 145 278 138, 140 63 Banal, and known from § 4. C

n2 +1 2

n=9 n = 13

41 85

C

2

34 50

2·

n2 +1 2

≠(

62 134

C

2

≠ 14)

–, – + 2 30, 32 66, 68

225 225 225 225 225 225 225 225 225 225

[[ [

]] ]

rr (46 D ,16 (14 14)) A: A: 13 13vv,9 ,9 -- 13 13vv,21 ,21 (ed. (ed. llll.. 610-623); 610-623); D:: 82 82 ,16 -- 82 82vv,1. ,1. Text and translation v v r v v r 14)) A: A: 13 13 ,9 ,9 -- 13 13 ,21 ,21 (ed. (ed. llll.. 610-623); 610-623); D : 82 ,16 - 82vv,1. ((206 14 ]] corr. A ]] om. ]] A 206 corr. ex ex A |||| 206 206 om. D D |||| 206 206 A |||| 206 206 230 230 206 corr. ex ex A |||| 206 206 ] om. D || 206 ] A || 206 ]]]] [ ]] corr. A [ [ [ 230206 ] 230 ]] A, D ]] om. ] A A |||| 207 207 A, D |||| 207 207 om. D D |||| 208 208 [[ ]] ]] A |||| 207 207 A, ]] A ]] A, D || 207 ] om. D || 208 ] ((16 16 A ]] (( pro sæpissime) D ]] om. A |||| 208 208 pro sæpissime) A, A, D |||| 209 209 om. )) ( 16 ) A |||| 208 208 A ]] (( pro sæpissime) A, D || 209 ] om. (pr.) A ]] om. (post.) A D |||| 209 209 (pr.) ]] A |||| 209 209 om. D D |||| 209 209 (post.) ]] A |||| 209 209 230 230 D 230 230 230 230 230 230 D |||| 209 209 (pr.) ]] A |||| 209 (pr.) A ] om. D || 209]] A ||]]209 230 D 230 ]] D ]] (sic) |||| 210 D 210 D |||| 210 210 (sic) A A 210 D ||||(post.) 210 ] D |||| 210 210 ]] D ]] (sic)] om. A || 210|| 211 ] D ||] 210 A || 211-2 ]((16 A ]] A 16)))) A |||| 211 211 A |||| 211 211 ] om. D D || 211 ] A || 211-2 ((16 16 A || 211 ] A || 211 ] om. D || 211 ] A || 211-2 ((16 16)) A || 211 ] A || 211 ]] A ]] A ]] A ]] A A |||| 212 212 A |||| 213 213 A |||| 214 214 A A |||| 212 212 ] A || 213 ] A || 214 ] A 270 A ] 270 56 268 262 262 60 180 146 178]]172 172 150 56 268 60 149 149 180 146 178 150 ]] A ]] A ]] 235 ||180215 215 A |||| 215 215 A |||| 217 217 235|| 270 259 268 262 262 178 159 172 154 56 254 60 171 149 158 146 164 150 261 68 69 268 180 178 172 56 60 149 146 150 261 270 259 254 171 169 164 68 69 64 64 158 169 159 154 || 215 A |||| 215 215 235|| 215 ]] A ] A || 217 A || 218 ] A || 218-9 235 A, om. D || 217 ] A || 217 ] ] 261 258 259 254 254 171 A, 169 164 165 68 259 69 263 64 152 158 155 159 173 154217 om. D ||173 ] A || 217 ] A || 218 ] A || 218-9 168 62 65 72 162 261 171 169 164 68 69 64 158 159 154 258 255 263 168 165 62 65 72 255 152 155 162 A, om. D ||174 217 A |||| 217 om. ]] A ] A ||(pr.) 218 ]D ||A219 || 218-9 258 256 255 263 151 168 166 165 || 173217 62 258 65 257 72 255 152 A, 155 162 D 167 174 61 71 66 161 156 263 168 165 173 62 65 72 152 155 162 256 257 264 166 167 61 71 66 264 151 161 156 17 ]] A ]] A ]] )) A |||| 219 219 A |||| 219 219 (pr.) ]] D || 219 ((17 256 257 257 264 166 157 167 170 174 61 253 71 256 66 264 151 163 161 160 156 148 267 70 67 58 166 167 174 61 71 66 151 161 156 267 253 260 177 163 170 70 67 260 58 177 160 157 148 ( 17] ) ]] A || 219 ] A || 219 (pr.) ] D || 219 A || 219 ((176 pr. ]] om. ]] A AD pr. scr. scr. et et del. del. A) A) |||| 219 219 om. A A |||| 219 219 A |||| 219 219 267 253 253 269 260 177 145 163 179 170 70 57 67 260 58235 160 AD 157 153 148 235 175 55 63 147 267 177 163 170 70 67 58 160 157 148 235 269 266 175 179 176 55 57 63 266 145 147 153 235 235 235 235 235 AD pr. scr. scr. et et del. del. A) A) || 219 AD ((176 pr. ] om. A || 219 ] A || 219 235 269 214 266235 175 179 176 55 110 57 208 63 114 145 179 147 280 153 216 288 286 41 38 42 269 266 175 55 57 63 145 147 153 216 214 208 288 286 280 110 114 41 (post.) 38 42 ]] (sic) ]] om. (post.) ]] om. om. A A |||| 220 220 (sic) A A |||| 221 221 om. A. A. 216 205 214 123 208 118 288 286 280 110 200 114 279 41 (post.) 38 272 42 A || 220 207 50 51 216 214 208 288 286 280 110 114 41 38 42 207 122 205 200 279 277 272 om. 122 123 118 50 277 51 46 46 (post.) ]] om. A || 220 ]] (sic) A || 221 ] om. A.

17 17)))) (((((17 17 (17 17))

207 122 205 200 200 123 279 276 277 272 272 273 122 119 123 209 118 279 50 277 51 281 46 116 44 47 207 205 118 50 51 46 204 201 209 281 116 204 119 126 126 201 44 276 47 54 54 273

204 119 201 209 209 44 276 274 273 282 281 116 204 119 203 126 120 44 53 47 275 54 273 115 48 201 116 126 47 54 202 203 210 43 275 282 115 125 125 202 120 210 43 276 53 274 48 281

99 9

202 203 203 120 210 43 274 275 275 278 282 115 125 125 202 120 210 43 53 53 52 48 282 213 40 115 48 213 199 199 124 206 112 285 271 271 274 124 121 121 206 112 285 52 49 49 278 40 213 109 199 124 206 112 285 37 271 287 278 284 124 121 121 206 112 285 52 39 49 45 40 213 199 52 49 40 215 111 212 283 283 271 109 215 111 117 117 212 37 287 39 278 45 284 283 234 287 232 284 109 215 111 117 117 212 37 287 39 226 45 284 324 22 322 66 283 95 92 96 109 111 37 39 45 324 215 322 316 316 212 226 95 234 92 232 96 324 22 308 322 15 316 10 234 92 232 226 226 96 95 104 92 232 96 315 234 14 66 225 95 315 324 313 322 308 316 225 223 218 218 105 14 313 15 10 104 223 105 100 100 315 312 313 18 308 309 225 222 223 218 218 105 14 313 15 317 10 225 104 223 105 100 100 315 8 11 98 14 15 10 317 222 101 219 227 227 8 312 11 308 18 309 98 104 101 108 108 219

999 9

( 15 ) A: 13v ,21 - 14r ,9 (ed. ll . 624-632); D 9 :9 982v ,1 - 82v ,6.

312 11 309 318 317 98 222 101 219 227 227 11 18 18 12 98 222 101 108 108 219 7878 312 17 310 311 311 309 318 97 220 221 221 102 228 17 310 12 317 97 107 107 220 102 228 310 13 311 314 318 220 221 221 102 228 17 310 12 318 97 217 107 220 102 94 240 77 307 17 12 97 107 321 231 16 44 240 321 307 231 217 106 224 228 16 311 13 314 106 103 103 224 94

222

] om. D || 222

323 11 323

320 229 229 91 233 93 230 91 233 93 99 99 230 99 320

33

224

D || 223 ]

AD || 223

]

321 307 307 323 314 231 91 217 233 224 230 16 13 106 93 103 224 94 321 231 16 44 229 94 1 313 99 320 320 229 217 233 103 1 323 3 314 91 106 93 99 99 230

]

A || 225

A || 223 ]

D || 226

]

] om. D || 223

A || 224

v ] 13vD ] AD ||: 82 228-9 ( 15 ) A: ,21||- 227 14r ,9 (ed. ll . 624-632); ,1 - 82v ,6.

222

] ] om. D || 222 AD ||] 229

AD ||] 223

D || 226

]

]

A,

]

]

A,

]

D ||

D || 226

AD || 229

D || 230 ] 223 A || 230 ] om. D || ]

A,]

D add. || 223D. ] A || 223 ] A || 224 ] A, D || )) ((18 18 ( 16 ) A: 14 ,9 - 14rr ,14 (ed. ll . 633-637); D : 82vv ,7 - 82vv ,10. 224( 16 ) A:] 14r ,9 -A 225(ed. ll] . 633-637); D || 226 ] D || 226 14||,14 D : 82 ,7 -]82 ,10. D || 226 tit. 245 192 61 71 256 257 66 264 166 D 167 :156 174 245151 161 tit. ] D D: vv|| 227 rr ] A || 228-9vv ] AD || 229 177 163 170 ,21 130 267 253 70 67 260 58 ( 160 157 148 -- 14 15 ) A: ( 15 ) A: 13 13 ,21 14 ,9 ,9 (ed. (ed. llll.. 624-632); 624-632); D D:: 82 82 ,1 ,1 -- 82 82vv,6. ,6. 231 ] AD || 231 ] D || 232 ] A || 232 ] corr. ex A || 194 265 55 269 57 63 266 175 145 179 147 153 176 || 231 ] AD || 232 AD || 232 ] corr. A || AD || 229 ] ||||D223 D || 230 ]]] om. ]D 222 222 ]] |||| 223 ]] ex ] A, 222231] ]] om. om.] D D ||||AD 222 AD 223 om. 223A || 230 A, om. A || 233 ] A || 233 ] A || 234 ] A || 24 113 216 110 214 208 114 41 288232 38 286 280] 42 270 56 44268 262 60 9 172 150 149 9 180 146 178 232add. ]D. om. A ||]]233 ] A ||]] 233A ] A || 234 ] D A || A || 223 || 224 ] A, || D || 223 A || 223 A || 224 ] A, D || || 207 122 205 272 51r 223 46 8 4 200 123 1189 279 50 277 D r v v 9 (158 261 68 259 4254 69 64 171 164 159 16169 ) A: 14154 ,14 (ed. ll . 633-637); D : 82 v,7 - 82 v,10. 235 ]r,9 - 14A ] A. r || 235 276 273 281 - 14 116 204 119 126 201 209 44 (224 16 )47A:54 14 ,14 (ed. D226 : 82 ,7 - 82]] ,10. 235 ] ,9 A ||||||235 168 155 162 165 ]]173 62 258 65 72 255 263 152 224 A 225 ]] ] D D D A 225 ll . 633-637); D ||||A. 226 D |||| 226 226 ]] D |||| 226 226 tit.53 D274 : 275 48 282r 300 115 125 202 203 120 210 43 61 71 256 257 66 264 151 161 A:17414r ,14 - 14rr ,20 (ed. ll . 638-644); D : 82vv ,10 - 82vv ,16. tit.(16617 D167): 156 77 33 ] D || 227 ] A || 228-9 ] AD || 229 ( 17 ) A: 14 ,14 14 ,20 (ed. ll . 638-644); D : 82 ,10 82 ,16. ] A || 228-9 ] AD || 229 278||40227 22 213 199 124 121 206 112 285 271] 52 49 D 2 267 253 70 767 260 58 177 3 163 160 157 170 148 7 3 231 ] om. ADA||||231 ] ] DD||||232 ] ] AA|| ||232 ] corr. ex 236 237 237 ] add. DA ||||238 302 211 109 215 111 117 212 283 287 39] ] 37236 45 284AD 231 231 ] D || 232 ] A || 232 ] ex D A 179 55 269 57 63 266 175 145 147 236 A ||AD ||AD 236|||| 229 ] D || 237 ] A || 237 ]230 ]] 153 ] 176om. ] D || 230 ] A || ]]|| ||238 229 ] D || 230 ] A ||corr. 230 add. 226 96 A || 233 78 5 324 2 322 316 6 95 232234 92 232] om. ] A || A || 233 234 A ||A || 216 110 214 208 114 41 288 38 286 ]280 42A || 238 ] 239 ] ] AA |||| 239 ] ] (sic) 270 180 149 146 150 232 ] 105 om. A || 233 ]A || A 234 A || )||) 270 56 268 262 262 60 178 172 56 11268 60 250 146 178 150 6149 6 225180 ((19 D. 250 19 315 14 313 308 15 10 218 add. D. ]172 A ] 239|| 233 ] ] AA||||239 ] ] (sic) A 104 223 add. 100 || 238 207 122 205 1200 123 118 279 50 277 272 ] 51 46 235 A || 235 ] A. 5261 6 261 68 259 1254 254 69 169 164 164 159 68 259 69 64 64 171 158 169 159 154 154 6171 158 239-40 ( . . .) ] in marg. A || 240 ] (sic) A || 240 222 101 108 219rr 227 A || 8 312 11 18 309 317 98235 vv ] 14 235(ed. ] ] A. 116 204 119 126 201 209 44 ( 47) 54 16 A: ,9 llll.. 633-637); 82 ,10. (276 16155 ) 162 A:273 14281 ,9 -- 14 14rr,14 ,14 (ed. 633-637); D::31 82vv,7 ,7 --||82 82240 ,10. 31 239-40 ( . . .) in D marg. A ] (sic) A || 240 62 258 65 255 263 263 152 168 155 165 173 173 62 258 65 72 72 255 152 168 162 165 246 7 17 310 311 12 318 97 107 220 221 102 31 r 228 r v v 274 275 48 282 115 125 202 203 120 210 43 ( 53 31 ] A || 240 ] A || 241 A || 241 ] A 17 ) A: 14 ,14 14 ,20 (ed. ll . 638-644); D : 82 ,10 82 ,16. ] 166 167 174 61 156 r r v v 256 257 257 66 264 151 166 167 174 61 71 71 256 66 264 151 161 161 156 tit. D : : 14 ] ,14 240 (ed. ll . 638-644); ] A 241,10 - 82 ,16. ] A || 241 ] A (tit. 17 A: - 14|| ,20 D :|| 82 217 )D 224 76 321 307 16 13 314 4 231 106 103 94 A 213 199 124 121 206 112 285 271 52 49 278 40 267 163 157 267 253 253 70 260 58 177 236 163 160 170 148 70 67 67 260 58 177 160 241 157 148 A || ] 236 add. ]et del. AD ||||241 ] ]add.Ain|| marg. ]93170 om. 237 237 A || 242 ] add. D || ]238 A 248 319 1 323 3 9 320 229 91|| 233 99 230 284 A ]|| 236 add. et 109 215 111 117 212 283 231 37 287 39 ] ]45 || 241 236 ] ]]del. AD||D ||241 ] ]] AinA ||marg. ]]] corr. add.ex 238|||| A ||237 ||237 231 ]om. AD |||| 231 231 D || 232 232 ] add. A || 232 232A || 242 corr. exD ||] A A 179 176 55 269 57 266 175 175 145 179 147 176 AD 55 269 57 63 63 266 145 147 153 153 324 2 322 316 6 232 226 95 234 || 92 ]243 96 ] AD 243 ] D || 243 ] A || 243 A || 238 ] A || 239 ] A || 239 ] (sic) A || 216 216 110 214 208 208 114 288 38 286 280 280 42 110 214 114 41 41 288 42 ||38 243 AD ] ] DA ]] A(sic) 232 ]]Aom. A ]] || ||243 A |||| 234 ]]|| 243 A ]286 || 238 A ]||]|| 243 239A A 232223 om. A] |||| 233 233 ] A |||| 233 233 A239 234 A |||| 315 14 313 308 15 10 225 104 218 105 100 207 50 207 122 205 200 200 123 279 239-40 277 272 272 ] 51 122 205 123 118 118 279 50 277 51 46 46 A. ( . . .) ] in marg. A || 240 ] (sic) A || 240 227 A. 8 312 11 18 309 317 98 222 108 ]219 235 ]273 A ]] ] in A. 239-40 ( . . .) marg. A || 240 ] (sic) A || 240 235101 ] 281 A |||| 235 235 A. 116 204 119 201 209 209 44 276 47 281 116 204 119 126 126 201 44 276 47 54 54 273 7 17 310 311 12 318 97 107 220 221] 102 228A || 240 ] A || 241 ] A || 241 ] A 274 275 282 115 43 53 48 202 203 203 120 210 255 274 275 282 115 125 125 202 120 210 43 53 48 255 ]2241494rr,14 A || 240 ] A || 241 ] A || 241 ] A 321 307 16 13 314 4 231 ( 217 17 )) 103 A: (271 17106 A: 1440,14 -- 14 14rr,20 ,20 (ed. (ed. llll.. 638-644); 638-644); D D:: 82 82vv,10 ,10 -- 82 82vv,16. ,16. 213 52 213 199 199 124 206 112 285 || 271241 278 40 124 121 121 206 112 285 52 49 49 278 ] add. et del. A || 241 ] add. in marg. A || 242 ] A 233 93 99 230 ] 1 323 3 9 320 229 || 91 241 add. et del. 241 ] add.] inAmarg. A || 242 ] ] A 284 109 37 215 111 212 283 283 236 287 39 284 A 109 215 111 117 117 212 37 287 39 ]]45 45 om. ]] A ||D add. 236 om. A |||| 236 236 D |||| 237 237 ] A |||| 237 237 ] add. D D |||| 238 238 ||234243 ] AD || 243 ] D || 243 ] A || 243 324 92 324 22 322 322 316 316 66 95 232 226 226 96 95 234 92 232 96 || 243] 243 ] 243 ] ] A || (sic) 243 ]] || A ||||] 238 ]] AD ||A ]] D ||A ]218 105 A100 238 A |||| 239 239 A |||| 239 239 ] (sic) A A || 315 315 14 313 308 308 15 225 104 223] 218 14 313 15 10 10 225 104 223 105 A. 100 ] A. 219 227 88 312 101 108 312 11 309 317 317 98 222 219 227 11 18 18 309 98 222 101 108 239-40 (( .. .. .) ]] in ]] (sic) 239-40 .) in marg. marg. A A |||| 240 240 (sic) A A |||| 240 240 270

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

132 59

268 262

72

60 149 180 146 178 172 150 64

171 158 169 164 159 154r

47

Text and translation

for an excess if (the quotient) is more.64 So the arrangement of the even numbers [ after that ] in the empty cells of each row must be such that it compensates the amount of the deficit, if any, or falls short by the amount of the excess, if any. We shall show how to do this in the appropriate place, when discussing the question of the even (numbers).65 (§ 16. Quantity of cells remaining empty in squares of orders n = 4t + 3) For the class of seven and those of the same kind (the situation is as follows).

There remain as empty cells in the first border — which surrounds the inner square — only the fourScience cornerofcells; in thesquare second border, the corner 42 the magic cells and sixteen cells adjacent to them, four on each side; in the third border, the empty cells are N the corner thirty-two adjacent N2 cellsNand N4 Ncells 1 3 5 (n = 4t + 1) to them. It will always be like that: each border has, excepting the corner 4 + 8 4 + 24 4 + 40 4 + 56 4 + 72 cells, sixteen more (empty cells) than the preceding border. 158

277

196 154

271 253

11

267 247 237

35

7

182 172 144 256 263 243 231 229

51

31

194 184 150

15

170 286

83

95

91

87

84

82

130

98

134

94

116 114 138 106

96

100

3

2

213 215 219 223

79

142 118 108 4

120

25

93

111 119 115 185 187 191 107 197 265

21

45

89

117 131 135 165 167 127 173 201 245 269

17

41

57

85

113 133 143 141 151 157 177 205 233 249 273

39

55

63

81

109 129 153 145 137 161 181 209 227 235 251 275

281 257 241 217 189 169 139 149 147 121 101

73

49

33

285 261 221 193 163 155 125 123 159

97

69

29

5

289 225 183 171 175 105 103

99

179

65

1

36

211 195 199 203

71

67

148

34

27

152 156

77

75

207 254

47

59

61

239 259 287 288 146

23

43

53

255 283

37

279

19

(n = 4t + 3) 160

13

9

174 176 140

N1 4

L

N2

N3

N4

N5

4 + 16

4 + 32

4 + 48

4 + 64

190 192 136

206 208 132

C

L

C

L

C

36 44

34 42

68 84

66 82

L

C

L

C

(§ 17. Excesses and deficits in1 each row for 1 2 2 the order 3 3n = 4t + 4 3) 4 5 5 In all the squares of this n= 5 12 10 kind, the first upper row has an excess of 4 over its sumn due, lower one = 9 and20the 18 36 has 34 a deficit of the same (amount); n = 13 28 26 50 of 276over74its sum due, and the the first right-hand (row) has an52 excess n = 17has a36 34 of the 68 same 66 (amount). 100 98 The 132second 130 upper left-hand (row) deficit = 21 of)44seven 42 has 84 82 124 164sum 162due,204 row of the n(square an excess of 20122 over its and202 the second right-hand (row an excess of) 18. In the square of eleven, the second upper (row) has Lan excess ofL28 over its Lsum Cdue, the second rightC C L C L C 1 26,1 the third 2 2 3 4 4 hand (row an excess of) upper (row 3an excess of) 52, 5the 5 n = 7 (row an 4 excess 2 2050,18 third right-hand of) and all their opposite (rows) have n = 11 4 2 28 26 52 the 50 second upper (row) a deficit equal to the excess. (Square of) fifteen: 64

n = 15 n = 19

4 4

2 2

100 98 124 122

164 162

This is confusing since the effect of the previous division is cancelled by the subsequent multiplication. To find the excess or deficit displayed by a border’s row, just take the difference between2the sum 2found in the m cells already filled in this row and L C n ≠1 n ≠5 2 + ( L1 ≠ 4) 4+( the product of m by the median number. 1 ≠4 1 ≠2 2 2 65 §§ 18–22 (equalization rules).

n=9 n = 13 n = 17

40 84 144

38 82 142

16 24 32

n2 +1 2

2·

18 26 34

n2 +1 2

≠ 12

16 24 32

–, – + 2

C

1

20 28 36

≠ 2)

--

.. .. . .... . om. ] ] AA|| ||240 ] ] AA|| ||241 ] ] AA||||241 ] ] AA 240 241 241 ] ]14 A,14 241 AA||||241 ] ] AA 235 ]1414,14 ||240 ]] A A.A||D A||-A 241 rr r235 vv vv vv vv ] ] r,20(ed. ((17 )) )A: 14 ,20 llllll .ll :: 241 ,10 -- 82 ,16. (17 17 A: -240 14 (ed. 638-644); :82 82 ,10 -82 82 ,16. A: ,14 -|| 14 ,20 (ed. . .638-644); 638-644); D||D 82 ,10 ,16. ( 17 ) A: 14 ,14 14 ,20 (ed. . 638-644); D : 82 ,10 82 ,16. || ||241 ]] add. ] ]add. ] ] AA 241 add.etetdel. del.AA|| ||241 241 add.ininmarg. marg.AA||||242 242 9 9 9 9 || 241 ] add. et del. A || 241 ] add. in marg. A || 242 ] AA || 241 ] add. et del. A || 241 ] add. in marg. A || 242 ] 238 r v ||237 99 v ,10 236 ]] om. 236 D ]] ]] -A ]] ]] add. ]om. 236 237 A 237 add. 238 236 AA 236 DD 237 A add. DD (236 17 ) A: 14 ,14 ,20 (ed.]] ]]ll . 638-644); D : 82 82 ,16. 236 ] om. om.A A||||-||||14 236 D||||||||237 237 A|||| ||237 237 add.D D||||||||238 238 48 Text and translation 240 311 220 221 228 7 307 17 310 12 318 97 || 107 102 314 231 217 224 16 13 44 240 106 103 94 || 243 ] AD || 243 ] D || 243 ] A || 243 321 314 231 217 224 16 13 106 103 94] 243 AD || 243 ] D || 243 ] A || 243 324 322 316 234 232 226 2 6 95 92 96 324324 3242 22322322 322316316 316 6 66 9595 2349292 232226226 2269696 95234234 92232232 96 || 243 ] AD || 243 ] D || 243 ] A || 243 || 243 ] AD || 243 ] D || 243 ] A || 243 r 233 v239 ]]AA 236 ]14 om. A238 ||(ed. 236 ll ] A D || D 237: 82 ] A A 237 ] ]] ]] add. DA ||A ]]238 ]218 A ]] ].] 633-637); ]] ]]v ,7 |||||||||| 239 (sic) |||||||| 321 307 231 217 224 16 13 4 229 106 94 320 230 11 (323 33 )314 99A: 91 93 99 A ||||238 239 (sic) ]218]218 A ||100 238 AA 239 A-A 239 (sic) 323 320 229 233 230 91 223 93 99 ]-103 Ar||100 238 A||||||||239 239 A 239 (sic) 16 14 ,9 ,14 82 ,10. 313 308 225 223 14 15 104 100 104 105 313313 308308 225225 223]223 1414 1515 1010 104 105 100 313 308 225 14 1510 10 104 105 A. ] 218105 A. 230 1 323 3 9 320 229 91 ]233] 93 99 A. A. 309 317 222 219 227 11 98 101 108 222 88312 101 312312 309309 317317 222 219219 227227 1111 1818 9898 101 108 312 309 317 222 219 227 1118 18 98 101 108 239-40 ]] ]in marg. 239-40 .. .) marg. 240 (sic) 240 239-40 .) marg. AA 240 (sic) A 240 ]108 A || 238 (( .(.(....) ].) A ||in ]||240 ]A (sic) 239-40 ]in in239 marg.A A|||||| 240A || 239 ]] ]] (sic) (sic)A A||||||||240 240 A || 309309 317317 222222 219219 227227 11 11 98 98 101101 108 8312312 324 322 316 234 232 226 2218 18 66 95 92 96 215 212 283 287 284 109 111 117 37 39 45 324 322 316 234 232 226 95 92 108 96 204 201 209 276 273 281 119 126 44 119 126 204204 201201 209209 276276 273273 281281 119 126 4444 4747 5454 204 201 209 27647 273 281 119 126 44 4754 54 310310 311311 318318 220220 221221 228228 12 12 107107 102102 7 313 308 225 223 218 324 322 316 234 232 226 14 15 10 104 105 100 2 697 97 95 92 96 315 17 313 308 225 223 218 14 15 10 104 105 100 202 203 210 274 275 282 120 43 120 43 255 202202 203203 210210 274274 275275 282282 120 43 5353 4848 255 202 203 210 274 27548 282 120 4353 53 48 255 255 rr rr 314314 231 217217 224224 16 16 13 13 106106 103103 94 94 231 4317 309 222 219 227 315 313 308 225 223 218 88 312 11 18 98 101 108 14 154 317 10 104 105 100 240 312 309 222 219 227 11 18 98 101 108 240 240 240 206 285 271 278 124 121 112 271 124 121 112 206206 285285 271 278278 124 121 112 5252 4949 4040 206 285 27152 27840 124 121 112 5249 49 40 323323 320320 229229 233233 230230 91 91 93 93 99 99 117 3311 309 317 222 219 227 310 318 220 221 113 311 189 9 98 101 108 7871 312 17 12 97 107 102 310 318 220 221 228 12 97 107 102 r228 215 212 283 287 284 109 111 117 37 109 111 117 215215 212212 283283 287287 284284 109 111 117 3737 3939 4545 215 212 283 28739 284 109 111 117 37 3945 45

tit. D :

310 311 318 220 221 228 12 97 107 102 17 107 102 310310 311311 318318 220220 221221 228228 1212 9797 107 102 310 311 318 220 221 228 17 12 97 107 102

77

]] ]] in A A 241 A 241 marg. A || 240 A||||||||241 241

A 240 AA 240 A||||||||240 240( . . .)

239-40]] ]]

314 217 224 16 106 103 106 103 314314 231231 217217 224224 1616 1313 106 103 9494 31444 44231 231 217 22494 1613 13 106 103 94

]] ]]

]] ]] 241 AA 241 ]A A||||||(sic) ||241 241A || 240

A AA A

r v v ] ] ||||(||||241 ]del. D ||242 232 15 ) AD A:] 13||]],21 ,9 (ed. ll .A 624-632); D ,1 -in 82 ,6. add. et ]82 ]]]]] corr. A 241 ]]231 add. del. 241 add. in marg. A 242 241 add. etet AA 241 ] ]add. in marg. A 242 AA 241 add. etdel. del. A|||| 241 ]add. add. inmarg. marg.A A|||||||| 242 A ex A ||- 14 240 ] ||||241 A|| || :232 241 ]A A || 241 [ 260 [ 260 [ 260 [ 260 ( 18 )18 ) ))) (A 18 (A, 18 222 ] om.]] D ] ||||||||243 ]marg. om. DA |||| 223 || 241 ]]] || 222 add. et del. ||AD 241 ] add. 242 ] ( (18 AD D ]] 234 243 AD 243 243 ]] A 243 243 AD 243 DD 243 AA 243 ||243 243 A AD 243 D||||||in ||243 243 A||||]||||243 243 ]||||om. || 233 ]A A|| ]]223 ||]] 233 ] A || ] 9 9 D A. ||A. 223 ] A || 223 ] ] D A || || 243 224 ] ] A, A || 243 ]]D || )) ] ] ( 18 ] ( 18 ] ] A. ] A. || 243 ] AD || 243 245 245 245 3333 ] 245224 A ||] 235 A || 225 ] ] A. D || 226 ] D || 226 ] D || 226

231

v 229 233 230 11 11323 91 323323 320320 229229 233233 230230 9191 9393 9999 32333 33 99 99320 320 229 23393 230 91 9399 99

232 4 4 235

A || A ||

A. D vv||r 227 rr ] A || 228-9vv v vv ] AD || 229 ( 15 )155 13 ,9 .. 624-632); :: 82 ,1 82 (417 ) A: - 14 ,20--14 (ed. ll . llll638-644); :8282 - 82v ,16. 4 9 9 14 15,14 ) A: A: 13173,21 ,21 14 ,9 (ed. (ed. 624-632); D DD ,1 --,10 82 ,6. ,6. 62 258 65 72 255 263 152( 168 162 165 AD || 229 ] |||| 223 ] ] A, 61 71 256 257 66 264 151222 161 166 ]167 156 174D || 222 om. AD 223 222 om. D || 222 AD 223 D || 230 ]] om. 223A || 230 ]] A, 236 ] 6149 om. A157]] || 236 ] ]] D || 237 ] om. ADD||||||237 ] add. D || 238 267 270 253 56 260 60 163 146 170 150 67 262 58 160 178 148 172 9177 180 170414268 6 9 add. D. D || 223 ] A || 223 ] A || 224 ] A, D |||| D || 223 ] A || 223 ] A || 224 ] A, D r176 r v v 9175(158 74254 269 266 179 55 7 57 69 63 64 145 147 153 3 3171 259 169 164 68 159 154 16 ) A: 14 r,9 - 14 r,14 (ed. ll . 633-637); D : 82 v,7 - 82 v,10. (224 1615538) 238 A: 1442,9 - 14A . 633-637); : 82 ,7]- 82]] ,10.A || D ] 263265 A224 ],14|||| (ed. 239 216 65 214 255 208 288|| 286 165 280]]173 110 72 114265 41 168 62 258 152 162 225 ]]|| 239 D |||| 226 D A 225 llA D ||||D226 226 D 226 ]] D ||||] 226 226 (sic) A || 265 265 tit. D 272 : 51 46 207 71 205 257 200 66 279 161 277 122 256 123 264 118 151 50 166 167 174 61 156 tit. D : 77 33 ]]47 15754 170 D ]]] in marg. A ]] 229 D || || 227 227 A |||| 228-9 228-9 AD ||||(sic) 229 A || 240 A || ] 240 ]AD 201 58 2099177 276 160 281 ( . . .) 116 204 239-40 119 126 260 163 4147 67 41470 9 69 69344 231 ] 273 148 AD || 231 ] D || 232 A || 232 ] corr. ex ( (19 A(19 ||)19 250 250 250 250 ((19 19) ))) 231 ] ] D || 232 ] A || 232 ] corr. ex A 202 57 203 63 210 175 274 147 275 153 282 AD || 231 115 55 125 269 120 266 43 145 53 179 48 176 ]] AD ]] D ]] A ]] || AD |||| 229 229 D |||| 230 230 A |||| 230 230 ] A || 240 ] A || 241 ] A || 241 ] A 232 ] om. A || 233 ] A || 233 ] A || 234 ] A || 213 216 199 110 206 114 285 288 271 38 278 42 124 214 121 208 112 41 52 286 49 280 40 10 10 10 270 180 149 146 150 232 ] om. A || 233 ] A ||10 233 ] A || 234 ] A || )) 270 56 268 262 262 60 180 178 172 172 56 11268 60 250 146 178 150 6149 6 ( 19 add. D. 250 ( 19 add. D. 215 200 212 279 283 50 287 272 284 109 205 111 123 117 118 37 277 39 51 45 46 122 235 ]159 A || 235 ] A. 259 68 154 259771254 254 69 169 164 164 159 68 7 69 64 64 158 169 154 33 33171 6171 158 7 || 241 ])) A: add. et || in marg. A || 242 ] A r96 235 ]22614 235del. ] 241 A. D r,9 324 119 322 201 316 209 234 232 273 2 126 6 44 95 ( 92 204 47 54 16 (ed. .. 633-637); 82 -- 82 (276 16155 A: 14281 ,9 -A - 14 14||rr,14 ,14 (ed. llllA 633-637); D::31 82vv],7 ,7add. 82vv,10. ,10. 31 162 258 65 255 263 263 152 168 155 165 173 173 62 258 65 72 72 255 152 168 162 165 3231 32 r100 r v v 315 14 202 313 203 308 120 225( 53 223 A: 218 48 15 210 10 43 104 274 105 275 282 17 ) 14 ,14 14 ,20 (ed. ll . 638-644); D : 82 ,10 82 ,16. r v 256 257 257 66 264 151 166 167:: 156 61 71 256 66 264 151 161 161 156 174 tit. D167 (tit. 17166 )]D A: 14174 ,14 - 14r ,20 (ed.||ll .243 638-644); D] : 82v ,10D - 82|| ,16. AD 243 ] A || 243 309 112 317 285 222 52 219 40 227 8 312 || 11 243 18 206 98 271 101 108 278 124 121 49 163 170 70 67 260 260 58 177 236 163 160 170 148 58 6177 160 157 157 148 A || 236 ] om. ] D || 237 ] A || 237 ] add. D || 238 117011 67 66 6 310 111 311 117 318 283 220 39 221] 45 228 A || 236 7 109 17 215 12 212 97 236 107 287 102 284 37 om. ] D || 237 ] A || 237 ] add. D || 238 231 ] AD || 231 ] D || 232 ] A || 232 ] corr. ex A 231 ] 176 ] D || 232 ] A || 232 ] corr. ex A |||| 55 145 147 269 57 176 AD || 231 55 269 57 63 63 266 145 147 153 153 ]314 11426664 17541175 A.17938179 321 324 307 216 322 231 234 217 224 13 316 106] 232 103 226 94 238 95 92 96 A || ] A || 239 ] A || 239 ] (sic) A || 216 214 208 288 286 280 110 42 216 110 214 208 114 41 288 38 286 280 42 232 ]99 A ]] ||A |||| 234 ]] A ] 93 105 || 238 A ]]|| 239A A ] (sic) A 232 ]Aom. om. A |||| 233 233 ] A |||| 233 233 ] A239 234 A |||| 323 308 320 225 229 104 233 218 230 3 159 10 91 223 141 313 100 255 255 279 277 272 122 50 51 46 255 205 200 200 123 279255 277 272 122 205 123 118 118 50 51 46 239-40 ( . . .) ] in marg. A || 240 ] (sic) A || 240 227 8 312 11 18 309 317 98 222 108 219 235 ]273 A ]] ] in A. 239-40 ( . . .) marg. A || 240 ] (sic) A || 240 235101 ] 281 A |||| 235 235 A. 204 204 119 201 209 209 44 276 47 281 119 126 126 201 44 276 47 54 54 273 7 17 310 311 12 318 97 107 220 221] 102 228A || 240 ] A || 241 ] A || 241 ] A 32 32 32 32 202 43 202 203 203 120 210 255 274 275 275 48 282 120 210 43 53 53 274 48 282 255 ]2241494rr,14 A ||- 14 240 ] A ||D241 ] A || 241 ] A rr vv vv 217 16 13 314 4 231 106 103 ( 17 ) A: ,20 (ed. ll . 638-644); : 82 ,10 82 ,16. ( 17 ) A: 14 ,14 14 ,20 (ed. ll . 638-644); D : 82 ,10 82 ,16. 240 240 240 271 124 52 206 112 285 || 271 278 40 124 121 121 206 112 285 52 49 49 278 40 ] 241 add. et del. A || 241 ] add. in marg. A || 242 ] ] ] ]]A ] 233 93 99 230 ] 1 323 3 9 320 229 || 91 241 add. et del. 241 ] add.] inAmarg. A || 242 ] ] A 284 109 37 215 111 212 283 283 236 287 39 284 A 109 215 111 117 117 212 37 287 39 ]]45 45 om. ]] A ||D add. 236 om. A |||| 236 236 D |||| 237 237 ] A |||| 237 237 ] add. D D |||| 238 238 ||234243 ] AD || 243 ] D || 243 ] A || 243 33 324 92 33 324 22 322 322 316 316 66 95 232 226 226 96 95 234 92 232 96 || 243] 243 ] D ||A243 ] ] A || (sic) 243 ]] || A ||||] 238 ]] AD ||A ]218 105 A100 238 A |||| 239 239 3333 ]] A |||| 239 239 ] (sic) A A || 14 313 308 308 15 225 104 223] 218 14 313 15 10 10 225 104 223 105 A. 100 ] A. 219 227 88 312 101 108 312 11 309 317 317 98 222 219 227 11 18 18 309 98 222 101 108 239-40 (( .. .. .) ]] in ]] (sic) 239-40 .) in marg. marg. A |||| 240 240 (sic) A A |||| 240 240 33A 33 270

261

268 262

56

]

60 149 180 146 178 172 150

245

3 245 3171 158 169 164 159 154 64

7 2597 254 69

68

r ]

17 310 311 311 12 318 97 220 221 221 102 228 17 310 12 318 97 107 107 220 102 228

77

16 314 16 13 13 314 323 323

11

33

99

56

268

132 59

270

261

68

62

258

65

61

71

256 257

192

4259 254 72

130 267 253 7 70

67

554 269

57

194 265

r [ [ A ]] A A ]] A A-||||14240 240 AD||||: 241 241 A |||| 241 241 A ( 16 ) A:]] 14r ,9 82v ,7 - 82v ,10. ]] [,14 [(ed. ll . 633-637); 320 233 91 320 229 229 || 233 93 230 ]] 91 241 93 99 99 230 add. et del. A || 241 ] add. in marg. A || 242 ] A || 241 add. et del. A || 241 ] add. in marg. A || 242 ] A tit. D : v ] ](]]18 ( 18 ) A: 14r ,20 - [14 ,3 (ed. ll . 645-650); D : 82v ,16 - 82v ,19. ] ((18 260 [ 260 )18)) 262 60 149 180 146 178 172 ] 150 ||231 ]232 D ]]] corr. A || 243 243 ]AD || 231 AD AD] |||| 243 243 D ||33 ] D ]|||| 243 243 A || || 243 243 A || 33 33 ] A || 232 ex 33 tit. (hab. ante infra) D : 9 69 64 171 158 169 164 159 154 ] ]] A. A. 232 A ||33233D || 245 ] A || ]234 ] || 246 A] || ] 245168 155 162 255 245 263 152 165 173A || 233 245 244 ] om. ] A || 245 ] ] A, A, D 66 264 151 161 166 167 156 174 235 AD] || 247A || 235 A.|| 247 ] ] D ] A || 247 ] A || 248 ] 260 260 260 231 217 44260 231260 217 106 224 94 106 103 103 224 94

260 358

177 163 160 157 170 148

r v A ||- 14 248 add.DD: || ] om. D. ( 17 ) A:add. 14 ,14 ,20 (ed.] ll . 638-644); 82249 ,10 - 82v ,16.

r 176 63 9266 175 145 179 147 153

24 113 216 110 214 208 114 41

288

38

286 280

44

276

47

54

43

53

274 275

42

236(5019 ] A: om. A ] D || 237 A ||r 237 v || 236v v] 277) 272 51 14 46 ,3 - 14 ,13 (ed. ll . 651-661); D : 82 ,19 - 83 ,4.

265 265 207 122 205 200 123 279 118 265 265 265 1

6

116 2047 119 126 201 3209 300 115 125 202 203 120 210 22

265 265

78

246 76

A ]||om. 238 D || 250 ]

] 250

239-4052]

213 199 124 121 206 112 285 271

302 211 109 215 111 117 212 283

414

324

5

322 316

2

969

37

273 281

287

48

278 45

251 ]

A || 240 ]

14

313 308

15

312

11

309 317

7

77 33 17 310 311 12 318 97 107 220 221 102 228

98 222 101 || 243

218 105

108 219 227

] ]

D A.|| 257 A || 259

]

D || 254 AD || 243 ]

270 56 268 262 60 149 180 146 178 172 150 321 307 16 13 314 4 231 217 106 103 224 94

261 68 259 254 69 64 171 158 169 164 159 154 248 319 1 323 3 9 320 229 91 233 93 99 230 62

258

65 1 1

61

71

72

]

255 263 152 168 155 162 165 173

66

A,

256 257

66

264 151 161 166 167 156 174

267 253

70

67

260

58

55

269

57

63

255 266255 175 255 145 179 147 153 176

177 163 160 157 170 148

216 110 214 208 114 41

288

38

207 122 205 200 123 118 279

50

277 272

51

46

] D ||A252 || 241

A,

]

A, ) A || 262

A || 259

A || 259

] ]

D || 260 ]

A, 32 32

] (sic)DA|| ||251 ] ]

A ||

A A || ( 19 )19)) ((19

] om. ] A ||A253

] rescr. DA||||256 ] 243

D |||| 243 255 D ]

] D || 258 32

D || 259

(corr. ex 42

286 280

10 10

]

add. D || 238

] || (sic) A 251 A || 240 ]

]

et del. ] A || 241 ] add. in marg. A. ||. . 242 A add. || 252 A || 253-8

]100

8

10

10 10 10] ] A || A || 10 241 252

284

315

18

A || ]250 A ||] 239 D || 251 ]

( . . .)] add. in ]marg. in marg. D || D || A 251|| 240 40 251

49 39

250 250234 92 232 226 96 250 6 95

|| 241 252 225 104 223

A ||] 239

282

]

]

]

A || 259

A,

]

] D || 259-60

A,

D || 262

]

D || 262-3

(quater ) ]

D ||

L

C

1

1

n2 +7 − 2 L

2

4jC

L

2

3

n2 +3 − 2 C

3

4j

L

=⇒ nC2 + 1 + 4 − 8j L

C

4

4

5

5

n=5 12 10 Text and translation 49 n=9 20 18 34 n236 −5 n2 −1 =⇒ n2 + 1 − 4 + 8j + 4j + 4j 2 2 n = 13 28 26 52 50 76 74 n = 17 36 34 68 66 100 98 132 130 n = the 21 first 44(upper 42 row) 84 by82 124third 122 the 164 162by 32. 204 And 202 so exceeds 32, the second

on for the others, (with excesses and deficits) increasing each time by 8.66 L

C

1

n = 7n + 7 n = 11 2 n = 15 n = 19 2

L

1

2

4 2 − 4j 4 2 4 2 4 2

n2 − 1 2

20 28 36 44

C

L

2

3

C

18 n + 3 − 4j 26 252 50 34 68 66 42 84 82 n2 − 5 L ≠ 4 22 + (

n ≠5 write2the

L

C

4

2

4jeven 2numbers by pairs) + 4j (§ 18. Grouping + the 2 n ≠1 need2to

L

3

L

4

5

C

5

=⇒ n2 + 1 + 4 − 8j

100 98 124 122 ≠ 4)

164 162

=⇒ n2 + 1 − 4 + 8j 4 + ( C1 ≠ 2) 1 ≠2 C

1 Thus we even (numbers) 1in the rows so as to equalize ⇓ 40 = 9 (therefore 16know ⇓ 16 in searching 20 for them,n and we38need) to how18to proceed 2 2 n + 1 − 2 n + 1 + 2 n = 13 displaying 84 82 required 24 amount 26 of the excess 24 or displaying 28 the numbers the n = 17 144 142 32 34 32 36 the (required) amount of the deficit. The way to determine this is (first) to associate 2 and the last even 2 n2 +1 2and · n 2+1 –, – reach + 2 the two terms 2 term, then 4 and its complement, so≠on12until you 67 which are then two = 9 medians 41 of these even 70 numbers.34, 36 2 2 n =4 13 | . . . 85 . .80 . | | n 2+3 − 4j 158 n 2+7 − 4j |78, n = 17 145 278 138, 140 2 2 n2 − 1 n2 − 3 | . . .n2|+1n 2−1 + 4jn2 +1 n 2−5 + 4j | . . . | 2 · ≠ 12 –, – + 2 2 2

2

n2 −5 2

n2 −1 2

n2 +7 2

n2 +3 2

n =placing 9 70 numbers34, (§ 19. Effect of a 41 pair of small even in 36 the same row) n = 13

85

158

78, 80

Once you have done that, you will find that if the last small number n = 17 145 278 138, 140 and the preceding one are placed on one side,68 and opposite to them their complements, the side containing the two small numbers will be less 2 2 C C 2 · n 2+1more ≠ ( by 14)Placing –, –the + 2next two than its sum duen 2+1 by 4, and2 the other 2 ≠ 4. small nnumbers on41one side34and opposite to the =9 62 them their complements, 30, 32 side containing the two small numbers will be less than its sum due by n = 13 85 50 134 66, 68 12, and more by66the same (amount). Doing the118, same n =the 17 other145 238 120 with the next pair, excess and deficit will be 20. And so on always until you reach 4 and 2.69 66

The differences of the excesses, both between successive borders and successive squares, are successive multiples of 8. As above (§ 14, n. 61), some further explanation might be missing. 67 As seen from the table, the pairs of consecutive smaller numbers are of the general 2 2 form n 2+3 − 4j, n 2+7 − 4j, with j taking successive natural values (j = 1 making the 2 2 last pair n 2−5 , n 2−1 ). 2 2 68 As seen above, the ‘last small (even) number’ is n 2−1 and the one before, n 2−5 . 69 See the figure below, with the pairs written in their general form (§ 18), and from right to left, as appropriate here.

]] om. D || 222 AD || 229 ] || 223A || 230]]] ] ]A, ] AD] || 223 D || 230] om. D A A, A, ] || ] add. D ||A238 add. D. D || 223 ] A || 223 ] A || 224 ] A, D |||||| )) D ||||154 223 ]] AD ||A |||| 223 ]]v D A ||||v243 224 ]] ] A, D ( 18 32 D 223 A 223 A 224 A, D ( 18 32 243 ] 243 ] || A || 243 r176 r 32 9175|| 2694254 266 171 179 55 259 57 69 63 64 145 147 153 32 169 164 68 158 159 ( 16 ) A A: || 14 235 ,9 - 14 r,14 (ed. ll]. 633-637); D : 82 v,7 - 82 v,10. r 235 ] 15241A A. (224 1615538) 238 A: 14 ,9 - 14A (ed. . ]633-637); : 82 ,7]- 82]]] ,10.A ||D ] 263 ],14 239 (sic) A || 224 ]A. ||||||225 D 226 ||||||226 216 56 214 262 208 288|| 286 172 280 110 268 114 149 42 258 255 168 165 173 65 72 162 224 ]]150 A 225 ]]|| 239 D 226 226 D 226 A 225 llA D||||||D 226 D 226 ]]] D D||||||]]226 270 180 178 60 146 50 Text and translation D ] ]]]] tit. D : ] 226 245 245 205 254 200 69 279 161 277 272 122 259 123 264 118 171 50 169 51 154 46 256 257 166 167 174 66 151 156 tit. D : 164 68 64 159 77 33 158 D 227 ]]]] in A 228-9 ]]] AD 229 r )]]] A: 13 r,21 r.) vv D || 227 A 228-9 229 r v,1 v Dvv|| ||,20 227 A||||D ||D AD||||||(sic) 229 A || 240 (--.14 .(ed. marg. 240 ]AD 33 ((15 (ed. .. ]624-632); :228-9 82 ( 239-40 17 )255260201A: 14 - ]14 ll . llll638-644); D :82vA 82 -A82|| v232 ,16. 204 209 152 276,14 273 173 281 119 72 126 54 15 )47A: 13 ,21 14 ,9 (ed. 624-632); : 82 ,1 --||,10 82 ,6. 177 163 170 70 58 160 157 148 33 258 263 168 165 65 155 162 7 67 344 231 AD ||,9 231 D || 232 ] ,6. ] corr. ex A || 33 33 231 ] AD || 231 ] D || 232 ] A || 232 ] corr. ex || 202 257 203 63 210 175 274 ]167 282 120 264 43 161 53 166 48 174 269 266 179 176 55 256 57 145 147 153 AD ]]] |||| 223 D ]D A ]A 66 151 156 ]]275 AD 229 D 230 ]] |||| 223 A 230 ]]A, AD||||||229 229 D||||||230 230]] om. A||||||230 230]] 222 om. D || 222 ] AD D 222 ] om. D || 222 ] AD 223 om. 223 A, ] A || 240 ] A || 241 ] A || 241 A 232 ] om. A || 233 ] A || 233 ] A || 234 ] A || 236 ] om. A || 236 ] D || 237 ] A || 237 ] add. D] || 238 206 285 271 278 124 121 112 52 49 40 216 214 208 288 286 280 110 114 41 38 42 260 177 180 163 170 270 268 178 172 70 67 58 160 157 148 149 146 9149 270 262 180 172 56 60 146 150 232 ] 150 om. ] A 33 || 233 ] A || 234 ] A || 270 56 268 262 262 60 180 178 172 564411268 60 9 146 178 150 add. D. 33 6149 6 250 ((19 add. D. 250 19 add. D. A r || 233 D ||||154 223 ]] A |||| 223 ]]v A ||||v 224 ]] A, D |||| )) D 223 A 223 A 224 A, D 215 2129279 283 50 287 272 284 109 269 111 123 117 266 37 179 39 45 r176 205 200 277 122 118 51 46 4 175 235 ] A || 235 ] A. 55 57 63 145 147 153 259 254 171 169 164 68 69 64 158 159 ( 16 14 ,9 - 14 r,14 ll .A 633-637); D : 82 v,7 - 82 v,10. 169 164 68 158 154 259 1254 254 69 169A: 164 159 68 259 69 64 64 171 158 ) rradd. rret (ed. v add. vv in marg. A || 242 6171 r154 || 241 241 ] A 16 A: 14 14 (ed. 633-637); D 82 ,7 82 r96 v ,10. 260 235 ]159 ||r],14 235 ] 239 A. D [[del. 260 r,9--A r[,14 (224 16 )92])))238 A: 14 ,9 ::::31 82 324 119 322 201 234 232 273 226 2 126 6 95260 204 209 276 281 44 47 54 ,14 (ed.llA 633-637); D 26041 ]316 114 A ||155 |||| A || DD239 260 (288 16 A: 14 (ed. llllll....633-637); 82 ,7 82 ,10. 260 ((168 16 A: 14173 ,9 --14 14A 82vvv],7 ,7]----82 82 ,10. 31 216 214 208 286 28014 110 38 42 ,9 258 255 168 165 ||||[[(ed. 225 ]]633-637); D 226 ]] v,10. |||| 226 D A[,14 225 D ||||D 226 226 ]] D ||||] 226 226 (sic) A || 258 263 65 72 152 162 258 65 255 263 263 152 168 155 165]]173 173 65 72 72 255 152224 155162 162 165 r r v v tit. D : 313 203 308 120 225( 53 223 A: 218 48 14 202 15 210 10 43 104 274 105 100 275 282 31 17 ) 14 ,14 14 ,20 (ed. ll . 638-644); D : 82 ,10 82 ,16. 205 200 66 279 161 277 272 122 256 123 264 118 151 50 51 174 46 257 166 167 tit. D 256 167 156 r r v D ::::156 256 257 66 264 33151 166 167 174 66 264 151 tit. 161 156 174 tit. D 77 257 tit. (161 17166 )]D A: 14|| ,14 ,20 (ed.||ll]]].243 638-644); D] : A 82v ,10 - 82|| ,16. ]]]]||||240 A 229 || AD 243]] ] ]]AA 243 D 227 A |||| 228-9 228-9 AD ||||(sic) 229 309 112 317 285 222 ]]52 219 || 227227 8 312239-40 11 243 18 206 98 271 101 108 D 278 124 121 49 40 ( .||-.14 .) in Dmarg. ||] ]D 240 ]AD 204 70 201 58 209 177 276 273 281 119 126 44 163 47 157 54 260 170 163 170 70 67 160 148 260 58 177231 163 160 170 148 707 67 67 260 58 3177 160 157 157 148AD ] || 231 ] D || 232 A 232 ex |||| 236 ] om. A 236 ] || 237 A ||||237 ] ]corr. add. D ||A 238 33 231 ] AD || 231 ] D || 232 ] A || 232 corr. ex A 33 310 111 311 117 318 283 220 221 228 7 109 17 215 12 212 97 231 107 102 287 284 37 39 45 ] AD || 231 ] D || 232 ] A || 232 ] corr. ex A || 33 236 ] 48 A || AD 236 D || ||237 ] ]] AA || ||237 238|||| || 231 231 D ||33 232 202 57 203 63 210 175 274 275 153 282 120 266 43 145 53 269 179 176 55 231 ]153 || 232 231 ]om. AD A ||]]232 232 A corr. exD || ]A 179 176 55 ]]147 |||| 229 ]] D D ||]] corr. 230add.ex 269 57 266 175 175 145 179 147 176 AD 55 269 57 63 63 266 145 147 153 AD 229] ]] D |||| 230 230 ]A ]314 112 A.A 231 217 224 13 316 4 95 106] 232 103] 226 94 238 || 240 ]A] ||] 239 AAA||||||233 241 ] A || ]230 A ||](sic) 241AAA ] A 232 om. AA||||233 ]A ||||234 |||||| 324 124 322 234 216214 6] 92 96 A || ] ] ||AA 239 206 285 271 278 121 52 49 280 40 216 208 288 286 41 38 42 216 214 208 288 286 280 110 114 41 38 42 232 ] om. 233 233 ] 234 ] 216 110 214262 208 114 288 286 172 280 110268 114149 41 180 38 178 42 A 270 56 60 146 150 232 ] om. || 233 ] A || 233 ] A || 234 ] A |||||||| )) 270 268 262 180 178 172 56 60 149 146 150 232 ] om. A || 233 ] A || 233 ] A || 234 ] A ]99Aom. A || 233 ] ]add. || 238 A ]|| 239A || 233 ] A || 239 ] (sic) (A 11 3 66 229 232 250 D. 250 (19 19 add. D. 323 320 233 218 230 91 223 93 105 313 308 117 225 104 141 205 159 212 10 100 215 283 287 284 109 111 37 39 51 45 200 279 277 272 122 123 118 50 46 205 200 279 277 272 122 123 118 50 51 46 235 ] A || 235 ] A. 205 200 279 277 272 122 123 118 50 51 46 169 68 239-40 ( .del. . .) A || in A. marg. Av]||add. 240v in marg. ] A (sic) || 240 259 1254 254 69 169 164 164 159 68 259 69 64 64 171 158 159 154 6171 158 235 ]] 154 A 235 A. 241 ])47) A: add. et || AA 242 ] A r96 r|| 312||11 309 317 222 219 227 18 316 98 235 101 108 ]226 235 ]]]]]] 241 r,9 r,14 235 A. 235 ]273 235 324 322 234 232 2 126 6 44 95 92 239-40 ( (ed. .(ed. . .) llll.. 633-637); in A. marg. Av,7 || --240 ] (sic) || 240 235 A || 204 201 209 276 273 281 47 54 16 14 14 D 82 82 (168 16 A: 14173 ,9 -A -A 14|||| ,14 633-637); D::31 82 ,7 82v,10. ,10. 204 201 209 276 273 281 119 44 47 54 31 204 119 201263 209152 276 281 119126 126255 44( 54 165 258 65 72 155 162 258 255 263 168 165 173 65 72 152 155 162 r r v v 310 311 318 220 221 228 12 97 107 102 ] A || 240 ] A || 241 ] A || 241 ] A 313 308 225 223 218 14 202 15 210 10 43 104 105 100 203 274 275 282 120 53 48 31 ( 17 ) A: 14 ,14 14 ,20 (ed. ll . 638-644); D : 82 ,10 82 ,16. r r v v 202 203 210 274 275 282 120 43 53 48 255 202257 203 66 210151 274 167 275156 120264 43 161 53 166 48 174 255 256 r282 r rr,20 (ed. ll . 638-644); 256 257 264 166 167 174 66 151 161 ]224 A ||- 14 240 ] AD ||D || 241 A (217 17 A: 14 ,14 14 D 82vvvv,10 ,10---82 82vvvv,16. ,16. ] r,14 :: 156 rr,14 D (tit. 17 )]D 14 (ed. ,20 (ed. 638-644); D 314 317 16 243 13 309 4 231 106 103 94 (tit. 17 )))A: A: 14 ,20 (ed. ..638-644); 638-644); ::: 82 82 AD ||llllll.243 ] :241 243A || A ] ]] A || 243 (271 17 A: 14 ,14---14 14r,20 82 ,10 ,10D- 82 82|| ,16. ,16. 240271 312 || 219 227 11 18 98 101 108 206 285 278 124 52 49 40 240 240 206 285 278 124 121 112 52 49 40 206 112 285163 271 278148 124121 121260 112177 52 157 49 170 40 ] ||222 241 add. et del. A ||D241 ] add. inAmarg. 242] A 70 67 58 160 260 177 163 170 70 67 58 160 157 148 236 ] om. A || 236 ] || 237 ] || 237 add. D || 238 323 311 320 229 233 221 230 ]A || 236 1 310 3 12 9 318 91 93 ]102 99 236 om. ] D || 237 ] A || 237 ] add. D || 238 || 241 add. et del. A || 241 ] add. in marg. A || 242 ] A 220 228 97 107 215 212 283 287 284 109 111 117 37 39 45 236 236 ]]] ]] D ||||||||237 ]]] ]] A ||||||||237 ]]] corr. add. 238 215 212 283 287 284 109 111 117 37 39 45 236 DD 237 215 57 212 283265 287147 284 A 231 AD 231 232 232 ex |||| 109269 111 63 117266 37 179 39]153 45 265 265 236 ]]153 om. A D A add. D 238 231 ]om. AD ||236 231 D ||237 232 A ||237 232 corr. exD A 236 om. A|||||||| AA 237 add. D||||||A 238 175 176 55 145 269 266 175 179 176 55 57 63 145 147 265 265 265 ]314 A. ||217 243 ] || 243 ] ||A243 ] ] A ||(sic) 243 A || 231 224 13 316 106 103 94 324 234 232 226 216 64 95 92 96 324 322 316 234 232 226 2 322 66 41 95 92 96 A || ] ] AD A || 239 ]]D ||||239 324110 322208 316114 234 38 232 280 226 42 95 288 92] 286 96 238 216 216 214 208 288 286 280 1102 214 114 41 38 42 ] A || 238 A || 239 A 239 ] (sic) A || 243 ] AD || 243 ] D || 243 ] A || 243 232 ]99 om. A A ]]A |||| 234 ](sic) A ]218 ||||||238 A A ]]] (sic) A]]|||||| 239 239 ]] |||||||| 232 ]A om. A |||| 233 233 ]]] A |||| 233 233 ]]] A239 234 ](sic)A A ]218 A 238 A 239 239 ]93218 105 A 238 A||||||A 239 A 323 320 225 229 233 230 1 313 3 9 10 91 223 308 14 313 308 225 223 14 15 104 105 100 313200 308 15 225 104 223]272 14 205 15 10 10 279 104 277 105100 100 [[. .) 122 50 A. 200 123 272 51 122 205 123 118 118 279 50 277 51 46 46 239-40 ( . ] in marg. A || 240 ] (sic) A || 240 [ 239-40 .) in marg. A |||| || 240 240 (sic) A || 240 ]108 A. 312 309 222 219 101 312 317 222 227 11 18 98 101 ]] ]]]] in 239-40 ((((.........).) marg. ]]]] (sic) 312 11 309 317 317 98 222 219 227 227 A 11 18 18 309 98 235 101108 108]]219 235 A |||| 235 235 A. .) inA. marg. A 239-40 in marg. A (sic) A 240 204 47 239-40 A || 240 240 (sic) A A |||||| 240 240 204 119 201 209 209 44 276 273 281 281 119 126 126 201 44 276 47 54 54 273 310 311 318 220 221 228 12 97 107 102 10 ] A || 240 ] A || 241 ] A || 241 ]] A 10 10 310 311 318 220 221 228 12 97 107 102 310203 311120 318 43 220 275 221 228 12 210 97 107 102 202 274 282 53 48 ] A || 241 ] A || 241 A 10 10 10 202 203 120 210 255 43 53 274 275 ]48 282 A || 240 255 ] A || 240 ] A || 241 ] A || 241 ]] (A ] A || 240 ] A || 241 ] A || 241 A r r v v r,14 r,20 (ed. ll . 638-644); ]224 A -|| 240 ] AD || A 314 231 217 224 16 106 94 20 17 )) 103 A: 14 (271 17 A: 1440 ,14 - 14 14 ,20 (ed. ll . 638-644); D:241 : 82 82v,10 ,10 -- 82 82v,16. ,16. 231 217 16 13 106 103 94 240 314 444 285 231( 217 224 16 13 13 314 106 103278 94 240 240 (A 20))) 124 52 49 ( 20 206 112 271 278 124 121 121 206 112 285 || 52 49 40 241 ]] add. etetdel. AA||||241 ] ]add. ininmarg. AA||||242 ]] A || 241 add. del. 241 add. marg. 242 229 233 230 111 323 333 999 320 91 93 99 || 241 ] add. et del. A || 241 ] add. in marg. A || 242 ] A 323 320 229 233 230 91 93 99 || 241 ] add. et del. A || 241 323 320 229 233 230 91 93 99 || 241 ] add. et del. A || 241 ] add. in marg. A || 242 ] A A 284 109 37 215 111 212 283 283 236 287 39 284 A 109 215 111 117 117 212 37 287 39 ]]45 45 om. ]] D ]] inA |||| 237 add. 236 om. A |||| 236 236 D |||| 237 237 ] add. Amarg. 237A || 242]] add. D D] |||| 238 238 243 ] ||||243 ]] DD||||243 ]] AA|||| 243 [[ AD 260 260|| 324 234 232 92 ( 18 )18)) 324 22 322 322 316 316 66 95 234 232 226 226 96 95 92 96 ] || 243 AD 243 243 243 |||||| ] AD || 243 ] D || 243 ] A || 243 ||243 243 ] AD || 243 ] D || 243 ] A || 243 243 || 243 ]238 AD A || 243 243 ] ]] A ||(sic) 243 A ]] ||||((18 ]]218 105 A |||| ]238 ]] AD |||| 239 ]] D ||A |||| 239 A100 A 239 ]] A243 239 (sic) A 14 313 308 308 225 223] 218 14 313 15 10 10 225 104 223 105 A. 100 4 15 9 104 270 56 268 262 60 149 180 146 172 A. 150 ]] 178 270 56 268 262 60 149 180 146 172 150 ] ]]108 A. A. A. 312 101 ] 178219 A. 312 11 309 317 317 98 222 219 227 227 11 18 18 309 98 101 108 239-40 (( .. .. .) ]] in ]] (sic) 245222 239-40 .) in marg. marg.33A A |||| 240 240 (sic) A A |||| 240 240 ] 245 245 261 68 259 254 69 64 171 158 169 164 159 154 256 257

232 236 44

66

255 255 255 264 161 166 167 156 174 255151 255 255

||222 241 A ]D add. et del. 241 ] add. in marg. A 242 om. AD 222 om. D |||| 222 222 AD 223 om. DA|||| 223 223234 om. ||236 233 A ||223 233 ] AD || ] ]om. A]] || ] ]] ]A ||D ||||||237 ] ]] om. || 237 222

260 60 163 146 170 150 70 268 67 262 58 9 160 178 157 172 148 56 149 9177 180

270

261

68 310 310 258 258 16 16 61 71 61 71 11 323 323 267 253 267 253 62 62

55 55

259 254 69 64 171 158 169 164 159 154 311 270 311 12 318 97 220 221 221 102 228 12 318 97 107 107 220 102 228 270168 155 162 165 173 270 65 72 255 263 152 65 72 255 263 152 168 155 162 165 173 13 314 44 231 231 217 217 106 224 94 13 314 106 103 103 224 94 7 3 256 257 66 264 151 161 166 167 156 174 256 257 66 264 151 161 166 167 156 174 320 320 229 229 91 233 93 230 91 233 93 99 99 230 433 6799 260 70 589 177 163 160 157 170 148 70 67 260 58 177 163 160 157 170 148

A A |||| 240 240

]]

269 269

57 57

63 63

|||| 241 241

260 260

]]

A A |||| 241 241

add. add. et et del. del. A A |||| 241 241

]]

[[

266 175 145 179 147 153 176 266 175 145 179 147 153 176

]]

A A |||| 241 241

|||| 243 ]] AD ]] D ]] 243 AD |||| 243 243 D |||| 243 243 ( 19 ) A: 14v ,3 - 14v ,13 (ed. ll . 651-661); D : 82v ,19 - 83r ,4. 280 42 38 286 38 286 280 42 ] A. ] 272 51A.46 118 50 277 3 279 245 245 33A || 250 277 272 51 ] 46om. D || 250 118 279245 50 250 ] ] D || 251

216 110 41 214 208 114 9 41 288 216 110 214 208 1146 41 288 207 122 205 7 200 123 207 122 205 200 123

116 204 119 126 201 209 44 276 265 26544 276 116 204 119 126 201 209 115 125 202 203 120 210 7 203 120 210 3 115 125 202

43 43

53 53

47 47

54

] 54 274 275

273 281 273 281

274 275

D || 251 48 282 48

282

213 199 124 121 206 112 271 52 49 278 40 9969 285 414 121 206 112 213 199 4124 285 271 52 49 278 40 250 250 250 109 215 111 117 212 283 37 287 39 45 284 275 109 215 111 117 212 283 27537 287 39 45 284 275 324 2 322 316 6 95 234 92 232 226 96 v 324 12 322 316 66 95 234 92 232 226 96 315 14 313 308 15 10 225 104 223 218 105 100 315 14 7313 308 15 10 333 225 104 223 218 105 100 77 8 312 11 18 309 317 98 222 101 108 219 227 8 312 11 18 309 317 98 222 101 108 219 227 265 265 7 17 310 311 12 318 97 107 220 221 102 228 7 17 310 311 12 318 97 107 220 221 102 228 321 307 16 13 314 4 231 217 106 103 224 94 321 307 16 13 314 4 231 217 106 103 224 94 270 149 180 146 178 172 150 4141 56 268 262969660 270 1 6 229 91 233 93 99 230 1 323 3 9 320 250 250 1 323 3 9 320 229250 91 233 93 99 230 261 68 259 254 69 64 171 158 169 164 159 154

251

]

[

] add. in marg. D || 251

[[[

A || 252

]

]

A,

]] A A |||| 243 243 ]

A || 251

D || 252

A A

]]

]] add. add. in in marg. marg. A A |||| 242 242

] ]

]A ]]A ( 18 )18)) ((18 ]]

D || 251 A ||

A || ( (19 )19 ((19 20 ) )) ] ] ] A || 253

252 ] A || 252 ] A || 253-8 v ... ] om. ( 19 ) A: 14 ,3 - 14v ,13 (ed. ll . 651-661); D10 : 82 ,19 - 83r ,4. 10 ] D || 254 ] D || 255 ] rescr. D || 256 ] 250 ] om. D || 250 ] A || 250 ] D || 251 ] D || 251 D || 257 ] D || 258 ] A || 259 ] ] D || 251 ] add. in marg. D 32 || 251 ] A || 251 ] A || ] A || 259 ] A || 259 ] [[[ A || 259] 251 A || 252 ] A, D || 252 ] A || ( 19 )19)) ( A, D || 259 ] A, D || 259-60(19 252 ] A || 252 ] A || 253-8 ... ] om. A || 253 ] 10 62 258 65 72 255 263 152 168 155 162 165 173 10 ] A, D || 260 ] A, D || 262 ] ] 174 D || 254 ] D || 255 ] rescr. D || 256 ] 255 151 255 61 71 256 257 66 264 161 166 167 156 255 77 33 (corr. ex ) A] || 262 D ||] 258A, D || 267 253 70 67 260 58 280 177 163 160 157 170 21 D ||148257 ] D || 262-3 A || 259 (quater )] ] (((21 280 21))) 280 32 32 32 55 269 57 63 266 175 145 179 147 153 176 262 ] A || 263 ] A || 263 ] A || 263 ] A || 259 ] A || 259 ] A || 259 ] 216 110 214 208 114 41 288 38 286 280 42 ] ] 11 66 ] ] om. D || 264 ] corr. ex A || 265 D || 266 A || 263 ] 27020556 200268123 26211860 279 13220759122 14950180277 146 272178 51172 46150 A, D || 259 [ ] A, D || 259-60 26120468119 259126 25420169 20964 44171276 15847169 54164273 116 33 A, melius rescr. in marg. D. ]159281154 A || 266 ] ] A, D 33 || 260 ] A, D || 262 ] 62 258 65 72 255 263 152 168 155 162 165 173 115 125 202 203 120 210

43

53

274 275

48

282

] v r r ( 20 ) A: 14v ,13 (corr. ex - 14 ),20 A (ed. || 262ll . 662-669); ] 33 A,D : 83 ,4 - 83 ,9. D || 262-3 (quater ) ] D || 2532157011167117 26021258 283177 37163287 130 267109 1603915745170284 148 32 271 ] r ] D ||r A 272|| 263 ] om. ]A || 32 272 ] v A || 273 ] AD] v 262 A || 263 ] A || 263 194 26532455 226932257 31663 260 175 176 - 14 [,14[[(ed. ll . 633-637); D : 82 ,7 - 82 ,10. 14592 147226 15396,9 232 6266 95 (234 16 )179A: 14 260 260 ||288223 273 ]] om. D ||D264 || 273 corr. utAvid. ex 28010042 24 315 11314216313 110308214 15208 10114 38218286105 22541104 ]] ] ] corr.] ex || 265 A || 263 ]D || 273 D ]|| 266 4 9 285 285 tit. D : 4 9285 277108 2722195122746 12211205 18200309 123317 1189827922250101 8207312 ] ]] ] corr. ex A || 274 ] om. A || 275 ] A || 276 33A, melius rescr. in marg. D. 33 266 ] [ 4722154]]102273228281ADA||||231 33 276220 119311 12612201318209 9744231 33 711617204310 107 33 ] D || ] A || 232 ] corr. ex A || 33232 D || 277 ] D || 277-8 ... ] om. A || 278 ] 300321 274103 27522448 94282 115307 12516202 13203314 120 421023143 21753106 7 3 232 ] om. A || 233 ] A || 233 ] A || 234 ] 7 3 33 271233 22 213 1199323 124 3121 9206320 112229285 91D 49 9927823040 ||52 93 278 ] D || 279 ] om. A || 279 ] A || 279 ] A. ]A || 255 275 255 255

19221361 19971124 256121 25720666112 264285 151271 16152166 49167278 15640174

37 287 39 45 284 ] A. D : 82v ,7 - 82v ,10. r 235 ( 235 16 ) A: ]14r ,9 -A 14|| [,14[[(ed. ll . 633-637); ( 21 ) A: 14 ,20 - 15r ,12 (ed. ll . 670-682); D : 83r ,9 - 83r ,17. r 100 r v v tit. 225 223 : 218 105 ,14 104D ( 17 ,20 (ed. ll 280) A: 14 ] A- 14 || 282 ] . 638-644); A || 282D : 82 ,10 ] - 82 ,16. D || 283 98 222 101 108 219 227 236 om. 236 237 237 33 231 231 232 33232

302 211 109 215 111 117 212 283 78

5 315 8

324

2

1

260 260 260 v 322 316 628095 234 92 232 226 96 6 610

14 1313 308

15

312

309 317

11

18

]]

( 21 )

] A ]|| 283 ]] add. 238 corr. ex

51

Text and translation

n2 +7 2

− 4j

n2 −5 2

+ 4j

n + 1 − (α + 4) 2

n2 + 1 − (α + 2)

α

n2 +3 2

− 4j

n2 −1 2

+ 4j

α+6

=⇒ n2 + 1 + 4 − 8j

=⇒ n2 + 1 − 4 + 8j

α+2

n2 + 1 − (α + 6)

[ Thus you find that (the sum of) each pair of (small) numbers is less 70 by successive (additions of) 8: the first pair is in deficit than that before α+4 n +1−α by 4, the next pair is in deficit by 12, the subsequent pair by 20, then successively (by) 28, 36, 44, 52, 60, and so on always till 4 and 2. ]71 n2 + 7 n2 rows + 3 which require arranging [ Now you find that the (numbers) =⇒ n2 + 1 + 4 −even 8j − 4j − 4j 2 2 in them have indeed, on pairs of (opposite) sides, this succession of exα n + 1 − (α + 2) =⇒ n + 1 − 2 cesses or deficits, namely 4, 12, 20, 28, 36 and so on always in this way.72 Thus, placing the excess of these even (numbers) where the deficit is hand n2 − 1 opposite to each n2 − 5n +number α + 2 (placed) 1−α =⇒ 1 + 2 among the large writing its=⇒ complement n2 +n 1+− 4 + 8j + 4j + 4j 2 2 (numbers)i, you will have equalized the (horizontal) rows on two sides, and you two (vertical) sides, which you will ⇓ will be left with equalizing ⇓ 2 2 n + 1 − 2 n + 1 + 2 do with the remaining even numbers. ]73 2

2

2

2

2

n2 + 1 − (α + 2s)

=⇒ n2 + 1 − 2s

(§ 20. ‘Neutral placings’ by means of two consecutive pairs)74 α

h If you take four (even) consecutive small numbers and put the first α + 2s n +1−α =⇒ n + 1 + 2s and the last on one side and the two middle on the other, opposite side, 2 2 n2 +7 the large one n2 −1is its complement, | . . . | nto2+3each − 4j number − 4j | . . . | n 2−5which 2and put 4 opposite 2 2 you will have equalized these two sides. i75 2

n2 − 1

n2 − 3 | . . . |

n2 −1 2

+ 4j

2

n2 −5 2

+ 4j | . . . |

n2 +7 2

n2 +3 2

α+6

n + 1 − (α + 4)

n + 1 − (α + 2)

α

=⇒ 2(n2 + 1)

n2 + 1 − (α + 6)

α+4

α+2

n2 + 1 − α

=⇒ 2(n2 + 1)

2

2

[ If you take four small (even, consecutive) numbers, the deficit of the 70

Rather: than the first (largest) pair. n +7 +3 − 4m − 4m and merely =⇒ extends n2 + 1 + 4 − 8m list of differences. Early reader’s addition: it adds nnothing the 2 2 It is true that the original text as preserved did not mention that the differences increase by successive additions of 8. n −5 n −1 2 72 + 4m + 4m 2 As seen in §§ 14 and2 17 for the horizontal rows. =⇒ n + 1 − 4 + 8m 73 Early reader’s addition, perhaps the same reader as before. First, we are here in the domain of general equalization rules; their application will follow (§§ 24–35). Furthermore, the alleged equalization is not applicable as such since equalization of two opposite horizontal rows depends on the quantities put beforehand in their end cells. 74 Neutral placings provide exactly the sum due for four cells. 75 This sentence is missing. The two subsequent interpolations must be some attempt to restore the meaning. 71

2

2

2

2

258 65 72 255 263 152 168 155 162 165 173 16 314 44 231 231 217 217 106 224 94 16 13 13 314 106 103 103 224 94

62

61 71 256 257 66 264 151 161 166 167 156 174 11 323 99 320 323 33 320 229 229 91 233 93 230 91 233 93 99 99 230 267 253

70

67

260

55

269

57

63

58

260 260

|||| 241 241

|||| 243 243

9

]]

52]

286 280

54

288

245

38

]

245 207 122 205 200 123 118 279245 50

51

273 281

116 204 119 126 201 209

44

276

47

43

53

274 275

48

282

9 285 271 213 199 4124 121 206 112

52

49

278

40

109 215 111 117 212 283 275 27537

287

39

45

284

275

322 316

10 3

225 104 223 218 105 100

14 7313 308

15

11

309 317

7

17

310 311

12

318

16

13

314

4

323

3

9

321 307

414

1

251 [[

250

233

93

252

94

A || 252

]

]

33

D || 257

280 280 280

66

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

264 151 161 166 167 156 174

130 267 253

70

67

260

58

194 265

269

57

63

266 175 145 179 147 153 176

192

55

268 262

72

60 149 180 146 178 172 150

A,

(corr. ex

262

38

207 122 205 200 123 118 279

50

277 272

51

116 204 119 126 201 209

44

276

47

273 281

300 115 125 202 203 120 210

43

53

274 275

48

282 40

3 213 199 7124 121 206 112 285 271

22

78

324

5 315

246

]

54

46

52

49

278

287

39

45r 284

234

92

232 226

6

95

A || 266

]

A || ( 19 )19)) ((19

]

] om. A || 253

...

] rescr. D || 256

A || 259

]

A,

D || 262

D || 262-3

(quater ) ]

] ]

A || 265

A || 263

8

312

11

309 317

7

17

310 311

12

318

16

314

290 v 290231 217 106 103 224 94 290 v 4275

222 101 108 219 227

] 221

220

]

D ||

]

]] D ]|| 266

]

3333A, melius rescr. in marg. D. 33

]

33

610 225 104 223 218 105 100

231 97 107

( 20 )

D || 259-60

]]

v

96

15

98

((21 21))

]

A,

v

]

A || 259

]

] corr. ex

r

tit. D :

14 1313 308 18

]

] om. D || 264

D || 252

]

A, 32 32 A || 263

A ||

]

( 16 ) A: 14 ,9 - 14 [,14[[(ed. ll . 633-637); D : 82 ,7 - 82 ,10.

260 260 260

322 316

2

280 42 A 286 || 263

37

302 211 109 215 111 117 212 283

A || 263

]

288

270 285 285 9285

A,

D || 251

]

A || 251

]

D || 260 ]

D || 251

]

]

) A || 262

177 163 160 157 170 148

24 113 216 110 214 208 114 41

4

]]

]]

D || 255

D || 258 32 A || 259

]

A,

]

255 255 255

10 10

D || 259

171 158 169 164 159 154

64

]

]

[

]

A || 253-8

]

D || 254

A || 259

270

132 59

A || 252

]

230

99

A || 250

]

] add. in marg. D || 251

D || 251

]

97 107 220 221 102 228 231 217 106 103 224

969

]A ]A ( 18 )18)) ((18

[[

] om. D || 250

250

250 320 229250 91

77

11

33

98 222 101 108 219 227 265 265

18

A A |||| 243 243

]]

v v v r ( 19 232A: 226 14 92 ) 96 ,3 - 14 ,13 (ed. ll . 651-661); D : 82 ,19 - 83 ,4.

95

312

]]

234

66

8

D D |||| 243 243

]]

Text and translation

A. A.46

277 272

]] add. add. in in marg. marg. A A |||| 242 242

AD AD |||| 243 243

42

7 203 120 210 3 115 125 202

315

[[

266 175 145 179 147 153 176

4

216 110 214 208 114 41

324 12

add. add. et et del. del. A A |||| 241 241

]]

177 163 160 157 170 148

102

AD || 231 228

]

[

] ]]

D ||33 33232

]

] corr. ex

A || 232

A ||

v r r ] || v,20 (ed. ll . 662-669); (( 20 14 :: 83 232 ] om. A ||-- 14 233 ] A || D 233 A || 234 ] A 20 )) A: A: 14 ,13 ,13 14 ,20 (ed. ll . 662-669); D 83r,4 ,4 --]83 83r,9. ,9. 248 319 1 323 3 9 320 229 91 233 93 99 230 271 D 272 ]] om. |||| 272 ]] A ]] AD 235 A ] A 271 ] ]] D |||| || 235 272 om. AA. 272 A |||| 273 273 AD || 273 ]] r D || 273 ]] vcorr. ut ex D |||| 273 ]] || 273 D || 273 corr. ut vid. ex D 273 r vvid. [[ ( 17 ) A: 14 ,14 - 14 ,20 (ed. ll . 638-644); D : 82 ,10 - 82 ,16. corr. ex A || 274 ] om. A || 275 ] A || 276 ] corr. ]exom. A || 236 A || 274 ] ] om. || 275 ] || 276D || 238 ] 236 DA || 237 ] A || 237 ] A add. [ 265 265 265 D ]] D .. .. .. ]] om. ]] D || || 277 277 D || || 277-8 277-8 om. A A |||| 278 278 ] A || 238 ] A || 239 ] A || 239 ] (sic) A || 270 56 268 262 60 149 180 146 178 172 150 D ]] [ D ]] om. ]] A ]] A. D || || 278 278 D || || 279 279 om. A A |||| 279 279 A |||| 279 279 A. 261 68 259 254 69 64 171239-40 158 169 164 159 154 ( . . .) ] in marg. A || 240 ] (sic) A || 240 r r 173v 62 258 65 72 255 263 152 168 155 )162 v ,20 - 15rr,12 (ed. ll . 670-682); 10 ( 21 A:165 14 D11 ::10 83 10 - 15 ,12 (ed. ll . 670-682); 83r,9 ,9 -- 83 83r,17. ,17. ] 11 11 280 ( 21 ) A: ] 14A,20 || 240 ] A || D 241 A || 241 ] ( 21 A) 61 71 256 257 66 264 151 161 166 167 156 174 ( 20 ) 280 ] A || 282 ] A || 282 ] D || 283 ] A || 283 280 ] 148] A || 282 ] A ||A241 || 282 ] add.] in marg. DA || 283 ] A 267 253 70 67 260 58 177||163 160 157 170 241 add. et del. || 242 ] || 283 A ]] 153 176 A ]] A ]] add. ]] pr. 55 269 57 63 266 175 145 179 147 A || || 283 283 A |||| 284 284 add. A A |||| 285 285 pr. || 243 ] AD || 243 ] D || 243 ] A || 243 216 110 214 208 114 41 288 38 286 280 42 scr. et del. A || 285 ] A || 286 ] (sic) A || 285-6 ( . . .) ] iter. et del. A || 285 ] A || 286 ] (sic) A || 285-6 ( . . .) ] iter. in in 8207 122 205 200 4 123 118 279 9 50scr. 277 51 A. 46 270 56 268 262 178 172 60 149 180 146 150 ] 272 marg. A, sed pro scr. , pro vero scr. (sic) || 287 ] D || 287 ]] 204 201 209 276 273 281 116 119 126 44 47 54 marg. scr. , pro vero scr. (sic) || 287 ] D || 287 261 68 259 254 69 64 171 158 169 164 A, 159 sed 154 pro 270 210 152 274 162 275 165 115 125 202 120 263 43 168 53 155 48 282 173 62 258 65 203 72 255 om. (pr.) A D ]] A ]] om. D D || || 287 287 (pr.) ]] A (saepius), (saepius), D |||| 287 287 A |||| 287 287 321 307

76

13

285 271 166 278 174 124 257 121 112 151 52 167 49 156 40 2213 7 206 3 161 61 199 71 256 66 264

A, A, (pr.) (pr.) ]]

283 163 287 157 284 109 215 117 212 37 160 39 170 45 148 267 253 70 111 67 260 58 177 324 2 55 269

285

322 234 179 232 153 226 176 6 175 95 145 92 147 96 57 316 63 266

315 216 313 214 308 208 14 110 15 114 10 225 288 223 286 105 280 100 416 104 38 218 42 1

5

D D || || 287 287 A || 289 A || 289

A A || || 290 290 ] A || 292 228 102 273 ]281 A || 292

]]

312 205 309 118 317 279 8 122 11 200 18 123 98 222 207 277 108 272 219 50 101 51 227 46 310 311 201 318 7 204 17 119 116 126 12 209

291 291

97 107 276 220 44 47 221 54

321 307 16 13 314 4 231 217 106 103 224 94 115 125 202 203 120 210 43 53 274 275 48 282 1 323 3 9 320 229 91 233 213 199 124 121 206 112 285 271 52

93 49

109 215 111 117 212 283 27537

287

39

324

92

232 226

271

]

2

322 316

6

315

14

313 308

15

8

312

11

309 317

7

17

310 311

18

12

10

318

95

99 230 278 40

98

A A |||| 289 289 ]] A A |||| 290 290

96

222 101 108 219 227

|| 273

97 107 220 221 102 228

D || 272 ]

D D |||| 291 291

] ]] A A |||| 289 289 ]] A A |||| 290 290 ]] A A |||| ]

[

284v ( 20 ) A:45 14 ,13 - 14v ,20 (ed. ll . 662-669); D : 83r ,4 - 83r ,9.

234

225 104 223 218 105 100

290

]] add. ]] add. A A |||| 288 288 ]] A ]] A A || || 289 289 A |||| 290 290 ]] A || 291 ] corr. A || 291 ] corr. ut ut vid. vid. ex ex ]] AD. 35 AD.

] om. A || 272

D || 273

]

A || 273

] corr. ut vid. ex

] ] D || 273

AD

]

53

Text and translation

sum of the first and the second is a certain number, and the deficit of the sum of the third and the fourth is this number less 8.76 Then adding the first and the fourth you will find that they have a deficit of half the sum of the two deficits; likewise, adding the second and the third gives the same deficit. ]77 [ For instance, the sum of the last two small numbers is less than their 2 α of) the two preceding numbers is less by 12;78 + 14, − and (α + 4)(the sum sum duenby then, adding the last and that separated by a pair will make a deficit of 8, and, likewise, adding the one preceding the last one and the next one n2 + 1 − (α + 2) α+2 will make a deficit of 8;79 now this equals half the sum of 12 and 4. ] α+6

n2 + 1 − (α + 6)

The knowledge of this is again necessary, for you will use it frequently.80 α+4

n +1−α (§ 21. Effect of placing two small even numbers in opposite rows) 2

When you write the first small number,81 or any small number, on one side and then the subsequent number on the other, opposite side, and you write the two large numbers which are their complements opposite to them, the side haveαwritten the first=⇒ small number will be n2 +where 1 − (α +you 2) n2 + 1 − 2 less than its sum due by 2, whereas the other (side) will be in excess by 2. If you write some small number on one side and you write on the 2 α + 2 small (even) n2 + 1 −number α =⇒ nfrom + 1 + 2this one, the opposite side the third (counted) side containing the first small number will be less than its sum due by 4. And so on: whenever you increase the distance between these two (small numbers) by one number, the deficit is always increased by 2. n2 + 1 − (α + 2s)

α

=⇒ n2 + 1 − 2s

α + 2s

n2 + 1 − α

=⇒ n2 + 1 + 2s

[ For instance, if 4 is put on one side and 6 on the opposite side, and opposite to each its complement, the side containing 4 will be less than 76

See § 19, in particular interpolations (the same glossator may thus be at work 2 α α+6 n2 + 1 − (α + 4) n2 + 1 − (α + 2) =⇒ 2(n + 1) here). Indeed, n2 + 1 − [ (α + 4) + (α + 6) ] = n2 + 1 − [ α + (α + 2) ] − 8. 77 The inference being that since placing them on opposite sides makes equal deficits while2 their complements placed opposite provide the same amount as excess, the two 2 n + 1 − (α + 6) +4 =⇒ 2(n + 1) n2 + 1 − α opposite rows will thusαbe equalized. α + 2 2 2 2 78 2 2 n −1 n −5 n −9 n2 −13 See § 19; indeed, 2 + 2 = n + 1 − 4, 2 + 2 = n + 1 − 12. 2 2 2 79 n2 −1 + n −13 = n2 + 1 − 8, n 2−5 + n 2−9 = n2 + 1 − 8. 2 2 80 Use of neutral placings in §§ 31, 33, 34; again, this time for even-order squares and consecutive natural numbers, in §§ 42–43 & 51–54. 81 We thus now consider the sequence n2 +7 n2 +3 beginning with 2. 2 2

− 4m

2

− 4m

=⇒ n + 1 + 4 − 8m

n2 −5 2

+ 4m

n2 −1 2

+ 4m

=⇒ n2 + 1 − 4 + 8m

54

Text and translation

( 22 ) ( 22 ) 295

295

[ [

300

300

] ]

( 23 ) ( 23 )

305

305

192

( 22 ) A: 15r ,12 - 15r ,17 (ed. ll . 683-688); D : 83r ,17 - 83r ,21.

270

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

70

67

260

58 310177 163 160 157 170 148

269

57

63

266 175 145 179 147 153 176

267 253

194 265

55

268 262

72

60 149 180 146 178 172 150 64

293-4

...

171 158 169 164 159 154

264 151 161 166 167 156 174

|| 294-5

310 41 24 113 216 110 214 208 114

D (

288

38

286 280

207 122 205 200 123 118 279

50

277 272

116 204 119 126 201 209

44

276

47

300 115 125 202 203 120 210

43

53

274 275

51

hab. A , pro

AD (pro

scr. D

) || 294

] om. (homoeotel.) A || 295

...

rescr. in marg.) || 296 42

D || 296

]

]

46

A || 273 281 54 297 48

]

D || 297

]

282

A || 296

A || 297

]

] om. D || 298

]

D || 296

]

(sic) A,

A || 299

marg. A || 299 ] om. D || 299 (pr.) ] D || 299 52 49 278 40 A, om. D. 302 211 109 215 111 117 212 283 37 287 39 45 284 ( 22 ) A: 15r ,12 - 15r ,17 (ed. ll . 683-688); D : 83r ,17 - 83r ,21. 78 5 324 2 322 316 6 95 234 92 232 226 96r 23 178 ) A: 15 ,18 - 15v ,8 (ed. ll . 690-699); D : 83r ,21 - 83v ,2. 270 56 268 262 60 149 180( 146 172 150 ... ] 315 14 313 308 15 10 225 293-4 104 223 218 105 100 261 68 259 254 69 64 171 158 169 164 159 154 tit.101 ad hab. A (l. 300, ) 219 23-35 227 8 312 11 18 309 317 98 222 108 §§ 315 hab. A , pro scr. D ) || 294 62 258 65 72 255 263 152 168 155 162 165 173 AD (pro 246 7 17 310 311 12 318 97 107 220 221 102 228 166 167 61 71 256 257 66 264 151 161 tit. (ad156§§17423-31) . . .D : ] om. (homoeotel.) A || 295 ] 217 294-5 76 321 307 16 13 314 315 4 231 || 106 103 224 94 213 199 124 121 206 112 285 271

22

267 253 70 67 260 58 177 163 160 157 170 148 248 319 1 323 3 9 320 229 91 233 93 99 230 55 269 57 63 266 175 145 179 147 153 176

D301 (

]

286 280

rescr.] in marg.) 296 D |||| 301

216 110 214 208 114 41

288

38

207 122 205 200 123 118 279

50

277 272

A 54|| 297 273 281 47

]

51

42

D || 296A || 302 ] om. 46

]]

116 204 119 126 201 209

44

276

115 125 202 203 120 210

43

marg. A48 ||282[299 D274 ||275304 52 49 278 40 A, om. D. ] 37 287 39 45 284 [ 53

213 199 124 121 206 112 285 271 109 215 111 117 212 283 324 315

14

8 7

2

322 316

10

225 104 223 218 105 100

15

312

11

309 317

17

310 311

12

318

98

92

232 226

95

313 308 18

234

6

222 101 108

add.

96

A || 297 (sic) A, ]] add. supra lin. D || ]302-3

] om.] D || A, 299

307

AD || 306

(pr.) ]

A, ]

] add. ( 24in)

( 24 )

]

12

D ||D299 || 304

AD

] ]

D || 306

307

]

A || 299 ) hab.] in add. in marg.

)

ut vid. A || 307

307 12

D || 297

]

(corr. supra lin. ex

A,

]

]

A ||] 296 (sic) ] A || D 301 || 296 ] om. D ]|| 302A,

D || 297 A; ult. ] om. || 298 part.D(sc.

]

] 219 227

97 107 220 221 102 228

]

A,

]

D || 297

]

AD

]

]

D || 305 ] AD ;

A || 305 A || 306

55

Text and translation

its sum due by 2 and the side containing 6 will be more by 2. If you write 8 instead of 6 and do the same, the side of 4 will be less by 4 and the side of 8 more by 4. And always likewise: whenever you increase the distance between these two, the amount of their augmentation (will vary) accordingly. ]82 (§ 22. Effect of placing pairs of small numbers in corners) If you place two small (even, consecutive) numbers in consecutive corners and their large complements diagonally opposite, and you put n +7 n +3 =⇒ n2 + 1 + 4 − 8j − 4j − 4j the two small numbers on 2the top and2 (therefore) the two large ones on the bottom, the deficit of the two small numbers will depend on their n −1 n2 + 1 − 4 ones, + 8j + 4j 12 if they + are 4j rank: 4 if they are the lastn 2−5 two, the two=⇒ previous and 2 so on always, with regular additions of 8, until you reach 2 and 4.83 (But) the right-hand side will always have, relative to its sum due, an excess of 2, or a deficit of 2,84 without any increase and decrease. 2

2

2

2

n2 + 7 − 4j 2

n2 + 3 − 4j 2

=⇒ n2 + 1 + 4 − 8j

n2 − 1 + 4j 2

n2 − 5 + 4j 2

=⇒ n2 + 1 − 4 + 8j

⇓ n2 + 1 + 2

⇓ n2 + 1 − 2

You must understand all this: it belongs to what you need for writing the even (numbers) in (squares of) this kind. n +7 n −5 n −1 . . . | have | . . . | n 2+3 − 4j [ Examples of4 treatments for all that explained ]85 − 4j | we 2 2 2 2 2

n2 − 1

2

+ 4j + 4j | . . . | (Square of order 5)

n2 − 3 | . . . |

n2 −1 2

n2 −5 2

2

n2 +7 2

2

n2 +3 2

(§ 23. Filling the square of order 5) Treatment for the square of five by five. You put the odd numbers in the central three by three square as we have explained.86 Those remaining are 1, 25, 3, 23. You put 1 in the lower middle cell and 25 opposite to it 82

This example, with the next even numbers after 2, must also be interpolated — if only because we are to remain in the domain of purely theoretical equalization rules. 83 According to the equalization rule of § 19. See the figure, with the smaller pair (in its general form, see § 18) placed at the top (the lesser on the right). 84 Depending on whether it contains the larger small number or not. 85 Appropriate, though only in MS. A: the equalization rules will now be applied to the two order classes: in §§ 24–31 (orders n = 4t + 1, t > 1) and §§ 32–35 (orders n = 4t + 3); the particular case n = 5 will be considered first. 86 See § 3 (and § 8, n. 41).

300 300

[

]

56

(( 23 23 )) ( 23 )

Text and translation

305 305 305

310 310 310

15rr ,12 - 15rr ,17

(( 24 24 ))) ( 24

83rr ,17 - 83rr ,21.

( 22 ) A: (ed. ll . 683-688); D : ( 22 ) A: 15 ,12 - 15 ,17 (ed. ll . 683-688); D : 83 ,17 - 83 ,21. 293-4 ... ] 293-4 ... ] AD (pro hab. A , pro scr. D ) || 294 ] AD AD (pro hab. A , pro scr. D ) || 294 ] AD || 294-5 ... ] om. (homoeotel.) A || 295 ] || 294-5 ... ] om. (homoeotel.) A || 295 ] v ( rescr. in marg.) || ll296 A83||r ,21 296 v ] D || 296 ] A, (D23 15 ,18 in - 15marg.) ,8 (ed.|| . 690-699);]] D :A D ( ) A: rescr. 296 || 296- 83 ,2. ] D || 296 ] A, D || 296 ] A || 297 ] (sic) A, D || 297 ] tit. ad §§D 23-35 hab. A (l. 300, ) || 296 ] A || 297 ] (sic) A, D || 297 ] ] A ]|| 297 ] D || 297 ] om. D || 298 ] A || 299 ] add. in ] A (ad || 297 D || 297 ] om. D || 298 ] A || 299 ] add. in tit. §§ 23-31) ]D : marg. A || 299 ] om. D || 299 (pr.) ] D || 299 ] marg. A || 299 ] om. D || 299 (pr.) ] D || 299 ] 301om. D. ] D || 301 ] (sic) A || 301 ] om. D || 302 A, A, om. D. [ ][[om. A || 302 ] add. supra lin. D || 302-3 ( 23 ) A: 15 ,18 -] 15v ,8 (ed. ll . 690-699); : 83r(sc. ,21 - 83v ,2. A; ult.Dpart. ) hab. in marg.

270 56 268 262 60 149 180 146 178 172 150 270 56 268 262 60 149 180 146 178 172 150 261 68 259 254 69 64 171 158 169 164 159 154 261 68 259 254 69 64 171 158 169 164 159 154 62 258 65 72 255 263 152 168 155 162 165 173 62 258 65 72 255 263 152 168 155 162 165 173 61 71 256 257 66 264 151 161 166 167 156 174 61 71 256 257 66 264 315 151 161 166 167 156 174 267 253 70 67 260 58315 177 163 160 157 170 148 267 253 70 67 260 58 315 177 163 160 157 170 148 55 269 57 63 266 175 145 179 147 153 176 r 55 269 57 63 266 175 145 179 147 153 176 216 110 214 208 114 41 288 38 286 280 42 216 110 214 208 114 41 288 38 286 280 42 207 122 205 200 123 118 279 50 277 272 51 46 207 122 205 200 123 118 279 50 277 272 51 46 116 204 119 126 201 209 44 276 47 54 273 281 116 204 119 126 201 209 44 276 47 54 273 281 115 125 202 203 120 210 43 53 274 275 48 282 115 125 202 203 120 210 43 53 274 275 48 282 213 199 124 121 206 112 285 271 52 49 278 40 213 199 124 121 206 112 285 271 52 49 278 40 109 215 111 117 212 283 37 287 39 45 284 109 215 111 117 212 283 37 287 39 45 284 324 2 322 316 6 95 234 92 232 226 96 324 2 322 316 6 95 234 92 232 226 96 r 315 14 313 308 15 10 225 104 223 218 105 100 315 14 313 308 15 10 225 104 223 218 105 100 8 312 11 18 309 317 98 222 101 108 219 227 8 312 11 18 309 317 98 222 101 108 219 227 270 310 268 12 262 318 180 220 178 102 172 228 56 311 60 97 149107 146 221 150 7 17 7 17 310 311 12 318 97 107 220 221 102 228 261 307 259 13 254 314 171 217 169 103 164 224 68 16 69 464 231 158106 159 94 154 321

tit. §§ 23-35 hab. A (l. 300, D ||ad304 ]

A,

) D || 304 12 12 12 lin. ex (corr. supra )

]

A || 305

D || 305 tit. (ad §§ 23-31)] D : A, ] A, D || 306 ] A || 306 301 ] D || 301 ] (sic) A || 301 ] om. D || 302 267 253 70 67 260 58 177 163 160 157 170 148 ] AD || 306 ] ut vid. A || 307 ] AD ; ] om. A || 302 ] add. supra lin. D || 302-3 55 269 57 63 266 175 145 179 147 153 176 add. A || 307 ] ] A 307part. (sc.] A || 307 216 110 214 208 114 41 288 38 286 280 42 A;||ult. ) ]hab. in marg. 207 122 205 200 123 118 279 50 272 51 46 D 277|| 308 ] om., hic hab. seq. A || 308 ] add. A || 308 D || 304 ] A, D || 304 ] A || 305 116 204 119 126 201 209 44 276 47 54 273 281 270 56 268 262 60 149 180 146 178 172 150 A || 309 ] ] A || 310 ] in textu, (sic) 115 125 202 203 120 210 43 53 274 275 48 282 ] A, (corr. supra lin. ex ) D || 305 add. in

321 307 16 13 314 4 231 217 106 103 224 94 263 229 168 233 165 230 173 62 1258 323 65 372 9255 320 152 91 155 93 162 99 1 323 3 9 320 229 91 233 93 99 230 61 71 256 257 66 264 151 161 166 167 156 174

261 68 259 254 69 64 171 158 169 164 159 154 213 199 124 121 206 112 285 271 52 49 278 40

marg. A ||] 310 ... ] add.A, in marg. D || 310 ] A ]|| 311 D || 306 A || 306] AD. 61 71 256 257 66 264 151 161 166 167 156 174 ] 96 AD || 306 ] ut vid. A || 307 ] AD ; 324 2 322 316 6 95 234 92 232 226 267 253 70 67 260 58 177 163 160 157 170 148v r v v 315 14 313 308 15 10 225 104 223) 218 100 ,8 - 16 ,6 (ed. ll . 700-716); D : 83 ,3 - 83 ,11. ( 24 A:10515 add. A 153 || 307 ] A || 307 ] A || 307 ] 179 147 176 55 269 57 63 266 175 145 8 312 11 18 309 317 98 222 101 108 219 227 ] A || 313 ] 216 110 214 208 114 41 288312 286 280 42 221 102 228 7 17 310 D 38220 || 308 ] om., hic hab. seq. A || 308 ] add. A || 308 4 311 12 3189 97 107 207 122 2054 200 123 118 9279 50 277 272 51 46 321 307 16 13 314 4 231 217 106 103D 224 ||94315 ] A || 316-7 ... ] om. (homoeotel.) D || 316 ] ] 281 A || 309 ] A || 310 ] in textu, (sic) add. in 116 204 119 126 201 209 44 276 47 54 273 1 323 3 9 320 229 91 233 93 99 230 317 ] A || 318 ] (& in marg.) D || 318 115 125 202 203 120 210 43 53 274 275A48|| 282 marg. A || 310 ... ] add. in marg. D || 310 ] A || 311 ] 7 3 213 199 1247 121 206 112 3285 271 52 49 278 40 ] ( corr. ex , et hab. in marg.) A, AD. 109 215 111 117 212 283 37 287 39 45 284 319 ] A, D || 319 ] AD || 320 ] A || 320 ] 324 2 322 316 6 226 96 95 234D92|| 232 62

258 65 72 255 263 152 168 155 162 165 173 109 215 111 117 212 283 37 287 39 45 284

315

14

3131 308

15

8

312

11

309 317

7

17

310 311

12

318

16

13

314

4

323

3

9

321 307 1

1

18

10 6225 104 223 218 105 100

6

98

om. D || 321

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

320 229

D || 326

91

233

93

99

94 230

A,

]

A || 323

D || 325

] deinde iter. verba

] ]

AD || 323

( . . .)

]

A || 324

A || 325

] ]

(cum erroribus

Examples of treatments for all that we have explained (A.II.23)276 Treatment for theand square of 5 by 5 (Fig. a 16). You put the57 Text translation odd numbers in the inner square of 3 as we have explained. Those remaining are 1, 25, 3, 23. You put 1 in the lower middle cell and 25 opposite to onitthe top;3you putleft-hand 3 in themiddle left-hand cellthe and 23 opposite to the it on above, in the cell middle and 23 on opposite side, on 87 the right. you 2put 2 inupper the upper right-hand and opposite right. Next,Next, you put in the right-hand cornercorner and opposite to it 277 todiagonally, it diagonally, in the lower left-hand corner, its complement, namely complement, namely in the lower left-hand corner, its even 24.24.You corner and andopposite oppositetotoititdiagonally, diagonally, Youput put4 4ininthe theupper upper left-hand left-hand corner ininthe corner, its itscomplement, complement, namely put 6 thelower lower right-hand right-hand corner, namely the22. evenYou number on22. theYou right-hand and its 20, opposite to it ontothe left. put 6 on side the right sidecomplement, and its complement, 20, opposite it on theput left.8 You 10 the and bottom, 8 on the 12 bottom, on the right, put You and put 10 on on the12right, and youand putyou opposite each its complement.278 toopposite each its to complement.

n −5 2 2

4

18

25

16

2

20

11

9

19

6

3

21

13

5

23

14

7

17

15

12

2 24

8

1

1 − ∆C 10n2 + 22 1

Fig. a 16

4

n2 − 1 2 n2 + 1 − 4

(Squares of orders n = 4t + 1, t > 1)

(A.II.24) You pass now from 5 to 9, 13, 17 and those of this kind.279

You pass now from five to nine, thirteen, seventeen and those of this n + 1 − ∆ it contains the larger small number ∆ or not. Depending on whether kind. 276 Construction of the particular magic square of the type n = 4t + 1, the square of 275

2

L 1

n2 + 3

L 1

n +1−2 (§5.24. Filling all first borders) 2 2

∆C 1

277

n2 + 7 2

No need for this last indication. Same in the next sentence. None of the previous rules applied since this square is a particular case (see n2 − 1 n2 − 5 A.II.13). n2 + 1 − ∆C 2 1 2 2 279 A.II.24–A.II.31 teach how to place the even numbers for the orders n = 4t + 1 (t ≥ 2 n + 1 − 4 2). In this section, how to fill the first border. For order 9 4(Fig. a17), once all odd numbers have been placed, there remain to be placed the even numbers from 2 to 40 and their complements. The first one will be 40 since they are taken in reverse order. 278

∆L1 n2 + 7 2

n2 + 1 − ∆L1

[email protected] ∆C n2 + 1 − 2 1

n2 + 3 2

Put the last small even term in the upper left-hand corner of the first border — that is, the (border) following the inner square which you have (completely) filled with odd numbers — and opposite to it diagonally, in the lower right-hand corner of the first border, its complement. Put the preceding small term in the upper right-hand corner of the first border, and opposite to it diagonally, in the lower left-hand corner, the term which is its complement.88 [ Once you have done that, you will find that the 87

As taught in § 9. C With this, the former excesses ∆L 1 and ∆1 (§ 14) are changed, according to the L* L * = ∆C equalization rule seen in § 22, into ∆1 = ∆1 − 4, ∆C 1 1 − 2 (see table). 88

... AD (pro hab. A , pro scr. D ) || 294 ] AD AD (pro hab. A , pro scr. D ) || 294 ] AD || 294-5 ... ] om. (homoeotel.) A || 295 ] || 294-5 ... ] om. (homoeotel.) A || 295 ] 58 translation D ( rescr. in marg.) || 296 Text and ] A || 296 ] D || 296 ] A, D ( rescr. in marg.) || 296 ] A || 296 ] D || 296 ] A, D || 296 ] A || 297 ] (sic) A, D || 297 ] ] A || 297 ] (sic) A, D || 297 ] ]] D || 296 ] A || 297 ] D || 297 ] om. D || 298 ] A || 299 ] add. in A ||] 297 ] D || 297 ] om. D || 298 ] A || 299 ] add. in marg. A || 299 ] om. D || 299 (pr.) ] D || 299 ] marg. A || 299 ] om. D || 299 (pr.) ] D || 299 ] A, om. D. A, om. D. [

261 261 62 62 61 61 267 267

68 259 254 69 64 171 158 169 164 159 154 68 259 254 69 64 171 158 169 164 159 154 258 65 72 255 263 152 168 155 162 165 173 258 65 72 255 263 152 168 155 162 165 173 71 256 257 66 264 151 161 166 167 156 174 71 256 257 66 264 315 151 161 166 167 156 174 315 253 70 67 260 58315 177 163 160 157 170 148 253 70 67 260 58 315 177 163 160 157 170 148 55 269 57 63 266 175 145 179 147 153 176 55 269 57 63 266 175 145 179 147 153 176 216 110 214 208 114 41 288 38 286 280 42 216 110 214 208 114 41 288 38 286 280 42 207 122 205 200 123 118 279 50 277 272 51 46 207 122 205 200 123 118 279 50 277 272 51 46 116 204 119 126 201 209 44 276 47 54 273 281 116 204 119 126 201 209 44 276 47 54 273 281 115 125 202 203 120 210 43 53 274 275 48 282 115 125 202 203 120 210 43 53 274 275 48 282 213 199 124 121 206 112 285 271 52 49 278 40 213 199 124 121 206 112 285 271 52 49 278 40 109 215 111 117 212 283 37 287 39 45 284 109 215 111 117 212 283 37 287 39 45 284 324 2 322 316 6 95 234 92 232 226 96 r 324 2 322 316 6 95 234 92 232 226 96 r 315 14 313 308 15 10 225 104 223 218 105 100 315 14 313 308 15 10 225 104 223 218 105 100 320222 101 108 219 227 8 312 11 18 309 317 98 320 320222 101 108 219 227 8 312 11 18 309 317 98 7 17 310 311 12 318 97 107 220 221 102 228 7 17 310 311 12 318 97 107 220 221 102 228 321 307 16 13 314 4 231 217 106 103 224 94 321 307 16 13 314 4 231 217 106 103 224 94 1 323 3 9 320 229 91 233 93 99 230 1 323 3 9 320 229 91 233 93 99 230

[[[

( 23 ) A: 15 ,18 - 15v ,8 (ed. ll . 690-699); D : 83r ,21 - 83v ,2. ( 23 ) A: 15 ,18 - 15v ,8 (ed. ll . 690-699); D : 83r ,21 - 83v ,2. tit. ad §§ 23-35 hab. A (l. 300, ) tit. ad §§ 23-35 hab. A (l. 300, ) tit. (ad §§ 23-31) D : tit. (ad §§ 23-31) D :

12 12 12 12

] D || 301 ] (sic) A || 301 ] om. D || 302 ] D || 301 ] (sic) A || 301 ] om. D || 302 ] om. A || 302 ] add. supra lin. D || 302-3 ] om. A || 302 ] add. supra lin. D || 302-3 ] A; ult. part. (sc. ) hab. in marg. ] A; ult. part. (sc. ) hab. in marg. D || 304 ] A, D || 304 ] A || 305 D || 304 ] A, D || 304 ] A || 305 270 56 268 262 60 149 180 146 178 172 150 ] A, (corr. supra lin. ex ) D || 305 270 56 268 262 60 149 325180 146 178 172 150 ] A, (corr. supra lin. ex ) D || 305 261 68 259 254 69 64 325 171 158 169 164 159 154 325 261 68 259 254 69 64 171 158 169 164 159 154 ] A, D || 306 ] A || 306 62 258 65 72 255 263 152 168 155 162 165 173 ] A, D || 306 ] A || 306 62 258 65 72 255 263 152 168 155 162 165 173 61 71 256 257 66 264 151 161 166 167 156 174 ] AD || 306 ] ut vid. A || 307 ] AD ; 61 71 256 257 66 264 151 161 166 167 156 174 AD || 306 ] ut vid. A || 307 ] AD ; 267 253 70 67 260 58 177 163 160 157 ] 170 148 267 253 70 67 260 58 177 163 160 157 170 148 add. A 153 || 307 ] A || 307 ] A || 307 ] 179 147 176 55 269 57 63 266 175 145 add. A 153 || 307 ] A || 307 ] A || 307 ] 179 147 176 55 269 57 63 266 175 145 216 110 214 208 114 41 288 38 286 280 42 D || 308 ] om., hic hab. seq. A || 308 ] add. A || 308 216 110 214 208 114 41 288 38 286 280 42 ] om., hic hab. seq. A || 308 ] add. A || 308 207 122 2054 200 123 118 9279 D 277 308 272 51 46 50 || 207 122 2054 200 123 118 9279 50 277 272 51 46 ] 281 A || 309 ] A || 310 ] in textu, (sic) add. in 116 204 119 126 201 209 44 276 47 54 273 ] 281 A || 309 ] A || 310 ] in textu, (sic) add. in ) 116 204 119 126 201 209 44 276 47 54 273 ( 25 115 125 202 203 120 210 43 53 274 275 48 282 ... ] add. in marg. D || 310 ] A || 311 ( 25]) 274 275 A 282310 115 125 202 203 120 210 43 53marg. 48 || marg. A || 310 . . . ] add. in marg. D || 310 ] A || 311 ] 213 199 1247 121 206 112 3285 271 52 49 278 40 330271 52 49 213 199 1247 121 206 112 3285 278 40 AD. 283 37 287 AD. 109 215 111 117 212 330 39 45 284 330 109 215 111 117 212 283 37 287 39 45 284 301 301

315 315 8 8 7 7 321 321

324 324 14 14 312 312 17 17 307 307 1 1

2 322 2 322 313 308 313 1308 1 18 11 11 18 310 311 310 311 16 13 16 13 323 3 323 3

316 316 15 15 309 309 12 12 314 314 9 9

6 95 234 92 232 226 96 6 95 234 92 232 226 96 10 225 104 223 218 105 100 6 10 225 104 223 218 105 100 3176 98 222 101 108 219 227 317 98 222 101 108 219 227 318 97 107 220 221 102 228 318 97 107 220 221 102 228 4 231 217 106 103 224 94 4 231 217 106 103 224 94 320 229 91 233 93 99 230 320 229 91 233 93 99 230

335 335 335

24)) A: A: 15 15vv,8 ,8--16 16rr,6 ,6 (ed. (ed. llll.. 700-716); 700-716); DD:: 83 83vv,3 ,3--83 83vv,11. ,11. ((24 312 312 ]]

A |||| 313 313 A

]]

A |||| 316-7 316-7 om. (homoeotel.) (homoeotel.) DD |||| 316 316 A ...... ]] om. ]] A |||| 318 318 (& in marg.) marg.) DD |||| 318 318 ]] A ]] (& in ] ( corr. ex , et hab. in marg.) A, ] ( corr. ex , et hab. in marg.) A, v 270 56 268 262 60 149 (180 178 172 24146) A: 15150 ,8 - 16r ,6 (ed. ll . 700-716); D : 83v ,3 - 83v ,11. 37 D || 319 ] A, D || 319 ] AD |||| 320 320 A |||| 320 320 ( 26 ])] 261 68 259 254 69 64 171 158 164 159 154 D 169 || 319 ] A, D || 319 AD ]] A 37 ] ] 312 A || 313 ( 26 ]) 62 258 65 72 255 263 152 168 155 162 165 173 om. DD |||| 321 321 A |||| 323 323 AD |||| 323 323 A |||| 324 324 om. ]] A ]] AD ]] A ]] 61 71 256 257 66 264 151 161 166 167 D156|| 174315 ] A || 316-7 ... ] om. (homoeotel.) D || 316 ] A, 325 A |||| 325 325 DD |||| 325 ]] A ]] 267 253 70 67 260 58 177 163 160 157 170 148A, A 153 || 317 ] A || 318 ] (& in marg.) D || 318 340 176 55 269 8 4 57 63 2669 175 145D179|| 147 326 ] deinde iter. verba ( . . .) (cum erroribus D || 326 ] deinde iter. verba ( . . .) (cum erroribus 340 340 4 9 24 113 216 110 214 208 114 41 288 38 286 280 42 ] ( corr. ex , et hab. in marg.) A, sed sine ) A || 327 ] A || 328 ] D. sed sine ) A || 327 ] A || 328 ] D. 207 270 205 268 200 262 279 180 277 178 272 172 122 56 123 60 118 149 50 146 51 150 46 D ||47 319 ] A, D || 319 ] AD || 320 ] A || 320 ] 204 259 201 64 209 171 276 169 273 154 281 116 68 119 254 126 69 44 158 54 159 261 164 2 7 3 rr ,6 - 16rr ,13 (ed. ll . 717-724); D : 83vv ,11 - 83vv ,16. ( 25 ) A: 16 202 203 255 210 274)162 275 282 ,6 - 16 ,13 (5325 A: . 717-724); D :] 83 ,11 - 83|| ,16. 115 258 125 65 120 263 43 168 7 72 3 152 165 173 62 155 om. D ||48 16 321 ] (ed. llA || 323 AD 323 ] A || 324 ] 315 DD |||| 315 A |||| 317 317 A

213 71 199 256 206 264 285 161 271 166 278 174 22 61 124 257 121 66 112 151 52 167 49 156 40

]]

|| 330 330 A ||||]330 330 A, 331 DD ||D ]] A ]] D || 325 A ||A, 325 ]D |||| 331 5 1 6 ] 226 17696 A A |||| 331 331 A |||| 331 331 post ( . . .) AD ||(cum || 331 331erroribus]] ]] verbaA ]] post AD D ||92 147 326 ] deinde iter. 324 2692 57 322 63 316 2666 175 234 179 232] 153 78 5 55 95 145 1 6 315 216 313 214 308 208 225 288 223 286 218 280 14 110 15 114 10 41 104 38 105A, 100 42 D || 333-4 . . . ] om. D, hic hab. : D 327 || 333-4 ] . . . A || 328 ] om. sed sine A, ) A || ] D, hic D. hab. : 312 205 309 118 317 279 11 200 18 123 98 222 101 108 219 227 2078 122 38 345 50 277 272 51 46 333 |||| 333 38 v 345 345 310 126 311 201 318 220 54 221 273 228 7 204 17 119 12 209 97 276 107 47 102 281 116 44 r r v ( 25 ) A: 16 ,6 16 ,13 (ed. ll . 717-724); D : 83 ,11 83 ,16. 321 125 307 202 314 2104 43 231 53 217 274 76 115 16 203 13 120 106 275 94 || 333 ] 48224 A A A |||| 334 334 om. A A |||| 335 335 add. in in marg. marg. DD |||| 282 ]103 || 333 ]] A ]] om. ]] add. 323 121 320 285 229 271 233 49 230 3 2069 112 91 52 93 278 99 ]40 329 D || 330 ] A || 330 ] A, D || 331 213 1991 124 336 A |||| 336 336 D. 336 ]] A ]] D. 109 215 111 117 212 283 37 287 39 45 284 ] A || 331 ] A || 331 ] post AD || 331 ] 329 329

]]A,

215 67 212 177 283 163 287 157 284 109 70 111 260 117 58 37 160 39 170 45 148 267 253

324

2

322 316

6

95

234

92

232 226

96

Texttreatise and translation An ancient Greek on magic squares

73

59

excess of the upper row is 16, the excess of the right-hand row 16, thus the excess of the upper row equals the excess of the right-hand row. ]89 L

1

n=9 n = 13 n = 17

n2 ≠1 2

C

1

n2 +3 2

n2 ≠5 2

L*

1

20 18 40 42 38 16 28 26 84 86 82 24 146magic squares 142 32 An36 ancient34 Greek 144 treatise on

C*

1

16 24 32 73

Put then 2 in the upper row; consider the (new) excess of the upper row, take its half, and count after 2 as many small even (numbers) L* C* 22 ≠ C1 24 ≠ L1 1 1 2 as this half, and, having arrived there,n put numberL * reached in the L C C* n ≠1 +3 the n ≠5 1 1 1 1 2 2 n = 9 to each 16 of2 16 ≠16numbers ≠16 its complementary lower row; put opposite these two n=9 2013 1824 4024 42 38≠24 16 16 = ≠24 after term. Put 4 onn the right side; then count 4 as many small even n = 13 28 26 84 86 82 24 24 n= 17excess, 32 and,32 ≠32 ≠32 numbers nas=half of the having arrived there,32put 32 the number 17 36 34 144 146 142 reached in the left-hand side; put opposite to each of these two numbers its complementary term.90 n2 +1 2

L*

1

nn==97 11 nn==13 15 nn==17

C*

1

16 25 16 24 61 24 32 113 32

n2 ≠1 2 2≠

C

1

n2 ≠5 L

4 ≠2

1

24 22 ≠16 ≠16 60 58 ≠24 ≠24 112 110 ≠32 ≠32 n = 19 181 180 170 Once you have done that, you will have equalized all the sides of the

first border in this kind of square. 2

n +1 2

n2 ≠1 2

(§ 25. First steps for2 filling all borders) C 1 second n +n 1= 7

2

i

n2 ≠5 2

+2 25 —i + “i 24 —i , —i 22

“i ≠ 2, “i

(i) Put then the pairn of small terms the sum of 58 which is less than their n=7 50 = 11 8 61 42 60 10, 12 30, 32 sum due by 12 in then = right-hand corners of the second border, with the n = 11 122 15 12 113 110 112 42, 110 44 66, 68 lesser one above; putnin= the opposite corners, on the left, their 19 diagonally 181 180 170 122 24 98 26, 28 70, 72 complements.91 n = 15

226 16 210 90, 92 118, 120 226 32 194 70, 72 122, 40, 12442, 38. As Early reader’s gloss 2illustrating situation for n = 9 after placing C 1 the n +1 —i + “i —i , —i + 2 “i ≠ 2, “i i 2 a matter of fact, this equality is verified generally for the new excesses (the 226 48 178 50, 52 126, 128terms thus 89

C placed cancel out7 the difference ∆L n= 50 422). 10, 12 30, 32 18 − ∆1 = * L* 90 1 · ∆ even numbers after 2 or68∆L First, itncomes same to count = 11 to the 122 12 110 42, 44 66, 1 1 (natural) N1 N22 N31 C * N4 C * N5 numbers after same holds · ∆26, 4. Second, the 24for counting 98 28 ∆1 after 70, 72 1 and 2 (n =2,4tand + 1)the122 C L two numbers thus reached will be equal to ∆ in the first case and ∆ in the second; 4 + 8 4 + 24 4 + 40 4 + 56 4 + 72 1 1 n = 15 226 16 210 90, 92 118, 120 it could therefore have been simply said that we are to put, in the rows 226 32 194 70, 72 122, 124 opposite to L those of 2 and 4, quantities equal to the initial excesses ∆C 1 and ∆1 , respectively, the 226 48 N2 178N3 50,N52 128 L N126, 1 4 by |2 − 5∆C placement of the complementsNthus reducing the excesses 1 | and |4 − ∆1 |. (n = 4t + 3) By the way, using the initial excesses 4 4 +and 16 deficits 4 + 32would 4 + be 48 in 4keeping + 64 with their use N2 N3 thatNthe N5 text was corrupt in the subsequent paragraphs. N It1is therefore likely 4 original = 4tlater + 1) attempts to restore it. at this place,(nwith 4 + 8 4 + 24 4 + 40 4 + 56 4 + 72 91 See the figure below, showing the placing of the numbers mentioned in §§ 25–27. The subsequent table gives, for the order considered, the sum due for two cells, then N1 from N2 whichNwe this sum due less 12, as required, findNthe pairN5of consecutive small 3 4 (n + = 24tattributed + 3) numbers α2 , α to the right-hand corners of the second 2 4 4 + 16 4 + 32 4 + 48 4 + 64 border.

207 122 2054 200 123 118 9279

50

277 272

51

116 204 119 126 201 209

44

276

47

273 281

115 125 202 203 120 210

43

53

274 275

213 199 1247 121 206 109 215 111 117 324 315

14

8 7

6

15

312

11

309 317

17

310 311

12

16

13

314

323

3

9

321 307 1

18

234

95

313 308

1

48

A || 309

282

marg. A || 310 112 3285 271 52 49 278 40 330 330 60 AD. 212 283 37 287 39 45 284

322 316

2

46

]

54

10

6

92

232 226

A || 310

]

in textu,

]

] add. in marg. D || 310

...

(sic) add. in

Text and translation

( 25 )

A || 311

]

]

96

225 104 223 218 105 100 98

318

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

4

320 229

233

91

93

99

94 230

335 335

( 24 ) A: 15v ,8 - 16r ,6 (ed. ll . 700-716); D : 83v ,3 - 83v ,11. 37 ] 312 D || 315

340 340

4

9 268 262

56

261

68

259 254

69

62

258

7 65

255 2633 152 168 155 162 165 173

61

71

256 257

66

267 253

70

67

260

58

55

269 1

57

63

2666 175 145 179 147 153 176

72

D || 319

om. D || 321

264 151 161 166 167 156 174

D || 326 sed sine

288

38

286 280

207 122 205 200 123 118 279

50

277 272

43

213 199 124 121 206 112 285 109 215 111 117 212 283 324 14

8 7

322 316

2

6

95

225 104 223 218 105 100

15

312

11

309 317

17

310 311

12

318

16

13

314

4

323

3

9

321 307 1

18

98

AD || 323

] ]

A || 328

]

222 101 108 219 227

]

]

corr. ex

(

) A || 327

A || 325

]

A || 328

A,

270

56

261

68

2594 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

267 253

70

67

260

58

55

269

57

63

266 175 145 179 147 153 176

64 9171 158 169 164 159 154

A || 333

]

264 151 161 166 167 156 174

336

]

177 163 160 157 170 148

3

216 110 214 208 114 41

288

38

50

277 272

51

116 204 119 126 201 209

44

276

47

273 281

43

53

115 125 202 203 120 210

286 280

54

42 46

274 275

48

213 199 124 121 206 112 285 271

52

49

278

40

109 215 111 117 212 283

37

287

39

45

284

324

234

92

232 226

96

322 316

6

95

282

]

A || 336

A || 330

A || 331

]

D || 333-4

60 149 180 146 178 172 150

207 122 205 200 123 118 279

2

A || 331

]

D.

]

A || 334 ]

]

] post

A,

AD || 331

] om. D, hic hab. :

...

] om. A || 335 D.

]

(cum erroribus

( . . .)

]

D || 330

]

]

6

(cum erroribus

hab. in marg.) A, || 333 ] inAmarg. || 320 D ||] ]] om. A ||AD 335|| 320 ] add. ] D.AD || 323 ] A || 324 ]

] deinde iter. verba

329

1

] ]

D.

]

( 25 ) A: 16r ,6 - 16r ,13 (ed. ll . 717-724); D : 83v ,11 - 83v ,16.

7

]

et

,

13

D ||] 319 A || 333] A, || 319 ] A ||D334 94 om. D 99|| ]230 321 A || 336] A ||] 323 233 93 91 336 A, D || 325 sed sine

72

A || 324

A || 325

D 38 : 83vv,3 - 83v ,11. v

r

231 217 106 103 224

D || 326

268 262

A || 320

]

( . . .)

97 107 220 221 102 228

320 229

AD || 320

( 24 273 15 281 ,8 - 16 ,6 (ed. ll . 700-716); 47 )54A: ( 25 ) A: 16r ,6 - 16r ,13 (ed. ll . 717-724); D : 83 ,11 - 83 ,16. 53 274 275 48 282 312 ] A || 313 ] 32952 49 278 ]40 D || 330 ] A || 330 ] A, D || 331 271 37 287 39 D . om. (homoeotel.) D |||| 331 316 ] 45 ||284 315 A || ]331 A || 316-7 ] A ||. .331 ]] post AD ]] 234 92 232 226 96 A A, || 317 ] A || 318 ] (& in marg.) D || 318 D || 333-4 ... ] om. D, hic hab. : v

10

313 308

) A || 327

46

345 116 204 119 126 201 209 345 44 276 115 125 202 203 120 210

hab. in marg.) A,

] deinde iter. verba

42

51

A || 323

]

in marg.) D || 318

(&

et

]

D || 325

( 26 ])

] om. (homoeotel.) D || 316

,

D || 319

]

A,

177 163 160 157 170 148

216 110 214 208 114 41

315

A,

]

171 158 169 164 159 154

]

corr. ex

(

60 149 180 146 178 172 150

...

A || 318

] ]

270

64

A || 316-7

]

A || 317

A || 313

D || 331

|| 333

]

] add. in marg. D ||

n=5 12 n(n == 9 4t + 20 3) n = 13 28 n = 17 36 L 1 n = 21 44 n=5 12 n=9 20 L1 n = 13 28 n 1=−7(α + 2) 4 n2 + n = 17 i 36 n = 11 4 n = 21 44 n = 15 4 + 19 1 − βi nn2= 4 L n2 + 1 − (βi + 2)

n=7 n = 11 n=9 n= 2 15 + 113 − αi nn = n = 19 n = 17

10 N4 N5 18 N1 36N2 34 N3 26 4 52 4 + 1650 4 + 76 32 474+ 48 4 + 64 Text and66translation 34 68 100 98 132 130 C L C L C L C L C 1 2 2 3 3 4 4 5 5 42 84 82 124 122 164 162 204 202 10 L C L C L C 18 C1 36 L2 34 C2 3 3 4 4 5 5 26 52 50 76 74 γi αi 2 202 18 γi + 34 68 66 100 98 132 130 2 28 26 52 50 42 84 82 124 122 164 162 204 202 2 36 34 68 66 100 98 βi 164 2C 44 42 84 C82 124 122 162 L C L L C L C

1

1

2

2

3

3

4

4

5

βi + 2

61

5

4 2 20 18 ≠ 4 2 + ( 4 + ( C1 ≠ 2) 1 1 ≠ 4) 1 ≠2 2 2 4 2 28 26 52 50 40 38 16 18 16 20 4 2 2 36 34 68 66 100 98 α + 2 1 − (γi + 2)24 n2 + 1 − γ26 i 84 n +82 24 i 28 4 2 44 42 84 82 124 122 164 162 144 142 32 34 32 36 n2 ≠1

n2 ≠5

n2 ≠1 2

L

n2 ≠5

n=9 40 n = 13 84 n = 13 n = 17 144 n = 17 n2 + 1 − β

C

C ≠ 42 2 + ( L1 ≠ 4) ≠ 2 4 + ( C1 ≠ 2) (n + 1) ≠ 12 –2 , –12 + 2 16 18 16 20 70γ + 2 34, 36 α 24 26 24 28 158 78, 80 32 34 32 36 278 138, 140

L

1

n22+ 1 38 γ 82 82 170 142 290

n2 + 1 − (αn+ = 2) 9

L

2

β

2

n + 1 C (n numbers + 1) ≠ 12 2 + 2 a sum less than C –2 , – (ii) Thenn2look for 2 the pair have α (n2 + 1) ≠ (which 1 2−+ (α1+ 4) of small 2 2 ≠ 14) β—2+, 2—2 + 2 + 1 − (β +n2)+n n= 9 amount 82 equal to) 70 34, 36 second right-hand their sum due by (an the excess of the n=9 82 34 62 30, 32 n= 13 in the 170 second border 158 80 and, opposite to row2 less 14; put them on the 78, right n = 13 170 50 134 66, 68 n + 1 − (α + 2)2 α+2 n = 17 290 2 92 n278 + 1 − (γ + 2) 138, 140 α 118, + 2 120 n2 + 66 1−γ 1 − α their them on the nn =+left, 17 290complements. 238 α+6

n=9 n=9 n = 13 n = 13 n = 17 n = 17

n2 2+ 1 n +1 82 α 82 +4 170 170 290 290 n2 + 1

n=9 n = 13 n = 17

7

48

(n2 + 1) ≠ (

L

2

46

82 52 170 50 290

n2 + 1 − (α + 6)

(n2 2+ 1) ≠ ( 2L (n + 1) ≠ ( 2 34 62 2 36 54 50 n + 1 − α 134 52 126 66 238 68 230 C

78

36 71 52 2768 31

9

29

40

64

39

77

28

26

69

3

61

63

54 38 126 232304

37

47

53

73

2

15

25

49

41

33

57

67

81

65

35

45

43

17

1

20

59

51

21

19

55

62

44

18

11

13

79

80

42

54

56

5

≠ 14) ≠ 8)

C 2L

2

L

2 ≠ 34

30 32

8)

—2 , —2 + 2 “2 , “2 + 2 30, 32 26, 28 66, 68 62, 64 118, 120 114, 116 “2 , “2 + 2 26, 28 62, 64 114, 116

75

36

n=9

(iii) Then look for the pair of small numbers which have a sum less than their sum due by (an amount equal to) the excess of the second upper 92 L For the values of ∆C 2 and ∆2 in the tables to (ii) and (iii), see § 14. Same for §§ 26–27.

335

62

Text and translation

( 26 )

340

345

13

270

56

261

68

259 254

69

62

258

65

255 263 152 168 155 162 165 173

61

71

256 257

66

264 151 161 166 167 156 174

130 267 253

70

67

260

58

194 265

269

57

63

266 175 145 179 147 153 176 r

192

55

268 262

( 26 ) A: 16r ,13 - 16r ,21 (ed. ll . 725-732); D : 83v ,17 - 83v ,22.

132 59

72

60 149 180 146 178 172 150 64

338

]

171 158 169 164 159 154

D || 342

345

]

177 163 160 157 170 148

24 113 216 110 214 208 114 41

286 280

42

50

347277

272

51

46

116 204 119 126 201 209

44

276

47

54

273 281

300 115 125 202 203 120 210

43

53

274 275

37

287

39

234

92

22

213 199 124 121 206 112 285

78

5

324

2

322 316

6

95

A || 339

]

A || 344

] D || 345

]

]

A || 339

] om. A || 344

A || 345

]

( 27 ) A: 16 ,21 - 16v ,7 (ed. ll . 733-742); D : 83v ,22 - 84r ,2.

207 122 205 200 123 118 279

302 211 109 215 111 117 212 283

D || 338

288

38

]

A, 48

282

45

284

232 226

96

A || 352 271 52 49 278

40

]

(sic) add. supra lin. D || 348

D || 350

]

] om. D || 354 A || 356

D || 350 ]

]

]

D || 351

D || 354 A || 357

] corr. ex

A,

]

A,

] D.

A ||

A || 348 ] ] ]

] (sc.

)

A || 356 A || 357

n = 13 n = 17

170 290

n2 + 1 n=9 row less 8; n =put 13 n = 17 complements.

82 them 170in 290

158 278

C (n2 + 1) ≠ ( Text2 and translation

34 this50row 66

n2 + 1 n=9 n = 13 n = 17

78, 80 138, 140

and,

2

2

36 52 68

≠ 14)

62 opposite 134 238

(n2 + 1) ≠ (

L

82 170 290

C

L

2

54 126 230

to

—2 , —2 + 2 30, 32 them66,below, 68 118, 120

≠ 8)

63 their

“2 , “2 + 2 26, 28 62, 64 114, 116

(§ 26. First steps for filling all third borders) (i) Put then the pair of small numbers which have a sum less than their 76 by 20 in the right-hand Translation sum due corners of the third border, with the lesser one above; put, opposite to them diagonally, in the corners of the third border on the left, their complements. 76

n2 + 1 (n2 + 1) ≠ 20 170 Translation150

n = 13 n = 17

290

–3 , –3 + 2 74, 76 134, 136

270

(ii) Then look for the2 pair which, –have a sum less than 2 of small 2numbers + 21)+≠1)20≠ ( C3 – n + 1n + 1 C3 (n (n ≠3 22)3 + 2 —3 , —3 + 2 their sum due by (an amount equal to) the excess of the third right-hand n = 13 170 170 74 150 74, 76 n = 13 118 row less 22; put them in290this row and, opposite to136them58,in60 the third n = 17 270 134, n = 17 290 98 214 106, 108 left-hand row, their complements. 2 n2 + 1

nn= =13 13 nn= =17 17

n = 13 n = 17

112

n2 + 1

n +1 17

290 13 29 n = 17 11

74 76 98 100

27 35 2

n +1

155 145

7

23

59

C 55

n572 +71 1

75

130

49

69

2 + 149 n165 1 129 109C4

n = 17 n = 17

290 290169 28

4

53 29073

83

L 93

4

22) ≠ 8)

“3—,3 ,“3—3++2 2 58,5260 50, 106, 108 98, 100

L 74 “ , “ + 2 ≠ 8) 32 3 (n +1) 1)≠ ≠(283 52 –50 4 , –4 +

144 100 151 139 137 51

33

161 (n22 +

L

76 290

C L

118 102 214 198

n2 +31

90

n =170 17

110 104 290 84 1022 166

n = 17

3 3

170 290 94

(n22 + 1) ≠ (

C L

79

132 130 95

133 103

64 262102 3198 2

62

78

82

66

125 127 131 47 4 68 (n2 + 1) ≠ ( C4 ≠ 30) 2 105 107 67 113 153 (n + 1) ≠ 28 –4 , –4 81

91 26297

85

77

190

123 111 115

45

43

15

25

163

+2

117 141 130, 157 132

101 121 135 143 159 L 4 ≠ 8) C 21 5 4 ≠ 30)

(n2 + 1) ≠ ( 1)61≠ (41 166 65 63 99 37 190 89 (n2 87 +

58

130, 13250, 52 60 98, 100

39

1

—4 , —4 + 2 94, 96

“4 , “4 + 2 —4 , —4 + 2 82, 84 94, 96

119 142

n2 ≠ 5 ≠ 2m · 4 n2 ≠ 1 ≠ 2m · 4 + = n2 ≠167 3 ≠ 2m · 4 = n2 + 1 ≠ 4(1 + 2m) n2 + 88 1 26 219L4 31 33 (n2 147 + 1) ≠ (168 L4 86 ≠ 8) “4 , “4 + 2 2 92

106 108

80

166 82, 84 n2n+=7 17 + 2m · 4 290 n2 + 3 132 + 2m · 4 + = n2 + 5 + 2m ·118 4 =120 n2 + 1 + 4(1 + 2m) 9 76 2 96 2 n = 13 2

2

n ≠ 5 look ≠ 2m for · 4 the n ≠ 1 ≠ of 2msmall ·4 2 (iii) Then pair + = nnumbers ≠ 3 ≠ 2m · 4which = n2 +have 1 ≠ 4(1a+sum 2m) less than 2 2 (m = 0, 1, 2, . . . ) their sum due by (an amount equal to) the excess of the third upper row n2 + 7 + 2m · 4 n2 + 3 + 2m · 4 + = n2 + 5 + 2m · 4 = n2 + 1 + 4(1 + 2m) 2 2 n2 + 3 n2 + 7 –, – + 2, – + 4, –+6 =⇒ n2 + 1 + 4 − 8j − 4j − 4j 2

2

(m = 0, 1, 2, . . . )

64

Text and translation

( 27 )

( 27 )

350

350

355

( 26 ) A: 16r ,13 - 16r ,21 (ed. ll . 725-732); D : 83v ,17 - 83v ,22. 338

]

D || 342

355

270

56

261

68

259 254

69

62

258

65

255 263 152

61

71

256 257

66

267 253

70

67

260

55 270 216 68 122 258 204 71 125 253 199

269 56 110 259 205 65 119 256 202

57 268 214 254 200 72 126 257 203

261 207 62 116 61 115 267 213

55 109 216 324

268 262

72

60 149 64

171

360

63 262 208 69 123 255 201 66 120 70 67 260 124 121 206 269 57 63 215 111 117 110 214 208 2 322 316

207 122 205 200 123 315 14 313 308 15

264 151 58

177

266 175 60 149 114 41 64 171 118 279 263 152 209360 44 264 151 210 43 58 177 112 285

D || 338

A || 339

]

A || 344

]

]

A || 339

] om. A || 344

A,

] ] corr. ex

A ||

345 ] D || 345 ] A || 345 ] A, D. ( 26 ) A: 16r ,13 - 16r ,21 (ed. ll . 725-732); D : 83v ,17 - 83v ,22. ( 28 ) ( 27 ) A: 16r ,21 - 16v ,7 (ed. ll . 733-742); D : 83v ,22 - 84r ,2. 338 ] D || 338 ] A || 339 ] A || 339 ] A, 180 146 178 172 150 347 ] (sic) add. supra lin. D || 348 ] A || 348 ] 158 169 164D 159||154 342 ] A || 344 ] om. A || 344 ] corr. ex A || A,173 D || 350 ] D || 350 ] D || 351 ] (sc. ) 168 155 162 165 345 ] D || 345 ] A || 345 ] A, D. 161 166 167 156 174 A || 352 ] om. D || 354 ] D || 354 ] A || 356 ( 28 ) 163 160 157 170 148 ( 27 ) A: 16r ,21 - 16v ,7 (ed.All .||733-742); D : 83v],22 - 84Ar ,2. 356 || 357 ] A || 357 145 179 147 153 176]

180 288 158 50 168 276 161 53 163 271

146 178 286 169 164 277 272 155 162 47 54 166 167 274 275 160 157 52 49 179 147 287 39 38 286 92 232

172 280 159 51 165 273 156 48 170 278

150 42 154 46 173 281v 174 282 148 40 153 176 45 284 280 42 226 96

] (sic) add. supra lin. D || 348 ] A || 348 ] A || 357 ] A. A, D || 350 ] D || 350 ] D || 351 ] (sc. ) ( 28 ) A: 16 ,7 - 16v ,11 (ed. ll . 743-747); D : 84r ,2 - 84r ,5. ( 29 A || 352 ] om. D || 354 ] D || 354 ] A || 356 ) 358 ] D || 359 ] A || 359 ] A || 360 ] (sic) A || 361 ] ] A || 356 ] A || 357 ] A || 357 266 175 145 212 283 37 A || 361 ] om. D || 361 ] (sic) A, D || 361 ] 114 41 288 6 95 234 ] A || 357 ] A. 365 118 279 50 277 272 A 51 ||46 362 ] A. 10 225 104 223 218 105 100 34738

]

( 29 )

116 204 119 126 201 209 44 276 47 54 273 281v 8 312 11 18 309 317 98 222 101 108 219 227

( 28 ) A: 16 ,7 - 16v ,11 (ed. ll . 743-747); D : 84r ,2 - 84r ,5. v v ( 26 ) A: 16 - 16||r ,21 ,22. 358 ] ,13 D 359 (ed. ]ll . 725-732); A || 359 D : 83 ] ,17 A -||83360

115 125 202 203 120 210 43 53 274 275 48 282 r 7 17 310 311 12 318 97 107 220 221 102 228

] (sic) A || 361 ] A || 339 D || 361 ] (sic) A, ] om. A || 344 ] corr. ex

213 199 124 121 206 112 285 271 52 49 278 40 321 307 16 13 314 4 231 217 106 103 224 94

338 D || 338 A || 361] ] om. D] || 361 A || 339] 234 92 232 226 96 342 AD |||| 362 ] ] A. A || 344 104 223 218 105 100 345 ] D || 345 ] A || 345 222 101 108 219 227

109 215 111 117 212 283 37 287 39 45 284 1 323 3 9 320 229 91 233 93 99 230 324 315

14

8 7

2

322 316

6

313 308

15

312

11

309 317

17

310 311

12

318

16

314

4

321 307

18

13

10

95

365

225 98

r v v r ( 27 ) A: 16 94 ,21 - 16 ,7 (ed. ll . 733-742); D : 83 ,22 - 84 ,2.

347

]

261 68 259 254 254 69 171 158 169 164 164 159 68 259 69 64 64 171 158 169 159 154 154 261

A,

258 65 255 263 263 152 168 155 165 173 173 62 258 65 72 72 255 152 168 155 162 162 165 62 256 257 257 66 264 151 166 167 167 156 174 61 71 71 256 66 264 151 161 161 166 156 174 61

A || 352

267 253 253 70 260 58 177 163 163 160 170 148 70 67 67 260 58 177 160 157 157 170 148 267

(sic) add. supra lin. D || 348

D || 350

]

9

D || 350

A || 357

]

]

A || 357

( 28 ) A: 16 ,7 - 16v ,11 (ed. ll . 743-747); D : 84r ,2 - 84r ,5.

202 203 203 120 210 43 274 275 275 48 282 115 125 125 202 120 210 43 53 53 274 48 282 115

2

358

]

7

3

215 111 212 283 283 37 287 39 284 109 215 111 117 117 212 37 287 39 45 45 284 109

D || 359

D || 351

A || 348 ] ] ]

]

(sc. ( 30 ))

A || 356 A || 357

A.

]

204 119 201 209 209 44 276 47 273 281 281v 116 204 119 126 126 201 44 276 47 54 54 273 116

213 199 199 124 206 112 285 271 271 52 278 40 124 121 121 206 112 285 52 49 49 278 40 213

]

D || 354

]

A || 356

]

207 122 205 200 200 123 279 50 277 272 272 51 122 205 123 118 118 279 50 277 51 46 46 207

]

] om. D || 354

269 57 266 175 179 147 176 55 269 57 63 63 266 145 179 147 153 153 176 55 370175 145 216 110 214 208 208 114 288 38 286 280 280 42 110 214 114 41 41 288 38 286 42 216

4

D.

231 217 106 103 224

370

270323 268 9262 262320 180233 178 99 172 230 56 3268 60 229 149 91 146 93 150 180 178 172 56 60 149 146 150 1270

8

A,

]

97 107 220 221 102 228

] A,] A ||) ( 30

]

A || 359

14

]

A || 360

]

(sic) A || 361

]

n2 + 1

(n2 + 1) ≠ 20

–3 , –3 + 2

n = 13 170 150 74, 76 An ancient Greek treatise on magic squares n = 17 290 270 134, 136

127

Text and translation

n2 + 1

C

3

n2 + 1 n = 13 170 74 = 13 less 8; put nthem this and, = 17nin 290row170 98 n = 17 290 their complements. n2 + 1 n2 + 1 170 170 290 290

n = 13 = 17 13 nn = n = 17

(n2 + 1) ≠ ( C3 ≠ 22) —3 , —3 + 2 (n2 + 1) ≠ 20 –3 , –3 + 2 118 58, 60 150 to214 76 third opposite them in74,the lower 106, 108 270 134, 136 (n2 + 1) ≠ ( L3 ≠ 8) (n2 + 1) ≠ ( C3 ≠ 22) 102 118 198 214

L

3

65

C

3

76 74 100 98

(§ 27. First steps for filling all fourth borders) 2 2

row,

“3 , “3 + 2 —3 , —3 + 2 50, 52 98, 58, 10060 106, 108

n +1

(n + 1) ≠ 28

–4 , –4 + 2

130 n2 + 1

190 (n2 + 1) ≠ 28

–4 , –4 + 2

n2 +of1 small L3numbers (n2 +which 1) ≠ ( L3have ≠ 8) a sum “3 , “less (i) Put then thenpair 3 + 2than their = 17 290 262 130, 132 sum due by 28 in the right-hand corners of the fourth border, with the n = 13 170 76 102 50, 52 n = 17 290 100 198 98, 100 lesser one above, and oppositeC to them diagonally on the left their comn2 + 1 (n2 + 1) ≠ ( C4 ≠ 30) —4 , —4 + 2 4 plements. n = 17

290

n = 172 290 n +1 158

n = 17

290 n2 + 1

262 (n2 + 1) ≠ (

L

4

132 C 4

277(n2

166 + 1) ≠ (

L

4

C

4

94, 96

130, 132 ≠ 8) “4 , “4 + 2 ≠ 30)

84

82, 84 —824 , —130 4+2

196 n154 100 98 13494,94 = 17 290 2 130271 253 11 190 96 n2 ≠ 5 ≠ 2m · 4 n ≠ 1 ≠ 2m · 4 2 2 + = n35 ≠ 37 ≠ 2m · 4116= n 1 ≠ 4(1 194 184 150 267 247 237 114+138 106 +962m) 2 2 L L 2+1 2 n , “4 + 2 182 172 144 256 263 243 31 ≠ (3 4 ≠ 2 8) 142 118“4108 4231 229(n 51+ 1) n2 + 7 + 2m · 4 n2 + 3 + 2m ·4 2 + = n + 5 + 2m · 4 = n2 + 1 + 4(1 + 2m) n =217 170 286 290 83 95 91 213287 213 215 219166223 79 4 120 82, 84 25

93

111 119 115 185 187 191 107 197 265

n2 ≠ 5 ≠ 21 2m ·454 89 n2117 ≠ 1131 ≠(m 2m · 4165 0, 1, 167 2, . . . ) 173 201 245 135= + = n2 ≠127 3 ≠ 2m · 4 = n2 +269 1 ≠ 4(1 + 2m) 2 2 17

41

57

85

113 133 143 141 151 157 177 205 233 249 273

n2 + 7 + 2m · 4 n2 + 3 + 2m · 4 2 2 = n137 +161 5 + 2m = n227 +235 1 + 4(1 2m) 15 39 55 63 +81 109 –, 129 – 153 145 181 · 4 209 251 +275 2 2 + 2, – + 4, – + 6 281 257 241 217 189 169 139 149 147 121 101

73

49

33

285 261 221 193 163(m 155 1251, 123 = 0, 2, . .159 .)

97

69

29

5

289 225 183 171 175 105 103

99

179

65

1

36

211 195 199 203

77

75

71

67

207 254

148

34

152 156 160

27

–, – + 2, – + 4, – + 6 47

59

61

239 259 287 288 146

23

43

53

255 283

19

37

279

13

9

174 176 140 190 192 136 206 208 132

n = 17

(226 look ; D, fol. (ii) Then for 103 the)pair of small numbers which have a sum less than their sum due by (an amount equal to) the excess of the fourth righthand row less 30; put them in this row and, opposite to them in the fourth left-hand row, their complements. r

315 7114 256 313 257 308 6615 264 225 161 223 167 218 156 10 151 104 166 105 174 100 61

350

321 216 307 110 314 114 231 288 217 38 224 42 16 214 13 208 4 41 106 286 103 280 94

A || 357

]

350 116 204 119 126 201 209 44

276 53

43

274 275

48

282

49

278

109 215 111 117 212 283

37

287

39

45

284

324

234

92

232 226

96

322 316

2

6

95

365

358 271 52

D || 359

A || 362

315

14

313 308

15

10

225 104 223 218 105 100

8

312

11

309 317

355 355 98 222 101 108 219 227 355

7

17

310 311

12

318

16

13

314

4

323

3

9

321 307 1

18

320 229

355

9

370 7

3

360 360 360

233

91

93

338 345

62

258

65

255 263 152 168 155 162 165 173

71

256 257

66

264 151 161 166 167 156 174

267 253

70 4 67

260

58 9177 163 160 157 170 148

269

63

266 175 145 179 147 153 176

347

A,

38

207 122 205 200 123 118 279

50

277 272

116 204 119 126 201 209

44

276

43

53

7

3

365 365 365

115 125 202 203 120 210

1

6

8

322 316

6

15

312

11

309 317

17

310 311

12

318

16

314

4

321 307

18

13

95

365

10

225 98

286 280

68

62 61

]

51

A || 357

52

49

278

338 ] D |||| 338 338 338 D 287 39 45 ]284 A || 361 ] om. D] || 361 A || 339] 234 92 232 226 96 D || 342 342 AD |||| 362 ] ] A. A || 344 104 223 218 105 100 345 D || 345 ]] D ] A || 345 222345 101 108 219 227 347 347

65

255 263 152 168 155 162 165 173 66

264 151 161 166 167 156 174

267 253

70

67

260

58

55

269

57

63

266 370175 145 179 147 153 176

370

A, A,

D |||| 350 350 D

177 163 160 157 170 148

]]

288

38

286 280

]

42

A || 357

276 158 4716954 53

256109 257 66 215 111 14

8

312

269

264 151 117 212 283

322 260 31658 6

313 4 308

15

11

309 317

57

63

18

274 275

48

282

358 ] 52 49 278 40 166 167 161 156 37 287 45 A 39|||| 361 361284 A

A |||| 362 362 A

266 175 145 179 147 153 176 98 222 101 108 219 227 v

7

323

13 314 123 1184 3

202 203 120 210

363

231 217 224 279 277103 272 50 106

3

91

44

276

43

53

233

93

99

A, 54 47

230

273 281om. 48

282

52

49

278

40

215 111 117 212 283

37

287

39

45

284

234

92

232 226

96

322 316

313 308

15

A || 360 ] (sic) A,

A || 348 ]

]

]

(sc. ( 30 ))

A || 356 A || 357

]

]

(( 30 30 ))

(sic) A || 361 D || 361

] ]

14

42

274 275

6

v

]9451 A 46 || 363

124 121 206 112 285 271

1

A.

]

D.

] A,] A ||

( 29 ) A: 16 ,12 - 16 ,16 (ed. ll . 748-752); D : 84r ,5 - 84r ,7.

320 229

9

119 126 201 209

2

]]

14 14

]

A || 357

]

359 D |||| 359 ] A || 359 om. D || 361 ]] om.

232 157 226 170 95 234 96 177 163 92160 148

208 114 41 288 38 286 280 11017 214 310 311 12 318 97 107 220 221 102 228 1

174

10 9 225 104 223 218 105 100

7

321 16 205307 200

A ||(( 29 357)) 29

A,

D || 354

A.

255 263 152 168 358155 162 165 ] 173 D

213 199 124 121 206 112 285 271

70324 672

A || 356 A

)

A || 356

]

D || 351

]

]

273 281 vv ,7 164 159 28)) A: A: 16 16vv,11 ,11 (ed. (ed. ll . 743-747); D : 84rr ,2 - 84rr ,5. ((28 16 ,7154 -- 16

116 119 126 44 259204 254 69 20164209 171 115 125 202 203 120 210

315

]]

A 150 || 357 60 149 180 146] 178 172 50 277 272 51 46

207 122 205 200 123 118 279

D || 350

]

om. D D |||| 354 354 ]] om.

A |||| 352 352 A

43

(sc.

( 29 )

]

(sic) add. add. supra lin. D || 348 (sic)

]]

171 158 169 164 159 154 64 370 370

216 110 214 208 114 41

]

]

]

rr ,21 - 16vv ,7 (ed. 27 103 ) A: A: 16 ((27 : 83vv,22 - 84 r,2. 217 224 16 106 ) 94 ,21 - 16 ,7 (ed. ll . 733-742); D40

256 257

72

D || 351

97 107 220 221 102 228 231

A || 348

A || 357

]

A ||

( 28 )

D || 354

37

71

65

14 40

D.

] (sic) A || 361 ] A,A || 339 D || 361 ] (sic) ] om. A || 344 ] corr. ex

258

268 262

]

A, (( 28 28 ))

] ] corr. ex

A,

]

]

40

69

56

A || 339

]

]

A.

]

46

259 254 72

A || 345

D || 350

]

A || 356

42

180 233 178 99 172 230 56 3268 9262 320 60 229 149 91 146 93 150 1270 323 261

]

47 54 273 281v ( 28 ) A: 16 ,7 - 16v ,11 (ed. ll . 743-747); D : 84r ,2 - 84r ,5. 274 275 48 282 rr ( 26 A: 16 16 ,13 ,13 -- 16 16rr ,21 (ed. (ed. ll . 725-732); D : 83vv ,17 - 83v ,22. ( 26 )) A: 358 ] D || ,21 359 ] A || 359 ] A || 360

213 199 124 121 206 112 285 271 109 215 111 117 212 283

]

] om. D || 354

]

288

7

D || 350

A || 352

216 110 214 208 114 41

2

]

( 30 )

] om. A || 344

(sic) add. supra lin. D || 348

]

171 158 169 164 159 154

360

313 308

(sic) A || 361 D || 361

( 27 ) A: 16r ,21 - 16v ,7 (ed. ll . 733-742); D : 83v ,22 - 84r ,2.

61

6

A || 344

D || 345

60 149 180 146 178 172 150 64

A || 339

]

]

]

69

14

D || 338

]

56

324

A.

]

D || 342

259 254

315

(sic) A,

( 26 ) A: 16r ,13 - 16r ,21 (ed. ll . 725-732); D : 83v ,17 - 83v ,22.

68

57

]

]

94

270

72

( 29 )

A || 360

230

99

261

55

268 262

]

97 107 220 221 102 228 231 217 106 103 224

4

1

A || 359

]

] om. D || 361

A || 361

A || 357

]

47 54 273 281v ( 28 ) A: 16 ,7 - 16v ,11 (ed. ll . 743-747); D : 84r ,2 - 84r ,5.

] 40

213 199 124 121 206 112 285

A || 357

]

A || 356

]

A.Text and translation

]

323 2003 123 320 279 229 50 233 272 230 9 118 91 277 93 51 99 46 207 1221 205

115 125 202 203 120 210

D || 354

]

A || 356

]

310 57 311 63 318 175 220 147 221 153 228 7 5517 269 12 266 97 145 107 179 102 176

66

.

] om. D || 354

A || 352

312 7011 6718 260 309 58 317 350 222 160 219 148 227 98 163 101 157 108 170 2678 253 177 350

D || 365

] om. A || 364 ]

]

A.

D || 364

( 30 ) A: 16v ,16 - 16v ,21 (ed. ll . 753-759); D : 84r ,8 - 84r ,11. 368

]

]

D.

D || 369

6

95

10

v r 225 104 218 16 105 ,21 100 - 17 ,4 ( 31223) A:

]

A || 369-70

(ed. ll . 760-765); D : 84r ,11 - 84r ,14. 40

...

]

] om. A || 370

n2 + 1

L (n2 + 1) ≠ ( L ≠ 8) “ , “ +2 n2 + 1 3 (n2 + 1) ≠ 28 3 –4 , –4 + 2 3 3 n = 13 170 76 102 50, 52 n =ancient 17 290 262 130, 132 An Greek treatise on magic squares 69 Text and translation n = 17 290 100 198 98, 100

n2 + 1 n = 17

290 n = 17

67

(n2 + 1) ≠ ( C ≠ 30) —4 , —4 + 2 (n2 + 1) ≠ 28 4 –4 , –4 + 2 190 94, 96 262 130, 132

C

n2 + 41 130 290

(iii) Then look fornthe numbers which have a sum less than 2 + 1pair ofL small (n2 2+ 1) ≠ ( L4 C≠ 8) “4 , “4 + 2 4C n2 + amount 1 (n to) + 1)the ≠ ( excess —4 , —fourth 4+2 4 equal 4 ≠ 30) of the their sum due by (an upper n 22≠1 +7 ≠5 + ≠ =∆ n + 44 ≠ =∆ n2+3 + 1290 1≠ +4m + 8m 8m 132 n = 17 166 82, 84 n =put 17 them 290 in this 130 row and, opposite 190 96 row less 8; to them94,below, their complements. 2 2 2

n ≠ 5 ≠ 2m · 4 2 n ≠ 1 ≠ 2m ·4 2 L + +1 =(n n22 ≠ 2m n +31)≠≠ ( · 4L4 = ≠n 8) + 1 ≠“4(1 +2 4 , “+ 4 2m) 2 2 4 n = 17 2902 132 166 82, 84 n2 + 7 + 2m · 4 n + 3 + 2m · 4 + = n2 + 5 + 2m · 4 = n2 + 1 + 4(1 + 2m) 2 2 for the right-hand row(s), by successive addi2 +3 n2 +3 ≠5 ≠1 +7 ≠5 ≠1 +7 Proceed likewise: | n2. 42.≠ | +. always 14m 3 2n 2 n2 ≠ 5 ≠ 2m · 4 n2 ≠ 1 ≠ 2m · 4 tions of 8, and, as for+ the upper one(s), always subtraction = n2 ≠ 3 ≠ 2m ·with 4 = n2a+uniform 1 ≠ 4(1 + 2m) 2 2(m = 0, 1, 2, . . . ) 93

of 8.

n2 + 7 + 2m · 4

n2 + 3 + 2m · 4

2

2

+ = n + 5 + 2m · 4 = n + 1 + 4(1 + 2m) (§ 28. Remaining excesses, deficits, and empty cells) 2 2 –, – +the 2, –excess + 4, – + Once you have done that, of6 each upper row over its sum (m = right-hand 0, 1, 2, . . . ) one over its sum due will be due will be 6, the excess of each 2, and the number of remaining empty cells will be four in each second row, eight in each third row, twelve in each fourth row, and so on always n2 ≠ +7 31 5 =∆ =∆ nnn222+ + 111≠ + ≠ 4m 424 ≠ + 8m 8m +» + –, – + 2, – + 4, – + 6 2 by adding successively four.

n=9 n = 13 n = 17

L **

C **

i

i

N2

6 6 6

2 2 2

4 4 4

**

N3

n2 + 1 − (δ + 2) δ+6

L*

1

n=9 16 n = 13 24 n = 17δ + 4 32

⇓

18 26 34

δ

C*

1

**

12 12 12

2

n2 +3 2

n2 ≠5 2

40 84 144

42 86 146

38 82 142

1

n = 92 20 n + 1 − (δ + 4) n = 13 28 n = 17 36

N4

8 8 8

L (§ 29. Reducing these differences toC ± 2)n2 ≠1

1

**

2+

16 24 2 n32 +1−δ

18 26 34

L*

1

=⇒ n2 + 1 − 4

δ+2

C*

4+

1 + 6) n + 1 − (δ 2

20 28 36

=⇒ n2 + 1 + 4

⇓

n2 + 1 + 4

n2 + 1 − 4

You will now turn your attention to the remaining numbers. You take 2 2 groups of four consecutive nsmall numbers) and put the first in ϵ + 1 − (ϵ(even + 2) =⇒ n + 1 −one 2 93 For determining the βi , the quantity to be subtracted from the excesses ϵ+2 n2 + 1 − ϵ =⇒ n2 + 1increases +2 by 8, whereas, for determining the γi , this quantity remains invariably equal to 8.

n2 − 5 2

2

n2 + 1 − ∆C 1

n2 − 1 2 2

365

68

Text and translation

( 30 ) 370

14

375 375

((31 31))

[[

]]

380 380

((32 32))

385 385

29 )) A: A: 16 16vv,12 ,12 -- 16 16vv,16 ,16 (ed. (ed. llll .. 748-752); 748-752); D D :: 84 84rr,5 ,5 -- 84 84rr,7. ,7. (( 29 363 363 A, A,

268 262

A |||| 363 363 om. A A |||| 364 364 A ]] om. om. D D |||| 365 365 A. om. ]] A.

]]

56

68

259 254

69

258

65

v r ( 31155 ) A: v ,21 255 263390 165 173 - 17r ,4 152 168 162 16 390

72

256 257

55

64

66

264 151

67

260

58

269

57

63

266 175

[[ (ed. llll .. 760-765); 760-765); D D :: 84 84rr,11 ,11 -- 84 84rr,14. ,14. ( 31 ) A: 16 ,21 - 17 ,4 (ed. 174 D || 376 161 375166 167 ]156 ] om. om. 375 D || 376 ]]

216 110 214 208 114 41

204 119 126 201 209

44

202 203 120 210

43

395 395

50

277 272

51

46

49

278

40

45] 284

109 215 111 117 212 283

37 384287

39

324

234

232 226D 96 ||

313 308

15

]]

((33 33))

D |||| 376 376 D A |||| 377 377 A

32 )) A: A: 17rr,4 ,4 -- 17 17rr,11 ,11 (ed. (ed. llll .. 766-773); 766-773); D D :: 84 84rr,15 ,15 -- 84 84rr,19. ,19. (( 32 276 273 281 47 54 17 tit.274D D 275 tit. :: 48 282 53

38252 124 121 206 112 285 271 382

14

om. A A |||| 370 370 ]] om.

.. .. ..

A |||| 376 376 A ] A || 377 ] A, D ] A, ] A || 377 ] A, D 145 179 147 153 176 377 (pr.) ]] om. A A |||| 377 377 (hic et et infra) infra) D D |||| 378 378 |||| 377 (pr.) ]] om. ]] (hic 288 38 286 280 42 v D (v. (v. enim enim D, D, fol. fol. 33 33 v )) |||| 378 378 ]] om. om. A A |||| 380 380 D. ]] D ]] D.

177 163 160 170 148 ] 157 A,

122 205 200 123 118 279

322 316

]]

171 158 169 164 159 154

70

2

D |||| 364 364 D

30 )) A: A: 16 16vv,16 ,16 -- 16 16vv,21 ,21 (ed. (ed. llll .. 753-759); 753-759); D D :: 84 84rr,8 ,8 -- 84 84rr,11. ,11. (( 30 368 D |||| 369 369 A |||| 369-70 369-70 368 ]] D ]] A 180 146 178 172 150 ]] D. D.

270

60 149

]]

384

92

]

6

95

10

225 104 223 218 105 100

om. D D ]] om. add. add.

388 D || 388

]]

382-3 |||| 382-3 AD |||| 385 385 AD

D |||| 388 388 D

D |||| 383 383 A, D |||| D ]] A, D D |||| 387 387 om. A A |||| 387 387 ]] ]] D ]] om. AD |||| 388 388 D |||| 389 389 D. ]] AD ]] D ]] D. ]]

69

Text and translation

any upper row, the third in the opposite row below, the second on the right and the fourth on the left, and you put opposite to each of these An ancient Greek treatise on magic squares 69 94 four its complement. (You proceed likewise) until you have done this for each border. n 4) = n2 + 1 − (δ +

9

L ***

C ***

i

i

N2

***

N3

***

N4

***

δ2

≠2

2

6

10 n2 + 1 − 4 =⇒

2

≠2

2

6

10

n= 13 2 ≠2 2 magic6 squares 10 An ancient Greek treatise on n = 17

n2 + 1 − (δ + 2)

71

δ+2

(§ 30. Eliminating these last differences) 2 n9 +**** 1 − (δ δ+6 **** **** 13 **** 15 C7 n = 2k + 1 3 L **** 5 11+36) 17 19 N2 4 i Once you have done that, youi consider a Npair ofNsmall numbers; put 3 50 7 6 7 9 9 n = 9 1row,0 the 03 4 the the first in any upper second below5 (in same border), and 2 2 0 n = 13 0 0 4 6 δ + 4 =⇒ n + 1 + 4 n + 1 − δ opposite to them their complements. Then turn your attention to another n = 17 0 0 0 4 6 95 2 +3 ⇓ pair; put the first on the ⇓ border, the second on n2 ≠1 left side n2 ≠5 of the n2 +1same n n2 +7 2 2 22 2 2 2 n + 1the + 4 right, and opposite to them their complements. n +1−4 Do the same for all 26 28 25 n=7 24 22 borders.96 n = 11 60 n = 15 112 2 n + 1 − (ϵ + 2) n = 19 180

58 110 ϵ 178

L ***

ϵ+2

n=9 n = 13 have done n = 17

2 +3 n2 ≠5 +3 +7 ≠1 ≠5 ≠1 +7 | n2. 42.≠ | +. 3 14m 2n 2

i

2 2 that, 2

C ***

i

n2 + 1 − ϵ

≠2 ≠2 each row ≠2

61 113 181

N2

***

62 64 114 116 2 =⇒ n + 1 − 2 182 184

N3

2 2 and 2

Once you its and none will exceed the other in any way. n2 − 5 2

C*

2i

n = 7 n =16 9 n 4= 11 n = 24 13 n = 48 17 n = 15 32 Completing all64the n2 + 1 − ∆L1 96

nL2**** +1 i

050 122 0 122 0 226 226 rows) 226

***

N4

6 10 6 10 opposite 6 10

2 1 C* **** n − 1 **** 2 **** nC + 1i − ∆ NC12—i + “iN3 “i i2 2

08 012 024 16 32 48

***

=⇒ n2 + 1 + 2

0 422 4 n +1−4 0 110 4 0 98 4 210 194 178∆L1

will be equalized

≠N 2, **** “i 4 30, 32 8 66, 68 8 70, 72 8 118, 120 122, 124 126, 128

—i , —i + 2 10, 42, 26, 90, 70, 50,

12 44 28 92 72 52

(§ 31. If there are remaining empty cells, it can (only) be four facing four, n2 + 3 n2 + 7 eight facing eight,n2twelve hand so on always by adding suc∆C + 1 − 2 facing 2twelve 1 2 2 n2 +1 n ≠1 n2 ≠5 2 2 2 cessively fouri. You will then equalize them by groups of four, (either) n ≠5 +3 +7 ≠1 ≠5 ≠1 +7 +3 | n2. 42.≠ |of still n +. 314m =7 24 22 numbers 25 using tetrads available consecutive in the way already 2n 2 n2 − 5 n2 − 1 C n2 + 1 −n∆= 2 60 section 58]97 , or using 61 1 11 of the explained [ at the beginning tetrads of numbers 2 2 1121 C * 110 113 * = 15 2 Cn consisting in pairs ofi consecutive ones, putting the n +1 —i + “4i the “i ≠first 2, “i of — + 2 pair i , —ifirst 2 i n2 + 1 − 4 2

2

n = 19 180 170 181 n=7 16 50 8 42 30, 32 10, 12 94 n = 11rule of24§ 21, with 122 s = 2.12 110 66, 68 42, 44 Equalization 95 122 24 98 70, 72 26, 28 The left side is now48in excess. 96 L 2 21, this L 118, Applying once again the equalization rule of § time with s = 1. 92 n∆= 15 32 226 16 210 120 90, n + 1 − ∆ 1 1 97 § 20, at the beginning section 32 on equalization rules of even 64 of the226 194 122, by 124means 70, 72 numn2 + 7 n2 + 3 bers. ∆ 96C1 226 126, 128 50, 52 n2 48 + 1 − 2 178 2

2

380 380 380

14

( 32 ) ( 32 )

380 380

( 32 ) 385 385 385

385 385

390 390 390

390 390

395 395 395

395 395

56

268 262

60

259 254

69

65

255 263

72

64

268 66 262 264 56 257 60 256

259 67 254 260 68 70 69 58 64

55

[ Text and translation ]

70

( 29 ) A: 16v ,12 - 16v ,16 (ed. ll . 748-752); D : 84r ,5 - 84r ,7.[ 363

A || 363

]

] om. A || 364

]

[

D || 364

] ]

]

( 33 ) ( 33 )

A, om. D || 365 ] A. (ed. llll.. 748-752); 748-752); D D:: 84 84rr,5 ,5 -- 84 84rr,7. ,7. ( 29 ) A: 16vv,12 - 16vv,16 (ed. v v r r [ ] ( 33 ) ( 30 ) A: 16 ,16 - 16 ,21 (ed. ll] .om. 753-759); D : 84 ,8 84 ,11. A |||| 364 364 D |||| 364 364 363 ] A || 363 ] om. A ]] D ]] 368 ] D || 369 ]] A || 369-70 ... ] om. A || 370 365 A. A, om. D || 365 ] A. ] D. (ed. llll.. 753-759); 753-759); D D:: 84 84rr,8 ,8 -- 84 84rr,11. ,11. ( 30 ) A: 16vv,16 - 16vv,21 (ed. ( 31 ) A: 16v ,21 - 17r ,4 (ed. ll . 760-765); D : 84r ,11 - 84r ,14. 368 ] D || 369 ]] A || 369-70 om. A A |||| 370 370 A || 369-70 ...... ]] om. 375 ] om. D || 376 ] A || 376 ] D. ] A, D || 376 ] A || 377 ] A, D vv rr ll.. 760-765); 760-765); D D:: 84 84rr,11 ,11 -- 84 84rr,14. ,14. (||31 ) A: 16 ,21 17 ,4 (ed. ll 377 (pr.) ] A || 377 ] om. A || 377 ] (hic et infra) D || 378 ] A |||| 376 376 375 ] ] om. D || 376 ] v D (v. enim D, fol. 33 ) || 378 ] om. A || 380 ] D. A

] ( 34 ) 376 A |||| 377 377 A, ] A, D || 376 ]] A ]] A, ] ( 34DD) ( 32 ) A: 17r ,4 - 17r ,11 (ed. ll . 766-773); D : 84r ,15 - 84r ,19. 377 om. A A |||| 377 377 (hic et et infra) infra) D D |||| 378 378 || 377 (pr.) ] A || 377 ]] om. ]] (hic tit. D : 149 180 146 ] 178 172 150 D (v. enim D, D, fol. fol. 33 33vv)) |||| 378 378 ]] om. om. A A |||| 380 380 ]] D. D. 382169 164 159 154 ] om. D || 382-3 ] D || 383 ] A, 171 158 ] (D34||) r r r r 766-773); D D:: 84 84r ,15 ,15 -- 84 84r ,19. ,19. ( 32 ) A: 17r ,4 - 17r ,11 (ed. llll.. 766-773); 15 384155 162 165] 173 add. AD || 385 ] D || 387 ] om. A || 387 ] 152 168 15 tit. D 178 : 172 146 167 150 180 166 178 156 172 149 161 146 150 174 151 D || 388 ] D || 388 ] AD || 388 ] D || 389 ] D. 382-3 D |||| 383 383 A, D |||| 382160 ]] D ]] A, D 171 163 169 157 164 170 158 159 148 154 ] om. D || 382-3 169 164 159 154 177 r r ( 384 33 ) A: 17r ,11 ll . ||||774-780); D :]] 84r ,19 - D 84 AD 385 D ||||,23. 387 om. A A |||| 387 387 ]] ] - 17 ,17 add.(ed. AD 385 387 ]] om.

165 176 173 155 147 162 153 255 266 263 175 168 179 165 173 65 57 72 63 152 145 155 162 269

]

A || 391 ]

15

] A || 391 ] D || 392 ] A || D |||| 388 388 ]] AD |||| 388 388 ]] D D |||| 389 389 D. D AD ]] D. 392277 ] A, D || 170 46 160 272 157 51 148 260 118 177 50 163 170 70 200 67 123 58 279 160 157 148 205 ( 33 ) A: 17r ,11 - 17r ,17 (ed. ll . 774-780); D : 84r ,19 - 84r ,23. 179 54 176 147 153 281 39347 ] add. in marg. D || 393 ] om. A || 394 ] A || 395 269 126 266 44 175 276 179 176 57 201 63 209 145 147 153 273 119 391 ] A || 391 ] A || 391 ] D || 392 ] A || 286 48 280 282 38 275 42] 214 120 208 210 288 274 286 280 110 203 114 43 41 53 38 42 202 A, om. (homeotel.) D || 395 ] A || 396 392277 ] A, D || 277 49 272 278 51 40 46 205 121 200 206 279 271 123 112 118 285 50 52 51 124 ] 272 A ||46 396 ] A || 396 ] A || 396-7 ] 393287 add. in marg. D || 393 ] om. A || 394 ] A || 395 273 284 281 47 39 54] 45 201 212 209 283 276 273 119 111 126 117 44 37 47 54 215 D (v. enim D, 281 fol. 33 v ). 274 232 275 226 282 ] 202 203 316 274 275 282 48 96 120 210 43 234 53 92 48 A, om. (homeotel.) D || 395 ] A || 396 2 322 6 95 r (271 34 )52 A:49 17278 ,1740 - 17v ,6 (ed. ll . 781-790); D : 84r ,23 - 84v ,5. 278 206 285 52 49 40 124 121 112 313 308 15 10 225 104 223 ] 218 105 A 100 || 396 ] A || 396 ] A || 396-7 ] 398 D || 398 ] add. in marg. D || 398 ] A || 398-9 215 18 212 98 283 222 287 284 287 108 284 111 309 117 317 37 39 45 39] 219 45D, 227 11 D 101 (v. enim fol. 33 v ). ] om. (ter ) ] D || 399 ] A || 399 322 12 316 318 234 220 232 226 2 311 6 97 95 107 92 96 232 102 226 92 221 96 (homoeotel.) A || 399 310 228 ( 34 ) A: 17r ,17 - 17v ,6 (ed. ll . 781-790); D : 84r ,23 - 84v ,5. 313 308 225 223 218 15 10 104 105 100 223 218 105 100 ] A || 399 ] AD || 400 ] A 224 94A || 399 16 13 314 4 231 217 106 ]103 (sic) 398101 ] 219 D || 398 ] add. in marg. D || 398 ] A || 398-9 317 229 222 227 11 318 309 98 91 108 219 230 227 101 93 108 99 323 233 9 320 || 400-1 ] A, 391

166 286 167 280 174 156 256 214 257 208 264 41 166 167 174 66 114 151 288 161 38 156 D42 || 388 110

71

Text and translation

and the second of the second pair on one side, the second of the first pair and the first of the second pair on the facing side, and opposite to each its complement.

(Squares of orders n = 4t + 3)98 Treatment for (the squares of) seven, eleven, fifteen, nineteen and the like.99 (§ 32. Completing all first borders)

71

An ancient Greek treatise on magic squares

You put the last of the small even terms in the upper left-hand corner of the first border, which surrounds the inner square which you have filled with odd numbers, and opposite to it diagonally, in the lower right-hand n = 2k + 1 3 5 7 9 11 13 15 17 19 corner of the same border, its complement. Put the preceding small term 1 corner 3 9 to it 9 in the upper right-hand of3 the 5same 5border7 and, 7opposite 100 diagonally, in the lower left-hand corner, its complement. n=7 n = 11 n = 15 n = 19

n2 ≠1 2

n2 ≠5 2

n2 +1 2

n2 +3 2

n2 +7 2

24 60 112 180

22 58 110 178

25 61 113 181

26 62 114 182

28 64 116 184

*** *** *** *** L ***you C Once you have done that, will have N2 equalized N3 N4the first border for i i all squares of thisn kind. =9 2 ≠2 2 6 10

= 13 2 rows ≠2 of all2other 6borders) 10 (§ 33. Equalizing nthe horizontal n = 17

2

≠2

2

6

10

Then consider [ after that ]101 the amount of the excess of each (remaining) upper row and Llook forC **** the Npair numbers such that **** **** of small **** **** N3 N4 2 i i their sum is less (than their sum due) by the same amount. Put them n=9 0 4 in the corner cells of this 0row, the lesser0 on the right,6 and opposite to n = 13 0 0 0 4 6 them diagonally below their complements. You complete in this way all n = 17 0 0 0 4 6 corners of the remaining borders.102 98

Subject of §§ 32–35. For the orders n = 4t + 1, the largest explicitly mentioned order was n = 17; here, it is n = 19, which is indeed the largest constructed (p. 87). By the way, we may observe that no square larger than 20 × 20 is represented in the treatise. 2 2 2 100 n2 −1 & n 2−5 (on the top), and n 2−5 & n 2+3 (on the right), then their Placing 2 2 +3 n2 ≠5 +3 +7 ≠1 ≠5 ≠1 +7 |2nn2. 2 .3 | L 42.≠ + 4m 1 complements, will eliminate the uniform differences ∆1 = 4, ∆C 1 = 2 (§ 17). 101 Either the gloss already seen (§§ 6 & 9) or to be inserted after ‘remaining empty cells’ below (as in § 34 — not 1a gloss there). C * if so then C* n2 + 1 —i + “i “ ≠ 2, “i —i , —i + 2 102 i The pair αi , αi + i2 is put in the 2upper corners of thei ith border (i > 1). As L 2 n= 7 sum162αi + 2 50 8 their sum 42 due n30,+32 we are told, their is less than 1 by ∆i 10, (see12table); L 2 n = 11 24 122 12 110 66, 68 42, 44 upper thus n + 1 − ∆i = 2αi + 2, and this determines their individual values. The horizontal rows will then 48 no longer 122 display 24 any excess 98 (or, for 70,the 72 opposite, 26,deficit). 28 n = 15 32 226 16 210 118, 120 90, 92 64 226 32 194 122, 124 70, 72 96 226 48 178 126, 128 50, 52 99

390 390 385 385 385 385

363 A,

72( 30 ) A: 368

390 390 390 395 395 395 390 390 390 390

268 262

56

68

55

259 254

69

65

255

72

256 257

66

70

67

260

269

57

63

110 214 208

56 268 262 205 200 123

68

259 254 69 56 201 119 126 268 262 262 56 268

65 72 255 69 202 203 259 254 254 120 68 259 69 256 257 66 65 72 206 124 255 65 121 72 255 70 67 260 256 66 215 111 256 257 257 117 66

A || 363

]

om. D || 365

] om. A || 364

D || 364

]

A.

]

andDtranslation 16v ,16 - 16v ,21 (ed. ll . Text 753-759); : 84r ,8 - 84r ,11. D || 369

]

A || 369-70

]

] om. A || 370

...

] D. ( 29 ) A: 16v ,12 - 16v ,16 (ed. ll . 748-752); D : 84r ,5 - 84r ,7. 33)) ( 31 ) A: 16vv,21 - 17rv,4 (ed. ll . 760-765); D : 84r ,11 - 84rr,14. [[ ]] ((33 v ] v ,12 A --||16 363 A || 364 ] 29)) A: A: 16 16 ,12 16 ,16 (ed. (ed. ll]ll.om. . 748-752); 748-752); D:: 84 84rr,5 ,5]--84 84r ,7. ,7.D || 364 ((363 29 ,16 D 375 ] om. D || 376 ] [[ A || 376 33)) ]] ((33 D || 365]] om. A. 363A, ]] A Aom. 363 om. ]A A |||| 364 364 364 363 |||| 363 ]] DD |||| 364 ]] ] A, D || 376 ] A || 377 ] A, D v A, om. D |||| 365 365 ll . 753-759); A.D : 84r ,8 - 84r ,11. A, D ]] A. (||30 ) A: 16v(pr.) ,16om. - 16 377 ] ,21 A ||(ed. 377 ] om. A || 377 ] (hic et infra) D || 378 ]vv,16 || (ed. 369 ] v ) || 378 A:: ||84 ] om. A || 370 venim r rom. ]A: 16 D--(v. D, llfol. 33 ]369-70 Arr,11. || 380 . . . ] D. 30)) A: 16 ,16 16vD ,21 (ed. ll.. 753-759); 753-759); D 84 ,8--84 84 ,11. ((368 30 16 ,21 D ,8

34 ((||34 )) om. AA || 370 ] ] om. 370

] ]] r D. r D ]-]84r ,19. ...... 368 D ||||(ed. 369ll . 766-773); A 369-70 369 ]] ||||r369-70 (368 32 ) A: 17 ,4 - 17 ,11 DA : 84 ,15 D. ] : 16v ,21 D. (tit. 31 )D]A: - 17r ,4 (ed. ll . 760-765); D : 84r ,11 - 84r ,14.

60 395 149 180 146 178 172 150 395 395

375 ] vvom. ],14.D r|| r 376 ]17 om. D 31 )) A: --D 17 ,4 .. 760-765); (382 31169 A: 16 ,21 ,4 (ed. (ed.|| llll382-3 760-765); D D:: 84 84r]r,11 ,11--84 84rr,14. 171 ( 16416 158 159,21 154

A || 383 ] A,|| 376D || [ D ||AD 376 || 385 ] ] A || 377D ]]|| 387 ] 384 ] 173D ] om.A,AA||||387 375 ]]165 om. 376 376 375155] 162 A, om. D ||||add. 376 376 D] 263 152 168 15 15 || 377 (pr.) ] A 377 ] om. ||AD ] ] ]]D (hic et infra) 264 151 161 166 156D 174 ]] 167 A, ||]|| 376 ]] ] A A ||377 DD A, D || 376D || 388 A || 377 377 A, ] D || 378 || 388 D || 388 || 389 A, D. v ] D (v. enim D, fol. 33 ) || 378 ] om. A || 380 ] D. 170 58 177 163 160 157 148 |||| 377 (pr.) ] A || 377 ] om. A || 377 ] (hic et infra) D || 378 377 (pr.) ] A || 377 ] om. A || 377 ] (hic et infra) D || 378 ( 33 ) A: 17r ,11 - 17r ,17 (ed. ll . 774-780); D : 84r ,19]- 84r ,23. 34)) ]- 84 ((34 r D 266 175 145 147 17 153 (v. enim D, 33 om. A |||| 380 ]] D. D- 17 (v.r ,11 enim D,llfol. 33vv)) |||| 378 378 om. Ar ,19. 380 D. ( 32179 ) ]]A: ,4176 (ed. .fol. 766-773); D : 84]r],15 391 ] A || 391 ] A || 391 ] D || 392 ] A || 114 41 288 38 286 280 42 34)) ]]84rr,19. ((34 tit. : 17rr,4 32 ))DA: --17 (392 32146 A: 17rr,11 ,11 (ed. (ed. llll.]. 766-773); 766-773); D D:: 84 84rr,15 ,15--84 ,19. 17817 172,4150 60 149 (180 A, D || 118 279 50 277 272 51 46 382 ] D || 383 ] A, D || tit. D 64 171 158 159 154 ] om. D || 382-3 tit.169 D :164 : add. 393 in marg. D || 393 ] om. A || 394 ] A || 395 60 149 146 209 276 273 281 44 180 47 178 54] 172 180 178 172 150 60 400 149 146 150 [ 405 384 add. || 385 ] D || 387 ] om. AA,|| 387DD ||||] 263 152 168 155 162 165] 173 ] om. 382 DDD||||||383 382169 D ||AD || 382-3 382-3 383 64 210 274 275 282 43 158 53 48 154 171 169 164 164 159 64 171 158 159 154 ]] om. D 15 A, om.]] (homeotel.) 395 ]]] A A, || 396 15 264 151 161 166 167 156D 174 || 388 ] D || 388 ] AD || 388 ] D || 389 384 ] add. AD || 385 ] D || 387 ] om. A ||]|| 387 ]] 384 ] add. AD || 385 ] D || 387 ] om. A 387 D. 263 168 165 173 152 155 162 285 271 112 52] 162 49 278 40396 263 152 168 155 165 A ||173 ] A || 396 ] A || 396-7 ] 15 15 58 177 163 160 157 170 148 rD r v ]] r |||| 388 264 166 167 156 212 283 161 287 284 39 17 45 264 151 166 167 174 151 161 156 |||| |||| 388 AD ]] D. D174 388 388 ]]D : 84 AD 389 D. D33(v. enim D, fol. [84r ,23.]] DD]||||389 (37 ) A: ,11 -388 1733 ,17).(ed. llD .D774-780); ,19 -388 64

400 395 395 395 395

269 57 63 266 175 145 179 147 153 176 70 322 316 234 232 226 2 67 6 177 95 163 92 157 96 260 58 177 163 160 170 148 70 67 260 58 160 157 170 148 rr 391 A 391llllll...781-790); ] A 391 r,19 - 84]vrr,23. 33 ,11 (ed. 774-780); D ::84 84 33)38 ,1142--17 ,17 (ed. 774-780); D:|| 84rr,23 ,19- 84 - 84,5. ,23. 34 ) A: A: 17]280 ,17 17rvr,17 ,6 ||(ed. D 288 28617 110 214 208 114 41 (( 313 308 225 223 218 14 15 10 175 104 105 100 269 57 266 175 145 179 147 176 55 269 57 63 63 266 145 179 147 153 153 176

55

]

392

D || 392

A ||

]

D |||| |||| 391 ]] in marg. A ]] A, DD ||||]392 AA A398 391 ]] add. A |||| 391 391 392 A || ]398-9 ] || ||A D || 398 ( 35 ) in marg. D || 393 ] om. A || 394 ] A || 395 ] A, D || 273 281 54] add. ] (ter ) ] D A, || 399 ] A || D 399|| 272 51 46 221 228 272 102 51] om. 46 (homoeotel.) A || 399 275 48 282 (homeotel.) D |||| 395 ] A || A 396 ] add. in 393 ]A om. A ]] add. in marg. D ||||A, 393 ]om. om. A |||| 394 394 395 ]103 (sic) A] marg. || 399 D ] || 399 ] AD 400 ]A |||| 395 A 54 224 94 273 281 281 54] 273

39138 205 200 123 118 279 391 277 272 ]]]51 46D 50 398 110 309 317 41 222 219 227 11 214 18 208 98 108 214 208 114 288 101 286 280 280 42 110 114 41 288 38 286 42 393 392 119 126 201 209 44 276 47 392220 205 279 50 310 311 318 405 12 118 97 107 205 200 200 123 279 277 123 118 50 277

202 203 120 210 410 43 393 53 274 393106 119 47 314 231 217 16 126 13 201 4 44 201 209 209 276 119 126 44 276 47 1

14 14

1 11

124 121 206 112 285 271 52] 49 278 40396 A || 202 274 323 320 229 233 230 3 120 9 210 91 93 48 99 || 400-1 202 203 203 210 43 274 275 275 282]] 120 43 53 53 48 282

215 124 124 56 56 2 215 215 259 259 313 22 65 65 11 313 313 256 256 310 11 11 70 70 16 310 310 269 269 323 16 16 110 110 323 323 205 205

111 121 121 268 268 322 111 111 254 254 308 322 322 72 72 18 308 308 257 257 311 18 18 67 67 13 311 311 57 57 3 13 13 214 214 33 200 200

117 206 206 262 262 316 117 117 69 69 15 316 316 255 255 309 15 15 66 66 12 309 309 260 260 314 12 12 63 63 9 314 314 208 208 99 123 123

212 283 112 285 112 285 60 60 149 149 6 95 212 212 283 283 64 171 64 171 10 225 66 95 95 263 263 152 152 317 98 10 225 10 225 264 264 151 151 318 97 317 317 98 98 177 58 177 58 410 231 4 415 318 318 97 97 266 266 175 175 320 229 44 231 231 114 114 41 41 320 320 229 229 118 279 118 279

322 322 316 316

313 313 308 308 15 15 11 11

7

66

9

] ] D (v. enim ]D, fol. 33 D). || 403 D A ]] DD ||||] 402 D |||| 401 401 A || 403 ]] A |||| 402 402 402 A] ]|| 403 ]D |||| 402 D || 404 om. (homoeotel.) A 404 D ]] A ]] DD ||||]403 ]] A 402 A |||| 403 403 403 A |||| || 403 403 275 275 48 282 48 282 ] ] D || 405 ] add. A, ]49 ]] om. ]49 278 D |||| 404 404 om. (homoeotel.) (homoeotel.) AA |||| 404 404 278 40 40D 41 add. D || 406 ] D || 406 ] (corr. ex ) A, ] D || 405 ] add. A, 284 39 45 39] 45 284 D || 405 ] add. A,

A,

A, 274 53 53 274

54 273 281 281 54 273

D || 407

18 108 309 317 317 98 222 219 227 18 309 98 222 101408 108 219 A 101 || ] 227

]

310 221 310 311 311 12 318 97 220 221 102 228 12 318 97 107 107 102 228 A ||220 408 11

16 16

13 314 13 314

323 323

33

99

44

41 D]] || 407 D ] D || 41 ]] 407 D |||| 406 406 A || 407 ] A || 407-8 ] ]] D ]] DD |||| 407 D |||| 407 407 407 v (v.||||enim A 407-8 A |||| 407 407 ]] DA A 407-8D, fol. 33 ).]]

D226 || D 407 95 96 234 92 232add. 226 95 234 92 232 96 || add. D || 406 406]

10 100 ]|| 407 225 104 223 218 218D105 10 225 104 223 105 100

3

]

A A, || 408D || 402

215 283 215 111 212 415 283 37 287 111 117 117 212 37 287 420

4

[

[

124 206 112 285 271 271 52 124 121 121 206 112 285 52

22

A || 396-7 om. DD |||| 395 ]] (homeotel.) om. (homeotel.) 395A,]]] AA |||| 396 396

] ( 35 )

119 201 209 209 44 276 47 119 126 126 201 44 276 47 202 202 203 203 120 210 43 120 210 43

] A, A || 396 A,

287 enim 37 39 45D,284 D (v. 33 ||v ).401 271 52 40 271 278 52]] 49 49 278 D ]||] 402 ] A 396 ]] A ]]A || 402 A A ||150 ||40fol. 396D A |||| 396 396 ] A |||| 396-7 396-7] 180 180 146 178 172 172 150 146 178 234 92 232 226 96 v 287 284 37 39 45 v 287 284 37 39 45 r v r v A, D || 402 || 403 D : 84] ,23 - 84 ,5. D || 403 ] A || 403 enim D, fol. D34(v. (v. enim D, fol. 33 (D ) A: ,17 - 1733 ,6]).). (ed. ll .A781-790); 169 164 158 159 154 169 164 17 158 159 154 104 223 218 105 100 234 234 92 232 226 226 96 92 232 96 ] 17 || 404 ] om. ](homoeotel.) A || 404 168 173 rr,17 162 398 ]165 398 add. in marg. A || 398-9 165 173 r,23||- 84 155 162 (168 34 )) A: --D 17 (ed. D (222 34155 A: ,17 17vv,6 ,6 (ed. llll.. ]781-790); 781-790); D:: 84 84rD ,23 -398 84vv,5. ,5. 219 227 101 10817 104 223 218 218 105 104 223 105 100 100 174 161 156 ] || 405 ] add. A, 166 167 167 174 D 161 166 156] om. A || 399 A || 399 398 D |||| 398 ]] add. in D AA ||||] 398-9 220 221]] 102 228 107 398101 D (homoeotel.) 398 add. in marg. marg.(ter D)||||] 398 398D || 399 ]] 398-9 222 108 222 101 219 227 227 108 219 41 163 170 163 160 170 148 160 157 157add. 148 D || 406 ] D || 406 ] (corr. ex A, || 399 ] A(ter || 399 AD ] ||||)399 A 217 106 ]103 (sic) 224 94A(homoeotel.) ]] om. A || 399 ))]] DD] |||| 399 ]] AA om. (ter 399 || 400 399 228 107 220 221 221 102 228 (homoeotel.) A || 399 107 220 102 176 145 179 147 176 145 179 147D153 153 || 407 ] D || 407 ]] D || 407 ] A, A || 407 233 93 99 230 91 || 400-1 A ]] A ]] AA 217 224 ]103 (sic) (sic) A |||| 399 399 A |||| 399 399 ]] AD AD |||| 400 400 217 106 224 94 106 ]103 94 288 288 38 286 280 280 42 38 286 ] 42 D ||A401 || 407 ] A ]|| 407-8 ] ] A A, ||D408 ]] A || 402 || 402 || 400-1 ] 233 91 233 93 230 91 93 99 99 230 || 400-1 ] A, 50 277 272 272 51 50 277 51 46 46 v

A A |||| 408 408

231 231 217 217 106 224 94 106 103 103 224 94

320 320 229 229 91 233 93 91 233 93

99 230 99 230

] ]]

v D D (v. (v. enim enim D, D, fol. fol. 33 33 v).).

](corr. ex A || 407 )) A, (corr. ex [ A, A || 408 ]] AA |||| 407 407] A ] ] ( 36 ) A |||| 408 408

n = 11 n = 15 n = 19

n=7 n = 11 n = 15 n = 19

(n2 + 1) ≠

L

2

50 ≠ 20 = 30 122 ≠ 28 = 94 226 ≠ 36 = 190 362 ≠ 44 = 318

61 113 181

58 110 178

73

Text and translation

–2 , –2 + 2 14, 16 46, 48 94, 96 158, 160

60 112 180

(n2 + 1) ≠

L

3

122 ≠ 52 = 70 226 ≠ 68 = 158 362 ≠ 84 = 278

–3 , –3 + 2 34, 36 78, 80 138, 140

(n2 + 1) ≠

L

4

226 ≠ 100 = 126 362 ≠ 124 = 238

–4 , –4 + 2

62, 64 118, 120

Once you have done that, you will have equalized all the upper and lower rows, and the remaining empty cells will be four facing four, eight facing eight, and so on by adding successively 4. You will then equalize each group of four by means of four numbers in the way we have explained previously.103 (§ 34. Equalizing the vertical rows of all other borders) αi + 2

αi

n2 + 1 − γi

γi

n2 + 1 − (γi − 2)

γi − 2

n2 + 1 − βi

βi

n + 1 − (βi + 2)

βi + 2

n2 + 1 − αi

n2 + 1 − (αi + 2)

2

There remains (to deal with) the right-hand rows [ (which) exceed their sum due by 16, 24, 32, 40, 48, and what results from successive 10 100 2 98 5 94 88 15 84 9 (additions of) 8 ]104 and there remain as empty cells four facing four, 18 26 74 73 72 71 31 32 25 83 eight facing eight, and so on by successive additions of 4.105 You then 85 77 38 68 34 66 60 37 24 16 look for a pair of large numbers such that their sum exceeds their sum due 14 23 42 47 52 57 46 59 78 87 by an amount which, when added to the excess of the right-hand row, 79 61 of 53 the 50 sum 43 56 of 40 two 22 small 12 equals the89 deficit numbers. Put then the two 90 21 on 62 the 44 right-hand 55 54 49 39 side, 80 11 large numbers and the two small numbers on this same side, 8and on 45the48left of the four numbers.106 20 put 36 58 51 the 65 complements 81 93 95 82 64 33 67 35 41 63 19 6 See §§ 20, 31. 4 76 27by28the29same 30 70 69 reader 75 97 as in § 19. For these values, see table Gloss, perhaps early below. 92 1 99 3 96 7 13 86 17 91 105 Thus, in order to be left in the vertical rows with neutral placings, we shall have to fill four more cells (see figure), this being indeed applicable from the smallest order n = 7. C 106 Let, for the right-hand row of the ith border, ∆i * be the new excess (after filling the corners), Bi the sum of the required pair of small numbers βi , βi +2, and Gi the sum C of the required pair of large numbers γi , γi − 2. We are told that Gi − (n2 + 1) + ∆i * = C* 2 2 (n +1)−Bi , that is, Bi +Gi = 2(n +1)−∆i (which indeed eliminates the remaining C excess), and therefore βi + γi = (n2 + 1) − 12 ∆i *. See the table, with the values set for βi and γi . 103 104

, , . . , , . ] D. 368 ] D || 369 ] A || 369-70 ... ] om. A || 370 ( 29 ) A: 16vv,12 - 16vr,16 (ed. ll . 748-752); D : 84r r ,5 - 84rr,7. ] D. ( 31 ) A: 16 ,21 17 ,4 (ed. ll . 760-765); D : 84 ,11 84 ,14. 390 390 33)) [[ ]] 390 ((33 v 395 363 395 ] v ,12 A -||16 363 Aand || 364 ] 395 29 ) A: 16 ,16 (ed. ]llom. . 748-752); D : 84r r ,5] - 84rr],7.D || 364 74 ( 375 Text translation ] om. D || A || 376 v r 376 ( 31 ) A: 16 ,21 - 17 ,4 (ed. ll . 760-765); D : 84 ,11 - 84 ,14. 390 [ ] 390 ( 33 ) D || 365 ] om. ] A || 364 A. 395 363A, 395 ]A, Aom. || 363 ] D || 376 ] A || ]377 ] D || 364 ] A,A] || 376 D 375 ] om. D || 376 405405 405 v (pr.) r A, om. || ||365 ] ] om.A.DA: ||84377 ] vD A D || 378 ( || 30377 ) ]A: 16 - 16 ,21 ll . 753-759); - 84r ,11.] A,,16 D (ed. || 377 376 ] A ||,8377 ] (hic et infra) A, D v ] D (v. enim D, fol. 33 ) || 378 ] om. A || 380 ] D. ] v ,16 || 369 ] ] om. D A om. A v A r || (pr.) ]D || 377ll . 753-759); A: ||||84369-70 377 D |||| 370 378 (368 30377 ) A: 16 - 16 ,21 (ed. ,8 - 84r ,11.] . . . (hic et ]infra)

( 34 )

( 34 ) D. ] om. A || 370

] [ ][ Ar ,19. v r D.(v. r] om. D,ll .fol. 33 || 380 .]. .] ] ( 32 ) ]A: 17 - 17r ,11 766-773); DA: 84 ,15]- [84 368 ] ,4D Denim ||(ed. 369 ] ) || 378 || 369-70

270

68 259 254 270 56 268 258 65 72 68 259 254 258

256 257 65 72

]

r : 17vr,21 ]A: D. 31 ))DA: 17r ,11 ,4 (tit. 32146 --17 17816 172,4150 60 149 (180 395 395 395

15

170 D 70 67 260 58 177 163 160 148(v. enim D, fol. 33 v ) || 378 ] 157 ] rom. A r|| 380 r(pr.) 256 257 66 264 151 161 166 167 17 156 377 ] r ,17 A ||(ed. ] om. ]D A: ||84 377 ] ] D 174 ||- 388 ] 377ll .D774-780); || 388 AD (||33 ) A: ,11 17 ,19||-388 84 ,23.

]] ]

] D. et infra)] D || 378 D(hic || 389 D.

34)) ((34 ( 34 )

63 266 175 145 179 147 153 176 v r 148(v. r] om. Ar || 380 260 58 177 (163 enim D,ll .fol. 33 ] D. ] 32 160 ) ]A: A:15717 17]170 - 17r ,11 (ed. D : : 84 ,15 - 84] ,19. rr,4D rA r r 391 391 ] ) || 378 AD 391 A || 33 ,11 llll.766-773); 33)38 ) A: ,1142--17 17r,17 ,17||(ed. (ed. . 774-780); 774-780); D|| : 84 84r,19 ,19--84 84r,23. ,23. D || 392 216 110 214 208 114 41 (( 288 286 17 280 55 269 57 63 266 175 145 179 147 153 176 tit. D : r r r r 392 A, DD |||| 392 D |||| (391 32146 ) A: -4617 ,11 (ed. D ,15 - 84]] ,19. ]]51,4150 A 391 ]] A 391 ]] AA 270 178 172 391 A |||| 391ll .] 766-773); A :||||84 391 392 || 205 268 200 262 279 180 277 27217 122 56 123 60 118 149 50 216 110 214 208 114 41 288 38 286 280 42 382 ] om. D || 382-3 ] D || 383 ] A, D 393 add. in marg. D ||]] 393 ] om. A || 394 ] A ||DD 395 169 68 tit. D 164 :54] 159 392 A, |||||| 204 259 201 64 209 171 276 273 154 281 119 254 126 69 44 158 47 392 A, 270 149 205 268 200 262 279 180 277 178 272 172 122 56 123 60 118 50 146 51 150 46 405 400 384 ] 173 add. AD || 385 D || 387 387 D ||] 258 65 165 202 72 203 255 210 152 274 162 275 282 120 263 43 168 53 155 48 395 ] AAA, ||||A 396 393 ] add. in 393 ]]]om. om. A ]] ] om. 382 om. || ||382-3 ] (homeotel.) 383 393 ] add. in]] marg. marg.DD D ||A, 393 om. A |||| 394 394D D|| || A |||| 395 395 68 204 259 201 64 209 171 276 169 273 154 281 119 254 126 69 44 158 47 164 54 159 415415 415 256 174 206 264 285 161 271 166 278 124 257 121 66 112 151 52] 167 49 156 40396 D || 388 ] D || 388 ] AD || 388 ] D || 389 ] A || ] A 396 ] A || 396-7 ] 384 ] 173 add. ADA, || 385 ]om. (homeotel.) D || 387DD |||| 395 387 D.] om. ]] AAA||||||396 258 65 202 72 203 255 210 152 274 162 275 165 282]] 120 263 43 168 53 155 48 A, (homeotel.) 395 ] om. 396 170 160 157 148 v 215 67 212 177 283 163 287 284 109 70 111 260 117 58 37 39 45 D (v. D,|| rD r 256 167 174 156 A |||| 396 396-7 206 264 285 161 271 278 124 257 121 66 112 151 4917 388 ] ]] llA 388 AD 388 ] D || 389]] ] D. ] enim A ||||40fol. 396 A 396 ]D : ]]84 396-7 (180 33166 )52]A: ,11 -396 17r33 ,17 ).(ed. .D774-780); ,19||A -A 84||||r ,23. 270 269 262 60 175 178 172 176 56 268 63 149 145 146 147 150 55 324 2 57 322 316 266 226 96 6 95 234 179 92 232 153 v 170 160 157 148 v 215 67 212 177 283 163 287 284 109 70 111 260 117 58 37 39 45 r v r v D (v. enim D, fol. 33 ). rr D, r ,23 - 84 D enim 33 391 ]159 A 391 llllll... 774-780); ] AD 391 ] rr,23. ] A || (33 34)(v. )A: A: 17 ,17 17rr,17 ,6||). (ed. 781-790); D 84 ,5. 259 254 208 171 (288 169 16417 ,11 -fol. (ed. ::: 84 68 110 69 64 41 158 154 33 )223 A: ,11 --17 17 ,17 (ed. 774-780); D|| 84r,19 ,19--84 84 ,23. D || 392 216 286 280 38 42 308 15 114 21817 14 313 214 10 225 ( 104 105 100 55 324 269 322 63 316 266 234 179 232 153 226 176 2 57 6 175 95 145 92 147 96 r v r v 392 A, D |||| 398 ||A add. marg. D ]392 A ||] ]398-9 AA 258 65 72 255 263 152 391 168 173 r,17 v,6398 r,23||- 84 155 16217 34 )) A: --D 17 D ]]]165 391 ]] in A 391 ]] v,5. 391 391llll.]. ]781-790); A 391 || 272 122 50 51 46 (222 34277 A: ,17 17A ,6||||(ed. (ed. 781-790); D||:||: 84 84 ,23 -398 84 ,5. DD |||| 392 312 205 309 118 317 279 219 227 11 200 18 123 98 ( 101 10817 216 313 214 308 208 225 288 223 286 218 280 14 110 15 114 10 41 104 38 105 42 100 256 257 66 264 151 392 166 167 174 161 156 393 ] ]]add. in marg. D ||]] ]393 ] om. || D 394 A 395 ] om. (homoeotel.) A || 399 (ter )||A ] 398 || 399 ]]] A ||||DD 399 398 D |||| 398 add. in marg. D A ||||] 398-9 A, |||| 204 119 273 281 47 310 126 311 201 318 44 220 54 221 228 12 209 97 276 107 102 392 A, 398 D 398 ] add. in marg. D || 398 A 398-9 279 122 312 205 309 118 317405 222 277 219 46 227 11 200 18 123 98 50 101 272 108 51 177 163 160 157 170 148 70 67 260 58 410 ]103 (sic) || 399 D ] || 399 ] |||| 399 AD ] ||||||||395 A 202 282 53 48 314 210 231 393 217 274 224 16 203 13 120 4 43 106 275 94 ] marg. om. (homeotel.) D ||||395 ] ]A 396 ] om. (homoeotel.) A (ter ))A ]A A 399 ] add. in 393 ]]A om. 394 ]400 393 ] add. inA marg. D ||||A, 393 om. 394 ] A 395 ] om. (homoeotel.) A |||| 399 399 (ter ] |||| D D 399 ] || A A 399 204 119 310 126 311 201 318 220 54 221 273 228 12 209 97 276 107 47 102 281 42044 266420175 145 179 147 153 176 55 269 57 63 420 52 40 323 121 320 285 229 271 233 230 1 124 3 206 9 112 91 93 278 99 || || 400-1 ] ]49 A 396 A || 396]] ] |||| 399 A ||]] 396-7 A A AD 400 ]] AA ]] |||| 399 om. (homeotel.) DD ||||||||395 (sic) A 399 ] A, A 399 AD 400A,]]] AA |||| 396 202 48 314 210 231 53 217 274 224 282 16 203 13 120 4 43 106 ]275 103 (sic) 94 A, om. (homeotel.) 395 396 216 110 214 208 114 41 288 38 286 280 42 v 287 284 109 215 111 117 212 283 37 39 45 D || 401 ] ] D] || 402 ] D (v. D,||||40 fol. ||91 A, ]] enim A 396 ]] A || 402 A 52 49 323 121 320 285 229 271 233 230 1 124 3 206 9 112 93 278 99 || 400-1 400-1 A, A 39633 ). ]] A A |||| 396 396 A |||| 396-7 396-7 ] 270 205 268 200 262 279 180 277 178 272 172 122 56 123 60 118 149 50 146 51 150 46 324 2 322 316 6 226402 95 234 92 232 96 A, D || ] A || 403 ] D || 403 ] A v D 401 A |||| 402 DD |||| 402 ] ]|| 403 287 enim 109 215 111 117 212 283 D 39 45 rD,284 vD v|| D, fol. ||). 401ll . 781-790);]] D : 84 Ar ,23 402 402 D34(v. (v. enim fol. 33 (37 )47A: ,17 - 1733 ,6 ). (ed. - 84v ,5. ]] 169 164 68 159 154 204 259 201 64 209 171 276 273 281 119 254 126 69 44 158 54 17 14 313 308 15 10 225 104 223 218 105 100 ]D |||| om. (homoeotel.) A 404 A, D ]] A |||| 403 ] DD ||||]403 ] A 324 2 322 316 6 226402 95 234 92 232 96 D || 404 A,155 402 Aadd. 403 403 A |||| || 403 403 r48 v 398 rr] || 398 v 398 in marg. ] A] || 398-9 258 65 173 r,17 162 202 72 203 255 210 152 275 ] 165 282 120 263 43 168 53 34 )274 --D 17 D ,23 34101 ) A: A: 17219 ,17 17||v,6 ,6 (ed. (ed. llll.. ]781-790); 781-790); D:: 84 84D ,23--84 84v,5. ,5. 312 11 18 309 317 98 (( 222 227 10817 14 313 308 15 10 225 104 223 ]218] 105 100D D || 405 ] add. A, || 404 ] om. (homoeotel.) A || 404 ]49 156 D||||398 404 ] om.] (homoeotel.) A |||| 399 404 256 174 206 264 285 161 271 166 278 124 257 121 66 112 151 52 167 40 ] om. A || 399 (ter A 398 D ]] add. in D )||] 398D || 399 A 310 311 12 318 97 107 221]] 102 228 398220 D (homoeotel.) || 398 add. in marg. marg. ] A ||||] 398-9 398-9 41 D || 398 ] 312 11 18 309 317 98 222 101 108add. 219 227 D ||D 406 ] D || 406 (corr. ex ) A, 415 ]] 170 || 405 ] add. A, 215 67 212410 283 163 287 157 284 177 109 70 111 260 117 58 37 160 39 45 148 D || 405 ] A(ter add. A, || 399 ] || 399 AD ] |||| 399 A 224 16 13 314 4 231 217 106 ]103 (sic) 94A (homoeotel.) ]] om. A || 399 ))]] D ]] AA om. (ter D] |||| 399 399 || 400 399 310 311 12 318 97 107 220 221 102 228 (homoeotel.) A || 399 D || 407 ] D || 407 ] D || 407 ] A || 407 41 D 406 ] D || 406 ] (corr. ex ) A, 324 269 322 63 316 266 234 179 232add. 226 176 2 57 6 175 95 145 92 147 96 || 55 153 add. D || 406 ] D || 406 ] (corr. ex ) A, 233 93 99 230 1 323 3 9 320425229 91 || 400-1 A ]] A ]] AA (sic) A |||| 399 399 A]|||| 399 399 ]] AD AD |||| 400 400A, 231 217 106 ]]103 (sic) 224 94 16 13 314 425 4 425 ]|||| 407 || D 407 ] A || 407-8 ] A ] 313 214 308 208 225 288 223 286 218D280 14 110 15 114 10 41 104 38 105 100 216 42 ]]A |||| 407 ]] A || 402D |||| 407 ]] ||D408 AA |||| 407 D 407 D 407 D 407 407 D || 401 ] ] || 402 ] 270 56 268 262 60 149 180 146 178 172 150 ]] A, 233 93 99 230 1 323 3 9 320 229 || 91 || 400-1 400-1 A, 312 205 309 118 317 279 222 219 227 ] 11 200 18 123 98 50 101408 108 51 A 277 || DA (v.||||enim D, fol. 33 v ).]] 272 122 46 ]]402 A |||| 407 ]]403 407-8 A |||| 408 ]] A 407 A 407-8 A 68 259 254 69 64 171 158 154 D || A, 169 164 D ||159 ] A || ] D || 403 401 ]] A |||| 402 ]] DD408 ||||] 402 D || 401 A 402 402 A] ]|| 403 vv 310 126 311 201 318 44 220 221 228 12 209 97 276 107 102 204 119 273 281 47 54 A D 258 65 72 255 263 152 168 A || || 408 408 D (v. (v. enim enim] D, D, fol. fol. 33 33D).).|| ]403 155 162 165 173 ]] ]D |||| 402 om. (homoeotel.) A 404 A, D ]] A ]] A A, 106 275 40294D || 404 A |||| 403 403 ] D || 403 A |||| || 403 403 314 210 231 53 217 274 224 282 16 203 13 120 4 43 103 48 202 256 257 66 264 151 161 166 167 156 v 174 v v v ( 35 ) A: 17 ,6 - D 17D||,16 (ed. ll . 791-804); D] : 84 ,5 - 84 ,12. ] om. (homoeotel.) ] || 405 add. A, ] 404 A || 404 ] om. (homoeotel.) A || 404 323 121 320 285 229 271 233 ]49 230D || 404 1 124 3 206 9 112 91 52 93 278 99 40 70 67 260 58 177 163 160 157 170 148 55

269 70

( 34 )

] r ,19. ( 34 ) r (ed. (ed. llll.. 760-765); 766-773); D D:: 84 84r,11 ,15 - 84 r,14. 382 ] om. D ||ll382-3 ] 375 om.154-D17 ||r ,4376 AA,|| (376 v 169 69 64 171 (158 tit. : ] 159 31 146 )DA:164 (ed. . 760-765); D : 84r],11 - 84r],14.D || 383 ( 35 )D))|| 262 60 400 17816172,21 149 180 150 (35 35 [ 395 395 384 ] add. AD || 385 ] D || 387 ] om. A || 387 ] ] A, D || 376 ] A || 377 ] A, 255 263 152 168 155 162 165 173 382169 164 ] 159 ] 375 om.154D] om. || 376D || 382-3 ] D || 383 AA,|| 376DD|| 69 64 171 158 15 ] 15 410410 166 167 156 66 264410151 161 ||384 377 ] add. A ] om. || AD 377 ] ] ] D(hic infra) || 378 || 388 388 || D 388|| 387 || 389 D. ]D 174 AD || ||385 ] et om. ] 162A,165(pr.) D ||||] 377 376 D ] ]A] A || 377 A,A ]||D387 D] 255 263 152 168 173 155

268 262

56

57 67

[

15 15 15

[

] ]]

[ [[

]

( 35 )

[

]

((36 ( 36 ) )) 36

]( 35 ) [

((37 ( 37 ) )) 37

[ [[

] ]]

]

41 A ||||D410 ] A ||D410 ] ] (sic) A (corr. || 410A, A || 406 ex ] ) A, |||| 405 ]||]16 add. 16 D406 405 add. A, 16 410 A,406 ] D D ||] 411 ] 41 D || 411 (corr. D]226|| D 407 ] A, D || ](corr. ex A] || 407 )) A, ]] 407 234 92 232add. 96 || ] || 407 D D |||| 406 406 ex A, [ A, 288 38 286add. 280 D 42 || 406 ||105 412-3 . . .|| D ]A om. A] || 412 D || 407 ] ] D || 413 ] ] A 407 ] || 407-8 A 408 ] 218D 104 223 D 100 || 407 ] || 407 ] A || 407 || 150 407 ] D || 407 ] D || 407 ] A || 407 277 178 272D 50 146 51 46 180 172 A, D 219 ||] 413 ]|| 407 ]DD|| 413-4 ]33 v ).]A || 414 ] ] (A || 222 227 ] 101408 108 A || (v. enim D, fol. A A || 407-8 A || 408 ] 281 A || 407 ] A || 407-8 ] A || 408 ] 36 ) 276 47 54 273

D 420 ]]add. 283 109 215 4111 117 212 415 37 287 39 45 284 9 145 179 147 153 176 55 269 57 63 266 175 409

324 2 322 216 110 4214 14 313 308 205 268 200 122 270 56 312 11 18 204 259 119 254 68 7126

]

316 6 95 208 114 41 9 15 10 225 279 123 60 118 149 262 309 317 98 201 64 209 171 44 69 3 158 169 164 159 154 310 311 12 318 97 107 221 102 ]228 414||220 A 202 72 203 255 210 152 275 165 282 120 263 43 168 53 48 173 258 65 A 155 ||274408 408 162 7 3 16 13 314 4 231 217 106 103 224 94 || 417 ] 206 264 285 161 271 166 278 174 124 257 121 66 112 151 52 167 49 156 40 256

1 323 3 9 320 229 91 233 93 99 230 ] 39 170 A 215 67 212 177 283 163 287 157 284 109 70 111 260 117 58 37 160 45 148

1

6

420

995 145 324 269 322 63 316 266 234 179 232 153 226 176 2 457 6 175 92 147 96 55 425

1 6 313 214 308 208 225 104 223 218 105 100 14 110 15 114 10 41 216 4 9 288 38 286 280 42]

312 205 309 118 317 279 222 277 227 11 200 18 123 98 50 101 272 108 51 122 add. D.219 46 310 7126 311 201 318 44 220 54 221 273 228 12 209 107 47 102 281 204 119 397 276

]](corr. ex A || 417

|| 419

) AD|| (v. 415 ] v (sic) A || 415 D (v. enim enim D, D, fol. fol. 33 33 v).). ] (sic) A || 418 ]

]

D || 419

]

add. D || 420-21

] hab. A,

]

D

D || 419 A || 421 (sic)

( 37 ) ( 36 )

Text and translation An ancient Greek treatise on magic squares

i

n=7 n = 11

16 24 48 32 64 96 40 80 120 160

n = 15

n = 19

1 2

n2 + 1

C*

50 122 122 226 226 226 362 362 362 362

C*

—i + “i

i

8 12 24 16 32 48 20 40 60 80

42 110 98 210 194 178 342 322 302 282

“i ≠ 2, “i 30, 66, 70, 118, 122, 126, 186, 190, 194, 198,

32 68 72 120 124 128 188 192 196 200

75

73

—i , —i + 2 10, 42, 26, 90, 70, 50, 154, 130, 106, 82,

12 44 28 92 72 52 156 132 108 84

Once you have done that, nyou have equalized each side of this pair 2 ≠1 will n 2 ≠5 n2 +1 2 2 2 of sides with its conjugate, and the remaining empty cells [ after that ] will = 7further24multiples 22 of 4. 25 be four facing four,nor Equalize them as we have n = 11 60 58 61 explained previously. (§ 35. Example)

n = 15 n = 19

112 180

16

110 170

113 181

45

14

18

24

39

37

3

22

32

20

9

23

21

31

41

30

7

15

33

25

17

35

43

40

49

19

29

27

1

10

38

28

11

13

47

26

12

2

n 22+7 ≠1 +3 + =∆ =∆ n n2≠5 + + 1≠ 1+ ≠4m 44 ≠ + 8m 8m

36

5

34

n=7

Example of the treatment for the right-hand side of the square of seven. of the (second) upper row13 is 820 and the 5 10 15 4 1 12 9 7 14You 4 find that the excess 107 excess of 15the right-hand 11one,8 18. (which 1 14 Putting the 15two6 numbers 3 10 6 12 1 have a sum) less than their sum due by 20 in the two upper corners,108 2 13 12 7 4 9 16 5 +7 31 510 3 13 n82222≠ =∆ =∆ the nnn + + 1lesser 11≠ + ≠ 4m 424 ≠ + 8m 8m +» + with one on the right, and, opposite to them diagonally, their 2 16 3 6 9 14 7 2 11 16complements, 2 11 5 you will have equalized the top and bottom (rows) and the 109 excess on the right-hand side will be 16. You consider then the two large numbers with an excess of 12. [ For5 if you add 12 to 16, which is 6 36 2 34 28 the right-hand excess, this gives an excess of 28, equal to the deficit of 107

10

15

20

25

14

27

29

21

18

11

24

8

These values are in fact known from § 17. 108 30 12 23 22 17 According to § 33. 109 As noted by the interpolator § 34.16 19 4 26 in 13 32

1

35

3

9

7 33 31

(v.r ,11 enim 33 v ) || 378 om. Ar ,19. || 380 ( 32 ) ]A: 17r ,4D- 17 (ed.D,ll .fol. 766-773); D : 84r],15 - 84

270

268 262

56

68 259 254 69 270 56 268 262

]

: r (tit. 32146 )DA: - 17r ,11 178 17 172,4150 60 149 180

(ed. ll . 766-773); D : 84r ,15 - 84r ,19. ] D || 383 154 ] om. D || 382-3

76

Text and translation

382 64 171 158 tit.169 D 164 : 159 60 400 149 180 146 178 172 150

add. || 385 om. D AD || 382-3 156D 174 || 388 add. ] AD D 388 || ||385 165] 173

258

256 257 66 264 151 161 166 167 384155 162 65 72 255 263 152 168

70 67 260 58 177 163 160 157 170 148 r D 174 256 257 66 264 151 161 167 17 156 ( 33166 ) A: ,11||- 388 17r ,17

269 70

r ] ll .D774-780); || 388 ]D : 84 AD (ed. ,19||-388 84r ,23. ]

63 266 175 145 179 147 153 176 260 58 177 163 160 157 170 rr 148 rrA || 391 r 391 ] AD 391 ] rr,23. 33 ,11 :: 84 33)38 ) A: A: 17]280 ,1142--17 17 ,17 ,17 (ed. (ed. llll.. 774-780); 774-780); D|| 84r,19 ,19--84 84 ,23. 216 110 214 208 114 41 (( 288 28617 55 269 57 63 266 175 145 179 147 153 176 55

57 67

392

A A |||| 391 391 ] ]] in marg. D ||]] 393 273 281 54] add.

391277 272 ]]51 122 205 200 123 118 279 391 50 216 110 214 208 114 41 288 38 286 280

46 42

51

46

393 392 204 119 126 201 209 44 276 39247 279 50 277 272 122 205 200 123 118405 202 203 120 210 204 119 126 201 209

43 44

420 420

124 121 206 112 285 202 203 120 210 43

109 215 124 56 2 215 259 313 2 65 11 313 256 310 312 11 70 16 310 55 269 1 323 16 216 110

111 121 268 322 111 254 308 322 72 18 308 257 311 18 67 13 311 57 3 13 214

270 324 109 68 14 324 258 312 14

1

117 206 262 316 117 69 15 316 255 309 15 66 12 309 260 314 12 63 9 314 208

212 283 112 285 60 149 6 95 212 283 64 171 10 225 6 95 263 152 317 98 10 225 264 151 318 97 317 98 177 58 410 4 231 318 97 266 175 320 229 231 425 4 425 114 41

202 203 120 210

43

215 111 117 212 415 283

324

2

14

313 56 11 259 310 65 56 16 256 259 323 70 65

68 1 258

49 278 A ||40396 275 48 282]]

322 316 308 268 18 254 311 72 268 13 257 254 3 67 72

15 262 309 69 12 255 262 314 66 69 9 260 255

6

95

10 60 317 64 318 263 60 4 264 64 320 58 263

225 149 98 171 97 152 149 231 151 171 229 177 152

420

A ]] A, A |||| 391 391 ] om. A || 394 A, A,

A,

] ]

D ||

AA,|| 387 D ||] A ]|| 387 D.]

[ [

D || 389

]

D || 392

]

DD |||| 392 392 ]

]]

D. A ||

D |||| AA || A ||DD 395 ||||

] A || A 396 ]]om. om. A ]] om.(homeotel.) A |||| 394 394 D || 395 A |||| 395 395 ] A || 396-7 ] om. om. (homeotel.) (homeotel.) DD |||| 395 395 ]] AA |||| 396 396 [ ] ]] A || 396-7 ] A || 396-7 ]

( 36 )) ( 36

( 35 )

D || 401

] D || 402 ] ]] A || 402 A, A, ( 37 )) ( 37 A || 403 D || 403 v] |||| 402 DD ||||] 402 A 402 A] ]|| 403 791-804);]]D : 84A ,5 - 402 84v ,12. ]] om. A 404 403 ]] ||||]403 ] ]] DD (sic) A |||| A 403 403 A |||| ||A403 403 ]A || 410 A (homoeotel.) || 410]] ] A || ] ] add. A, ] om. (homoeotel.) A || 404 || 411 ] A, D ||] 411 ] om. (homoeotel.) A || A, 404 41 ] D16 406 ] (corr. ex ) A, ]||]A add. A, ] om. || 412 ] D || 413 ] add. A, 16 D]] || 407 ] D || 407 ] A || 407 41 |||| 406 )) A, D || D 413-4 ] ]] A || 414(corr. ||A, D 406 (corr. ex ex ] [ A

]D |||| 402 || 404 A, D ]] [402 A, [ ||D410 409 282 53 274 275] 48 A ] 405D 410 A,D|||| ||404 D 404 271 52 ]]49 ] 278 40D 37

287

234 A,

92

D ||D |||| 405 45 284 D39]]||add. 412-3 ... D406 405 D || 407 ] D 232 96 || Dadd. ||226413 add. D || 406 406 ]

]

]|| 407 407 ]|| 415 A || 407-8 ] 408 223 218D105 104 100 (corr. ]]Aex|| D 407 ]] D ]]|| ||415 A 414146 (sic) AA ] ] D 407 D ||)|| A 407 D] |||| 407 407 A |||| 407 407 180 178 D 172|| ]150 v v v 222 227p. 310) 101408 10869219 A417 || ] (v.||||(236 enim 33 ).p. 67 ; A,A (D, fol. ]]; A, A 407-8 ]] ||309) || ] ; D,D,fol.fol. (sic) 418A ] D ] ]|| 419 A |||| 407 407 ]] DA A 407-8 A |||| 408 408 169 164 159 158 154 A || 417 vv 220 221 228 107 102 A ]162 A ||] 419 ] DD || (v. 419 ] 33 168 A || || 408 408 D (v. enim enim D, D, fol. fol. 33 ).). add. D || 420-21 155 180 178 165 172 173 146 150 ] 217 106 103 224 94 161 ] A || 421 169 167 164 156 158 166 159 174 154 91 233 93 99 230 163 170 160 157 148 168 155 162 165 173 ] hab. A, (sic)

4 9 269 147 153 176 256 57 257 63 264 175 166 66 266 151 145 161 179 add.167D.156 174

341 216 119 214 201 208 209 110 7126 114 44 7

288

38

286 280

40 204 124 201 112 209 285 276 273 281 119 121 126 206 44 271 47 tit. 52 A 49 :54 278

18 322 309 316 317 6

313 311 308 12 14 310 15 318 10

tit. A 108 : 98 234 101 226 227 95 222 92 232 219 96

97 225

tit. 223 D 221 :218 107 104 220

4

16 323

7

9

314

3

9

4

231 217 106 103 224

320 229

3

91

233

93

p. 309); D : 84v .

( 37 )

[

9 320 91 233 93 102 99 230 318 229 220 221 228 12 97 107

13

cod.

]

102 105 228 100

4 231 106 103 94 312 16 309 317 222 (6, 219 11 13 18 314 98 217 1014) 108 ] 224 D. 227 323 3 310 311

v p. 309); cod. D : || 84425 .

37 39 45 274 282 53 tit.287 D 275 : 48 284

2 322 6 95 206 112 285 234 271 92 278 96 124 121 316 52 232 49 226 40 425 1 6 (7 ⇥ 7) A : 23r284(ed. 313 15 10 215 308 212 225 283 104 287 218 111 117 37 223 39 105 45 100

324 11 2

( 36 )

42

3

215 202 1111 203 117 210 283 120 212 643

(ed. ll . 806-809).

tit. A :54 codd. hic 47 omm. 276 273 281

202 274 r46] 205 203 200 120 279 53 2775)275 272 424 123 210 118 43 50 51 (5 ⇥ A :48 23282 (ed.

1

[

( 34 )

v 287 enim 37 39 45D,284 D (v. A 396 271 52]] 49 278 A ||||40fol. 39633 ). ]] A A |||| 396 396 180 146 178 172 150 234 92 232 226 96 v v). 287 284 37 39 45 r v D (v. enim D, fol. 33 enim D, fol. (D34(v. ) A: ,17 - 1733 ,6 ). (ed. ll . 781-790); D : 84r ,23 - 84v ,5. 169 164 17 158 159 154 104 223 218 105 100 234 92 232 226 96 rr 173D v rr || 398 v 398 ]165 398 add. in marg. ] A || 398-9 16217 (168 34 )) A: ,17 --17 (ed. D ,23 (222 34155 A: ,17 17||v,6 ,6 (ed. llll.. ]781-790); 781-790); D:: 84 84D ,23--84 84v,5. ,5. 219 227 101 10817 104 223 218 105 100 166 167 174D (homoeotel.) 161 ] om. A || 399 A || 399 398 ]] add. in D A 228D || 107 102 398220 221]]156 || 398 398 add. in marg. marg.(ter D)||||] 398 398D || 399 ]] A ||||] 398-9 398-9 222 101 108 219 227 (319 ; D, fol. 84v ; A, p. 309) 163 160 157 170 148 || 399 ] A(ter || 399 AD ] |||| 399 A 217 106 ]103 (sic) 224 94A (homoeotel.) ]] om. A || 399 ))]] D ]] AA om. (ter D] |||| 399 399 || 400 399 228 (homoeotel.) A || 399 107 220 221 102 145 179 147 153 176 233 93 99 230 91 || 400-1 A ]] A ]] AA (sic) A |||| 399 399 A]|||| 399 399 ]] AD AD |||| 400 400A, 217 106 ]]103 (sic) 224 94 288 38 286 280 42

110 38 286 42 9 288 260 114 177 163 160 170 148 70 4214 67 208 58 41 157 280 (50 36⇥ ) 3) A: 1751v ,1746- 17v ,20 (3 205 200 279 277 272 123 118 55 269 57 63 266 175 145 179 147 153 176

1

D 393 D ||||A, 393 ] A, A A, || 396

99 230 51 46 A, D ||v 402 v D 401 D ||]||(ed. 401ll . ( 35 ) A: 17 ,6 - 17 ,16 44 276 47 54 273 281

124 121 206 112 285

270

275 48 282 ] add. in in] marg. marg. 273 281 54] add.

323 3 233 93 9 320 229 || 91 || 400-1 400-1 205 200 123 118 279 50 277 272

204 119 126 201 209

312

53 274 393 39347 276 271 52] 53 274

]

] ] D || 387 D || 383 ] om. ] 15 15 ] ] AD || D 388|| 387] D || 389 ] om. 15

384 258 65 72 255 263 152 168 162 165] 173 382155 169 164 159 154 ] 68 259 254 69 64 171 158 415 415

D.

]

99

94 230

] 42 42 42 16 (319 ; D, fol. 84v ; A, p. 309)

124 110 120 21

35

25

4

18

8

141 127 137

114 118 122 31

27

23

14

10

6

131 135 139

116 126 112 29

19

33

12

2

16 133 143 129

58

73

83 106 92 102 39

Text and translation

72

62

87

53

77

43

two68small ]11085You96then 64 numbers. 60 77 81 100 put 104 these 49 45two41numbers

the sum of on the right-hand side, as also the two small numbers which have a sum less 79 89 75 98 108 94 47 37 51 than their sum 66 due56by 70 28, and you put on the left, opposite to each one, its complement. 105 91 101 40 54 44 57 71 61 88 74 84 You will proceed likewise for all other (squares of this kind). 95

99 103 50

46

42

67

63

59

78

82

86

13

9

11

1

2

9

42

16

25

18

423

5

13

21

3

7

5

36

19

9

11

20 12

15

17

7

14

6

1

823

5

13

21

322

10

1

8

24

12

15

17

7

10

1

8

2

n 46 =5

45

44

8

16

22

8

37

29

24

18

&

#22 $ ('

14 14 24 32

14

2

46

45

44 30

841 16 31

21

23

9

20

)

*

32

22

3

37

39 43 24 35 18 17

25

33

15

7

30

41

31

21

23 10

91

20 27

29

19

49

40

43

35

17

25

33 12 15 26

747

13

11

28

38

10

1

27

29

19 34 49 48 404

5

6

42

36

12

26

47

13

11

28

38

34

48

4

5

6

42

36

I have represented the squares obeying this condition successively from 97 107 93 48 38 52 65 55 69 80 90 76 (the order) three to nineteen; you will thereby become acquainted with 111 them and rely on the128 subsequent ones. 3 them 17 7for142 138 123 109 119 22 36 26

n=3

! %$ %

"

5

132 136 140 113 117 121 32 28 24 2 16 25 18 4 15 134 144 130 115 125 111 30 20 34 6 19 9 11 20

n=7 Our previous Gi − (n2 + 1) + ∆i * = (n2 + 1) − Bi . This sentence, which attempts to complete the (obviously lacunary) reasoning of § 34 (see below, p. 199), must be an interpolation. 111 We add the square of order three, missing here (in fact, superfluous), and repeat that of order 5, already represented (§ 23). For all, left to right orientation, thus different from the orientation used above (§§ 9–11, 23, 25–27, 35); this will facilitate the comparison with the squares constructed in our Commentary. For the squares of order n = 4t + 3, the place of the two ‘larger numbers’ given in § 34 (our γi − 2, γi ) may vary, either in their succession (with γi − 2 above, see case n = 11 with i = 2) or within the border (see case n = 15; same for the pair βi , βi + 2). See also p. 27, n. 37. 110

C

78

Text and translation

(307 ; D, fol. 84v ; A, p. 310)

(308 ; D, fol. 85r )

268 262 69

(D, fol. 61v ; A, p. 311)

(9 ⇥ 9) A : 23v (ed. p. 310); D : 84v . tit. A :

60 149 180 146 178 172 150 64

tit. 169 D :164 159 154 171 158

255 263 152 168 155 162 165 173

D.

(8, 2) ]

264 151 161 166 167 156 174 (11 ⇥ 11) A : 24v 260 58 177 163 160 157 170 148 66

57

63

tit. A :

266 175 145 179 147 153 176

214 208 114 41

tit. D : 38 286

280

42

277 272

51

46

288

123 118 279

50

201 209

44

276 273 281 47 54quadrati A latere

120 210

43

53

206 112 285 271 111 117 212 283

274 275 (3, 11) ]

A282 ||

48

52

49

278

40

287

39

45

284

92

232 226

96

(6, 4) ]

A ||

37

(9, 8) ]

(2, 5) ]

A ||

6

95

15

10

225 104 223 218 105 100

309 317

314 3

9

318 4

98

234

A ||

322 316

12

(ed. p. 311); D : 85r .

222 101 108 219 227

231 217 106 103 224 91

233

93

99

94 230

(= n2 + 1). A ||

(8, 10) ]

(6, 7) ] (3, 5) ]

(6, 3) ]

97 107 220 221 102 228

320 229

in A :

A ||

A || A ||

A ||

(6, 9) ] (7, 7) ] (7, 5) ]

(11, 3) ]

A ||

A ||

A ||

A ||

(5, 8) ]

(4, 6) ] (8, 5) ] (6, 2) ]

A ||

A ||

A.

(8, 8) ] (11, 6) ] (4, 4) ]

A || A ||

A ||

79

Text and translation

34

26

28

6

77

72

14

66

46

30

38

2

3

69

71

64

40

52

34

26 32 28 4

6 77 23 63 72 61 14 31 66 27 46 78

50

30

38 8

3 69 53 47 71 37 64 39 40 29 52 9

74

32

4 23 75 67 63 57 61 33 31 41 27 49 78 25 50 15

7

8

73 70 53 1

12

75

67 60 57 62 33 55 41 19 49 21 25 51 15 59

7 20

22

70

1 17 24 42 43 80 45 79 35 13 65 11 81 18 12 44

58

60

62 36 55 56 19 54 21 76 51 5

48

24

42 80 79 34 2 118 56 54 76 72 46 18

36

2 73

47 17 37 43 39 45 29 35

13 11 n= 116 8 5 10 102 7

9 74 65 81

59 10 20 68 22 16 18

44 10 68 16 97 107

9 113

58 110 108 16 48 100 24 48

50

36

34

2 118 70 66 116 58

83 113 23 10 89 110 91 108 103 16 60 36 56

52

72

46 7 97 26 18 68 102 105 43 83 107 81 100 51 24 47 48 17 50 54

96

70

66 58 93 3 23 28 109 73 89 67 91 57 103 59 60 49 56 29 52 13

94

26

68 105 111 95 43 87 83 77 81 53 51 61 47 69 17 45 54 35 96 27

11

28 109 30 93 5

73 13 117 94 92 21 67 37 57 63 59 65 49 55 29 85 101

111 95 90 87 42 77 1

53 27 11 75 61 39 69 41 45 71 35 79 121 80

32

30

5 21 63 65 84 44 37 62 119 99 55 33 85 31 101 19 117 64 92 78

38

90

42 40

82

84

44 62 119 86 120 4 99 6

40

74 104 20 115 25

86 120

1 104 75 39 41 71 74 20 115 25 79 15 121 22 80 98 32 76

4

6

114

33 31 114 9

9

15

19 64 38 88 112 12 78 14 106 22

112 12

n = 11

98

76

82

14 106 88

80

Text and translation

(317 ; D, fol. 85v )

268 262 69

60 149 64

(13 ⇥ 13) D : 85r . tit.146 : 178 172 150 180

171 158 169 164 159 154

(3, 6) ]

255 263 152 168 155 162 165 173

57

(5, 1) ]

D.

(15 ⇥ 15) D : 85v . 163 tit.160 : 157 170 148

66

264 151 161 166 167 156 174

260

58

63

266 175 145 179 147 153 176

177

D ||

214 208 114 41

288

38

286 280

123 118 279

50

277 272

42

(1, 10) ] corr. ex 51

46

(v. loculum infra) ||

(7, 6) ]

mut. ut vid. in

||

81

Text and translation

74

50

52

14 152 34 161 134 42 126 124 48

58

78

62

64

6

7

60

66

82

2

3

23 137 139 151 144 84 104 110

16

68

4

47 131 127 125 55

59

51 166 102 154

150

8

153 113 67 107 105 75

71

57

17 162 20

132 157 141 117 97

94

145 155 160 22 146 90 112

91

81

83

73

53

29

13

38

159 143 135 121 101 77

85

93

69

49

35

27

11

40

5

21

41

61

87

89

79 109 129 149 165 130

1

37

99

63

65

95 103 133 169 12 116

114 140 142 119 39

43

45 115 111 123 28

54 158

100 32 72

86 168 167 147 33

31

19

80 108 106 164 163 25

15

10 148 24

76 120 118 156 18 136

11

18

13

74

16

14

12

15

10

56

9

26

36 128 44

n = 13

30

56

88 138 70 92

98

46 122 96

81

76

29

36

31

79

77

75

34

32

30

17

78

73

80

33

28

35

63

58

38

45

40

20

27

22

61

59

57

43

41

39

25

23

21

60

55

62

42

37

44

24

19

26

47

54

49

2

9

4

65

72

67

52

50

48

7

5

3

70

68

66

51

46

53

6

1

8

69

64

71

82

Text and translation

(309 ; D, fol. 86r )

(13 ⇥ 13) D : 85r . tit. : D ||

(3, 6) ]

268 262 69

(15 ⇥ 15) D : 85v . tit. :

60 149 180 146 178 172 150 64

171 158 169 164 159 154

(1, 10) corr. 255 263 152 168 155 173 ex 162 ] 165

57

66

264 151 161 166 174 156 loculum (3, 3) ] 167 (v.

260

177 163 160 157 170 148 (17 ⇥ 17) D : 86r . 266 175 145 179 147 153 176

63

D.

(5, 1) ]

58

tit. :38

214 208 114 41

288

123 118 279

50

277 272

201 209

276

281 ex 47 54 ] 273 (2, 15) corr.

44

286 280 51

(v. loculum infra) ||

supra) ||

(7, 6) ]

mut. ut vid. in

(8, 8) ]

.

.

(2, 2) ]

42 46

||

(7, 8) ]

||

||

83

Text and translation

62

2

222 220

66

78

26 198 196 32

11 189 207 34 190 188 40

94

7

35 173 183 203 180 48

31

51 165 167 179 199 112 106 56

158 54

8

42 182

152 170 120 110

3

10 214 213 212 16

76 168 118 201 75 159 155 153 83

18 206 204 24

64

80 160

96 172 68 74

87

79

25 108 58 150

60 205 181 141 95 135 133 103 99

85

45

21 166 144

142 209 185 169 145 125 119 109 111 101 81

57

41

17

84

211 187 171 163 149 129 105 113 121 97

63

55

39

15

82

140

9

77

33

49

69

5

29

65 127 91

93 123 131 161 197 221 102 138

128 122 90

1

147 67

73 143 139 151 225 136 104 98

88 124

126 70 50

89 115 117 107 137 157 177 193 217 86

71

92 114 223 195 175 61

59

47

27 116 134 156 100

72 130 184 44 219 191 53

43

23

46 178 132 154 176

52 146 200 28

30 194 215 37

19 192 36

162 224

218 216 12

14 210 208 20

4

6

13

n = 15

38 186 148 174 22 202 164

11

18

13

74

81

76

29

36

31

16

14

12

79

77

75

34

32

30

15

10

17

78

73

80

33

28

35

56

63

58

38

45

40

20

27

22

61

59

57

43

41

39

25

23

21

60

55

62

42

37

44

24

19

26

47

54

49

2

9

4

65

72

67

52

50

48

7

5

3

70

68

66

51

46

53

6

1

8

69

64

71

84

Text and translation

(13 ⇥ 13) D : 85r . tit. : D ||

(3, 6) ]

D.

(5, 1) ]

(15 ⇥ 15) D;: D, 85vfol. . 86v ) (318 tit. :

(1, 10) ] corr. ex (3, 3) ]

268 262 69

(v. loculum supra) ||

(7, 6) ]

mut. ut vid. in

(8, 8) ]

.

.

(2, 2) ]

(17 ⇥ 17) D : 86r . tit. :

60 149 180 146 178 172 150 64

171 158 169 164 159 154

255 263 152 168 155 173 (2, 162 15) ]165 corr.

57

(v. loculum infra) ||

66

264 151 161 166 167 156 174

260

58

63

266 175 145 179 147 153 176

177 163 160 157 170 148

214 208 114 41

288

38

286 280

42

ex

||

(7, 8) ]

||

||

85

Text and translation

130 82

84

22 264 42 246 58 277 230 228 64

66 222 220 72 158

94 134 98 100 14 272 38

11 253 271 250 74 214 212 80 154 196

96 106 138 114 116

6

7

35 237 247 267 280 30 258 150 184 194

24 108 118 142

2

3

31

51 229 231 243 263 256 144 172 182 266

262 16 120

79 223 219 215 213 87

236 270

8

4

91

95

83 286 170 274 28

265 197 107 191 187 185 115 119 111 93

25 282 20

54

56 240 269 245 201 173 127 167 165 135 131 117 89

45

21

50 234

86 273 249 233 205 177 157 151 141 143 133 113 85

57

41

17 204

275 251 235 227 209 181 161 137 145 153 129 109 81

63

55

39

202

15

9

33

49

73 101 121 147 149 139 169 189 217 241 257 281 88

200 52

5

29

69

1

65 179 99 103 105 175 171 183 225 289 12 168 198

92 122 278

97 159 123 125 155 163 193 221 261 285 238 90

102 166 244 254 207 67

71

75

77 203 199 195 211 36

46 124 188

186 164 48 146 288 287 259 239 61

59

47

27

180 128 140 176 174 284 283 255 53

43

23

10 260 32 152 162 110

112 136 192 190 276 18 252 279 37

19

40 216 76

132 208 206 268 26 248 44 232 13

60

62 226 224 68

n = 17

34 148 242 126 104

78 210 156 178 70 218 160

86

Text and translation

(217 ; D, fol. 96v )

268 262 69

(19 ⇥ 19) D : 86v . tit. :

60 149 180 146 178 172 150 64

171 158 169 164 159 154

(17, 12) ]

255 263 152 168 155 162 165 173

57

66

(14,166 6) ] 167 156|| 174 264 151 161

260

58

63

266 175 145 179 147 153 176

||

288

38

123 118 279

50

277 272

51

201 209

44

276

47

273 281

120 210

43

53

274 275

48

282

52

278

40

206 112 285 271

286 280

54

49

42 46

||

(11, 5) ]

177 163 160 (2, 2) ] 157 170 || 148(11, 2) ]

214 208 114 41

||

(2, 9) ]

||

(v. loculum infra, 13, 6) ||

(12, 7) ] (13, 5) ]

(14, 1) ]

||

ut vid.

(19, 3) ]

ut vid. ||

87

Text and translation

98

2

358 356

8

10 350 348 16 345 18 342 340 24

200 118 34 326 324 40

42 318 15 313 339 316 48

26 334 332 32 100

50 310 308 56 120 162

198 196 138 58 302 300 64

11

47 289 307 335 66 294 292 72 140 166 164

82 194 192 158 74 286

7

43

71 273 283 303 331 284 80 160 170 168 280

84 106 190 188 178

39

67

87 265 267 279 299 327 180 174 172 256 278

3

122 108 130 186 329 115 259 255 251 249 123 127 131 119 33 176 232 254 240 238 94 132 333 301 233 143 227 223 221 151 155 147 129 61

29 230 268 124

236 266 337 305 281 237 209 163 203 201 171 167 153 125 81

57

25

96 126

128 341 309 285 269 241 213 193 187 177 179 169 149 121 93

77

53

21 234

343 311 287 271 263 245 217 197 173 181 189 165 145 117 99

91

75

51

19

134 13

45

69

85 109 137 157 183 185 175 205 225 253 277 293 317 349 228

226 260

9

41

65 105 133 195 159 161 191 199 229 257 297 321 353 102 136

220 104 86

5

37 101 215 135 139 141 211 207 219 261 325 357 276 258 142

144 110 274 154

1

243 103 107 111 113 239 235 231 247 361 208 88 252 218

146 250 272 156 182 359 323 295 275 97

95

83

63

35 184 206 90 112 216

214 248 92 202 288 76 355 319 291 89

79

59

31

78 282 204 270 114 148

212 116 222 304 60

62 298 351 315 73

55

27 296 68

152 242 328 36

38 322 320 44 347 49

23

46 314 312 52

262 360

354 352 12

4

6

14 346 17 344 20

n = 19

70 290 224 246 150 54 306 244 210

22 338 336 28

30 330 264

420 420

88

Text and translation

36)) ((36

425 425

37)) ((37 [[

]] 16 16 [

430

[

]

[

]

]

]

435

( 35 ) A: 17v ,6 - 17v ,16 (ed. ll . 791-804); D : 84v ,5 - 84v ,12. 409

410

268 262

445

A || 410

]

]

] D || 412-3

A,

tit. A :

255 263 152 168 155 162 165 173 v v ) ( 36

171 158 424169 164 159 154]

[ (sic) A || 410

D || 411

A ||

] ]

[]

A,

] hab. A, ]

[

]

cod.

17vr,17 - 17r v ,20

(ed.llll. .810-822). 806-809). ( 37 ) A: 17 ,20 - 17 ,21A: & 19 ,1 - 19 ,11 (ed. tit.cod. A :|| 429 (post.) ] quasi cod. || 431

264 151 161 166 167 156 174 268 262 60 149 180 146 426178 172 150 ] 70 67 260 58 177 163 160 157 170 148 269

A,

]

] v] om.

add. D.cod. || 425

65

66

]

]

60 149 180 146 178 172 150 64

A || 410

( 36 ) A: 17v ,17 - 17v ,20 (ed. ll . 806-809).

69

256 257

D || 411

. . [.

259 254 72

]

A || 412 || 413 v ( 35 ) A: 17v ,6 - 17 ,16 (ed. ll . 791-804); ]D : 84v ,5D- 84 ,12. D || 413 409 ] ] A ||D410 || 413-4 ] ] A || 414 ] A || 410 ] (sic) AA||||410 ] A || 414 ] (corr. ex ) A || 415 ] (sic) A || 415[ ] D] 410 ] A, D || 411 ] A, D || 411 ] A, || 417 ] A || 417 ] (sic) A || 418 ] D || 419 D || 412-3 ... ][om. A || 412 ] ] D || 413 ] ] A || 419 ] D || 419 ] add. D || 420-21 A, D || 413 ] D || 413-4 ] A || 414 ] A || ] A || 421 414 ] (corr. ex ) A || 415 ] (sic) A || 415 ] D ] || 417 hab. A, (sic) ] A || 417 ] (sic) A || 418 D || 419 ( 38 ) ] add. D. ] A || 419 ] D || 419 ] add. D || 420-21 A,

56

]

] [

440

[

164 159 154] 69 64 171 158 169 || 435 57 63 266 175 145 179 147 153 176

255 263 152 168 155 162|| 165 cod. 439173 110 214 208 114 41 288 38 286 280

42

424cod. || 438

]

cod. || 439

]

66 264 151 161 166 167 156 174 ] 46 cod. 205 200 123 118 450 279 50 277 272 51 260 58 177 163 160 157 170 148 r 281 r 119 126 201 209 44 276 47 54 273

cod. ||cod. 438 || 425

] ]

|| 441

]

]

( 38 ) A: 19 ,12 - 19 ,15 (ed. ll . 823-826).

57 63 266 175 145 179 147 153 176 202 203 120 210 43 53 274 275 48

282

214 208 114 41 288 38 286 280 42 124 121 206 112 285 271 52 49 278

40

444

[

]

cod. || 444

]

]

cod.

cod.

]]

cod. || 440

cod.

]

cod. || 439 ]

( ]39 ) cod. || 440

[

A || 421 (sic)

89

Text and translation

16

14 14

14

3 32 22 18 1824 18 2439 24 3937 39 373 37 322 22 32

45

32

(Bordered even-order squares) 16 16

45 45

9 21 23 9 20 41 31 (§ 36. The three categories 20 of 20even-order squares) 923 23 2131 21 31 4130 41 30 30 7 15 33 25 17 35 7 715 1533 3325 2517 1735 3543 43 43 The treatment for squares of even (orders) depends on their three 19 1 10 112 40 4049 40 19 49 1 27 49 1929and 2927 29 27 110 10 categories: evenly-even, evenly-odd evenly-evenly-odd. 38 3828 38 2811 28 1113 11 1347 13 4726 47 2612 26 12 12 The first (square of even order) is the square of four; indeed, the first 5 36 36 36 5 5 34 34 34 even (order) is 2 and this is not possible for it, that is, to place in it numbers in magic arrangement. 5 15 10 510 10 154 15 4

4

1

8114 14 1

14

15 156 15 63

2 12 13 213 13 127 12 7

7

4

9

14 147 14 72

9

97

9714 14 7 4 14 4

4

5

6

6 1 12 1 612 12 115 15

15

11 118 11 81

3

3 8 13 8 313 13 810 10

10

2

2 5 11 16 162 16 211 11 5

5

16 163 16 36

6

36 9

n=4

69

6 2 36 2 28 34 636 36 234 34 285 28 5

1 13 12 112 12 138 13 8

49

8

6310 10 3

10

4916 16 9 5 16 5

5

7211 11 2

11

5

113 (§ 37. Construction of the square 10 15 25 10 1015 of 1520 order 2025 20 2514 4) 1427 14 27 27 29 21 18 11 24 8 29 2921 2118 So let us turn our attention to1811the ofi four.114 You place 1124 hsquare 248 8 12 23 7 12 30 7 17 1223 of 2322 its 2217 22 17 7 median (the numbers as follows).11530 30One two numbers (8) in ha 4 26 13 16 19 33 4 426 (9) 16 16 19 1933 2613 13 33 cell ini its centre and the other in its corner [ like the bishop’s move 1 3 35 9 321 32 135 35 3 9 3931 31 in chess ], namely the third 32 (corner), hwhich is31the third celli diagonally from it.116 Then, the number preceding the small median (7) next to the large median, on its right117 , and that following the large median, 112

10 2 100 2 5 98 5 88 94 10 10 100 100 298 98 594 94 8815 88 1584 15 849 84 9

9

‘Evenly-even’ (ἀρτιάκις ἄρτιος, Nicomachos’ Introduction 83I.8.4, or ἀρτιάκις ἄρτιος 18 18 18 83 83 μόνον, Euclid’s Elements IX, 32), thus (here) of the form n = 8t, t natural ≥ 1, such 68 34 66 60 37 85 85 8538 3868 38 34 66 60 37 16 16 16 68 (ἀρτιοπέριττος, 34 66 60 37 as 8, 16, 24, 32, 40, 48, . . . ; ‘evenly-odd’ Nicomachos I.9.1, or ἀρτιάκις 14 42 47 52 57 46 59 14 14 42 4247 46 the περισσός μόνον, Euclid Elements IX, 33), form87n87= 874t + 2, such as 6, 10, 14, 4752thus 5257 57of 4659 59 61 56 18, 22, 26, . . . ; ‘evenly-evenly-odd’ (περισσάρτιος, 89 89 8961 6153 5350 53 5043 50 4356 43 5640 Nicomachos 40 4012 12 12 I.10.1), thus (here) of the form n = 8t + 4, such as 12, 20, 28, 36, 44, 52, . . . . The author will now examine 90 62 44 55 54 49 90 90 62 6244 4455 5554 5449 4939 39 3911 11 11 the squares with these even orders, after dealing with the particular case n = 4. 8 36 3658 36 6593 93 93 51 48 113 8 5845 58 4548 45 5165 51 Since there is no magic8 square of order 2,48as just65stated, the square of order 4 will 6 35 67 63 41 6 6 95 9564 6433 64 3367 67outer 3541 35 41 63 63separately. be filled as a whole without95considering its33 border The description of 4 medians, thus 8 and 9. 97 the placing will begin with 4its4 two 97 97 114 We have added the first correspond with the subsequent instructions. 1 3to99 3 7 96 7 86 13 92 92figure, 1 92 199 99 396 96 713 13 8617 86 1791 17 91 91 The manuscript (A) has the second (same, but turned around the ascending diagonal), used in all subsequent even-order squares. The third is also found in the manuscript but not mentioned in the text (probably an earlier reader’s addition); it starts with the larger median in the centre. 115 The text being corrupt, all our additions to it are explicitly indicated. 116 From the text, it is clear that ‘third’ refers to a corner, not a cell. In what follows, the original text must have had the words (added by us) ‘third cell diagonally’ — their replacement by a chess move must be Arabic. These modifications may explain the present corruptedness of the text. 117 As seen by the reader (p. 19, n. 25). From here on the instructions are no longer applicable to the second figure.

37)) ((37 90

[[

]] Text and translation 16 16 [[

430 430 430

[[

]]

[[

]]

[[

]]

]] [[

]]

]]

435 435 435

[[ ]] [[

]]

( 35 ) A: 17v ,6 - 17v ,16 (ed. ll . 791-804); D : 84v ,5 - 84v ,12. 409

440 440 440

A || 410

]

410

] D || 412-3

A,

D || 413

414

]

|| 417

A,

D || 411 ...

]

A,

]

][[om. A || 412

[

(corr. ex

A || 419

A || 410

[[ ]]

D || 413-4

]

D || 419

D || 411

]]

D || 413

[[

]

38)) ] (sic) A || 418 ((38

add. D || 420-21

]

]]

A ||

]

(sic) A || 415

] add. D.

A,

]

] 445 445 445

A ||

]

A || 414 ]

] ]

(sic) A || 410

] ]

) A || 415

A || 417

] ]

]

[[

hab. A,

]

D || 419 A || 421

( 36 ) A: 17v ,17 - 17v ,20 (ed. ll . 806-809).

56

v v r : ( 37 )60A: ,20178- 17 &tit. 19A ,1 - 19r ,11 (ed. ll . 810-822). 180 146 172 ,21 149 17 150

268 262

259 254

69 426

64

171 158 169 164 159 154

39)) ((39

] (post.) ] cod. || 425 ] cod. cod.|| 431 quasi ] cod. A: 173 17v ,20 - 17v ,21 & 19rr ,1 - 19rr ,11 (ed. ll . 810-822). 65 72 255 263 152 168 155( 37 162) 165 ( 37 ) A: 17v ,20 - 17v ,21 & 19r ,1 - 19r ,11 (ed. ll . 810-822). ( 37 17 - 17438 ,21 ,11 (post.) (ed. ll .] 810-822). 435 ],1 - 19cod. || 438 ] || 431cod. ||] 439 cod. ] 426 cod.&||19 429 quasi cod. 256 257|| 66 264 151 161 ]166 167) A: 174 ] ,20|| 156 cod. 450 426 ]] ] cod. || 429 (post.) ] quasi cod. || 431 ] cod. 450 450 426 ] cod. || 429 (post.) ] quasi cod. || 431 ] cod. 70 67 260 58 177 163 160||157 435170 ]148 ] cod.cod. || 438 ] cod. ||] 438 ] 440 cod. || 439 ] ] cod. || 440 cod. || 439 || 439 cod. || || 435 ] cod. || 438 ] cod. || 438 ] cod. || 439 [ ] 435 153 cod. 438 || 439 ] cod. || 438] 269 57 63 266 175 145 179||cod. 176 ] 147 [ ||] 440 || 439 ] || cod. cod. ||] 440 cod. || 439 ] cod. || 439|| 441 ] cod. ||] 439 cod. || 440 ] cod. || 440 ] cod. cod. cod. ]] 286 ||280 110 214 208 114 41 288 38cod. 439 cod. || 439 ] cod. || 440 ] cod. || 440 cod.] ||]] 441 cod. 270 56 268 262 60 149 180 146 178 [[ 42172]] 150 cod. || 441 ] cod. 205 200 123 118 279 50 277 272 51 46] ] cod. r 158 169 164 r 159r 154cod.r || 441 259 171 68 ( 69 6419 38254) A: ,12 - 19 (ed. . 823-826). ( 38 ) A:,15 19r ,12 - 19rll,15 (ed. ll . 823-826). 281 119 126 201 209 44 276 47( 38 54) 273 A: 19 ,12 19 ,15 (ed. ll . 823-826). r 173 r 258 65 72 255 263 152 (168 38 ) A:16219165 ,12 ,15 (ed. ll . 823-826). 444155 ] - 19 cod. || 444 ] cod. 202 203444 275 48 282 ] || 444 120 210 43 53 ] 274444 cod. ] ] cod.cod. 256 257 66 264 151 161 166 167 156 174 cod. || 444 444 ] r cod. || 444 ] cod. 124 121 206 112 285 271 52( 39 49 ) 278 A: 4019 ,15 - 19r ,18 & 18vv ,2 - 18vv ,20 & 18rr ,1 - 18rr ,5 (ed. ll . 827-854); de hac confu70 67 260 58 177 163 160 157 170r 148 r ( 39 ) A:r 19r ,15 - 19rr ,18 v & v18v v,2 - 18v ,20 & r18r ,1 - 18rr ,5 (ed. ll . 827-854); de hac confu( 39 ) A: 19 ,15 19 ,18 18 - 18 18 & 18 ,1,1- -18 ,5 (ed. (ed. . 827-854); hac confusione scr. alio cod.) ,1,2:,20 ( 39 19(ex ,15& - 19 ,18,2&18 - 18 ,20 & 18 18 ,5 ll . ll 827-854); de hac de confu215 111 117 212 283 37 287 284 39) A: 45 179 scr. 55 269 57 63 266 175 145 147 153 sione (ex 176 alio cod.) 18vv ,1 : scr. (ex alio 316 6 scr. 232 226 2 322sione 95 234 96 (ex92sione alio cod.) 18vcod.) ,1 : 18 ,1 : 17

cod.v || 424 429 v

]

216 110 214 208 114 41

288

38

286 280

42

17

cod. || 452 ] cod. || 452 ] post 449 ] 46 cod. || 452 ] cod. || 452 ] post 11 18 309 317 98 222 101 108 219 227 449 ] cod. || 452 ] cod. || 452 ] post (l. 452) hab. cod. || 463 ] cod. || 464 ] cod. || 464 (pr.) ] 204 119 126 201 209 44 276 47 54 273 281 (l. || 452) hab. cod. ||]463 ] cod. cod.||||452 464 ] cod. || 464 (pr.) ] ] 452 310 311449 220 221 cod. 228452) 12 318 97 107 102 (l. hab. cod. || 463 ] cod. || 464 ] cod. || 464 (pr.) ] cod. 46448 282 ] cod. || 467 ] quasi cod. || 468 ] cod. || 471 ] post 202 203 120 210 43 53 274 ||275 cod. ||224464 ] cod. || 467 ] quasi cod. || 468 ] cod. || 471 16 13 314 4 231 217 106 103 94 cod.52 || 464 cod. ] || 467 ] ] quasi cod. || 471 cod.|| ||464 475cod. || ]468] cod. ]cod. || 475 ] 271 124 121 206 112 49 278 40] || 463 (l.285452) hab. cod. cod. || 464 (pr.) ] ] cod. || 475 ] cod. || 475 ] 323 3 9 320 229 91 233 93 99 230 ] cod. || 475 ] cod. || 475 ] 284 109 215 111 117 212 283 37 287 cod. 39 45 || 476 ] cod. cod. || 464 ] cod. cod. ] quasi cod. || 468 ] cod. || 471 || 476 || 467] cod. ] cod. 324 2 322 316 6 232 || 226476 95 234 92 cod. 96 313 308 15 10 225 104 223 449 218 105 100] 122 205 200 123 118 279 50 277 272 51

D

(sic)

Text and translation

91

[ its complement118 ] (10), in the [ bishop’s cell ] hthird cell diagonallyi as well, next to the small median. Then, the (next) number following the small median (6) underneath the large median, and that following the large (median), [ its complement119 ] (11), in the [ bishop’s cell ] hthird cell diagonallyi. Then, the (next) number following the small median (5) in the lower right-hand corner and its complement (12) in the [ bishop’s cell ] hthird cell diagonallyi from it; the (next) number following the small median (4) in the upper right-hand corner and its complement (13) in the [ bishop’s cell ] hthird cell diagonallyi from it; the (next) number following the small median (3) in the cell next to (that of) the larger number which you have reached [ thus 3 ]120 , and the complement of it, thus 14, in the [ bishop’s cell ] hthird cell diagonallyi from it; 2 in the cell underneath (that of) 13, and 15 in the [ bishop’s cell ] hthird cell diagonallyi from it; 1 underneath 14, and 16 in the [ bishop’s cell ] hthird cell diagonallyi from it. This square is finished.

(Bordered squares of orders n = 4k + 2)

(§ 38. Filling the inner square) As to the treatment for the evenly-odd square, it is as follows.121 You draw the square, and then write the two medians of the numbers in the central square, where you had put the two medians of the previous square, the subsequent (numbers) where were the corresponding subsequent (numbers) of these two medians, (and so on) until you have finished with the inner square [ as you had done with the odd (orders) ].122 You are then left with the outer (part of the) square and the remaining numbers. (§ 39. Filling evenly-odd borders) Once you have done that, you take the two terms you have reached;123 you put the small one — which will always be even if the sequence (of the numbers) proceeds from 1 to the last by successive additions of 1 — on the left-hand side underneath the corner cell, and its complement on the 118

Complement of 7 and not of the ‘large median’. Again, complement of 6, in spite of the Arabic form. 120 Interpolation, whether we leave it as it is or change it to ‘thus 13’. 121 After considering the square of order 4, the author treats the case of the subsequent even order 6 and its category. Next (§§ 40–43) he will deal with the other even orders. In later treatises, the study of squares with orders n = 4k (including both n = 8t and n = 8t + 4) precedes that of evenly-odd ones (n = 4k + 2). 122 Just as, for odd-order squares, we filled first the inner square (see §§ 4–5) —but this last remark seems superfluous. We are then left with filling the border. 123 That is, the next small number in descending sequence, and the next in ascending sequence for the large numbers; thus, 10 and 27 for order 6, 18 and 83 for the border of order 10 (how to fill the preceding border of order 8 will be taught in § 41). 119

( 37 ) A: 17v ,20 - 17v ,21 & 19r ,1 - 19r ,11 (ed. [ ll . 810-822). 426 ] cod. || 429 (post.) ] quasi 92

] cod. || 431

Text and translation

||A:435 ] v ,21 &cod. || -438 ] cod. || 438 17v ,20450- 17 19r ,1 19r ,11 (ed. ll . 810-822). ]

( 39 ) ]

cod.

cod. || 439 ] ( 39 ) cod. cod. cod. ||||440 431 ]][ cod.|| 440 ] cod. || 439 ]

cod. ||] 439

]

cod. ||] 429 cod. || 439 (post.) ] quasi ] ] cod. cod.|| ||438 441 ] cod.] || 438cod. ] [ ] ] 450 439 cod.(ed. || 439 cod. || 440 (||38 ) A: 19r ,12]- 19r ,15 ll . 823-826). ] 441 || 444 ] ] cod.cod. 444 ] ]cod.[ || cod. ]

]

cod. || 440

[

r r ,1219 - 19 . 823-826). r ,15 (ed. 455 (A: 39 )19A: ,15 - 19r ,18ll& 18v ,2 - 18v ,20 & 18r ,1 - 18r ,5 (ed. ll . 827-854); de hac confu] 444 18v ,1 ]: cod. sione scr. (excod. alio ||cod.) 17 455 r A: 19 ,15 - 19r ,18 & 18v ,2 - 18v ,20 & 18r ,1 - 18r ,5 (ed. ll . 827-854); de hac confu449 (ex alio ] cod.)cod. ] ] post 18v||,1452 : 17 cod. || 452 (l. 452) hab. cod. || 463 ] cod. || 464 ] cod. || 464 (pr.) ] [ ] cod. ||] 464460 cod. ] || 452cod. || 467 ] quasi cod. || 468 ] ] cod. || 452 ]cod. post|| 471 ] || 475 ] cod.] || 464 cod. (pr.) || 475] ] (l. 452) hab. cod. || 463 ] cod. cod. || 464 [ ] ] cod. || 464460cod. ||] 476 cod. || 467 ] quasi cod. || 468 ] cod. || 471 ] cod. || 475] ] cod. || 475 ]

cod. || 476

[

cod.

]

465

] [

465

( 37 ) A: 17v ,20 - 17v ,21 & 19r ,1 - 19r ,11 (ed. ll . 810-822). 426

|| 435

470

cod. || 429

]

cod. || 438

]

cod. || 439

(post.) ] quasi

cod. || 441

]

cod. || 438

]

cod. || 439

]

cod.

( 38 ) A: 19r ,12 - 19r ,15 (ed. ll . 823-826). 444

470

cod. || 444

]

]

cod. || 440

]

]

]

cod. || 431

cod.

]

cod. || 439 ]

] cod. || 440

cod.

( 39 ) A: 19r ,15 - 19r ,18 & 18v ,2 - 18v ,20 & 18r ,1 - 18r ,5 (ed. ll . 827-854); de hac confusione scr. (ex alio cod.) 18v ,1 : 475

449

cod. || 452

]

68

cod. 464 259 69254 64 171 169 169|| 164 69 17164 158 158 159 154] 164 159 154 475

258

255152 263 168 168162 165 173 65 25572 263 152155 155165 162173

cod. || 476 256 6625726466 151 264161 166156 167 174 174 151166 161167 156 56 268 262 60 149 180 146 178 172 150 177160 163157 170 148 70 26067 5826017758 163 160170 157148

[

259 254 69 64 171 158 169 164 159 154 266145 175 179 179153 55 57269 6357 26663 175 145147 147176 153 176 65 72 255 263 152 168 155 162 165 173 216214 214114 208 41 286 42 280 42 110208 11428841 38288 28638 280

480 161 256 257 167 66 264 151 156 174 180 178 166 172 60 149 150 200118 277 51 272 4651 46 122 205123 123279 118 146 50279 27750 272

70

67

260

58

177 163 160 157 170 148

cod. || 467 ]

]

cod. || 452

]

452) 27026856 262 268 60262149 180 178 178(l. 172 60 180 149146 146172 150 hab. cod. || 463 150

cod.

]

cod. || 464

] quasi

cod. || 475

] post cod. || 464

]

cod. || 468 ]

]

]

(pr.) ] cod. || 471

cod. || 475

( 40 )]

( 40 ) ( 41 )

93

Text and translation

9 6 3

7 12 13

14 1 8

right-hand side [ facing it (= this cell) ], opposite to it (= the number). Next you take the two numbers following them;124 you put the small one, which is odd, on the lower side, the large one on the upper side [ facing 16 14 you take the two numbers following them, it ], opposite to 45it. hNext which will always 18 24 39 37 be 3 even, 22 32 and you put the small one on the right-hand side and20 the opposite to it on the left-hand side.i125 Then you 9 large 23 21 one 31 41 30 continue7 placing this35 way the small even numbers alternately on the 15 33 25in 17 43 right and on the left — but beginning on the left — and you place the 40 49 19 29 27 1 10 small odd (numbers) alternately on the top and on the bottom — but 38 28 11 13 47 26 12 beginning on the bottom — until you attain, on the right-hand side, the 36 5 34 small even number which precedes the even number corresponding to the denomination of the side of the square126 [ with an odd number ]127 , this occurring always in the cell completing half of the side. 5

10

15

4

1

15

11

8

1

14

15 10 6 1003 2 10 98

5

94

88

15

84

9

10

2

13

12

7

4 18 9 2616 74 5 73

72

71

31

32

25

83

16

3

6

9

14 85 7 77 2 3811 68

34

66

60

37

24

16

4

12

13

8

14

23

42

47

52

57

46

59

78

87

6

36

2

34

28

5

89

79

61

53

50

43

56

40

22

12

10

15

20

25

14

27

90

21

62

44

55

54

49

39

80

11

29

21

18

11

24

8

8

20

36

58

45

48

51

65

81

93

30

12

23

22

17

7

95

82

64

33

67

35

41

63

19

6

76

27

28

29

30

70

69

75

97

1

99

3

96

7

13

86

17

91

4

26

13

16

19

33

4

32

1

35

3

9

31

92

n=6

n = 10

Once you have written it there, put the next small odd number again 100 2 5 94 88 15 84 9 7one, 14 and 13 53 54 55 56 8 on the left on 10this same98 side, thus the right-hand opposite to it 18 83 6 19 50 16 48 42 20 59 its complement. Put then the preceding small number, namely that cor85 38 to 68 the 34 66 60 37 16 of the60side 41 [28 39 if 34 the 29 denomination 24 5 responding denomination thus of 47 it52 will 57 be 46 the 59 87 the14 side 42 is 6, number 6;4 if 22it is 10, it will be the number 38 25 32 35 43 61 10 89 ]128 in 61the53 upper corner, and opposite in the 50 43 left-hand 56 40 12 62 21 31 36 to 37 it 26 diagonally, 44 3 (lower) right-hand corner, its complement. Put then the small odd num90 62 44 55 54 49 39 11 63 47 33 30 27 40 18 2 ber8 preceding upper right-hand corner, and opposite 36 58 this 45 48number 51 65 in the 93 1 45 15 49 17 23 46 64 124 95

64 33 67 35 41 63 57 and 51 17 52 and 12 84, 11 respectively. 10 9 58 The next in each sequence, thus6 9 and 28,

125 4 This addition, at least, seems necessary. 97 126 The even number preceding the order, thus 92 1 99 3 96 7 13 86 17 91

8 and 12. The expression ‘corresponding to the denomination’ (al-muw¯ afik li-ism) must be the Greek ὁμώνυμος. 127 This gloss may have originally referred to ‘beginning on the bottom’ above. 128 Hardly the author’s since he would (presumably) have put it before, at the first occurrence of ‘denomination’.

450

|| 439

]

]

cod. || 438

cod. || 439

94 ]

465 465 465

cod. || 441 [

]

cod. || 438 ]]]

] cod. || 439 ] [ [[ || 440 ] translation cod. ] cod. || 440 Text and [ cod.

] ]

]

A: 19r ,12 - 19r ,15 (ed. ll . 823-826). cod. || 444

]

cod.

]

455 r A: 19 ,15 - 19r ,18 & 18v ,2 - 18v ,20 & 18r ,1 - 18r ,5 (ed. ll . 827-854); de hac confuv (ex alio 17 470 cod.) 18 ,1 : 470 470

cod. || 452

]

cod. || 452

]

(l. 452) hab. cod. || 463

|| 464460

] quasi

cod. || 475

]

475 475 475

cod. || 476

cod.

]

cod. || 464

]

cod. || 467

]

] post

cod. || 464 [ cod. || 468 ] ]

(pr.) ] ] cod. || 471

cod. || 475

]

]

40))) 40 (((40

[ 56

465

268 262

178 17 172v ,20 60 149 180 150- 17v ,21 ( 37146 ) A:

259 254

69

65

255 263 152 168 155 162 165 173

72

64

171 158 169 164 159 154

426

480

]

|| 435

66

480 161 166 167 156 174 480 264 151

70

67

260

58

269

57

63

266 175 145 179 147 153 176

cod. || 439

177 163 160 157 170 148

( 3838) 288

]

444277 272 50

51]

119 126 201 209

44

276

54

273r 281

202 203 120 210

43

52

49

278

40

37

287

39

45

284

234

92

232 226

96

6

95

449 [

cod. || 441

cod. || 444

]

44

cod. || 431 ]

cod. || 440

]

]

]

cod. || 452

cod.

cod.

]

cod. || 439 ]

]41 ) 41)) (((41

cod. || 440

cod. 18

18 18

cod. || 464

475 309 171 317 158 222 164 219 227 11 69 18 64 98 169 101 159 108 154

]

cod. || 467 ]

310 255 311 263 318 168 220 165 221 173 12 152 97 155 107 162 102 228

cod. || 476 268264 262151 180167 178174 172 94 314 231 217 224 56 66 149166 146156 150 16 13 460161 106 103

r r 40 A: 18 r ,5 - 18r ,7 170 160 157 ((158 40 )) A: 18 ,5230 - 18 ,7 490 91 259260325458969177 171 169 164148 64163 159 154 320 229 233 168 323 93 99 147 477 25857 65 63 72266255175263145 152179168 155153 162176165 477 ]] 173

]

]cod.

]

(l. 452) hab. cod. || 463 150

268 313 262 308 60 225 146 223 172 218 105 100 15 149 10 180 104 178

270

cod. || 438

]

]]]

( 39 ) A: 19 ,15 - 19r ,18 & 18v ,2 - 18v ,20 & 18r ,1 - 18r ,5 (ed. ll . 827-854); de hac confu( 42 ) 53 274 275 48 282 sione scr. (ex alio cod.) 18v ,1 :

215 111 117 212 283 322 316

(post.) ] quasi

cod. || 439

]

46

124 121 206 112 285 485 271

2

cod. || 429

r r A: 286 19 280,12 42 - 19 ,15 (ed. ll . 823-826).

205 200 470 123 118 279

47

& 19r ,1 - 19r ,11 (ed. ll . 810-822).

cod. || 438

]

256 257

110 214 208 114 41

[[[

]

cod.

]

] post

|| 452

cod. || 464

] quasi

cod. || 475

cod. || 464

]

cod. || 468 ]

]

(pr.) ] cod. || 471

cod. || 475

(hinc pag. pag. vacat) vacat) 19rr,18 ,18 -- 19 19rr,19 ,19 (ed. (ed. llll.. 855-858). 855-858). (hinc [ && 19 cod. || 478 ] cod. cod. || 478 ] cod.

( 40 )

21425620825711466 412642881513816128616628016742156 41 A: 19rr ,19174- 19vv ,3 (ed. 859-862). ((41 )) A: 172 60 149 180 146 178 150 19 ,19 - 19 ,3 (ed. 859-862). 279 277 272 123 118 50 51 46 70 67 260 58 177 163 160 157 170 148 480 cod. |||| 482 482 cod. |||| 482 482 cod. |||| 482 482 69 64 171 158 169 164 159 154 ]] 480 cod. ]] cod. ]] cod. 55 2692015720963 442662761754714554179273147281153 176 255 263 152 168 155 162 165) A: 173 19v ,3 - 19v ,17 (ed. ll . 863-877). 42 v 275)38 48286282 ((288 42 A: 19280 ,3 -4219v ,17 (ed. ll . 863-877). 216 110120214210208431145341274 480 161 166 167 156 174 66 264 151 484 ] cod. |||| 484 484 post hoc hoc del. (atramento (atramento rubro) rubro) cod. cod. |||| 485 485 206 285 271 278 112 52 49 40 122 205 200 123 118 279 484 50 277 272] 51 46 cod. ]] post del. 260 58 177 163 160 157 170 148 495 cod. cod. |||| 485 485 cod. |||| 486 486 3947||45485 204111 119117 1262122012832093744287276 54284273 281 cod. || 485 ]] cod. ]] cod. ]] 266 175 145 179 147 153 176 322202316203 6 1209521023443 9253 23227422627596 486 ] cod. || 488 ] cod. || 488 ] add. || 491 282 || 488 486 ] 48cod. ] cod. || 488 ] add. || 491 ]] 114 41 288 38 286 280 42 2185210549100 491 ]278 cod. || 493 ] cod. || 493 ] cod. || 493 1241512110206225112104285223271 40

[

]

]]

( 41 )

44

]

cod. cod. ]] cod. |||| cod. cod. |||| cod. ]

]

95

Text and translation

to it diagonally, in the lower left-hand corner, the number which is its complement. Once you have done that, you place the remaining small even numbers alternately on the right and left sides, beginning on the left, until you attain 4. Then you put opposite to each number its complement. This results in the completion of the right-hand and left-hand cells. You place (then) the hremainingi small odd numbers alternately on the top and on the bottom, beginning on the bottom, and you put opposite to each number its complement, until you reach 3. Once you have done this, you put 3 on the bottom, and opposite to it on the top its complement; 2 on the top, and opposite to it on the bottom its complement; 1 on the bottom, and opposite to it on the top its complement. In this way you will have finished what you wanted.

(Bordered squares of orders n = 4k) (§ 40. Filling the inner square) As to the treatment for the evenly-even and evenly-evenly-odd, it is as follows. You put the two medians and the numbers next to them in the square inside these squares until you complete it [ as we have explained for the odd and evenly-odd ].129 (§ 41. Filling the inner part of the horizontal rows for the square of order 8) If your treatment is for the square of eight, you take the numbers attained.130 Put two consecutive small (numbers) on the top, excluding the corner cell; and you put their complements opposite to them on the bottom. Then, put four consecutive small (numbers) on the bottom and, opposite to them on the top, their complements.131 8

56

55

54

53

13

14

7

59

20

50

16

48

42

19

6

5

24

29

34

39

28

41

60

61

43

35

32

25

38

22

4

3

44

26

37

36

31

21

62

2

18

40

27

30

33

47

63

64

46

15

49

17

23

45

1

58

9

10

11

12

52

51

57

n=8

129

In §§ 4–5 and 38. But, as in § 38 (n. 122), this reference seems superfluous. Thus we shall use the descending sequence from 14 and ascending from 51. 132 131 130 129 17 18 19 125 124 22 11 131 How to fill the 12 vertical rows will be taught later on (§ 43). 130

135

32

106

37

110 116

27

120

24

122

31

9

40

48

96

95

94

93

53

54

47

105 136 38

137 107 7

36

10

99

60

82

88

56

90

59

46

45

64

69

74

79

68

81

100 109 138

8

96

Text and translation

( 42 ) 485

[

]

490

495

( 43 )

]

[

500

( 40 ) A: 18r ,5 - 18r ,7 (hinc pag. vacat) & 19r ,18 - 19r ,19 (ed. ll . 855-858). 477

cod. || 478

]

( 41 ) A: 19r ,19 - 19v ,3 (ed. 859-862). 480

cod. || 482

]

505

cod. || 482

]

( 42 ) A: 19v ,3 - 19v ,17 (ed. ll . 863-877). 270

56

68

259 254

258

65

268 262

72

256 257

55

69

484

64

cod. || 485

486 168 155

] cod. || 488 162 165 173

491166 167 ]156 174 cod. 264 151 161

70

67

260

58

57

63

cod.179|| 147 496153 176 266 510 175 145

177 163 160

216 110 214 208 114 41

122 205 200 123 118 279

288

38

286 280 ex 42 ] corr.

50

277 272

51

204 119 126 201 209

44

276

47

273 281

202 203 120 210

43

53

54

506

274 275

]

48

cod.

cod. || 506 282

52

49

278

40

37

287

39

] 45

284

324

234

92

232 226

96

95

]

]

]

cod. || 486 ]

cod. || 494

cod. || 496

]

cod. || 497

]

cod. || cod. ||

]

cod. || 493

cod. || 494

cod.

]

]

add. || 491

cod. || 493

]

]

]

]

(sic)

]

cod. || 497

]

cod. || 507

46

109 215 111 117 212 283 6

|| 493

]

cod. || 488

]

cod. || 482

]

del. (atramento rubro) cod. || 485

cod. || 485

( 43 ) A: 19v ,17 - 20r ,8 (ed. ll . 878-890).

124 121 206 112 285 271

322 316

] post hoc

cod.170||148 493-4 157

269

2

]

171 158 169 164 159 154

255 263 152 66

cod. || 484

]

60 149 180 146 178 172 150

cod.

]

cod. || 508

(post.) ]

cod. || 506

]

cod. || 506

(sic) in textu, et add. in marg.:

] ]

cod.

|| 507

97

Text and translation

(§ 42. Filling the inner part of the horizontal rows for higher orders) If your treatment is for (an order) higher than eight,132 you have as (quantity of) cells in each row, excepting the corner cells, ten cells, or fourteen cells, or eighteen cells, and so on always by successive additions of 4 [ according to what you have placed in the square of 8 ]133 . Now you are to know that, for any group of four consecutive numbers and four 56 55puts 54 on 53 one 13 14side 7 the first and the last of complementary numbers, 8if one 59 20 50 16 48 42 19 6 the four small numbers and, on this same side, the two middle of the four 5 24 29 to 34 them 39 28 on 41 the 60 other side the numbers large ones, and one puts opposite which are their complements, will4 be equalized.134 Thus if, 61 43 the 35 two 32 25sides 38 22 dismissing the corner cells, you will equalize four 3 there 44 26 remain 37 36 31ten 21 cells, 62 on the top with the four 2 facing them on the bottom by means of four 18 40 27 30 33 47 63 (consecutive) numbers of64those you have reached after filling the inner 46 15 49 17 23 45 1 135 Likewise, if fourteen (cells) remain, square and their complements. 58 9 10 11 12 52 51 57 you will equalize eight with eight. And always like that until you are left with six (empty) cells on the top and six on the bottom. 12

132 131 130 129

17

18

19

125 124

22

11 10

135

32

106

37

110 116

27

120

24

122

31

9

40

48

96

95

94

93

53

54

47

105 136

137 107

99

60

82

88

56

90

59

46

38

8

7

36

45

64

69

74

79

68

81

100 109 138

6

111 101

83

75

72

65

78

62

44

34

139

33

5

140 112

43

84

66

77

76

71

61

102

141

42

58

80

67

70

73

87

103 115

3

117 104

30

86

63

57

89

55

85

41

28

2

26

98

49

50

51

52

92

91

97

119 143

144 114

39

108

35

29

118

25

121

23

113

134

14

15

16

128 127 126

20

21

123 133

13

(216 ; D, fol. 85v ; A, p. 314)

4 142

1

n = 12

Once you have done that, put, of the small numbers you have reached, two consecutive numbers on the top and, opposite to them on the bottom, their complements. Then put, of the small numbers you have reached, 132

Considering thus horizontal rows for both orders n = 8t, t 6= 1, and n = 8t + 4. Misplaced gloss (see below). 134 This ‘neutral’ arrangement has been already mentioned for a sequence of even numbers (§ 20, text corrupt), and used several times since (§§ 31, 33, 34). 135 For order 12, we begin with 22 and 123, respectively. The numbers left will be consecutive since that is what we have placed so far. 133

68

55

495

98

Text and translation

( 43 )

]

[

500

505

510

19

( 44 )

( 40 ) A: 18r ,5 - 18r ,7 (hinc pag. vacat) & 19r ,18 - 19r ,19 (ed. ll . 855-858). 477

515

cod. || 478

]

( 41 ) A: 19r ,19 - 19v ,3 (ed. 859-862). 480

cod. || 482

]

cod. || 482

]

( 42 ) A: 19v ,3 - 19v ,17 (ed. ll . 863-877). 484

cod. || 484

]

cod. || 485 486

cod. || 488

]

491 520

]

] post hoc

cod. || 493-4

cod. || 496

]

] corr. ex

cod.

]

cod. || 488

]

]

268 262

506

cod. || 506

]

60 149 180 146 178 172 150

259 254

69

65

] 173 255 263 152 168 155 162 165

72

64

171 158 169 164 159 154

256 257

66

264 151 161 166 167 156 174

70

67

260

58

269

57

63

266 175 145 179 147 153 176

cod. || 508

(post.) ] ]

38

286 280

205 200 123 118 279

50

277 272

51

42 46

cod. || 506

cod. || 494

cod. || 497

]

cod. || 506

] cod. || cod. ||

]

cod. || 493

cod.

]

]

(sic)

]

cod. || 497

]

cod. || 507 || 507

( 45 )

525

288

]

]

cod.

177 163 160 157 170 148

110 214 208 114 41

]

(sic) in textu, et add. in marg.:

]

]

]

add. || 491

cod. || 494

cod. || 496

( 43 ) A: 19v ,17 - 20r ,8 (ed. ll . 878-890). 56

]

cod. || 486

cod. || 493

]

cod. || 482

]

del. (atramento rubro) cod. || 485

cod. || 485

cod. || 493

]

cod.

]

]

Text and translation

99

four consecutive ones on the bottom and, opposite to them on the top, their complements.136 (§ 43. Filling the corners and the vertical rows) Once you have done that, put the small number you have reached — which [ is always even ]137 , for this succession, corresponds to the denomination of the side — in the upper left-hand corner cell and, opposite to it diagonally, in the right-hand corner, its complement. Put then the subsequent small number, which is odd, in the upper right-hand corner and, opposite to it diagonally, in the left-hand corner, its complement. Put then the subsequent small number on the right-hand side and its complement on the left-hand side, and the subsequent small number on the left-hand side and its complement on the right-hand side. Once you have done this, you will have equalized these two sides as well.138 What you are left with (as empty cells) in these two (vertical rows) is four cells facing four,139 or eight facing eight, or twelve facing twelve, and so on with successive additions of 4. You will then equalize each group of four cells with their four opposite as we have explained,140 that is, putting (for each group) on one side the first and the last small and the two large middle, and their complements opposite to them on the other side, until you have finished with all (empty cells).

136

The above gloss fits here. What comes next makes this information superfluous. 138 Indeed, the four cells thus filled in each vertical row make their sum due. 139 Case n = 8, thus here completed. 140 § 42. See also n. 134. 137

(A, p. 312 ; a21, p. 165)

100

Text and translation

(A, p. 312 ; a21, p. 165)

(A, p. 312 ; a23, p. 165)

(A, p. 312 ; a22, p. 165)

(A, p. p. 312 312 ;; a23, a23, p. p. 165) 165) (A,

(A, p. p. 312 312 ;; a22, (A, a22, p. p. 165) 165)

(A, p. 312 ; a23, p. 165)

(A, p. 312 ; a22, p. 165)

(A, p. 312 ; a24, p. 167 ; D, fol. 63v ) (A, p. 312 ; a25, p. 167 ; D, fol. 63v ) (A, p. 312 ; a21, p. 165) (A, p. 312 ; a25, p. 167 ; D, fol. 63v )

(A, p. p. 312 312 ;; a24, a24, p. p. 167 167 ;; D, D, fol. fol. 63 63vv )) (A, (A, p. p. 312 312 ;; a25, a25, p. p. 167 167 ;; D, D, fol. fol. 63 63vv)) (A, (A, p. p. 312 312 ;; a21, (A, a21, p. p. 165) 165) (A, p. p. 312 312 ;; a25, a25, p. p. 167 167 ;; D, D, fol. fol. 63 63vv)) (A,

(A, p. 312 ; a24, p. 167 ; D, fol. 63v ) (A, p. 312 ; a25, p. 167 ; D, fol. 63v ) (A, p. 312 ; a21, p. 165)

(A, p. 312 ; a25, p. 167 ; D, fol. 63v )

(A, p. 312 ; a23, p. 165)

(A, p. p. 312 312 ;; a23, (A, a23, p. p. 165) 165)

(4 ⇥ 4) A : 25r (ed. p. 312). tit. :

(A, p. 312 ; a23, p. 165) (A, p. 312 ; a24, p. 167 ; D, fol. 63v )

Eandem fig. præb. D, fol. 63 v , cum tit. : 270

56

68

259 254

258

65

55

268 262

72

r (4 (4 ⇥ ⇥ 4) 4) A A :: 25 25r (ed. (ed. p. p. 312). 312). tit. :

60 149 180 tit. 146 : 178 172 150

69 64 171 158 169 164 159 154 v Eandem præb. r r D, fol. 63 , (4 Afig. :A25 p. p. 312). (10⇥⇥4)10) : 25(ed. (ed. 312). 255 263 152 168 155 162 165 173

256 257

66

70

67

260

269

57

63

58

177

231 217 106 103 224

94

202 203 120 210

43

124 121 206 112 285 109 215 111 117 212 283

312

6

95

10

225

313 308

15

11

309 317

98 97

18

310 311

12

318

16

314

4

13

50

277 272

46

44

322 316

(10 : 25 (ed. p. 312). (6 ⇥⇥6) A (ed. p.p.312). (4 4)10) A :A : 25 25 (ed. 312). 38 286 280 42 tit. : tit.

288

51

204 119 126 201 209

2

(A, p. 312 ; a24, p. 167 ; D, fol. 63v )

p. 312). Eandem fig. præb. D, fol. 63 v ,excum. tit. : (1, 6) ] || (4, 6) ] corr. tit. 160 : 157 170 148 163

266 175 145 179 147 153 176r vr

122 205 200 123 118 279

14

cum tit. :

tit. :: tit.166 264 151 161 (4 ⇥ 4)167 A :156 25r174(ed.

216 110 214 208 114 41

324

v (A, p. p. 312 312 ;; a24, (A, a24, p. p. 167 167 ;; D, D, fol. fol. 63 63v))

6)A] : 25v|| (ed. ] corr. ex . r (4, (8 ⇥(1, 8)10) p. 6) 312). 273: 25 281 (ed. p. 312). 54 A tit. :: tit. 53 (6 ⇥2746) 275 A : 48 25v 282 (ed. p. 312). Alteram fig. præb. cod. 271 49] 278 ||40 tit. :52 (1, 6) (4, 6) ] corr. ex . r (ed. p. 313). (12 ⇥ 12) A : 26 37 287 39 45 284 (8 ⇥ 8) A : 25vv (ed. p. 312). (6 tit.⇥: 6) A : 25 (ed. p. 312). 234 92 232 226 96 tit. tit. :: Eandem fig.105 præb. D, fol. 63 v , cum tit. : 223 218 104 100 cod. Alteram fig. præb. v (ed. p. 312). (8 ⇥ 8) A : 25 222 101 108 219 227 (12 A : 26r (ed. p. 313). tit. :⇥ 12) 22112) 107 220(10, 102 ] 228|| (1, 9) ] || (8, 8) ] tit. : Alteram fig. præb. cod.

(10 47 ⇥ 276

(4, 6) ]

||

(6, 5) ]

||v

(10, 5) ]

||

||

(10, 8) ]

(3, 4) ]

||

||

||

(12, 8) ] (6, 1) ]

||

(3, 7) ] (7, 1) ]

|| ||

16

45

14

7 7 1515 3333 2525 1717 3535 4343 18

24

39

37

3

22

32

25

17

35

43

29

27

1

10

13

47

26

12

4040 4949 1919 2929 2727 1 1 1010

101

Text and translation 20 9 23 21 31 41 30

3838 2828 1111 1313 4747 2626 1212 3636

7

15

33

40

49

19

38

28

11

5 5

3434

36

9 9 7 7 1414 4 4

5 5 1010 1515 4 4

6 6 1212 1 1 1515

1111 8 8 1 1 1414

5

34

1 1 1212 1313 8 8 1515 6 6 3 3 1010

3 3 1313 8 8 1010 9

7

14

4

2 2 1313 1212 7 75

10

15

4

4 4 9 9 1616 51 5 12

13

8

1616 2 2 1111 5 5

6

12

1

15

1616 3 3 6 6 9 911

8

1

14

1511 6 1414 7 7 2 2 11

3

10

3

13

8

10

2

13

12

7

4

9

16

5

16

2

11

5

16

3

6

9

14

7

2

11

n=4

6 6 3636 2 2 3434 2828 5 5

1010 1515 2020 2525 1414 2727 6

36

2

34

28

5

10

15

20

25

14

27

29

21

18

11

24

8

30

12

23

22

17

7

4

26

13

16

19

33

32

1

35

3

9

31

2929 2121 1818 1111 2424 8 8 3030 1212 2323 2222 1717 7 7

4 4 2626 1313 1616 1919 3333 3232 1 1 3535 3 3 9 9 3131

n= 6 100 2 2 9898 5 5 9494 8888 1515 1010 100 8484 9 9 1818

5

83 94 8388

8585

3838 6868183434 6666 6060 3737

1616

1414

4242 4747855252 5757 38 4646 68 5959 34

8989

15

84

9

88

87 66 8760

15

84

83 9

31

37 32

25

16 83

42 5656 47 4040 52 6161 5353145050 4343

34

12 57 1246 66 60

59 37

24

87 16

9090

61 49 53 39 50 55 54 6262 444489 1455 2354 4249 47 39 52

43 1156 11 57 46

40 59

78

12 87

8 8

62 5151 44 55 3636 585890 894545794848 61 53 6565 50

54 9349 93 43 56

39 40

22

11 12

9595

8 36 58 63 45 6464 3333906767213535624141 44 63 55

48 51 6 54 6 49

65 39

80

93 11

35 41 97 48 9751

63 65

81

6 93

33 8686 67 1717 35 9141 91 9292 1 1 9999 3 3 95969682 7 7641313

63

19

97 6

4 4

10

100

2

98

10

100

2

98

5

18

26

74

73

72

85

95 8

77

20

38

64 36

68

33 58

67 45

94

71

4

92 4

1 76

99 27

3 28

96 29

7 30

13 70

86 69

17 75

91 97

92

1

99

3

96

7

13

86

17

91

n = 10 7

14

13

53

54

55

56

8

6

19

50

16

48

42

20

59

60

41

28

39

34

29

24

5

4

22

38

25

32

35

43

61

62

21

31

36

37

26

44

3

63

47

33

30

27

40

18

2

102

Text and translation

(A, p. 312 ; pr´ec´edente retourn´ee, selon a26 ; D -)

(A, p. 312)

( ; D, fol. 63v ; A, p. 313 (ver.) ; a27, p. 169)

(A, p. 312 ; pr´ec´edente retourn´ee, selon a26 ; D -)

(4 ⇥ 4) A : 25r (ed. p. 312). tit. : Eandem fig. præb. D, fol. 63 v , cum tit. : (4 ⇥ 4) A : 25r (ed. p. 312). tit. : (10 ⇥ 10) A : 25r (ed. p. 312). tit. : (4, 6) ] corr. ex

||

(1, 6) ]

.

(6 ⇥ 6) A : 25v (ed. p. 312). tit. : 56 68

268 262

259 254

69

(8 ⇥ 8) A : 25v (ed. p. 312). tit. :

60 149 180 146 178 172 150 64

171 158 169 164 159 154

Alteram fig. præb. cod.

255 263 152 168 155 162 165 173 (12 ⇥ 12) A : 26r (ed. 256 257 66 264 151 161 166 167 156 174 65

55

70

67

260

58

57

63

266 175 145 179 147 153 176

177 163 160 157 170 148

Eandem fig. præb. D, fol. 63 v , cum tit. :

110 214 208 114 41

288

38

205 200 123 118 279

50

(10, 12)51] 277 272

119 126 201 209

44

276

47 6)54 (4, ]

202 203 120 210

43

53

274 275

286 280

42 46||

48

282

49

278

40

(11, 1) ]

.

52

215 111 117 212 283

37

287

39

45

284

234

92

232 226

96

6

95

10

225 104 223 218 105 100

313 308

15

11

309 317

18

98

222 101 108 219 227

||

(1, 9) ]

273|| 281 (6, 5) ]

124 121 206 112 285 271

322 316

p. 313).

tit. :

269

2 14

72

||

(8, 8) ]

(10, 5) ]

||

||

(10, 8) ]

(3, 4) ]

||

||

||

(12, 8) ] (6, 1) ]

||

(3, 7) ] (7, 1) ]

|| ||

103

Text and translation

8

56

55

54

53

13

14

7

8 59

56 20

55 50

54 16

53 48

13 42

14 19

76

59 5

20 24

50 29

16 34

48 39

42 28

19 41

6 60

5 61

24 43

29 35

34 32

39 25

28 38

41 22

60 4

61 3

43 44

35 26

32 37

25 36

38 31

22 21

4 62

32

44 18

26 40

37 27

36 30

31 33

21 47

62 63

2 64

18 46

40 15

27 49

30 17

33 23

47 45

63 1

64 58

46 9

15 10

49 11

17 12

23 52

45 51

1 57

58

9

10

11

n = 8141

12

52

51

57

17

18

19

125 124

22

11

12 132 17 135 32 131 106 130 37 129 110 116

18 27

19 125 120 24 124 122

22 31

11 10

135 9

12

132 131 130 129

32 40

106 48

37 96

110 95 116 94

27 93

120 53

24 54

122 31 136 10 47 105

9 107 40 137

48 99

96 60

95 82

94 88

93 56

53 90

54 59

47 46

137 7 107 36

99 45

60 64

82 69

88 74

56 79

90 68

59 81

46 109 38 138 8 100

36 101 45 111

64 83

69 75

74 72

79 65

68 78

81 62

100 44 109 34 138 139

6 111 140 112 101 43

83 84

75 66

72 77

65 76

78 71

62 61

44 102

140 141 112 30

43 42

84 58

66 80

77 67

76 70

71 73

61 87

102 33 103 115

141 30 104 42 3 117

58 86

80 63

67 57

70 89

73 55

87 85

103 4 41 115 28 142

86 49

63 50

57 51

89 52

55 92

85 91

41 97

28 142 119 143

52 118

92 25

76

32

117 26 104 98

34 33

139 5 54

2 114 26 144

98 39

49 108

50 35

51 29

91 121

97 23

119 113 143 1

144 134 114 13

39 14

108 15

35 16

29 118 25 121 128 127 126 20

23 21

113 1 123 133

134

14

15

16

128 127 126

21

123 133

13

(216 ; D, fol. 85 ; A, p. 314) (216 ; D, fol. 85v ; A, p. 314) v

141

105 38 136 8

20

n = 12

In MS. A this figure is rotated (thus 8, 59, . . . , 58 at the top).

104

Text and translation

510 510

((44 44))

515 515

520 520

270 56 268 262 262 56 268 270

rr r r,9 - 20rrrr,18 (ed. ll . 44 A: 20 44))A: A: 20 20rrr,9 ,9 -20 20,18 ,18(ed. (ed. 891-901); 891-901); D D : 99 99rr,7 ,7 -99 99rr,13. ,13. (((44 44))) A: A: 20 20rr,9 ,9 ---20 20rr,18 ,18 (ed. (ed. llllll... 891-901); 891-901); D D::: 99 99rr,7 ,7 ---99 99rr,13. ,13. ((44 525 in summa pag. praeb. D tit.: insumma summa pag. praeb. D D tit.: tit.: in pag. praeb. 180 178 172 172 60 525 149 180 146 178 150 60 149 146 150 in summa pag. praeb. D D tit.: tit.: in summa pag. praeb.

259 254 254 69 171 158 169 164 164 159 68 259 69 64 64 171 158 169 159 154 154 68

post quae: quae: ,, post post quae: quae: ,, post

((45 45))

]]

(.(....).) (. .. .) .) (. 256 257 257 66 264 151 166 167 167 156 174 66 264 151 161 161 156 174 511 511166 A, D |||| 511 511 256 511 ]] A, D 511 A, D |||| 511 511 511 ]] A, D 270 268 262 180 178 149 146 150 260 177 163 170 7056 67 5860 160 157 27070 268 26258 180 178 172 5667 60177 149 150 260 163 170 160 157 D 511 D172 ||148 511 A |||| 512 512 D |||| 512-3 512-3 om. A A ]]]146 D ||||148 511 ]]] A ]] D ...... ]] om. D153 511 A |||| 512 512 D |||| 512-3 512-3 om. A A ]] 169 D ||||176 511 ]] A ]] r D .. .. .. ]] om. 259 254 171 164 158 159 154 269 26664 175 179 176 5568 57 6369 145 147 r159 rr rr 25957 25463 171 169 164 68269 69266 64175 158 154 179 55 145 147 153 r r (||(44 )) A: llll513 ..513 891-901); ,7 ,13. 512 D 512 D|||| AD ||99513 513 A, 44 A: 20 20 ,9 ,9 20 ,18 ,18 (ed. (ed. 891-901);]] D D:: 99 99AD ,7--||99 ,13. |||| 512 ]]] --20 D ]] A, ||288512 512 D |||| 513 513 AD |||| 513 513 A, 258 255 263 168 165 65 152 155 162 216 21472 208 286 280 110 114 41 38 42173 ||288 ]] 42 D ]] AD ]] A, 258 255 26341 16838 165 173 65214 72208 152 155 162 216 286 280 110 114 [ [ in ,, post in summa summa pag. praeb. D tit.:D post quae: quae: D |||| 513 ]] tit.: D ||pag. 513praeb. D |||| 513 513 (post.)]] om. om. A A |||| 513 513 om. D D |||| 514 514 D 513 ]D (post.) ]] om. 256 257 264 166 167 205 200 279 277 272 66 151 161 122 205 123 118 50 51156 46174 D 513 ] D |||| 513 513 (post.) om. A A |||| 513 513 om. D D |||| 514 514 256 257 264 166 16751 174 66 151 161 156 200 279 277 272 122 123 118 50 46 D |||| 513 D ]] om. ]] om. r r ] r (post.) r ( 44 ) A: 20 ,9 20 ,18 (ed. ll . 891-901); D : 99 ,7 99 ,13. 530 530 A 514 510 A281 514 A, D |||| 514 514 AD |||| 514 514 510 ]]] 54157273 A |||||| 514 ]] A, D ]] AD ]] 204 119 201 20958 276 273 281 119 126 44177 47 260 163 170 70 67 160 148 260 177 16347 170 204 209 276 70 67201 5844 126 A 514praeb. D ]]tit.: A, A, D |||| 514 514 AD |||| 514 514 summa pag. , post ]]quae: ]]16054in157 A |||| 148 514 D AD ]] (.(....).) D ||||274 514-5 202 20357 210 275 282 120 43175 53 48153 269 266 179 176 55 63 145 147 D 514-5 A |||| 515-6 post (l. ]515-6 203 275 282 269 26643 17553 179 176 [[ ]]] 120 48 55202 57 63210 145 147 153 D ||274 514-5 A ...... ]] post (l. ] ((44 44)) D511 514-5 A 515-6 . post (l. D |||| 514-5 ]] A |||| 515-6 .. ..A, .A, ]] post (l. ] D || 511 511 ] D || 511 (. . .) 206 28541 271 278 124 121 112 5238 49286 40 216 214 208 288 280 110 114 42 206 285 271 278 216 214 208 28852 28040 124 121 112 110 114 41 3849 42 513-4) A, om. (homoeotel.) D |||| 515 515 pr. scr. scr. et et del. del. A A |||| 517-8 517-8 513-4) A,286 om. (homoeotel.) 513-4) A, om. (homoeotel.) D ]] pr. ] ] 513-4) A, om. (homoeotel.) D || 515 ] pr. scr. et del. A || 517-8 ] A, D || 511 511 513-4) om. (homoeotel.) ] del. ]]A, D ||284 511 ]] D || 515 A ]]pr. scr. D |||| 512-3 ...... ]] om. 215 212 283 287 109 111 117 3750 39272 45 205 200 279 277 122 123 118 51 D || 46 511 A |||| 512 512 D et 512-3A || 517-8 om. A A 215 212 283 287 284 109 111 117 39 205 200 27937 277 27245 122 123 118 50 51 46 A |||| 517 517 . . . om. D || 518 ......... ] ]]] || 511 ]] om. D ] A || 512 ] DA|| 512-3 ] om. A D || [518 [ . . . ] A || 517 ] om. D || 518 324 322 316 234 232 226 2 6 95 92 96 276 47 ]]281 D ]] AD ]] D || 518 A, .232 . .54 ]281 A || ||517 ] om. 324204 322126 316201 234|| 226273 92 2042 119 2016 209 20995 44 27396 119 126 44 47 54 ||276512 512 D |||| 513 513 AD || 513 513 A, || 512 D || 513] om. ] D || AD 518|| 513 AA,|||| 518-9 518-9 ....]..... ] om. D || 518 ]]] A 313 308 22543 223 218 14 313 15 10 104 10548 100 202 203 210 274 275 282 120 53 308 225 223 218 14 10 104 105 100 202 20315 210 274 275 282 120 43 53 48 . . . ] om. D || 518 ] A || 518-9 D ]] ] D ||||513513 om. . . 513 . D ||(post.) 518(post.) || ]]518-9 D ||||A513 513 D] ||om. 513 (post.) ]] om. om. A A] ||||] 513 513 D ||A514 om. D D |||| 514 514 D D om. |||||| 519 519 ] 52 10849 219278 A 519 om. D D |||| 519 519] om. A || 513 A |||| 520 520 ]]515 A227 ||40 ]] om. ]] A ]] 312 11 309 317 222 227 11 18 98285 206 271 124 121 112 515 312 309 317 222 219]278 18 98 101 108 49 206 285 271 124 121 112 52 40A ]]101 A || 519 ] om. D || 519 ] A || 520 ] || 514 ] A, D || 514 ] AD || 514 ] A || 519 ] om. D || 519 ] A || 520 ] ] A || 514 ] A, D || 514 ] AD || 514 ] ] A || 514 ] A, D || 514 ] AD || 514 ] 535 535 D ||||520-1 520-1 D||221 520-1 A, D |||| 523 523 D |||| 523 523 310 311 318 220 228 12 97283 107 10245 D ]]] A, D ]] D ]] 215 212 287 284 109 111 117 37 310 311 318 220 22139 228 12 107 102 215 21297 283 284 109 111 117 37 39 45 || 514-5 ]]] A || 515-6 D287 520-1 A, D .||||. . 523 523 . . .] post Dpost 523 D |||| D520-1 A, D ]] D |||| (l.523 ]] D || 514-5 ] A || 515-6 ] D || 514-5 ] A || 515-6 . . . ] post (l. D ||||106 523 (pr.) ]] A ||||||523 D234 523 (pr.) 523 ] hab. hab. in textu, textu, (sic) suprascr. suprascr. D. D.(l. 314 23195 217 224 162213 13 103 94 324 322 316 232 226 92 96 D ||106 523 (pr.) A in (sic) 513-4) A, om. 515 ] pr.] scr. et del. || 517-8 314 217 224 103 94 32416 322 31644 66231 232 226 95 92 96 ](homoeotel.) D 234 523 (pr.) AD 523 hab. in A textu, (sic) suprascr. suprascr. D. D. D |||| 523 (pr.) ]] A |||| 523 ]] hab. in textu, (sic) 513-4) A, om. (homoeotel.) D ]] A || 517 pr. 517-8 323 32010 229 233 230 91 93 99 513-4) A, om. (homoeotel.) D |||| 515 515 pr. scr. scr. et et del. A ||||||518 517-8 313 225 223 218 104 105 .230 . .100 ] ]del. om. A D 233 1114 31333308 3089915 22591 22393 21899 14323 15320 10229 104 105 100 258 65 255 263 263 152 168 155 165 173 173 65 72 72 255 152 168 155 162 162 165 258

312 .... 219 ]] 312 11 309 317 317 98 222 101 219 227 227 11 18 18 309 98 222 101 .108 .108

]

310 310 311 311 12 318 97 220 221 221 102 228. . . 12 318 97 107 107 220 102 228 16 314 16 13 13 314 11

323 323

33

99

44

520 520

D || 520-1. . .

231 103 224 231 217 217 106 224 94 94 || 523 ]]106 D103 A 519 A |||| (pr.) 519]

320 320 229 229 91 233 93 230 91 233 93 99 99 230

D D |||| 520-1 520-1

] om. D || 518

20 20

...

] om. D || 519

A || 519 ]

]] om. D || 518 om. A, D ||D518 || 523

A || 523 ]] om. om.

]]

A A |||| ]517 517 ]

]

hab. D D ||]|| 519 519 in textu,

A, A,

D D |||| 523 523

]] om. om. D D |||| 518 518

A || 518-9

A || 520

]

D || ]523

]

]

A A |||| 518-9 518-9

(sic)]]suprascr. D.A A |||| 520 520

]]

D D |||| 523 523

]] ]]

105

Text and translation

(Composite even-order squares)142 (§ 44. Composite squares for orders divisible by 4)143 As for the division of the even (order squares), it proceeds as follows. One draws the main square, which arises from the multiplication of an even number by itself; then its inner part is divided into squares all equal.144 For instance, it is divided into four squares with the side of each equal to half the side of the main square;145 or it is divided into nine squares with the side of each equal to a third of the side of the main square;146 or it is divided into sixteen squares with the side of each equal to a fourth of the side of the main square;147 hor it is divided into twentyfive squares with the side of each equal to a fifth of the side of the main square.i148 (218 ; D, fol. 97r )

(219 ; D, fol. 97v )

(232 ; D, fol. 107r )

Or the main square is divided into unequal squares.149 For instance, some of these (subsquares) will have the size 8 by 8 and others the size 6 by 6 and others the size 4 by 4.150 Then you will arrange the numbers from 1 to the end of their quantity in all the cells of the main square in such manner that in the (main) square the sums are everywhere the same, and that in each inner square consid142

Subject of §§ 44–54. All subsquares must be of even order and filled with pairs of complements (filling described in § 48). In our transcription of the examples (pp. 127–161) the squares will have the left to right orientation. 143 Thus of the form n = 4k, k ≥ 2 (that is, both orders n = 8t and n = 8t + 4, see § 36). Even orders divisible by 2 only will be considered in the next paragraph. 144 ‘Squares having equal sides’, in the text; perhaps translating ἰσόπλευρα τετράγωνα (although τετράγωνον has mostly our sense of ‘square’). 145 Main order n = 4t, t ≥ 2. Examples given for n = 12 (p. 127) and n = 16 (p. 131). 146 Main order n = 6t, t even. Example for n = 12 (p. 129). 147 Main order n = 8t, t ≥ 2. Example for n = 16 (p. 133). 148 Main order n = 10t, t even. We have added this last case, omitted in the two manuscripts; for the text constructs one example of such a square (p. 137, n = 20). 149 ‘Squares having unequal sides’, in the text; perhaps translating ἀνισόπλευρα τετράγωνα. 150 Examples below, pp. 143 (n = 16), 147 & 149 (n = 20).

520

106

268 262

56

259 254 65

72

Text and translation

( 44 ) A: 20rrr ,9 - 20rrr ,18 (ed. ll . 891-901); D : 99rrr ,7 - 99rrr ,13. 44)) A: A: 20 20 ,9 ,9 -- 20 20 ,18 ,18 (ed. (ed. llll.. 891-901); 891-901); D D:: 99 99 ,7 ,7 -- 99 99 ,13. ,13. ((44 525 525 in summa pag. praeb. D tit.: 172 150 60 149 180 146 178 pag. in summa summa praeb. D D tit.: tit.: in pag. praeb.

]]

69 64 171 158 169 164 159 154 ( 44 ) A: 20rrr ,9 - 20rrr ,18 (ed. ll . 891-901); D : 44)155 ) A: A:16220 20165,9 ,9 -173 - 20 20 ,18 ,18 (ed. (ed. llll.. 891-901); 891-901); D D:: 255 263 152 ((168 44 525 525

in summa pag.150 praeb. D tit.: 268 262 180 178 156 172 56 257 60 151 149 146 167 525 in 256 166 174 66 264 161 in summa pag. praeb. D D tit.: tit.: 511summa pag. praeb.

((45 45))

, post quae: post quae: quae: ,, post

99rrr ,7 - 99rrr ,13. 99 ,7 -- 99 99 ,13. ,13. 99 ,7

(. . .) (. .) (((.45 )) .. .) 45 45 ) D ||(511

, post quae: post quae: quae: ,, post

A, A, D |||| 511 511 A, D ] D || 511 ] ] D || 512-3 ... ] om. A D 511 D |||| 512-3 512-3 om. A ]] 169 D ||||176 511 ]] ]] D .. .. .. ]] om. (. .A .) 68 154 255 26364 168158 165159 173 65259 72254 152171 155 162164 r159 25957 254 171 169 164 68269 69266 64175 158 154 rr,18 (ed. ll . 891-901); D : 99rr,7 - 99rr,13. 179 63 69 145 147 153 (. ..A, .) (. .) 44)) A: A: 20 20r,9 ,9 20 ,18 (ed. . 891-901);] D : 99AD ,7 -||99513 ,13. ||((44 512 ] --20 D ||ll513 ] || 512 ] D || 513 ] AD || 513 ] A, 168 25665 25772 264263 166 167162 66255 151 161 156 ||288 512 ] 174 D || 513 ] AD || 513 ] A, 511 ] A, D || 511 258 255 263 168 165 173 65214 72208 152 155 162 286 280165 110 114 41152 38155 42173 [,[, post 511 A, D |||| 511 511 in summa pag. praeb. tit.:] 511 A, in summa pag. praeb. D post quae: quae: D ||156 513 ]D tit.:]D || 513 (post.) ] om. A || 513 ] om. D D || 514 270 268 180 178 172 56 60 149 146 150 270 268 262 180 178 172 56 60 149 146 150 260262 177151 163 170 70256 67257 58264 160166 157 148 167 174 66 D || 513 ] D || 513 (post.) ] om. A || 513 ] om. D || 514 256 257 264 166 167 174 66 151 161 156 205 200 279 277 272 123 118 50161 51 46 || 513 ]] D || (post.) ]D om.|| 512-3 A || 513 . . . ] om. D || 514 ] DD || 511 A 513 || 512 ] ] om. A D 159 511 A |||| 512 512 D |||| 512-3 512-3 om. A A 530 D ||||176 511 ]] ] A ]] 514 D .. .. .. || 514 ]] om. 530 171 158]] 169 164 68 69 64 154 r159 rr rr rr ]160 A ||-148 514 A, D: ||99 ] AD ] 259 254 171 169 164 68 69 64175 158 154 269259 26658 179 57254 63260 145 147 153 163 170 70 67 160 157 r,9 260 177 163 170 209 276 273 281 70 67201 58 157 148 119 126 44177 47 54 ( 44 ) A: 20 20 ,18 (ed. ll . 891-901); D ,7 99 ,13. 44]]) A: 20 20 ,18 (ed. . 891-901); ,7 -||99513 ,13. ] A,9 514 A, D :||||99 514 AD] |||| 514 514 ||(512 ]|||| -514 D ] ||ll513 ]D AD A,. .) A 514 AD ]] (. (. . .) || 145 512 D] || 513A, AD |||| 513 513 ] A, 258 263 168 165 173 65 72 152 155 162 ||288 512 ]]282 ]] AD ]] A, 258 255 263 168 165 173 [ ] D || 513 65 72 152 155 162 21457 208255 286147 280 110 114266 41175 38514-5 42176 269 179 55 63 153 [,[, post ] 203 275 269 266 175 179 176 120 43 53 48pag. 55202 57 63210 145 147 153 Din ||274 A || 515-6 ... ] post (l. [ ] summa praeb. D tit.: quae: in summa pag. praeb. D tit.: post quae: D || 514-5 ] A || 515-6 . . . ] post (l. [ D || 513 ] D || 513 (post.) ] om. A || 513 ] om. D || 514 D || 514-5 ] A || 515-6 . . . ] post (l. 511 ] A, D || 511 511 ] A, D || 256 257 264 166 167 174 66 151 161 156 D || 513 ] D || 513 (post.) ] om. A || 513 ] om. D || 514 256 257 264 166 167 174 66 151 161 156 205 200 279 277 272 123 118 50 51 46 214 208 288 286 280 110 114 41 38 42 || 280 513 ] 513 ] (post.) om. etAdel. || 513 ] om. D || 514511 206 208 285 41 271 28852 3849D 278 216 124110 121214 112114 40 42 513-4) A,286 om. (homoeotel.) DD|| ||515 pr.] scr. A || 517-8 530 ] 530 513-4) A, om. (homoeotel.) D || 515 pr. scr. scr. et del. A ||||AD 517-8 ]160 A 514 ] || 515 A, D || 514 ]] A || 514 260 177 163 170 70 67 58 160 513-4) om. (homoeotel.) ]] D del. 517-8 D ||||||148 511 ]]]D A ]]pr. D || .....|| ]] ]]om. A 260 177 163 170 201123 209118 276 273 281 70 67 58 157 148 119 126 44 47]277 54157 205 200 279 272 530 51 ]A, D 511 A |||| 512 512 D et || 512-3 512-3 A 215 212 283 287 284 111 117 37 50 39 45A 205 200 279 272 122 123 118 50 51 46 A 514 A, 514 AD 514 ] 277 || 46 514 ] A, D |||| 514 ]] AD ||. 514 ] om. (.(....).) . . . ] A || 517 ] om. D || 518 [ 269 266 175 179 176 55 57 63 145 147 153 [ . . . ] A || 517 ] om. D || 518 [ ] 202 203 210 274 275 282 269 266 175 179 176 120 43 53 48 55 57 63 145 147 153 D || 514-5 ] A || 515-6 . . . ] post (l. 201 209 276 273 281 119 126 44 47 54 || 512 ] D || 513 ] AD || 513 ] A, . . . ] A || 517 ] om. D || 518 [ ] 322126 316 2016 20995 44 234|| 232 54 226 27396] 281 2042 119 27692 47 513 515-6 A, D511 514-5 ] D ||A A 515-6 . post ] (l. D ||||512 514-5 ..A, .A, ]] post ] D ||] 518 AD || ..513 ]] ]||||om. 511 511 D ||||(l. 511 . . . ][ ] A || 518-9 D 216 214 208 286 110 114 41 38 42 206 28543 271288 278280 216 214 208 288 286 280 124 121 112 52274 49275 40282 110 114 41 38 42 202 203 53 48 313 308 225 223 218 15120 10210 104 105 100 202 203 210 274 275 282 120 43 53 48 . . . ] om. D || 518 ] A || 518-9 513-4) A, om. (homoeotel.) D || 515 ] pr. scr. et del. A || 517-8 D || 513 ] D || 513 (post.) ] om. A || 513 ] om. D || 514 . . . ] om. D || 518 ] A || 518-9 ]] ||A D ||(homoeotel.) 513 ] D D515 || 513 ]] (post.) ] om. A 513 517-8 ] om. D || 514 513-4) A,272 om. pr. scr. scr. et del. 513-4) om. del. ]]A, D ||(homoeotel.) 511 ]] D |||| 515 |||| 512 D || ......A || 520 ]] om. 205 200 279 122 123 118 50 51 46 ||||40 511 A 512 D et || 512-3 512-3 om. ]A A ] 277 A227 519 ]A om. D || 519]]pr. ] ]]A |||| 517-8 215 212112 283 284 111 117 37271 39 49 45D 205 200 279 277 272 122 123 118 50 51 46 206 278 309 317 222 219 11124 18121 98285 101 108 206 285 271 278 124 121 112 52 49 40 ]28752 A || 519 ] om. D || 519 A|| ||||514 520 ] A || 519 ] om. D || 519 ] A 520 . . . ] A || 517 ]AD om. D || [518 ]]. . . A ||]|| 514 ]] A, D |||| 514 ] AD ]] ]] A 514 A, D 514 ] || 514 535 [ A || 523 517 ] om. om. D |||| 518 518 535 204 201 209 276 273 281 119 126 44 47 54 || 512 ] D || 513 ] AD || 513 ] A, . . . ] A || 517 ] D 322 316 234 232 226 2 6 95 92 96 204 201 209 276 273 281 119 126 44 47 54 D || 520-1 ] A, D || ] D || 523 ] || 512 ] D || 513 ] AD || 513 ] A, 215 212 283 287 284 111 117 37 39 45 [ 310215 311111 318 21297 283 220 287 221 39 228 284 12 117 107 37 102 45 109 D514-5 520-1 A, D |||| 523 523 . . . ]]] DA 523 |||| 520-1 A, D |||| || 523 ... ] om. D || 518 D 518-9 ]] D ||D ]] ]] A |||| 515-6 (l. D ||523 514-5 A 515-6 .textu, . .A]] || 513 ]] post post (l. 202 203 210 274 275 282 120 43 53 48 22595 223 218232 15316 10 104 105 100 203 274 275 282 120 43 53 48 . . . ] om. D || 518 A || 518-9 226 22308 66231 92 96 D234 ||106 (pr.) ] A || 523 ] hab. in (sic) suprascr. D. D || 513 ] D || 513 (post.) ] om. ] om. D 514 . . . ] om. D || 518 A || 518-9 314 217 224 16 202 13322 4 210 103 94 324313 322 316 234 232 226 95 92 96 513 ] ] A || D || 513 om. A || 513 (sic) ]suprascr. om. D ||||D. 514 D |||| ]523 523 D ||A (pr.) 523 ](post.) hab.] in in textu, D (pr.) ] A || 523 ] hab. textu, (sic) suprascr. D. || 519 ] om. D || 519 ] A || 520 ] 206 285 271 278 124 121 52 49 40 513-4) A, om. (homoeotel.) D || om. 515 pr. 517-8 309 317112 222 219 227|| 11313 18 98225 101 206 285 271 278 124 121 112 52 49 40519 513-4) A, om. (homoeotel.) 515 pr. scr. scr. et et del. A ||||AD 517-8 218 14 10 105 100 ] 223 A D ||]]D 519|| 514 ] A A || 520 520 ] ]] 230 3 308 9 15 91104 313 308 225 223 14323 15320 10229 104 100 ]233 A ]108 A 514 ]]D ]]||om. A, ]del. ]93 21899 105 A|||| ||519 514 A, D || 519 D || 514 ]] ADA||||||514 514 ] 535 535 D || 520-1 ] A, D || 523 ] D || 523 ] 215 212 283 287 284 109 111 117 37 39 45 310 311 318 220 221 228 12 309 107222 215 21297 283 287 284 109 11 111 117 317 37 D 45 227 535 98 .39 ]] |||| 517 ]||] om. 312 317 219 227 11 18 18 309 98 222 101 108 520-1 A, 20 D DA || 523 ] Dpost 523 D .520-1 ..102 . 219 A523 517 om. D ||||]]518 518 (l. D101 ||||.108 ]] A, || ] D || 523 D || 514-5 ] A || 515-6 . . . ] D|| ||523 514-5 ] A || 523 A || 515-620 ] hab. in .textu, .. ] (sic) post suprascr. D.(l. 324 322 232 226 2213311 95 92 96 D234 (pr.) 314316 217 224102 4 6623197 106220 103221 32416310 322 316 234 232 22694 95 92 96 ] 228 12 310 311 318 97 220 221 228 12 318 107 102 D107 523 (pr.) A |||| 523 523 hab. in in textu, textu, ]] (sic) suprascr. D. ..]].... ]] om. A 518-9 D |||| 523 (pr.) A ]] hab. (sic) om. D D]|||| 518 518 A ||suprascr. || 518-9 D. 513-4) A, om. (homoeotel.) D || 515 pr. scr. et del. A || 517-8 513-4) A, (homoeotel.) D || 515 ] pr. scr. et del. A || 517-8 223 218 14 104 105 229225 233106 230100 3 308 9 15 313 308 22591217 22393103 21899om. 14323313 1532010 104 105 100 224 16 410 314 231 217 224 94 16 13 13 314 4 231 106 103 94 ]] A ]] om. ]] A ]] A227|||| 519 519 om. D D |||| 519 519 A |||| 520 520 312 309 317 11 98 101 .93 ]] A ]] om. 312 317 229 222 219 227 11 18 98 222 101 .108 108 .93 ... 219 A |||| 517 517 om. D D |||| 518 518 11 323 318 99 320 91 99 323 320 229 233 230 3 309 91 233 99 230 20 20 D ]] A, ]] D ]] D |||| 520-1 520-1228 A, 20 D D |||| 523 523 D |||| 523 523 310 310 311 311 12 318 97 220 221 221 102 12 318 97 107 107 220 102 228. . . ]] om. ]] A ... om. D D |||| 518 518 A |||| 518-9 518-9 ] D || 523 (pr.) ] A || 523 ] hab. in textu, (sic) suprascr. D || 106 523103 224 (pr.) ] A || 523 ] hab. in textu, (sic) suprascr. D. D. 16 94 314 44 231 231 217 217 106 16 13 13 314 94 ]] 103 224A ]] om. ]] A ]] A |||| 519 519 om. D D |||| 519 519 A |||| 520 520

511 511

171149 169146 164178 69262 6460 158180 159 154150 270259 268 26258 180 178 172 56254 60177 149 146 150 260 163 170172 70 56 67268 160 157 148

11

323 323

33

99

68

55

60 149

D ]] A, D ]] D |||| 520-1 520-1 A, D |||| 523 523 r v r r (D45 ) A: 20 ,18 20 ,15 (ed. ll . 902-919); D : 99 ,13 99 ,24. (pr.) A ]] hab. D[ |||| 523 523 (pr.) ]] A |||| 523 523 hab. in in textu, textu, 525146 178 ]172 150add. D || 526 ] A, D || 526 180

259 254

69

65

255 263 152 168 155 162 165 173

72

64

171 158 169 159||154 ] 164 D 527

66

264 151 161 166 167 156 174

4 70

67

260

589 177 163 160 157 170 148

269

57

63

266 175 145 179 147 153 176

|| 530

288

38

42

46

] om. A || 535

A || 536

D || 536

]

99 168 155 162 165 173 65 4472 255 263 152 215 77111 117 212 3283 37 287 39 45 284 3 A || 537 256 257 66 264 151 161 166 167 156 174 2 322 316 6 95 234 92 232 226 96 14

70 67 260 313 1 308 15 269 7757 11 18

1

A 58 177 163 160 ]157 170 148 106 225 104 223 218 105 100

63 266 175 145 147 153 176 539179 309 317 33 98 222 101 108 ]219 227

] || 538-9

6

1 6 110 214 208 114 550 41 288 38 286 280 42 310 311 12 318 97 107 221 102 228 D ||220540 ]

205 200 123 118 279 50 277 272 51 46 16 13 314 4 231 217 106] 103 224 add.94A.

119 126 201 209 44 276 47 323 11 3 229 9 320 6 91 233 6 202 203 120 210

43

53

54 273 281 93 99 230

274 275

48

282

A || 538

] 46 ] 46 46

A || 534

A || 535

A,

( 46 )

]

] corr. ex

] A || 538

] D || 539

A ||

]

A || 540

A || 541

A, A,

D || 537

A,

A || 533

]

A || 538

D

]

]

A || 536

]

46 46

D || 532

]

]

D || 540

]

A || 530

]

]

A || 539-40 A,

D || 529

(pr.) ] post hoc

]

(post.) ] 46 46 ] A || 537 46

(& hoc

]

]

A || 530

] om. D || 534

D || 534

259 69 643 171 158 169 164 159 154 7 254 285 271 52 49 278 40 124 121 206 112

55

]

ut vid. A || 532

]

56 268 262 60 149 180 146 178 172 150 202 203 120 210 43 53 275 48 ]282 A ||274535 om.

6

AD || 529

]

D ]] D |||| 523 523 (sic) (sic) suprascr. suprascr. D. D. ] iter. A || 527

A || 528

]

(v. ll. 450-1) add. et del. A || 532

286 280

|| 534 273 281 11944126 201 209 99 44 276 47 D54 1

AD || 528

A || 530

]

A || 532

4 9 ] 205 279 50 277 272 51 7 200 123 118 3 545

1

(post.) ]

melius rescr. in marg.) D || 529

256 257

110 214 208 114 41

68

]] A || 512 A |||| 512 512] A

320 320 229 229 91 233 93 230 91 233 93 99 99 230 540

268 262

56

] ]]

] ]

D || 541

Text and translation

107

ered by itself the sums are everywhere the same.151 Such (constructions) are (always) possible for the evenly-even and evenly-evenly-odd orders.152 (§ 45. Division of squares of orders n = 4k + 2, k ≥ 2, into equal parts)

As for the evenly-odd orders, it is not possible to divide the (main) square into four parts, then to arrange the numbers in them in such a way as to satisfy the above condition.153 [ Indeed, it is absolutely impossible for the number in question to have an integral fourth, without fraction; now, as a rule, a fraction could not possibly be put in these squares. That is why this (kind of division) is impossible for evenly-odd (orders, that is) for the specific case of division (of this order) by two. ] As for an evenly-odd order having a third or some part other than a half, as (for the orders) eighteen or thirty or the like, it will be possible to divide it according to the part found in it [ other than the half ] which corresponds to an even order [ we leave out 2 which, as noted by us before, offers no possibility ].154 Arranging then there the numbers, it will lead to what we have explained.

Thus, the square of eighteen is divisible into nine squares, each with size 6 by 6, where the numbers will be arranged in such a way that the sums in the main (square) are the same and also (those) in each of these (smaller) squares.155 The square of thirty is divisible into nine squares, each with size 10 by 10, and also into twenty-five squares, each with size 6 by 6, [ and also, in the four corners, into four squares, each with size 12 by 12, separated by nine squares, each with size 6 by 6, ]156 where the numbers will then be arranged in such a way that the sums be everywhere the same, in the main (square) as well as in each of the small ones. 151

How to arrange the numbers will be explained in § 48. Thus the main square as well as the subsquares all display the magic property. In the case of subsquares of different sizes, the arrangement of the subsquares within the main square must be such that the latter’s diagonals make the magic sum. The somewhat odd ‘everywhere’ (min jam¯ı‘ al-jih¯ at; cf. pp. 30–31) might translate πάντη. 152 Thus for orders of the general form n = 4k, k ≥ 2 (above, n. 143). The possibility for the remaining category, that of evenly-odd orders, will be now examined. 153 The four subsquares will be of odd order, whereas filling a subsquare exclusively with pairs of complements requires its order to be even, thus the order of the main square to be divisible by 4. This statement was misunderstood by an early reader. 154 The subsquares, being of even orders, may then be filled with pairs of complements. The glosses are repetitive. 155 Example below, p. 135. 156 This ought to be in the next paragraph. Furthermore, it must be interpolated: the orders of the subsquares considered here do not exceed 10 (see p. 149).

......

20

310 310 311 311 12 318 97 220 221 221 102 228 12 318 97 107 107 220 102 228 310 310 311 311 12 318 97 220 221 221 102 228. ........ 12 318 97 107 107 220 102 228 ... 314 217 224 16 13 44 231 106 103 94 314 231 217 224 16 13 106 103 94 16 94 ]] 103 519 314 44 231 231 217 217 106 224 A 16 13 13 314 103 224 94|| A 519 ]]106 A ||||||519 A 519 320 229 233 230 1 323 91 93 99 323 333 999 320 320 229 229 91 233 93 230 91 233 93 99 99 230 111 323 323 3 9 320 229 91 233 99 230 D D ||93 || 520-1 520-1

om. D || 518 om.D 518 ]]]]om. om. DD||||||518 518 om. D || 519 om.D 519 ]]]]om. om. DD||||||519 519

]]]] Text and translation A, D 523 A, 20 D ||||523 523 ]]]] A, ]]]] A, 20 DD |||| 523 ] A || 523 ] hab. in textu, A ||523 523 hab.in textu, ]] ] hab. hab. inintextu, textu, ]]AA |||| 523

108 D || 520-1 D || 520-1 D || 523 523 D DD||||||523 523

540 540 540

4 4

7 7

(pr.) (pr.)]]]] (pr.) (pr.)

(46 46) ) (( 46 )

545 3545 3 545 9 999

4 444

56 268 262 60 149 180 146 178 172 150 56 268 262 60 149 180 146 178 172r 150

259 254 69 64 171 158 169 164 159 154 259 254 69 64 171 158 169 164 159 154

1

6

( 45 ) A: 20 ,18 - 20v ,15 (ed. ll . 902-919); D : 99r ,13 - 99r ,24. 46 46 46 A, 46 525 ] add. D || 526 ] 46 D || 526 46 168 155 162 165 173

6 152 258 165 72 255 263 65 77 72 255 263 152 33 168 155 162 165 173

77

33

D || 527

]

melius rescr. 170 in 70 67 260 58 177 163 160 157 148marg.) 70 67 260 58 177 163 160 157 170 148

269 57 63 266 175 145 179 147 153 176 || 530 269 57 63 266 175 145 179 147 153 176

11

D || 534

] om. A || 535

124 121 206 112 285 271 52 49 278 40 124 121 206 112 285 271 52 49 278 40

A || 535

] om. A || 536

D ||226536 2 322 316 6 95 234 92 232 96 2 322 316 6 95 234 92 232 226 96

313 308 15 10 225 104 223 14 313 105 100 A 218 ||218537 308 15 10 225 223 105 100 555 104

310 310 270 56 16 16 68 259 323 11 323 258

55

311 311 268 13 13 254 33

12 12 262 314 314 69 99

256 257

66

70

67

260

269

57

63

65

72

318 318 60 44 64 320 320

220 221 102 228 228 97 107 107 220 97 539 221 102 ] 149 180 146 178 172 150 231 217 224 106 103 94 231 217 106 103 224 94 D || 540 ] 171 158 169 164 159 154 229 91 233 93 230 91 233 93 99 99 230 229

|| 538-9

add.173A. 255 263 152 168 155] 162 165

A || 539-40

v 174 264 151 (161 46 166 ) A:16720156 ,15 - 21r ,8 58

177 163 148 tit. 160 D :157 170 (iter.

560 560

543

ante

38

122 205 200 123 118 279

50

277 272

204 119 126 201 209

44

276 A47|| 547 54 273 281]

202 203 120 210

43

286 280

]

51

46

A || 541

]

A || 543

45A 284 || 553

] ]

D || 541

A || 546

] D || 547

]

pr. scr. et del. A || 544

] ]

A || 548

A || 547

234

39

A ||

]

A || 540

324

]287

A || 538

D || 539

109 215 111 117 212 283

37

] ]

] 275 A || 549 ] D || 549 ] 274 282 (5346 ) A: 2048v ,15 - 21r ,8 (ed. ll . 920-933); D : 99r ,24 - 99v ,11. 40 ] A || 550 , l. ]548) D || 551 ... tit. D : (iter. ante

124 121 206 112 285 271 A52|| 550 49 278

A,

]

D || 537

]

]

42

D || 546

] corr. ex

A,

, l. 548)

] om. A || 543

288

]

A || 533

(ed. ll . 920-933); D : 99r ,24 - 99v ,11.

266 175 145 179 147 153 176

216 110 214 208 114560 41

A || 535

A,

D || 540

D

]

]

A || 534

]

A || 538

]

A,

D || 532

]

A,

]

]

A || 530

]

A || 536 ]

A || 538

]

555

D || 529

(pr.) ] post hoc

]

A || 537

]

(& hoc

]

]

A || 530

(post.) ]

109 215 111 117 212 283 37 287 39 45 284 215 111 117 212 283 37 287 39 45 284

219 227 227 11 18 309 317 555 98 222 101 108 219 11 18 309 317 98 222 101 108 ] A

]

450-1) add. et del. A || 532 46 46 46 46 || 534 ] om. D || 534

204 119 126 201 209 44 276 47 54 273 281 119 126 201 209 44 276 47 ] 54 273 281 D 202 203 120 210 43 53 274 275 48 282 202 203 120 210 43 53 274 275 48 282

AD || 529

]

] iter. A || 527

A || 528

]

ut vid. A || 532

]

122 205 200 123 118 279 50 277 272 51 46 205 200 123 118 279 50 277 272 51 (v. 46ll.

D || 529

A || 530

]

66

66550 216 11011 214 208 114 550 286532 280 42 41 288 38 || 280 42 110 214 208 114 550 41 288 38A 286

AD || 528

(post.) ]

256 257 66 264 151 161 166 167 156 174 256 257 66 264 151 161 166 167 156 174

312

D || 523 523 D ]] ]] DD||||||523 523 (sic) suprascr. D. (sic)suprascr. suprascr.D. D. (sic) (sic) suprascr. D.

9 9

68

324

]] ]]

[ [[

270

55

A || 518-9 518-9 A AA||||||518-9 518-9 A || 520 520 A AA||||||520 520

]]]]

(pr.) ] A || 550

] D || 549

] om. A || 552

]

] A || 553 ] om. D || 553 ] A || 554-5 ] om. A || 543 ] A ] pr. scr. et del. A || 544 21|| 543 21 96 . . .232 226] om. (homoeotel.) A || 555 21(pr.) ] om. D || 555 ] A || 555 ] 218178105172D || 546 ] A || 546 ] A || 547 ] 180223146 150 1427031356308268 15262 1060225149104 100 ] A || 556 ] A || 556 ] A, D || 556 164219159227154 31268 11259 182543096931764998171222158 A101169 || 108 547 ] D || 547 ] A || 548 (pr.) ] D || 549 44 9 ] 155 om. D102165 || 228 557 ] A || 557 ] A, D 2583106531172 12255318263 97152107 168220 173 221 162 ] A || 549 ] D || 549 ] A || 550 ] ||161 557-9 . .94 . 174 ] om. (homoeotel.) A || 560-2 ... ] om. A || 166103 167224156 16256 1325731466 4264231151217 106 A || 550 ] A || 550 ] D || 551 ... ] om. A || 552 add. D. 1 32370 3 67 926032058229177562 911632331609315799]170230148 2

322 316

6

95

543

92

109

Text and translation

(220 ; D, fol. 98r )

(§ 46. Division of squares of orders n = 4k + 2, k ≥ 158 2, into unequal parts)

It is (also) possible with these two orders157 as well as with other evenly-odd orders to carry out a division into different parts and to place in them the numbers so as to be led to the constancy of the sums in compliance with the condition.158 Thus (for the square of) ten it is possible to have in its four corners four squares each with size 4 by 4 displaying each equal sums everywhere, with the sums in the whole square being (themselves also) everywhere the same.159 It is likewise 158 possible for the (square of)277fourteen to have in84its 82 four corners four squares of size 6 by 6, each displaying sums everywhere 196 154 271 253 11 100 98 134 the same, with the sums in the main square being (themselves also) ev194 184 150 267 247 237 35 7 116 114 138 106 erywhere the same;160 it is moreover possible to have in the centre of this 144 256 263 243 231 229 51 31 142 118 108 square a square of size 6 182 by 172 6 surrounded by eight squares3 of 2size 4 by 170 286 83 95 91 87 213 215 219 223 79 4 120 4, each displaying individually same sums, with the sums in the whole 161 93 111 119 115for 185 the 187 square 191 107 197 265 square being the same. It is 25 again possible of (order) eighteen to have in its centre21 a 45square of size 10 by 10 surrounded by 89 117 131 135 165 167 127 173 201 245 269 162 twelve squares of size 4 by 4, each displaying also same sums; it is 17 41 57 85 113 133 143 141 151 157 177 205 233 249 273 moreover possible to have in its four corners four squares of size 8 by 8 15 39 55 63 81 109 129 153 145 137 161 181 209 227 235 251 displaying equal sums.163 And the same for all such 160 squares. 157

281 257 241 217 189 169 139 149 147 121 101

73

49

33

130 94 96

275

9

Namely n = 18 and n = 30. 285 261 221 193 163 155 125 123 159 97 69 29 5 158 Namely, constancy of the sums in the main square and in its various parts, as will 289 225 183 171 175 105 103 99 179 65 1 be specified below. 159 Example below, p. 155. How to fill the remaining central cross will be explained 36 211 195 199 203 77 75 71 67 207 254 in detail further on (§§ 50–54). 160 148 34 27 47 59 61 239 259 287 288 146 Example below, p. 157. 161 Example below, p. 139 — with, as seen here in 152 23 the 43 illustration, 53 255 283 strips 174separating 176 140 the 4 × 4 squares. 162 156 strips separating the19 279 190 192 136 Example below, p. 145, with 4 ×374 squares. 163 Examples below, with 160 a central cross and four 8×8 squares or sixteen 4×4 squares, 13 206 208 see pp. 159 and 161. Although the square of order 30 is mentioned at the beginning, there is no example for it here (see p. 107 & n. 156). The author may have confined himself to the examples actually constructed by him.

132

124 121 206 112 285 271

52

49

278

40

215 111 117 212 283

287

39

45

284

92

232 226

96

322 316

2

1

110 234

6

95

10

225 104 223 218 105 100

313 308

15

11

309 317 555 98

18

310 311

12

16

13

314

323

3

9

37

Text and translation

555

222 101 108 219 227

31855597 107 220 221 102 228 231 217 106 103 224

4

320 229

233

91

93

94 230

99

560 560

560

270

56

68

259 4 254

69

65

255 263 565 152 168 155 162 165 173 v

258

55

268 262

72

60 149 180 146 178 172 150

( 46 ) A: 20 ,15 - 21r ,8 (ed. ll . 920-933); D : 99r ,24 - 99v ,11.

66

264 151 161 166 167 156 174

70

67

260

58

269

57

63

266 175 145 179 147 153 176

3

tit. D :

(iter. ante

] om. A || 543

543

38] 286 280 D42||

288

122 205 200 123 118 279

272 51 50A 277 || 547

6

, l. 548)

177 163 160 157 170 148

216 110 214 208 114 41

1

( 47 )

649 171 158 169 164 159 154

256 257

7

21 21 21

204 119 126 201 209

44

276

202 203 120 210

43

53

]47

A || 543

]

546

A || 546

]

46]

D || 547

281 A 273 || 549

54

274 275 48 282 A ||v 550 r]

A || 547

]

A || 548

]

]

tit. D : ]

109 215 111 117 212 283 324

2

322 316

270 56 268 262 14 313 308 15 68 259 254 69 312 11 18 309 258

65 72 255 310 311 12

37

52

[ 287

49

278

40

39

45

284

|| 553 (iter.Aante

A || 553 , ]l. 548)

] [

1

70 67 260 323 3 9

269

57

63

...

A 226 || 543 234] om. 92 232 96

A || 550 ] 177 163 160 157

A || 550add. D. ] 170 148

] om. A || 552

] om. D || 553

58 320 575 229 91 233 93 99 230 r 47 179 ) A: ,9 176 - 21r ,20 ] (145 A147||21153 553 ] 266 175

D || 551

122 20569 20064123 50164277 51 . . 46 . 171118 158279169 159 ] 563 A272||154556

204 119 281 255126 263201 152209168 44155276 162]47165 54173273 A ||

] om. D || 557

20266 203264120 151210 16143166 53167274 156275174 48

282

124 26012158 206177112 163285 160271 15752170 49148278

40

]

A || 556

]

564

A || 554-5

]

A add. || 567D.

580 ] 324214 2208322 11431641 6288 9538234286 9228023242226

A 96 || 569

]]

284

D || 570-1 313 2722235121846105 100 123308 11815279 1050225277104 11201 1820930944 317276 984722254101 ] 281219 227 273108

310 27522048221282102 228 120311210 124331853 97274107 16206 13112314285 42712315221749106 27810340224

94

A || 567

]

D || 570

] ] D || 571-3

...

A || 554-5

]

A || 555

]

D |||| 556 563 D

A,

A || A, 565

]]

] D

] om. A || 567 A,

D || 570

] def. in D, hab. hic

(v. enim cod. D, fol. 60 v - 61 v ) || 572

A || 567

]

A,

]

D

om. AD|||| 566 ] ]om.

...

]

( 48 )

] om. A || 552

]

A ||AD 557|| 565

]

]

10957 21563111 266117 175212 145283179 37147287 15339176 45

562

...

(pr.) ] om. D || 555

|| 566 . . . ] om. ] D || (homoeotel.) 566 ] 566-7 || 557-9 om. AD||||560-2

14

]

v (ed. . 934-944); 99553 ,11 - 99v ,17. A || ll553 ] om.DD: ||

tit. D172]:286 . 288 . .178 38 om. A || 555 216268110 42 26221460 208 180 41146 149114 150280(homoeotel.)

312

]

] om. (homoeotel.) A || 555 || 555 ] || 544 A || 555 ] A || 543(pr.) ] ]om. D pr. scr. et del. A 60 149 180 146 150 ] 178 A546 ||100 556 ] A] || 556 A, 218 10 225 ]104 223 D 172 ||105 A] || 546 A || 547 ] ] D || 556 64 171 158 169 164 159 154 || 557 || 557 (pr.) ] ] A, D 317 98 222 ]101 227 A || 547 om.108D] 219 D || 547 ]] AA || 548 D || 549 263 152 168 155 162 165 173 318 97 || 228 107 220 221 102 ] om. || 560-2 ] A || 550 . . . ] 557-9 A || 549 . . . ] (homoeotel.) D ||A549 ] ] om. A || 654395

256 257 66 264 151 161 166 167 156 174 217 106 103 224 94 16 13 314 4 231 562 ] 55

D || 549

A || 550

A ll||. 550 || 551 (570 46 ) A: 20 ,15 - 21 ,8 (ed. 920-933);] D : 99Dr ,24 - 99v ,11. . . .

124 121 206 112 285 271

]

(pr.) ]

D || 549

]

pr. scr. et del. A || 544

]

D || 571 A.

]

( 49 )

111

Text and translation (222 ; D, fol. 102v )

(225 ; D, fol. 102r )

(D, fol. 102r )

(228 ; D, fol. 104v ) (227 ; D, fol. 103v )

Squares admitting of (such) a division, (of orders) from eight to twenty, have been constructed, each being divided in several possible ways; you will thereby rely on that for any desired division of other squares.164 (§ 47. No division into odd-order subsquares) As for the odd(-order) squares, it is impossible for such a square, being divided into squares with the order of each being odd, to satisfy this condition.165 Indeed, a square of odd order necessarily has a single median number; so if one places in the main square a certain quantity of squares, each of odd order and supposed to display the same sum, the median number of any of these squares will have to be the same as that of any other in order that the equalization with the remaining numbers might be performed.166 [ Now of the numbers (to be placed) none may ever occur twice, and it may be put only in a single place; so if you put the median in all (these) squares and (then) omit their places altogether whilst putting consecutive 164

See p. 124 seqq. (and n. 142). The filling of their parts will be explained later (§§ 48 seqq). There is no 8 × 8 square in the extant text (see p. 125). 165 This, incidentally, anticipates any reader’s question about the division of the 18×18 and 30 × 30 squares into 36 squares of order 3 and 5, respectively. Generally, it is impossible for the whole square to be magic with its (equal) subsquares displaying equal sums (this being the ‘condition’). 166 Since each subsquare must contain the same magic sum, their central cells should all contain half the sum of two complementary numbers. What follows is a few disorderly glosses attempting to complete the argument; namely, that either putting the same quantity in all central cells or leaving them all empty whilst filling in both cases the remainder of the squares with pairs of complements will in no way satisfy the magic condition. Argument reproduced in an early 11th-century text (Un traité médiéval, pp. 80 & 166, ll. 656–658).

258

55

65

72

255 263 565 152 168 155 162 165 173 565 565

256 257

66

70

67

260

58

269

57

63

266 175 145 179 147 153 176

7

264 151 161 166 167 156 174

112

216 110 214 208 114 41

288

38

122 205 200 123 118 279

50

277 272

51

273 281

1

Text and translation

177 163 160 157 170 148

3

6

286 280

42 46

204 119 126 201 209

44

276

47

202 203 120 210

43

53

274 275 v

48

282

52

278

40

54

(570 46 ) A: 20 ,15 - 21r ,8 (ed. ll . 920-933); D : 99r ,24 - 99v ,11.

570 285 271 124 121 206 112570

tit. D :

37

654395

234

2

322 316

14

313 308

15

312

11

309 317

1

18

12

318

16

13

314

4

323

3

9

69

232 226

92

98

A || 547

A || 549

]

A || 550

575 575

]

233

91

93

99

230

A || 553

]

171 158 169 ] 164 159 A ||154556

260 57

63

|| 557-9 562 580

580 288 580 38

50

201 209

44

570 47 276

120 210

43

53

563

286 280

277 272

206 112 285 271

]

42

51

46

||54566 273

281

274 275

] 48

282

52

278]

40

49

[

50

204 119 126 201 209

44

57747 276

202 203 120 210

43

53

124 121 206 112590285 271 109 215 111 117 212 283

1

4

322 316

313 308 11

18

15

6

580

95

10

309 317

310 311

12

16

13

314

323

3

9

318 4

277 272

A || 567

om. 273 ] 281

54

A, 48 274 275 52]

37

287

234

92

46

49 278 om. D.

D || 566-7

]

] [

A || 569

51

AD || 565

]

] om. D || 566

AD || 576

122 205 200 123 118 279

312

D || 556 A,

] om. A ||

D || 570

]

]]] D || 571-3

40

A,

] A,

]

] om.

A || 567

A || 575

]

] in marg. A || 577

A.[[[

]

D || 578

D || 576

]

D || 577

]

D || 580

]

22 22 22

( 49 ) A: 21 ,7 - 21v ,17 (ed. ll . 953-962); D : 99v ,22 - 100r ,4. 232 226

tit. D :

581 222 101

96

D || 581 219 227

] 108

D 228 || 97 107 220 ] 221 102

231 217 106 103 224

320595229

91

233

93

99

(ult.) ]

94

585

D || 585

A

582

A || 581

]

A || 582

]

48 A || 583

]

230

]

9

]

...

( 50 )

225 104 223 218 105 100 98

A ||

D || 578

]

( 48 ) ]

]

D || 578

]

]

D || 571

(ed. ll . 945-952); D : 99v ,18 - 99v ,22. D || 575

D

))) D(((49 ||49 566 49

D || 570

] def. in D, hab. hic

...

]

]

]

(v. enim cod. D, fol. 60 v - 61 v ) || 572

D || 579-80

282

A || 565

]

] om. A || 567

A || 567

]

45 284 v

39

D

D || 563

A || 564

111 117 4 212 283 37 9 287 39 45 284 570-1 270 56 268 262 60 149 180 146 D 178|| 172 150 322 316 6 95 234 92 232 226 96 ] 159 154 68 259 254 69 64 171 158 169 164 15 10 225 104 223 218 105 100 585 258 65 72 255 263 152 168 155 162 165 173 585 585 309 317 98 222 101 108 219 227 7 3 r 174 256 257 66 264 151 161 167 21 156 ( 48166 ) A: ,20 - 21v ,7 12 318 97 107 220 221 102 228 70 67 260 58575177 163 160 157 170 148 tit. D : 314 4 231 217 106 103 224 94 55 269 57 63 266 175 145 179 147 153 176 ] A, 3 9 320 229 91 233 574 93 99 230 216 110 1 214 208 114 641 288 38 286 280 42

14

A,

]

add. D. ... ]

]

266 175 145 179 147 153 176

123 118 279

2

A || 556

A || 555

]

] A || 557 ] v v (ed. ll . 934-944); D : 99 ,11 99 ,17. . . . ] om. (homoeotel.) A || 560-2

...

177 163 160 tit. 170 : 148 157 D

214 208 114 41

324

A || 554-5

]

264 151 161 166 ( 167 47 )156 A:17421r ,9 - 21r ,20 58

((48 48))) (48 47

]

] om. A || 552

...

(pr.) ] om. D || 555

]

] om. D || 557

D || 549

A || 550

]

] om. D || 553

A || 553

(homoeotel.) A || 555

565

255 263 152 168 155 162 165 173 66

D || 551

]

]

(pr.) ]

D || 549

]

94

A || 550

A || 547

]

A || 548

]

]]]

pr. scr. et del. A || 544

]

A || 546

D || 547

]

231 217 106 103 224

320 575 229

A || 543

]

97 107 220 221 102 228

. . .178 172] om. 60 149 180 146 150 64

96

222 101 108 219 227

]

268 262

, l. 548)

]]] [[[ ]

225 ]104 223 218 100 D ||105546

10

310 311

(iter. ante

[[[287 39 45 284 ] om. A || 543

109 215 111 117 212 283 324

49

A || 583

]

]

A || 583 ]

D (v. enim D, fol. 33 ) || 585 ]

v

D || 584 ]

A || 586

587

]

A || 583

] ]

A || 584

A || 581

]

587

A || 583

]

AD || 583

]

A || 584-5

] A,

) D(||49 581

]

D || 585

]

[

A

587

113

Text and translation

numbers [ thus omitting the place of each pair of medians ] [ each pair of complements ]167 , this will not meet the condition of magic square and hnoti go back to what we have shown for (squares of) odd orders hini the first hchapteri, and the main sum will be wrong because of the repetition of some (numbers) and the omission of others. ] (§ 48. Filling a square divided into even-order subsquares)168 If you wish to write the numbers in some square you have divided into squares not separated by strips insufficient to (contain) a complete square (proceed as follows). Take the two medians of the numbers allotted to this (main) square and put them in one of the squares into which you have divided the (main square), at the place of the two medians of the 4 by 4 square, and then (proceed) according to this (known way of) filling to the end of this square.169 Take then the two terms you have attained, attribute them the rôle of medians and put them in some other square among these, and proceed as before.170 (Do) always the same until you have finished with what you wanted. (§ 49. Filling rectangular parts surrounding a central subsquare) Magic squares in the tenth century 33 If (now) you have divided such a square into squares separated by strips insufficient to (contain) a complete square and these separations are in the sides without meeting the diagonal (proceed as follows). α4

α1

β4

β1

γ4

γ1

α1

γ3

γ2

α2

β3

β2

α3

α3

α2

α4

— If the separation is 6 by 4, you will put in each (group of) four− 4jcells − 4j with (a group of) four (cells) below it eight complementary terms;171 with — If the separation is 6 by 4 (Fig. a 36*), you will place in each group of them, each row will be equalized with its conjugate [ as we have explained four cells with a group of four cells below it eight complementary +terms 4j + 4j (Fig. a 37*); with them, each row will equalize its conjugate, as we have 167 Obscurum per obscurius. explained at168 the beginning of this section.151 You proceed like that until Thus the smallest subsquares, or inner squares within subsquares, are 4 × 4. you have completed the six2 (rows). 2 169 Taking thus n2 and n2 + 1, then the decreasing sequence from the first and the — If the separation is 6 by 8, you will theinfour in two and increasing one from the second. Fortreat example, the on steps, p. 137, the n2 +figure 7 n2 first + 3 numbers =⇒ − 4j − 4j 152 the six as here above. 2 put are 200 and 201, namely in the first subsquare (if, unlike the author,2 we consider n2 +7 2

n2 +3 2

=⇒ n2 + 1 + 4

n2 −5 2

n2 −1 2

=⇒ n2 + 1 −

n2 + 1 + 4 − 8j

putting the numbers from 1 on, we shall begin with the last subsquare).

It is not 170 possible that the square be divided into parts with the sepaSince each subsquare is to display the same sum, they can be dealt with in any 153 n −5 1 ration beingsequence. odd (in dimensions) because of what wen −have =⇒ + 4j explained; + 4j 2 2 171 in our above figure, each of the be four6αby with8,their as a matter of Thus, fact, the separation will always 4,βor or 2complements i (or i , γ6 i ) by ⇓ ⇓ lined up 2 vertically. by 4, or 2 by 6, or by 8, and so on always.154 If the nseparation is 2 by n4+ 1 − 2 +1+2 (Fig. a 38*) and is situated in the sides without meeting the diagonal, the treatment for its equalization will be as we have explained previously.155 2

2

2

1

4

12

n2 + 1 − 4 + 8j

2

4

| ... |

n2 +3 2

− 4j

n2 +7 2

− 4j | . . . |

n2 −5 2

n2 −1 2

592

AD || 593

] A,

580 580

D || 594

A114 || 595

570

[

][ 601

(post

AD || 597

]

]] [[]

A || 598

A, om. D || 600-1

A || 597

]

D || 597

(pr.) ]

602 575 575

( 51 ) 603

]

]

A,

AD A, || 593

]

D || 594

A || 602

A || 595

r A ||22 596 || 596ll . A: ,8 - 22] rom. ,11D(ed.

]

D || 597

(pr.) ] ]

]

A || 603

A || 598

|| 178 605 om.172 D ||150 600-1 60A,149 180 D 146A,

56

68

259 254

69

65

adn.— ponenda) || 601 590 255 263 590 152 168 155 162 165 173 590 602 ] A,

72

256 257

55

64

264 151 161 166 167 156 174

580 58580177 163 160 157r 170 r148 ( 51 ) A: 22 ,8 - 22 ,11

70

67

260

269

57

63

266 175 145 176 153|| 603 603 179] 147 A

216 110 214 208 114 41

A, 38 288

204 119 126 201 209

44

595 202 203 120 210 595 43 595

tit. 274 D 275 : 53 287] 39

324

234

92

14

313 308

15

585 585

10

95

55

A || 604

]

A || 605

A || 602

]

22 22

D || 603

AD ||

]

ut vid. D || 602

]

(50 50)) (((50 ) 49 ( 49 ))

]

A || 605-6

]

]

—v. pænult.

AD || 602

A.

]] ]

581

45D 284 || 582

232 226

96

225 104 223 218 105 100

]

A || 582

]

D (v. enim D, fol. 33 ) || 585

65 152 314 263 231 168 217 155 16 72 13 255 4 600 106 103 224 94 600 600 A,93 156 264 166 66 151 161 230 9 320 229 91 233 167 99 174

70

67

260

58

57

63

266 590 175 145 588179 147 153 176

D || 587

177 163 160 ] 157 170 D ||148 588

590

110 214 208 114 41

288

38

205 200 123 118 279

50

277 272

119 126 201 209

44

276

47

43

53

274 275

|| 589

286 280

A || 588

42

51

D || 590

]

273 281

54

D || 591

282

48

]

52

] om. D.

A,

A || 587

] corr. ex

A || 588-9

D || 585

]

A,

]

D || 587

A || 588

D ||

]

A

]

A || 589

]

52 || 590 AD

A || 584-5

]

]

22 22

A || 589

]

46

[

AD || 583 ]

]

AD || 588

]

]

]

A || 586

]

A [ || 583

] ]

D || 584

]

v

D || 581

]

A || 583 A || 583

]

A || 584

]

A || 581

]

]

A || 583

]

256 323 257 3

269

A || 581

]

A || 583

(ult.) ]

605 605

A add. || 605-6 A ||

]

add. (post

]

]

]

282

48

259 310 254 311 69 318 171 220 164 221 159 228 12 64 97 158 107 169 102 154

202 203 120 210

A || 604

46

D 585173 162 ||165

1

A || 605 D || 601 ]

A || 602

]

D || 598

]

D || 599

]

] A || 597

A, quasi

(alt.) ]

D || 599

]

312 56 309 60 317 149 222 146 219 150 227 11 268 18 262 98 180 101 178 108 172 68

D || 603

(ed. ll . 974-977); D : 100r ,11 - 100r ,13.

278 D40||

] 49

37

6

D:

AD || 597

[

v ( 4947 ) A:54 21273 ,7281 - 21v ,17 (ed. ll . 953-962); D : 99v ,22 - 100r ,4.

109 215 111 117 212 283 322 316

A || r597 ] 100 ,11 - 100r ,13.

48)) A || 602((48

]

]

] om. D || 596

A || 595

D || 602

]

ut vid. D ||

AD [||

]

]602

] om. D || 595

A || 594

]

—v. pænult.

AD || 602

A || ]593-4

602

]

]

add. A ||

]

276

58152 124 121 206 112 285 271 2

]

]

add. (post

] 974-977);

(post.) ]

D || 280 605 42 286

272 51 122 205 200 123 118 279 50 277 A.

A.

]

A || 602

]

D

||]

A post quod

601 (post159 154 ) ] A. 171 158 169 164

66

]

]

D || 595

]

270

258

268 262

592

49)) A || 597((49

D || 598

]

D || 599]

D || 601

r r 602 r adn.— ponenda) 601 ] D: A ( 50 ) A: 21v ,17||- 22 ,8 (ed. ll .(post.) 963-973); 100|| ,4 - 100 ,11. ] 585

585

A, quasi

]

]

]

]

(alt.) ]

D || 599

A post quod ]]

)]

]

|| 595 ] om. D || 596 Text ]and A translation

] om. D || 596

A 570|| 596

]

] om. D || 595

A || 594

]

D || 595

]

A || 593-4

]

(51 51)) ((51 )] (( 50 50))

] (sic) D || 589

A || 590

]

( 5052) A: 21v ,17 - 22r ,8 (ed. ll . 963-973); D : 100r ,4 - 100r ,11. 49 278 40

285 271 124 121 206 112605 215 111 117 212 283 2 14

1

322 316

590 590 6 595 95 595

313 308

15

11

309 317

18

10

310 311

12

318

16 4 13

314

4

323

3

7

9

592287 37

39

]45

284

234

A, 226 232

96

92

225 104 A 223 ||218 595105 100 ] 98

AD || 593

D || 594

] om. D || 596

A || 596

97 107 220 221 102 228

(pr.) ]

231 9 217 106 103 224

610 610 320 610 229 595 595 600 3 600

91

233

601

]

93

]

D || 597

99

94

230

A || 598

A, om. D || 600-1 (post

AD || 597

)]

adn.— ponenda) || 601

A || 597 (alt.) ]

D || 599

]

]

A post quod (post.) ]

A || 602

A, quasi

D || 599

AD || 602

A (||50 597 )

D || 598

] add. A ||

]

add. (post

]

(52 52)) ((]52 ) ( 50 )

]

D || 601

]

]

] om. D || 596

A || 595

] ]

]

] om. D || 595

A || 594

]

D || 595

222 101 108 219 227

A || 593-4

]

—v. pænult. ]

AD ||

115

Text and translation

at the beginning of this section172 ]. You proceed like that until you have completed the six (rows). — If the separation is 6 by 8, you will treat the remaining 4 by 6 in the same way. It is not possible that the square be divided into parts with the separation between them being odd (in dimensions) because of what we have explained;173 as a matter of fact, the separation will be 6 by 4, or 6 by 8, or 2 by 4, or 2 by 6, or 2 by 8, and so on always. If the separation is 2 by 4 and is situated in the sides without meeting the diagonal, the treatment for its equalization will be as we have explained previously.174 α4

α1

β4

β1

γ4

γ1

α1

γ3

γ2

α2

β3

β2

α3

α3

α2

16

α4

n2 +7 2 14

45

n2 +3 2

− 4j

− 4j

=⇒ n2 + 1 + 4 − 8j

(§ 50. Filling the central part of a cross placed in the middle) 18

24

39

37

3

20

9

23

21

22

32 30

31

41

7

15

33

25

17

35

43

40

49

19

29

27

1

10

38

28

11

13

47

26

12

If this separation is in the middle diagonal, n + 1 − 4 +there 8j + 4j and thus + 4jmeets the =⇒ appears in the centre of the (main) square a square with size 2 by 2. This for instance occurs in the (square of size) 10 by 10 when are placed in its corners four squares each of size 4 by 4, or in the square of fourteen when are placed in its four corners four squares each of size 6 by 6.175 36

172

5

10

n2 −1 2

n2 −5 2

34

9

7

14

4

5

15

4

1

13

8

6

12

1

15

11

8

1

14

15

6

3

10

3

13

8

10

2

13

12

7

4

9

16

5

16

2

11

5

16

3

6

9

14

7

2

11

n2 + 7 − 4j 2

12

n2 + 3 − 4j 2

2

=⇒ n2 + 1 + 4 − 8j

In § 42, or else in § 20. (See § 31, n. 97, similar interpolated reference.) 173 The case of odd-order subsquares having been ruled out (§ 47), the sides of the strips are necessarily even. If not (case of strips 1 × 4, generally 1 × 4m), the two n2 − 5 n2 − 1 =⇒ n2note). + 1 − 4 + 8j + 4j (see examples in + 4j subsequent opposite strips will be associated the 2 2 174 Using a neutral placing (§ 42), see our figure. The separations 6 × 4 and 6 × 8 ⇓ been treated, leaving⇓ the cases 2 × 2m. A pair of opposite (generally, 6 × 4m) have just n2 + 1 + 2 n2 + 1 − 2 rectangles 2 × 6 may be reduced to the case 6 × 4 by combining them. A strip 2 × 8, or 2 × 4m, is reducible to two, or m, rectangles 2 × 4. Note also that the separations 2m × 4 and 2m × 6, m odd, can be reduced to rectangles 2 × 4 by removing 4 × 4 squares. Thus we find among the examples of larger squares: a 6 × 4 rectangle reduced to one 4 × 4 square and either a 2 × 4 rectangle (pp. 141, 153) or two ndisjunctive 1×4 n +7 −1 | . . . | n +3 − 4j − 4j | . . . | n 2−5 2 4 2 2 strips (p. 139); a 6 × 8 rectangle reduced2 to filling either two 4 × 4 squares and two 2 × 4 −1 n −5 +7 n +3 | . . . | nand +two 4j disjunctive + 4j | .1. .×| 6nstrips, rectangles (p. 153) or one n2 − 1 6 n2× −63 square 2 2 2 2each associated with its opposite horizontally and vertically (p. 151); a 10 × 4 rectangle reduced to two 4 × 4 squares and two disjunctive 1 × 4 strips (p. 145). The examples with a central cross (see below) also lead to filling 2 × 4 rectangles. 175 See pp. 155 & 157. These examples have been mentioned in § 46, together with the example of an 18 × 18 square. What follows is generally applicable to evenly-odd squares. 2

2

2

2

2

2

2

2

286 280

42

205 200 123 118 279 50 277 A. 272 51

46

110

114 41

119 126 201 209

44

288 , 38

276

116 53

274 275

48

282

124 121 206 112 285 271

52

49

278

40

215 111 117 212 283

37

287

39

45

284

234

92

232 226

96

322 316

2

1

273 281

54

202 203 120 210 595 43 595

14

47

6

95

10

225 104 223 218 105 100

313 308

15

11

309 317

18

310 311

12

16

13

314

323

3

9

318

98

-

Text and translation

222 101 108 219 227

97 107 220 221 102 228

231 217 106 103 224 4 600 600

320 229

91

233

93

99

94 230

( 51 ) ( 51 )

52 52 605 605

( 52 ) ( 52 ) 4

9

610 610

7

50)) A: A: 21 21vv,17 ,17--22 22rr,8 ,8 (ed. (ed. llll.. 963-973); 963-973); D D:: 100 100rr,4 ,4 -- 100 100rr,11. ((50

3

592 592

AD |||| 593 593 AD

]]

A, A, 1

615

A |||| 596 596 A

56

268 262

om. D D |||| 596 596 ]] om.

]]

68

259 254

69

65

255 263 152 168 155 162 165 173

256 257

55

64

171 158 169 164 159 154

601 601

66

(post (post

adn.— ponenda) adn.— ponenda) 264 151 161 166 167 156 174 177

602160 602 163

170 ]] 148

)) ]]

601 |||| 601

70

67

260

58

269

57

63

A |||| 266 175 145 179 147 ]] 153 176 A

157

286 280

A, A,

602 602

]]

A post post quod quod A (post.) ]] (post.)

A || || 602 602 23 A

23

D |||| 602 602 D

A. A.

]]

A, quasi

D || 599

]

288

51)) A: A: 22 22rr,8 ,8--22 22rr,11 ,11 (ed. (ed. llll.. 974-977); 974-977); D D:: 100 100rr,11 ,11 -- 100 100rr,13. ((51

38

50

277 272

51

204 119 126 201 209

44

276

47

273 281

202 203 120 210

43

53

603 603

]]

A, A, 274

275

124 121 206 112 285 271

52

54

46

A |||| 603 603 A

D |||| 605 605 D 48 282

A.278 A. 49

40

109 215 111 117 212 283

37

287

39

45

284

324

234

92

232 226

96

14 312

2

322 316

6

95

10

225 104 223 218 105 100

313 308

15

11

309 317

18

310 311

12

318

16

314

4

13

98

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

94

]]

]]

D |||| 603 603 D

A |||| 605 605 A

—v. pænult. ]

ut vid. D || 602 ut

216 110 214 208 114 41

]

]

A || 604

A || 605-6

]

add. A ||

]

AD || 602

]]

122 205 200 123 118 279

42

D || 598

add. (post

]]

] A || 597

]

D || 601

]]

]

] om. D || 596

(alt.) ] (alt.)

D |||| 599 599 D

]]

]

]] om. D || 595

A |||| 597 A

AD |||| 597 597 AD

]]

A |||| 593-4 A

A |||| 595 595 A

]]

A |||| 598 598 A

A, om. om. D D |||| 600-1 600-1 A,

258

72

D |||| 595 595 D

]]

60 149 180 146 178 172 150

A |||| 594 594 A

]]

D |||| 597 597 D

(pr.) ]] (pr.)

615

270

D |||| 594 594 D

A |||| 595 595 A

6

]]

]

AD ||

A || 602

]

]

117

Text and translation

+3

(D, fol. 102r )

n2 2 n2 2

≠3

n2 2

+3 +1

n2 2

≠2

+2 ≠2

(You will proceed in this case) as follows. You write the two medians of the whole set of numbers176 in two diagonally opposite corners of the square with size 2 by 2 in the centre of the main square. Omitting two numbers after the two medians,177 you take the next two and place them in the two opposite corners left.178 At this point, you will find that the row160 containing the two small numbers has a century deficit of 3 relative 35 to its squares in the tenth [ 51sum ] due, that the Magic row parallel to it has an excess of 3, that the row the larger of the two small (numbers) and the larger of the by containing 6 (Fig. a 40 the four sides, (strips of size) 2 by 4 which you will equalize in the manner large (numbers) has an excess explained. of two two cells 163 on each side;162of 2, and that (the row) parallel to it has a deficit of 2. (§ 51. Case of order n = 14) 156

Reducing a square of evenly odd order n to a central cross of size 2 even-order squares; how to fill the central square of the cross. 157 These examples have been mentioned in A.II.46 (see Fig. a29* and a follows is generally applicable to evenly odd squares. 158 Thus the two medians of the main square. 159 One on either part. 160 How to equalize the four remaining branches of the cross. 161 On each what side remains of the 2in◊the 2 central If then cross is,square. in its four directions,179 2 by 162 180 byhave means of twothe pairs of cells on by each sideofoftwo the 2◊2 6,That youis, will to equalize (central) square means cells •

• •

• •

•

• •

◊ ◊ ◊ ◊ • •

•

• •

• •

•

on each side;181 in this way, the central (then filled part of the) cross will be (of size) 6 by 2 and there will remain, to complete the cross on the 164

[ 52 ] the (In to equalize thesquare central proceed as follows. and n2 you + 1), which are complemenThus twoorder medians of the main ( n2square) After putting in the centre the four numbers which we have indicated, tary numbers. 177 you consider the row with a deficit of 3 (Fig. a 39*). Put in the pair of One after each. ‘Omitting’: taraka, perhaps translating ὑπερβαίνειν. 178 cells to it, onlarger one ofone theabove. two sides, the lesser of the two numbers In ouradjacent figure, with the 2 179 On of the after 2 × 2 (placing) central square. youeach haveside omitted the two medians (thus n2 ≠ 1) and the 180 Case the 14 ×following 14 square, seetwo figure. largeofnumber the large numbers you have written in the 181 n2 That (considering also the complements) by means of two of cells on each + 4); and put in the two cells adjacent to itpairs (= the central centreis (thus 2 side of the 2 ×on 2 central square. square), the other side, the lesser of the two small numbers following 176

2

2

2

the small number you have reached (thus n2 ≠4) and the larger of the two 2 large numbers following the large number you have reached (thus n2 + 6); and put opposite to each its complement. Once you have done that, you will have equalized these six (pairs of) cells.

( 51 )

600

118

Text and translation

605

( 51 ) ( 52 ) 605

610

( 52 )

610

615

23

615

620

( 53 )

23 620

( 53 ) 625

( 52 ) A: 22r ,11 - 22v ,4 (ed. ll . 978-992); D : 100r ,13 - 100r ,21. ]

56

68

259 254

69

65

AD 610165 173 255 263 152 168 155||162

258

268 262

607

270

72

60 149 180 146 178 172 150 64

609 ] 171 158 169 164 159 154

AD || 608 ]

625

]

A || 610 A,

]

D || 608

D || 611

]

D || 610

]

A || 608

A || 610

A || 611

]

A || 612 ] A, D || 613 ( 52 ) A: 22 ,11 - 22v ,4 (ed. ll . 978-992); D : 100r ,13 - 100r ,21. 70 67 260 58 177 163 160||157 AD 615170 148 ] A || 616 ] (sc. ) AD || 607 ]172 150 ] D || 608 ] A 270 5656 268 268 262 262 6060 149 180 178 172 149 180 146 178 150 AD || 608 146 55 269 57 63 266630175 145 179 147] 153 176 D || 618 ] A || 618 ] 609 ] A || 610 ] D || 610 ] 259 254 254 6969 6464 171 171 158 169 164 164 159 68 259 158 169 159 154 154 216 110 214 208 114 41 288 38 286 280 42 619 ] om. D || 619 ] A || 619 ] AD ||162 610 A, D || 611 ] A 258 6565 7272 255 255 263 263 152 168 165 173 173 ] 152 168 155 162 165 155 122 205 200 123 118 279 50 277 272 51 46 612 ] D. A || 612 ] A, D || 613 256 257

66

612 ] 174 264 151 161 166 167 156 r

256 257 257 6666 264 264 151 166 167 167 156 174 151161 161 166 156 174 256 204 119 126 201 209 44 276 47 54 273 281 v148 v 260 5858 177 177 163 163 170 160 157 170 7070 6767 260 160 157 148 ( 53 ) A: 22 AD || 615 ] - 22A,9|| (ed. 616 202 203 120 210 43 53 274 275 48 ,4 282

269 5757 6363 266 266 175 175 145 179 147 176 55 269 145621 147 153 179 176 A ] ]153 278 40 124 121 206 630 112 285 271 52 49

||D621 || 618

205 200 200 123 279 5050 277 277 272 51 4646 122 205 123118 118 279 324 2 322 316 6 226 96 95 234 62492272232]51 om. D D.

|| 624

216 110 214 208 208 114 288 3838 286 286 280 280 4242 110 214 114 4141 288 A284 622 109 215 111 117 212 283 619 37 287 ] 39 om.45D |||| 619

]]

204 119 201 209 209 4444 276 276 4747 5454 273 273 281 281 119126 126 201 14 313 308 15 10 225 104 223 218 100 ] 105 enim D, v D (v. (5353 ) A: - 22v ,9 (ed. ll . 202 203 203 120 210 4343 53 274 275 22 282 120 210 202 274 275 282 4848,4 312 11 18 309 317 98 222 101 108 219 227 635

6215252 4949 278 206 112 285 271 271 278 4040 124121 121 206 112 285 124

621

]

A] || 618

623-4 AA|| ||619

]

r

A ||] 622 .] . .

AD || 625

]

fol. 33 v ). 993-998); D : 100r ,21 - 100r ,24. 622

( ]54 )

AD ||

]

] om. A || 614

616 ] || 608 ] D || 618 A || 610 A || 619-20 || 611 ]

] om. A || 614

ll . 993-998); D : 100 ,21 )- 100 ] (sc. AD ,24. || 616 r

D ||

]

D || 618 ]

]

]

A || 617 D || ] om. D || ] ( 54 ) ] AD || ]

A || 617

AD || 622 ] om. ||

om. (homoeotel.) A] || A ||] 619-20

A || 625

( 53 )

] om. D || 626 622

119

Text and translation

four sides, (strips of size) 2 by 4 which you will equalize in the manner explained.182 (§ 52. Equalizing the cross) (In order to equalize the central square) you proceed as follows.183 After putting in the centre the four numbers which we have indicated, you consider the row with a deficit of 3.184 Put, in the pair of cells adjacent to the (central square), on one of the two sides, the lesser of the 2 two numbers you have omitted after the two medians (thus n2 − 1) and the large number following the two large numbers you have written in the 2 centre (thus n2 + 4);185 and put, in the two cells adjacent to the (central square) on the other side (of this same vertical row), the lesser of the two 2 small numbers following all that (thus n2 − 4) and the larger of the two 2 large numbers following all that (thus n2 + 6); and put opposite to each its complement. Once you have done that, you will have equalized these six (pairs of) cells.186 †

n2 2

n2 2

−9

n2

+ 10

n2 2

2

n2 2

−3

n2 2

+4

n2 2

+2

n2 2

−1

+9

n2

+3

−8

n2 2 n2 2

2

n2 2

n2 2

+1

n2 2

−2

+5

n2 2

−4

−5

n2

+6

2

n2 2 n2 2

+7

n2 2

−7

−6

n2 2

+8

⋆

Next, you examine the row in deficit of 2.187 Put on one of the two sides, in the two cells adjacent to the (central square), the lowest 2 of the terms you have reached ( n2 − 6), then the larger of the two large 182

Using neutral placings, as seen in §§ 42–43. But, as said, we are first to eliminate the differences displayed by the central square. 183 As told in the previous section, we must now equalize the central square, already filled (§ 50), by means of two pairs of cells in each branch. Our addition, in brackets, of the numbers alluded to (and of the figure) should make the text less abstruse. 184 Marked with † in our figure. 2 2 185 Thus, the smallest number placed is now n2 − 2 and the largest, n2 + 4. 186 ‘these six cells’: we may consider the equalization on just one row since ipso facto the complements will equalize its conjugate. 187 Marked with ? in our figure. With the complements just placed, the next available 2 2 numbers are n2 − 6 and n2 + 7.

610 615

( 51 ) 120

Text and translation

605 615 615

605

23 615

( 52 )

2323

( 52 )

23

( 53 ) ( 53 ) ( 53 ) ) ( 53 ( 53 ) ( 53 )

620 620 610

620 620 620 620 610 620

620

625 615 625

625 625 625

( 52 ) A: 22r ,11 - 22v ,4 (ed. ll . 978-992); D : 100r ,13 - 100r ,21.

607 ] AD || 608 ] D || 608 ] A || 608 ] 270 56 268 262 60 149 180 146 178 172 150 r r v v r r r r 615 625 ( 52 ) A: 22 ,11 22 ,4 (ed. ll . 978-992); D : 100 ,13 100 ,21. ( 52 ) A: 22 ,11 22 ,4 (ed. ll . 978-992); D : 100 ,13 100 ,21. 609 ] A || 610 ] D || 610 ] A || 610 169 164 159 154 68 259 254 69 64 171 158 625 607 ] 150 AD || 608 ] ] D || D || 608 || ||608 ]] ] 607 ] 150 || 608 || 608 ] ] ] AAA ||611 608 268 268 262 262 180 180 178 172 172 56 56 60 60 149 149 146 AD || 610 ] AD A, 611D 270 178 146

( 53 ) D ||

( ]54 ) D54 || )|||| D (AD 258 65 72 255 263 152 168 155 162 165 173 609 ] A || 610 ] D || 610 ] A || 610 ] 609 ] A || 610 ] D || 610 ] A || 610 ] 259256 254257 171151 169166 164167 69 6669 6426464 158161 159 ) ) 259 254 171 169 164 68 158 159 154 A || 612 ( ]54 612 ]154174 ] A,23 D || 613 ] om. A || 614 ( 54 156 AD || 610 ] A, D || 611 ] A || 611 ] AD ( 54 )|| || AD || 610 ] A, D || 611 ] A || 611 ] AD 255260 26358 168 165 173 65 7065 72 6772 152177 155 162 258 255 263 168 165 173 152 155 162 163 160||157 AD 615170 148 ] A || 616 ] (sc. ) AD || 616 ] A || 617 625 ( 54 ) 612 ] A || 612 ] A, D || 613 ] om. A || 614 ] 612 ] A || 612 ] A, D || 613 ] om. A || 614 ] 25757 264266 166179 167147 66 6366 151175 161145 156 256 257 264 166 167 174 vv 151 161 r 174176 55256269 153 23 r r156 r

52)) A: A: 22 ,11 -- 22 22 ,4 ]22 ,11 D || 618ll . 978-992); ] AD||: 618 ] ,21. D || 618 ] om. D || ((52 (ed. 100 ,13 - 100 ]] ] (sc. ) AD || A 616 ] ]] A A || D 617 (sc. ) ]AD || ||616 || ||617 D || 608 608 A || 619 ] A || 619-20 ] ] ] ] DD ||A618 ] om. DD || || ] A] || A 618 || 618 ||||618 ] ]om. D || 610 610 ( 54( )53 ) 619|| ]610 ] om. || 619 A, || 619 A || 619-20 619 om. D ||D]619 ] ] A ||A619 || 619-20 AD 610 D || 611 r ] ] ]r A ||A611 ] AD ||] ] AD 165 v173 ] v 65 72 255 263630 152 168 155||162 (50 53 )272A: ,4 - 22 ,9 (ed. ll . 993-998); D : 100 ,21 - 100 ,24. 205 200 279 277 272 122 123 118 46 205202 200203 27943 277 123120 118210 50 53 5122 46 274 275 282 4851 D.]] D. 612 (] 53 ) 612 A || 612 ] A, D || 613 ] om. A || 614 256 257 66 264 151 161 166 167 156 174 204 201 209 276 281|| 621 119 126 44 201206 209112 276271 273 28140 119124 126121 44285 47 5247 54 4954 621 ] 278273 A ] A || 622 ] A || 622 v ] v v v 148 r r r r 170 70 67 260 58 177 163 157 AD ||275 615 (160 )615 A: 22 ,4 22 ,9|| (ed. . 993-998); : 100 100 ,24. ( 53 )53 22 ,4 22 ,9A(ed. ll] . ll993-998); D :D 100 ,21,21 - 100 AD ||A: ]||- 622 616 (sc. )-. AD A || 617 202 203 210 274 282 120 43 48 203111 210212 274 282 120117 43283 53 48 45 ]39275 A-284 A ]|| 623-4 . .,24.|| 616 ] om. ](homoeotel.) A || 287 109202215 3753 269 57 63 266630175 145 179 176 147 153 621 ]226278 ||D621 ||] 622 || )||622 || 618 ] || 618 A A D || 618] ] ] om. D 621 ] ]]49 A96 ||A || 621 ] A]AD || 622 A (A ||53 622 206 285 271 121 40 206316 28595 271 121322 112 6112 52 49232 40 324124 2124 234 9252 624 ]278om. D 624 ] || 625 ] A || 625 ] om. D || 626 (A53A ) 280 42 110 214 208 114 41 288619 38 286] om. D ||||619 A284 622 A||v623-4 ||619 623-4 ] 619-20 om. (homoeotel.) 619 ]] om. || ]] A|| A] || 215 111 212 283 287 109 117 37 223 ]39218 A45D ||150 622 . . .]. . . om. (homoeotel.) ||] || 215313 21210 117262 14 15 100 270 268 180287146 178 172284 56111308 60283225 14937104 ] 3945105 D (v. enim] D, fol.A33 ). 625 ( 53 ) ( 53 ) 205 200 123 118 279 50 277 272 51 46 D. 324 11322 322 316 234 232 226 2 18 6 9895 92 234 232108 226 92 96 624 ]D. om. || 624 ||: 625 ] Ar ,21. A || 625 ] om. || 626 r 96 v 624 D154 ||D 624 ADAD || D 625 || 625 ] om. DD || 626 312 222 219 227 254316309 169 68 2 259 69 6 317 6495171 158 159 (101 52 )]164om. A: 22 ,11 - 22 ,4 (ed.] ll ]. 978-992); 100r ,13] - 100 630

620 260208 177630 163 170 6721467 58114 160 157 14842 260 177 163 170 70 58 AD ||160 615 ]148AD AD ||157 615 ] A ||A 616 || 616 21670110 288 280 41 607 61938 286] om. || 619|| 608] 607 ]] D AD 56 268 262 60 149 180 146 178 172 150 269 266 175 179 176 57 63 145 147 153 269 266 175 179 176 55 57 63 145 147 153 630 279 50 277 272 122 205 200 123 630 118 630 ] ]51 ]] 46 D ||D 618 ||A 618 609 609 || 610] D.154 259 254 69 64 620 171 158 169 164 159 630 216 214 208 288 286 280 214126 208201 288276 28654 280273 110 114 42 110119 114209 41 4441 38 4738 42281 204

( 53 ) ( 53 ) ( 54 )

119 126 201 209 635 44 276 47 54 273 281 313 308 225 223 218 14 10 104 105 100(v. 313310 225152 223155 218162 10318 104107 100173 ]105 D enim 220 221 1215 97 102 (v. enim D, fol.fol. 33 v33 ). v ]). 258 263 165 65308311 7215255 vvD vv AD rr || 608 rr 625635 (53 53 A: 22 ,4 - 22 607 ] 228 || D, 608 D ] A || 608 ] D || (168 )) ]146 A: 22 270 180 172 150 ,9 (ed. ll . 993-998); D : 100 ,21 - 100 ,24. 202 56 203 268 210 60 274 275 178 282- 22 120 262 43 149 53 48 ,4 r v 312 309 317 222 219 227 11 18 98 101 108 309 317 222 219 227 11 256 18 98 101 108 231 217 224 16 257 13 314 4 635 106 94 - 22 ,4 (ed. ll . 978-992); D : 100r ,13 - 100r ,21. 264 166) 103 167 156 174 ( 52 A: 22 ,11 66 635 151 161 635 60949 ]] 164 A || 610 ] ] 610 ] A] || 610 A || 622 ] 621 A 154 621 A ||||] 621 A D|| ||622 68 158 206 69 285 171 271 278 159 124 259 121 254 112 64 52 169 40 310 3311 318 220 221 228 97 107 102 318320 220160 221157 228148 12260 97229 107163 102170 233 230 1 310323 912 91 93 99 177 70311 67 58635 607 ]A AD || 608 ] 623-4 D || 608. . . ] ] AA||] ||om. 608 ]] D|| || || AD || 610 ] A, D || 611 611 AD 270258 268 262 180 178 172 56 60 149 146 150 ] || 622 ] A || (homoeotel.) A 165 ]39 162 622 215 65 212 263 283 152 287 155 284|| 173 111 72 117 255 37 168 45 A 314 231 217 224 16 4 175 106 103 94 314 63 231 217145 224153 13 5713 4 266 106179 103147 94176 635 5516 269 609 ] A || 610 ] D || 610 ] A || 610 ] 254257 169166 164 68 2259256 158 159 154174 612 ]D A || 612 ] ] AD D ]|| 613 || 614 ] 624 om. AD ||||A,625 A || || 625] om. A ] om. D || || 626 167 161 156 322 31669 66 234 232 226 664264 95171151 92 96 624 ]]93om. D |||| 624 A D 323 320 229 233 230 1 110 9 114 91 99 22941 23338 23042 3 21439 208320 91288 93286 99280 216323 AD || 610 ] A, D || 611 ] A || 611 ] AD || v 258 31365 70 255 263 168 165 173 72 152 155 162 170 160 148enim 308 67 225 177 223 218 15 260 10 58 104 163 105 100(v. D (v. enim D, fol. 33 33 v ). ). ] AD 615D ] AD, || fol. 616 (sc. ) AD || 616 ] A || 617 ]] ||157 122 205 200 123 118 279 50 277 272 51 46 612 ] A || 612 ] A, D || 613 ] om. A || 614 ] 256269 264266 166179 167147 174176 v 151222 161145 156 55 153 30966 63 317 219 227 11 1825757 98 101 630175 || 618 ] AD||: 618 ] v ,3. D || 618 ] om. D || ( 54 ) 108 A: 22]v ,9 - 22 ,17D(ed. ll . 999-1006); 100r ,24 - 100 635 44 276 204 47 54 270 119 268 201 262 209 180 160 178 273 172 281 56 126 60 177 149 163 146 157 150 260 170 70 67 58 148 216 214 208 288 286 280 110 114 41 38 42 || 615 A || 616] ] || 619 (sc. ) AD || 616 A || 619-20 ] A || 617 ] 310 311 12 318 97 107 AD 220 221 102 228 ] 619 ] om. D || 619 A ] 627 ]159150 A || 627 ] D || 627 ] A || 627 ] 640 43 180 202 203 210 274 275 282 120 53 48 27068 56 259268 254262 171 145 169178 164172 69 60 64149 158146 154 269 175 55122 147 153 205 279 272 118 50179277 51 31463123 231 217 106 224 16 1357200 4266630 103 9417646 D || 618 ] ] A || 618 ] D || 618 ] om. D || 640 D. A172 ||165150 627 ] A || 628 ] AD || 628 ] A || 628 206 285 271 278 124 121 112 52 49 40 259 254 171 169 164 6825856 69 64 158 159 154 25560 263149 168146 173 65268 72262 152180 155178 162 280 110 114209 41 42281 276 273 119 44 323 320 229 23338 47 230D 1216204 3214126 9208201 91288 93286 99 619 ] 54 om. || 619 ] A || 619 ] A || 619-20 ] 640 640 640 215 212 283 287 284 109 259 111 117 37 39 45 v v r r ] A, D || 628-9 ] A 254 171 169 164 68 69 64 158 159 154 258 255 263 168 165 173 65 72 152 155 162 256 257 66 264 151 161( 53 166 ) 167 156 174 A: ,4 272 122 205202200203123120 11821027943 50 53277274 2755122 282- 22 ,9 (ed. ll . 993-998); D : 100 ,21 - 100 ,24. 48 46 640 D.174 324 65 322 316 234 232 226 27072 658152 95 92 96 162 256 257 264 166 167 66 151 161 156 ||163155 629 ]]170173 AA||||629 ] A || 629 ]A || 622 D || 629 260263 177168 67255 160 157165 148 621 621 ] ] ] A || 622 204 119124 126121201206209 47 52 54 49273 27828140 11244285276271 640 v174 v r r 313 308 225 223 218 1455256 1563264 10 104 105 100 166 167 156 260 177 163 170 70 67 58 160 157 148 269257 266151 175161 179 176 5766 145 147 153 ( 53 ) A: 22 ,4 22 ,9 (ed. ll . 993-998); D : 100 ,21 100 ,24. AD || 630 ] A, (sic) D || 630 ] D (v.A || 202 203 210 274 275 282 120 43 53 48 ] A || 622 ] A || 623-4 . . . ] om. (homoeotel.) 109 215 111 117 212 283 37 287 39 45 284 312 309 317 222 219 227 11 18 9841163 101 108 v 170 269 266 175 179 176 160 55 57 63 145 147 153 21670 214260 20858 288 286 280148 11067 114177 38157 42 ||33 621 ] A 621 ] A || 622 ] A enim cod. D, fol. ) || 631 ] ( corr.) A || 631 ] D || 52 92 49232]278 324124 2121322206316112 6 28595271234 22640 96 624 om. D || 624 ] AD || 625 ] A || 625 ] om. D||631 ||622 626 310 311 318 220 221 228 12 97 107 102 216 214 208 288 286 280 110 114 41 38 42 55 20557 20063 279145 277147 272153 122269 123266 118175 50179 51176 46 ]39218 A284A || 622 ] A || . . . ] om. (homoeotel.) A || ]] 45105 || 631 ] A || 632 ] post haec add. A : 1091421531311130811715212 v623-4 6401028322537104287223 100 D (v. enim D, fol. 33 ). 314 231 217 224 16 13 420941 106 103 94 205 200 279 277 272 122 123 118 50 51 46 204110 201114 27638 27342 281 119214 126208 44288 47286 54280 324312 2 1132218316309 6 31795 98234 22621996 ; D|| hab. 624 ] om. D227|| 624 ] AD 625 ], post A quod || 625add.: ] om. D || 626 22292101232108 635 4126 944 323 320 229 233 230 1 205 3203123 9120 9153277 93 9948 46 204 201 209 276 273 281 119 47 54 118 202200 210279 274272 27551 282 43 50 v 14 31331030831115 121031822597104107223 . 220218221105102100228

( 53 )) ( 54

( 54 ) ( 54 ) ( ( 5454 ) ) ( 54 ) ( 54 )

( 54( )53 )

]

D (v. enim D, fol. 33 ).

24 24 24 24 24

( 53 )

Text and translation

121

2

(numbers) you have reached ( n2 + 8); and (proceed) in like manner (as in the previous case) for the other side.188 Put opposite to each number its complement. Once you have done this, you will have equalized these six (pairs of) cells. At that point, you will equalize the remainder on the (four) sides, (of size) 4 by 2 on each side, using what I have explained to you.189 (§ 53. Case of order n = 10)190 (But) if what is left (in the cross), excepting the central square, is on each side 2 by 4, you will place that which will equalize the row in deficit of 3 in four cells on (just) one of the two sides of the (central square), and that which will equalize the row in deficit of 2 in four cells on (just) hone Magic squares in the tenth century 37 ofi its two sides: you do not need to do this with sharing out on its four sides. Once you have done that, you will equalize each remaining (group of) four cells in the two other sides using what I have explained.191 •

•

•

•

• •

• •

◊ ◊ ◊ ◊

• •

• •

• •

• •

(§ 54. Extension to higher orders) You will proceed likewise for remainders of 2 by 8 or 2 by 10.192 You [ 54 ]171 You will proceed likewise for a remainder of 2 by 8, or of 2 by will 172 (first) equalize the centre; (namely,) when there remain eight (cells), 10. You will (first) equalize the centre:173 when there remain eight you equalize the centre on two sides only, using two groups of four cells, (cells on each side), you will equalize on two sides only, each time with and when there remain ten (cells), you do it on all four sides, using pairs four cells; when there remain ten (cells), you will equalize on the four ofsides, cells.each (Then) what you remains, (consisting strips of time you withwill twoequalize cells. (Then) will equalize whatofremains, (consisting of strips of size) 2 by 4, as I have explained to you before. 188

As for the other side of the vertical branch, see above. We shall thus place the 2 2 n2 be for you,pair after thethus squares which we higher have filled lower It of will the next small ( n2 examining − 7, n2 − 8), − 8, and the of theand next 2 2 2 n acquainted n becoming with how to place the numbers and arrange them, large pair ( 2 + 9, 2 + 10). 189 to Each deduce from it,part with God and valuable assistance, whatp. remaining of the the help cross of receiving onehis neutral placing. See example, 157 (n = omitted 14). I have to explain; thinking about it will lead you to the purpose, 190 With,isonthe each side the central (2 × 2)174 square, a 2 × 4 rectangle; the equalization if such will ofofGod Most High. of the central square may then be performed on two sides only, see our figure (this is grouped together, the illustrations for what we have exby noHere meansare, necessary). 191 plained about odd and even (orders).175 Example on p.the 155. 192 Disregarding the 2 × 2 central square, thus case of, respectively, order 18 (with equalization of the central square on two of its sides only) and 22 (distributing the equalization among its four sides). Once again, this distinction is superfluous since a neutral placing does not need to occupy consecutive cells.

620

AD 610165 173 255 263 152 168 155||162 620 610

66

57

612 ] 264 620 151 161 166 167 156 174

122

260

58

63

266630175 145 179 147 153 176 ]

177 163 160||157 AD 615170 148 ]

620 214 208 114 630 41 288

38

619

286 280

42

50

277 272

201 209

276

47

51

D.

D || 611

A || 612

A || 616

D || 618

] om. D || 619

279 123 118 630

(AD 54 )|| ( 54 ] A,23 D || 613 ] om. A || 614 ( 53] ) ) (A53 ) Text translation ] and (sc. ) AD || 616 ] || 617 53 )D || ] A || 618 ] D || 618 ] (om.

A,

]

A || 619

]

46

A || 611

]

A || 619-20

]

273 281 v v r r ( 53 ) A: 22 625 282- 22 ,9 (ed. ll . 993-998); D : 100 ,21 - 100 ,24. 120 210 43 53 274 275 48 ,4 v r 615 625 ( 52621 ) 52A: 4922] r278 ,11 -40 (ed. ll . 978-992);] D : 100r ,13 A22|| ,4 621 A -||100 622,21. 206 112 285 271 44

54

]

]

( 53 )

A || 622 607 ] 150 ] || 623-4D || 608 . . ]. A || 608 ] DA || || A284||AD 622|| 608 ] A ] om. (homoeotel.) 111 117 180 37 178]39172 45 60 212 149283 146287 ( 53 ) 609 ] A || 610 ] D || 610 ] A || 610 ] 32269 31664 6171 95 226 169 158234 154 96 62492164232159 ] om. D || 624 ] AD || 625 ] A || 625 ] om.( 54 D ||) 626 ( 53 ) AD ||162 610 ] A || 611 ] (AD 54 )|| 218 104 100](v. enimA, 255 15263 10 168 152225 155223 ] 165105173D D, fol. 33 vD). || 611 625 ( 53 ) (]54 ) r A || 612 612 ] 219174 A,23D : 100 D ||r ,13 613- 100r ,21. ] om. A || 614 222 264 317 166(101 167108 66 309 151 98 161 52 ) 156 A: 22227 ,11 - 22v ,4 (ed. ll] . 978-992); ]

625

635

220 221 97163 107 260 1258 318 177635 170102 160607 148 AD ||157 615 ] AAD || 616 ) AD || ]616A || 608] ] 228 || 608 ] ] (sc. D || 608 ]A || 617 D || 56 268 262 60 149 180 146 178 172 150 314 231 224 4175635 266630 179106 176 94 145217 147103 153 609 ] D || 618 A ||] 610 A || 618 ] D ||] 610 D] || 618A || 610 ] om. D] || 259 254 69 64 171 158 169] 164 159 154 620229 91 233 93 99 230 630 3 9114320 41 288 38 286 280 42 ||162 610D 611 ] AD 619168AD ] om. 619] ] A,A || 619 D || 611 ] ] AA||||619-20 ] || 165|| 173 65 72 255 263 152 155 279 50 277 272 51 46 123 118 630 612 ] 174 A || 612 ] A, D || 613 ] om. A || 614 ] 256 257 66 264 151 161 166 D. 167 156 201 209 44 276 47 54 273 281 70 67 260 58 177 163 160||157 615 A || 616 ] (sc. ] A || 617 v 170 148 v ] r r ) AD || 616 ( 53 )AD A: 274 275 22 282- 22 ,9 (ed. ll . 993-998); D : 100 ,21 - 100 ,24. 120 210 43 53 48 ,4 269 57 63 266630175 145 179 147 153 176 ] A || 621 D || 618 ]] A || 618 D ||] 618 621 A || 622 ] A ]||om. 622 D || 206 262 271 278 172 112 285 52 146 49 ] 178 40 150 268 180 60 149 110 214 208 114 41 288 38 286 280 42 ] om. D || 619] ] A || A || 619 ] A ||(homoeotel.) 619-20 ] 640 37 287619 622 623-4 ... ] om. A || 212 284|| 154 39 164 45 A 69 283 64 171 158 ]169 159 205 200 123 118 279 50 277 272 51 46 640 D.173 234 232] om. 226 165 6 263 95 152 92 155 96 624 D || 624 ] AD || 625 ] A || 625 ] om. D || 626 255 168 162 119 126 201 209 44 276 47 54 273 281 640 v v v r r 225 151 223( 53 218 15 66 10 264 104 161 105 100(v. 166 174enim 156 ]274) 167 D D, fol. 33 ). A: 202 203 120 625 210 43 53 275 22 282- 22 ,9 (ed. ll . 993-998); D : 100 ,21 - 100 ,24. 48 ,4 r v r 309 260 317 58 222 (163 219 227 98 177 101 108 170 160 15722 52621 ) 52A: ,11148-A22|| ,4 (ed. ll . 978-992);] D : 100r ,13 621 A -||100 622,21. ] A || 622 635 112 285 271 124 121 206 49 ] 278 40 220 221 228 12 318 97 107 102 175 179 176 57 63 266 145 147 153 635 607 ] 150 ] || 623-4D || 608 . . ]. A || 608 ] DA || || A284||AD 622|| 608 ] A ] om. (homoeotel.) 215 268111 262117 180 37 178]39172 45 60 212 149283 146287 314 208 231 217 224 280 4 635 106 103 94 42 214 286 114 41 288 38 609 ] 96 ] AD ||D625 || 610 ] ] A || A || 610 ] om. ]D || 626 234 226 2 32269 31664 6171 95 62492164232159 ] om. D || 624A || 610] 625 169 158 154 320 118 229 279 3 9 123 91 233 93 272 99 230 50 277 51 46 vD || 611 AD ||162 610 A, ] A || 611 ] AD || 313 308 218 104 255 15263 10 168 165105 173100] 152225 155223

( 54 )

( 53 )

( 54 ) ( 54 ) ( ( 5354 ) ) ( 53 ) ( 53 )

( 53 ) ( 54 )

D (v. enim D, fol. 33 ). 24 24 D || 613 ] A, ] om. A || 614 ] 999-1006); D24 : 100r ,24 - 100v ,3. ] (sc. ) AD || 616 ] A || 617 ] D || 627 ] A || 627 ] 54 )D || ] D || 618 ] A || 618 ] D || 618 ]( om. 39172 ||45150 627 ] A || 628 ] AD || 628 ] A( 54 || 628 268111 262 180 178 60 149 146 171283 169287 164A 69117 64212 15837 159 154284 ) ] 323 214 3208 911432041 229 288 9138 233 286 93280 9942 230 619 ] om. D || 619 ] A || 619 ] A || 619-20 640 3 95 234 92 232 226 96 27 32269 A, D || 628-9 ] A 25531664 263 6171 168 169 165 154 173 152 158 155 164 162 ]159 ( 54 ) 123 118 279 50 277 272 51 46 640 D. 255 168 165 152 155 162 26410 166223 167218 174100 66 15263 151225 161 156 ||104 629 ] 105173 A || 629 ] A || 629 ] D || 629 ] 201 209 44 276 47 54 273 281 ]

201 209 44 276 47 54 273 281 222 11 1866 309 612 ] 219174227vA || 612 264 317 166101 167108 151 98 161 156 4 9635 A:27522v48 ,9 -282 22 ,17 (ed. ll . 120 210 43 ( 54 53 )274 310 311 318 220 221 12 97 107 260 58 177 163 160||157 148228 AD 615170102 ] A || 616 627 ] 27815040 A || 627 285180 2711465217849172 268 13 26220660112 149 16 4175 231 266630 179106 176 94 57 63 314 145217 147103 153224

640

219 108 v174 v 264 166 66 151 161 156 260309 17798 163 170 58317 160101 157 148227 (222 53 )||167 A: 22 AD 630 ] ll . 274 275 282- 22 ,9 (ed. 120 210 43 53 48 ,4 318 220 221 228 12 97 107 102 v 266 177 175 179 157 176 160 63 58 145 147 ]170 153 157 260 6 163 621 A ||33 621 enim D,148 fol. ) || 631 52 cod. 49 278 40 268 206 262 112 180 146 178 172 150 60 285 149 271 231 217 224 106 103 94 214 63 208314 288 286 280 266 179 176 114 4175 41 38 42 57 145 147 153 640 37 287 ]39] 45 A284 || 622 A || 631 ] 111 117 171 158 169 164 159 154 69 212 64 283 279229 277233 27293280 123 9114 11832041 50 9138 51 9942 46230 214 3208 288 286 322 316 226 6 152 95 234 92 232 96 624155 ] om. D || 624 255 263 168 165 173 162

201 118 209 279 276 277 273 46 281 44 50 47 272 54 51 123 15 100 . 218 ]167 105 264 225 166 174(v. 66 10 151 104 161 223 156 D 210 44 274 54 275 273 282 120 209 43 276 53 47 48 281 201 309 98 222 260 317 177 163 101 170 227 58 160 108 157 219 148 Post tit. 9635 206 210 285 53 271 274 278 282 112 43 52 275 49 48 40 120 318 220 221 228 12 97 107 102 179 147 153 176 57 63 266 175 145 quadratorum. 212 285 283 271 287 49 284 111 206 117 112 37 52 39 278 45 40 314 231 217 4 106 214 208 114 41 288 38 103 286 224 280 94 42 322 117 316 212 234 287 232 45 226 284 6 283 95 37 92 39 96 111 3 123 9 320 93 99 279 91 277 272 118 229 50 233 51 230 46 225 234 223 232 218 226 15 10 104 92 105 100 322 316 6 3 95 96 201 209 44 276 47 54 273 281 309 10 317 225 222 223 219 100 227 98 9104 101 218 108 105 4 15 120 210 43 53 274 275 48 282 318 220 221 228 12 97 107 102 309 317 98 222 101 108 219 227 206 112 285 271 52 49 278 40

993-998); D : 53 100r ,21 - 100 A, (sic) Dr ,24. || 630 ]

A( || 622 corr.) A || 631 ]

]

] A || 623-4 A || 632

]

enim D, fol. 33 v ).

; D|| hab. AD 625

24 24 24

24 53

]

]

D (v.

( 54 )

622 DA|| ||631

. . . ] post haec ] om. (homoeotel.) A || add. A: ], post A quod || 625add.:

( 53 )

] om. D || 626

praeb. A (fol. 23 r - 27 r ) figuras

Text and translation

123

size) 2 by 4, as I have explained to you before.193 After examining the squares which I have constructed and being acquainted with how to place the numbers and arrange them, you will (be able) to rely on it for what I have explained to you.194

193

Two examples, see pp. 159 & 161 (n = 18). Rather (and against the MSS.): ‘what I have not explained to you’. That is, if this specifically refers to the method of the cross, extension of it to orders n = 8t + 2 and n = 8t + 6 with t ≥ 3. 194

80

Translation

124

Text and translation

Excerpta e cod. D, fol. 95 v -96 v . Inc. (D, 95 v )

80

Translation Excerpta e cod. D, fol. 95 v -96 v . Inc. (D, 95 v )

(. . .) (. . .)

Excerpta e cod. D, fol. 95 v -96 v . Inc. (D, 95 v )

Excerpta e cod. D, fol. 95 v -96 v .

(. . .)

Inc. (D, 95 v )

(A, p. 312 ; a22, p. 165)

Excerpta e cod. D, fol. 95 v -96 v . Inc. (D, 95 v )

Excerpta e cod. D, fol. 95 v -96 v . (. . .)

(215 ; D, fol. 96 ) v

Inc. (D, 95 v )

(A, p. 312 ; a21, p. 165) Excerpta e cod. D, fol. 95 v -96 v . Inc. (D, 95 v )

Excerpta e cod. D, fol. 95 v -96 v . Inc. (D, 95 v )

(A, p. 312 ; a22, (. . .)p. 165) (. . .)

(215 ; D, fol. 96v )

(215 ; D, fol. 96v )

(A, p. 312 ; a23, p. 165)

(A, p. 312 ; a22, p. 165) 61

(A, p. 312 ; a21, p. 165) (A, p. 312 ; a24, p. 167 ; D, fol. 63v )

(A, p. 312 ; a21, p. 165)

61

(A, p. 312 ; a23, p. 165)

60

125

Text and translation

Division into equal subsquares There are six examples of composite squares with equal subsquares all displaying the same magic sum. Their filling always follows the same pattern, namely that of the bordered squares of even orders already seen; we have added each time, as a reminder, the arrangement of the small numbers in the borders (except for the 4×4). Since subsquares displaying same sums can be filled in any order, we have added a table showing this order for the transcription (remember that the squares are filled with the descending sequence of smaller numbers, thus the first subsquare to be 2 filled is that containing n2 ).

In MS. D, all these squares are given a heading, later on also a short ‘notice’. Their characteristics are pointed out, together, at times, with further remarks. Here the Arabic text will be given below the figures of 60 87 82 61 28 119 114 29 68 79 74 69 the squares. 86

57

64

83

118

25

32

115

78

65

72

75

In his introduction to this part, the author of the compilation of MS. 63 84 85 58 31 116 117 26 71 76 77 66 D tells us that he finds this arrangement ‘more general’ than the arrange81 62 59 88 113 30 27 120 73 70 67 80 ment with subsquares displaying unequal magic sums, for, in this case, 12 135 130 95 90 whereas 53 20 127it 122 21 a 2 × 2 arrangement of subsquares13is 52possible would be impossible with subsquares134displaying sums. On17the hand, 9 16 131 unequal 94 49 56 91 126 24 other 123 he adds, the other type of15 arrangement admits subsquares of odd orders, 132 133 10 55 92 93 50 23 124 125 18 with sums in arithmetical progression (see Commentary, below p. 213). 129 14 11 136 89 54 51 96 121 22 19 128 He also notes the main characteristic of the present type of arrangement: 111 106 37 4 143 138 5 98 45 the configuration of the 36 basic subsquare (as¯ as) will44be103 repeated in each 33 be 40 filled 107 142with 1 8 139 102 41 48 sequence 99 subsquare, half of which110will the continuous of small numbers and the other half with the continuous sequence of their 39 108 109 34 7 140 141 2 47 100 101 42 complements. As an example, he presents the case of the 4 × 4 square 105 38 35 112 137 6 3 144 97 46 43 104 producing the 8 × 8 composite square. 28

39

34

29

12

55

50

13

38

25

32

35

54

9

16

51

31

36

37

26

15

52

53

10

33

30

27

40

49

14

11

56

4

15

10

5

4

63

58

5

20

47

42

21

14

1

8

11

62

1

8

59

46

17

24

43

7

12

13

2

7

60

61

2

23

44

45

18

9

6

3

16

57

6

3

64

41

22

19

48

4

14

7

9

15

1

12

6

10

8

13

3

5

11

2

16

126

Text and translation

(D, fol. 94v )

(12 ⇥ 12) A : 26v (ed. p. 314); D : 96v . tit. A :

tit. D : A latere quadrati in A inven. (2, 11) ]

A ||

A || (5, 2) ] (id.) D.

(sc. 2 quadrata).

(2, 8) ] corr. ex

A ||

(3, 1) ]

A ||

(10, 6) ]

A ||

(v. præced. loculum) D ||

(12 ⇥ 12) A : 27r (ed. p. 315); D, 96v . tit. A : tit. D :

In ima pag. A rec. manu: 12 ⇥ 12 = 144.

(4, 4) ] (4, 1) ]

A ||

(6, 3) ] quasi

(id.) D || (5, 1) ]

127

Text and translation

23

126

20

124 118

24

59

90

56

88

82

60

117

32

115 110

33

28

81

68

79

74

69

64

26

114

29

111 119

62

78

65

72

75

83

25

35

112 113

30

120

61

71

76

77

66

84

123 109

34

31

116

22

87

73

70

67

80

58

121

19

125

21

27

122

85

55

89

57

63

86

41

108

38

106 100

42

5

144

2

142 136

6

99

50

97

92

51

46

135

14

133 128

15

10

44

96

47

54

93

101

8

132

11

129 137

43

53

94

95

48

102

7

17

130 131

12

138

105

91

52

49

98

40

16

13

134

4

103

37

107

39

45

104 139

1

143

3

9

140

7

14

13

53

54

55

8

36

59

60

5

62 63

3

1

1 57 2 14

13

51 53

54

52 55

56

6

412

11

10

v

61.

ii

.

.

vi

.

3iv

.

.

v

x

viii 2 .

i

viii

.

.

vii 64vii

.

.

ii

.

9.

58.

vi

iii

.

iv

.

i

.

iii

ix

.

8 59

60

5

4

61

62

3

63

2

1 57

56

6

4

7

141 127

18

64 51

52

12

11

10

9

58

10

100

2

98

5

94

88

15

84

9

18

25

32

31

71

72

73

74

26

83

85

24

37

60

66

34

68

38

77

16

Commentary in D: Square of 14 order × 623subsquares 78 n59= 12, 46 with 57 52four 47 6 42 87 (musammat.¯ at). (Here and in89 what follows, the various magic sums are 22 40 56 43 50 53 61 79 12 indicated.) 90

80

39

49

54

55

44

62

21

11

8

81

65

51

48

45

58

36

20

93

95

19

63

41

35

67

33

64

82

6

4

75

69

70

30

29

28

27

76

97

128

Text and translation

(12 ⇥ 12) A : 26v (ed. p. 314); D : 96v . tit. A :

tit. D :

(216 ; D, fol. 85v ; A, p. 314)

A latere quadrati in A inven. A ||

(2, 11) ]

A || (5, 2) ] (id.) D.

(sc. 2 quadrata).

(2, 8) ] corr. ex

A ||

A ||

(10, 6) ]

A ||

(4, 4) ]

(v. præced. loculum) D ||

(3, 1) ]

(4, 1) ]

A ||

(6, 3) ] quasi

(id.) D || (5, 1) ]

(12 ⇥ 12) A : 27r (ed. p. 315); D, 96v . tit. A : tit. D :

In ima pag. A rec. manu: 12 ⇥ 12 = 144. (10, 12) ] A || ex

(1, 7) ] A ||

A ||

(2, 1) ]

(16 ⇥ 16) D : 97r . tit. :

(1, 9) ] corr. ex

A ||

(7, 5) ]

A ||

(iter. in marg.) A ||

A ||

(5, 1) ]

(8, 5) ]

A ||

A ||

(9, 1 ]

(5, 9) ] (4, 4) ] A.

A ||

A ||

(6, 8) ] (9, 4) ] corr.

129

Text and translation

60

87

82

61

28

119 114

29

68

79

74

69

86

57

64

83

118

25

32

115

78

65

72

75

63

84

85

58

31

116 117

26

71

76

77

66

81

62

59

88

113

30

27

120

73

70

67

80

12

135 130

13

52

95

90

53

20

127 122

21

16

131

94

49

56

91

126

17

24

123

15

132 133

10

55

92

93

50

23

124 125

18

129

14

11

136

89

54

51

96

121

22

19

128

36

111 106

37

4

143 138

5

44

103

98

45

110

33

107 142

139 102

41

48

99

39

108 109

34

105

38

112 137

134

9

40

35

7

1

8

140 141 6

3

2

47

100 101

42

144

97

46

104

43

28

39

34

29

12

55

50

13

38

25

32

35

54

9

16

51

31

36

37

26

15

52

53

10

33

30

27

40

49

14

11

56

4 8 63

58

35

20

477

42

21

62

1

8

59

46

17

24

43

60

61

2

23

44

45

18

19

48

2

7 57

5

6

3

6

9

64

41

1

4

22

Commentary in D. Square n = 12, with 4 × 4 subsquares. Remark. Every other subsquare is filled in turn; same below, pp. 135 & 137. This example is mentioned by B¯ uzj¯an¯ı (together with the 8 × 8 square seen above, pp. 124–125), see edition of his text, pp. 185–186 and 241, or Magic squares in the tenth century, B.26.

130

Text and translation

(218 ; D, fol. 97r )

(12 ⇥ 12) A : 26v (ed. p. 314); D : 96v . tit. A :

tit. D : A latere quadrati in A inven.

(2, 8) ] corr. ex

A ||

(2, 11) ]

(sc. 2 quadrata).

A || (5, 2) ] (id.) D.

A ||

A ||

A ||

(10, 6) ]

(4, 4) ]

(v. præced. loculum) D ||

(3, 1) ]

(4, 1) ]

A ||

(6, 3) ] quasi

(id.) D || (5, 1) ]

(12 ⇥ 12) A : 27r (ed. p. 315); D, 96v . tit. A : tit. D :

In ima pag. A rec. manu: 12 ⇥ 12 = 144. (10, 12) ] A || ex

(1, 7) ] A ||

A ||

(1, 9) ] corr. ex

A ||

(2, 1) ]

(7, 5) ]

A ||

(iter. in marg.) A ||

A ||

(5, 1) ]

(8, 5) ]

A ||

A ||

(5, 9) ] (4, 4) ]

(9, 1 ]

A.

||

(8, 6) ]

A ||

A ||

(6, 8) ] (9, 4) ] corr.

(16 ⇥ 16) D : 97r . tit. :

(9, 13) ] quasi ut vid. (16 ⇥ 16) D : 97v . tit. :

||

(15, 11) ]

||

(3, 7) ]

ut vid. ||

(5, 5) ]

131

Text and translation

103 110 109 149 150 151 152 104 39

46

45

213 214 215 216

40

102 115 146 112 144 138 116 155

51

210

48

156 137 124 135 130 125 120 101 220 201

60

100 118 134 121 128 131 139 157

36

54

158 117 127 132 133 122 140

222

53

38

99

208 202

52

219

199 194

61

56

37

198

57

195 203 221

63

196 197

58

204

35 34

64

159 143 129 126 123 136 114 98

223 207 193

62

59

200

50

97

33

47

209

49

55

206 224

153 147 148 108 107 106 105 154 217 211 212

44

43

42

41

7 6

141 111 145 113 119 142 160

14

13

245 246 247 248

19

242

16

252 233

28

8

71

240 234

20

251

231 226

29

24

5

227 235 253

ix .

78 .

70

77

ii

83

vii

xiv

vi

v

.

178 xiii

181 182 183 184 v

80 .

.

.

176 170 .

.

xv

84 .

72 .

187

92

167 162

93

68

166

89

163 171 189

95

164 165

ii

.

69

vi

.

v

x

v

90

172

67

.

230

25

254

21

31

228 229

26

236

3

i

viii

30

27

232

18

2

94.

91 .

168 ii

82 .

15

241

17

23

238 256 . 65 . 173. 79. 177 vi

81 iii

87 .

174 192 viii iv ii

249 243 244

12

11

10

76.

75 iii

74 ix

. 73

. . 186

.

xii

xi

x

ix

.

iv

vii

xiii

.

xvii

.

1

237

9

xii xi

.

4

3

2

iv

.

85

viii

iv .

191 . 175 161 vii

vii

250 vi 185 i 179. 180i .

1

.

190

.

96

.

22

.

86

.

x

viii xviii

188 169

xvi

88

218

4

255 239 225

32

205

.

.

.

.

.

66 . iii

.

.

i

.

iii

.

vii

xiv

xiii

.

.

.

.

viii

vi

v

.

ii

.

.

vi

.

.

.

iv

.

.

v

x

v

iv

viii

.

i

viii

.

.

.

.

vii

vii

.

.

ii

.

iii

.

.

.

vi

iii

.

iv

ii

i

.

i

.

iii

ix

.

.

.

.

.

xii

xi

x

ix

.

Commentary in D. Square n = 16, with 8 × 8 subsquares.

xiv

132

Text and translation

(219 ; D, fol. 97v )

(12 ⇥ 12) A : 26v (ed. p. 314); D : 96v . tit. A :

tit. D : A latere quadrati in A inven.

(sc. 2 quadrata).

(2, 8) ] corr. ex

A ||

(2, 11) ]

A || (5, 2) ] (id.) D.

A ||

A ||

A ||

(10, 6) ]

(4, 4) ]

(v. præced. loculum) D ||

(3, 1) ]

(4, 1) ]

A ||

(6, 3) ] quasi

(id.) D || (5, 1) ]

(12 ⇥ 12) A : 27r (ed. p. 315); D, 96v . tit. A : tit. D :

In ima pag. A rec. manu: 12 ⇥ 12 = 144. (10, 12) ] A || ex

(1, 7) ] A ||

A ||

(1, 9) ] corr. ex

A ||

(2, 1) ]

(7, 5) ]

A ||

(iter. in marg.) A ||

A ||

(5, 1) ]

A ||

(8, 5) ]

A ||

(5, 9) ] (4, 4) ]

(9, 1 ]

A.

||

(8, 6) ]

A ||

A ||

(6, 8) ] (9, 4) ] corr.

(16 ⇥ 16) D : 97r . tit. :

(9, 13) ] quasi ut vid.

||

(15, 11) ]

||

(3, 7) ]

(16 ⇥ 16) D : 97v . tit. :

(7, 11) ] e corr. (forsan ex

) ||

(9, 5) ] quasi

.

ut vid. ||

(5, 5) ]

133

Text and translation

52

207 202

53

206

49

167 162

93

199 194

61 124 135 130 125

203 166

89

163 198

57

64

195 134 121 128 131

55

204 205

50

164 165

90

196 197

58 127 132 133 122

201

54

208 161

94

168 193

62

200 129 126 123 136

76

183 178

77

231 226

29 108 151 146 109 44

215 210

45

182

73

179 230

25

32

227 150 105 112 147 214

41

48

211

79

180 181

74

228 229

26 111 148 149 106 47

212 213

42

177

78

184 225

30

232 145 110 107 152 209

46

216

20

239 234

21

175 170

85

238

17

235 174

81

23

236 237

18

233

22

68

56

51

80

75

92

95

28

31

91

27

63

43

37 116 143 138 117

171 222

33

40

219 142 113 120 139

172 173

82

220 221

34 119 140 141 114

240 169

86

176 217

38

224 137 118 115 144

191 186

69

255 250

190

65

187 254

71

188 189

66

185

70

192 249

19

72

67

87

4

7

1

88

83

8

5

3

39

2

35

100 159 154 101 12 97 104 155 246

251 158

252 253 6

36

59

223 218

24

84

96

60

103 156 157

256 153 102 99

10

5

9

1

7

13

3

11

14

6

12

2

8

16

4

15

98

15

160 241

247 242 9

13

16

243

244 245

10

14

248

11

Commentary in D. Square n = 16, with 4 × 4 subsquares. It is noted that the 8×8 and 12×12 subsquares formed by associating contiguous 4 × 4 subsquares are magic as well.

134

Text and translation

(220 ; D, fol. 98r )

(18 ⇥ 18) D : 98r . tit. :

(13, 18) ] (20 ⇥ 20) D : 98v . tit. :

||

(13, 17) ]

||

(14, 14) ]

||

(16, 7) ]

||

(10, 5) ]

.

135

Text and translation

131 198 128 196 190 132 59

270

56

189 140 187 182 141 136 261

68

259 254

69

134 186 137 144 183 191

62

258

65

255 263 152 168 155 162 165 173

133 143 184 185 138 192

61

71

256 257

66

264 151 161 166 167 156 174

195 181 142 139 188 130 267 253

70

67

260

58

193 127 197 129 135 194 265

269

57

63

266 175 145 179 147 153 176

72

60 149 180 146 178 172 150 64

171 158 169 164 159 154

177 163 160 157 170 148

23

306

20

297

32

295 290

33

26

294

29

291 299 116 204 119 126 201 209

25

35

292 293

30

300 115 125 202 203 120 210

303 289

34

31

296

22

301

19

305

21

27

302 211 109 215 111 117 212 283

77

252

74

250 244

243

86

241 236

87

80

240

83

79

89

238 239

249 235

88

247

251

73

2

8

5

304 298

55

268 262

36

24 113 216 110 214 208 114 41

288

38

207 122 205 200 123 118 279

50

277 272

51

44

276

47

273 281

43

53

274 275

48

282

52

49

278

40

37

287

39

45

284

234

92

232 226

96

28

213 199 124 121 206 112 285 271

78

5

324

82

315

14

313 308

15

237 245

8

312

11

309 317

84

246

7

17

310 311

12

318

85

242

76

16

13

314

4

75

81

248 319

323

3

9

90

6

3

9

321 307 1

2

322 316

18

1

4

54

42 46

6

95

10

225 104 223 218 105 100 98

222 101 108 219 227

97 107 220 221 102 228 231 217 106 103 224

320 229

233

91

93

99

.

ii

.

.

vi

.

iv

.

.

v

x

viii

.

i

viii

.

.

vii

vii

.

.

ii

.

.

.

vi

iii

.

iv

.

i

.

iii

ix

.

v

7

286 280

94 230

10

100

2

98

5

94

88

15

84

9

18

25

32

31

71

72

73

74

26

83

90

80

39

49

54

55

44

62

21

11

8

81

65

51

48

45

58

36

20

93

95

19

63

41

35

67

33

64

82

6

4

75

69

70

30

29

28

27

76

97

Commentary in D. Square n = 18, with 6 × 6 subsquares. It is noted 85 24 37 60 66 34 68 38 77 16 that the 12 × 12 squares comprising four adjacent squares are magic 78 59 46 57 52 47 42 23 87 as well; the next figure has a similar14remark. Observe that in both the 89 above, 22 40 p. 56 129). 43 50 53 61 79 12 subsquares are filled alternately (see

136

Text and translation

(221 ; D, fol. 98v )

(18 ⇥ 18) D : 98r . tit. :

(13, 18) ]

||

(13, 17) ]

||

(14, 14) ]

||

(16, 7) ]

||

(10, 5) ]

.

(20 ⇥ 20) D : 98v . tit. :

(12, 20) ]

||

(17, 12) ] quasi (10 ⇥ 10) D : 102r . tit. :

(20, 18) ] ||

(13, 9) ]

||

||

(13, 17) ] ||

(13, 4) ]

||

||

(9, 14) ] (3, 1) ]

.

(12, 14) ]

||

137

Text and translation 196 207 202 197 92

311 306

93 188 215 210 189 84

319 314

85 180 223 218 181

206 193 200 203 310

89

96

307 214 185 192 211 318

81

88

315 222 177 184 219

199 204 205 194 95

308 309

90 191 212 213 186 87

316 317

82 183 220 221 178

201 198 195 208 305

94

312 209 190 187 216 313

86

320 217 182 179 224

91

83

76

327 322

77 172 231 226 173 68

335 330

69 164 239 234 165 60

343 338

61

326

73

80

323 230 169 176 227 334

65

72

331 238 161 168 235 342

57

64

339

79

324 325

74 175 228 229 170 71

332 333

66 167 236 237 162 63

340 341

58

321

78

328 225 174 171 232 329

70

336 233 166 163 240 337

62

344

75

67

59

156 247 242 157 52

351 346

53 148 255 250 149 44

359 354

45 140 263 258 141

246 153 160 243 350

49

56

347 254 145 152 251 358

41

48

355 262 137 144 259

159 244 245 154 55

348 349

50 151 252 253 146 47

356 357

42 143 260 261 138

241 158 155 248 345

54

352 249 150 147 256 353

46

360 257 142 139 264

51

43

36

367 362

37 132 271 266 133 28

375 370

29 124 279 274 125 20

383 378

21

366

33

40

363 270 129 136 267 374

25

32

371 278 121 128 275 382

17

24

379

39

364 365

34 135 268 269 130 31

372 373

26 127 276 277 122 23

380 381

18

361

38

368 265 134 131 272 369

30

376 273 126 123 280 377

22

384

35

116 287 282 117 12 286 113 120 283 390

391 386 9

16

27

13 108 295 290 109

4

387 294 105 112 291 398

119 284 285 114 15

388 389

10 111 292 293 106

281 118 115 288 385

14

392 289 110 107 296 393

11

7

399 394 1

8

396 397 6

1

14

2

15

3

16

4

17

5

18

6

19

7

20

8

21

9

22

10

23

11

24

12

25

13

3

5

100 303 298 101

395 302 2

19

97 104 299

103 300 301

98

400 297 102 99

304

Commentary in D. Square n = 20, divided into twenty-five 4 × 4 subsquares, with 8 × 8, 12 × 12 and 16 × 16 subsquares also magic.

138

Text and translation

(225 ; D, fol. 102r )

(18 ⇥ 18) D : 98r . tit. :

||

(13, 18) ]

||

(13, 17) ]

(14, 14) ]

||

(16, 7) ]

||

(10, 5) ]

.

(20 ⇥ 20) D : 98v . tit. :

||

(12, 20) ] (17, 12) ] quasi

(20, 18) ] ||

(13, 9) ]

||

||

(13, 17) ] ||

(13, 4) ]

||

||

(9, 14) ] (3, 1) ]

(12, 14) ]

||

.

(10 ⇥ 10) D : 102r . tit. :

(1, 8) ]

(227 ; D, fol. 103v )

ut vid .

(14 ⇥ 14) D : 102r . tit. :

(14, 10) ] corr. ut vid. ex corr.

et postea

||

(12, 5) ]

pr. scr., mut. in

quod iterum

139

Text and translation

Division into unequal parts 78

121 116

79

120

75

113 108

87

189

94

105 100

95

117 190 112

83

90

109

7

104

91

98

101

81

118 119

76

191

89

110 111

84

6

97

102 103

92

115

80

122

5

107

88

85

114 192

99

96

93

106

12

186 187

9

21

180

18

178 172

22

16

182 183

13

62

137 132

63

171

30

169 164

31

26

70

129 124

71

136

59

66

133

24

168

27

165 173 128

67

74

125

65

134 135

60

23

33

166 167

28

174

73

126 127

68

131

64

61

138 177 163

32

29

170

20

123

72

69

130

185

11

10

188 175

17

179

19

25

176 181

15

14

184

38

161 156

39

46

153 148

47

193

54

145 140

55

160

35

157 194 152

43

50

149

3

144

51

58

141

41

158 159

36

195

49

150 151

44

2

57

142 143

52

155

40

162

1

147

48

154 196 139

56

146

82

77

42

37

1 2

3

8

4

1 2

2

45

1s4 −α

4

9

5

s1 −α

10

11 1 3

8

18

33

34

10 s2 +3α

11

16

86

1 3

7

63

s3 −α

6

53

v

s −2α . 3 ii .

.

vi

.

iv

.

.

v

x

viii

.

i

viii

.

.

vii

vii

s2.

.

ii

.

.

.

vi

iii

.

iv

.

iii

ix

.

.

i

s1

s4 +2α

s2 +2α

s1 +α

s4

s3 −3α

s3

s4 +α

s1 −2α

s2 +α

10

100

2

98

5

94

88

15

84

9

18

25

32

31

71

72

73

74

26

83

85

24

37

60

66

34

68

38

77

16

vii Square n = 14, 14 59 22 37 in. 12 Commentary D. with 6 ×476 square, four 78 one 59 central 46 57 52 42 23 87

. ix 4 × 4 subsquares in the corners, separated by 4 × 4 subsquares (and 89 22 40 56 43 50 53 61 79 12 . x 54 strips). 27 iii 44 5. . i

1

48

.

iv

31

viii

.

vi

50

.

v

ii

.

90

80

39

49

54

55

44

62

21

11

8

81

65

51

48

45

58

36

20

93

95

19

63

41

35

67

33

64

82

6

4

75

69

70

30

29

28

27

76

97

140

Text and translation

(226 ; D, fol. 103r )

(16 ⇥ 16) D : 103r . tit. :

(7, 15) ] vid. ||

||

(1, 3) ]

(16 ⇥ 16) D : 103v . tit. :

(12, 13) ] ut vid. ||

|| (1, 1) ]

(13, 7) ] ut vid.

||

(1, 4) ]

||

(2, 4) ]

ut

1

97

94

2

Text and 5 translation

223 218

37 115 146 112 144 138 116

155 106 153 148 107 102 222

33

40

219 137 124 135 130 125 120

100 152 103 110 149 157

220 221

34 118 134 121 128 131 139

99 109 150 151 104 158 217

38

224 117 127 132 133 122 140

161 147 108 105 154

242 243

13

95 101 160 241

15

244 141 111 145 113 119 142

164

162 156

159

93

28

231 226

29

230

25

31

163

96

36

39

16

35

14

143 129 126 123 136 114

249

88

171 166

89

12

245

44

215 210

45

227 250

7

170

85

167 246

11

214

41

48

211

228 229

26

251

6

91

168 169

86

247

10

47

212 213

42

225

30

27

232

5

252 165

90

172

9

248 209

46

43

216

53

208

50

206 200

54

4

254 255

1

71

190

68

188 182

72

199

62

197 192

63

58

253

256 181

80

179 174

81

76

56

196

59

193 201

20

239 234

21

74

178

77

175 183

55

65

194 195

60

202 238

17

24

235

73

83

176 177

78

184

205 191

64

61

198

52

236 237

18

187 173

82

79

180

70

49

207

51

57

204 233

22

240 185

67

189

69

75

186

203

32

66

8

98

141

7

2

3

23

92

87

2

19

1

10 8

12

3

11

6

13 5

9

4 10

100

84

v

.

ii

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.

vi

.

iv

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.

v

x

viii

.

i

viii

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vii

vii

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ii

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vi

iii

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iv

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i

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iii

ix

.

2

98

5

94

88

15

84

9

25 four 32 31 73 74 26 in83the Commentary in D. Square n = 16, 18with 6 ×71 6 72subsquares 85 24 (one 37 60central, 66 34 labb) 68 38and 77 216× 4 corners, separated by 4 × 4 subsquares rectangles (faw¯ a.sil). But, unlike 14 what is said, extending the 78 59 46 57 52 47 42 23 687× 6 corner squares to 10 × 10 squares89will not satisfy the (usual) magic 22 40 56 43 50 53 61 79 12 conditions (one diagonal will not be magic). Similar assertions are 90 80 39 49 54 55 44 62 21 11 found in the third part, that of the central cross (pp. 154–161). 8

81

65

51

48

45

58

36

20

93

95

19

63

41

35

67

33

64

82

6

4

75

69

70

30

29

28

27

76

97

92

1

99

3

96

7

13

86

17

91

142

Text and translation

(16 ⇥ 16) D : 103r . tit. :

||

(7, 15) ] vid. ||

(1, 3) ]

||

(12, 13) ] ut vid. ||

(13, 7) ]

(1, 1) ]

||

(1, 4) ]

||

(2, 4) ]

ut vid.

(16 ⇥ 16) D : 103v . tit. :

(10, 16) ]

||

(11, 16) ]

||

(9, 7) ]

(v. seq.) ||

(13, 7) ]

.

ut

143

Text and translation

28

231 226

29

230

25

175 170

85

37

92

167 162

93

227 174

81

171 222

33

40 219

166

89

96

163

31

228 229

26

172 173

82

220 221

34

95

164 165

90

225

30

232 169

86

176 217

38

224 161

94

91

168

76

183 178

77 103 110 109 149 150 151 152 104 44

215 210

45

182

73

179 102 115 146 112 144 138 116 155 214

41

48

211

79

180 181

74

212 213

42

177

78

184 100 118 134 121 128 131 139 157 209

46

43

216

20

239 234

21

207 202

53

238

17

235 159 143 129 126 123 136 114 98

49

203

32

27

80

75

24

84

87

88

83

36

223 218

39

35

156 137 124 135 130 125 120 101 47

158 117 127 132 133 122 140

.

ii

52 206

.

v

236 237

18

233

22

240 153 147 148 108 107 106 105 154 201

68

191 186 72

.

xvi

69

4

255 250

190

65

71

188 189

66

185

70

192 249

67

141 111 145 113 119 142 160

.

23

19

97

ix

99

187 254 7

1

8

252 253 6

3

5

2

9

8

4 1

7

11

13

5

6

12

1

62

51

16

17

46

35

32

36

31

18

45

52

15

2

61

xiii

v

.

ii

.

54

.

251 198xii 57iv

viii 64

196 197

.

vii

vii

.

62

.

.

200 241

vi

iii

i

.

iii xi

2

.

.

204 205

. i246viii 195

63

xi

256 193

.

vi

.

3

xiv

vi

55

xv

.

51 vi

60 . 199. 194 61 . 12 . 247v 242 . x iv

.

10

vii

.

56 .

i

59 .

58

.

.

15

.

16.

ii

.

9

244 245 14

.

iv

.

50

x

viii xviii

208 .

.

13 v xiv . . 243 iii

.

ii

viii

10

11

248 .

.

x

ix

.

iv

xvii

.

ix

.

.

.

.

xii

i

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iii

.

vii

xiii

.

vii

xiv

xiii

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viii

vi

v

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ii

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.

vi

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.

.

iv

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v

x

v

iv

viii

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i

viii

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vii

vii

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.

ii

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iii

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vi

iii

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iv

ii

i

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i

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iii

ix

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.

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.

xii

xi

x

ix

.

14 49 64 30 33 48 19 Commentary in D.3 Square n = 16, with one central 8 × 8 square sur47 20 29 34 63 4 50 rounded by twelve 4 × 413subsquares. 28

39

42

21

12

55

58

5

57

6

11

56

41

22

27

40

23

44

37

26

7

60

53

10

54

9

8

59

38

25

24

43

144

Text and translation

(228 ; D, fol. 104v )

(18 ⇥ 18) D : 104r . tit. :

(5, 18) ]

||

(10, 18) ]

||

(6, 15) ]

||

(7, 4) ]

.

(18 ⇥ 18) D : 104v . tit. :

(10, 16) ]

||

(14, 14) ]

||

(6, 11) ]

||

(10, 4) ]

.

145

Text and translation

92

235 230

93

234

89

231 310 282

95

232 233

90

311

47

280 281

42 103 224 225

98

229

94

236

13

277

46

284 221 102 99

228 312 213 110 107 220

12

314 315

9

121 212 114 210 117 206 200 127 196 122

8

318 319

5

84

243 238

85

195 137 144 143 183 184 185 186 138 130 36

291 286

37

242

81

239 128 136 149 180 146 178 172 150 189 197 290

33

40

287

87

240 241

82

288 289

34

237

86

244 124 134 152 168 155 162 165 173 191 201 285

38

35

292

28

299 294

29 123 192 151 161 166 167 156 174 133 202

275 270

53

298

25

295 205 193 177 163 160 157 170 148 132 120 274

49

56

271

31

296 297

26 118 131 175 145 179 147 153 176 194 207

272 273

50

293

30

27

300 209 187 181 182 142 141 140 139 188 116 269

54

51

276

313

11

10

316 203 113 211 115 208 119 125 198 129 204 317

7

6

320

76

251 246

77

250

73

247 322 258

79

248 249

74

323

71

245

78

252

1

253

4

96

91

88

83

32

80

75

1 4

16

44

283 278

45 100 227 222 101 309 108 219 214 109

41

279 226

48

43

4

68

3

10

259 254

69

14 111 216 217 106

65

55

321

60

267 262

61

255 306

17

24

303

3

266

57

64

263

256 257

66

304 305

18

2

63

264 265

58

70

260 301

22

308 324 261

62

268

1 4

72

67

2

11 1

12

9

15

15 13

52

21

5

7

218 105 112 215

307 302

15

1 6

15

199 190 171 158 169 164 159 154 135 126 39

15

6

97 104 223

1 6

8

20

23

19

59

ix

.

ii

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v

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.

xv

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.

vii

xiv

xiii

.

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.

xvi

vi

v

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ii

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.

vi

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x

viii xviii

.

.

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iv

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v

x

v

xiv

xii

iv

viii

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i

viii

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xi

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vii

vii

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ii

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iii

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vi

iii

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iv

ii

viii

vi

i

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i

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iii

ix

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xii

xi

x

ix

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iv

i

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iii

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vii

xiii

.

xvii

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vii

xiv

xiii

.

.

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viii

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.

iv

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v

x

v

iv

viii

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i

viii

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.

vii

vii

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ii

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iii

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.

vi

iii

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iv

ii

i

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i

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iii

ix

.

.

.

Commentary in D. Square n = 18, with one central 10 × 10 square, and . . . . v vi ii vi twelve corner and lateral 4 × 4 subsquares with separating strips.

146

Text and translation

(229 ; D, fol. 105 ) v

(20 ⇥ 20) D : 105v .

tit. :

(13, 20) ] ut vid. ||

|| (11, 9) ]

(20 ⇥ 20) D : 106r . tit. :

(13, 14) ]

||

(9, 13) ]

||

||

(5, 8) ]

||

(13, 2) ]

(1, 11) ]

.

ut vid. ||

(2, 9) ]

147

Text and translation 143 150 149 253 254 255 256 144 28

375 370

29 175 182 181 221 222 223 224 176

142 155 250 152 248 242 156 259 374

25

371 174 187 218 184 216 210 188 227

260 241 164 239 234 165 160 141 31

372 373

26

140 158 238 161 168 235 243 261 369

30

376 172 190 206 193 200 203 211 229

262 157 167 236 237 162 244 139 60

343 338

61

263 247 233 166 163 240 154 138 342

57

64

339 231 215 201 198 195 208 186 170

137 245 151 249 153 159 246 264

340 341

58 169 213 183 217 185 191 214 232

62

344 225 219 220 180 179 178 177 226

63

257 251 252 148 147 146 145 258 337 4 398 7

5

399 394 1

8

44

395 358 2

396 397

47

27

59

228 209 196 207 202 197 192 173

230 189 199 204 205 194 212 171

359 354

45 132 271 266 133 52

351 346

53

41

48

355 270 129 136 267 350

49

356 357

42 135 268 269 130 55

46

360 265 134 131 272 345

393

6

3

400 353

71

78

77

325 326 327 328

72

70

83

322

80

332 313

92

68

86

334

85

43

36

383 378

21

347 382

17

24

379

348 349

50

380 381

18

54

352 377

22

384

56

51

20

23

19

367 362

37 103 110 109 293 294 295 296 104

320 314

84

331 366

33

40

363 102 290 112 . 288. 282 116 299 . 115 . . v xv x ix ii

311 306

93

88

69

364 365

. 120 34. 300 124 . 279 . 274. 125 101 vii 281 xiv xiii viii xviii

310

89

307 315 333 361

38

. . . v xvi 100 vi 118 ii121 . 128. 275 vi 283 368 278 301

95

308 309

90

391 386

96

316

67

39

12

335 319 305

94

91

312

82

66

390

65

79

321

81

87

318 336

15

329 323 324

76

75

74

73

317

32

330 385

9

35

16

.

13

xii

5

4 12

v

x

v

xiv

i

viii

.

.

.

iv

.

.

.

ii

.

iii

.

99

.

14

. .108iii 107 . 105 . . 392vi 297 291 292 298 i i ix 106

11

9

vii

.

97.

11

.

7

vii

.

xi

6

3

viii

.

10.

1

8

iv

388 389

10

13

.

387 303 287 273 126 123 280 114 98

.

2

.

302 117 127 276 277 122 284

285 111 289 113. 119 286 304 . . vi

.

.

.

xii

i

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iii

.

vii

xiv

xiii

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vi

v

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iv

iv

viii

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vii

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i

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.

iii

iv

ii

viii

x

ix

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iv

xiii

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xvii

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viii

ii

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vi

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v

x

v

i

viii

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vii

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ii

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iii

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vi

iii

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iv

ii

i

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iii

ix

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.

xii

xi

x

ix

.

xi vii

Commentary in D. Square n = 20, with four 8 × 8 squares in the corners and 4 × 4 separating subsquares.

148

Text and translation

(230 ; D, fol. 106 ) r

(20 ⇥ 20) D : 105v .

tit. :

(13, 20) ] ut vid. ||

|| (11, 9) ]

(13, 14) ]

||

(9, 13) ]

||

||

(5, 8) ]

||

(13, 2) ]

(1, 11) ]

.

ut vid. ||

(2, 9) ]

(20 ⇥ 20) D : 106r . tit. :

.

(12, 19) ] (15, 14) ] ||

(19, 9) ]

ut vid. (v. 9, 19) || (v. infra) || ||

(7, 16) ]

(15, 13) ]

(1, 4) ]

||

||

(15, 15) ]

(v. infra) || (12, 1) ]

.

(3, 10) ]

(v. loculum infra) || ||

(14, 10) ]

149

Text and translation

144 259 254 145 28

375 370

29 152 251 246 153 96

307 302

258 141 148 255 374

25

32

371 250 149 156 247 306

93 100 303 242 157 164 239

147 256 257 142 31

372 373

26 155 248 249 150 99

304 305

94 163 240 241 158

253 146 143 260 369

30

27

376 245 154 151 252 301

98

308 237 162 159 244

20

383 378

21

71

334

68

332 326

382

17

379 325

80

323 318

81

23

380 381

18

74

322

77

319 327 190 206 193 200 203 211

377

22

384

73

83

320 321

78

328 189 199 204 205 194 212 309

112 291 286 113 331 317

82

79

324

70

290 109 116 287 329

333

69

75

330 213 183 217 185 191 214 266 133 140 263

24

19

67

84

97 160 243 238 161

95

72 187 218 184 216 210 188 88

315 310

89

76

85

92

311

312 313

86

90

316

209 196 207 202 197 192 314 91

215 201 198 195 208 186 136 267 262 137

115 288 289 110 169 236 166 234 228 170 53

352

50

285 114 111 292 227 178 225 220 179 174 343

62

341 336

63

44

359 354

45 172 224 175 182 221 229

56

340

59

337 345

358

41

355 171 181 222 223 176 230

55

65

338 339

60

346 390

47

356 357

42

64

61

342

52

353

46

360 231 165 235 167 173 232 347

351

51

57

348 385

48

43

233 219 180 177 226 168 349 335 49

104 299 294 105 36

367 362

37 120 283 278 121

298 101 108 295 366

33

363 282 117 124 279 398

107 296 297 102 39

364 365

34 123 280 281 118

293 106 103 300 361

38

368 277 122 119 284 393

5

17

18

40

35

4

13

11

7

2

15

16

14

8

66

58

5

399 394 1

8

6

261 138 135 268 12

15

13

391 386 9

16

387

388 389

10

14

392

11

128 275 270 129

395 274 125 132 271 2

396 397 3

131 272 273 126

400 269 130 127 276

7

.

ii

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.

vi

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iv

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v

x

viii

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i

viii

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vii

vii

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ii

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.

vi

iii

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iv

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i

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iii

ix

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v

19

20

54 139 264 265 134

350 344

3

6

9

10

4

12

1

87

10

100

2

98

5

94

88

15

84

9

18

25

32

31

71

72

73

74

26

83

85 24 37 60 66 34 68 38 77 16 Commentary in D. Square n = 20, with four 6 × 6 squares (not: one 12 × 14 78 59 46 57 52 47 42 23 87 12, see p. 107n) in the centre surrounded by sixteen 4 × 4 subsquares. 22

40

56

90

80

39

49

8

81

65

51

95

19

63

41

35

89

43

50

53

61

79

12

54

55

48

45

44

62

21

11

58

36

20

93

67

33

64

82

6

150

Text and translation

(231 ; D, fol. 106 ) v

(20 ⇥ 20) D : 106v . tit. :

(5, 14) ]

||

(20 ⇥ 20) D : 107r .

tit. :

(10, 13) ]

||

(19, 13) ]

||

(20, 7) ] quasi

.

151

Text and translation

151 254 148 252 246 152 12 169 236 166 234 228 170 389 187 218 184 216 210 188 245 160 243 238 161 156 390 227 178 225 220 179 174 11 154 242 157 164 239 247 391 172 224 175 182 221 229

209 196 207 202 197 192

10 190 206 193 200 203 211

153 163 240 241 158 248

9

171 181 222 223 176 230 392 189 199 204 205 194 212

251 237 162 159 244 150

8

233 219 180 177 226 168 393 215 201 198 195 208 186

249 147 253 149 155 250 394 231 165 235 167 173 232

7

213 183 217 185 191 214

24

378 379

351 346

53

383

79

326

76

347 133 272 130 270 264 134

317

88

82 81

20

382

36

367 362

37

324 318

80

366

33

363 350

49

315 310

89

84

39

364 365

34

348 349

50

314

85

311 319 361

38

368 345

54

51

352 136 260 139 146 257 265

91

312 313

86

320

44

359 354

45

375 370

29 135 145 258 259 140 266

323 309

90

87

316

78

358

41

355 374

25

371 269 255 144 141 262 132

321

75

325

77

83

322

47

356 357

42

372 373

26

267 129 271 131 137 268

377

23

22

380 381

19

353

46

43

360 369

30

27

376

18

61

344

58

342 336

62

395

97

308

94

306 300

98

6

335

70

333 328

71

66

5

299 106 297 292 107 102 396 281 124 279 274 125 120

64

332

67

329 337

4

100 296 103 110 293 301 397 118 278 121 128 275 283

63

73

330 331

68

338 398

341 327

72

69

334

60

57

343

59

65

340

339

21

92

74

3

1 4

35

48

55

28

31

399 305 291 108 105 298 1

56

32

99 109 294 295 104 302

303

1 4

2

93

307

96

11

4 10

12 13

13 1 4

6

1 4

16

386 387

13

263 142 261 256 143 138

384 385

15

14

388

115 290 112 288 282 116

3

117 127 276 277 122 284

2

287 273 126 123 280 114

1

9

7

17

95 101 304 400 285 111 289 113 119 286

13

13

8

40

52

5

v

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ii

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.

vi

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iv

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v

x

viii

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i

viii

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vii

vii

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vi

iii

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iv

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i

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iii

ix

.

10

100

2

98

5

94

88

15

84

9

18

25

32

31

71

72

73

74

26

83

66 34 (or 68 one 38 778 × 168) Commentary in D. Square n = 20, with 85 four24 4 37 × 460squares in the centre and eight surrounding 14 6 ×786 59subsquares, separated 46 57 52 47 42 23 87by strips. 89 22 40 56 43 50 53 61 79 12 90

80

39

49

54

55

44

62

21

11

8

81

65

51

48

45

58

36

20

93

95

19

63

41

35

67

33

64

82

6

4

75

69

70

30

29

28

27

76

97

152

Text and translation

(232 ; D, fol. 107 ) r

(20 ⇥ 20) D : 106v . tit. :

(5, 14) ]

||

(10, 13) ]

||

||

(19, 13) ]

(20, 7) ] quasi

.

(20 ⇥ 20) D : 107r .

tit. :

(3, 11) ] Expl. D, 107v :

||

(1, 8) ]

||

(5, 6) ]

||

(20, 3) ]

||

(1, 1) ]

. .

153

Text and translation 169 236 166 234 228 170 92

311 306

93

353

47

358

44 187 218 184 216 210 188

227 178 225 220 179 174 310

89

96

307

46

356

41

359 209 196 207 202 197 192

172 224 175 182 221 229

308 309

90

43

357

48

354 190 206 193 200 203 211

171 181 222 223 176 230 305

94

312 360

42

355

45 189 199 204 205 194 212

233 219 180 177 226 168 25

375 374

28

351 350

52

231 165 235 167 173 232 376

26

373 352

50

349 213 183 217 185 191 214

95

91

27

49

51

215 201 198 195 208 186

32

371 366

33

337

64 139 146 145 257 258 259 260 140 361

40

370

29

36

367

63

338 138 151 254 148 252 246 152 263

362 322

35

368 369

30

62

339 264 245 160 243 238 161 156 137 38

363

365

34

31

372 340

61 136 154 242 157 164 239 247 265 364

341

59

346

56

389

12

58

344

53

347

55

345

60

342

348

54

101 304

343 98

57

323 318

81

77

84

319

83

320 321

78

37

317

82

79

324

266 153 163 240 241 158 248 135 313

88

381

19

386

16

11

390 267 251 237 162 159 244 150 134 87

314

18

384

13

387

10

391 133 249 147 253 149 155 250 268

315

15

385

20

382

xv 14

. 383

x17

392

9

261 255

302 296 102 73

104 292 107 114 289 297

399 394

301 287 112 109 294 100 299

97

303

99 105 300 393

2

16

7

6

14

17

13

1

75

3

5

9

18

21

10

xiii

.

.

.

.

xvi

vi

v

.

ii

.

.

vi

.

.

.

iv

.

.

v

x

v

xiv

.

i

.

.

.

.

379 378

viii xviii

24 119 286 116 284 278 120

22

23

377 277 128 275 270 129 124 68 122 274 125 132 271 279

.

.

334

395

70

332 xi

65 335 vii 121 . 131. 272ii 273 280 . . 126 . vii iii

2

67

333.

. 72

. .283 vi 269iii 130 . 127 iv 276 ii viii 330 118

vi 331 i 66

. . . . . ix 117 69 i281 115iii 285 123 282

400 336

8

19

4

xiv

71

1

7

20

vii

329

3 12

.

5

396 397 6

21

325 380

8

15

11

76

327 326 74

103 113 290 291 108 298 398

86

. .316 v 85 . 388. ix 141 ii 256 144 143 142 262

295 110 293 288 111 106 328 4

39

80

xii

iv

viii

viii

.

.

.

.

xii

xi

x

ix

.

iv

.

i

.

iii

.

vii

xiii

.

xvii

.

vii

xiv

xiii

.

.

.

.

viii

vi

v

.

ii

.

.

vi

.

.

.

iv

.

.

v

x

v

iv

viii

.

i

viii

.

.

.

.

ii

.

iii

iii

.

iv

ii . .

.

vii

vii

.

.

.

.

vi

i

.

i

.

iii

ix

.

.

.

.

xii

xi

x

ix

Commentary in D. Square n = 20, with one central 8 × 8 square, 6 × 6 squares in the corners, separated by 4 × 4 subsquares (not noted: our subsquares 10, 12, 14, 19 are rotated) and 2 × 4 rectangles.

154

Text and translation

Excerpta e cod. D, 102v :

.(sc. n = 4k + 2, k

2)

.

(14 ⇥ 14) D : 102v .

n2 2

1 2

[4n

4] + 1

(D, fol. 102r )

tit. : (18 ⇥ 18) D : 98r . tit. :

(2, 9) ] corr. ut vid. ex (13, 18) ]

||

.

(13, 17) ]

||

(14, 14) ]

||

(16, 7) ]

||

(10, 5) ]

.

(20 ⇥ 20) D : 98v . tit. :

(12, 20) ]

||

(17, 12) ] quasi

(20, 18) ] ||

(13, 9) ]

|| ||

(13, 4) ]

(10 ⇥ 10) D : 102r . tit. :

(1, 8) ]

ut vid .

(14 ⇥ 14) D : 102r . tit. :

||

(13, 17) ]

63

||

||

(9, 14) ] (3, 1) ]

.

(12, 14) ]

||

155

Text and translation

Cross in the middle As indicated in MS. D after the above 20 × 20 square (see p. 152, bottom), we have reached the end of the examples in our treatise. Indeed, the four examples with the central cross, which we have chosen to present separately at the end (since they are treated last in the theory), are found among those displaying unequal parts, all these examples being in order of size. Note that this first example even heads the sequence of examples with unequal parts, whence the remark in MS. D that what follows ‘from here on’ is taken from al-Mufad.d.al’s text. As an introduction to the first case, the commentator indicates a way to find the smallest number in the 2 × n cross when it is filled first: adding 1 to half the difference between n2 and the whole number of cells in the cross — which is indeed evident. He further remarks in the first three cases below that these subsquares when extended with ‘part’ of the cross will display equal sums (he fails to note that they will not be found everywhere; see also above, p. 141). In addition to the tables supplemented with the previous transcriptions, we have indicated how the cross has been filled (disregarding neutral placings). 28

75

70

29

40

61

20

83

78

21

74

25

32

71

62

39

82

17

24

79

31

72

73

26

63

38

23

80

81

18

69

30

27

76

37

64

77

22

19

41

59

43

57

50

53

36

66

60

42

58

44

48

51

65

4

99

94

5

49

52

98

1

8

95

54

7

96

97

2

93

6

3

100

.

i

.

iii .

.28

75

ii70

29

40

61 iv

20

74

25

32

71

62

39

82

31

72

73

26

63

38

23

83 viii

78

17

24

69

30

27

76

37

64

77

41

59

43

57

50

53

36

84

60

42

58

44

48

51

65

4

99

94

5

49

52

12

67

33

98

1

8

95

54

47

90

7

96

97

2

46

55

15

35

34

68

93

6

3

100

56

45

85

12

91

86

13

47

90

9

16

87

46

55

15

88

89

10

56

45

85

14

11

92

4

. ix

iii

.

x

.

iv

7

ix80

79

. 18

22

19

84

66

67

33

vi35

34

vii

91

.

68

.9

86

13

16

v 67

88

89

10

14

11

92

5

. .

.

viii

.

ii

vi

.

.

v

1

81

. 21

.

vii

2

.

x

3

i .

6

Commentary in D. Square n = 10, with 4×4 subsquares and a separating . v cross. vi

.

.

vii

ix

.

x

.

.

iii

.

i

156

Text and translation

(222 ; D, fol. 102v )

Excerpta e cod. D, 102v :

.(sc. n = 4k + 2, k

2)

.

(14 ⇥ 14) D : 102v . tit. :

(2, 9) ] corr. ut vid. ex

.

n2 2

1 2

[4n

4] + 1

126 70 50

92 114 223 195 175 61

59

47

27 116 134 156 100

72 130 184 44 219 191 53

43

23

46 178 132 154 176

Text and translation

52 146 200 28

30 194 215 37

19 192 36

162 224

218 216 12

14 210 208 20

4

41

160

38

151

50

44 43

42

88

109

59

142

56

149 144

51

46

110

87

133

68

148

47

145 153 111

86

62

53

146 147

48

154

85

112

61

157 143

52

49

150

40

102

95

155

37

159

39

45

156

97

77

119 118

80

91

105

120

78

79

117 106

5

196

2

187

14

8 7

60

131 126

69

64

130

65

127 135

71

128 129

66

136

139 125

70

67

132

58

100 137

55

141

57

63

138

98

101 107

89

81

115 114

84

92

96

99

90

108 116

82

83

113

194 188

6

94

103

23

178

20

176 170

24

185 180

15

10

104

93

169

32

167 162

33

28

184

11

181 189

76

121

26

166

29

163 171

17

182 183

12

190 122

75

25

35

164 165

30

172

16

13

186

4

123

74

175 161

34

31

168

22

195

3

9

192

73

124 173

19

177

21

27

2

7

4

94

8

.

.

v

iii

vii

vii

.

.

ii

.

.

.

.

vi

iii

.

.

i

.

iii

60

66

34

68

38

77

16vi

.

78

59

46

57

52

47

42

23

. 87

vii

79

12ix

.

x

. .

249

88

171 166

89

12

245

44

215 210

45

7

170

85

167 246

11

214

41

211

92

ix

.

37

227 250

32

.

.

24

25

230

vi

viii

v

29

89 48

22

40

56

43

50

v

iv

i

. 83

231 226

.

.

.

26

14

.

i

viii

74

143 129 126 123 136 114

vii

vi

ii

x

73

13

.

.

v

72

244 141 111 145 113 119 142

14

.

viii

.

71

15

8

16

ix

iv

.

31

242 243

28

96

.

.

iv

32

95 101 160 241

viii

.

.

25

161 147 108 105 154

iv

x

vi

18 85

.

174

.

.

9

224 117 127 132 133 122 140

.

.

.

84

38

x

ix

ii

15

99 109 150 151 104 158 217

.

vii

.

88

34 118 134 121 128 131 139

iii

.

v

94

220 221

163

ii

5

100 152 103 110 149 157

93

.

.

98

219 137 124 135 130 125 120

159

i

2

40

35

36

100

37 115 146 112 144 138 116

33

39

72

10

223 218

155 106 153 148 107 102 222

162 156

36

18

3

5

98

54

6

1

9

22 202 164

140 134

1

191

164

13

158 152

193 179

97

6

157

38 186 148 174

53

61

.

.

90 80n = 39 14, 49 with 54 55 6 × 44 662subsquares 21 11 Commentary in D. Square and iiia cross.i

31

228 229

26

251

225

30

27

232

5

168 169

86

247

252 165

6

91

90

172

9

248 209

254 255

1

53

208

50

206 200

54

4

199

62

197 192

63

58

253

56

196

59

193 201

20

55

65

194 195

60

205 191

64

61

49

207

51

203

66

3

87

2

10

47

212 213

42

46

43

216

8

71

190

68

188 182

72

256 181

80

179 174

81

76

239 234

21

74

178

77

175 95

183

202 238

17

235

73

83

176 177

78

184

198

52

236 237

18

187 173

82

79

180

4

70

57

204 233

22

240 185

67

189

69

75

186

23

24

19

84

92

81

65

51

48

45

58

36

20

viii 93

19

63

41

35

67

33

64

82

6

75

69

70

30

29

28

27

76

97

1

99

3

96

7

13

86

17

91

iv

.

ii

.

158

Text and translation

(223 ; D, fol. 104r )

(18 ⇥ 18) D : 104r . tit. :

(5, 18) ]

||

(18 ⇥ 18) D : 104v . tit. :

(10, 18) ]

||

(6, 15) ]

||

(6, 11) ]

||

(10, 4) ]

.

159

Text and translation

71

78

77

249 250 251 252

72 148 177 103 110 109 217 218 219 220 104

70

83

246

80

244 238

84

255 178 147 102 115 214 112 212 206 116 223

256 237

92

235 230

93

88

69

68

86

234

89

231 239 257 145 180 100 118 202 121 128 199 207 225

258

85

95

232 233

90

240

67

259 243 229

94

91

236

82

66 158 167 227 211 197 126 123 204 114 98

241

79

245

81

87

242 260 166 159 97

253 247 248

76

75

74

73

65

96

179 146 224 205 124 203 198 125 120 101

168 157 226 117 127 200 201 122 208

99

209 111 213 113 119 210 228

254 161 164 221 215 216 108 107 106 105 222

137 187 186 140 141 183 182 144 162 165 169 155 171 153 152 174 175 149 188 138 139 185 184 142 143 181 160 163 156 170 154 172 173 151 150 . 176x . i iii 39

46

45

. .284 iv 40 ii viii 281 282 283

319 190 135 38

51

278

48

7

14

13

313 314 315 316

6

19

310

16

320 301

28

299 294

29

295 303 321 133 192

308 302

4

22

298

25

322

21

31

296 297

323 307 293 305

30

32

27

26

.

vii

xiv

xiii

.

xvi

.

ii

304.

.

vi

v

3 .

132 . . iv193. 290

53 v

. 131i

.

.

.

.

.

vi

iii

3 6

2

306 324 195 130 33 .

.

. viii 275 291

.

9

.

54

vii

10

.

36

vii

11

8

.

viii 194

12

5

v

.

317 311 312

9

.

191 134 288 269

iv 2

23

1

ii

xi

17

4

.

5

xii 18

309

7

ix

24

136 189

300

15

1

20

8

.

ix

.

.

vii

vi

.

64

263 271 289.

v

x63 v 264xiv

265

58

. . 261

59

268 . 50 vii 34

49

. 55 ix274 292

.

xv

60

.

276 270 x

267 262

viii xviii

266

57

.

vi

.

iii

.

iv

ii .

277

47

318 129 196 285 279 280

.

62 .

ii

273

.

.

viii iii

44

43

56

37

272

.

x

iv

42

287

41

35

.

.

i

ii

.

286

i

.

i

.

iii

ix

.

viii

.

.

.

.

.

xii

xi

x

ix

.

iv

vi

.

.

i

.

iii

.

vii

xiii

.

xvii

.

.

v

vii

xiv

xiii

.

.

.

.

viii

vi

v

.

ii

.

.

vi

.

.

v

.

.

iv

.

.

v

x

v

vi

.

iv

viii

.

i

viii

.

.

.

.

vii

.

vii

vii

.

.

ii

.

iii

ix

.

.

.

.

vi

iii

.

iv

ii

x

.

.

iii

.

i

i

.

i

.

iii

ix

.

.

viii

.

iv

.

ii

.

.

.

.

xii

xi

x

ix

.

vi

.

61

52

.

Commentary in D. Square n = 18, with 8 × 8 subsquares and a cross.

160

Text and translation

(224 ; D, fol. 105r )

(18 ⇥ 18) D : 105r .

tit. :

||

(11, 18) ] || ut vid. ||

(4, 8) ] (10, 6) ]

(10, 13) ] , corr. ex ||

|| (sc.

(13, 1) ]

(11, 13) ] ) ||

||

(16, 13) ]

(10, 7) ]

pr. scr. (v. seq.).

(=

ut vid. || ) ||

(8, 9) ]

(1, 6) ]

161

Text and translation

64

263 258

65 108 219 214 109 72

253 140 187 182 141 76

262

61

68

259 218 105 112 215 254

71

67

260 261

62 111 216 217 106 255

257

66

264 213 110 107 220

63

69

251 246

77

73

80

247

70 143 184 185 138 79

248 249

74

256 181 142 139 188 245

78

252

186 137 144 183 250

100 227 222 101 48

279 274

49

226

45

275 158 167 270

97 104 223 278

103 224 225

98

221 102 99

228 273

51

52

168 157 56

276 277

46

50

280 161 164 265

47

166 159 59

75

271 266

57 132 195 190 133

53

60

267 194 129 136 191

268 269

54 135 192 193 130

58

272 189 134 131 196

55

40

286 287

37 152 174 175 149 162 165 169 155 171 153 44

282 283

41

285

39

288 173 151 150 176 160 163 156 170 . 154 172 281 43 . 42 . ii iv viii

284

92

235 230

93

234

89

95

38

295 290

33 148 177

231 294

29

291 178 147 302

232 233

90

292 293

30

229

94

236 289

34

296 145 180 297

16

311 306

17

243 238

85

310

13

307 242

81

239 314

19

308 309

14

240 241

82

315

305

18

312 237

86

244

9

96

91

20

15

.

i

14

32

35

84

87

8

9

16

18

2

10

19

11

21

13

36

31

88

83

4

15 1

3

22

24

179 146 27

12

7

.

x

25 124 ix203 . 198 125

21

28

299 202 .121vii128 199

300 301

22 127 200 201 122

26

304 197 126 123 204

23

313 116 211 206 117 .

11

.

303 298

4

vii

vi

.

.

v

323 318

210 113 120 207 322 . .7

8

319

.320 i

321

2

. . iv 212 viii 317 ii 6 . 3 316 205 118 115 vi

.

.

v

.

iii

.

i

iv

.

ii

.

324

12

5

6

.

v

vi

.

.

vii

ix

.

x

.

viii

.

23

Square = 18, 4 × 112 4 (or1448 ×858) subsquares 36 223 n218 37 with 115 116 17 10 15 in the corners and a cross. As noted by the commentator, the inner 10 × 10 5 7 298 302 306 289 293 297 271 275 279 82 86 90 12 14 16 square is also magic.

in D. 97 Commentary 164 94 142 156 98 3

5

1

ix

10 119 208iii 209. 114 x

17 20

iii

4

9

2

303 304 299 294 295 290 276 277 272 87

88

83

13

18

11

62

57

58

98

93

94 215 210 211 188 183 184 161 156 157 269 264 265

55

59

63

91

95

99 208 212 216 181 185 189 154 158 162 262 266 270

60

61

56

96

97

92 213 214 209 186 187 182 159 160 155 267 268 263

IV. General commentary Introduction (§ 0) Translator’s introduction The translator, who had previously no knowledge of magic squares, tells us in detail how he came to know them. First, he says, there was the 3×3 square mentioned (or ‘reported’, dhakara) by Nicomachos. Now there is no mention of a magic square in the Introduction to arithmetic, nor in its Arabic translation by Th¯ abit ibn Qurra (the translator’s father?). But a 3 × 3 square may have appeared as a marginal addition. Still, the translator’s allusion is problematic. All other scholars mentioned in this introduction belong to the ninth century; they are known from contemporary writings or also from the Fihrist of Ibn al-Nad¯ım, written in the end of the tenth century, which in particular lists contemporary Arabic books or translations. The first scholar mentioned is Ab¯ u’l-Q¯asim al-H . ij¯az¯ı, of the first half of the ninth century, who saw a 4 × 4 square with the first sixteen natural numbers and is said to have been impressed by its realization with these numbers. A mathematician would have admired the arrangement rather than the use of a continuous number sequence; but then al-H . ij¯az¯ı was a theologian. Ish.¯aq ibn H . unayn (d. 910/1), a contemporary of Th¯abit ibn Qurra, translated among others Euclid’s Elements, a translation then revised by Th¯abit himself. According to our author, Ish.¯aq represented a 6×6 square on the ‘back’ of a copy of Euclid’s book. In an unspecified book our translator saw a few magic squares, all of orders less than ten. Knowing that already in early Islamic times the first seven squares were attributed to the (then) known planets,195 one wonders if that was not also the case in ancient writings. The two exemplars used by our translator for his work were found, he says, ‘in the Library, among the books of the caliphs’ collection’. Now this library must be that of the Bayt al-h.ikma, the ‘House of wisdom’, 195 Above, pp. 6–7; see also Magic squares, pp. 11–12, 16–19 (and earlier editions: Les carrés magiques, pp. 10, 14, 253–266; , pp. 18, 23, 263–272).

N xki N N

164

General commentary

indeed known to have preserved works collected by the early caliphs.196 Since there were two exemplars, the translator could complete his work, using the parts preserved in one to make up for the parts of the other destroyed by termites.197 But our translator was not the first to have paid attention to this treatise. As he reports, there was a note by (Muh.ammad b. ‘¯Is¯a b. Ah.mad) al-M¯ah¯an¯ı, who was a Persian mathematician and astronomer working in Baghdad in the middle of the ninth century, known to have commented various Greek works; it is therefore hardly surprising to see him examining the two manuscripts of our treatise. His note was not the only one: more extensive was that by H us¯a al-Nawbakht¯ı, a . asan (as commonly) b. M¯ theologian and philosopher of the second half of the ninth century, who is said by Ibn al-Nad¯ım to have been in relation with translators of (Greek) philosophical works.198

(§ 1) Definition of a magic square The first leaf of a manuscript is generally that which is less well preserved. This may explain why the definition of a magic square (§ 1) is somewhat imprecise, in particular without any mention of the constant sum being also found in the two main diagonals.

(§ 2) Two ways of constructing magic squares The next paragraph tells us about the two main kinds of magic squares, namely ordinary ones and bordered ones, and the difference between the ways in which they are constructed. But there is no allusion to their main difference, that is, that all the inner squares of a bordered square are also magic, despite such characteristic being essential to the subject treated. As for the construction of an ordinary magic square, it is based on the natural square, that is, a square of the same order as the one to be constructed but containing the first numbers in their natural order. Such a square has two notable properties: first, its main diagonals already contain the magic sum; second, as implied in our text, pairs of symmetrically placed lines or columns display differences with the magic sum which have 196

Eche, Bibliothèques, pp. 26–27, 36, 56–57. Ibn al-Nad¯ım also mentions the damage done by termites to manuscripts (Fihrist, ed. Flügel et al., I, 243, 27; tr. Dodge, II, p. 585). The impressive result of such damage to a book, after termites first attacked its book case, is shown in T. E. Snyder’s study, p. 114 (however, our two MSS. cannot have been in such a state); see also Ghidini’s article. Some known Italian libraries suffered likewise, both in the south (Palermo, Catania) and in the north (Florence, Venice). 198 Fihrist, I, 177, 11–12; tr. Dodge, I, 441. 197

General commentary

165

the same amount but with a different sign. To obtain a magic square, one would then normally, without modifying the diagonals, proceed to exchange numbers between symmetrically placed lines and columns until reaching the magic sum. Having stated that constructing a magic square from exchanges in the natural square of the same order ‘is a method presenting difficulty for the beginner’, our text turns to the ‘different, and easier, way’ for the construction of bordered squares. This is the only allusion of our text to ordinary magic squares. It at least indicates that ordinary magic squares were also studied in Greece. Note on the construction of ordinary magic squares As the study of B¯ uzj¯ an¯ı’s treatise, the earliest originally Arabic one on magic squares (Ant.¯ ak¯ı’s is in this respect irrelevant), shows, his exchanges between opposite rows of the natural square turned out initially to be successful only for small orders; that is, they did not give rise to general methods. B¯ uzj¯ an¯ı indeed considers exchanging numbers in natural squares of odd order first for the (banal) case of 3, then for the order 5; and the two ways he then suggests for the latter are particular to that order and cannot be extended to higher orders. The exchanges for even orders turn out to be easier, but here again B¯ uzj¯an¯ı did not reach a general method after considering the first even orders.199 For, to reach a general method, one has to consider exchanges in the natural square as a whole, and not by considering one order after the other. That is why general methods were not found before the 11th century, when individual treatments were abandoned and general properties of the natural square set out.200

Part I: Construction of odd-order bordered squares (§§ 3–6) The method found in our treatise We shall now thoroughly examine the method our text presents; first, explaining the general placing to be memorized; then how this placing must have been found empirically; and ending with the mathematical foundation of this general placing. 199 200

See Magic squares in the tenth century, pp. 30–36. See above, p. 5 and n. 3.

166

General commentary

1. Description of this method This method may be generally described, when one (unlike our text) takes the numbers from the smallest on, here 1, and puts them in the outer border first, as follows. Starting next to a corner cell, we put in the rows meeting there the natural sequence, alternatively in one and then the other, till we reach the middle cells. We write the next number in the cell next to the cell just reached (7 in the figure), then move to the opposite corner (that of the row where we started), then put the next in the middle cell on the other side, the next in the corner of the same row (just one possibility since the other corner is for a complement), whereupon we resume placing on either side of this corner, but this time towards the corner cell. Opposite cells are each time left empty for the complements. A characteristics of this method is to separate the numbers by parity: the vertical rows contain, except for the corner cells, only odd numbers, while in the horizontal rows are found, except for their middle cells, even numbers. The subsequent mathematical justification will provide the reason for this anomaly. 8

12

14

16

10 15 13 11 9

5 3 1 2

4

6

7

Fig. 6

Since our treatise gives the above general method but does not explain its foundation, we shall first consider how it must have been obtained and, second, provide its mathematical justification.

2. Discovery of this general method Since the construction of bordered squares is described in our treatise, but not justified, B¯ uzj¯ an¯ı attempted to trace back the origin of this method. In the present case of odd-order squares, he succeeded. So let

167

General commentary

us follow, in our writing, his path towards the empirical finding of this method.201 The outer border of a square of order m comprises 4m − 4 cells, to be filled with 2m − 2 pairs of complements. Consider then, as B¯ uzj¯an¯ı does, a square of order m = 5, with 16 cells in its border, thus to be occupied by the numbers 1, . . . , 8 and 18, . . . , 25. We are first to fill the central square of order 3. To do so, we start by placing in the central cell the 2 median number, m 2+1 = 13, and then take the 2 · 3 − 2 = 4 numbers on each side of 13 and arrange them in pairs of complements, 9 10 11 12 17 16 15 14, which we write in the border surrounding the central cell in the known way (Fig. 7–8).202 10

17

12

2

9

4

15

13

11

7

5

3

14

9

16

6

1

8

Fig. 7

Fig. 8

The sum of opposite cells is thus 26, and, as seen (p. 2), this is the sum we are also to find in opposite cells of the outer border of order 5, namely using the pairs 1 2 3 4 5 6 7 8 25 24 23 22 21 20 19 18. The first placing to be considered is that of the corners, for it will determine the filling of two rows. B¯ uzj¯an¯ı chooses to put in the two upper corner cells the smaller numbers 4 and 6, with their complements opposite (Fig. 9). But he also notes that there would be other possibilities —though not many for this case of the 5 × 5 square. As he asserts, the only possible pairs of smaller numbers for two consecutive corner cells with the first numbers are 1, 3; 3, 7; 5, 7; 2, 8; 4, 6; 6, 8, which would give altogether ten borders.203 His choice is obviously made in conformity 201 See the edition of his treatise, pp. 150–157 and (Arabic) 209–216; or Magic squares in the tenth century, B.15–16. 202 Putting 2 in the upper left-hand corner cell and 4 in the right-hand one fits the general method to be explained. 203 Given by us in Magic squares in the tenth century, pp. 44–46 and Magic squares, p. 144 (Les carrés magiques, pp. 120–121; , p. 131).

N xki N N

168

General commentary

with the Greek method as, too, with the case of the 3 × 3 square, with 2 and 4: the upper corner cells of the outer border of a square of odd order n, filled with the first small numbers and their complements, will be n − 1 and n + 1. Accordingly, for an inner border of order m, they will be the (m − 1)th and the (m + 1)th of the small numbers which are to occupy this border. B¯ uzj¯an¯ı is then left with completing the border, or, rather, two contiguous rows of it since complements will occupy the cells opposite. Their choice must complete the still required sum. But their arrangement, he writes, is irrelevant; thus the configuration chosen should be convenient for memorization. He then completes the required sum for this order, 65, first horizontally (Fig. 10), then vertically (Fig. 11), and puts the complements opposite. Here again, his choice is to fit the general arrangement he has in mind. 4

6

4

24

23

8

6

4

24

23

8

6

10

17

12

10

17

12

19

10

17

12

7

15

13

11

15

13

11

21

15

13

11

5

14

9

16

14

9

16

1

14

9

16

25

2

3

18

20

2

3

18

22

20

22

Fig. 9

20

Fig. 10

22

Fig. 11

To construct the square of order 9, B¯ uzj¯an¯ı proceeds in the same manner. He first takes the median number, 41, which he puts in the central cell (Fig. 12). Then he arranges the four numbers preceding it and the four numbers following it in the border of order 3. To the border of order 5 he allots the eight numbers from 29 to 36 and from 46 to 53, placed in the same way as were the corresponding ones in the border already constructed (Fig. 11). He then considers the two sequences of twelve numbers for the border of order 7, 17 18 19 20 21 22 23 24 25 26 27 28 65 64 63 62 61 60 59 58 57 56 55 54. He begins by writing 22 and 24 in the corners, which is in accordance with his previous choices: he takes here the sixth and the eighth smaller numbers. He is left with finding five numbers making, together with 22 and 24, 287, which is the sum found in the main diagonals (and thus the required magic sum for the 7 × 7 inner square); with the remaining numbers, he completes the vertical rows. There remain the numbers from 1 to 16 and 66 to 81 for the border of order 9, the eighth and tenth

169

General commentary

of the first sequence being for the corner cells, and the others used to complete, by trial and error, the sums in the border’s horizontal and vertical rows. 8

80

78

76

75

12

14

16

10

67

22

64

62

61

26

28

24

15

69

55

32

52

51

36

34

27

13

71

57

47

38

45

40

35

25

11

73

59

49

43

41

39

33

23

9

5

19

29

42

37

44

53

63

77

3

17

48

30

31

46

50

65

79

1

58

18

20

21

56

54

60

81

72

2

4

6

7

70

68

66

74

Fig. 12

This work being done, B¯ uzj¯ an¯ı infers from the arrangement found, as he writes, a method to improve the student’s skill and (also) intended for those who prefer to save themselves the trouble of working out which numbers to arrange in the square.204 This is the general way of constructing odd-order bordered squares found in the Greek treatise.

3. Mathematical basis of this method The principle for constructing a border, of whatever order, is as follows. Consider we have a magic square of order n − 2, in which are arranged the (n − 2)2 first natural numbers. Its magic sum is therefore Mn−2 =

n−2 2





(n − 2)2 + 1 .

We now wish to surround it with a border of order n. Since this border comprises 2n + 2(n − 2) = 4n − 4 cells, it has to be filled with 2n − 2 small numbers and their complements to n2 + 1. As usual, we shall consider putting in this outer border the 2n − 2 first numbers and their complements. So we shall, to begin with, uniformly increase each number of the square already constructed by 2n − 2. The sum in any row of this square (including the two main diagonals), being increased by n − 2 times this quantity, becomes 0 Mn−2 =

n−2 2





(n − 2)2 + 1 + 2 (2n − 2) =

n−2 2

 2  n +1 ,

204 Edition, pp. 157–158 & 216–217; Magic squares in the tenth century, B.17; Magic squares, pp. 146–147 (Les carrés magiques, pp. 121–122; , pp. 132–133).

N xki N

170

General commentary

with the smallest number in the square now being 2n − 1 and the largest, (n − 2)2 + 2n − 2 = n2 − 2n + 2. We are now to fill the 4n − 4 cells of the new border with the two sequences 1, 2, . . . , 2n − 2

n2 − 2n + 3, n2 − 2n + 4, . . . , n2 . Since the magic sum must be increased by 0 Mn − Mn−2 = n2 + 1, we shall arrange in pairs of complements the numbers to be placed, that is, we shall superpose the above two sequences with the second taken in reverse order: 1 2 3 ... 2n − 2 n2 n2 − 1 n2 − 2 . . . n2 − 2n + 3, and put these pairs at either end of each horizontal row, column and main diagonal. We are now left with the problem of obtaining the magic sum in each of the four border rows. Let us apply that to our case of odd orders. Since the outer border of our square of order n = 2k + 1 contains 4n − 4 = 8k cells, it will be filled with the following four sequences of k smaller numbers k X 1

ag ,

k X 1

bh ,

k X 1

ci ,

k X 1

dj ,

and the corresponding four sequences of larger numbers, namely their complements. Note too that, since two of the smaller numbers are to occupy two consecutive corner cells, and therefore their complements the two opposite corners, each of two consecutive rows will contain k + 1 smaller numbers and k complements, whereas the other two will accordingly contain k smaller numbers and k + 1 complements. Consider now taking the smaller numbers in natural order. These four sequences of smaller numbers will have the form {ag } {bh } {ci } {dj }

1, 2, ..., k

with sum

k(k+1) 2

k + 1, k + 2, ..., 2k

with sum

k2 +

2k + 1, 2k + 2, ..., 3k

with sum

3k + 1, 3k + 2, ..., 4k

with sum

k(k+1) 2 k(k+1) 2 2k + 2 3k 2 + k(k+1) . 2

We could choose to consider placing these sequences as they are and modify the places of some of their elements to arrive at the required

171

General commentary

sum.205 The same holds if we take the four sequences of small numbers displaying the same parity, namely206 {ag } {bh } {ci } {dj }

1, 3, ..., 2k − 1 2, 4, ..., 2k 2k + 1, 2k + 3, ..., 4k − 1 2k + 2, 2k + 4, ..., 4k

with with with with

sum sum sum sum

k2 k2 + k 3k 2 3k 2 + k.

Indeed, we may again consider putting, to begin with, each of the above sequences in each of the border rows, and the complements opposite, and then modify the places of some elements. But we may also begin by leaving out, of each sequence, some elements and use them afterwards to obtain the required sum. A simple possibility is thus to take the k − 1 first terms of the two lesser sequences and the k − 1 last terms of the other two, write these terms continuously within the rows (opposite rows thus containing terms of same parity), and use the four terms left out to equalize the rows. Such is the way adopted to produce our general method (Fig. 13). IV {dj } D

A

{ci } I

C

III

{ag } {bh } B

Let then the {ag } {bh } {ci } {dj }

II Fig. 13

incomplete sequences be 1, 3, ..., 2k − 3 2, 4, ..., 2k − 2 2k + 3, 2k + 5, ..., 4k − 1 2k + 4, 2k + 6, ..., 4k

with with with with

sum sum sum sum

k 2 − 2k + 1 k2 − k 3k 2 − 2k − 1 3k 2 − k − 2,

the omitted numbers thus being 2k − 1, 2k, 2k + 1, 2k + 2, respectively. The sum in column I of Fig. 13 will be X

205

ag + A + (k − 1)(n2 + 1) −

X

ci + (n2 + 1) − C + (n2 + 1) − D

See Magic squares, pp. 151–153 (Les carrés magiques, pp. 127–129; , pp. 138–140). 206 Grouping by parity the numbers to occupy a row is merely a matter of preference. N xki

N k N

172

General commentary

= k 2 − 2k + 1 + A + (k + 1)(n2 + 1) − 3k 2 + 2k + 1 − C − D

= A − C − D − 2k 2 + 2 + (k + 1)(n2 + 1). C +D−A+

and therefore




1 2 2k 2

Since this must equal Mn = k +

(n2 + 1), we shall have

− 2 = 2k 2 + 2k + 1

C + D − A = 2k + 3.

(∗)

On the other hand, the sum in the upper row IV will be X

dj + A + D + (k − 1)(n2 + 1) −

X

bh + (n2 + 1) − B

= A + D − B + 3k 2 − k − 2 − k 2 + k + k(n2 + 1)

= A + D − B + 2k 2 − 2 + k(n2 + 1),

and we shall therefore have

A + D − B + 2k 2 −2 2 =9 2k42 + 2k + 1,

whence

7

5

A + D − B = 2k 6 +13.

3

8

(∗∗)

Now using the four still unplaced numbers, 2k − 1, 2k, 2k + 1, 2k + 2, 24 23 4 8 6 we have the two combinations (2k + 2) + (2k + 1) − 2k = 2k +193

(2k + 2) + 2k − (2k − 1) = 2k +213.

10

17

12

7

15

13

11

5

14 16 25 1 9 Comparing the terms in these expressions with our two relations (∗) and (∗∗) we shall take A = 2k, B = 2k − 1, C = 2k + 1,20D =2 2k3 + 18 2. 22 10

120 118 116 114 113

14

16

18

20

12

103

28

100

98

96

95

32

34

36

30

19

105

87

42

84

82

81

46

48

44

35

17

107

89

75

52

72

71

56

54

47

33

15

109

91

77

67

58

65

60

55

45

31

13

6

48

46

45

10

12

8

111

93

79

69

63

61

59

53

43

29

11

39

16

36

35

20

18

11

7

25

39

49

62

57

64

73

83

97

115

41

31

22

29

24

19

9

5

23

37

68

50

51

66

70

85

99

117

43

33

27

25

23

17

7

3

21

78

38

40

41

76

74

80

101 119

3

13

26

21

28

37

47

1

92

22

24

26

27

90

88

86

94

121

1

32

14

15

30

34

49

110

2

4

6

8

9

108 106 104 102 112

42

2

4

5

40

38

44

Fig. 14

Fig. 15

vi

.

.

.

x

xii

viii

.

iv

.

.

viii

vi

xi

173

General commentary 10

14

120 118 116 114 113

16

18

20

12

103 28 100 98 96 95 32 34 36 30 19 This thus gives the placing method of our treatise and shows, inci105small 87 42 84 82 in 81the 46upper 48 44 35 17and middle cells, dentally, why four numbers, corners break the otherwise regular succession, either of numbers or of parity (see 107 89 75 52 72 71 56 54 47 33 15 in Fig. 14, with k = 5, the places of 10, 9, 11, 12; the place of 11 explains 91 77 67 58 65 60 55 45 31 13 why the numbers109following 12 were placed towards the corner cell, see p. 111 93 79 69 63 61 59 53 43 29 11 166).

We have seen 7the25 cases = 362(Fig. = 583(Fig. 39 n49 57 7), 64 n 73 97 11) 115 and n = 9 (Fig. 12); our treatise also has the two cases n = 11 and n = 7 (Fig. 14–15). 5 23 37 68 50 51 66 70 85 99 117

As already said (p. 13), this section on the construction of odd-order 3 21 78 38 40 41 76 74 80 101 119 magic squares is omitted in MS. D; for, we are told, it is wordy and may 92 table, 22 24 that 26 27 be replaced by a 1single set 90 out 88by 86 Kh¯a94zin¯121 ı, who starts with 207 the numbers in each border afresh. With the numerals 110 2 4 6 8 9 108 106 104 102 112 adapted to our writing, this table corresponds to Fig. 16: the Roman numerals indicate 2 the sequence of smaller numbers for filling each border, ◦ stands for n 2+1 , and dots for complements. vi

.

.

.

x

xii

viii

.

iv

.

.

viii

vi

xi

.

.

ii

.

iv

vii

ix

.

.

.

◦

iii

v

vii

iii

i

.

i

.

.

.

i

.

ii

iii

.

.

.

.

ii

iv

v

.

.

.

Fig. 16

207

See MS. D fol. 60v , 11 (or p. 16 n) for the Arabic text and fol. 61r for the figure.

174

General commentary

Part II: Odd-order bordered squares with separation by parity We now come to the longest (§§ 7–35) and most complicated construction of the treatise, that of bordered squares displaying separation by parity, the odd numbers all being located in a rhomb (rather, an oblique square) within the main square. Because of its difficulty, this construction left few traces later. Thus, an eleventh-century author, speaking on the subject of separation by parity for odd-order squares, says that he left out ‘difficult methods’ as being beyond the scope of his work. For there are, he adds, various simpler methods. But he omits to say that the difficulty is only for bordered squares.208 Note on this separation for ordinary magic squares In the case of ordinary magic squares, the construction is very simple, and one way to obtain such a configuration is the following.209 We take an empty square of the considered order and draw in it an oblique square the corners of which will bisect each side. With the parallels drawn through the points of intersection of that square’s sides with the main square’s rows, there appears a square of the same order as the original one. After filling it with the natural sequence of numbers (Fig. 17), we observe that some numbers, the odd ones, fall in cells of the main square while others, the even ones, are on the crossings. Let us now move the latter ones, divided into four groups, into the empty corner triangles opposite (Fig. 18). We thus obtain a magic square with the desired separation (Fig. 19): the even numbers are in the corners whereas the others, which have stayed in place, are within the rhomb.

21

16 22

11 17 23

6 12 18 24

1 7 13 19 25

14

2

3

8 14 20

9 15

Fig. 17

4 10

20 5

21

10 16 22

11 17 23

6 12 18 24

1 7 13 19

2

3

8

9 15

25

4

5

14

10

1

22

18

20

11

7

3

24

21

17

13

9

5

2

23

19

15

6

8

4

25

16

12

Fig. 18

Fig. 19

208

See our Un traité médiéval, pp. 36 & 201. See Magic squares, pp. 34–36 (Les carrés magiques, pp. 33–35; pp. 42–44). 209

,

N xki N k

97 107 93

48

38

52

65

55

69

80

90

76

3

17

7

142 128 138 123 109 119 22

36

26

13

9

5

132 136 140 113 117 121 32

28

24

15 134 144 130 115 125 111 30

20

34

175

General commentary 11 1 (§ 7) The main square and its parts 2

9

4

7

5

3

6

1

8

Fig. 20

For the construction of bordered displaying this separation, #$ ! " squares we are first to consider the different parts of the main square. As an %$ & (' example, we can consider the first case, that of order 3, which naturally % and displays the required arrangement, ) *this may have given rise to the idea of extending it to larger orders. As seen (Fig. 20), the main square contains an oblique square, which itself includes a smaller square (in this case the central cell). And this situation occurs in all larger squares (Fig. 21), and displays some peculiarities which will turn out to be essential for the construction.210

n=5

n2 + 1 − (α + 4)

158

n=7 277

α

196 154

n2 + 1 − (α + 2) α + 6 194

271 253

α+2

184 150

182 172 144 256 263 243 231 229 α+4

170 286

83

n2 + 1 − α

95

91

87

84

82

130

98

134

94

116 114 138 106

96

11

100

267 247 n2 237 + 1 − (α35 + 6) 7 51

31

3

213 215 219 223

2 79

142 118 108 4

120

n=9

n = 11

n = 13

Fig. 21

The main square, of odd order n = 2k + 1, contains n2 = 4k 2 + 4k + 1 cells. The inner oblique square will contain as many cells as there are odd numbers, thus 2k 2 + 2k + 1, and the quantity of 2k 2 + 2k even numbers will be equally divided among the outer triangles, thus 12 k(k +1) elements in each — an integer, since the product of k by k + 1 is even. Now, of these 2k 2 + 2k + 1 odd numbers, a part will be contained in the largest possible square within the rhomb while the remainder will occupy the equal triangles completing the rhomb. From Fig. 21 and the table below it appears that the inner square is the same for two consecutive odd orders 4t + 1 and 4t + 3, namely 25

15

93

111 119 115 185 187 191 107 197 265

21

45

89

117 131 135 165 167 127 173 201 245 269

17

41

57

85

113 133 143 141 151 157 177 205 233 249 273

39

55

63

α+2

n2 + 1 − (α + 2)

81

=⇒ n2 + 1 − 2

α

n + 1153 − α 145 137 161 =⇒ n181 + 1 + 209 2 109 129 227 235 251 275 2

2

281 257 241 217 189 169 139 149 147 121 101 285 261 221 193 163 155 125 123 159 n2 + 1 − (α + 2s)

α

289 225 183 171 175 105 103 36

148

152

156

160

211 195 199 203 2

α + 2s

34

n +1−α

27

77

75

97

73

49

33

69

29

5

=⇒ n2 + 1 − 2s

99

179

71

67

65

207 254

=⇒ n2 + 1 + 2s

47

59

61

239 259 287 288 146

23

43

53

255 283

37

279

19

13

9

1

174 176 140

190 192 136

206 208 132

210

This is one of the parts amply commented by B¯ uzj¯ an¯ı; see Magic squares, pp. 177– 182 (Les carrés magiques, pp. 153–157; , pp. 163–168); or also Magic squares in the tenth century, B.21, i-ii, or ll. 757–784 in the edition of the Arabic text. But there is no contribution of his to the later, more N difficult parts — rather the opposite since he turns to an empiric way of filling xki (ibid.).

N N k N

176

General commentary 76

Translation

(4t + 1) + (4t + 3) = 2t + 1. 4 t=1

t=2

t=3

t=4

t=5

n = 4t + 1 n = 4t + 3

5 7

9 11

13 15

17 19

21 23

2t + 1

3

5 Translation

7

9

11

t=1

t=2

t=3

t=4

t=5

10 t=1 18 263 345 42 4·1

t=2 36 34 52 7 50 68 9 66 844 · 482

t=3

t=4

t=5

76

Fig. 22

On the other hand, the second table below shows that the four trin = 4t ≠the 1 same 3 7 11 15 19 angles will contain for odd t = 1number t = 2 of elements t=3 t= 4 two t =consecutive 5 n = 4t + 1 5 9 13 17 orders, but just not the same order pairs as before; indeed, 21 each of the four 2 +1 n =4t4t 13 4 ·51 4 ·94 ·9 417 · 16 421· 25 triangles willncontain t2 cells for the two 4orders 4t ∓ 1. The above Fig. 21 = 4t + 3 7 11 15 19 23 shows how the 2tsame triangles can be found in two9 different (consecutive) +1 3 5 7 11 L C L C L C L C L C squares. n = 4t + 1 1 1 2 2 3 3 4 4 5 5 t=1 12 t=2 20 n = 4t ≠ 1 t=3 28 t =n 4= 4t + 136 t = 5 4t2 44 = 4t 4t+ +13 nn =

LL

C C

Fig. 23

LL

CC

11 76 13 100

19 74 15 17 21 98 132 130 4124 · 9 1224 · 16 164 4162 · 25 L

C

LL

CC

204 202 LL

CC

55 11 11 22 22 3 Summarizing, of three consecutive odd 3orders 4t +441, 4t44+ 3, 4t + 5,55 the = 11 of the 2 two20contain 18 tt = 124 first 10 oblique squares on the one hand (largest) inner t= = 22 4 18 2 28 34 26 52 50 t 20 36 squares of equal size, and, on the other hand, triangular parts surround= 33 2 36 50 34 68 74 66 100 98 tt = 284 26 52 76 ing these equal inner squares which are unequal since these squares are = 44 2 44 66 42 84 98 82 124 tt = 364 34 68 100 132 122 130 164 162 surrounded by, respectively, t and t + 1 borders. For the last two orders, t=5 44 42 84 82 124 122 164 162 204 202 the largest inner squares will be of unequal size, with sides 2t + 1 and 2(t + 1) + 1 respectively, while the triangular parts of the oblique square L C L C L C L C L C n = 4t since + 3 both 1 1 2 2 are 3surrounded 3 4 by 4t + 1 borders. 5 5 will be equal inner squares

t=1

4

2

20

18

Remark. As we shall see, arranging the odd numbers supposes knowledge t=2 4 2 28 26 52 50 of the tabove method for filling odd-order bordered squares, both for =3 4 2 36 34 68 66 100 98 filling the the 42 triangular of the square. t = 4inner squares 4 2 and44 84 parts 82 124 122oblique 164 162 Furthermore, for placing the even numbers, a specific preliminary arrangement is followed by straightforward placings of consecutive numbers, just as in the construction of even-order bordered squares (below, pp. 206–207). This indicates intimate knowledge in Greek antiquity of how to construct common bordered squares.

General commentary

177

A. Placing the odd numbers (§ 8) Filling the inner square 2

Placing the numbers in the inner square is easy: we shall put n 2+1 (odd) in the central cell, then fill the borders successively with the subsequent smaller and larger numbers, just as we did with bordered squares, but here taking only odd numbers. The largest inner square will thus be occupied by two sequences of consecutive odd numbers on either side of n2 +1 2 , and it will display its sum due as well as the characteristics of a bordered square. This is exemplified in Fig. 24, of order n = 11 (see above, Fig. 11 or 16). The smallest term reached, and last placed (in the outer border since the text begins with filling the centre) will be 37. Remark. If we wish to start, as we did with common bordered squares, in the outer border of this inner square and place ascending sequences of smaller numbers (that border being at least of order 5), we shall put the smallest number above the lower left-hand corner cell; this number will be 4t2 + 1 if n (≤ 9) has the form 4t + 1, but 4(t + 1)2 + 1 for the form 4t + 3. Next we shall put the subsequent smaller odd numbers alternately around the corner, then in the middle and corner cells, finally alternately on both sides of the opposite corner, just as we did for usual bordered odd-order squares.

43

83

81

51

47

73

55

69

59

49

77

65

61

57

45

37

63

53

67

85

75

39

41

71

79

Fig. 24

The inner square being thus a bordered square filled with two continuous sequences, we may proceed with placing the next odd numbers in the triangular parts.

178

General commentary

(§§ 9–11) Filling the remainder of the rhomb The way to place the remaining odd numbers is clearly seen from the text. After filling the inner square with the median number and the next numbers on either side of it, the subsequent odd numbers are placed in the successive borders of the main square, but within the rhomb, beginning with the border closest to the inner square. We shall thus have filled the larger halves of the triangles on the right (left in the Arabic text) and below, and the smaller halves of the triangles opposite, namely the parts left empty after placing the complements of the numbers placed first (Fig. 25). This is done progressing from inside to out with the decreasing sequence of the smaller numbers. 213 11 189 207 7

35 173 183 203

3

31

51 165 167 179 199

75

159 155 153

83

87

79

25

135 133 103

99

85

45

21

209 185 169 145 125 107 121 111 101

81

57

41

17

77

63

55

39

201

205 181 141

95

211 187 171 163 149 129 117 113 109 9

97

33

49

69

89

115 105 119 137 157 177 193 217

5

29

65

127

91

93

123 131 161 197 221

1

147

67

71

73

143 139 151 225

223 195 175

61

59

47

219 191

53

43

23

215

37

19

15

27

13

Fig. 25

This procedure may look simpler when we consider placing the smaller numbers in increasing order, beginning thus with 1. Indeed, consider the oblique square as an entity, with its lateral rows as borders and the starting point, 1, being the cell adjacent to the middle of its lower lefthand side, or else, if preferred, the lateral cell next to the lower left-hand corner of the inner square. We shall place the odd numbers alternately in the upper halves of the two contiguous left-hand lateral rows until we arrive at the two corner cells, which we leave empty whereas we fill the other two, starting with the one below. Then we resume the alternate

179

General commentary

placing from the cell next to the one last reached, this time in the righthand lower halves, until we reach the sides of the inner square. We do the same for each inner border of the oblique square. The placing will end with the two cells on the diagonals of the oblique square and next to the inner square, which we shall fill, following the same movement, first the one at the bottom then the one on the right. Finally, we shall place the complements of the numbers already put in — considering here, of course, the borders of the main square. From Fig. 25 & Fig. 26 — with n = 15, thus order 4t + 3 with t = 3, and n = 17, thus order 4(t + 1) + 1 — the perfect regularity of the procedure becomes apparent. As we have seen (p. 176), the triangular parts of the oblique squares are the same for these two consecutive orders and will therefore contain just the same smaller odd numbers. 277 11 253 271

3 79

7

35 237 247 267

31

51 229 231 243 263 95

83

265 197 107 191 187 185 115 119 111

93

25

269 245 201 173 127 167 165 135 131 117

89

45

21

273 249 233 205 177 157 139 153 143 133 113

85

57

41

17

275 251 235 227 209 181 161 149 145 141 129 109

81

63

55

39

9

223 219 215 213

87

91

33

49

73

101 121 147 137 151 169 189 217 241 257 281

5

29

69

97

159 123 125 155 163 193 221 261 285

1

65

179

99

103 105 175 171 183 225 289

207

67

71

75

77

203 199 195 211

287 259 239

61

59

47

283 255

53

43

23

279

37

19

15

27

13

Fig. 26

Seen that way, this placing indeed recalls the method taught earlier for odd-order bordered squares, with its alternate movement placing the successive numbers in two contiguous rows (Fig. 6, p. 166), and might have been inferred from it. This, though, does not appear from the extant text, which merely states which odd number will occupy which border,

180

General commentary

these borders being taken in turn from inside; nevertheless, the general placing will be inferred from it, just as for common bordered squares the general placing was inferred from successive examples.

(§ 12) Different treatments for placing the even numbers The treatment for all types of (odd) orders has so far been the same. But it now becomes necessary to differentiate between the two order types n = 4t + 1 and n = 4t + 3. Indeed, consider the cells remaining empty in each of the successive borders’ rows of the main square (including the corner cells, common to two rows); their number is a multiple of 4 in the first case and an evenly-odd number in the second: see the table below, with p numbering the incomplete borders from the smallest, thus the one surrounding the square already filled (or above, Fig. 21). Thus the procedure cannot be the same if we wish to be left, after the equalization, 122 with straightforward placings,Translation as said above (p. 176). We shall therefore consider the orders n = 4t + 1 with t ≥ 2 (since n = 5 is a particular case) and n = 4t + 3 separately. p=1

p=2

p=3

p=4

p=5

4 2

8 6

12 10

16 14

20 18

n = 4t + 1 n = 4t + 3

Fig. 27 t = 1the tsituation = 2 t = in 3 each t = border 4 t =for 5 the order Our text next considers n = 4t + 1, nbeginning = 4t + 1 with5 the smallest. 9 13 17 21

(§ 13)

n = 4t + 3

2t +of 1 Quantity

7

11

5 empty 3border cells

in

15

19

7 9 squares of

23

order 11 n = 4t + 1

t=1

t=2

t=3

t=4

t=5

n = 4t ≠ 1 n = 4t + 1

3 5

7 9

11 13

15 17

19 21

4t2

4·1

4·4

4·9

4 · 16

4 · 25

n = 4t + 1

L

C

1

1

Fig. 28C L 2

2

84

82

L

3

C

L

3

C

4

L

4

C

5

5

As known from § 7, if the main square has the order n = 4t + 1, the t=1 12 10 inner square will have the order 2t + 1. That is, there will be a quantity t=2 20 18 36 34 of t borders partly filled with odd numbers; from the informations given t=3 28 26 52 50 76 74 in the text we infer that the first six will leave the following number of t=4 36 34 68 66 100 98 132 130 empty cells Np : t=5

n = 4t + 3 t=1

44

42

L

C

1

4

1

2

L

2

20

C

2

18

124 122 L

3

C

3

164 162 L

4

C

4

204 202 L

5

C

5

181

General commentary Translation

122 N1

N2

N3

N4

N5

N6

12

28

44

60

76

92

N1

N2

N3

N5

N6

Fig. 29 N4

Generalizing, and having in mind the previous results (§ 12), we may 4 20 36 52 68 84 say that In a square of order 4t + 1, with an inner–psquare of2 order—p 2t +—1p +completely –p + 2 “p “p + 2 filled and thus with t nborders partly filled, the pth border starting from = 9, p = 2 34 36 30 32 26 28 the inner square contains 16p − 4 empty cells, with 4p in each of its rows n = 13, p = 2 78 80 66 68 62 64 (p = 1, . . . , t). p=3 74 76 58 60 50 52 n = 17, p = 2

138

140

118

120

114

(§ 14–15) Excesses and deficits in each row for the order n= 4t + 198 p=3 134 136 106 108 2

116 100 84

4 130 Since the magic sum for pa =square of order132n is M94n = n 96 · n 2+1 , 82 the sum due for m cells in one row in order to satisfy the magic condition is 2 m · n 2+1 (above, p. 2). Let us examine the each –p situation –p + 2 in “p ≠ 2 row “p after—the —p + 2 p previous placing of odd numbers —disregarding the inner square since = 7,sum p =due. 2 14 16 30 32 10 12 each of its rows makesnthe n = 11, p = 2 46 48 66 68 42 44 Relative to the cells already filled, the upper rows in each incomplete p=3 34 36 70 72 26 28 border display an excess over the sum due, as do also the (for us) left-hand n = 15, p = 2 94 96 118 120 90 92 columns, while the opposite rows show a78deficit 80 of equal122 amount since they p=3 124 70 72 are filled with complements. p = Our all necessary information 4 text 62 gives 64 126 128 50 52 about the excesses or ndeficits for2 the 158 orders 160 n = 4t 186 + 1, starting from = 19, p = 188 154 156 the border surrounding the inner in fact,190 in the192 absence p = 3 square 138 — 140 130of 132 symbolism, it merely gives the values for the first orders and notes the p=4 118 120 194 196 106 108 way they increase, but this isp = sufficient out and 5 98 to set100 198 extend 200 at will 82 a 84 table of the differences ∆ displayed by the horizontal rows (lines, L) and 76 Translation 2 4 to both 6 8 value 10 of12t for14the the vertical ones (columns, C ) =according the ú úúof theú border: ú order n = 4t + 1 (we include nI = 5)úand the number II

n = 4t + 1 t=1 t=2 t=3 t=4 t=5

L

C

1

12 20 28 36 44

1

10 18 26 34 42

L

2

C

L

2

36 34 n = 4t + 1 52 50 n = 4t + 3 68 66 84 82

ú

3

C

ú

L

3

p=1

4

p=2

4 76 74 2 100 98 124 122

8 6

ú

C

ú

L

4

p=3

12 10 132 130 164 162

ú

C

5

ú

5

p=4

p=5

16 14

20 18

204 202

Fig. 30 n = 4tdata + 3 are1sufficient 1 2 us2 to express 3 3 the differences 4 4 5 These for in5 a formula. L, 4t+1 t = 1 4 2 20 18 Let us designate by ∆p the excess of the upper row belonging to 2 26 main 52 square 50 the ptht =border if4 the2 order28of the is n = 4t + 1, and by C, 4t+1 t =that 3 of the 4 left-hand 2 36 row 34 of the 68 same 66 border. 100 98We infer from the ∆p L

t=4

4

C

2

L

44

C

42

L

C

84

82

L

C

124 122

L

C

164 162

182

General commentary

4t+1 4t+1 4t+1 table that ∆L, = 8t + 4, ∆L, = 16t + 4, ∆L, = 24t + 4, 1 2 3 L, 4t+1 ∆4 = 32t + 4, thus, generally, 4t+1 = 8pt + 4 = 2p (n − 1) + 4. ∆L, p

As for the lateral rows of these borders, their differences happen to be less than those of the horizontal rows by 2, and we shall thus have 4t+1 = 8pt + 2 = 2p (n − 1) + 2. ∆C, p

(§ 16) Quantity of empty border cells in squares of order n = 4t + 3

122 158

277

196 154

271 253

11

100

Translation

Fig. 31

84

82

130

98

134

94

The number Np ofN1empty p cut by the sides of N2 cells N3 in N4 each N5 border N6 the oblique square, starting with the innermost (p = 1), is as indicated 12 28 44 60 76 92 in the table below, which merely extends the data of the text. 194 184 150

15

267 247 237

35

7

182 172 144 256 263 243 231 229

51

31

170 286

83

95

91

87

116 114 138 106

3

213 215 219 223

96

142 118 108 4

120

25

93

111 119 115 185 187 191 107 197 265

21

45

89

117 131 135 165 167 127 173 201 245 269

17

41

57

85

113 133 143 141 151 157 177 205 233 249 273

39

55

63

81

109 129 153 145 137 161 181 209 227 235 251 275

281 257 241 217 189 169 139 149 147 121 101 285 261 221 193 163 155 125 123 159

152 156 160

2

79

N1

97

73

49

33

69

29

5

N2

289 225 183 171 175 105 103

99

36

211 195 199 203

77

75

71

148

34

61

239 259 287 288 146

27

47 23

59

4

43 19

53

255 283

37

279

13

179

65

67

207 254

20

1

9

N3

N4

N5

N6

36

52

68

84

174 176 140

190 192 136

206 208 132

Fig. 32 –p

–p + 2

—p

—p + 2

“p

Generalizing, and having in mind the previous results (§ 12), we may n = 9, p = 2 34 36 30 32 26 say that n = 13, p = 2

78

80

66

68

130

132

94

96

64

82

52 116 100 84

(§ 17) Excesses and deficits in each row for the order n = 4t + 3 –

– +2

“ ≠2

“

—

p p As before, the text first indicates the pexcesses and the deficits pfor eachp horizontal and vertical row the borders for30the first n= 7, ofp = 2 successive 14 16 32 orders, 10 and then the values ofn the increments both the = 11, p=2 46 for successive 48 66rows within 68 42 same square and corresponding rows in successive squares. This enables p=3 34 36 70 72 26 us to set out a tablen of thep differences by the horizontal = 15, =2 94 ∆ displayed 96 118 120 90 (L) and vertical (C ) rows according to both the value of t for the order p=3 78 80 122 124 70 n = 4t + 3 and the number ofp = the4 border: 62 64 126 128 50

n = 19, p = 2 p=3 p=4 p=5

158 138 118 98

160 140 120 100

186 190 194 198

188 192 196 200

28

62

In a square of order 4t + 3, with again an inner square of order 2t + 1 comp=3 74 76 58 60 50 pletely filled and thus t + 1 borders partly filled, the pth border starting from n = 17, p = 2 138 140 118 120 114 the inner square contains 16p − 12 empty cells, with 4p − 2 in each of its p=3 134 136 106 108 98 rows (p = 1, . . . , t + 1). p=4

“p + 2

154 130 106 82

—p + 2 12 44 28 92 72 52 156 132 108 84

t=2 t=3 t=4 t=5 n = 4t + 3 t=1 t=2 t=3 t=4

20 28 36 44

18 26 34 42

L

C

1

4 4 4 4

1

2 2 2 2

36 34 52 50 76 74 68 66 100 98 General commentary 84 82 124 122 L

2

20 28 36 44

C

L

C

52 68 84

50 66 82

2

18 26 34 42

3

3

132 130 164 162 L

4

C

4

100 98 124 122

204 202 L

5

183

C

5

164 162

Fig. 33

From this we may infer, as before, what these differences are in gen4t+3 and ∆C, 4t+3 the excesses of eral. Let us thus designate by ∆L, p p the upper row and of the left-hand column in the pth border when the main square has the order n = 4t + 3. Then, according to the table, 4t+3 4t+3 4t+3 ∆L, = 4, ∆L, = 8(t + 1) + 4, ∆L, = 16(t + 1) + 4, 1 2 3 L, 4t+3 ∆4 = 24(t + 1) + 4, whence, generally, 4t+3 = 8(p − 1)(t + 1) + 4 = 2 (p − 1)(n + 1) + 4. ∆L, p

As for the columns of these borders, their differences are again less by 2 than those of the horizontal rows, so 4t+3 = 8(p − 1)(t + 1) + 2 = 2 (p − 1)(n + 1) + 2. ∆C, p

B. Placing the even numbers Introductory note We now know, for each incomplete row of a border, the (even) number of empty cells it contains and its difference to the sum due. We are then to proceed as follows. An initial placing must first eliminate the difference in each incomplete border’s row (‘equalization’); that is, the cells already occupied by odd numbers and those about to receive even numbers during this initial placing must then display their sum due, thus, for the order n, as many 2 times n 2+1 as the number of occupied cells. But this initial placing must meet certain conditions: — it must be uniform so that it may be applied to all squares of the same kind of order, either n = 4t + 1 (t ≥ 2: order 5 has a treatment of its own) or n = 4t + 3; — it must involve the least possible number of cells in order to be applicable to their lowest order, namely n = 9 and n = 7, respectively; — it must fill the corner cells, each common to two rows; — it must leave in each row a number of empty cells divisible by 4;

184

General commentary

— the remaining even numbers must form groups of four, or an even number of pairs of, consecutive numbers. If all these conditions have been met in this preliminary placing, completing the treatment is an easy matter: the rows are equalized and completing them will involve only neutral placings, that is, putting in the empty cells repeatedly and without further reflection tetrads of numbers making their sum due (see § 20).

(§ 18) Grouping the even numbers by pairs Let us first consider the set of even numbers to be put in the square considered, as does our text. For the order n = 2k + 1, there
are 2k 2 + 2k 2 even numbers, namely k(k + 1) smaller ones less than n 2+1 and k(k + 1) larger ones, their complements; furthermore, since k(k + 1) is even, all these numbers, smaller and larger, may be arranged in pairs of consecutive even numbers. See the table below; here these pairs are numbered starting from the highest smaller numbers, the jth pair of smaller numbers having 2 2 then the form n −8j+3 , n −8j+7 , with j = 1, 2, . . . , 12 k(k + 1). 2 2 2 n2 − 1 1 k(k 2

4

| ... |

n2 − 3 | . . . | + 1)

| ... |

n2 −8j+3 2

n2 −8j+7 2

n2 +8j−1 2

n2 +8j−5 2

j Fig. 34

| ... | | ... | | ... |

n2 −5 2

n2 −1 2

n2 +7 2

n2 +3 2

1

We are, as far as possible, to equalize the rows without breaking these pairs of numbers, for that will facilitate later neutral placings. Furthermore, since the difference to the sum due is the same (but with opposite signs) in two opposite rows, we may carry out the equalization for just one of two opposite rows. Note on the equalization rules We are now to see two main equalization rules and particular cases of them. As seen before, all the differences displayed by the rows are even, and follow one another in regular succession. We already know that we are to use, as far as possible, pairs of successive numbers, in a quantity as little as possible, to eliminate the differences left. The author’s purpose is thus to show how placing pairs of even numbers may modify these differences.

185

General commentary

(§ 19) Effect of placing a pair of small even numbers in the same row n2 − 8j + 3 2

n2 − 8j + 7 2

=⇒ n2 + 1 − (8j − 4)

n2 + 8j − 1 2

n2 + 8j − 5 2

=⇒ n2 + 1 + (8j − 4)

Fig. 35

What we shall call Rule I tells us that placing the jth (j ≥ 1) pair on one side and the complements opposite will produce a deficit of 8j − 4 on the side of the jth pair and an excess of the same amount opposite (Fig. 35). That is, all remaining differences of the form 4, 12, 20, 28,. . . may be eliminated using two cells provided the appropriate pair of successive even numbers is available. Particular cases

(I0 , § 22) If these pairs are written in the corners (Fig. 36), that will also produce a constant difference of 2 in the two columns, which will be a deficit in that containing the lesser element. n2 − 8j + 3 2

n2 − 8j + 7 2

=⇒ n2 + 1 − (8j − 4)

n2 + 8j − 5 2

n2 + 8j − 1 2

=⇒ n2 + 1 + (8j − 4)

⇓ n2 + 1 − 2

⇓ n2 + 1 + 2

Fig. 36

(I00 , § 22) Thus, for instance, placing the largest pair in the upper corners

will reduce the excess of the upper row by 4 and that of the left-hand column by 2.

(§ 21) Effect of placing two small even numbers in opposite rows As said in the text, with an even number α written in one row then α + 2s in the opposite row, and next, facing them, their complements, the row of α will display a deficit of 2s and the opposite row an excess of 2s (Fig. 37).

always increased by 2.176 That is, with a number – written in one row α+2 then –+2s in the opposite row, and next, facing them, their complements, 2 α + 6 the n + 1 − (α + 6) 186row of – will display a General deficit ofcommentary 2s and the opposite row an excess of 2s (Fig. 294).

n2 + 1 − (α + 2)

α+4

–

⇓

n2 + 1 − α n2 + 1 ≠ (– + 2s)

n2 + 1 + 4 – + 2s

n2 + 1 ≠ –

⇓

=⇒ n2 + 1 + 4 =∆ n2 + 1 ≠ 2s

n2 + 1 − 4

=∆ n2 + 1 + 2s

Fig. 294 37 Fig.

n2 −1 n2 −5 n2 +1 we may any difference, 2 eliminate 2 2 , we may eliminate anyeven even difference,

With rule, II thus, in in fact, fact, With this Rule any which may occur its use with s = ” 1 should be any difference difference which here. But its use with s = 6 1 should be n=7 24 22 25 avoided to keep keep all all pairs pairs of of consecutive consecutive avoided as as far far as as possible since we must try to n = 11 60 58 61 even numbers numbers unbroken unbroken in order to complete even complete the the placing. placing. 112 110 113 n = 15 Particular cases n = 19 –180 n2 +170 1 ≠ (– + 4) 181 =∆ n2 + 1 ≠ 4

(II0 , § 30) With s = 1 (Fig. 38), we may eliminate an excess or a difference

of 2. Since the two even numbers placed are consecutive, no pair is broken –+2 n2 + 1 ≠ (– + 2) . 2 n + 1 ≠ (– + 6)

–+6

n + 1 − (α + 2) 2

α n2 + 1 ≠ – »

n +1≠4 2

n2 + 1 − α

=⇒ n2 + 1 − 2

–+4

α+2

Fig. Fig. 295 38

=∆ n2 + 1 + 4 » n2 + 1 + 2 =⇒

n +1+4 2

17600 , § 29) We may also consider placing two consecutive pairs, thus four (II Magic squares in the tenth century, A.II.21, and MS. London BL Delhi Arabic

110, fol. 83r , 9-14; or numbers, below, Appendix 35. the border. With the complements, consecutive even around

this will produce uniform differences of 4, which will be a deficit in the two rows containing the lesser pair (Fig. 39). α

n2 + 1 − (α + 4)

=⇒ n2 + 1 − 4

α+2

n2 + 1 − (α + 2)

n2 + 1 − (α + 6)

α+6

n2 + 1 − α

α+4

⇓

n2 + 1 − 4

=⇒ n2 + 1 + 4 ⇓

Fig. 39

n2 + 1 + 4

Having thus given the two main equalization rules, the text concludes (§ 22): You must understand all this: it belongs to what you need for writing the even (numbers) in (squares of) this kind.

General commentary

187

Remark. As seen above, the differences to be eliminated are always even. Now by applying once, twice or at most three times the above rules using pairs of consecutive numbers placed on one or the other side, we can eliminate differences of the form 8u, 8u ± 2, 8u ± 4, 8u ± 6, that is, any difference which might occur. To do this, however, the number of empty cells available must be sufficient. If that is not the case, we shall be obliged to eliminate the difference by means of Rule II, thus using two even, non-consecutive numbers (α, α + 2s, with s 6= 1); but then we shall have to apply it a second time in order to use the other two terms of the two broken pairs. See example below, § 24, in the equalization of all first (that is, smallest) borders for the order n = 4t + 1 (excess of the form 8u + 4 but with four cells to be filled211 ). It may also happen that the difference can be eliminated with fewer cells than the quantity which is required to leave a number of empty cells divisible by 4; then we shall just use more cells than necessary (one application may only partly eliminate the difference, or even increase it). See example below, §§ 25–31, in the equalization of the other borders’ horizontal rows for the order n = 4t + 1 (once again excess of the form 8u + 4 but with this time eight cells available, this enabling us to use only consecutive pairs). In all other cases we shall be able to apply these rules to the least possible number of cells, and so as to leave a number of empty cells divisible by 4, thus paving the way to the neutral placings.

(§ 20) Neutral placings Once the equalization has been effected, completing the arrangement is easy: we are to place groups of four — or groups of two pairs of — even consecutive numbers, with the extremes on one side and the means opposite. The four numbers will thus have the form α, α + 2, α + 4, α + 6 if they are consecutive (Fig. 40) or α, α + 2, β, β + 2 if they form two pairs (§ 31). α

n2 + 1 − (α + 2)

n2 + 1 − (α + 4)

α+6

=⇒ 2(n2 + 1)

n2 + 1 − α

α+2

α+4

n2 + 1 − (α + 6)

=⇒ 2(n2 + 1)

Fig. 40 211

With two cells the choice of the pair j = u + 1 would be appropriate: see §§ 32–33.

188

General commentary

C. Completing the squares for the three order types Introductory note As seen before (§ 12), there are for the filling of odd-order squares displaying the separation by parity three categories. That of order 5, which is a particular case, that of the further squares with n = 4t + 1, and that of the squares with n = 4t + 3. The first will be treated in the coming paragraph, the second category in §§ 24–31, and the third in §§ 32–35.

(§ 23) Square of order 5

As observed in § 12, the general construction for orders of the type n = 4t + 1 (represented in our Fig. 49 further below, p. 194) cannot apply 2 to the smallest order, with t = 1. Indeed, since in this case n 2−5 and n2 −1 happen to be numerically equal to, respectively, 2n and 2n + 2, the 2 even numbers supposed to occupy the upper corners would also be found within the border. Our text gives therefore a particular construction for this square, with the result as in our Fig. 41 (orientation changed and inner square inverted). 2

16

25

18

4

6

7

21

11

20

23

17

13

9

3

12

15

5

19

14

22

10

1

8

24

Fig. 41

Whereas our text merely indicates where to put each number, without any justification, B¯ uzj¯ an¯ı observes that if we place 2 in a corner, the next corner (horizontally or vertically, if we keep 1 below) must be occupied by the (smaller) number 4, 6, 8 or 12, which leads to ‘numerous figures’.212 As a matter of fact, this gives eleven figures. But if we do not keep 2 in a corner, there are altogether twenty-one possibilities with the first twenty-five numbers.213 212

Magic squares in the tenth century, B.22, ii; or edition of the full text, pp. 177 & 233–234. 213 Listed (including the previous eleven) in Magic squares, p. 209 (Les carrés magiques, p. 178; , p. 190). Furthermore, considering permutations of the even numbers within the border rows together with the eight aspects resulting from inverting and rotating the inner 3 × 3 square will lead to 21 · 4 · 8 = 672 configurations N for this 5 × 5 square. xki N N k N

189

General commentary

Squares of orders n = 4t + 1, t > 1 Since the bottom rows are to receive the complements, we shall consider only the top ones, thus those in excess; likewise, we shall consider the left-hand rows, also in excess. As we have seen (§ 13), the rows of the pth border (p = 1, . . . , t, counting from the inner square already filled) comprise 4p empty cells. Eliminating the differences must thus be effected with four cells for the rows of all first borders, whereas we may use eight cells for the rows of the other borders, this being applicable to the outer border of the lowest order n = 9 as well.

(§ 24) Filling all first borders As just said, only four cells are available, and the excess in the upper 4t+1 rows is of the form ∆L, = 8t + 4 (§ 14). We are thus to eliminate this 1 excess by means of two pairs of numbers. This will be done as follows (Fig. 42). n2 − 5 2

n2 + 1 − ∆C 1

2

n2 − 1 2

4

n2 + 1 − 4

n2 + 1 − ∆L1

∆L1

n2 + 3 2

n2 + 1 − 2

n2 − 1 2

n2 + 1 − ∆C 1

∆C 1

Fig. 42

n2 + 7 2 n2 − 5

2

Let us take for the first pair the largest2 dyad (j2 = 1), namely n 2−5 2 and n 2−1 , and put and right-hand 4 n2 +it1 −in 4 the corners, say in the left-hand 2 corners, respectively. Since their sum is n − 3, they display a deficit of 4 relative to their sum due n2 + 1 (according to Rule I0 ), and this leaves us with eliminating L8t by means of two numbers. The only way is, in ∆1 n2 + 1 − ∆L1 conformity with Rule II, to use two numbers from different dyads, say n2 +27 and thus nn2 2++ 3 1 − (8t + 2), which C 2 the smallest number the larger number ∆1 n +1−2 2 2 happens to be n2 + 1 − ∆C 1 ; their sum eliminates the excess completely and fills the last two available cells. This is just what the text says, except that it is 8t+2 (the 4tth even number after 2, as the extant text expresses it) which is said to be put in the opposite row. See, for the values, the table in Fig. 43 and, for the squares, Fig. 46–48. Consider now the columns. The left-hand corner cells are now occupied by the smaller of the numbers placed first and the complement of 2 2 the other, thus by n 2−5 , n 2+3 , the sum of which is n2 − 1; the initial

190

General commentary

36 excess on the left-hand side, ∆C 1 = 8t + 2, is thus reduced to 8t (again in 0 accordance with Rule I ). Since there are only two empty cells available, display a deficit of 4 relative to their sum due n2 + 1, and this leaves weusmust again eliminate thismeans excess,ofaccording to Rule II,only by two with eliminating 8t by two numbers. The waynumbers is, in from different dyads. Putting 4 in the (for us) left-hand column and, conformity with Rule II, to use two numbers from different dyads, sayin L 2 + the one, 8t 2+ and 4 (thus not only shall nwe have 1 ), larger theright-hand smallest number thus∆the number 1 ≠ eliminated (8t + 2), the remaining excess 8t, since the left-hand column now contains and the sum of which eliminates the excess completely and fills the last 4two 2 n available + 1 − (8t + 4),This butiswe shall also the remaining cells. just what thehave text placed says, except that it is terms 8t + 2of the twois,pairs broken, avoiding therow. problem of using (that 2n) previously which is said to be thereby put in the opposite See, for the subsequently two single numbers. See Fig. 43 and Fig. 46–48. values, the table below and, for the squares, Fig. 269-271. n = 9 ( t = 2) n = 13 (t = 3) n = 17 (t = 4)

n2 ≠5 2

n2 ≠1 2

2

8t + 2

4

8t + 4

38 82 142

40 84 144

2 2 2

18 26 34

4 4 4

20 28 36

III

IV

V

VI

I

Fig. 43 II

(§§ 25–28) Obtaining uniform excesses and deficits in all other borders

Considern2 ≠8p+7 now the columns. (§§ The left-hand corner cells are occuAccording to what precedes 13–14), the excesses in the now upper rows n2 ≠8p+3 n2 ≠2pn+10p≠5 n2 ≠2pn+10p≠1 n2 ≠2pn+2p+3 n2 ≠2pn+2p+7 2 4t+1 2 2 2placed first 2and the complement 2 L, pied by the smaller of the numbers of are ∆p = 8pt + 4,2 and there are 4p empty cells. A direct elimination n2 +3 n = 9, p = 2 the other, 34 36 by n ≠530 32 26 is n2 ≠ 1;28the initial thus , , the sum of which by one dyad, thus with 2 two2 cells (in this case corner cells), would be C, 4t+1 n = 13, p = 2 excess 78 on the80 66 68= 8t + 2, is62thus reduced 64 left-hand 8t (in possible, namely taking jside, = pt + it would be unsuitabletosince the 1 1; but p = 3 accordance 74 76 58 60 50 52 Õ with Rule II , just as before). Since there are only two empty number of cells left empty would not be divisible by 4. On the other n = 17, p = 2 140 118 120 114 116 cells 138 available, we by must again eliminate this excess, according to possible. Rule I, hand, elimination two dyads, thus using four cells, is not p=3 134 136 106 108 98 100 by two numbers from different dyads. Putting 4 in the (for us) left-hand shall eliminate these excesses by means of eight p = 4Therefore 130 we 132 94 96 82 84 cells, this column in the right-hand one, 4order + 8t n(the number being alsoand, applicable to the smallest = 9.4tth Theeven same will beafter done 4), not only shall we have eliminated the excess of 8t since the left-hand C, 4t+1 with the left-hand columns, originally ∆p VI = 8pt + 2. I excesses II in the III IV V column now contains 4 and n2 + 1 ≠ (4 + 8t), but we shall also have placed All that will be reached in five steps (i-v below). In the first (§ 25i, § 26i, the remaining terms of the two pairs previously broken, thereby avoiding 2 ≠2(p≠1)n≠2p≠3 2 two corner cells § 27i), nwe shall fill the of each pthn2column in excess (for ≠2(p≠1)n≠10p+5 n2 ≠2(p≠1)n≠10p+9 n2 +8p≠5 n2 +8p≠1 the problem of usingn ≠2(p≠1)n≠2p+1 subsequently two single numbers. See above table, 2 2 2 2 2 us the left-hand ones) by means of a consecutive even pair αp , αp + 2. In2 and 14 269-271. n = 7, p the =V-VI 2 second 32 10 β , β + 2 put12 (§ 25ii, § 26ii, 16 § 27ii), two30consecutive numbers p p n = 11, p =(§§ 2 25-31)46Filling all other 48 borders 66 68 42 within these columns will reduce their excesses to a uniform quantity. In44 p=3 34 36 70 72 26 28 4t+1 = the third step (§seen 25iii, 26iii, §the 27iii), this in same will be L,attained (§§§13-14), excesses the situation upper rows n = 15, p = 2 As already 94 96 118 120 90 are p 92 for the horizontal rows, placing pairs γ , γ + 2 in two of the six cells still p p 8pt + 4, and there are 4p empty cells. A direct elimination by one dyad, p=3 78 80 122 124 70 72 available. relies on thepossible, use 126 of Rule I and Rule j50 I0= . The two52 thus withAll twothis cells, would namely putting pt + last 1; but p= 4 62 64 be 128 equalization applied together to horizontal rows andwould columns, will it would be steps, unsuitable since the number of cells left empty not be remove, thethe remaining fourelimination cells, the differences left. VI divisiblebybyusing other by two dyads, thus using I 4. On II hand, III IV constant V four cells, is not possible. Therefore we shall eliminate these excesses (i) Filling the corners of the column in excess by means of eight cells, this being also applicable to the smallest order We first look for two even numbers, αp and αp + 2, adding up to n = 9. The same will be done with the excesses in the left-hand columns, (n2 + 1) − (8pC,−4t+1 4). Therefore, these numbers will be originally p = 8pt + 2. All that will be reached in five steps (i-v n2 − 8p + 3 § 26i), we shall n2 − 8p + 7 below). In the αp first = (§ 25i, § 26i, , αp + 2 = fill the two, corner cells 2 2

n2 − 5 2

n2 − 1 2

n + 1 − ∆1 2 General commentary 2

C

191

the lesser of them4 being put in the upper cornern2(Fig. + 1 − 4 44). For these placings, the text explains that we are to put, at either end of each of n2 + 1 − ∆L ∆L1 these columns, a pair 1of consecutive small numbers the sum of which is, respectively, less than their sum due by 12, 20, 28, that is, by our 8p − 4 2 (this subtractive nquantity being thus the same for same +3 n2 + 7borders). See the ∆C n2 + 1 − 2 1 resulting values for2 the first orders in Fig. 45, and, for2 their placing, Fig. 46–48. Note that the above formulas are valid for the first border as well (p = 1), but with the pair being placed in the upper row. γp + 2

γp

αp

n2 + 1 − (αp + 2)

βp

43 n2 + 1 − β p

βp + 2

n2 + 1 − (βp + 2)

of consecutive small numbers the sum of which is, respectively, less than their sum due by 12, 20, 28, thus by our 8p ≠ 4 (this subtractive quantity being thus the same for same2 borders).24 2See the resulting values for the αp + 2 n + 1 − (γp + 2) n2 + 1 − αp n + 1 − γp first orders in Fig. 268, I and II, and Fig. 269-271. Note that the above formulas are valid for the first border as well (p = 1).

Fig. 44

–p

–p + 2

—p

—p + 2

“p

“p + 2

n = 9, p = 2

34

36

30

32

26

28

n = 13, p = 2

78

80

66

68

62

64

p=3 n = 17, p = 2 p=3 p=4

74 138 134 130

76 140 136 132

58 118 106 94

60 120 108 96

50 114 98 82

52 116 100 84

Fig. 45

(ii) With the placing of these two numbers, adding up to (n2 +1)≠(8p≠4), C, uniform 4t+1 = 8pt (ii) the Reducing the excesses the columns value initial excess of the pthinleft-hand columnto a + 2 has p become 8p(t ≠ 1) + 6. Now we know from the second main equalization 2 With these two numbers, adding up to (n + 1) − (8p − 4), the initial rule how to reduce an excess of such a form: putting 8p(t≠1)+6 = 8j ≠4, 4t+1 excess of the left-hand ∆C, 8pt + has≠ become p closest= we have 8j pth = 8p(t ≠ 1) + 10.column Taking the value, j =2 p(t 1) + 1, 8p(t − 1) +we 6.shall Nowputwein infer from the first main rule how an excess the pth left-hand column (p Øequalization 2) the pair

of such a form may treated: −+1)10p +≠6 5= 8j − 4, we have n2 ≠be 8[p(t ≠ 1) + 1]putting +3 n28p(t ≠ 2pn —p = = 8j = 8p(t − 1) + 10; taking 2 the closest integral 2 value, j = p(t − 1) + 1, n2 ≠remainder 8[p(t ≠ 1) +of 1] + 7 We n2 shall ≠ 2pn therefore + 10p ≠ 1 put in the pth will leave a— uniform 2. = , p+2= 2 pair 2 left-hand column (p ≥ 2) the the values of which for the first orders are seen in Fig. 268, III and IV, or Fig. 269-271.n2This then leaves an1] excess left-hand − 2pn + 10pcolumn −5 − 8[p(t − 1) + + 3 of 2nin2 each p = to complete the equalization. = and fourβcells

2

2

For the above placings, the text explains that we2 are to put in the (for n2 − 8[p(t − consecutive 1) + 1] + 7smallnnumbers − 2pnadding + 10pup − to 1 us left-hand) columns pairs of βp + 2 = = . their sum due less an amount 2equal to the (initial) excess of2the column minus, respectively, 14, 22, 30, thus minus the quantity 8p≠2; that is, the sum of these two numbers —, — + 2 must be (n2 + 1) ≠ [8pt + 2 ≠ (8p ≠ 2)] = n2 ≠ 8[p(t ≠ 1) + 1] + 5, whence the above pair of values.25 24

Magic squares in the tenth century, A.II.25i, 26i, 27i and MS. London BL Delhi v

192

General commentary

These values for the first orders are seen in the above table, and, for the corresponding squares, in Fig. 46–48. We are then left with an excess of 2 in each left-hand column and have four cells to complete the equalization. For the above placings, the text explains that we are to put in the (for us left-hand) columns pairs of consecutive small numbers adding up to their sum due less an amount equal to the (initial) excess of the column minus, respectively, 14, 22, 30, thus minus the quantity 8p−2; that is, the sum of these two numbers βp , βp +2 must be (n2 +1)−[8pt+2−(8p−2)] = n2 − 8[p(t − 1) + 1] + 5, whence the above pair of values. (iii) Reducing the excesses in the upper rows to a uniform value

Consider now the upper rows. Their initial excesses have changed since one number has been placed in each pth left-hand corner while the right-hand one is occupied by a complement. Since these two numbers, namely n2 − 8p + 3 n2 + 8p − 5 , , 2 2 add up to n2 − 1 = (n2 + 1) − 2, the initial excesses in the upper rows have been reduced from 8pt + 4 to 8pt + 2, and there are 4p − 2 empty cells. Putting, in the same manner as before, 8pt + 2 = 8j − 4, we are led to take j = pt and thus place within the pth upper row, the ptth pair of smaller numbers n2 − 8pt + 3 n2 − 2pn + 2p + 3 γp = = 2 2 n2 − 8pt + 7 n2 − 2pn + 2p + 7 γp + 2 = = . 2 2 These values for the first orders are seen in the above table, and in Fig. 46–48 for the squares. This then leaves an excess of 6 in each upper row and four cells to complete the equalization. The text expresses the choice of these numbers in the same way as before: we are to put in the pth upper row two consecutive even smaller numbers, say γp and γp + 2, the sum of which is less than their sum due by the (initial) excess of the upper row minus 8 (thus minus a quantity independent of border and order); that is, their sum must be 2γp + 2 = (n2 + 1) − [(8pt + 4) − 8], which gives the above two values. With this and what we have found for the first border, we can now set out the table in Fig. 49 (the asterisk marks cells to receive complements). Although it gives the required quantities for only five borders, it clearly shows how it may be extended: the terms belonging to the same oblique line can be found using arithmetical progressions —the only exceptions

193

General commentary

being in the first border. That is therefore sufficient for constructing any square of order n = 4t+1 with separation by parity; indeed, what remains to complete its rows is straightforward. 50

52

58

78

62

66

82

2

68

4

47

60

34

26

77

28

161

74

64 3

153 113

67

107 105

75

71

57

17

97

79

83

73

53

29

13

49

35

27

69

71

64

40

52

32

4

23

63

61

31

27

78

50

159 143 135 121 101

36

35

49

39

29

9

45

41

37

25

15

166 102

46

3

57

137 139 151 144 84 104 110 51

2

53

23

90 112

59

38

67

145 155

55

30

73

7

131 127 125

157 141 117

75

94

5 7

93

89

85

81

69

21

41

61

87

77

91

109 129 149 165

1

37

99

63

65

95

103 133 169

39

1

17

43

33

47

65

81

142 119

43

45

115 111 123

28

62

55

19

21

51

59

20

86 168 167 147

33

31

26

88

42

80

79

13

11

18

44

163

25

15

56

54

5

80 108 106

Fig. 46 130 82

92

9

76 120 118

48

19

96

Fig. 47

84

158

277 11

253 271

7

35

237 247 267

31

51

229 231 243 263 256 144 172 182

94 134 98 100 96 106 138 114 116 108 118 142

2

120

79

3

87

91

150 184 194

83

265 197 107 191 187 185 115 119 111

93

25

269 245 201 173 127 167 165 135 131 117

89

45

21

273 249 233 205 177 157 139 153 143 133 113

85

57

41

17

275 251 235 227 209 181 161 149 145 141 129 109

81

63

55

39

9

223 219 215 213

154 196

95

4

286 170

33

49

73

101 121 147 137 151 169 189 217 241 257 281

5

29

69

97

159 123 125 155 163 193 221 261 285

1

65

179

99

103 105 175 171 183 225 289

254 207

67

71

75

77

203 199 195 211

146 288 287 259 239

61

59

47

283 255

53

43

23

279

37

19

140 176 174 136 192 190 132 208 206

13

Fig. 48

27

11

15

36

34 148 152 156 160

n2 −29 2

∗

Fig. 49

∗

∗

n2 −13 2

n2 −6n+9 2

∗

∗

∗ ∗

∗

∗

n2 −4n+19 2

n2 −6n+29 n2 −4n+15 2 2

n2 −8n+39 n2 −6n+25 2 2

n2 −21 2

n2 −8n+11 n2 −8n+15 2 2

n2 −10n+13 n2 −10n+17 2 2

n2 −10n+49 n2 −8n+35 2 2

n2 −10n+45 2

n2 −37 2

∗

∗

∗

∗

4

n2 −5 2

n2 −4n+7 2

n2 −6n+13 2

∗

∗

2

n2 −4n+11 2

2n

∗

∗

2n + 2

∗

n2 −1 2

∗

∗

∗

n2 +11 2

∗

∗

∗

n2 +19 2

∗

∗

∗

n2 +27 2

∗

∗

∗

n2 +35 2

194 General commentary

195

General commentary

(§§ 29–31) Completing the equalization α +2

p As stated in the text (§ 28) there pnow remains — except for the rows of the first border, which are completely filled (and equalized) — a number of cells divisible by 4 in each row and uniform excesses of 6 in the upper n2 + 1 − γp γp ones and of 2 in the left-hand columns (Fig. 50). In order to have an 2 γp − 2 n + 1 − (γp − 2) equalization valid from the smallest order (here n = 9), we are to fill four βp that the numbers still available all n2 + 1 − βp cells in each row. We further know form groups of four, or pairs of dyads βp + 2 of, consecutive even numbers, forn2 + 1 − (βp + 2) we have already placed the two pairs which had been broken.

α

2 Eliminating the remainingnexcesses + 1 − (αp +will 2) be carried out in the next two steps.

⇒ +2

⇒ +6 ⇒ −6 ⇓ +2

n2 + 1 − (ϵ + 2) ⇓ ⇓ −2 +2

ϵ

⇓ −2

Fig. 50

⇒ −2

Fig. 51 ϵ+2

n2 + 1 − ϵ

(iv) First (§ 29), using Rule II00 (p. 186), we put four consecutive (even) numbers around each border, beginning on the top and turning (for us) towards the left. δ

n2 + 1 − (δ + 4)

δ+2

n2 + 1 − (δ + 2)

n2 + 1 − (δ + 6)

δ+6

δ+4

n2 + 1 − δ

Fig. 52

As we know from this rule, and from Fig. 52, with the complements in place, that will change the excesses and deficits, formerly ±6 in the horizontal rows and ±2 in the columns (Fig. 50), to ±2 and ∓2 (Fig. 51). In Fig. 54 (n = 9, with p = 2) we have thus placed, starting in the upper row, the groups of four numbers 6, 8, 10, 12; in Fig. 55 (n = 13) the groups 6, 8, 10, 12 (p = 3) and 30, 32, 34, 36 (p = 2); in Fig. 56 (n = 17) the groups 6, 8, 10, 12 (p = 4), then 22, 24, 26, 28 (p = 3) and 42, 44, 46, 48 (p = 2). Our text also constructs the three examples with

n2 + 1 − αp

196

γp

n + 1 − γp

γp − 2

n + 1 − (γp − 2) 2

n2 + 1 − βp

General commentary

βp βp + 2

n2 + 1 − (βp + 2)

n2 + 1 − (α + 2)

n2 + 1 − α

n = 9, 13, 17.214 p p (v) Second (§ 30), using Rule II0 , thus the simplest case (s = 1) of the second equalization rule, take any two pairs of available consecutive numbers and place each in a pair of opposite rows (Fig. 53), with the lesser term on the side of the excess.

ϵ

n2 + 1 − (ϵ + 2)

n2 + 1 − ϵ

ϵ+2

Fig. 53

With that done for all borders, and the complements written in, all remaining differences will be eliminated. In Fig. 54 (n = 9), we have thus placed 14, 16 and 22, 24 (p = 2); in Fig. 55 (n = 13) 14, 16 and 18, 20 n2 + 1 − (δ + 4) δ (p = 3), then 22, 24 and 38, 40 (p = 2); in Fig. 56 (n = 17) 14, 16 and 18, 20 (p = 4), then 30, 32 and 38, 40 (p = 3), finally 50, 52 and 54, 56 (p = 2). n2 + 1 − (δ + 2) δ+2 n2 + 1 − (δ + 6)

74

50

52

6

160 δ 14 + 6 161 154 42 126 124 48

58

78

62

64

30

7

60

66

82

2

3

23 137 139 151 144 84 104 110

8

68

4

δ+4

n2 + 1 − δ

145 155 136 22 146 90 112

47 131 127 125 55

59

51 166 102 162

158 32 153 113 67 107 105 75

71

57

17 138 12

152 157 141 117 97

34

26

28

6

77

72

14

66

46

30

38

2

3

69

71

64

40

52

32

4

23

63

61

31

27

78

50

8

73

53

35

49

39

29

9

74

75

67

57

45

41

37

25

15

7

54 134

79

93

83

73

53

29

13

18

159 143 135 121 101 89

85

81

69

49

35

27

11

20

5

21

41

61

87

77

91 109 129 149 165 150

1

37

99

63

65

95 103 133 169 36 116

43

45 115 111 123 28

70

1

17

43

33

47

65

81

12

114 132 142 119 39

60

62

55

19

21

51

59

20

22

100 40

24

42

80

79

13

11

18

44

58

72

36

56

54

76

5

10

68

16

48

76 120 118 164 10 156

Fig. 54

94

86 168 167 147 33

31

19

80 108 106 140 163 25

15

34 148 24

9

26

16 128 44

38

56

88 130 70 92

98

46 122 96

Fig. 55

As asserted (§ 31) we have now completed the first two borders around the central square and are left in the subsequent borders (p ≥ 3) with numbers of empty cells all divisible by 4. Since the rows of these borders 214

The difference with the squares actually presented by our text is due to the fact that the author fills the cells from the first inside border and by taking the available small numbers in decreasing order. In both cases the numbers chosen just fill up progressively the gaps left by the former placings in the continuous number sequence.

197

General commentary

display neither excesses nor deficits, we may fill them repeatedly with neutral placings, thus either four consecutive even numbers or two separate pairs of even numbers, of which we write the extreme terms on one side and the middle terms on the other, the complements providing then the sum due for four cells (above, § 20). In Fig. 55 (n = 13), 42, . . . , 48 and the pairs 54, 56 & 70, 72 complete the outer border (50, 52 and 58, . . . , 68 have been used for the preliminary placing); in Fig. 56 (n = 17), the two upper rows and the ones opposite have been completed with the tetrads 58, . . . , 64, 66, . . . , 72 and 74, . . . , 80, respectively, the columns with 86, . . . , 92 and the pairs 102, 104 & 110, 112, and 122, . . . , 128, respectively. 130 82

84

6

280 14 274 58 277 230 228 64

66 222 220 72 158

94 134 98 100 22 264 30

11 253 271 258 74 214 212 80 154 196

96 106 138 114 116 42

7

35 237 247 267 244 50 238 150 184 194

8

31

51 229 231 243 263 256 144 172 182 282

108 118 142

278 24 120

4

2

3

79 223 219 215 213 87

91

95

83 286 170 266 12

272 262 44 265 197 107 191 187 185 115 119 111 93

25 246 28

20 252 269 245 201 173 127 167 165 135 131 117 89

45

21

38 270

86 273 249 233 205 177 157 139 153 143 133 113 85

57

41

17 204

275 251 235 227 209 181 161 149 145 141 129 109 81

63

55

39

202

18

15

9

33

49

73 101 121 147 137 151 169 189 217 241 257 281 88

200 40

5

29

69

1

65 179 99 103 105 175 171 183 225 289 48 168 198

92 122 242

97 159 123 125 155 163 193 221 261 285 250 90

102 166 236 254 207 67

71

75

77 203 199 195 211 36

54 124 188

186 164 56 146 288 287 259 239 61

59

47

27

34 148 234 126 104

180 128 140 176 174 248 283 255 53

43

23

46 240 52 152 162 110

112 136 192 190 268 26 260 279 37

19

32 216 76

132 208 206 284 10 276 16 232 13

60

62 226 224 68

78 210 156 178 70 218 160

Fig. 56

Squares of orders n = 4t + 3 We shall again consider only the rows in excess, thus the upper rows and the (for us) left-hand columns.

198

General commentary

(§ 32) Completing all first borders As already seen (§§ 16–17, pp. 182–183), the number of empty cells 4t+3 = in each row of the pth border is 4p − 2 and the excesses are ∆L, p 4t+3 = 8(p − 1)(t + 1) + 2. Thus, for all first 8(p − 1)(t + 1) + 4 and ∆C, p borders the equalization must be effected with two cells. 4t+3 4t+3 Since the excesses are ∆L, = 4 for the upper rows and ∆C, =2 1 1 for the columns, we shall apply the particular case I00 (§ 22, p. 185). That is, we put in the upper corners of the first row the largest pair of smaller numbers, thus n2 − 5 n2 − 1 , , 2 2

with the larger one in our right-hand corner, where the column displays a deficit. Then we shall have, as stated in the text, equalized all first borders. Note that the same pair had been used for the first borders in the case n = 4t + 1 —though without then completing the equalization. In Fig. 59 (n = 7) we have thus placed in the upper corners of the first border 22, 24; in Fig. 60 (n = 11), 58, 60; in Fig. 61 (n = 15), 110, 112.

(§ 33) Equalizing the horizontal rows of all other borders A single pair will turn out to be again sufficient for all upper rows of larger borders (p ≥ 2), while for the columns, of which two corner cells are now occupied, at least four cells will be available. αp

αp + 2

γp

n2 + 1 − γp

γp − 2

n2 + 1 − (γp − 2)

βp

n2 + 1 − βp

βp + 2

n2 + 1 − (βp + 2)

n2 + 1 − (αp + 2)

n2 + 1 − αp

Fig. 57 4t+3 = 8(p − 1)(t + 1) + 4, we Since the excess in the upper rows is ∆L, p shall choose, as stated in the text, a pair of (consecutive) small numbers ϵ 1 −1, (ϵ +by 2) that amount: less than their sum due, nn2 2++

αp + αp + 2 = n2 + 1 − [8(p − 1)(t + 1) + 4]. n2 + 1 − ϵ

ϵ+2

199

General commentary

Thus, applying Rule I0 (p. 185; cf. p. 187n), we shall take j = (p − 1)(t + 1) + 1 and write in the end cells of the pth upper row (p ≥ 2) the corresponding pair, that is, n2 − 8[(p − 1)(t + 1) + 1] + 3 n2 − 2(p − 1)n − 2p − 3 = , 2 2 n2 − 2(p − 1)n − 2p + 1 n2 − 8[(p − 1)(t + 1) + 1] + 7 = , αp + 2 = 2 2 51 the pair with the lesser term in the column with an excess. (Note that for p = 1 relies on the same formula.) See Fig. 58, and Fig. 59–61 for the 279-281. As then stated in the text (§ 33, in fine), we are left with filling squares of orders n = 7, 11, 15. As then stated in the text (§ 33, in fine), the incomplete horizontal rows with neutral placings. we are left with completing the horizontal rows with neutral placings. αp =

–p

–p + 2

n = 7, p = 2

14

16

n = 11, p = 2

46

p=3

34

n = 15, p = 2 p=3 p=4 n = 19, p = 2 p=3 p=4 p=5

“p ≠ 2

—p + 2

“p

—p

30

32

10

12

48

66

68

42

44

36

70

72

26

28

94 78 62

96 80 64

118 122 126

120 124 128

90 70 50

92 72 52

158 138 118 98

160 140 120 100

186 190 194 198

188 192 196 200

154 130 106 82

156 132 108 84

Fig. 237*** Fig. 58

34 (§ 34) §Equalizing the vertical rows of all other borders

(ii) With the above placing, the situation in the left-hand columns has

With thesince above placing, the has changed: the corners are the now situation occupied byinthe firstleft-hand number ofcolumns the 2 changed, for and their cellsof are now the occupied by the number of pair above the corner complement the other, sum of which is (nfirst +1)≠ thereabove remains on the left-hand side an of 8(p≠1)(t+1) the 2,pair and thepthcomplement of excess the other, the sumwhile of which is number emptyremains cells is now 4 (p on = 2)the or apth multiple of 4 (p side Ø 3). an We excess of (n2 the + 1) − 2. of There then left-hand are then to eliminate the differences with four numbers. 8(p − 1)(t + 1) while the number of still empty cells is now 4 (p = 2) or But here (§ 34) the text preserved is imprecise, for it gives a single a multiple of 4 (p ≥ 3). We shall then eliminate the differences with four condition for determining the two pairs of consecutive numbers to be put numbers. in the (for us) left-hand columns, two larger and two smaller, leaving

thus one number (or pair) optional. As a matter of fact, since the corners But here the text preserved is imprecise, for it gives a single condition opposite to those filled in the first border are occupied by the two larger 2 +7 for determining twon2pairs of consecutive numbers to be put in the +3 complements n the and 2 2 , we shall just take, as the pairs of larger 2 +11 smaller, 2 2 +19 (for numbers, us) left-hand columns, two larger and ntwo n2 +15 nleaving the subsequent ones, namely ; n +23 ; . . . thus one 2 , 2 2 , 2 number (or optional. a matter of fact, corners opposite . That is, pair) generally, the pairsAs of larger numbers to besince put inthe the left-hand columns of the be are occupied by the two larger completo those filled in pth theborder first will border 2 2 n2, +we 8p shall ≠ 1 just take, as n2 + 8p ≠ 5 ments n 2+7 and“pn=2+3 the pairs , “p ≠ 2 = , of larger numbers, 2 n2 +15 n2 +11 n22+23 n2 +19 the subsequent ones, namely ; 2 , ; . . . . That is, 2 2 2, 2 2 the sum of which is n + 8p ≠ 3 = (n + 1) + 8p ≠ 4. (iii) The two larger numbers being thus determined, the pair of smaller numbers to be put with them is easy to find. Since the excess in the pth

200

General commentary

generally, the pairs of larger numbers to be put in the left-hand columns of the pth border will be n2 + 8p − 1 n2 + 8p − 5 γp = , γp − 2 = , 2 2 the sum of which is n2 + 8p − 3 = (n2 + 1) + 8p − 4. 34

14

16

45

70

46

72

66

58

26

68

28 111

52

60

56

50

47

17

54

96

59

49

29

13

94

61

57

45

35

27

11

63

53

67

85

101 117

75

39

41

71

79

121

80

119

99

33

31

19

64

78

115

25

15

97

107

3

23

89

91

103

105

43

83

81

51

109

93

73

55

69

95

87

77

65

5

21

37

42

1 62

22

3

37

39

24

20

32

41

19

33

23

9

18

43

35

29

25

21

15

7

10

1

27

17

31

49

40

44

12

26

47

13

11

28

38

74 86

36

5

48

7

30

34

76 88

9

Fig. 59

Fig. 60

62

64

213

126 78 128 122 94

189 207

7

35

173 183 203

51

165 167 179 199 112 108 102 176

3

31

52

75

159 155 153

70 120 201

80 100

11

50 124 118 110

96 104 98

106 156 174

83

87

79

25

135 133 103

99

85

45

21

154

209 185 169 145 125 107 121 111 101

81

57

41

17

77

63

55

39

72

205 181 141

95

211 187 171 163 149 129 117 113 109 9

146 162

36

113

97

33

49

69

89

115 105 119 137 157 177 193 217

5

29

65

127

91

93

123 131 161 197 221

90

1

147

67

71

73

143 139 151 225 136

92 114 223 195 175

61

59

47

130

219 191

53

43

23

215

37

19

13

Fig. 61

27

15

116 134 132 148 164

52

201 53

General commentary

52

The two larger numbers being thus set, the pair of smaller numbers 118 Since 116 8 the 113 excess 10 110 in 108the 16 pth 36 left-hand to be put with them is easy34to 2find. column has now increased to 8(p−1)(t+1)+8p−4 = 8[(p−1)(t+1)+p]−4, 70 46 18 102 7 97 107 100 24 48 52 we shall just put, according72to 66Rule I (p. 185), j = (p − 1)(t + 1) + p, and 58 3 23 89 91 103 60 56 50 thus take as the pair of smaller numbers 26 68 105 43 83 81 51 47 17 54 96 14 2 46 45 2 44 8 16 n − 8[(p − 1)(t + 1) + p] + 3 n2 − 2(p − 1)n − 10p + 5 β = = , 28 109 93 73 55 69 59 49 29 13 94 30 22 p3 37 39 24 20 2 2 2 −98[(p 111 95 + 87p]77 45 − 351)n 27 −1110p + 9 32 41 19 33 n 23 18 − 1)(t + 1) + 765 61n2 57 − 2(p βp + 2 = = . 230 5 21 37 63 53 67 85 101 2117 92 43 35 29 25 21 15 7 these two79smaller 42 two 1 larger 75 39 and 41 71 121 80 — 32 have been 5290 — 10 1 Once 27 17 31 four 49 40numbers placed in the (for us) left-hand columns, all their excesses are eliminated, 84 44 62 119 99 33 31 19 64 78 38 12 26 47 13 11 28 38 and so also in the right-hand columns after placing of the complements. 40 74 104 20 115 25 15 22 98 76 82 34 48 4 5 6 42 36 See the values in the table above, or Fig. 59 (n = 7), Fig. 60 (n = 11), Fig. 61Fig. (n = 52 86 120 4 6 114 9 112 12 14 106 88 Fig. 62 61 15). Knowing all these relations we may now set out Fig. 62 (p. 203), which placings for45 the five borders. As before 14 44 first 8 16 34 2 shows 118 116 the 8 113 10 110required 108 216 4636 (Fig. 49), that enables us to fill the squares for any order of this type 70 46 18 102 7 97 107 100 3024 2248 352 37 39 24 20 since determining the terms of the preliminary placing becomes evident. 72

66

58

23

89

91 103 3260 4156 1950 33

84

44

62 119 99

33

31

19

64

40

74 104 20 115 25

15

22

9834 76 2 82 118 116

3

23

9

18

Since now each row displays the sum due for the number of cells filled, 7 26 68 105 43 83 81 51 47 4317 3554 2996 25 21 15 it can be completed with neutral placings. In Fig. 63 (n = 7), a single 28 109 93 73 55 69 59 49 1029 113 2794 17 31 49 40 group of four numbers 2, . . . , 8 completes the square; in Fig. 64 (n = 11), 4711 13 11 28 38 111 95 87 and, 77 65 45 1235 2627 2, . . . , 16, for61the57columns, the pairs 30, 32 & 38, 40 (p = 3), next 48 49215), 5 6 . 42 36 and, for the columns, 18, 65 2, . . , 24 30 . .5. , 24 21 (p 37 =632); 53 in67Fig. 85 34 101 (n 117= 54, . . . , 60, then the pairs 66, 68 & 74, 76 (p = 4), next 26, . . . , 40 and 90 42 1 75 39 41 71 79 121 80 32 Fig. 61 82, . . . , 88 (p = 3), finally, 42, . . . , 48 (p = 2).

86 120

4

6

114

9

112 12

78

38

8

113 10 110 108 16

1470 106 46 8818 Fig. 102 64 7 97 107 100 24 58

36

48

52

72

66

3

23

89

91 103 60

56

50

68 105 43

83

81

51

47

17

54

96

14

2

46

45

44

8

16

26

30

22

3

37

39

24

20

28 109 93

73

55

69

59

49

29

13

94

32

41

19

33

23

9

18

111 95

87

77

65

61

57

45

35

27

11

43

35

29

25

21

15

7

30

5

21

37

63

53

67

85 101 117 92

10

1

27

17

31

49

40

90

42

1

75

39

41

71

79 121 80

32

12

26

47

13

11

28

38

84

44

62 119 99

33

31

19

64

78

38

34

48

4

5

6

42

36

40

74 104 20 115 25

15

22

98

76

82

Fig. 63

86 120

4

6

114

9

112 12

14

2

46

45

44

8

16

30

22

3

37

39

24

20

32

41

19

33

23

9

18

14 106 88

Fig. 64

202

General commentary

62

2

222 220

66

78

26 198 196 32

11 189 207 34 190 188 40

94

7

35 173 183 203 180 48

31

51 165 167 179 199 112 106 56

158 54

8

10 214 213 212 16

42 182

152 170 120 110

3

76 168 118 201 75 159 155 153 83

18 206 204 24

64

80 160

96 172 68 74

87

79

25 108 58 150

60 205 181 141 95 135 133 103 99

85

45

21 166 144

142 209 185 169 145 125 111 121 107 107 81

57

41

17

84

211 187 171 163 149 129 109 113 117 97

63

55

39

15

82

140

9

77

33

49

69

5

29

65 127 91

93 123 131 161 197 221 102 138

128 122 90

1

147 67

73 143 139 151 225 136 104 98

88 124

126 70 50

89 119 105 115 137 157 177 193 217 86

71

92 114 223 195 175 61

59

47

27 116 134 156 100

72 130 184 44 219 191 53

43

23

46 178 132 154 176

52 146 200 28

30 194 215 37

19 192 36

162 224

218 216 12

14 210 208 20

4

6

13

38 186 148 174 22 202 164

Fig. 65

(§ 35) Conclusion 41 160 38

158 152

42

88

109

59

142

56

140 134

60

The theory ends our51 text46with that126of constructing the 151 50 149in144 110 an 87 example, 133 68 131 69 64 column of order 7. 44 148 47 54 145 153 111 86 62 130 65 72 127 135

The treatise then represents the squares from n = 5 to n = 19 (above, 43 53 146 147 48 154 85 112 61 71 128 129 66 136 215 pp. 76–87). They may differ from those we have just constructed: 49 150 40 102 95 139 125 70 67 132 58 as already 157 said143(p.52196n), the author fills the incomplete borders from 155 37 159 one 39 and 45 156 97 100 the 137 available 55 141 57 small 63 138 the first, innermost by taking numbers in decreasing 77 order. 119 118 80 91 105 98 101 107 89 81 115 114 84 In our transcription of these examples, we have been faithful to the 120 78 79 117 106 92 96 99 90 108 116 82 83 113 Arabic, except for changing the orientation (reading from left to right, 5 196 the 2 194 103 23 text). 178 20Note 176 170 as was probably case188in 6the 94original also24that in all these text’s187examples inverted 14 185 the 180 inner 15 103 × 1043 square 93 169 appears 32 167 162 33 28around the ascending diagonal, with its smallest element on the vertical side; this 8 184 11 18 181 189 76 121 26 166 29 36 163 171 must have been made to be in keeping with the situation in the larger 7 17 182 183 12 190 122 75 25 35 164 165 30 172 odd borders. 215 179 13 the 186 squares 4 123for74 161 n 34 Of these,193Ant ¯k¯ı 16 reports n =175 5 and = 9, 31 then168 n =227 and n = 11 .a (those for n = 5, 7 with Indo-Arabic numerals, see p. 27n).

191

1

195

3

9

192

73

124 173

19

177

21

27

174

n2 +27 2

n2 +31 2

n2 +39 2

n2 −8n−45 2

∗

n2 +23 2

n2 +19 2

n2 −4n−9 2

∗

∗

n2 −4n−21 2

n2 −6n−31 n2 −4n−25 2 2

n2 −8n−41 n2 −6n−35 2 2

n2 −6n−11 2

n2 +35 2

n2 −8n−13 2

Fig. 62

∗

n2 −2n−11 2

n2 −2n−15 2

n2 +15 2

n2 +11 2

n2 −2n−7 2

∗

n2 −5 2

∗

n2 −1 2

∗

∗

∗

∗

∗

n2 −2n−3 2

∗

∗

∗

∗

∗

n2 −4n−5 2

∗

∗

∗

∗

∗

n2 −6n−7 2

∗

∗

∗

∗

∗

n2 −8n−9 2

General commentary

203

204

General commentary

Part III: Construction of even-order bordered squares (§ 36) The three categories of even-order squares There are, apart from the particular case of order 4, three categories of even squares. Those of evenly-odd orders (n = 4k + 2), those of (socalled) evenly-even orders (n = 8t), and those of evenly-evenly-odd orders (n = 8t + 4), with k, t natural. Their constructions will be taught now. But, as we shall see, the constructions of the last two, which group the even orders divisible by 4 (n = 4k, usually called properly ‘evenly-even orders’), are (slightly) different only for the smallest square n = 8. On the other hand, neither construction applies to the 4 × 4 square: since there is no magic square of order 2 with different numbers, the border of order 4 cannot be filled separately, that is, the 4 × 4 square must be filled as a whole.

(§ 37) Construction of the square of order 4 As already said, just as the smallest inner square in the bordered squares of odd orders is that of order 3, the smallest inner square for even orders is that of order 4. In our treatise, its construction is taught, but not justified. The resulting square (here from left to right) is as in Fig. 66. 4

15

10

5

14

1

8

11

7

12

13

2

9

6

3

16

4

Fig. 66 14

7

9

Note on this construction 15 1 12 6 After placing 1 in the second horizontal row, we descend first by the 10 8 13 3 knight’s move, then move away from the side by a queen’s move, then 16 (here top) row. For the other resume with the knight’s move5 to 11the 2next sequence of four numbers, beginning with 8, we descend symmetrically 8 13 12 1 with decreasing numbers. 3 6 15 It then appears that each 10horizontal row contains the same sum, 9, and two alternate columns also 5 the 16 same 9 4 sum, 7 or 11. It also appears that each bishop’s cell of a cell filled has14remained empty. We shall fill 11 2 7 them in such a way that the sum of two conjugate cells — that is, joined by a bishop’s move — make n2 + 1 = 17. Since all horizontal rows

General commentary

205

contain, after the preliminary filling, the same sum, the complements of any horizontal row will produce the magic sum in any other, in particular in its conjugate. As for the columns, the sums in them are not always the same but they are in two conjugate columns, so when filled they will display the magic sum. The remaining case, that of the diagonals, is clear since they contain pairs of complements. As noted earlier (p. 3), the possibilities for a square of order 4 are numerous. With this construction, the resulting square will be ‘pandiagonal’, which is to say that not only the two main diagonals will make the required sum, but also any broken diagonal, that is, two complementary parts on either side of a main diagonal. Thus, in Fig. 66, the sets 7, 1, 10, 16 and 10, 11, 7, 6 will each add up to M4 = 34. Now the property of a pandiagonal magic square is that it will remain magic if we move any lateral row to the other side: the main diagonals of the new square will be magic since they were already magic as broken diagonals. Thus, by repeating such vertical and/or horizontal moves we are able to place any chosen element in any given cell of the square; for example, the cell containing 1 can be made to occupy any place in the square. Now, as we shall see, the configuration adopted by our treatise for squares of even orders is to have the order number in an upper corner cell. This will also be the case for the square of order 4 by putting 1 (and the smaller median) in the second horizontal row, whence this particular choice.

A. Bordered squares of evenly-odd orders (n = 4k + 2) (§§ 38–39) Method of construction The method of constuction described in our treatise is illustrated by the first two squares, for n = 6 and n = 10 (Fig. 67–68). To place the smaller numbers in this way, our text begins by filling the inner 4 × 4 square, starting, as usual, with the two medians, and then continuing for the 6 × 6 border with the (smaller) numbers taken in decreasing order. The differentiation by parity is pointed out: the two columns are filled with sequences of small (mainly) even numbers, and the two horizontal rows with sequences of small (mainly) odd numbers. If, instead of taking the numbers in decreasing order, as does the text, we start in the outer border with the smallest number, say 1 here, the general method can be described as follows. We write 1 in the bottom row, next to the lower left-hand corner, then 2 above, 3 below, and the next ones cyclically around the border until we reach 4k = n − 2 (thus, here, 4 and 8, respectively), which will be in a cell of the right-hand row

206

General commentary 5

36

2

34

28

6

27 14 25 20 15 10 if we have adopted the anti-clockwise movement. We then write the next 24 in11 21 29 two numbers, n − 1 and n (here 5, 6 and 9,8 10) the18upper left and right corners. So far, we have written 4k + 2 small numbers. Next, we put the 7 17 22 23 12 30 two following numbers on the left, then the subsequent ones around the 33 19 16 13 26 4 border, in the same rotating movement as before, until we reach on the 31 1 numbers, 35 3 98k + 32 2 = 2n − 2 right side (for us), after thus placing 4k more (10 and 18 here). Half the border cells are thus filled, and putting their complements opposite will fill the border completely. 9

100

2

98

5

94

88

15

84

10

83

25

32

31

71

72

73

74

26

18

16

24

37

68

34

66

60

38

77

85

87

78

59

46

57

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47

42

23

14

5

36

2

34

28

6

12

22

40

56

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50

53

61

79

89

27

14

25

20

15

10

11

80

39

49

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44

62

21

90

8

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11

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21

29

93

81

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45

58

36

20

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7

17

22

23

12

30

6

19

63

33

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41

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95

33

19

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26

4

97

75

69

70

30

29

28

27

76

4

31

1

35

3

9

32

91

1

99

3

96

7

13

86

17

92

Fig. 67 9 83 16 87 12 11 93 6 97 91

Fig. 68

Remark. Our choice for the place of 1 being here the same as in the Arabic, right-to-left squares, the horizontal rows of evenly-odd orders 100 2 98 94 88 15 84 10 5 are, except for their end elements, the same in the original version 2 25 32 31 71 72 73 74 26 18 α + 1 − (α +to 1) start, n2 + 1as − (αfor + 2)the other α+3 and in the transcription. (We haven chosen 66 60 77 85 24 squares, 37 68 34 38 next to the lower left-hand corner.) of this n2 + 1 − (α + 3) α + 1 method α+2 61 79 89 borders of even orders, whether or not 22 The 40 56aim 43 when 50 53constructing divisible is in44principle the same: to reach a situation where only 80 39 49by544, 55 62 21 90 ‘neutral placings’ are needed, that is, taking tetrads of consecutive num81 65 51 48 45 58 36 20 8 34 middle 28 5 ones in the bers of which the extremes are put in one 6row36and2 the 63 67 64 82 95 19 33 35 41 opposite row: indeed (Fig. 69), with the10complements of these tetrads 27 15 20 25 14 in place, not only will the sum due be obtained in cells facing one an75 69 70 30 29 28 27 76 4 29 21 18 11 24 8 other, but also in all groups of four cells in each row. The same principle 99 96 1 3 7 13 86 17 92 12 23 22 squares 17 7 displaying had been used for completing the borders 30in odd-order separation by parity (above, p. 184, top). 4 26 13 16 19 33 78

59

46

57

52

47 42 232 14 Mathematical

n + 1 − αbasis

35 31 3 of 9 even For such a situation to be attained in32 the1 borders order, an initial placing is required which must satisfy the following conditions.

α

n2 + 1 − (α + 1)

n2 + 1 − (α + 2)

α+3

n2 + 1 − α

α+1

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207

General commentary

(i) The initial placing must settle the question of the corner cells, each common to two rows. (ii) The initial placing must be such that, after addition of the comple2 ments, each row displays its sum due, thus m · n 2+1 if m cells are filled. (iii) This number m of cells filled must be as low as possible in order for the method to be applicable, as far as possible, to the smallest orders.

(iv) The initial placing must be uniform to be applicable to any order of the same type, excepting at most small orders. (v) The number of cells left empty thereafter must be divisible by 4 for subsequent neutral placings. Note also that, ideally, the numbers placed preliminarily should be the first natural numbers, for that will make the neutral placings straightforward. But this condition may be disregarded if the preliminary placing made with other numbers is particularly easy to remember. α

n2 + 1 − (α + 1)

n2 + 1 − (α + 2)

α+3

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Fig. 69

Consider, in accordance aim 5of being left with neutral plac36with 34 28 6 2 the ings, the situation for an order n = 4k + 2. The smallest possibility, also 10 15 20 25 14 27 applicable to the smallest order, 6, is to put ten smaller numbers, three 29 21 18 11 24 8 in each row (including two consecutive corners); with the complements 23 22 in17question 12 order 7 in place, this will leave,30if the is n = 4k + 2 with k > 1, 4(k − 1) empty cells in 4each 26 row. 13 16 19 33 32 35 ten 31 numbers as in Fig. 70. Let 1the 3 smaller 9 Suppose then we place us designate by a prime their complements, that of a being thus a0 = (n2 + 1) − a. After writing in the complements, we must have the sum 10 100 thus, 2 98 88 15 9 due for six cells filled, for 5the94upper and84left-hand rows, respectively, 18

31 f 032= 3(n 25 283+ 1), a26+ b74+ c73+ d720 +71e0 +

85

68 +34 60 j 037 24 216+ 1). a77+ g38+ h c0 +66i0 + = 3(n

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Since d0 +e0 +f 0 = 3(n2 +1)−d−e−f and c0 +i0 +j 0 = 3(n2 +1)−c−i−j, 79 61 become 53 50 43 56 40 22 12 the two previous89relations (

39 80 11 a44+ 55 b + 54 c =49d + e+f 65 i + 81 j.93 a58+ g45+ 48 h =51 c +

208

General commentary

a b g

c

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Fig. 70

This could be easily satisfied with the first ten numbers, for there are 140 possibilities.216 13

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Fig. 71 34 28 2 Now the method used in our text also 6relies on the preliminary placing 27 14 25 20 15 10 of ten numbers, but, except for n = 6, discontinuously. Those of the upper 24 11 18 21 29 8 row are 2, n − 1 and n, those7 of17 the lower row 1, n − 3 and n + 3, thus 2, 22 23 12 30 9, 10 and 1, 7, 13 in Fig. 68 33or192,1613, 26 13 14 4& 1, 11, 17 in Fig. 71 (n = 14);

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Listed in our Magic squares in the tenth century, pp. 63–64. There are several examples of such possibilities in later Arabic treatises, see Magic squares, p. 172.

209

General commentary

since their sums are equal, we shall have the sum due after adding the complements. The three numbers in the columns are n − 1, n + 1, n + 2 on the left and n, n − 2, n + 4 on the right, thus 9, 11, 12 and 10, 8, 14 in Fig. 68 or 13, 15, 16 & 14, 12, 18 in Fig. 71, both sets making equal sums. The cyclical placing distributes the remaining numbers appropriately: the upper row associates 5 and 2n − 5, 9 and 2n − 9, and so on to n − 5 and 2n − (n − 5), whereby the row is increased by k − 1 times 2n; the lower row associates the other odd numbers, thus 3 and 2n − 3, 7 and 2n − 7, and so on to n − 7 and 2n − (n − 7), whereby here too the sum in the row is increased by (k − 1)2n. The columns will likewise comprise, together with the initial numbers, the pairs 6 and 2n − 4, 10 and 2n − 8, ..., n − 4 and 2n − (n − 6) for one of them, and for the other the remaining even numbers, thus 4 and 2n − 2, 8 and 2n − 6, ..., n − 6 and 2n − (n − 8); this adds to each column two sequences of k − 1 numbers making the same sum, namely (k − 1)(2n + 2). Since the sums in these two opposite rows are equal, the complements of one will produce in the other the sum due. 5

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Remark. If n27= 4k 14 + 252 with 20 15k ≥ 10 2, we could just keep the configuration seen in the 86×6 square for the first ten numbers and complete the rows 24 11 18 21 29 with neutral placings. But then the order number would not appear 17 22 23 12 30 in an upper7 corner cell (see p. 205). 33

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General commentary

(§§ 40–43) Method of construction For a general evenly-even square, our author takes the smaller numbers to be placed in the outer border in descending order, for he supposes the inner part to have been filled first (Fig. 72, here from left to right, with n = 12, and 22 the first number to be placed since the outer border comprises 44 cells). He takes groups of four consecutive numbers of which he places the extremes (say) on the top and the two middle on the bottom, until he is left with six empty cells within the row. Of the next six numbers, the first two are put above and the others below — leaving, as usual, the opposite cells empty. He then fills, successively, the two corners, the cell below the corner just filled, then one cell on the other side. Returning to the opposite column, he completes the two columns by means of groups of four consecutive numbers, the two extremes on one side and the two middle on the other. These smaller numbers and their complements fill the outer border completely. The case of order 8 is similar, except that we directly begin by putting two numbers at the top and four at the bottom before filling the corner cells (Fig. 73). These are the two squares represented in our treatise. Mathematical basis of this method As before, we shall consider which preliminary placing could, in the case of evenly-even orders n = 4k, leave us with neutral placings, thus a quantity of empty cells equal to a multiple of 4. Consider then a square of order n = 4k with k ≥ 2.

n’est pas aussi simple que dans le cas précédent. a e

c

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d j

f

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e f g h

d

Fig. 74

Fig. 75

We shall be left with a quantity of empty cells divisible by 4 by placing two smaller numbers in each row, two of which must occupy consecutive corner cells (Fig. 74). Indeed, since with the complements in place four cells in each row will be occupied, 4(k − 1) will still be empty. Thus only six smaller numbers are needed for this preliminary placing.

General commentary

211

We might also place two more numbers in one row and two more in its opposite; this will leave 4(k − 2) empty cells in these two rows and 4(k − 1) in the other two, as before. Here we shall have placed ten smaller numbers (Fig. 75). This method will also be general since it is applicable to the smallest square, of order 8. This remains true if we place another pair of smaller numbers in the columns, thus altogether fourteen numbers. We could of course continue placing further pairs, but this would be at the cost of simplicity and no longer be applicable to smaller orders. Suppose we place six numbers as in Fig. 74. Let us once again designate by a prime their complements. After writing in the complements, we must have the sum due for four cells filled, thus, for the upper and left-hand rows, respectively, a + b0 + d0 + c = 2(n2 + 1),

a + c0 + f 0 + e = 2(n2 + 1).

Since b0 + d0 = 2(n2 + 1) − b − d and c0 + f 0 = 2(n2 + 1) − c − f , the two previous relations become (

a+c=b+d a + e = c + f,

a and c being common to the two equalities since they occupy the corner cells. We shall have attained our aim if we can solve this pair of equations by means of the first six natural numbers. The remainder of the border, each row of which contains 4(k − 1) empty cells, will be filled by means of tetrads of consecutive numbers. But the possibilities offered by this placing of the first six numbers are limited: we may either place 2 and 3 in the corners and 1 and 4 in the opposite row, or 4 and 5 in the corners and 3 and 6 in the facing row, the two remaining numbers being used in both cases to cancel out the difference of 1 left in the columns. At least this preliminary placing is particularly easy to remember. The placing of ten numbers offers notably more possibilities. In that case we have the relations (Fig. 75) (

a+b+c+d = e+f +g+h a + i = d + j,

which gives, with the first ten numbers, 128 solutions.217 The method described in our text relies on the preliminary placing of ten numbers, but which are just not the first ones, namely, in Fig. 72, 217

Listed in Magic squares in the tenth century, pp. 57–58.

212

General commentary

11 + 18 + 17 + 12 = 16 + 15 + 14 + 13, 11 + 10 = 12 + 9. Generally, we shall write in the top line n − 1, n, n + 5, n + 6 and at the bottom n + 1, n + 2, n + 3, n + 4, which make equal sums; in the columns, along with n − 1, we shall put n − 2 and, along with n, n − 3, which make equal sums as well. The remaining smaller numbers, from 1 to n − 4 and from n + 7 to 2n − 2, are then just written as tetrads of consecutive numbers, with neutral placings. As to the square of order 8 (Fig. 73), which differs in the description of the method (see above), it simply follows the same rule. By the way, its initial arrangement, with the same small numbers, could just be kept for any order n = 4k and the rows completed with neutral placings of the subsequent numbers. The remark made above for the evenly-odd orders (p. 209) also applies, mutatis mutandis, here. xi

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Fig. 76

This part on common bordered magic squares of even orders is, as Part I on common bordered magic squares of odd orders, omitted by MS. D and thus preserved only in Ant.¯ ak¯ı’s text. The reason is the same in both cases, namely the existence of a synthetic representation (Kh¯azin¯ı’s) displaying the arrangement in the first borders of even orders (Fig. 76218 ). Notable is the place of the smallest number, each time next to the lower left-hand corner (for us), while the order numbers occupy the opposite corners; the 4 × 4 square is the sole exception. 218

Here adapted: see, for the original, MS. D, fol. 63r .

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General commentary

Part IV: Construction of even-order composite squares Introductory note on composite squares Here the construction of a larger magic square is reduced to that of its individual parts, mostly squares, each of which displays its sum due. The possibility of constructing a square in such a way depends of course on the divisibility of its order. A prime order is thus excluded. (a) What is usually meant by composite square is a square divided into even or odd-order subsquares each of the same size, filled one after the other completely by a continuous sequence of numbers in magic arrangement. Since the sums in the subsquares are different, but they form an arithmetical progression, these subsquares must be arranged in such a way that the main square will itself be magic. Now, because of the impossibility of constructing a magic square of order 2 with different elements, the order of such a composite square cannot be twice a prime nor twice the smallest possible magic squares. This last restriction excludes both orders n = 6 and n = 8, and the smallest such composite square is thus of order 9, broken up into nine squares of order 3 (Fig. 77 & Fig. 78). The next possible order is n = 12, which may be divided into sixteen squares of order 3 arranged according to one of the ways for order 4, or into nine squares of order 4 placed according to the arrangement for order 3. These two examples are given in the tenth century by Ab¯ u’l-Waf¯a’ B¯ uzj¯an¯ı.219

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Fig. 77

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There is no mention of such squares in our treatise. On the other 11 18 13 74 81 76 29 36 31 α4 α1 hand, 16the14 two are fully explained in it. 12 79 subsequent 77 75 34 32 30 types 219

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4 1 See56the edition of his text,βpp. 183–185 βand (Arabic) 238–240. 63 58 38 45 40 20 27 22

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214

General commentary

(b) If the main square can be divided into subsquares of the same even order ≥ 4, these subsquares can be filled individually with pairs of complements in magic arrangement. Since they are all of the same size and display the same sum, they can be arranged in any manner. Note that in this case, an 8 × 8 square is possible, as observed, or reported, by the author of MS. D (see above, p. 125). Note also that such squares are generally of order n = 4k, but may be of evenly-odd order as long as that order has a factor (smaller than n2 ) other than 2.

(c) A less conventional possibility is to divide the main square into unequal parts, square or even rectangular, but here again filled with pairs of complements. Each part is to display its sum due, which of course will vary according to its dimensions. This extends the possibility of constructing composite squares to evenly-odd orders not satisfying the above condition, beginning in our treatise with n = 10.

A. Division into equal subsquares (§§ 44–45, 48) Consider thus the division of a given square into identical

subsquares of even orders which all display their sum due with respect to 2 the whole square, thus m · n 2+1 for m × m subsquares within a square of order n. In the examples given, the order of the main square is either of the form n = 4k, with subsquares of orders 4, 6, 8 (§ 44), or n is evenlyodd, but of the form 2 · r with r a composite (odd) number, say r = s · t with s, t ≥ 3, for the square may then be divided into s2 squares of even order 2t (§ 45). Thus we find: — A 12 × 12 square divided into four 6 × 6 (bordered) subsquares (see example p. 127). — A 12 × 12 square divided into nine 4 × 4 subsquares (p. 129).

— A 16 × 16 square divided into four 8 × 8 (bordered) subsquares (p. 131). — A 16 × 16 square divided into sixteen 4 × 4 subsquares (p. 133).

— A 18 × 18 square divided into nine 6 × 6 (bordered) subsquares (p. 135). In addition to considering (and constructing) the case n = 18, our text mentions the division of a 30 × 30 square (this being the next admissible evenly-odd order) into nine 10 × 10 subsquares or twenty-five 6 × 6 subsquares (§ 45). — A 20 × 20 square divided into twenty-five 4 × 4 subsquares (p. 137).

For filling the main square, as asserted in § 48, we shall take any subsquare and put in it the first set of smaller numbers, starting with the

215

General commentary

two medians of the main square and their neighbours, then proceed with any other subsquare, and so on until the main square is filled: since each subsquare makes the same sum, their arrangement is indifferent. The smallest possibility is the 8 × 8 square divided into four 4 × 4 subsquares, omitted in our text but supplied in MS. D (p. 125). Magic squares in the tenth century

B. Division into unequal parts

33

(§§ 44, 46–47, 49–54) A division into equal subsquares of even orders is

not always necessary and not always possible. The impossibility includes first all the squares of odd orders n = 2k + 1 (§ 47) and second those squares of evenly-odd orders divisible by 2 only, as seen previously; in this latter case a division is still possible, but either into unequal parts (square or rectangular) or with a cross in the middle. Division into unequal parts Magic squares in the tenth century is also possible for orders n = 4k, depending on the divisibility of33n.

— If the separation is 6 by 4 (Fig. a 36*), you will place in each group of

four cells with that a group four cells beloworders. it eight complementary (§ 49) Consider the less banal case, ofofevenly-odd Suppose terms

a 37*); with has them,the eachorder row will as we have thus that the square to be (Fig. constructed n equalize = 4k +its2.conjugate, Draw in explained at the beginning of this section.151 You proceed like that until its centre a square of orderyou4thave + 2completed (k > t the ≥ 1). The whole square is then six (rows). If theinseparation is 6 bythe 8, you will treat the four4t in + two2,steps, and broken up as follows (Fig.—79): its centre, square of order the six as here above.152 distant by 2(k − t) from the —sides; in each corner, one square the side the separation is 6 by 4 (Fig. a 36*), you will place in each group of Itfour isIfcells not possible that the square be divided intoterms parts with the sepawith[ aor group of four cells below it when eight complementary of which, 2(k − t), is divisible by 4 becomes sobecause taken together 153 (Fig. a 37*); odd with them, each row will equalize its conjugate, as wewe havehave explained; ration being (in dimensions) of what at the beginning of this section. You proceed like + that2) until lateral with that of the square opposite ]; finally, four 2(k − t) × (4t as a explained matter of fact, the separation will always be 6 by 4, or 6 by 8, or 2 you have completed the six (rows). 154 If the separation is 2 by 4 4,—orIfpart 2theby 6,ofor the 2is by 8,8, and sodiagonals on always. separation 6 bymain you will treat the four in two steps,with and rectangles, not containingbyany and one six*) as here (Fig.the a 38 andabove. is situated in the sides without meeting the 220 diagonal, the of their dimensions [ or that of two ] divisible by 4. It is not opposite that the rectangles square be divided into parts with the sepatreatment forpossible its equalization will be as we have explained previously.155 151

152

atteindre la somme due horizontalement et

ration being odd (in dimensions) because of what we have explained;153 as a matter of fact, the separation will always be 6 by 4, or 6 by 8, or 2 154 If the separation is 2 by 4 by 4, or 2 by 16, or 2 by 8, and so 4 on always. 1 the (Fig. a 38*) and is situated in the sides without meeting the diagonal, verticalement. treatment for1 its equalization will be as we have explained previously.155 4 1

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Fig. 80 Fig. 81 152 Above, A.II.42. Considering two strips like that of Fig. a37*. 153 The case of odd-order subsquares dismissed (A.II.47), the sides of the Considering two strips like having that been of Fig. a37*. 153 strips are necessarily even. The 154 case of odd-order subsquares having been dismissed (A.II.47), the sides of the The separations 6 ◊ 4 and 6 ◊ 8 have just been treated; a pair of rectangles 2 ◊ 6 reduced to theeven. case 6 ◊ 4 by combining them; a strip 2 ◊ 8 is reducible to stripsmay arebenecessarily 154 two 2 ◊ 4, to be discussed now. Note that Fig. a37* can be reduced to Fig. a38* by The separations 6 ◊ 4 and 6 ◊ 8 have just been treated; a pair of rectangles 2 ◊ 6 removing a 4 ◊ 4 square. 155 reduced to the case 6 ◊ 4 by combining them; a strip 2 ◊ 8 is reducible to may be Using a neutral placement. two 2 ◊ 4, to be discussed now. Note that Fig. a37* can be reduced to Fig. a38* by removing a 4 ◊ 4 square. 155 Using a neutral placement. 151 152

151

Above, A.II.42.

We shall fill the square parts, central or in the corners, as before, with pairs of complements. We are then left with the rectangles. We may either fill them with neutral placings only, since their length, 4t + 2, is even while their width, 2(k − t) [ or twice it ], is divisible by 4 (Fig. 80); or we can consider subsquares (which can always be 4 × 4) and fill them in such a way that their sum due appears, which will leave us again with 220 The cases between square brackets are not considered in our text. Thus its smallest example is n = 14 and not n = 10.

216

General commentary

filling rectangular strips using neutral placings (Fig. 81).221 Our text mentions (§ 46, p. 109) two examples of division, and constructs these and others, with subsquares of orders 4, 6, 8, 10 (interpolated mention of the case n = 30 in § 45). Thus we find: — A 14 × 14 square with a central 6 × 6 square surrounded by eight 4 × 4 squares and four pairs of four-cells-long strips (§ 46 & p. 139). — A 16 × 16 square with four 6 × 6 squares in the corners, a central 4 × 4 square and four 4 × 4 squares separated from the central square by four 2 × 4 rectangles (p. 141).

— A 16 × 16 square with a central 8 × 8 square surrounded by twelve 4 × 4 squares (p. 143). — A 18 × 18 square with a central 10 × 10 square surrounded by twelve 4 × 4 squares and four pairs of four-cells-long strips (§ 46 & p. 145).

— A 20 × 20 square with four 8 × 8 squares in the corners separated by nine 4 × 4 squares, one of which in the centre (p. 147).

— A 20 × 20 square with four 6 × 6 squares in the centre and sixteen 4 × 4 squares around (p. 149; note the reduction of the central square: we thus remain with even orders ≤ 10, see n. 156, p. 107). — A 20 × 20 square with four 4 × 4 squares in the centre and, separated by four pairs of six-cells-long strips, eight 6 × 6 squares (p. 151).

— A 20 × 20 square with a central 8 × 8 square, four 6 × 6 squares in the corners, and eight 4 × 4 squares each completed with a 2 × 4 rectangle (p. 153).

C. Cross in the middle (§ 50–54) Another possibility for evenly-odd orders is to eliminate a pair of median horizontal and vertical rows —in other words, to fill them separately (Fig. 82). Filling this cross (and taking into account that its centre meets the diagonals) will leave us with four squares of even order 2k, the filling of which is well known.

Since the order of the whole square is 4k + 2, it will contain the numbers from 1 to n2 = 16k 2 + 16k + 4. We shall use 16k + 4 of these numbers for the cross, for example those in the middle of the sequence, while the 8k 2 first and the 8k 2 last will fill the subsquares. How to fill the cross is explained in detail in the text (§§ 50, 52). For equalizing the cross, the branches of which are two cells wide, we shall put in each pair 221 There are two examples of the second case, none of the first (though the arrangement of Fig. 80 is treated in § 49).

217

General commentary

of such cells a pair of complements. This will give the sum due for two cells and we shall no longer need to consider the cross when filling the subsquares.

65

the cross is explained in detail in the text. For equalizing the cross, the branches of which are two cells wide, we shall put in each pair of such cells a pair of complements. This will give the sum due for two cells and we shall no longer need to take them into account when filling the subsquares. Fig. 82 We sum due due longitudinally longitudinallyininthe thebranches branches and Weare arestill stillto to obtain obtain the the sum and diametrically in the four central cells. Now it is already possible to attain diametrically in the four central cells. Now it is already possible to attain the six cells, cells, comprising comprisingthe thecells cellsininthe the middle thesum sumdue duewith with aa length length of of six middle ofofthe cross, thus altogether twenty cells, by using ten consecutive pairs the cross, thus altogether twenty cells, by using ten consecutive pairs among placed in in the the cross. cross. Consider Considerfirst firstthese these amongthe the8k 8k++22 pairs pairs to be placed 16k + 4 numbers, aligned vertically by pairs of complements with sum 16k + 4 numbers, aligned vertically by pairs of complements adding up 2 2 2 2 = 16k + 16k n to+n1 + = 116k + 16k + 5:+ 5: 2 2 2 8k + 2 8k2 2++11 8k22 + 22 ...... 8k 8k 8k 8k 2++8k 8k++1 1 8k8k+ + 8k + 2 2 2 2 2+ 2 2 2 2 8k + 16k + 4 8k + 16k + 3 . . . 8k + 8k + 4 8k 8k8k ++ 3. 3. 8k + 16k + 4 8k + 16k + 3 . . . 8k + 8k + 4 8k +

Let us for instance take the last ten pairs, which may also be expressed Let us for instance take the last ten pairs, which may also be expressed as as n2 ≠9 2 n2 +10 2

n2 ≠8 2 n2 +9 2

n2 ≠7 2 n2 +8 2

n2 ≠6 2 n2 +7 2

n2 ≠5 2 n2 +6 2

n2 ≠4 2 n2 +5 2

n2 ≠3 2 n2 +4 2

n2 ≠2 2 n2 +3 2

n2 ≠1 2 n2 +2 2

n2 2 n2 +1 2

X

IX

VIII

VII

VI

V

IV

III

II

I

We shall place I and III in the centre, with the elements of each pair diagonally opposite. Next, the elements of the pairs II, IV, V, VI n2 n2 +4 the 3 will occupy the vertical branch and 2 2 others the horizontal branch, as represented in Fig. 83. n2 n2 1

+2

2 2 With this arrangement, the occupied cells display in all three direc2 2 2 n n n2 n2 n2 n tions the sum due, namely: 7 +7 +3 +9 9 2

2

2

— diagonally: n2 + 1 for two cells, 2 2 2 — vertically:

3(n2

n n +8 2 2

6

n 2

2

+ 1) for six cells,

2

2

n2

n2

2

+1

2

2

8

n2 +10 2

n 4 +5 — horizontally: 3(n2 + 1) for sixn2 cells. 2 2

2

We are left with placing in nthe branches of the cross the 16k − 16 2 n2 +6 5 2 2 of 8(k − 1) complementary numremaining numbers, thus two sequences bers, each sequence being formed by consecutive numbers since the centre Fig. 172

We shall place I and III in the centre, with the elements of each pair diagonally opposite. Next, the elements of the pairs II, IV, V, VI

218

General commentary

of the cross has been filled with the last ten pairs. Filling the remainder of the cross can thus be completed using neutral placings. It is also mentioned in our text (§§ 53, 54) that the preliminary equalization of the cross may be performed for orders n = 4k + 2 with k even on just one side of the branch, the effect being that we shall have an integral number of neutral placings on each side.222 Filling the part outside the cross is easy: we shall fill the four squares with the remaining numbers, disregarding those already used for the cross. Remark. Thus, as usual, our text takes the numbers in decreasing order and starts with filling the cross.223

n2 n2 +4 −3 2 2 n2 n2 −1 +2 2 2 n2 n2 −7 +7 2 2

n2 2

n2 n2 n2 +3 +9 −9 2 2 2

n2 n2 n2 n2 n2 n2 +8 −6 −2 +1 −8 +10 2 2 2 2 2 2 n2 n2 −4 +5 2 2 n2 n2 +6 −5 2 2

Fig. 83

Our text includes several examples of squares with a central cross.224 — A 10 × 10 square with a cross and four 4 × 4 subsquares (§ 46 & p. 155). — A 14 × 14 square with a cross and four 6 × 6 subsquares (§ 46 & p. 157, with the remaining rectangular strips in the cross filled individually). 222 This is not necessary. Furthermore, there is one example of neutral placings separated by a square (p. 151). 223 Some later authors, instead of filling the cross with the numbers in the middle, do it with the first and the last ones; see Magic squares, pp. 101–102 (Les carrés magiques, p. 98; , pp. 109–110). In that case, the placing is similar to that for the outer border of evenly-odd squares (the four numbers in the centre of the cross are then the four numbers N in the corner cells). 224 The casexkn = 6 can also be treated in this way, the remaining 2 × 2 parts being i then filled as a 4 × 4 square. See Magic squares, p. N 101 (Les carrés magiques, p. 97; , p. 108), or N Magic squares in the tenth century, p. 112.

k

N

N

xki

N= N

X i

xki

N

General commentary

219

— A 18 × 18 square with a cross and four 8 × 8 subsquares in the corners (§ 46 & p. 159). Of course, we may also consider in this latter case a further division: — A 18 × 18 square with a cross and sixteen 4 × 4 subsquares (p. 161).

Remark. Knowing how to fill the cross as well as 4 × 4 squares displaying equal sums is sufficient to construct any even-order square. Indeed, if n = 4k, the square may be filled with squares of order 4, and if n = 4k + 2 the cross will leave strips of k squares of order 4, one of which will be cut by the cross when k is odd. But there are other possibilities in this last case of evenly-odd squares, according to the form of k: if k = 3t, thus n = 12t + 2, we may also consider strips of t squares of order 6 (displaying equal sums) on both sides of the cross (example of p. 157, n = 14); if k = 3t + 1, thus n = 12t + 6, we may also divide the square into strips of order 6 without using the cross (example of p. 135, n = 18, and mention of the case n = 30 in § 45). This may explain the assertion of an eleventh-century author who, after explaining the method of the cross, writes that knowing how to fill the cross and both the 4 × 4 and the 6 × 6 squares enables us to construct any square of even order.225

225

See Un traité médiéval, pp. 83 & 165. This (unknown) author’s knowledge of our text seems apparent from two other places: his repeating an argument in § 47 (above, p. 111, n. 166) and his possible allusion to Part II (above, p. 174).

◆ ˙˙˙˙◆◆◆

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Z@ P@ : ( Z@ PAK ⇣ @ (v.) : (II) 95, 525, 532, 543, 563. . @ (II) 95, 525, 532, 543, 563. 2. [487], @ (v.) (II) 95, 525, 532, 543, 563. @ (v.) : (II) 95, 525, 532, 543, 563. @ (v.) : (II) 95, 525, 532, 543, 563. @ @ Å kA”Ò ÆJ K : 2. : 2. ⇣ ⇣ @ 7.:@@:@@7. ⌦ ÆJ⇣⌦K : 2. :: 2.2. [489], 509, 566, 577,525, 532, 543, 563. 2. : 2. :[487], 7. @ : 7. È ìP@ @ @ : ( @ ÅÅkA”Ò (v.) : (II) 95, : @ @ @ : ( @ (v.) : (II) 95, 525, 532, 543, 563. @.@)(v.) : 525, (II) 95, 525, 532, 543, kA”Ò⇣ÆJ⇣ÆJ⌦⌦KK :: 2.2.:: @2.2.@:: 2.2. 95, 532, 543, 563. ⌘P@kA”Ò @ @@:Z@⇣@(v.) 26, 28, 30, 35, 38, 40, 43,563. . . .. ... . P@:( 7. :@ ✏ :( Z@)@(II) PAK 24, 35, 38, 40, 43, Å⇣flkA”Ò (II)28, 525, 532, 543, 563. 95,30, 526, 533, 544, :(v.) 7. ::26, :24, 7.@ (v.) ⇣ @ : 2. @ ˘ÆJ⇣ ⌦£A÷ : An 2. ✏ ⌘ : 2. @ : 2. : 2. 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ÆJm⇣PAj @ @ @ :BZ@‡@ (@ 7. @ ✏455, 24, 26, 28,456, 30, 35, 38,38, 40, 43, . . .. ... . @@✏:P@(v.) 525, 532, 543, 563. .456, 7.455, @ : 2. ¯ m '@ : 2. @ : 7. (v.) (II) 95, 525, 532, 543, 563. B @ : 455, 456, 568, . ✏ @ : 568, à ⌘ ⇣ @ @ : ( @ :: 4. :@@):(v.) @)✏455, 24,28, 26,30, 28,35, 30, 35,543, 38, 563. 40, .43, ... . :)26, (II) 95, 525, 532, ¯⇣ PAj '@ :: 2.2.@@ ::: @ @2.@2. flflPAj : 4. ˘˘ÆJÆJ⇣ ⌦£A÷ ⌘P@P@:..:::mmÃ2.2. 4. ✏Z@P@(P@:@@@:✏@(v.) @ @ :Z@‡@ 24, 38, 40, 43, 24, 26, 28, 30, 38, 40, 43,43, ..... ... . : 455, (II) 525, 532, 543, 563. (@(:B Z@✏@PAK 24, 26,95, 28, 30,35, 35, 38, 40, @((v.) ::7. 456, 568, : 4. 4. ≠À (II) 247. .B@456, @ : 456, 568, @2.2. ÄYJ : .4. : [487], ⇣ Ø@m⇣à :'@2.'@(line) ⌦£A÷ B 455, 456, 568, ¯¯ Ø@PAj :4.2.@ :: @2. @ : 7. ⌦ÄYJ @ ( ) 24, 26, 28, 30, 35, 38, 40, 43, ≠À @ (v.) : (II) 247. ✏ @ : ( @ @ B @ : 455, 568, ::2.4. @ : ( @ : 26, 24, 26, 24, 26, 28, 30, 35,28, 38,30, 40,35, 43, ...... .... . . @@::✏ ((40, @@@✏@ @B:43, ))@@:24, 28, 30, 35, 38, 40, 43, ⇣ :@@ [487], @ :: 5. ⌦ 7. ⇣ 51, 52, . . . . : 455, 456, 568, à B @ : 455, 456, 568, ÄYJ   Ø@ : 4. @ (v.) : (II) 247. @ 5. @ : 4. ¯ PAj m '@ : 2. @ (v.) : (II) 247. ⌦ Å kA”Ò ÆJ K : 2. ⇣ ( @ ) 24, 26, 28, 30, 35, 38, 40, 43, 43, ...... .. @ : ✏@ @(455, @:✏((40, )@456, 24, 568, : 5. 4. @38, :456, ÖÄYJ 43,26, 51,28, 52,30, . . .35,. 38, 40, ⇡ÄYJ @ @ :@B‡@ 568, @(@@::: @[email protected].✏455, @@::: 5.5.5. ⌦Ö :: Ø@⇣Ø@2.2.:: 4.4.@@:::: 4.@4. ÜAm ¯ PAj 5. ⇣ÜAm..@⇣:m:Ã⇡'@5. ✏ ✏ @ @ @ ) 24, 26, 28, 30, 35, 38, 40, 43, . .. . B @ : 455, 456, 568, ≠À (v.) : (II) 247. (v.) : (II) 247. @ B @ : 455, 456, 568, @ : 7. ⌦ : : 5.5. @ : 8. ⇣⇣ fl⌘⇡P@⇡:: Ö@Ö4.:4. ✏:)456, ⇣ £A÷ @@@:::@:✏(v.) (@455, 24, 26, 28, 30, 35, 38, 40, 43, . . . . ✏(v.) ‡@✏@ ::BÀ@BA455, 455, 568, ⇣ ⇣ 456, 457, [569]. @ B 568, ÜAm @ : @@456, 7. (II) 247. @ : 5. 5. @@ :: 8. ‡@ 456, 568, : (II) 247. ⇣ ÄYJ   Ø@ @ : ˘ ÆJ 2. 8. ≠J K 244, 244, 299, ✏ ÜAm @ : 5. ⌦ ⇣œØ@@ ⇡::: Ö8.4.@4.: 5.@ :: 5.5.@@ :: 8.8. :B455, 456, 457, [569]. ⌦ ÜAm @ :26, 455, 568, 456, 568, (v.) (II) 247. @(:@24, @ :247. 455, ≠J À28, A247. K456, :456, 244, 244, : 8. @@@ @B≠À 30,568, 35, 38,299, 40, 43, . . . . ˙ÄYJ GAÎA÷ :B✏)✏(II) ⌦(II) ✏@B455, B:✏(v.) 455, 456, 568, @@(:⌦:@@ @@455, @BB:(v.) œ @œ @:: 8.8.@ : 8. @ @:: 8.8. @456, (v.) :568, 247. :: (II) ˙⌦˙Ö GAÎA÷ 244, 244, 299, : 244, 244, 299, ✏ @ @ : 456, 568, @ : 455, 456, 568, (v.) : (II) 247. @ B @ : 455, 456, 568, GAÎA÷ ⇣ÜAm ≠À (v.) : (II) 8. @ : 8. (II) 247. @@GAÎA÷ ::✏)(II) 247. :(v.) (: 455, 24, 26, 28, 30, 35, 38, 40, 43, . . . . œ @2.2. 8. : 8.@@:: 8. ¯⇣Ê⇣mPAj m::à '@5.5. 455, 456, 568, (v.) (II) 247. :JÀ@ @ B455, (v.) :: (II) 247. ⇢⇡⇡'˙. ˙ÒÖ.GAÎA÷ (v.) (II) 247. @@ @B@≠J @(v.) 456, 568, :@::244, (II) 247. @B:@ ::)(v.) (:@ :24, @ A@@⇣K(v.) 244, 299, 5. ::œ:@8. :≠À 26, 28, 30, 35, 38, 40, 43, . . . . :(⌦À@244, 244, 299, ↵ : 8. : 8. @@ :: 8. ⇣ (II) 247. : 8. ˙ÜAm @ @ 456, 568, ✏ ⇢ ' ↵ @@(v.) :(: 244, 244, 299,299, ˙˙Êœ⇣Êm⇣:m⇢⇢.''Ò8.ÒJÀ@JÀ@:: 8.8.@@ :: 8. @≠À (v.) :@ (v.) (II) :247. 244, 244, @ (v.) (II) 247. ✏ ⇣ 8. A ” @ ✏ (v.) : (II) 247. : (II) 247. @ : (II) 247. A244, K455, 244, 244, 299, 244, 299. A244, ”299, @568, .8. @ @≠J @::⌦:À244, 456, ✏ @ 299, ˙˙ÄYJ GAÎA÷ Ò4.JÀ@JÀ@:: 8.8.@ :: 8.8. @B⇣(v.) (II) @ :::@:@B✏(v.) @:::@244, 455, 456, 568, ::247. 244, 244, GAÎA÷˙⌦˙ Ê⇣Ø@œÊ@@m⇢:m:::'..8. 8. :247. 299. 244, 244, 299, (II) 244, ✏✏(v.) ⇣ Ò (v.) (II) 247. : (II) 247. (v.) : (II) 247. ✏ @ ↵ ≠J À A K : 244, 244, 299, ⌦ ✏ I K@ : [261], 471. @ (v.) : (II) 247. @ ⇣˙ÜAmÊ⇣⇣m⇢⇡'ÖÒ@ JÀ@ : 244, 244, 299, @ B @ : 455, 456, 568, ✏ @@ @ AB≠J : 110, 244, 244, 299, ✏244, 244, 244, 299, @:@ :32, 455, 456, 568, :@:B ”(v.) @@ :⌦:@@✏À@ A@⇣@⇣K22, ✏(II) 197, 205, 231, 244, 299, 455, 456, 568, (v.) :247. (II) 247. ↵ ✏ @ B :244, 455, 456, 568, ˙Êm⇢'.. Ò:JÀ@5.::: 8.8. 8. @ @ : 22, 32, 110, 197, 205, 231, ⇣ : 244, 299, ≠J À A K : 244, 244, 299, ✏ : 244, 244, 299, ✏ : 244, 244, 299, I K@ :262-3, [261], 471. ”@ :301, 397-8, 484, 587, ⌦ : 244, 244, 299, : [261], 471. @ ✏ ⇣ ↵ ✏ A ” @ @ ✏ A ”@ 262-3, 397-8, 484, 587, .: . 262-3, ; :(247. ): 397-8, [262-3], [398-9], :@ .(v.) 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(II) 247. 397-8, 484, 587,587, :@244, 262-3, 397-8, 484, 66, 69, 74, 81, 255, 260, 263, 286, Ê⇣ÆJm⇢⌦£A÷ Ò66, JÀ@ : 69, 8. 74, @ K@ : [261], ✏ fl : [261], 471. ◆ A ” @ A÷ @ : 261, 375, 587. @ YK◆ . @ :@˙˘YK◆18, 81, 83, 95, 201, 220, 255, 260, 263, 286, . @ @ @ ✏ Q” (v.) : (I) 1, 19, 230, ✏ ◆ (s.) 1, 19, 230. ⇣ ✏ 201, 220, 255,. . . . @ : [261], 471. ✏ ◆@ YK A ”@ : 262-3, 397-8, 484, 587, @ @ : 18, 66, 69, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, : [261], 471. ✏ Q” @ (v.) : (I) 1,: 19, 230, @ @::@@ ◆18, 66, 69, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, ✏ ✏ ✏ I K@ : [261], 471. ↵ @ : 262-3, 397-8, 484, 587, ⇣ : 244, 244, 299, ✏ @ (v.) : (I) 1, 19, 230, @ : 262-3, 397-8, 484, 587, 244, 244, 299, @ : 18, 66, 69, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, : 1, 19, 230. ◆@¯◆YK. PAj :66, 261, 375, 587. @@:261, 262-3, 397-8, 484, @ (v.)587, (I) 1, 19, 230, Translation 18, 66, 69, 74,:74, 81, 83,83, 95,95, 201, 220, 255, 263, 286, I K@@: :✏255, :255, [261], 471. @220, 69, 81, 201, 263, 286, ✏✏@ @ @ 260, ✏260, @ 18, 66, 69, 74, 81, 83, 95, 201, 220, 260, 263, 286, ✏ fl 261, 375, 587. @ A÷ 375, 587. . @ : . 18, Ã@'@::108 ✏ ✏ 108 Translation 66, 69, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, @ : [261], 471. m :18, 2. ✏ » . A ”@ 262-3, 397-8, 484, 587, @ : [261], 471. @ : 262-3, 397-8, 484, 587, ⇣ @ (v.) : (I) 6, 92, 116, 118, 120, 122, 129, 145, 189, 392, ↵ ✏ @ : 261, 375, 587. ✏ @ YK @ : 18, 66, 69, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, fl : [261], 473. ✏ @ : 262-3, 397-8, 484, 587, : 244, 244, 299, A÷ @ : 261, 375, 587. @ : 262-3, 397-8, 484, 587, : 244, 244, 299, I K@ : [261], 471. : 18, 66, 69, 74, 81, 83, 95, A✏”@✏↵@:✏(v.) 262-3, 397-8, 484, 587, (v.) (I) 6, 92, 116, 118, 120, 122, 145, 189, 392, (v.)::: (I) 6, 120, . ⇣ @@@ (v.) 108 Translation 108 Translation Q” (I) 1, 19, 230, (v.) (I) 6, 6, 92, 92,116, 116,118, 118, 120, 122, 129, 145, 189, 392, @:261, @255, ✏✏@✏:@263, :244, [261], 473. 92, 116, 118, 120, 122, 129, 145, 189, 392, :@:145, 244, 299, 244, 244, 299, :262-3, [261], 471. (v.) : 189, (I) 1,392, 19, 230, @129, (v.) (I) 1, 19, 230, ✏255, @:122, (v.) (I) 1, 19, 230, fl A÷ @ :260, 375, 587. ⇣G◆◆ @. @(v.) @ (v.) :129, (I) 6, 92, 116, 118, 120, 129, :260, 261, 375, 587. @ 262-3, 397-8, 484, 587, » @ 108 Translation @˙@YK :ÄYJ 18, 66, 69, 74, 81, 83, 95, 201, 220, 286, : 397-8, 484, 587, A ”@ : 262-3, 397-8, 484, 587, @ : 262-3, 397-8, 484, 587, 201, 220, 255, . . . . ✏ : 261, 375, 587. @ 262-3, 397-8, 484, 587, : 18, 66, 69, 74, 81, 83, 95, 201, 220, 263, 286,   Ø@ : 4. ✏ 122, 145, 189, 392, 445, @ @ : 261, 375, 587. ↵ ✏ :⌦ (I) 6,69, 92, 116, 118, 120, 122, 129, 145, 189, 392, ✏ fl YK. @ ˙: ⇣G18, 66, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, (v.) : (I) 1, 19, 230, Q” A÷ @ : 261, 375, 587. ✏ ✏ @ (v.) : (I) 1, 19, 230, @ (v.) : (I) 6, 92, 116, 118, 120, 122, 129, 145, 189, 392, ✏ ✏ ✏ ✏ ✏ @ (v.) : (I) 1, 19, 230, @ ⇣ fl (v.) : (I) 1, 19, 230, A÷ @ : 261, 375, 587. @ (v.) : (I) 6, 92, 116, 118, 120, 122, 129, 145, 189, 392, @ (v.) : (I) 6, 92, 116, 118, 120, 122, 129, 145, 189, 392, 375, 588. 262-3, 397-8, 484, 587, ✏@✏✏397-8, »✏262-3, @:[261], :145, 262-3, 397-8, 484, 587, @:397-8, :189, 261, 375, 587. ✏✏@@::(v.) @280. 262-3, 397-8, 484, 587, :¯@ 262-3, 484, 587, :: (I) 6, 116, 118, 120, 122, 129, 189, 392, 476, 510, 580, 585. @@129, :129, 484, 587, (v.) (I)479, 6, 92, 92, 116, 118, 120, 122, 129, 145, 189, 392, ˙⇣⇣⇣GG@@⇡(v.) (v.) :◆ (I) (I) 6, 92, 116, 118, 120, 122, 145, 392, 261, 375, 587. ✏145, 262-3, 397-8, 484, :(v.) [261], 375, 588. ✏ Ö @ @: (v.) : 392, (I) 1, 19, 230, 587, Q” ::@ :189, (I) 1, 19, 230, @ ˙˙ÜAm :5. 6, 92, 116, 118, 120, 122, (v.) 6, 92, 116, 118, 120, fl ✏ : 245. » @ A÷ @ : 261, 375, 587. Q” @ (v.) (I) 1, 19, 230, ◆ (v.) : (I) 1, 19, 230, @ ✏ ✏ (v.) : (I) 1, 19, 230, @ ✏ 245. G @ (v.)118, : ◆◆(I) :::6, 92, 122, 116, 118, 120, 122, 145, 189, 392, 261, 375, 587.164, :@8,262-3, 397-8, 484, 587,172, ✏@✏✏392, 245. @129, :@ »¯@ 262-3, 484, 587, (I) 6, 92, 116, 120, 129, 145, 189, :@280. 141, 143, 150, 245. ◆6, 92, :(v.) (I) 1, 19, 230, @✏:397-8, @@ :@(v.) : :(I) 1, 19, 230, 122, 129, 189, 392, 445, :1, 141, 143, 150, 164, 172, :: 145, 245. @ (v.) :(v.) (I) 1, 19, 230, Q” @ (I) 1, 19, 230, ⇣G◆⇣G⇣A”@ @(v.) 245. ✏ ˙˙˙@◆YK (v.) : (I) 116, 118, 120, 122, 129, 145, 189, 392, @ (v.) : (I) 19, 230, » @ ✏ ✏ :: ◆◆(I) 6, 92, 116, 118, 120, 122, 129, 145, 189, 392, œ ✏ @ :479, 8. 18, 66, 69, 74, 81, 83, 95, 201, 220, 255, 260, 263, 286, :˙◆⇣GAÎA÷ 245. ◆ ✏ @ : 262-3, 397-8, 484, 587, » @ G @. (v.) (I) 6, 92, 116, 118, 120, 122, 129, 145, 189, 392, 186, 188, 206, . . . ; (s.) 1, [378]; ✏ : 245. @ 186, 188, 206, . . . ; (s.) @ : 262-3, 397-8, 484, 587, @ (v.) : (I) 1, 19, 230, ¯@ : 280. 476, 510, 580, 585. (v.) :◆ (I) (I) 1, 19, 230,230, 1, 245. :280. 19, 230, ✏ @@✏@:1,1, ˙˙◆G◆⇣⇣GA”A”::◆ 245. A¯@ í @ ✏:280. @ (v.) (v.) :@@ (v.) (I) 19, 230, @ (I) 19, 230, ::: 245. ✏ (v.) : 1, (I) 1, 19, 245. ⌦ ✏ 245. ✏ ✏ : » @ @@ ::(v.) 484, 587, ˙˙◆⇣GÊ⇣A”m⇢:':Ò245. 245. (v.) 69,: 71, 71, 74, 80, ✏124, 280. : 280. ◆ [378]; :: 69, 74, 80, 83, 84, 124, 321, 447, @@ (:[271], ([271], ))397-8, 580. @(I) 580. ✏í262-3, @110, :(v.) 397-8, 484, 587, ✏ 262-3, @280. :B 1, 19,363, 230, JÀ@ : 8.@@@@(v.) (v.) 69, 71, 74, 80, 83, 84, 110, 321, 363, 447, 262-3, 484,447, 587, 280. A¯@ :280. (I) 1, 19,397-8, 230, (v.)::: 69, 69, 71, 71, 74, 74, 80, 80, 83, 83, 84, 84,✏@ 110, 110, 124, [271], 321, 363, 447, :. 245. :[271], 280. :¯@ 397-8, 484,363, 587, (v.) 124, 321, 363, ⌦◆ :::@124, ✏✏ 262-3, ⇣◆ ⇣ ˙: G A”245. 110, 124, 321,80, 363, @ (v.) : 69,[271], 71, 74, 83,448, 84, @110, [271], 321, 447, 280. ✏ 280. ¯@ ✏◆ @@ :::@ 1,392, @321, (v.) :280. (I) 19, 447, 230, : 245. A124, í◆ :⌦:@ 280. :: (I) 6,71, 92, 116, 118, 120, 122, 129, 189, 280. Y˙˙◆GgGA”A”@ @ (v.) 69, 74, 80, 83,578, 84, 83, 110, 363, 447, @ :(v.) (v.) 69, 71, 74, 80, 83, 84,124, 110,[271], [271], 321, 363, @ 145, (v.) :280. (I) 19,1, 230, H. 451, 480, 575, 599. 245. :: 71, 69, 71, 74, 80, 84, 110, 124, [271], 321, 363, 447, ✏ (v.)@@ (v.) : 69, 74, 80, 83, 83, 84,84, Y:(v.) g @ (v.) 69, 71, 110, 124, [271], 321, 363, 447, : 69, 71, 74, 80, 110, 124, [271], 321, 363, 447, ✏ (v.) : 69, 71, 74, 80, 83, 84, 110, 124, [271], 321, 363, 447, A í @ ¯@ : 280. ◆ ◆ @ : 280. ⌦ YYgg@@ (v.) (v.) :: @ 69, 69, 71, 74, 80, 83, 84, 110, 124, [271], 321, 363, 447, @ (v.) : (I) 1, 19, 230, : [487], 71, 74, 80, 83, 84, 110, 124, 321, 447, @@Aí (v.) 1, 19,363, @[271], (v.) (I) 1,230, 19, 230, :: 416, 279, 416, 430, [487], :::@ (I) 279, 430, 462, 462, 505, 505,H 321,80, 363, . ◆◆ @321, [487], [488]. @ 124, Yg110, @ (v.)124, : 69, 71, 74, 83,448, 84, 110, 363, 447, ◆◆ ⌦⌦[271], (v.) (I) 1, 19, 230, @@@:::::[271], [487], A í @ @ [487], @ A í @ 536, 537, [538], 542, 543, H 451, 480, 575, 578, 599. 537, [538], 542, 543, ... . ⌦ (v.) (v.) 69, 71, 74, 74, 80, 80, 83, 83, 84, 84, 110, 110, 124, 124, [271], [271], 321, 363, 447, . @YK◆.536, 468; (VIII) 3, 16. ◆: @(I) ::: 69, 71, 321, 363, 447, [487], 245. QYY˙k◆↵⇣GggA”@ :Q@@ :(v.) [487], @@:::: [487], [487], @@ :: 15, 106, 166, 192, 252, 254, 283, 313, 370, 379, (v.) 69, 71, 74,21, 80,106, 83,106, 84, 110, 124, [271], 321,281, 363, 447, H @ A í @ @ . [487], ⌦ 15, 21, 106, 106, 166, 192, 252, 254, 281, 283, 313, 370, 379, k @ : [487], 15, 21, 99, 106, 166, H. [487], [488]. @YK. (v.) : (I) 468; (VIII) 3, 16. ↵ @@ ::: [487], ◆@ YK @QQQ:kkk↵↵18, 192, 252, 254, 281,. . . . H. 66,21, 69,106, 74, 81, 83, 95,192, 201,252, 220,254, 255, 260, 263, 286, 15, 106, 166, 281, 379, . Qk↵ @ @: : [487], ⇣JJ.”@ :283, 15, 21, 106, 106, 166, 192, 252, 254, 281, 313, 370, 379, @YK (v.) :283, (I)313, (VIII) 3, 16. . (v.) :468; (I)370, 468; (VIII) 3, 16. H . Z¯Y 20. Q k @ : 15, 21, 106, 106, 166, 192, 252, 254, 281, 283, 313, 370, 379, ::15, 106, 166, QQYkk↵g@@:@:: (v.) [487], : 99, 81,106, 145,83, 147, 247, 250, @ : 81, 145, 147, 247, 250, [275], [276], [277], 295, 383, 448, 508, @YK (v.) : (I) 468; (VIII) 3, 16. H [487], 69, 71, 74, 80, 84, 110, 124, [271], 321, 363, 447, . @ 254, : 81, 145, 147, 247, 250, [275], [276], [277], 383,(VIII) 448, 508, (v.) ::: 295, (I) 468; 3, 16. ⇣JJ.”@ ::: 20. (v.) (I) 468; (VIII) 3, 16. 16. . (v.)468; (I) (VIII) 469. (VIII) 3, ↵ [487], (v.) (I) 3, 192, 252, 281, . [277], . . . 295, 383, @YK (v.) (I) 468; (VIII) 3, 16. 16. Z¯Y . [275], [276], @@ ::: 16. (v.) :: (I) 468; @YK (v.) (I) 468; (VIII) 3, 16. 16.3, (v.) (I) :468; 469. (VIII) 3, ⇣JJ189, (v.) (I) (VIII) 468; (VIII) (VIII) 3, 16. 16. @YK (v.) (I) 3, 16. (v.) :120, (I) 468;129, (VIII) 3, 20. ⇣JJ..[277], QQ⌦g@ 449, 509. :383, Z¯Y ” : 295, 20. 145, 147, 247, 250, [275], [276], [277], 448, 508, 81, 145, 147, 247, 250, ˙⇣G @ Q(v.) (I) 6, 92, 116, 118, 122, 145, 392, g@::::81, 81, 145, 147, 247, 250, [275], [276], 295, 383, 448, 508, . @Y ” : 455, 456, 471. (v.) : (I) 468; (VIII) 3, 16. ⌦⌦g@ ⇣ . @ (v.) : (I) 468; (VIII) 3, 16. 81, 145, 147, 247, 250, [275], [276], [277], 295, 383, 448, 508, Z¯Y ::: 20. @YK⇣. (v.) (I) ::468; @@ (v.) :: (II) 95, 532, 20. ⇣JJJJ:..””455, [275], [276], 295,525, 383, 20. (VIII) 3, 16. (v.)[277], (II) 95, 525, 532, 543, 543, 563. 563. Qk↵ @ : [487], Z¯Y : 20. : 20. @Y JJ ” 456, 471. :: 20. 449, 509. Z¯Y 20. 20.456, 471. ⇣JJ. . ”⇣JJ⇣JJ:..:””@ 455, : 456, 455, Z¯Y ::456, 20. @Y 471. ⇣ 457, @ (v.) : (II) 95, 525, 532, 543, 563. ◆˙G A”¯X ¯X @ (v.) : (II) 95, 525, 532, 543, 563. …¢ (v.) : (I) : 20. : 572. 20.472. ⇣ ⇣ . :¯X 245. @Y JJ ” : 455, 456, 471. Z¯Y JJ ” : 20. @@ :: 7.7. @ (v.) 525, 532, 543, 563. 95,:526, @ : 455, 456, (v.) :: (II) 95, 533, 544, 20. . 455,:456, 455, 456, 471. 471. ⇣JJ..” :(v.) @Y 471. : 455, 456, 471. …¢ : (I) 572. (v.) : (I) [573]. . ⇣ @ : 455, 456, 471. 564. @455, 455, 456, 471. 455,:(v.) 456, 471. @Y…¢⇣JJJJ..”” ::(v.) : (I) 572. @Y 456, 471. ⇣ÈìP@ 572. ⇣ìP@: : 7.7. .(v.) @ :(I) 455, 456,[291]. 471. Y™K :::(III) 285, 291. (v.) (III) È ⇣ . ⇣ :: (I) 572. …¢ (v.) :. .(v.) (I) 572. @Y JJ ” : 321, 455, 456, 471. : 455, 456, 471. @@ :: (( 74, @@ ))80, 24, 26, 28, 30, 35, 38, 40, 43, . . Yg @ È(v.) 83, 84, 110, 124, [271], 363, 447, (v.) (I) 572. . . ìP@ :: 69, 7. @@ 71, 24, 26, 28, 30, 35, 38,…¢ 40, (v.) .(I) (v.) (v.).:::. (III) (I). 572. 572. Y™K 291. : 455, 456, 471. (v.) :: 285, (I) . .. 43, (v.) (I) 572. 572. …¢ (v.) : (I) :572. 572. (v.) (III) 285, 291. …¢ (v.) : (I) Y™K (v.) : (III) 291. (v.) : 285, (I) 572. . ✏ Y™K . Z@Z@P@P@: : ((Z@Z@PAKPAK. ).@)24, 30, 35, . .. ..(v.) . . : :(v.) ✏@ : 26, 24, 26,28, 28, 30, 35,38, 38,40, 40,43, 43, .(v.) (III) 285, Y™K (III) 285, (v.) (III) 291. 285, 291. 291. …¢ (I) ::572. B 455, 456, 568, . . P@ : ( Z@PAK.@) B24,@ : 455, 26, 28, 30,568, 40, 43, . . :: (III) Qk↵ @ :Z@[487], 456, (v.) :35,(I)38,572. Y™K.. .(v.) 285, 291. (v.) (III) 285, 291. (v.) : (III) 285, 291. (III) 291. 285, 291. Y™K. (v.) (v.) :: (v.) (III): 285, 285, (v.) : (I) 572. Y™K (III) 291.

˙◆ Arabic V. glossary ◆

:

@

@ V. Arabic @ glossary

@

@

@

@

@

@

@

I⇣ K@

@ @

✏ @ ✏ fl A÷ @ @ » @✏

@✏

✏ ¯@

✏@

◆ Aí⌦ @

◆

@

H.

@YK. @

Z¯Y⇣JJ.” @ @Y⇣JJ.”

✏✏

…¢ .

Y™K.

. . ⇣ ë™K. :: 10, 10, 18, 100, 100, 374, 374, 517, 517, 518, 572, 572. I⌧.⌘K 518, ë™K 18, 100, 100, 374, 374, 517, 517, 572, 572. . ⇣ ⌘ ⇣ ⌘ ⌘ ˘Y™K ÆK.. (v.) : (I) 45, 64, 89, 197, 199, 228, 231, [265], [266], HAJ K@ : 110, 190. 302, 359, ⇣ . I⌧ K HAJ K@ : 110, 190. (v.) : (III) 285, 291. . . 518, 572, 222 Arabic glossary ⇣ ë™K : 10, 18, 100, 100, 374, 374, 517, 517, 572. . : 10,: (I) ⌘. K@518, ˘˘ë™K ÆK (v.) 64,100, 89, 374, 197, 374, 199, 517, 228, 517, 231, [265], [266], ⇣ ⇣ÆK. (v.) 18,45, 100, 572, 572.302, 359, . HAJ : 110, 190. : (I) 64,100, 89, 374, 197, 374, 199, 517, 228, 517, 231, [265],572, [266], ë™K : 10, 18,45, 100, 572.302, 359, ✏⌘ 518, ⇣ AK. : . 59, ✏ ⌘ ÜY™K 65, 65, 94, 116, 378, 393, 446. ⇣ ⌘ I   J” : 105. . ⌘ [89], [132], 202, [228], [89], [228], 235, 235, 259, 321, [390], [406], 420, 480, HAJ K@518, 110, 190. I .J”324, :: 105. .. :.:(v.) hhh. . ë™K : 10,: (I) 18, 45, 100,64, 100, 572, 572.302, 359, ˘˘◆⇣⇣ÆKÆK⇣AK⇣259, 89,374, 197,374, 199,517, 228,517, 231, [265], [266], ✏ Ü : 59, 65, 65, 94, 116, 378, 393, 446. 321, 324, [390], . . . . . (v.) : (I) 45, 64, 89, 197, 199, 228, 231, [265], [266], 302, 359, ⌘ ⌘ . ⇣ An ancientë™K treatise on magic squares 111359, ◆ ÆKAK.. :. (v.) I  J”518, :[265], 105. ܢ◆Greek 65, 94, 116, 378, 393,517, 446. :59, 10,:65, 18, 100, 100, 374, 374, 517, 517, 518, 572, 572.302, h (I) 45, 64, 89, 197, 199, 228, 231, [266], . ë™K : 10, 18, 100, 100, 374, 374, 517, 572, 572. h ... :: 10, ë™K 10, 18,17, 100, 100,374, 374,374, 374, 517, 517, 517, 517, 518, 572, 572, 572. 572. 18, 100, 100, h.. © K : (I) 468. ë™K 18, 100, 100, 374, 374, ✏’Á✏⌘◆◆✏◆⌘'⌘'✏ ⌘J”518, . (v.) ✏ ⇣ ⇣ I : 105. ’Á ✏ .:. :::2,[265], ˘© K ÆK. (v.) : 65, (I) 65, 45, 94, 64, 116, 89,[573]. 197, 359,27, 29, @@Yg 9. 6, 18,302, 19, 25, :(v.) 59, 378,199, 393,228, 446.231, 518, [573], Yg 9.4, 5,[266], 9. (I)519, 17, 94, 468. ◆ h. ÜÜ⇣⇣◆◆ÆK⇣⇣⇣AKAK518, : 59,: 65, 65, 116, 378, 393, 446.✏@ Yg . . ✏ ⌘ . ◆ 33, 36, . . . . (v.) : (I) 17, 468. ✏ ˘˘© K (v.) : (I) 45, 64, 89, 197, 199, 228, 231, [265], [266], 302, 359, ’Á231, ' . ::[265], (I) 45, 64, 89, 197, 199, ◆Yg Ü◆ÆKÆK⇣AK... :(v.) 59,:: 65, 65, 94, 116, 378, 393,228, 446.231, @ Yg 9. (I) 45, 64, 89, 197, 199, [266], 302, 359, ✏ @ 9. ˘H (v.) (I) 45, 64, 89, 197, 199, 228, [265], [266], 302, 359, 20,: [196], [583], ˘⇣. ÆKAK228, (I) 45,[378], 64, 89, 197, 199, h 228, [266], 302, 359, @˙Yg 9. 9. ◆Ê⌘'Ê✏✏⌘J”⌘J”. . ::::[265], .. :(v.) . 231, h. ✏»Yg [568]. : [568]. [569]. 231, [265], [266],378, 302, ◆ ’Á ˙ ÜH AK : 59, 65, 65, 94, 116, 393, 446. © K (v.) : (I) 17, 468. ◆ : 1, 2, 4, 5. . ✏ »Yg : 1, 2, 4, 5. . : 20, [196], [378], © K : (I) 17, 468. [583], »Yg : 1, 2, 4, 5. 363, .65, . . .[378], ✏⌘. on Yg : ..9. ⇣⇣⇣◆◆⇣.. AKAK359, H AK...AK.:.:✏◆(v.) : 59, 20, [196], [583], ◆ 59, 65, 94, 116, 378, 393,treatise 446. ˙@»Yg An ancient Greek magic squares 111 h. © K (v.) : (I) 17, 468. Ê J” : [568]. ÜÜI⌧ 65, 65, 94, 116, 378, 393, 446. : 1, 2, 4, 5. h Ü :: : .59, 59, 65, 65, 94, 116, 378, 393, 446. . ◆ : 9. »Yg : 1, 2, 4, 5. ◆Ü◆◆ AKAK⌦..@KYg 28, 32, 34, 40, 41, 42, 45, 47, 48, 48, 50, 50, 53, 54, . . . . . ✏ 378, 393, 446. »Yg 65,65, 65,94, 94,116, 116, 378, . 59, 65, ✏⌘. : . .9.:: 1,1, 2,2, 4,4, 5.5. @ ◆Yg © K : (I) 17, [378], 468. [583], ˙ Ê J”. on : 529, [568]. H AKAKK.(v.) : 20, [196], ⇣I⌧ .H⇣. .393, Qk ::50, 530, ◆»Yg 447, 600, 626. An ancient Greek treatise magic squares Qk 529, 529, 530,. 530, 543. 111 28, 32,586, 34,[378], 40, 41, 42, 45, 47, 48, ZZZ48, 50,529, 53, 54, .530, . . 543. ✏ : 20, [196], [583], . ⌦ . Qk : 529, : 1, 2,53, 4, 530, 5. . .530, :50, 9. . I⌧ K :: 28, 32, 34,[378], 40, 41, 42, 45, 47, 48, @48, 50,529, 54, . . 543. © K (v.) : (I) (I) 17, 468. Yg . H AK 20, [196], [583], ⌦ . . . © K (v.) : 17, 468. .·© K 1, 4, 469. 5. (v.). ::: (I) (I) An ancient Greek treatise magic © K⌦.K.»Yg (I) 17, 468. ZZQk :: 529, 529, 530, ✏◆ 2,17, (II) 229, [571], 586, (v.) 17, 468. Qk.. .on 529, 529,squares 530, 530, 530, 543. 543. 111 Z Qk :: 529, 529, 530, 530, 543. 529, 529, 530, 530, 543. @ Yg : 9. ⇣ »Yg : 1, 2, 4, 5. . H :: 20, [583], I⌧ 28, [196], 32,An 34,[378], 40, 41, 42, 45, 47, 48, 48,on 50, 50, 53,squares 54, . . . . . magic [585]. ancient Greek treatise 111 .⇣⌦AKK.⌦⌦.KK..(v.) ·I⌧ 229, [571], 586, : 28,: (II) 32, 34, 40, 41, 42, 45, 47, 48, …™k 48, . .50, 50, 53, 54, . . . . …™k ·H ⇣I⌧ …™k K (v.) : (II) 229, [571], 586, H AK : 20, [196], [378], [583], 530, 530, 531, 531, 544. Z Qk : 529, 529, 530, 530, 543. . 50, K : 20, 32, 34,[378], 40,530, 41, 42, 45, 47, 48, »Yg 48,. on 50,2,53, 54, 28, ↵.. ⌦AK.AK⌦Z..Qk 1, 4, 5. . . . . An ancient Greek treatise squares 111 :28, 529, 529, 530, 543. [196], [583], . :magic H 20, [196], [378], [583], ·H …™k .:.::::.286, 291, 559, 586. An ancient Greek treatise on magic squares 111 .⇣⌦KAK.47, 20, [196], [378], [583], . …™k . .I⌧ 48, 48, . . . . . ··↵↵ ⌦⌦K.K. (v.) (v.) : (I) 23, 25, 26, 27, 29, . : 28,: 32, 41, 42, 48, 50, 50, 53, 54, . . . . Qk : 529, 529, 530, 530, 543. (II) 34, 229,40,[571], 586,45, 47, 48, Z…™k . on magic squares :(v.) 286, 291, 559, 586. ggg32, :40, 1, 2, 42, 4, 5. ::»Yg (II) 229, [571], 586, (v.) (II) 229, [572], [584], ©‘ An ancient Greek treatise 111 ⇣⇣⌦⌦K⌦K..…™k ·I⌧ . ©‘ 33, 36, .. . .. . ... :.50, :K (v.) 291, 559, 586. I⌧ K :: 286, 28, 32, 34, 41, 45, 47, 48, 48, 50, 50, 53, 54, . ..530, ⇣I⌧ (v.) : (I) 23, 25, 26, 27, 29, 32, 33, 35, 35, 36, . .35, . 53, . 35, ©‘ : (II) 229, [571], 586, . …™k . 28, 32, 34, 40, 41, 42, 45, 47, 48, 48, 50, 54, . . . ⇣ Z Qk 529, 529, 530, 543. ⌦ . K : 28, 32, 34, 40, 41, 42, 45, 47, 48, 48, 50, 50, 53, 54, . . . . ⇣I⌧ 587. gg. 50,(v.)50,: 53, ⌦K. : 28, 32, 34, 40, 41, 42, 45, 47, H 48, ©‘ 48, 54, .[272], . . . [273], ©‘ [272], g ·↵↵ ⌦K⌦. .:(v.) . ©‘ : (II) 229, [571], 586, 286, 291, 559, 586. : 286, [291], 560, 587. …™k H⇣⇣ [275], ·↵ ⌦K.©‘⇣: g286, 291, 559, 586. [274], gg.g. . [276],[275], ©J ‘ H (v.) :291, [6]??, [272], [272], [273], [274], [277], [276], 332, [277], 332, ©J ‘ ··⇣↵⌦⌦⌦KK.K.K⇢. K(v.) :: :(II) (II) 229, [571], 586, ⌦ ≠  (v.) (I) 10. ⇣ ··Im . ⌦ :(v.) 286, 559, 586. g . ©J ‘ 229, [571], 586, ⇣ ©‘ Z Qk : 529, 529, 530, 530, 543. ⌦ ' (v.) (II) 229, [571], 586, ·↵ ⌦⌦K.. (v.) : 26, 49,. 229, 50, 54, 586, 56, 58, 70, . . . …™k H 53,[571], :: (II) g. g.. 339, 341, . . . . ©J©J.⌦⌦‘335, ⇣ ‘ ·≠  ⇣ ↵Im H ⇣ K : 286, 291, 559, 586. ©J.⌦‘gg..Greek : (I) 10. 79, 92, 95,on 109, 122, squares :g 26, 49, 50, 53, 54, 56, 58, 70, .⇣ . 300, . ©‘ ↵↵⇣↵⇣⌦⇣.⇢©J⇢K⇣''⌦(v.) H An ancient magic ‘ : 19, 79, 92, 95, 109, 123, 129, 241, 327, 372, treatise 388, 393, ↵Im ⇣ ◊◊.◊: ::19, ··Ò  50, 53,586. 54,85, 56, 58, :286, 26, 49,63, 56,121, 58, 127, 70, H . 136, . . ®Ò“m . .138, .(v.) 273, [278]. 291, 559, ®Ò“m 273, [278]. . ⌦⌦⌦KK.⇣K.. ::⇣K:(v.) : (I) 71, 80, 140, 147, . . . ,500. ↵·≠  286, 291, 559, 586. ·ºQ 129, 241, [300], 327, 372, ... . g : 291, (I) 10. ®Ò“m 286, 559, 586. ©J ‘g.. .◊ : 273, [278]. 608. ©‘ ⌦ :(v.) 286, 559, 586. . .:. (I) .291,597, ⌦⇣ .70, ⇣ ↵Im ⇣ ®Ò“m : 273, [278]. (v.) (I)56, 23,58,25, 26, 27, 32, 33, 35, 35, 36, . . . . ⇢⇣K⇣''(v.) ®Ò“m 273, [278]. .10. : 26,:…™k 49, 50, 53,: 54, 70,H . . . ®Ò“m . .◊.◊:29, ≠  (I)597, [273], [278]. ⇣↵↵⇣K⇣⇣K®Ò“m 273, [278]. ::: 140, 273, [278]. ⇢ g ºQ (v.) ::(I) (I) 608. ◊ . (v.) 598, 609. Ò ºQ (v.) : 63, 71, 80, 85, 121, 127, 136, 138, 147, . . . ,500. Im : 26, 49, 50, 53, 54, 56, 58, 70, . . . . ©J ‘ ⇣ : 273, [278]. ⇣ ⌦ . ⇣⇣⇣K⇢⇣K⇣(v.) . : : (I) 597, 608. ' H ⇣ g Im : 26, 49, 50, 53, 54, 56, 58, 70, . . . . g ≠  (v.) (I) 10. …‘ ⇣ H …‘ 33. ».◆AKK™↵K:(v.) ... .◊(v.) H⇣136, Ò  :9.:(I) 140, 147,224, . . . ,500. …‘⌦‘gg138, I :: 9. ≠  (v.) (I)63, 10.71, 80, 85, 121, 127,H : (IV) 521, 522, ®Ò“m : 273, [278]. ©J ↵ºQ⇣⇢⇣K J”⇣'(v.) . g ⇣Im …‘ g : 26, 49, 50, 53, 54, 56, 58, 70, . . . . : (I) 597, 608. ⇣ 547, 548, 550, 551, 554, …‘ . ⇣ Ò  (v.) :: ©‘ (I) 63, 71, 80, 121, 138, 140, 147, . . . ,500. 554. g.: (v.) g550, . .◊ : 553. :g (IV) 224, 520,85, 521, 546,127, 547, 136, 549, I 9. ⇣'J”⇣(v.) …‘ ↵↵.A⇣⇣KK™⇢™K⇣…‘ ºQ 597, 608. ↵ ≠  (v.) : (I) (I) 10. 10. :⇣K⇣J” 33. »Im ⇣ºQ . ®Ò“m 273, [278]. ↵Im I : 9. ⇣ . : 26, 49, 50, 53, 54, 56, 58, 70, . . . . ⇣Ò  ⇣.. gg138, '⇣''(v.) (I) 597, 608. ◆⇣↵⇣. 'K⇣⇢K⇢⇢↵(v.) :: 26, 49, 50, 53, 54, 56, 58, 70, .. 136, .. .. È ‘ ✏Ò  (v.) : : (I) (I) 63, 71, 80, 85, 121, 127, 140, 147,107, . . . ,500. ⇣ : 13, [14], [222], 521, Im 26, 49, 50, 53, 54, 56, 58, 70, È ‘ ’ÁIm : 31, 44, 60, 440. ... ◊ : 273, :33. 26, 49,63, 50,71, 53,80, 54,85, 56,121, 58, 127, 70, . 136, . . …‘ . g138, »◆AK⇣ :(v.) È ‘ : (I) 140, [278]. 147, . . . ,500. ®Ò“m ⇣ ⇣ . g ⇣ ⇣↵ ⇣(v.) ⇣◆ g [526]. ºQ :13,(I)[14], 608. I⇣.A⇣⇣KK127, ™È ‘ J”g33. :: 136, 9. È ‘ g540, ⇣È ‘gg521, 522, 535,550, . . . ;; È ‘ [222], 520, 520, 521, 534, 547, 547, . 597, . . .107, See p. 21n. . : »’ÁI . ™ J” : 9. . ✏Ò  È ‘ ⇣ ↵(v.) ⇣J” : ::9.::(I) (I) 63, 71, 80, 440. 85, 121, 127, 136, 44, 60, ◆'..K⇣⇣KK⇣⇣™(v.) . 140, 147, . . . ,500.. …‘ 138, ºQ (v.) (I)31, 597, 608. I . g ⇣ ºQ (v.) (I) 597, 608. ↵ ©J ‘ ºQ K (v.) : (I) 597, 608. :::(v.) 33. I 33. ⇣I... KAg ⌦ 31, KAg... : 50, 51, 54, 59, 72, 75, ✏»’Á»–A÷◆A⇣'AK⇣KKfl(v.) 458, 604. ºQ (I) 597, . 44,608. gKAg : : (I) 60, 440. I 33. È ‘ …‘ .. . . 133, ◆✏ . ⇣ ™↵I:⇣J”. KAg I : . 9. : 50, 51, 54, 59, 72, 75, 105, 105, 109, .KAg 176, 178, 429, I 105, 105, 109, . .¶.¶181, .(III) . .(v.) PÒk : : (I) 196 531 . PÒk (v.) (I) 196 (III) 531 ¶¶ (VI) (VI)312. 312. I KAg ⇣ ’Á–A÷ ' (v.) : (I) 31, 44, 60, 440. . . (v.) ⇣ ⇣ ⇣ . I KAg (v.) : (I) 31, 44, 60, 441. g PÒk : (I) 196 ¶ (III) 531 ¶ (VI) 312. ⇣ I ™ J” : 9. fl . K : 33. »I A : 458, 604. . È ‘ . ⇣ .◆..⇣'⇣™™↵↵⇣↵(v.) J” : 9. PÒk (v.) : (I) 196 ¶ (III) 531 ¶ (VI) 312. . ⇣ ✏I ✏✏’ÁI PÒk 531 ¶ (VI) 312. J” :: :9. 9.(I) 31, 44, 60,. 440. (v.) :: (I) (I) 196 196.¶ (III) (III) [532]. . (v.) :J” 581. ⇣. KAg –A÷ :: 574, 458, 604. ’Á–A. ⇣'K™fl↵(v.) : (I) 31,.◊ 44, 60, 440. g(VI) 459, 605. I ®Ò“m : 273, [278]. È ‘ . 312. . hhh ✏–A’Á–A÷⇣'K fl:⇣(v.) : 458, 604. : (I) 31, 44, 60, 440. 574, 581. 575, 582. I KAg h h . . ✏ ⇣Kflfl⇣⇣::: 574, –A÷ 458, 604. H⌘ I✏⇣ ✏KAg –A–A÷ 458, 581. 604. .521, ✏⇣⇣ . 546, 547, 549, 550, 553. g. H ⌘ :✏⇣ (IV) 224, 520,˙˙˙Êk Êk ✏ ⇣K⇣: 574, …‘ (v.) Êk ✏ –A 581. ⇣ : 7, 11, 79, 92, 103, 112, ˙Êk –A÷ 458, 604. H⌘ ˙Êk ✏–A✏⇣⇣Kfl::⌘ 574, 116, 118, (v.):581. : (IV) (IV) 13, 13,39, 39,41, 41,47, 47,49, 51, (v.) 53, 54, . . 120, . . 122, 131, . . . . H⌘ 52, –AI⌧⇣K .:K 574, 581. ↵ ↵ I Çk ( ´ 227. I Çk 227. ˙Œ´´)):))227. 49, 51, ⇣52, 53, 54, . . . . ↵ (( ˙Œ˙Œ . .. Çk ⌘ I 227. ↵ ✏I⌧ H ⇣ ↵ ⌘ g I Çk ( ´ ) 227. ˙Œ È ‘ I Çk ( ´ ) 227. –A⇣ K .:K 574, 581. (v.) : (IV) . 41, 47, 49, 51,⌘ 52,. 53, 54,˙Œ. . . . . 13, 39, H ⇣ . ⌘K. ⌘K@(v.) HAJ : 110, 190.13, 39, 41, 47, 49, 51, I⌧ : (IV) 54,: :.() .(). 96, . 126, ⌘Çk H⌘ 52, ⌘53,(v.) ÒÒÒÇk (v.) 96, 126,130, 130,314, 314,384. 384. ⌘ ⌘ Çk (v.) : () 96, 126, 130, 314, 384. ⇣ . K⌘ (v.) : (IV) 13,ÒÇk ⌘ 41, ⌘ I⌧ 39, 52, 53, 54, . . . . (v.)47, : ()49, 96,51, 126, 130, 314, 384. ⌘ ⇣HAJ H ÒÇk (v.) : () 96, 126, 130, 314, 384. : 110, 190. ⇣I⌘ ✏ ..⌘KJ”⌘K@⌘ (v.) I KAg I⌧ : (IV) 13, 39, 41, 47, 49, 51, 52,⌘ ⌘53, 54, . . . . ⇣HAJ . K@ : 105..

I ´ ˙Êk ⇣K : 190. . ⌘Çk X ⇢'✏. ⌦ PY l ✏ ↵ »X (v.) : (X) 420, 560, 631. I Çk ( ´ ) 227. ÒÒ114 Çk (s.)˙Œ✏: 92, 94, 95, 492. Translation … g@X .Çk ⌘✏⇣⌘ (v.) »X (v.) :: (X) (v.) (X) 420, 420, 560, 560, 631. 631. :: () 96, 126, 130, 314, 384. ⇣K122, Ò˙Çk () 96, 126, 130, 314, 384. ◆ ⇢'.✏ancient g@X »X (v.) : (X) 420, 560, 631. Êk : (v.) 7, 11, 79, 92, 103, 112, 116, 118, 120, 131, 147, 169, 248, PY : 190. Anl… Greek treatise on magic squares fl ⌦ A ‹ @X : 95, 112, 122. ⌘ ↵ 114 Translation ÒIÇk : () 126, 130, 314, 384. ( ˙Œ ´ )96,227. Arabic glossary 223 g@X . ⌘Çk(v.) »X (v.) : (X) 420, 560, 631. An… ancient Greek treatise on magic squares ◆ 114 Translation ⇣ fl A‹◆⇢'‹. fl⌦ PY @X::K :95, 95, 112, 122. Ò…ík Çk (v.) : () 96, 126, 130, 314, 384. l 190. A @X 112, 122. ⌘⌘ ↵ (s.) (v.) : (I) 483. :⇣ 95, 112, 122. Ò114 :: 92, 94,469, 95, 492. Translation l✏⇢'. ⌦ PY ÒIÇk Çk (s.) 92, »X (v.) :190. (X) 420, 560, 631. Çk ( ˙Œ ´ ) 94, 227.95, 492. ◆ . ⌘ ⇣K:K :::95, fl(I)@X96, ⇢'A.✏‹◆⌦flPY ‹: 92, 95, 112, 122. 114 Translation ÒÇk (s.) 94, 95, 492. (v.) () 126, 130, 384. l116 190. Translation (v.)A::‡X 96, 126, 130, 314, :: 6, 47. 190. @X 112, 122. ‡X : 6,: 47. ⌘⇣ 384.(s.) : 92, 94, 95, 492. Translation »X (v.) (X) 420, 560, 631. ✏…ík Ò114 Çk ‡X : 6, 47. ✏ : 6,239, 47. 240, 252, . . . . Translation (v.) : (I) 469, 483. 116 Translation 207, 209, 222, [224], 226, 237,116 »X : (X) 631. ⌘⌘ : (v.) fl @X(v.) (v.) 421, 561, 632. …ík (v.) (I)96, 469, 483. ‹✏◆ 237, : 95, 112,420, 122.560, ÒÒák Çk :: () 126, 130, 314,Translation 384. A116 114 (s.) 92, 94, 95, 493. Translation »X …ík (v.): :92, 469,95, 483. Çk (s.) 492. ‡X(v.) : 6,: (X) 47. 420, 560, 631. 116 Translation X ‡X(I) :94, 6, 47. X A‹◆ fl @X : 95, Translation 112, 122. 114 Translation X ✏⇣✏ (v.) …ík 483. (v.) :: (I) (I) 469, 470,116 484. 116 Translation ⇣ ⌘Òk ::: 207, 222, [224], 226, 237, 240, 252, .. .. .. .. »Ag 357. 209, fl239, A◆‹◆ 239, @X261, 95, 112, ‡X ::: 6, 47. ák 207, 222, [224], 226, 237, 237, 237, 240, 252,122. 114 Translation hák (v.) : (VIII ) 191, 244, 245, [266], 279, 299, Òák (s.) : 209, 92, 94, 95, 492.227, 116 Translation ⇣✏Çk .…ík fl A ‹ @X : 95, 112, 122. X 207, 222, [224], 239, 240, 252, . . . . 532. 207,:209, 209, 222,483. [224],226, 226,237, 237,Qª (v.) (I) 469, 116 Translation X (v.) : (I) 1, [300], ✏⇣ ::: 357. »Ag ‡X : 6,240, 47. Qª X (v.) : (I) 1, [300], 532. X ák : 207, 209, 222, [224], 226, 237, 237, 239, 252, . . . . 237, 237, 239, 240, 252, . . . . (v.) : (I) 1, [300], 532. QªX (v.) : (I)227, 1, 244, [300], 532. ⇣ Òk hÈ J )) 191, [266], 279, ‡X117, : 6,118, 47. k : (v.) 531.:: ((VIII (s.)245, : 12,261, 44, 130,299, 146, 168, 192, 192, 195,X 300, »Ag 357. ..✏⇣⌦Òk h†Òk (v.) VIII 191, 227, 244, 245, 261, [266], 279, 299, …ík (v.) : (I) 469, 483. (v.) : : (VIII ) 191, 227, 244, (v.) (IV) 64, 552, 555. ‡X : 6, 47. Qª X (v.) : (I) 1, [300], 532. (v.) (s.) : 12, 44, 117, 118, 130, 146, 192, 192,Translation 195, P300, ⇣»Ag h Òk (v.) : ( VIII ) 191, 227, 244, 245, 261, [266], 279, 299, 116 ák : 207, 209, 222, [224], 226, 237, 237, 239, 240, 252, . . .168, . 533. . ⌦k245, È J 531. :: 357. PX [261], [266], 279, 299, 124 Translation h⇣.✏ Òk: 357. (v.) : (VIII) 191, 227, 244, 245, 261, [266], 279, 299, »Ag Qª X (v.) : (I) 1, [300], 532. X J : 230. P Qª X (v.) : (I) 1, [300], 532. 566, 566, 604, 624. †Òk :: (IV) 552, 555. Y†Òk :(v.) 108, 191,64, 363, 599, 618. È J k⌦k:::(v.) 531. ◆⇣J⌦⌧J565, (IV) 64, 552, 555. ák 207, 209, 222, [224], 226, 237, 237, 239, 240, 252, . . . . X ✏ P J : 230. »Ag 357. ©✏KK.XQ”Q”[266], †Òk (v.) :: (IV) 552, 555. ⇣°J 553, 556. h.J⌧JkÒk (v.) ((IV) VIII64, )64, 191, 227, 244, 245, 261, m⌦ ◊k:: ::357. 198, 232, 384. ⌦ Qª (v.) : 279, (I) 1,299, [300], 532. © Y»Ag 108, 191, 363, 599, 618. È J 531. . ⌦ ◆⇣ k : (v.) : 12 (adj.), 13, 23, 32, 35, †Òk : (IV) 64, 552, 555. ˙ (v.) : (I) [14], 424, ⇣ ⇣ Qª X (v.) :461, (I) [462]. 1, [300], 532. P È J 531. ◊ »Ag : 357. ⌦ ✏ °J m 198, 232, 384. p : 198, 232, I ⌧ KQ K P ⌦ Y°J ⌧Jkm◊⌦k:: :(v.) 108, 191, 363, 599, 618. 64, 93, 94, 95, 96, 97, . . . . ⇣◆. J⌦Òk . X✏K.⌦40, 198, 232, 384. ⇣ ˙ (v.) 245, : (I) [14], 424, 461, [462]. © K Q” Qª (v.) : (I) 1, [300], 532. ✏ h°J : ( VIII ) 191, 227, 244, 261, [266], 279, 299, © Q” . È J 531. ⇣È™444, ⇣KQ⇣Q”⇣K⇣ (s.) ✏K⌧⌦K.Q” 198, 384. †Òk : 232, (IV) 64, 552, 555. 324, 443, ⌘J⌦⌧J⌦k⌦m◊kk::: :(v.) P I ⇣»Ag 357. È™ : 31, 44, 45, 45, 61, 62, :531. 17, 34,37, 37, 63, 322, 576. p . YÈ J 108, 191, 363, 599, 618. : 17, 34, 63, 322, 324, . I ⌧ ◆ ⌦ ✏ ◊ . ◆YIJ ˙ : 11, 222.©K Q”⇣KQ⇣K ⌦⌘J⌦⌧Jm k: :198, °J 232, 384. ⇣IJ P ⌧⌦63, K⇣ 65,: ()111, 113, 122, 442. 108, 191, 599, 618. 443, 444,I 445,34, 577. È J k⌦kk::::(v.) 531. »Ag 357. 17, 37, 63, 322, ⇣.. 576. hIJ :34, () 398363, ; (X) 10. ◆.⌘⌦Q⌦444, ✏K⇣⌦KQ⇣KQQ”(v.) ˙◆324, : 11, 222. p ✏ ’ÊÖP I ⌧ K(v.) P 17, 37, 63, 322, 324, 443, 444, 576. ⇣ È™ ✏ †Òk (v.) : (IV) 64, 552, 555. ⌦ . ✏ © K Q” YIJ 108, 191, 363, 599, 618. I KP : : (II) 103, 519, 532,533, 540. 123, 441. ⌘J⌧Jk⌦m⌦◊kk:: ::357. ⇣ ⇣ (v.) (II)262, 103, 520, ⇣È J ⇣ . È™ K Q” (s.) : 31, 44, 45, 45, 61, 62, 63, 65, 111, 113, . © K Q” ⇣ ⇣ ◆h»Ag 17, 34, 37, 63, 322, 324, 443, 444, 576. °J 198, 232, 384. I ⌧ KQ K : 97, 191, 207, 227, 228, 261, 295, 631. . . I KP (v.) : (II) 103, 519, 532, 540. 531.: () 398 ; . (X) ⌦ 10.124 ⌧⌦541. KQ⇣ (v.) K⇣(v.)::: ()()(II) Translation ’ÊÖP 103, 519, 532, 540. . ✏K✏Q” p444, (v.) : (I) 571. ⇣’ÊÖP . ⌘JQ⌦⌧J⌦⌦kkk (v.) Y◆IJ 108,34, 191, (v.) © :: 17, 37,363, 63, 599, 322, 618. 324, 443, p 576. I ⌧ KQ K È™ K Q” ⇣hYÈ J . ⌦ . ⇣ . ✏ ⇣K(v.) ’ÊÖP (v.) : () Translation (v.) : (I) 571. :531. 108, 599,124 618. [532]. : 9. 191, ©’ÊÖP K.. Q”✏⌧⇣KP⌦⇣KQ(s.) I (v.) () 398363, ◆.⇣ J⌦@⌦Q⌧Jkm◊⌦kjk::JÉ@ : :97, 191, 207, 227, 228, 103, 519, 532, 540. 101. (v.) : :99, ()(II) ⇣ °J 198,⇣ :232, 384.; (X) 10. p ✏ 124 Translation ⌘YIJ 124 Translation È™ K Q” ✏ ⇣ . k : 17, 34, 37, 63, 322, 324, 443, 444, 576. ⇣ È J k : 531. È™ K Q” (s.) : 31, 44, 45, 45, 61, 62, 63, 65, 111, 113, 123, 441. [261], 295, :: 108, 363, 599, 618. (v.) :619. 18, 94, 101, 121,632. 147, 280, 281, : 104, 108, ⇣ 123. .191, I KP398 (v.) : 600, (II) 519, 532, 540. ✏K99, h◆. JQ@Q⌧J⌦⌦kjJÉ@ : 9. Translation ⌦() 12, (v.) :. () ;’ÊÖP (X) 10.124 ’ÊÖP (v.) ()106, ⇣ (s.) ::[262], 99, 101. p ✏103, È™ Q” I KP (v.) : (II) 103, 519, 532, 540. ’ÊÖP (s.) 99, 101. . ⇣’ÊÖP . ✏K⌧Q” 123. ⇣KQ(v.) ⇣K ::: 99, ’ÊÖP () Q⌧J@Q⌦kjkg⇣JÉ@ (v.) : 191, () 398363, ; (X) 10. ⌦ : 104, 108, p È™ 124 Translation (v.) (I) 12, 18, 94, 99, 101, Yhh◆..⌘JPA : 108, 599, 618. I 445. (s.) 115 ⌦ : 9.34, . . ’ÊÖP (v.) : () 101. ⇣ IJ k : 17, 37, 63, 322, 324, 443, 444, 576. X@ P (v.) : (IV) ’ÊÖP (s.) : 99, 101. hY. JQ⌧J⌦ k⇣(v.) : () 398 ; (X) 10. p 124 Translation I KP (v.) : (II) 103, 519, 532,281, 540. . ⇣106, 121, 147, 280, 281, : 445. 108, ⇣191, 363, 599, 618. p :(v.) ’ÊÖP (s.) : 99, 101. h..◆ @PAQkj⌦gJÉ@ :I 9.: () KP (v.) : (II) 103, 519, 532, 540. 398 ; (X) (v.)283, (II) 103, 532,92. 540. ’ÊÖP (s.) 99, 101. X@I (v.) ::: :(IV) ⇣ : 10. 284, . . .519, . (VIII) An ancient Greek treatise on magic squares 115 [email protected]⇣KPKP282, (v.) 230. hÈh⇣.K@@QPAkjg⇣JÉ@ : 9..:: () ’ÊÖP (s.) :::(IV) 99, 101. ⌦ 124 Translation ; (X) I (v.) (II) 103, 519, 532, 540. p (v.) (I)398 [399]. (X)10. 10. :⇣(v.) 6.445. . ’ÊÖP (v.) () : X@ P (v.) : (IV) 124 Translation .h. @QjJÉ@ : 9. (s.) : 99, 101, 577. 124 Translation 529, [530], 542,101. 581, 589. P X@’ÊÖP P (v.) : (IV) ⌦ : 196, 526, p 529, 124: 196,treatise Translation : () 398 ;X@P(X) 10. 398 ; (X) 10. An ancient Greek on magic squares 115 ::(v.) 6. hÈh⇣◆..K@@PAQQjkg⇣JÉ@ 445. 526, [530], 542, 581, 589. (v.) : ⌦ (IV) 17, 145,X@P189, 475, 561, 573, 579. : 9. (v.) : (IV) PP (v.)190, (IV) 17, 145, 189, 190, ✏@QQkjkgg⇣JÉ@ hÈ⇣.ìA 445. Ø (v.) 581;:: (IV) (II) [539]. : 9. QK@PA (v.) : () 398 ; (X) 10.: (I) 574, 580, X@’ÊÖP P476, (v.) :: 6. [528]. (s.) : 99, 101. 562, 574, 580. P X@. P P(v.) : (IV) :(v.) 445. 446.: () 398 ; (X) 10. h P ⇣Èhh◆hK@..ìA ✏. QPA Q@ kjkgg:⇣JÉ@ : 9. ⇣ [528]. h⇣.◆ PAg⇣:: 6.445. : 230. P ⌦ ⇣ hh. 77, PP 83, ⇣P 6. 445. hÈ⇣È⇣.K@ìA @PAQ✏⇣Jjkm◊ggJÉ@ : 9. 230. QÂî :::: 7. Ø (s.)116:⌦ Ø582, 584, 587, : :22, 69, 74,586, 91, 589. 95,Translation 99, 101, 114, . . . ; ( pl. È✏⌦P Ø ) . [528]. X@h77, : (IV) ⇣ P83,:h66, 68, 99, 72, 101, 78, 81, K@✏Q@Q⇣kj:⇣JÉ@6.: 9. . PP(v.) :: 230. Ø : ⇣Ø⇣⇣22, 69, 74, h 91, 114,85, . . . ;89, ( pl. È✏⌦ PØ ) 230. ⇣◆hÈhìA ⌦  P :95, 421, 476, 522. ⌦ 445. .È⇣◆K@. QPAJkgm◊g:::: 6.7.[528]. . . (v.) : (I) 574, 580, 581; (II) [539]. QÂî [529]. ⇣ : 230. ⌦ 68, 108, 122, 125, 207, 301, 562, 571. È⇣.✏J⌦522. k.. 375, ✏Q⇣k :: 6. 100, 106, .(II) . . ; [539]. (pl. ) 109,ÈJ⌦k.  P) h.  P⇣: 66, 72, 78,166, 81, 85, 89, 100, 106, .;P (pl. ✏°◆.ìA h  P Ø (v.) : (I) 574, 580, 581; h  P : 421, 476, ÈQÂî [528]. ⇣ P Ø : 199, 200, 201, 228, 232, 234, 360, 395, 399, 406, . . . . K@ . h PA g : 445. . k✏ Jm:◊ :8,7.8. h. 227. 581; P : (II) 421, 476, 522. : 230. : 3.⌦ P ØØ⇣::(v.) . 192, hhh201, PPP580, . : (I) 574, 580, [539]. (v.) : (I) 574, 581; (II) [539]. 199, 200, 234, 360, 375, 395, 399, 406, P ........ 228, 232, 234, 360, 375, 395, 399, 406, È⇣◆⇣⇣h✏ìA g : [528]. h  P : 421, 476, 522. :(s.) 230. .. Ø580, . P581; h584, Ph.586, ⌦ ØØ (v.) 582,574, 589. 445. XQ  :P587, 422, 442, [478], 524, [527], 528, 542. ÈÈ◆ìA K@. ✏kQPA⇣Jmk⇣g:◊g:8,::8,:7.6. h P h :: 421, 476, 522. :: (I) (II) [539]. 422, 477, 523. ° 8. 8. QÂî . . [528]. (s.) ::: 476, 582, 584, 586,619, 587,625, 589.629, 631. ≠  516, 543. h  P h  P :ØØØ ⇣(v.) ✏Q⇣JkJmg◊mg◊:::::6. 590, 605, 421, 522. (v.) (II) 590, QÂî 7. (v.) : (II) (I) 574, 580, 581; (II)442, [539]. . . h  P h  P :422, 421, 476,[478], 522. 524, [527], 528, 542. [528]. XQ Ø h  P : È⇣°K@✏ìA : 230. ¨C 192. . . . ⇣ ⌦ k : 8, 8. Ø(v.) :::(II) 582, 584, 589. XQ Ø605, h586, 116, P619, :587, 422, 442, [478], 524, [527], ØØ ⇣(s.) (s.) 582, 584, 586, 589. :3. 590, 625, 629, 631. :587, 423, 443, [479], 525, :⇣ 80, 230. Ø (v.) :⌦⌦ Ø(I) 96, 102, 118, 120, 122,524, 124, 129,528, 144,542. . h  P h  P : 421, 476, 522. (v.) (I) 574, 580, 581; (II) [539]. ◆⇣✏È⇣K@✏Q⇣J⇣mk◊◊:: 7.6. QÂî . . : :: 230.  P XQ Ø h  P :P587, 422, 442, [478], [527], 528, 542. Ø (s.) : 582, h 584, 586, 589. . ⇣ . h  P h  : 421, 476, 522. 517, 544. ÈQÂî ìA g : [528]. ⇣ ≠  J m 516, 543. : : 230. h586,  P589. : 442, 422, [478], 476, 522-3. :h.422, 524, [527], 528, 542. °¨C 8. 529, 543. ⌦ Ø (s.) 3. X XQ. Ø[528], . . P581; ◆✏kk✏⇣J⇣m◊::◊:8,8,:7.192. ::: 582, 584,580, 587,(II) ⇣QÂî ⇣ Ø (v.) (I) 574, ⇣ (v.) (I) [539]. ° 8. Ø h  P : 422, 442, [478], 524, [527], 528, 542. XQ :: 3. È◆✏ìAJgm g:::7.[528]. 3.:: 582, . XQXQ ØØ96,580, hh.586, PP581; :587, 442, [478], 524, [527], 528, 542. hh116, (II) P(II) ::118, 422, 476, 522-3. (v.) (I) 80, 574, [539]. 423, 477, 523-4. 192. X (s.) 584, ..422, ⇣(v.) ØØØ(v.) 102, 120, 122, 124, [527], 129, 144, P589. 422, 476, 522-3. (I) 574, 580, 581; [539]. : 3.: : (I) . ⇣ÈìA Ø h  P : 422, 442, [478], 524, 528, 542. XQ Ø (s.) : 96. °¨C k✏⇣ ◊:g8,: [528]. 8. . hXQ⇣XQ584, ØØ P580, h. PP581; P:587, :. 422, 421, 476, 522. X 80, Ø(v.) (v.) (I) 574, (II) [539]. h h  P : 442, 422, 476, (s.): : (I) 582, 586, 589. Ø 96, 102, 116, 118, 120, 122,522-3. 124, 129, 144, . QÂî J:mg8, ::: 7. . ✏°…kg@X ⇣ h [478], 524, [527], 528, 542. ¨C 192. ÈKXQ @Ø Ph102, 130, 232, 314, 384, : 3.: 198, 28, P 116, h.28,  P118, : 445, 422, 476, 522-3. 8. : 40, 40,108, 108,123, 123,126, 126,128, 128, 40, 42, 48, 49, 50,129,squares ⌦ancient ..:102, Ø(v.) (s.) 582, 584, 586, 587, 589. ✏ Ø : (I) 80, 96, 120, 122, 124, 144, An Greek treatise on magic ⇣ ⇣ Ø (v.) : (I) 80, 96, 116, 118, 120, 122, 124, 129, 144, ¨C g : 192. ◊ Ø (s.) : 582, 584, 586, 587, 589. ⇣ °k130, 8. 232, 314, QÂî Jm : :8,7.198, XQ⇣ Ø .h.. ..P h.ØØØ(v.) P::: 3.3.::422, 476, (s.) 96. 55, .treatise . 120, (s.) : (I) 582, 584, 586, 589. Ø@@52, 52, P Greek h587,  P .589. : . 422, 476, ÈKÈK⇣584, PPh522-3. 80,XQ 96, 102, 116, 118, 122,522-3. 124, 129, 144, ⌦ .586, . 384, An ancient magic squares ¨C g : 192. Ø (s.) : 582, 587, ⇣ ⌦ ⇣ XQ Ø h  P h  P : 442, 422, 476,on 522-3. Ø (s.) : 96. ✏°QÂî Ø : 195, 232, 627. ◊ …⇢'g@X : 40, 108, 123, 126, 128, 130, 198, 232, 314, 445, . . J m : 7. ⇣ Ø h  P : 422, [478], 524, [527], 528, 542. XQ ⇣ : 3.: : (I)(I)80,ÈKXQ @PØ P(v.) 8. Ø (s.) (v.) 116, 118, 120,476, 122,522-3. 124, 129, 144, ⇣ancient (v.) ..102, ⌦96, l . ⌦kPYg: K8,:: 192. 190. h  P h  P : treatise 422, ¨C ⇣ . Ø : 96. YK : (I) An Greek on magic squares ÈK @ P Ø (s.) : 96. ⇣ ⌦ : 3. ⇣195, (v.) : (I) 80,627. 96, 102, 116, 118, 120, 122, 124, 129, 144, ✏ k :g8,: 192. ⇣ @P : 28,⇣ØØ28, :(s.) 232, :40, ⇣48, °¨C 8. :3.96. ⇣195, ÈK 42, 49, 50, 52, 52, 55, . .on . . magic 67, 199, 202, AnYK ancient Greek treatise squares ⌦ ÈK @ : 3. ⇣ ⌦ Ø : 232, 627. ⇣ ⇢'✏.✏⌦kPY: K8,: 190. . Ø (s.) : 96. YK⌦⌦ PPP(v.) (v.) :: (I) (I) l¨C ÈK @ P ⇣ ° 8. Ø (v.) I : () 7, 299. ⇣ : 232, (I)An 80, 116, 120, 122, 124, 129, 144, g : :192. treatise on magic squares XQ  : Greek 422, 476, 522-3. »X (v.) (X) 420, 560, 631. ØØØ::(v.) YK P (v.) (I) 195, 627. 195, 232, 627. . 102, .  P :118, ⌦96, ÈK⇣⌦ancient

. ⌦. ÖQÂÖ ◆ ⌦ ÷fi⌦264, : 469. ⇣ËXAK ⇣ 124P : 18, 67, 97, 222, 227, 229, Translation ⇣ 246, 255, [292], 296, 298, . . . ; ËXAK P 571. ✏ ()419, ⌘Ö : ·93, ⌦ ⌘Ö⌘⌘Q Ȣ 525, 526, 544, ⌦563, Ö Z˙ Ê (v.) ::⌦™” :280. 104, 108, 123. ⌦ fi »A÷ : 469. ◆ Z˙ Ê Ö (v.) () 280. ⇣ P ⌦⌦ ⌘264, ⌘Ö40, ⇣55, 230. 18,»A÷ 67, 97, 222,48, 227, 246,Ä 255, [292], 296, 298, . . .544, ; ËXAK563, ⌘ÖQÂ(v.) ÈK⇣ËXAK @⌦PP:: 28, 28, 42, 49,229, 50, 52, 52, . ⌘Ö. :. .93, 67, 199, 202, ⌦ 571. È¢ 419, 525, 526, ⌦ : 224 124 Translation fi ⌦124 : 469. ºQÂ Ö : (III) 206, 591. ⌦ ⌘ fi »A÷ : 469. Arabic glossary Z˙ Ê Ö (v.) : () 280. ◆ Translation ⇣ËXAK ⇣ ⌘ Ä255, ⌦ fi⌘264, 124 Translation 196,67, 526, 542, 246, 581, 589. 97,529, 222,[530], 227, 229, [292], 296, 298, . . . ; ËXAK⌦ P »A÷ :(v.) 469. ºQ Ö⌘ÖÖ::(v.) :: (III) 206, 591. ⌦ P : 18, Z˙Ê 374, 375, 527, [568]. Z˙ Ê () 280. 124 Translation ◆◆ P Z˙Ê 374, 375, 527, [568]. ⌘ ⇣ËXAK ⇣ËXAK Ö⌘:(v.) : :196, 526,97, 529, [530], ⌦ fi ⌦P⌦ P(v.) »A÷ 469. 18, 67, 222, 227,542, 229,581, 246,Ä589. 255, 264, [292], 296, 298, . . . ; ⌘ ⇣YKËXAK ⇣. ⌦.⇣◆ P ¯QÂÑ” : [426]. ºQÂ Ö : (III) 206, 591. ⇣ ⌘ (v.) (III) 592. :574, 194, 201, 207, 212, 213, 214, ºPA Ç” : 211, 581, 589. (v.) : (I) (I) 193, : 18, 97,67, 222, 227, 229, 246, 255, [292], 296, 298, . . . ..; .. .ËXAK ⌦ ⌦ P(I) Z˙ Ê⌘fi210, Ö264, :: 375, () 280. Ø (v.)¯QÂÑ” 580, 581; (II) [539]. ËXAK P67, :⌘193, 18, 97, 222, 227,208, 229,209, 246, 255, 264, [292], 296, 298, ; ⌦ËXAK⌦ P Z˙Ê Ö ::(v.) 374, 527, [568]. ⌦ ◆ »A÷ 469. Z˙ Ê Ö (v.) : () 280. ⌦ ⇣ËXAK:208, ⇣ : 18, [426]. ⌘ ⌘ 209, 210, 211, . . . . Ä ◆ Z˙ Ê Ö (v.) : () 280. Q P : 67, 97, 222, 227, 229, 246, 255, 264, [292], 296, 298, . . . ; ËXAK ºPA Ç” : 581, 589. ⌘ ⌦ ( ⌦´ 375, ) : 582, 590. qJ Z˙Ê Ö :::⌦ P6. 374, 527, [568]. ⇣◆◆ ⌦ P ⌦⌦ÉÉ⇣ËXAK qJ 6. ⇣ËXAK⌦P : 18, 67, 97, 222, 227, 229, 246,Ä255, fi264, »A÷ 469. ⌘ ◆ ⇣i¢É ⇣ËXAK⌦ PP Z˙ Ê Ö (v.) : () 280. [292], 296, 298, ⌘: É374, ⇣ËXAK ¯QÂÑ” : :18, [426]. ⌘: ::581, 18, 67, ËXAK⌦⌦⌦ P ::(s.) 229, 246,Ä255, 264, [292], 296, 298, .. .. .. ;; ËXAK ⇢.⌦Ä 'ÉQ¢ : 67, 12. 97, 222, 227, 229, ºPA Ç” 589. 230. l [426]. ⌦ ⌘ [427]. Z˙Ê Ö 375, 527, [568]. P ⌘ qJ : 6. ⌦ ◆ Z˙ Ê Ö (v.) : () 280. ✏⌦ 116 Translation ⇣ ⌘ (s.) : 12. : 230. ⌘ Ø (s.)i¢É : 582, 584, 586, 587, 589. ⌘ Ø : :22, 69,.Ö. :74, 77, 91,362, 95, 99, . ;375, ( pl.527, Ø) ⇢. ⌦'114, ¯QÂÑ” [426]. ⌦242, ËXAK Ä101,lZ˙Ê ⌦ [568]. 246, . ; 374, 356, Q¢ É374, :. .[426]. Z˙Ê 375, 527, [568]. ⌦ P 83, ê qJ Éfi⌘Ö⌘⌘ÖÖ :::(v.) 6. Ä74, ✏ ⌦ 471n. »A÷ 469. Z˙ Ê : () 280. ê : 230. Ø : 22, 69, 77, 83, 91, 95, 99, 101, 114, . . . ; ( pl. Ø ) : 230. 118 Translation ⌦ Z˙Ê Ö : 374, 375, 527, [568]. 396, 402. Ä ⌦ ⌦ ⌘ ⌦ ¯QÂÑ” : [426]. ⇢ ' i¢É (s.) : 12. Ä ⌘ Ø (v.) : (I) 574, 580, 581; (II) [539]. l Q¢ É : [426]. Q¢É : 14, 15, 18, 19, 19, 574, 580, 583, 599, 614. ¯QÂÑ” : [426]. ¯QÂÑ” [426]. qJ : 6. : 230. ⌦ :Ø122, ê 118 Translation Z˙Ê :614. 374, 375, 527, [568]. (v.) : (I) 280. 108, 125, 166, 207, 301, 562, 571. 599, AJ✏⌦. É⌦⌘ÉÉ⌘Ö⌘375, Ä Ä (v.) : (I) 574, 580, 581; (II) [539]. Q¢É 14, 15, 18, 19, 19, 574, 580, 583, 120 Translation A Ø : 199, 200, 201, 228, 232, 234, 360, 395, 399, 406, . . . . i¢É (s.) : 12. qJ : 6. ¯QÂÑ” [426]. ï XQ Ø ⇡ ✏ ⌦ : 230. : [426]. ⌘ llZ˙ :: () 11. A : 118 3.⌦ ØØ (v.) ⌘ :574, ï⌘Ö (v.) ⇡375, ê (v.) () 280.527, Z˙Ê 374, 375, () 11. [528], qJ É200, 6. :: 12. (I) 580, 581; (II) [539]. Translation (v.) (I) 574, 580, 581; (II) [539]. A :Ø:::(s.) 199, 201, 234, 360, 395, 399, 406,[568]. ..... .. 228, 232, 234, 360, 395, 399, 406, .[569]. ⌦ qJ ÉÊ375, :::614. 6. ¯QÂÑ” :14, [426]. i¢É :15, :(s.) 230. ⌦ Q¢É 18, 19, 19, 574, 580, 583, 599, : [426]. i¢É (s.) : 12. : 582, 584, 586, 587, 589. ⌦ : [427]. …¯QÂÑ” ÆÉ@ 49, 56, 58, 68, 74, 79, 88, 90, . . . ; ✏ i¢ÉØ (v.) (s.) : 12. (I) 574, 580, 581; (II) [539].l⇡ï ⌘(v.) : () 11. ê qJ⌦É :614. 6. 118 Translation Ø (s.) : 582, 584, 586, 587, 589. … ÆÉ@ : 49, 56, 58, 68,580, 74,574, 79, 88, 90, . .↵.599, ;631. Q¢É : 14, 15, 18, 19, 19, 580, 583, Ø (v.) : (II) 590, 605, 619, 625, 629, ✏ (II) 590, i¢É (s.) 12. ï ê ⇡ ®PA Ø (s.) : (s.) : 12. (I) 574, 581; (II) [539]. (Ø …(v.) ÆÉ@ ·” ) 142, 144, 491, 493; ( … ÆÉ@ ) 157. ï : ⇡⌦⌦mmÉ⇡⌘Ö⌘:ï(v.) iJ lZ˙Ê : ()375, 11. 527, [568]. : 230. ↵ qJ : :374, iJ :6.526. 526. ⌦ Ø (s.) : 582, 584, 586, 587, 589. Ø (s.) : 582, 584, 586, 587, 589. Ø (v.) : (II) 590, 605, 619, 625, 629, 631. : 230. Ø (v.)Q¢É : (I) 80, 96, 102, 116, 118, 120, 122, 124, 129, 144, i¢É (s.) : 12. ê (v.) :56, 574, 580, 581; (II) [539]. :(I) 14, 15, 19, 19,88, 574, 580, 583, 599, 614. ⌦⌦ (:ØØ:A:…(s.) 118 Translation 14, 18, 19, 19, 574, 580, 583, 599, 614. …Q¢É ÆÉ@ 68,18, 74, 79, 90, . .ÆÉ@ ;599, ÆÉ@ ·” )58, 142, 144, 491, 493; ( .… 157. : 230. 3. i¢É :15, 12. :49, :Q¢É ⇡✏⌦mï⇡)ï(v.) ê (s.) :15, 582, 584, 586, 587, 589. 14, 18, 19, 574, 580, 583, 614. 14, 15, 18, 19,19, 19, 575, l120 : () 11. Translation iJ : 526. ↵ : 230. 118 Translation ê ✏ ï)(v.) A:…(s.) : 3.56, 120 Translation ⇣⌦581, ÆÉ@ 49, 58, 68,[570]. 74, 88, 90,583, . .ÆÉ@ ;⇡ 584, 600, 601, . . . 574, .79, (Ø:Ø(v.) ÆÉ@ ·” )574, 142, 144, 491, 493; (. … 157. l : Ø()(v.) 11.:91, 14, 15, 18, 19, 19, 580, 599, 614. QÂÑ (II)405, 583. : 582, 584, 586, 587, 589. ï ⌘ °…Q¢É ÆÉ : (IV) 569, ⇡ I kAì : 10, 90, 118 Translation iJ m : 526. (v.) : (I) 580, 581; (II) [539]. qJ É. .:614. 6.: 10, 90, 91, 405, 583. (v.) : (I) I ::…3. ê ↵90, 3. ⇣ ØA::ØA(v.) ⇡✏.. ⌦ï.kAì (v.) (I) 574, 580, 581; (II) [539]. 120 Translation ÆÉ@ : 18, 49, 56, 58, 68, 74, 79, 88,.120, ; 124, Q¢É 14, 15, 18, 19, 19, 574, 580, 583, 599, (s.) :::56, 582, 584, 586, 587, 589. l (v.) : () 58, 58, 68, 74, 79, 88, (I)11. 11. … ÆÉ@ : 49, 68, 74, 79, 88, 90, . . ; (v.) : (I) 80, 96, 102, 116, 118, 122, 129, 144, ° ÆÉ : (IV) 569, [570]. 118 Translation Q¢É 14, 15, 19, 19, 574, 580, 583, 599, 614. ( … ÆÉ@ ·” ) 142, 144, 491, 493; ( … ÆÉ@ ) 157. Ø (v.) (I) 574, 580, 581; (II) [539]. ï ⇡ A : 3. ✏ 56, Ø (s.) …: ⇣ÆÉ@ 96. : 49,⇡ 58, 68, 74, 79, 88, 90, . .↵ I .iJ :: 526. ↵ (v.) 10, 405, 583. Translation ïm122, ï582, Ø(v.) (v.) (I) 574, 581; (II) [539]. 120 l.⇡;✏(⌦kAì : 157. ()90, 11.91,144, (s.) 584, 586, 587, 589. l (v.) :)580, () 11. Ø :; :((I) 80, 96, 102, 116, 118, 120, 124, 129, … ÆÉ@ ·” ) 142, 144, 491, 493; … ÆÉ@ ) ↵ ⇣ ï 90, . . . ( 142, 144, : 49, 56, 58, 68, 74, 79, 88, 90, . . . ; ⇡ iJ m : 526. ( … ÆÉ@ ·” ) 142, 144, 491, 493; ( … ÆÉ@ ) 157. ° ÆÉ (v.) : (IV) 569, [570]. :49, 3.: 56, …ÆÉ@ ::… 58, 68, 586, 74, 79, 88, 90,(. … . .PYì ;✏ ⌦kAì :[583]. [527]. †A ÆÉ@ [572]. (s.) 582, 584, 587, 589. I :124, 10, 90, 91, 405, 583. Translation ï122, ÆÉ@ ·”An ) 142, 491, 493; ÆÉ@ )ï:(v.) 157. ⇡. 122, PYì :on [583]. ↵80, ancient Greek treatise magic 125ê ØØØ(AØØA(v.) :3. (I) 80, 96, 102, 116, 118, 120, 124, 144, l : ()129, 11.squares (v.) ::56, (I) 96, 144, 102,79, 116, 118, 120, 129, 144, 120 ⇣ ↵ ⇡ (s.) 582, 584, 586, 587, 589. ⇣ : iJ m : 526. … ÆÉ@ : 49, 58, 68, 74, 88, 90, . . . ; : 3. 492, 494; ( ) 157. …ÆÉ@ ·” ) 142, 144, 491, 493; ( … ÆÉ@ ) 157. (s.) : 96. ⌦ †A ÆÉ@ : [572]. … ÆÉ@ : 49, 56, 58, 68, 74, 79, 88, 90, . . . ; Ø (s.) : 582, 584, 586, 587, 589. ° ÆÉ (v.) : (IV) 569, [570]. 80, 96, 144, 102,491, 116,493; 118, 122, 124, 129, 144, …(s.) ÆÉ@⇣:: (I) ·” ) 142, (↵ 120, …PYì ÆÉ@ 157. …ÓDÖ@Ø(Ø:(v.) 21. I 91, 405, 584. 583. ↵ 10, 90, 91, 406, An ancient Greek treatise magic squares 125 [583]. . ⌦kAì ⇡):ïon 582, 586, 587, 589. iJ m :: 526. ↵ Ø (s.) :: 627. 96. : 195, 232, ° ÆÉ (v.) : 584, (IV) 569, [570]. ⇣ ⇣ …ÆÉ@) Ø157. ⇣ ï ( … ÆÉ@ ·” ) 142, 144, 491, 493; ( … ÆÉ@ ) 157. ⇡ ° ÆÉ (v.) (IV) 569, [570]. 10, 90,129, 91, 405, ⇣ (Ø(v.) …ÓDÖ@ :… 21. 80, 96, 102, 116, 118, 122, 129, 144, †A ÆÉ@ :(v.) (v.) :() [570], [571]. m(I)94. :ancient 526. : (I) 96, 102, 116, 118, 122, 124, 144, 583. 166, 189, ÆÉ@ ·” ) 80, 142, 144, 491, 493; ( 120, …120, ÆÉ@ ÈKI :::124, 20. Ω É ::iJ :[572]. 3. PYì ïkAì ⌦(IV) [584]. ï:157. ⇡)on °ÆÉ (v.) (IV) [570]. ÈK⇣⇡✏.. Ò™ì Ò™ì 20. liJ (v.) : () 11.squares An569, Greek treatise magic 125 ØØØAA(s.) m :[583]. 526. (s.) :::96. 96. ⌦ : 3. (v.) (I) 80, 96, 102, 116, 118, 120, 122, 124, 129, 144, I kAì : 10, 90, 91, 405, 583. (v.) : (IV) 569, [570]. ⇣ :(s.) 195, 232, 627.327, .ÈK Ò™ì: 373, Ω É ::3. ()96. 94.[318], 250, 356, 358, 368,PYì 371, [445], 447, 578, 579, . . . :[573]. ⇣ †A ÆÉ@ ::21. [572]. :A254, Ø(v.) °AÇ” ÆÉ (v.) :290, (IV) 569, [570]. …ÓDÖ@ 105, 402. [583]. ⇣ ::: 20. 20. ancient Greek treatise on magic 125 . . kAì :†AQ3. I 10, 90,squares 91, 405, 583. ÆÉ@ :An [572]. ⇣⇣ ØØ A:(v.) 195, 232, 627. .◆ AÇ” Æì@ :569, 93. ⇣ ⇣ ° ÆÉ : (IV) [570]. PYì : [583]. (s.) 96.  : 105, 402. ⇣ †A ÆÉ@ : [572]. Ω É (v.) : () 94. (v.) : (I) 94. Q ™ì@ ° ÆÉ : (IV) 569, [570]. É Ø (v.)’Œ…ÓDÖ@ I :ÆÉ@ 299. :(v.) :7, (II) 10. : 232, (I) 80, 96, 102, Greek 116, 118, 120, 129, 144, ÈK™ì@ ::124, 20. ⇡ïon An ancient treatise magic squares Q583. : :24, 25,129, 26, 27, 29, 30, :21. [572]. . ⌦Ò™ì iJ m122, 526. ØØ⇣(v.) 195, 627. I kAì 10, 90, 91, 405, 583.34, 125 ◆†A ØØØ:() :(v.) 195, 232, 627. ↵ I kAì :299. 10,102, 90,116, 91,118, 405, . :I. 96. (I) 80, 96, 120, 122, 124, 144, : [572]. (s.) 96. (s.) :(II) PYì : [583]. ⇣ ⇣ (v.) : () 7, É Ø (v.) : (I) 80, 96, 102, 116, 118, 120, 122, 124, 129, 144, ’Œ (v.) : 10. :: (II) AÇ” :::493; 105, 232, 627. Ω É (v.) ()402. 37,on 40, [44], .squares .. . ”) 142, 144, 491, (94. …10. ÆÉ@ ) 157. 116, 118, †A :195, [572]. AÇ⇣ÆÉ@ ⌧”ØØ(v.) 107, 511. : 124, 20. An ancient treatise magic 125 QÈK™ì@ Æì@ 93. QQPYì .✏Ò™ì …ÓDÖ@ :(v.) 21. Ω É (v.) : () 94. 102,Greek 80, 96, 120, 122, 129, 144, :: 93. [583]. ◆†A ⇣ Ø (v.) I: :(I) ()An 7, 299. ⇣ Æì@ : ⇣ ✏ ⇣ ancient Greek treatise on magic squares ÆÉ@ : [572]. ÈK Ò™ì : 20. Ø : 195, 232, 627. : 195, ’ÊÖ@ AÇ ⌧” : 107, 511. É Ω É (v.) : () 94. : (row) 49, 51, 57, 58, 58, 125 . kAì ’Œ…ÓDÖ@ :I (II) 10. 21. †A ÆÉ@ :105, [572]. AÇ” :457, 402. Ø(v.) (s.) 96. 462, 499. Q≠ì ™ì@ 458, 463, [463], 500. 604. ≠ì J  í :299. 602, ⇣ Ω É (v.) 94. I : 10, 90, 91,490, 405, 493. 583. Ø:Ø⇣:Ø(v.) I:::I462, :. :() () 7, 299. PYì : :[583]. (v.) () 7,57, ⌦ () 94. Ø : 27,◆…ÓDÖ@ . (s.) 96. 48, 52, 55, 60, . . . ; ( Ø ) 135,151, Q Æì@ 93. : 195, 232, 627. ÈK Ò™ì 20. 21. ✏ 60,):⇣64 (border, see493. also 201), Ø :(v.) 27, 48, 52, 55, 57, 60, . . . ; ( Ø 135,151, 490, . (s.) : 96. : 457, 462, 462, 499. É PYì : [583]. Ø (v.) I : () 7, 299. Q’ÊÖ@ AÇ ⌧” : 107, 511. ’Œ◆…ÓDÖ@ (v.) : (II) 10. I : () 7, 299. QI ™ì@ KAÉ 205, 216, 418, 558. () 94.: 243, J⌦ í  í:)⇣71, : 602, 604. ’Œ:É: 48, (v.) Ω É AÇ”⇣Ø: (v.) 105, 402. 96. 93. ÈK⇣Æì@ 20. :::(s.) 21. . ✏Ò™ì 70, 89, 604. 90,490, 91, 493. ... . 21. 27, 52,(II) 55,10.57, 60, . . . ; ( ØQ≠ì 135,151, (IV) 569, [570]. I J :: 602, ◆KAÉ ⌦ Q ™ì@ . Ø (v.) I : () 7, 299. Ω É : () 94. ’Œ’ÊÖ@ (v.) : (II) 10. QΩ É : 205, 216, 243, 418, 558. É ⇣ AÇ” : 105, 402. : 457, 462, 462, 499. AÇ ⌧” : 107, 511. (v.) : () 94. Ø 195, 232, 627. ≠ì ✏ ÉØØØ(v.) ⇣135,151, :(v.) 52, 55, 135,151, 490, 493. [583]. : I48, (II) 10. 27, 48, )))::135,151, ÈK™ì@ : 20. 490, 27, 48, 52, 55, 57, 57, 60, 60, ......;; (( ØØQPYì 490,493. 493. (II) 10. QI Æì@ 93. .✏Ò™ì ::::: 27, 195, 232, : 402. () 7,627. 299. J  í : 602, 604. ◆’ŒKAÉ AÇ” 105, J Ø : [426], [429], [431], [433], [435], [437], [439], [440]. 105, 403. ⌦ ✏ I  í” : 603. . ⌘ Ø 195, 232, 627. ⌦ Ø : 27, 48, 52, 55, 57, 60, . . . ; ( Ø ) 135,151, 490, 493. ⇣ Q’ÊÖ@ : 205, 216, 243, 418, 558. J⌦Ø : [426], [429], [431], [433], [435], [437], [439], [440]. .: Ò™ì 73, 76, 202, 217, 218, 257, [276], [277], 296, 316. 457, 462, 499. Ä 250, ÉJJ:Ø⌧” ≠ì ⇣ : 602, : ⇣462, (II) 10. ’ÊÖ@ 457, 462, 462, 499.221, 235, ’Œ◆ AÇ” AÇ 107, 511. ✏J⌦✏ í” I  í” 603. Ø(v.) 195, 232, 627. QI ™ì@ 603, 605. 105, 402. I  í[439], 604. ÈK : [431], 20. ::: :[426], [429], [435], [437], [437], [439], [440]. . [433], [435], [440]. . . ⌘ : 603. ◆KAÉ ≠ì ÉAÇ : 27, 48, 52, 55, 57, 60, . . . ; ( Ø ) 135,151, 490, JÉJ⌦ØØØ:Ø⌧”⇣(v.) :(v.) 73, 76, 202, 217, 218, 221, 235, 250, 257, [276], [277],493. 296, 316. ◆Q’Œ’Œ’ÊÖ@ Ä :: (II) (II) 10. . ⇣ : 107, 511. 457, 462, 462, 499. : 205, 216, 243, 418, 558. (v.) I : () 7, 299. 512. ✏ 10. ⇣ ✏ : [426], [429], [431], [433], [435], [437], [439], [440]. ÈK Ò™ì : 20. Q ™ì@ 457, 462, 462, 499. J : [426], [429], [431], [433], [435], [437], [439], [440]. ⇣ ⌦AÇ . I J  í : 602, 604. ◆’ÊÖ@ Ø (v.) I : () 7, 299. ⌦ :(v.) 27, 52, 55, 57, 60, . . . ; ( ⌘ Ø≠ì ) 135,151, 490, 493. ⌦ 257, ◆ÉJ⌦⌘Ø⇣⌧”Ø(v.) 604. I :: 603. :[426], 107, 511. I48, : [429], ()202, 7, 299. ..✏  í” [431], [433], [435], [437], [439], [440].[277], 296, 316. Ä[37], ⇣ 73, 76, 217, 218, 221, 235, 250, [276], ⇣ ÈJ :76. (IV) 383, 529. ◆ AÇ 457, 462, 462, 499. : .◆KAÉ Q’ÊÖ@ 205, 216, 243, 418, 558. Ø⌘ ⌧”:A ::(v.) I : () 7, 299. ËPÒì 17, 22, 31, [35], [39], 108, 109. ËPÒì 17, 22, 31, [35], [37], [39], 108, 109. ≠ì :: 205, 419, 559. 107, 511. ⇣ËPÒì I  í”[439], :17, 603. . 22, 31, [35], [37], 108, 109. ✏ ⌘ ::457, [426], [429], [431], [433], [435], [437], [440]. ◆ÈJKAÉ : (IV) 383, 529. : 17, 22, 31, [35], [37],. 316. ’ÊÖ@ 457, 462, 462, 499. Q ™ì@ : 24, 25, 26, 27, 29, 30, 34, 37, 40, [44], .[39], . . 436, 541. 73, 76, 202, 217, 218, 221, 235, 250, 257, [276], [277], 296, ✏ Q’ÊÖ@ 205, 216, 243, 418, 558. : 76. Ä . ÉJ⌦JØ::ØAØ(v.) : 27,462, 48, 52, 57, 60, . . . ; ( ØQ≠ì ) 135,151, 490, 493. 462,55, 499. ™ì@ ⇣ Ø : 27, 48, 52, 55, 57, 60, . . . ; ( Ø ) 135,151, 490, 493. ⌘ I  í” : 603. J Ø : [426], [429], [431], [433], [435], [437], [439], [440]. 108, 109. (v.) : (III) 79, 90, 91, 129, 134, 136, 138, 140, 142, 144, 149, ◆ÈJQ.KAÉ 94. (v.) : (III) 79, 90, 91, 129, 134, 136, 138, 140, 142, 144, 149, 151, . 257, 205, 216, 243, 418, 558. ⌘⌦ Ø:::: 104, 22, 31,[277], [35],493. [37],316. [39], 108, 109. Ä 27, 48, 52, 55, 57, 60,221, . . . ; 235, ( ⌘ 250, Ø ËPÒì ) 17, 135,151, 490, 76, 202, 217, 218, [276], 296, :76. (IV) 383, 529. 105. :73, ⌦◆É⌧.ÉJJAØ⌘ØA(v.) Ø⌘:(v.) :205, 48, 52, 55, 57, 60, . . .134, ; (. . Ä ) 135,151, 490, 493. :27, 253, [431], 437. : 156, (III) 79, 90, 91, 129, 138, 140, 142, 144, 149, Q153, KAÉ 216, 243, 418, 558. 154, 158, 160, 162, 164, .136, .ØQQ⇣ËPÒì ì (v.) : (I) 93, [319], 358, 373, 17,: 22, 31, [35], [37], [39], 108, 109. ⌘[437], Q79, ✏105. ⌘ÉØ⌦ØØA (v.) (v.) (I) :: 73, 76, 202, 217, 218, 221, 235, 250, 257, [276], [277], 296, 316. JØ (v.) ÈJ:⌦. É⌧J. J(III) 90, 91, 129, 134, 136, 138, 140, 142, 144, 149, 104, ì (v.) :437. (I) 93, [319], 358, 373, 411, 414, 591, Ä ⌦⌦ì✏138, 530. :76. (IV) 383, 529. [426], [429], [431], [433], [435], [439], [440]. ⌦ : 253, [431], (v.) : (III) 79, 90, 91, 129, 134, 136, 140, 142, 144, ⇣ J (v.) : (III) 79, 90, 91, 129, 134, 136, 138, 140, 142, 144, 149, 412, [415], 592, 604.149, (II) 316. 251, ≠ì ≠ì JJ⌦J⌘ØØØ:(v.) [433], [435], [437], [439], [440]. ⌘ 73, 76,[429], 202, [431], 235, 250, 257, [276], [277], 296, ◆ :: [426], 217, 218, 221, 235, 250, 257, [276], [277], 296, 316. ËPÒì 17, 22, 31, [35], [37], [39], 108, 109. Ä136, Q [426], [429], [431], [433], [435], [437], [439], [440]. ì (v.) : (I) : (III) 79, 90, 91, 129, 134, 138, 140, 142, 144, 149, ⌦ ÈJ ⌧ É : 104, 105. ⌦ 251, 252, 411. : 104, ⌘ hQÂ Ö (v.) : (I) 302, 378, 396. A : 76. ⌘ 10. 383, 529. ÈJ⌦◆.. É⌦JAØA (v.) : (IV) 383, 529. :J :[426], [429], [431], [433], [439], [440]. [431], 437. ê :253, 583, 596, 599. [435],Ä[437], Q (v.) : (I) 142, 144, 149, (v.) :[570], (III) 79, 378, 90, 91, 140, ê hQ ::(IV) (I) 396.129, 134, 136,⌦ì138, (v.) (I) 302, 302, :253, 383, 529. ::: 76. ÈJ⌦◆.⌧.ÉJÉJ⌘⌘ØØAA⌘Ö(v.) :J(v.) 104, 105. [431], 437. [570], 583, 596, 599. (v.) (III) 79, 90, 129, 134, 136, ê 144, 149, Q⌦ì138, : 76. (v.)140, : (I)142, ÈJ.É⌘ÖQÂA⌘(v.) ::: (IV) 383, 529.91, ⇣È¢ ê ⌘ hQ (v.) (I) 302, 378, 396. Ö : 93, 419, 525, 526, 544, 563, 571. : 93, 420, 526, [527], 545, ✏ ⌘ ⌦ A 253, [431], 437. : 104, 105.383, ÈJ⇣ÈJ.⌦É⌧Ø. ÉA:(v.) : (IV) 529. J : [570], 583, 596, 599. XA í” : 19. ✏ ê 6, 99, 114, 129, 145, 166, 189, 225, 327, 358, 373, 388, 394, 475, 504, ⌘Ö :104, È¢ 93, 419, 525, 544, : 19. XAí” [572]. J⌦ÉØØA⌘Q (III) 79, 90,526, 91, 129,563, 134,571. 136, 138, 140, 142, 144, 149, ÈJ559, 105. hQ Ö⌘ :(v.) (v.) : (I) 302, 378, 396. ⌦⌧Ø.564, 253, [431], 437. J :[570], (III) 79, 90, 91,599. 129, 189, 134, 225, 136,✏ 327, 138, 358, 140, 373, 142, 144, 394, 149, A J ::99, [570], 583, 596, : 6,.:104, 114, 129, 145, 166, ✏ ⇣È¢ ⌧ É : 105. ÈJ J Ø (v.) : (III) 79, 90, 91, 129, 134, 136, 138, 140, 142,388, 144, 149,475, 504,ê ⌘ : 93, 253, [431], 437. ⌦559, .⌦J⌘ÖQÂØ⌘A⌘(v.) XA í” : 19. Ö : 419, 525, 526, 544, 563, 571. XA í” : 19. ºQ : (III) 206, 591. (v.) :: (III) 79, 378, 90, 91, 140, 142, 144, 149, .104, (I) 302, 396.129, 134, 136, J(v.) :99, [570], 583, 596, ÈJ⇣hQ 105. ⌦⌧.ØÉ⌘Ö:A:⌘Ö:(v.) H327, QÂ138, ï: 358, (358, ·”))373, :373, ( 388, Õpo) 12, 6, 114,206, 145,599. 166, 189, 189, 225, 225, 327, 388,394, 394,101, 475,511, 504, 129,591. 145, 166, 475, 504, XA✏H í” 19. . ⌘ ºQ : (III) Q ï ( ·” : ( Õpo) 12, 101, 511, hQÂ Ö (v.) : (I) 302, 378, 396. È¢ QÂ Ö : 93, 419, 525, 526, 544, 563, 571. . ⌦ 302, 378, 396. A J : [570], 583, 596, 599. ✏ 559, . 559, . ⌘ :(v.) 295. Ø⌘Ö ØA:⌘Ö(v.) 99, 114, 129, 145, 166, 189, 225,XAí” 358, 373, 388, 394, 475, 504, hQ :(III) (I) 302, ⇣559, J6,⌘Ö.:[291], :581, 583,378, 596,396. 599. H.327, QÂï: (19. ·”) : ( Õpo) 12, 101, 511, ºQ : [570], 206, 591. ºPA Ç” 589. ⌘ È¢ Q : 93, 419, 525, 526, 544, 563, 571. : (I)295. 302, 378, 396. [291], ØØ⌦Ö :(v.)

✏ í” ÈØQ:Q: 19. An ancient on magic squares 123 19. (v.) : (V)414, [527], 562. Greek treatise XA✏XAí” ⇣K: :magic P»XA™” Y´ (v.) (V) [527], 562. 562. X : 373, ⇣ 373, 414, P Y´ (v.) :: on (V) [527], An ancient Greek treatise on squares 121 …K Y™ [266], [266], 590. êQ´ : 11. ⌦ Q : 11. An ancient Greek treatise magic squares 123 È ØQ 1, 247, 279. An ancient Greek treatise on magic squares 123 ✏ í” (v.) :An (V)ancient [527], 562. ⇣ : 19. † êQ´ : 11. Greek treatise on magic squares 121 ⇣ ✏XAXAí” 1, 247, 247, 279. P…K Y´ (v.) : on (V) [527], 590. 562. »XA™” 373, 414, An279. ancient Greek treatise magic squares 123 K:: 11. [266], [266], ::: 11. 1, êQ´ ⌦ Y™on Arabic glossary 225121 ⇣ÈÈØQØQïï:Q(19. : An (V)Õpo) [527], 562. An ancient Greek treatise magic squares H.. QÂQ ((v.) ·” ) : ( 12, 101, 511, @ : 108. An ancient Greek treatise on magic squares 123 ancient Greek treatise on magic squares 121 H ·” ) : ( Õpo) 12, 101, 511, 1, 247, 279. êQ´ 11. ⇣Y´ :: 11. PêQ´ (v.) : 247, (V)414, [527], 562. :::: 11. »XA™” 373, ⇣ÈØQQ (v.) : An (V)ancient [527], 562. Greek treatise on magic squares 121 È ØQ™” 1, 279. Greek treatise on magic squares 121 123 : 108. 1,: An 247, Greek treatise magic squares H. QÂ⇣ ïQ(@·” ( An Õpo) 12, 511, QÍancient £279. :ancient 4. 101, È⇣Y´ ØQ™”on : 11. 1,: 247, 279. 562. : ))11. ⇣ êQ´ : P (v.) (V) [527], H Q ï ( ·” : ( Õpo) 12, 101, 511, »XA™” : 373, 414, , Gr. ὁ ὑπό : 12, 101, @ :::: 226. 108. È⇣ØQ™” 1, 247, 279. squares ˘™J : 16. 1, 247, 108. An279. ancient Greek treatise on[415]. magic 123 ⌦ÈJ.ØQ£Qî” H... Q Q 226. H : 11. ⇣ ⇣ÈØQ î”(v.) 512, 518, 518, 519, 534, . . . . : ) 245, [486]. È ØQ™” : 1, 247, 279. @ :: 108. êQ´ : 11. È ØQ™” 1, 247, 279. P Y´ (v.) : (V) [527], 562. 1, 247, 279. —¢´@ : 108. ®[528], 563. ˘™J : 16. :: 11. H. Q 226. ⌦⇣J.£Qî” ⇣ØQ™” @ : 108. — ¢´@(v.) 108. (v.) : ) 245, [486]. H QÂ î” :: 226. È 1,: 134, 247, 279. 562. êQ´ ::: 11. È ØQ 1, 247, 279. P Y´ (V)514, [527], .˘™J ¨Q£ : (Ìroc) (number) 110, 113, 114, 124, 128, — ¢´@ 108. (s.)©  : 12, 44, 117, 118, 130, 146, 168, 192, 192, © ìì⇣⌦J.£:: 457, 459, 462, [462], 511, 512, 512, 513, 513, 514, :457, 16. (v.) : ) 245, [486]. An ancient Greek treatise magic squares 459, 462, [462], 511, 512, 512, 513, 513, 514, 514, .. ..195, .. .. on300, 458, 463, [486]. [463], 500, @ : 108. (v.) :460, ) 245, ⇣—È—ØQ™” È ØQ : 1, 247, 279. ¢´@ : 108.  1, 247, 279. : ¢´@ : 108. 11. @(v.) :16. 47, 52, 55, 58, 68, 72, 78, 82, 88, . . 168, .[486]. ´513, ˙⇣⌦J.12, ˘™J :(v.) :459, )() 245, ’ŒêQ´ (v.) : 192, ) .245, :´ 245, [486]. 401, 413. (Ìroc) (number) 113, 114, 124, 128, 134, (s.) 12, 117, 118, 130, 146, (s.)¨Q£ :512, 44, 117, 118, 130, 146, 168, 192, 300,195, 300, I j:. [486]. ´[462], (v.) : 511, () 4. @513, :: …‘ 108. ©  457, 462, 512, 512, 513, 514, 514, 192, . 195, . . . 192, 513, 514, .57, . .110, .44, . 279. ÈQ£ØQ:£:::457, 1,459, 247, ⇣ ´ ⇣áKìì ’Œ (v.) : ) 245, [486]. ©  462, [462], 511, 512, 512, 513, 513, 514, 514, . . . . (v.) : ) 245, [486]. — ¢´@ : 108. È ØQ™” 1, 247, 279. @ : 47, 52, 55, 57, 58, 68, 72, 78, 82, 88, . . . . 94. ˙ êQ´ : 11. ⌦ ⌦⌦ì⌦ìJ.£:(v.) ≠J (v.) (IV) 401, 413. ’Œ´ 130, (v.) : 134, ) 245,168, [486].192, 192, 195, 300, (v.) 402, [414]. ¨Q£ (number) 110, 113, 114, 124, 128, ´(IV) @16. : :::108. ˘™J :(Ìroc) ≠J (IV) 401, 413. (s.) : [486]. 12, 44, 117, 118, 146, @ :: …‘ 47, 52, 55, 57, 58, 68, 72, 78, 82, 88, . . . . (v.) : ) 245, : 47, 52, 55, 57, 58, 68, 88, . . . . ⇣ : 230. ´ ⇣áK ˙ 47, 52, 72, 78, 82, 88, . . . . 413. (v.) : (I) 222, [223]. ✏I.k(v.) ’ŒÈØQ™” (v.) 245, [486]. ´´@(v.) ¢´@ :: 108. ˘™J : @(Ìroc) 16. ’Œ—˙Œ ::: )192, [486]. (v.) ()247, I)245, 245, 487. ¨Q£ (number) 113, 114, 124, 128, 134, : 108. 1, 279. ⌦ìJ.£:(v.) 94. Y´ : 110, (I) 321, 324. ≠J : :(IV) 401, (s.) 12, 44,.88, 117, 118, 130, 146, 168, 192, 195, 300, ⌦˙Q£ (v.) () 65, 102, 363, 370, 425. ⌦ @ : 47, 52, 55, 57, 58, 68, 72, 78, 82, 88, . . . . † (v.) : ) 245, [486]. ⇣ ⇣ Y J´ : 230. ≠J ì (v.) : (IV) 401, 413. ˘™J J £ : 16. ´@ ⇣ ⇣ ˙Œ ⌦ ⌦ . @ : 108. : 230. ÈáK ÆK Q£ : 95. ´ ˘™J J £ : 16. (s.) : 12, 44, 117, 118, 130, 146, 168, 192, 192, 195, 300, ’Œ (v.) : ) 245, [486]. — ¢´@ : 108. ´ (s.) : 12, 44, 117, 118, 130, 146, 168, 192, 192, 300, † ⌦ ⌦ .J£: @:(v.) È˙Œ 1, 279. 94. …‘ØQ™” ()(number) 370, 425. :(v.) 47,:: 52, 55,88, 57,102, 58, 113, 68, 72, 88,247, . . .55, . 57,195, ¨Q£ (Ìroc) 110, 114, 124, 134, ´@82,128, † 78, () 65, 363, ˙ ⌦ Q£ : 47, 52, 58, 68, 72, ˘™J 16. (v.) : ) 245, [486]. ⌦ . ⇣ÆKQ£ :: 12, 44, 117, 118, 130, 146, 168, 192, 192, 195, 300, ´ ⇣ÈáK Y J´ : 230. ✏ ˘™J J £ :(s.) 16. ⇣ …‘ I k @  : [207]. (v.) () 65, 88, 102, 363, 370, 425. ´@ ¨Q£ : (Ìroc) (number) 110, 113, 114, 124, 128, 134, ⌦ . ´ 78, 82, 88, . . . . ˙Œ : 94. . @ : 47, 52, 55, 57, 58, 68, 72, 78, 82, 88, . . . . (v.) : () 65, 88, 102, 363, 370, 425. ’Œ (v.) : ) 245, [486]. 95. † 146, ⌦˙: (I) —¢´@ : 108. ˙Œ Ë Y´ : 105, 105, 106??, 565. Y‘ (v.) : 192, () 65,192, 88, 102, :(Ìroc) 230. (v.) : ::) 12, 245, [486]. ˙ (v.)I¨Q£ [14], 424, 461, [462]. (s.) 44,88, 117, 118, 130, 168, 195,363, 300,370, 425. ´´@´´[169], …‘ † (v.) () 21, 116, 131, 146, 168, [169], 418, 584, ::(ὅρος) (number) 110, 113, 114, 124, 128, 134, (number) 110, 113, (v.) : () 65, 102, 363, 370, 425. ⇣ȨQ£ (v.) : () 21, 116, 131, 146, 168, 418, 584, 589, 626; (X)425. 279. (v.) : (I) 65, 88, 102, 363, ⇣áK Y‘ (v.) : () 65, 88, 102, 363, 370,  £ (v.) : (I) 245, 335, 341, 344, 350, 353, 390, 400. @ : 47, 52, 55, 57, 58, 68, 72, 78, 82, 88, . . . . (number) 113, 114,461, 124, 128, 134, ˙Q£ ::(v.) ˙Ê´ (v.) : [486]. (I)110, [14], 424, [462]. .ÆK⌦114, Y¢´@ J´´´´´@168, :(v.) 95. ’ŒY‘ (v.) :: 192, )()245, [486]. 230. :(Ìroc) 230. ˙Œ : 128, )12, 245, 94. — :230. 108. (s.) 44, 117, 118, 130, 146, 192, 195, 363, 300,370, 425. 65, 88, 102, 124, 134, 134, 134, : (Ìroc) (number) 110, 113, 114, 124, 128, 134, …‘ 370, 426. (v.) : () 65, 88, 102, 363, 370, 425. ˙ (v.)IáKÈ⇣¨Q£ : (I) [14], 424, 461, [462]. (v.) : 52, ()(number) 21, 131, 146, 168, [169], 418, 584, ⇣ÆK⇣ Q£ YJ´´´@´22, :(v.) 230. ¨Q£ (Ìroc) 110, 113, 114, 124, 128, 134, 230. :95. 55, 57, 58, 68, 72, 78, 82, 88, . 24, .192, .88, Yg ,:116, (v.) :118, (I) ::::@(s.) 94. ˙Q£(v.) Y‘ :400. ().245, 65, 102, 363, 370, 425. . :47, (I) 245, 335, 341, 344, 350, 353, 390, .(v.) . . (syn. see ll. 158, ⌦140, 13, 13, [14], 16, 18, 22, 24, 25,102, 25, 363, 27, 27, 27, XY´ ˙ Ê´ (v.) : (I) [14], 424, 461, [462]. : 12, 44, 117, 130, 146, 168, 192, 195, 300, Y‘ (v.) : () 65, 88, 370, 425. ˙Œ (v.) : () 21, 116, 131, 146, 168, [169], 418, 584, ´ .áK ’Œ ) [486]. ⇣⇣ ⌦ £ (s.) : 12, 44, 117, 118, 130, 146, 168, 192, 192, 301, 382, …‘ (v.) : () : () 65, 88, 102, 363, 370, 425. (v.) : () 21, 116, 131, 146, 168, [169], 418, 584, (v.) : (I) 21, 131,195, 146, 300, 168, Y J´ : 230. ˙ (v.) : (I) [14], 424, 461, [462]. : 230. Q£ : 94. @ : 47, 52, 55, 57, 58, 68, 72, 78, 82, 88, . . . . † ⌦ ˙ 160); (outer part) 96, 100. ´ ⇣ ⇣ ◆ áK Q£ : 94. …‘ () ˙ Ê´ (v.) : (I) [14], 424, 461, [462]. ⌦˙ £ IÈáK (v.) :◆ (I) 245, 335, 341, 344, 350, 353, 390,: :419, 400. (v.) () 21, 116, 131, 146, 168, [169], 418, 11, 421, 476, 479, 483, 606, .’Œ .´´´@ .(v.) Q£222. 95. Y‘.425. 65,584, 88,627. 102,(X) 363, 370, 425. ::350, () 88, 102, 363, 370, .⇣⇣ÆK408, [169], 279. (I) [14], 424, [462]. ˙Œ :442, (I) [14], 424, 461, [462]. ⌦˙˙Q£ 245,˙ 335,: 341, 344, 353, 390, 400. 230. ˙ Ê™” :65, 11, 222. Y…‘ J´´Ê´ : (v.) 230. (v.) :)() ()245, :::@(v.) 94. :(v.) 47, 52, 55, 57, 58,461, 68, 72, 78, 82, 88, . . 586, .[14], .[486]. ⌦ ⇣ ⇣ ⇣ ˙ (v.) : (I) 424, 461, [462]. áK Q£ : 94. 94. (v.) () 21, 116, 131, 168, 584, È⇣ÆK.⇣˙⌦ £ 222. ::(v.) 95. †121, ÈÍk335, : 424, 72,341, 73, 75, 76, 78, 79, 80, .418, .(). 248, . 125, 127,427, 179,428. 180, 181, I (v.) : (I) 350, 353, 390, ::: (I) [14], 461, [462]. È¢É@ 344, : 146, 115, 225, 425, ´[169], . 88, ´223, …‘ ˙◆ : QÍ11, (v.) ()245, 65, 102, 363, 370, 425. Y‘ (v.) :::400. () 102, 363, 370, 425. (s.) 12,65, 44,88, 116, 117, 130, …‘ Y J´ :(v.) 230. 230. ◆ ´@ ˙ Ê´ (v.) : (I) [14], 424, 461, [462]. ˙Œ ´ ◆:Q£4.::: (v.) ⇣ÈÈÆK⇣£ÆK˙⌦˙Q£ …‘ ˙ Ê™” : 11, 222. 95. : 11, 222. (v.) (I)21, [14], 424, 461, [462]. :: () 116, 131, 146, 168, [169], 418, 584, ◆ 95. ˙ Ê™” : 11, 222. † 146, 168, 192, 192, 195, [300], ⌦ (v.) : () 65, 88, 102, 363, 370, 425. I : (I)222. 245, 335, 341, 344, 350, 353, 390, ◆◆Q£(v.) …‘ (v.) () 65, () 88,424, 102,461, 363,[462]. 370, 425. ˙YY‘ :: 400. (I) [14], J´Ê™” :(v.) 230. ::95. 11, ´@´301, ÈÈ⇣⇣.ÆKÆK⇣⇣:˙⌦˙ £ :(v.) 11, 222. ◆Ê´ ˙Œ :350, (I) [14], 424, 461, [462]. () 21, 116, 131, 146, 168, [169], 418, (v.) : () (I) [14], 245, 344, 353, 390, 400. ˙ : 11, 222. † 382, 409,584, ... . Q£ : 95. X 335,(v.)341, (I) 571. XÒ´ (v.) : (I) 571. † (v.) : 65, 88, 102, 363, 370, 425. I  £ (I) 245, 335, 341, 344, 350, 353, 390, 400. (v.) : (I) 245, 335, 341, 344, ⌦ QÍ116 ≠ì (v.) : Translation (I) 407, [478], 507, 606. .£˙◆: 4. : 11, 222.Èk.  : 247. ◆Y‘ ´[169], ˙Ê™” (I)65, [14], ˙Y…‘ :(v.) 11,::400. 222. () () 88,424, 102,461, 363,[462]. 370, 425. (v.) ::390, (I) [14], 424, 461, [462]. J´Ê´ : (v.) 230. (v.) ()245, 21, 116, 131, 146, 168, 418, 584, 353, 401. (v.) :11, (I) 335, 341, 344, 350, 353, 390, ® X £ (v.) : (I) 571. XÒ´ (v.) : (I) 571. ˙:◆ (I) ::571. 222.  £ (v.) :XÒ´ (I) 245, 335, 341, 344, 350, 353, 390, 400. X (v.)QÍIII..£:˙350, 4. 11, 222. (v.) : (I) 571. † ◆ ˙…‘ Ê™” :390, 11, 222. ´ (v.) Ê´´[169], (v.) [14], 424, 461, [462]. (I) [14], 425, 462, [463]. : ::(I) 245, 335, 353, 400. (v.) () 21, 116,341, 131,344, 146,350, 168, 584, () ◆ £: 4.(v.) Y‘ ::: 418, () 88, 102, 363, 370, 425. (v.) 571. ..£X˙X £ XÒ´ (v.) (I)65, 571. (v.) : (I) (I) 571. : 11, 222. ✏ ⇣ I (v.) : (I) 245, 335, 341, 344, 350, 353, 390, 400. † ✏ QÍ ® † ✏ : 104, ◆ Ë Yg : 521, 553. · …ì (v.) : (VIII) 198, 200, 201, 233, 234, 608, 610, 615. (v.)⌦::™” (): 21, 116, 131,123. 146, † 168, 584, ˙Ê´ :: 11, 222. 104, 108, (v.) (I) 571. XÒ´ (v.) :: 418, (I) 571. 11, 222. ✏Ê™” ˙·…‘ (v.) [14], 424, 461, [462]. ´[169], 108, (v.) : (I) ()108, : 11, 222. ⌦✏ IQÍ. £Xj˙.X◆ :´✏⌦(s.)4.(v.) : () 123. 4. † ™” : 104, 123. ®† XÒ´ ⌦✏ (v.) :: ·104, 108, 123. (v.) 571. 4.✏ : (I) : (I) 571. ˙·◆Ê™” :: 11, 222. [572]. 122 ✏✏ 108, Translation ™” : 104, 108, 123. ´ ™” 104, 108, 123. : 104, 123. ⌦ † …‘ (v.) : () ⌦ : 104, 108, 123. 104, 108, 123. £ : 4. ⌦ QÍI122 ⇣ ⇣ : 104, ® X (v.) : (I) 571. ⌦ : () 4. ÈØP : 8. ✏ .✏ j.✏´⌦ (v.) · XÒ´ (v.) : (I) 571. Translation © ì (v.) : () 22. ◆ ™” : 104, 108, 123. 104,222. 108,® ⌦ : 11, ˙Ê™” 104, 108, QÍ££j 4.:: (I) ® 571. ® Y´ 324.123. QÍ :⌦:´✏4. (v.): :()321, (I) ✏ I (v.) 4. ® · .QÍ£X.(v.) ™” : 104, : 104, 108, 123. XÒ´ (v.) : (I)108, 571.123. : 4. ⌦ ⌦ ® ✏ £j(v.) QÍ ® ·✏™” : 104, 108,® 123. I :: () 4. (v.) (V) 4.324. Y´ 321, .⇣ ✏ . ✏⌦:´ 4.(v.) :: (I) 104, 108, 123. ® ¯ P (v.)©ì : (III) 491, 601. (s.) 54, : 16, ⌦ :(v.) XÒ´ : (I) 571. ® [23]. Q⌦✏600, ✏Y´(v.) ´ 196, 526, 529, [530], 542, 581, 589. ËY´ :✏ 105, 105, 106??, 565. ® Q (I) 321, 324. (v.) ® : 104, 108, 123. I j ´ (v.) : () 4. · ´ :526, 196, 526, 529, [530], 542, 581, ® 4. ™” : 196, 104, 108,589. 123. .⇣ ✏ Q.⌦⌦ : 196, ⌦ Q 529, [530], 542, 581, 589. ® ⌦ : 196, [527], 530,[530], [531],542, 543,581, 589. ´ : 526, 529, ✏ ⌦ ® Q⌦ : 196, I j ´ (v.) : () 4. 526, 529, [530], 542, 581, 589. Y´ (v.) : (I) 321, 324. Q Ë Y´ 106??, 565. . Q.⌦⌦: 105, 566. Q·147, ::: 196, 529, [530], 542, 581, 589. 196, 542, 581, ® 591, . 196, 526, 529, [530], 589. 582, 590. : :196, 526, 529, [530], Q⌦105, ⌦✏⌦´542, ´ :526, 196, 526, 529, [530], 581, 589. °É : 45, 100, 103, 104, 112, 119, 167, 302, 426,542, 443,581, 551,589. 554, ™” 104, 108, 123. © ìÒ” : 39, 132, 230, [568], 630. I j ´ (v.) : () 4. ¨ ✏ Q ⇣ ✏ . . XY´ : 13, 13, [14], 16, 18, 22, 22, 24, 24, 25, 25, 27, 27, 27, : 196, 526, 529, [530], 542, 581, 589. Q : 196, I (v.) :105, ()321, 4.106??, 16, 18, 18, 22, 581, Q589. ËY´ Y´ :´526, 105, 565. . jj 529, [530], 542, : (I) ´ : 196, 526, ¨ 529, [530], 542, 581, 589. ®¨ I ´ (v.) : () 4.. . ;324. Qj.⌦(v.) ✏Y´ ..(v.) ✏. 22, 4. ⌦ .ËIY´ : 196, 526, 529, [530], 542, 581,⌦ 589. 24, 24, . (order) 525, ⇣XY´ ⌦::´13, (v.) : () 4. : (I) 321, 324. 105,13, 105, 106??, 565. [14], 16, 18, 22, 24, 24, Q25, 27,529, 27, [530], 542, 581,¨ ´589. : 25, 196,27, 526, 589. ⇣ 22, ⌦33, ✏ Q565. ® : See 196, 526,°É@ 529,,: 22 [530], 542, 581, ( XY´ ), 27, 32, 53, 56, 63, 63, 80, 84, 88, ✏Y´ also . á Ø (v.) : (III) 462; (VIII) 458, 520, 521, 540, ⌦ XQ Ø Y´ (v.) : (I) 321, 324. : 22, 69, 74, 77, 83, 91, 95, ¨27, [530],⇣542, ✏ (v.) ⇣Ë✏Y´ : (I) 321, 324. XY´ :: 13, 13, [14], 16, 18, 22, 22, 24, 24,QXQ 25, 25, 27, 27,529, ´Ø589. : 196, 526, 581, ¨ 589. 105, 105, 106??, 565. QXQØ⌦(v.) ⌦ : 196, 526, 529, [530], 542, 581, ✏Y´ Y´ : (I) 321, 324. ✏ XQ : 22, 69, 74, 77, 83, 91, 95, 99, 101, 114, . . .;.;(.; (;pl. pl.È⇣✏K⌦ÈXQ✏K⌦ XQ Ø ) 102, 102,⇣103, ⇣»Y´ 99, 101, 114, (pl. ✏K103, Ø : 22, 69, 74, 95, 99, 101, 114, . Ø ) 22, 69, 74, 77, 83, 91, 95, 99, 101, 114, (pl. ) (v.) : (II) 244, [265], 327, (v.) : (II) 244, 265, 327, 377, 388, 394, 395, 405, 407, (v.) : (I) 321, 324. Ë⇣Y´ :: 105, 105, 106??, 565. XY´ 13, 13, [14], 16, 18, 22, 22, 24, 24, 25, 25, 27, 27, 27, XQ Ø : 22, 69, 74, 77, 83, 91, 95, 99, 101, 114, . . . ; ( pl. È XQØ ) ¨ ⇣ Ø XQ ⇣ ✏ ✏ ⌦ Q ✏ ✏Y´ ´ :Ø 196, 526, 529, [530], 542, 581, 589. ⇣ ⇣ 122, 125, 166, 207, 301, 562, 571. XQ377, ::105, 22, 69, 74, 77, 83, 91, 95, 99, 101, 114, . . . ; ( pl. È K XQ Ø ) 102, 103, 108, 121, 125, 166, 22, 69, 74, 77, 83, 91, 95, 99, 101, 114, . . . ; ( pl. È K XQ Ø ) XQØ:Ø(v.) ⌦ 388, 394, 396, 406, 408, ✏Y´ ⌦ ®PA ⇣ËË108, ⌦ ÜA Æ K@ (s.) : 543. »Y´ : (II) 244, 265, 327, 377, 388, 394, 395, 405, 407, 105, 106??, 106??, 565. 565. ⇣✏ :Ø: :105, 105, XQ®PA Ø207, ✏✏XQ ⇣Ë⇣ Y´ 301, [572]. 22, 74, 77,18, 83, 91, 101, 114, .101, .. ..;.27, (; 114, pl. È⇣⇣✏K⌦✏XQ. .ØÈ⇣.✏K⌦);XQ(Øpl. [14], 16, 22, 22,95, 24, 24, 25, 27,563, 27, 412, 490, 491, 493, 505, 507, XQ13, Ø69, :77, 22, 69, 74, 77, 83, 91, 95, 99, XQØ : 22,XY´ 69, 74, 83, 91, 95, 99,99, 101, 114, ( pl. ) ¨ÈK⌦ XQØ ) Ø 25, :⇣ 13, 105, 105, 106??, 565. Ë⌦Y´ 106??, 565. Ø.:[266], 200, 201, 228, 232, 375, XQ::KØ13, 22, 69, 74, 77, 83, 95, 99,®PA 101, 114, .395, . . ;27, ( 399, pl.228, È406, K⌦ XQ⇣232, Ø✏ ). . . . ¨ Y™ ::105, [266], 590. 567, . .199, .105, 13, [14], 16, 18, 22,91, 22, 24, 234, 24, 27, 200, 27, XQ25, Ø360, Ø :25, 199, 201, ⇣ ⇣ ⇣ ✏ XQØ : 22,…KXY´ 69, 74, 77, 83, 91, 95, 99, 101, 114, . . . ; ( pl. È ®PA Ø ⇣ QÂÑ Ø (v.) : (II) 590, 605, 619, Ø13, 199, 200, 201, 228, 232, 234, 360, 375, 395, 399, ....Ø.. 629, .). 631. ::Ø:22, 199, 200, 201, 232, 234, 360, 375, .XQ :[266], 199, 200, 201, 228, 232, 234, 360, 375, 399, 406, XY´ 13, [14], 16, 18, 22, 22, 24, 24, 101, 25, 27,1, 27, 27, 69, 74,16, 77, 83, 95, 99, 114, .395, .395, . ; 27, (399, pl. È406, K⌦406, XQ.Ø.K⌦625, á228, Ø :91, 424, [571]; á25, ØÒÀ@ XY´ 18. …K [266], 590. 591. ⌦XQY™:Ø:ØK:Ø13, 234, 296, 360, 375, 395, .) . XY´ 13, [14], 18, 22, 22, 24, 24, 25, 25, 27, 27, ®PA Ø XQ Ø Ø (v.) : 27, (II)27, 590, 605, ⇣619, 625, : 199, 200,16, 201, 228, 22, 232, 360, 375, 395, 399, . . . .629, 631. ✏K406, XY´ : :13, 13, [14], 24,234, 24,QÂÑ 25, 25, ®PA Ø XQ Ø 22, 69, 74,16, 77,18, 83,22, 91, 95, 99, 101, 114, . 27, .590, . ; 27, (605, pl. 619, È XQ Ø ) 629, 631. XY´ : 13, 13, [14], 18, 22, 22, 24, 24, 25, 25, 27, 27, ⌦ ®PA Ø QÂÑ Ø (v.) : (II) 625, Q Ø (v.) : (II) 590, 605, 619, 625, 629, 631. Ø : 199, 200, 201, 228, 232, 234, 360, 375, 395, 399, 406, »XA™” : Ø373, 414,200, 201, 228, 232, 234, : 199, XQ 360, 375, 395, 399, 406, . . . . . . . . ⇣ (II) 590, 605, 619, 625, 629, 631. á228, Ø@590, Ò619, ”619, : 232, 14, 457, 499. (v.) (II) 590, 605, 625, 629, 631. QQQØ ØØ(v.) : ::(II) 590, 605, 625, 629, 631. :373, 199, 200, 201, 234, 395, 399, 406,406, . . . 629, . . . .631. QÂÑ Ø414, (v.) : 228, (II) 605, 619, 625, 629, 631. »XA™” :Ø(v.) QÂÑ Ø360, :375, (II) 590, 605, 619, 625, ®PA Ø(v.) Ø : 199, 201, 232, 234, 360, 375, 395, 399, . Q Ø Ø200, (v.) : (II) 590, 605, 619, 625, 629, 631. : 199, 200, 201, 228, 232, 234, 360, 375, 395, 399, 406, . . . . QÂÑ Ø (v.) : (II) 590, 605, 619, 625, 629, 631. (II) 590, 605, 619,Ø 625, 629, 631. Y´ (v.) [527],: 562. Ø :QÂÑ(V)Ø (v.)

K⌦ 126 Translation ⇣ P@Y 126Æ” : 226, 228, 229, 245, 246. Translation Translation 126 Translation Q⌦ÇÆ⇣K :: 230. ⇣⇣(v.) –Y Ø⇣Æ” : (II) [532]; (V) 102, 396, 407. 230. 126 P@Y ::: 226, 228, 229, 245, 226, 228, 229, 245, 246. 126 ®PA Ø : 199, 200, 201, 228, 232, 234, 360, 375, 395, 399, 406, .245, . . . 246. P@Y Æ” 226, 228, 229, 246. Translation ⇣ 226 Arabic glossary Q⌦ÇÆK : 230. ⇣ –Y Ø (v.) : (II) [532]; (V) 102, 396, 407. QQ…í ✏ ◆ 230. : (I) 574, 580, 581; (II) [539]. ⌦⌦ÇÇÆÆØ⇣K⇣K :(v.) : (I) 574, 580, 581; (II) [539]. A” Y⇣⇣Æ⇣ ⇣J” : 590, 629. :(v.) 230. –Y (V) 102, 396, (v.) (II) [532]; (V) 102, 396, 407. ⇣ QÂÑ Ø (v.) : 590, 605, 619,620, 625,[539]. 629, 631. (v.) (II) 591, 606, Q…í –Y✏ØØ (v.) (v.) (II) [532]; (V)(V) 102,102, 396, 407. 407. (v.)::::(II) (II)[532]; [533]. ÆØK⇣(v.) : 230. :: (II) (I) 574, 580, 581; (II) ⌦Q ÇÇ626, ◆ ⇣ ⇣ Æ K : 230. 630, 632.: 559, 561. ⌦ A ” Y Æ J” : 590, 629. 397, 408. 574, 580, 581; (II) …í ØØÆ⇣K(v.) (I) [539]. (s.) :: 582, 584, 586, 587, 589. (s.) 582, 584,580, 586,581; 587,(II) 589. …í (v.) (I) 574, 574, 580, 581; (II) [539]. °Ç Q⌦Ç[539]. :: 230. ✏ ◆ ⇣Ø⇣⇣⇣J”⇣: ::222. ⇣ Q…í ✏ ◆ Ç Æ K : 230. A ” Y Æ 590, 629. 591, 629. 630. ⌦ Ø ⇣(s.) (v.):: 582, (I) 574, 590, 629. 584,580, 586,581; 587,(II) 589.[539]. A” YÆJ” : 590, Q…í Ç Æ K : 230. ⌦ Ø (v.) : (I) 574, 580, 581; (II) [539]. ⇣ 575, 581, 582. (II) 80, 96, 102,…í 116, 120, 122, 124,589. 129, 144, ⇣K :: 118, °ÇØ : 222. ØØØÆ(s.) :: 582, 584, 586, 587, : 510. ……í ìA 3. …í (s.) 584, 586,581; 587,(II) 589.[539]. ⇣⇣ (v.)3. : 582, (I) 574, 580, [540]. :: (I) ’ÊÑØØØ⇣Ø⇣⇣(v.) ØØÆ(v.) (I) 574, [539]. ’ÊÑ ⇣K : 3.:: 582, (I) 224, 512, 513, 514, °Ç ::: 222. …í (s.) 584,580, 586,581; 587,(II) 589. 222. ’ÊÑ ……í ìA °Ç Ø(v.) 222. …íØ ⇣(v.) (I) 574, 580, 581; (II) [539]. 515, (s.)::: 582, 583, 585, 587, 588, ⇣ (s.) 584, 586, 587, 589. 517, 525, 531, . . . . (V) ⇣Ø⇣(v.)124, ’ÊÑ122, ………™ ìA K⇣K(s.) :: 3. ’ÊÑ Ø : (I)129, 144, Ø ØÆÆ(v.) ::: (I) 80, 96, 102, 116, 118, 120, 590. (v.) (I) 80,584, 96, 102, 116, 118, 120, 122, 124, 144, ìA 3. ’ÊÑ Ø ⇣ ⇣ …í 582, 586, 587, 589. ⇣ ⇣ (s.) 191, 561.129, (v.) : 586, (II) ⇣ ⇣ ⇣ ⇣ È“Ç Ø : 559, 561.421, 586 ; (quotient) 225. ’ÊÑ Ø : 191, 191, [194], [195], [206], 231, ’ÊÑ Ø …í Ø (s.) : 582, 584, 587, 589. ⇣ ’ÊÑ Ø ……í ìAØ ØÆ(v.) K(s.) : 3. È“Ç Ø : 559, 561. ⇣ …™ : (I) 80, 96, 102, 116, 118, 120, 122, 124, 129, 144, : 3. ’ÊÑ (v.) (I) (s.) :: (I) 191, : 582, 584, 586, 587, 589. ’ÊÑ (v.) (I) 191, [194], [195], ⇣ ØØ⇣on ⇣(v.) …ìA120, Æ⇣K : 3.122, 80, 96, 102, 116, 118, 124, 129, 144, È“Ç Ø : 559, 561. An ancient Greek treatise magic squares 125 ⇣ ⇣ ’ÊÑ Ø …™ ØØ (v.) (I) 102, 122, 129, 144, ⇣K : :3. (s.) [206], 231, 422, (s.) :: 96. 96. …™ (v.) (I) 80, 96, 102, 116, 118, 120, 120, ⇣È“Ç 122, 124, 129, 144, 587; (quo(v.) (I)80, 80, 102,116, 116,118, Ø :124, 559, 561. ⇣ 96, ⇣96, ⇣ ⇣ … ìA Æ ⇣ ⇣ ⇣ È“Ç Ø : 559, 561. ⇣ 510. An Greek treatise on magic squares 125 559, 561. …ìA K⇣ : 120, 3. 225. ØÆÆØ:⇣KK:::124, 559, 561. (v.) (I)122, 80,:ancient 96, 102, 129, 144, ⌦122, ’Ê’ÊÈ“Ç ÇÇtient) 510. …™ Ø Æ(s.) :: 96. 226. 118, 124, 129,116, 144,118, 120,È“Ç ⌦ ……™ìA Æ K : 3. ⇣ ⇣ Ø (v.)189, : (I)250, 80, 254, 96, 102, 120,’Ê 144, ⇣ 122, Çon ÆØ⇣K:⇣::124, 510.129, . . . Greek . 116, 118, 560, 562. È“Ç 559, 561. An 627. ancient treatise magic squares 125 …™ ØØØ:(s.) :: 96. ⌦ °…™Æ⇣166, 195, 232, ⇣ :(s.) 195, 232, 627. …™ ⇣ ’Ê Ç Æ K : 510. ⇣ (v.) : 96. (I) 80, 96, 102, 116, 118, 120, 122, 124, 129, 144, ⇣ ⇣K :124, ’Ê⌦Ç96, ÆK : 102, 510. 116, 118, 120,QÂî ⌦ ⇣ ⇣ ⇣ Ø (v.) : (II) ’Ê Ç Æ 510. …™ Ø (v.) : (I) 80, 122, 129, 144, ⇣ (s.) 96. ’ÊQÂî⌦ÇØÆon K(v.) : 510. : 96. : (II) squares An627. ancient Greek treatise magic 125 511. °…™ 195, 232, () 7, 299. …™Æ⇣ ØØ :(s.) (v.) (I) 80,: 96, 102, 116, 118, 120,⌦ 122, 124,(II) 129, 144, ⇣ (s.) :: 96. 110. ⇣ ⇣ QÂî Øon : (II) squares ⇣Æ⇣(v.) ’ÊQÂî Ç257, K(v.) : magic 510. °° ÆÆ⇣ØØØØ(v.) :(v.) 195, 232, 627. ⇣ (v.) : (II) 195, 628. ancient Greek treatise 125 —Í I:I ::96. () 7, 299. ()An 7, 299. ::(s.) 195, 232, 627. ⌦ Ø : : (II) (v.) (II) [275], 225, [276], 251, 253, ⇣ QÂî Ø 225, 251, 253, 258, 258, [277], …™ ⇣ Q ⇣ ⇣ í Æ K : 226. ⇣ QÂî Ø (v.) : (II) …™ Ø (s.) : 96. ⇣ ⌦ Q QÂî⌦ íØ[257], :[258], (II) squares °…™ÆØ⇣ØØ(v.) : 195, 232, Æ⇣(v.) K : magic 226. —Í ()An 7,627. 299. ::I :(I) 7, 299. ancient Greek treatise on [258], [275], [276],125 7, 299. (s.) 96. ⇣ °⇣ Æ :(v.) 195, 232, 627. ) 135,151, ⇣490, ⇣ Q 493. í Æ K : 226. [277], [278], 282, . .294, . . 306, 308, 315, ⇣ ⇣ ⌦ QÂî Ø (v.) : (II) ⇣K135,151, —Í ØÆØ⇣ØØ(v.) II 48, :48, () 7, 299. :52, 34,.....36, ; ( 490, Q⌦23, Q⌦í))⇣Æ82, 27, 55, 57, ÜÒ ::: 27, 57, 60, ÜÒØ 55, ·” 135,151, 55, 57, 60, ....;;; (( 52, 490, )493. 493. —Í (v.) : 232, () 52, 7, 299. í Æ⇣K :25, 226.60, : 226. ° 195, 627. ⇣ ⇣ ⇣ Q ⇣ Æ⇣Ø(ØØÜÒ ⇣ ˙Êî ˙Êî 110. ⇣ 195, 226. 490, 493. ° ⌦íÆ)Æ⇣Ø@Ø@KK135,151, í :::: 226. —Í (v.) :)232, 7,627. 299. 110. Ø195, ·”I48, )() 135,151, 490, 493. 135, 151, 491,60, 494. ÜÒ :: 27, 52, 55, 57, . . . ; ( ÜÒØ Q·” ⌦ ° Æ Ø : 232, 627. ⇣ —Í I : () 7, 299. ⇣⇣ Ø :(v.) ⇣⇣ Ø ·” ⇣ [432], ˙Êî ::: 110. Q·” 110. ÆØ@Ø@⇣⇣K135,151, 226. ÜÒ 27, 48, 52, 60, .. .. [435], ..[435], ;; (( ÜÒ 490, ⌦í⇣)[439], ˙Êî55, Ø@[431], : 57, 110. …J Ø ::(v.) [426], [429], [431], [433], [437], [440]. [427], [430], [434], ⇣ [426], [429], [433], [437], [440]. ÜÒ 27, 48, 52, 55, 57, 60, ÜÒ Ø )[439], 135,151, 490, 493. ˙Êî : 110. ⌦ ⇣ —Í Ø I : () 7, 299. ⇣ ◆403, …J Ø : 73, 76, 202, 217, 218, 221, 235, 250, 257, [276], [277], 296,91, 316. : 65, 67, 70, 72, 75, 78, 81, 83,34, 85,36,493. 89, . . .; (. @Q¢ ⇣ ⇣…J ⇣ . Q¢ Ø : 23, 25, 52, 55, 82, Ø ˙Êî Ø@ : 110. —Í Ø (v.) I : () 7, 299. ⇣ ⇣ ◆ ˙Êî Ø@ : 110. ÜÒ Ø : 27, 48, 52, 55, 57, 60, . . . ; ( ÜÒ Ø ·” ) 135,151, 490, 493. : 23, 25, 34, 36, 52, 55, 82, Q¢ Ø : 23, 25, 34, 36, 52, 55, 82, ; ( @ Q¢ Ø Ø : [426], [429], [431], [433], [435], [437], [439], [440]. [438], [440], [441]. ⇣⌦[436], ⇣ Ø ·”⇣257, —Í : ()202, 7, 299. …J Ø⇣Ø:(v.) 217,57, 218, [276], [277], 296, 316. ÜÒ : 73, 27,I76, 48, 52, 55, 60,221, . . . ; 235, ( ÜÒ⇣250, ):⇣ 135,151, 490, 493. ◆ Ø⇣⇣ . ⇣ ⇣))294, Q¢ Ø 23, 25, 34, 36, 52, 55, 82, ; ( @Q¢ ◆ . . . ; ( 306, 308, . . . . ⇣ ˙Êî Ø@ : 110. Q¢ Ø : 23, 25, 34, 36, 52, 55, 82, ; ( @ Q¢ Ø 294, 306, 308, 315, ⇣ ⇣ …J Ø : [426], [429], [431], [433], [435], [437], [439], [440]. ◆ ⇣Ø Ü …J [426], [429], [431], Q¢ Ø[439], 23, [440]. 25, 34, 36,493. 52, 55, 82, ; ( @Q¢ ⇣)::135,151, ÜÒ 27, 48, 52, 55, 57,[433], 60, . .[435], . ; ( ÜÒ Ø Q¢ ·”⌦⇣ Ø⇣257, 135,151, 490, ⇣:::Ü73, ⇣◆ ⌦⌦Ø⇣ØAØØÆ” ⇣:27, ⇣ [437], ◆ ⇣ ⇣ …J Ø È K 76. …J : 76, 202, 217, 218, 221, 235, 250, [276], [277], 296, 316. 23, 25, 34, 36, 52, 55, 82, ; ( @ Q¢ ◆ ÜÒ 48, 52, 55, 57, 60, . . . ; ( ÜÒ Ø ·” ) 490, 493. ⇣ Q¢ Ø :[439], 23, 25, 34, 36, 52, 55, 82, ; ( @Q¢ØØ …J [429], [431], [433], [435],⇣Ü [437], [440]. …J   Ø ⇣⇣ .⌦.Ø⇣Ø⇣:: [426], ◆ÜÒ : 65, 67, 70, 72, 75, 78, 81, ⌦ 48,[429], 52, 55, 57, [433], 60, . . [435], . ; ( ÜÒ[437], Ø …J ·” ⇣257, ) 135,151, 493. ⇣[439], …J⌦. ⇣ØØAÆ”:: 73, [426], [431], [440].490, ⇣217, ⇣⇣250, È K :27, 76.76, ◆ Ø⇣ …J 202, 218, 221, 235, [276], [277], 316. Ø …J   Ø :90, 65,91, 67,129, 70, 72, 75, 78, 81, 83, 85, 89, 91, . .296, . . 55, 403, ⇣ 83, . . . . ⌦ Q¢ Ø : 23, 25, 34, 36, 52, 82, ; ( @Q¢ Ü ⌦ …J Ø (v.) : (III) 79, 90, 134, 136, 138, 140, 142, 144, 149, ✏ (v.) : (III) 79, 91, 129, 134, 136, 138, 140, 142, 144, 149, Ü (v.) : (III) 79, 90, 91, 129, . …J   Ø ⇣ …J Ø : [426], [429], [431], [433], [435], [437], [439], [440]. ⇣:(v.) ⇣Æ” ⇣◆…K⌦.⌦Ø134, ⇣250, ✏138, 79, 90, 91, 129, 134, 136, 138, 140, [431], 142, 218, 144, 149, ……J Ø⇣: 149, …J ØA⇣AÆ” [426], [429], [437], [439], [440]. ⇣Ø@Ø@ ⌦⌦Ø ⇣257, 202, 217, 235,136, [276], 316. : 76. 253, [431], 437. È K :73, Ü …J 136, 138, 140, 142, . .221, . .[435], : 76, (III) 79, 90, 91,[433], 129, 134, 140, 142,[277], 144, 149, … 226, [272], 330,296, 332, 335, ⌦ ⇣ 79, 90, 91,…J 129, 134, 136, 138, 140, 142, 144, ✏ ✏ …J Ø : [426], [429], [431], [433], [435], [437], [439], [440]. ⇣ ◆…K ⇣ Ü ⇣ (v.) :226, (I) 577; (II) 11; (IV) 11, 264. ver. ⇣ ⇣È K⌦⇣Ø⇣A⇣Æ”:(v.) ✏ … Ø@ : [272], 330, 332, 335, 339, 341, 344, 348, 350, 353, 608, … Ø@ : 253, [431], 437. …J 73, 76, 202, 217, 218, 221, 235, 250, 257, [276], [277], 296, 316. 339, 341, 344, . . . . ⇣ 76. ⇣ …J :: 76, (III) 79, 90, 138, : 73, 218,129, 221,134, ⌦⇣Ø@ ⇣Ø 140, ...Ø (v.) …J (III)202, 79, 217, 90, 91, 91, 129, 134, 136, 136, 140, 142, 142, 144, 144, 149, 149, ✏138, … Ü ⇣ ✏ ◆…J ⇣ ⇣ ⇣ 235, Ü⇣ …–Ò –Ò (v.) (I) 577; (II) 11; (IV) ⇣Æ⇣(v.) …Ø@138, Ø@⇣ØØ (v.) (v.) (I) 11. (II) 11.(IV) (IV)11, 264. ver. ⇣J”:: :253, 250, [257], .90, .596, . .91,599. : (III) 79, 437. 129, 134, 136, 140, 142, 144, 149, È K 76. ::: (I) (I) 577; (II) 11; …K [570], 583, …K [431], (II) (IV) 11, 264. ver. .◆ ..AØØ⇣AƔƔ ⇣ Ü ✏ ⇣ …J (v.) : (III) 79, 90, 91, 129, 134, 136, 138, 140, 142, 144, 149, –ÒØ (v.) : (I) 577; (II) 11; (IV) ⇣ :: 76. ⇣ A⇣ÆÆ” Ø⇣11, (v.)264. :578. (I)ver. 577; (II) 11; (IV) 11, 264. ver. …–Ò138, Ø@⇣⇣[264], …K [570], 583, 596, 599. È K : 79, 11,437. 264, 578.134, 136,–Ò 253, [431], ..⇣Ø ⇣J”(v.) Ø (v.) : (I) 577; (II)149, 11; (IV) 11, 264. ver. …J : (III) 90, 91, 129, 140, 142, 144, ⇣ ⇣ : 11, 11, [264], 579. …J⇣. Ø⇣⇣ ⇣(v.) : (III) 79, 90, 91, 129, 134, 136,–Ò 140, 142, 144, –A–Ò138, Æ” :(v.) 264, 578. Ø : (I) (I) 577; (II)149, 11; (IV) (IV) 11, 264. 264. ver. ver. ⇣ Ø (v.) : 577; (II) 11; –A Æ” : 11, 264, 578. ⇣ 438. 253, [431], 437. [432], YØ. .AØÆ:Æ”J”(v.) 6,:: 99, 114, 129, 145, 166, 189, 358, 373, 388, 394, 475, 11, 504, …K [570], 583, 596, 599. …J : (III) 79, 91, 129, 134, 225, 136, 327, 138, 140, 142, 144, 149, ⇣ –A Æ” :90, 11, 264, 578. –A–Ò Æ”⇣Ø⇣ :(v.) 11,: 264, 578.(II) 11; (IV) 11, 264. ver. (I) 577; 559, Y…K Ø⇣. A:Æ” 114, 129, 145, 166, 189, 225, 327, 373, 388, 394, 475, 504, 253, [431], 437. –A Æ” : 358, 11, 264, 578. Æ⇣ ⇣J”6,.: ::99, [570], 583, 596, 599. [571], 584, 597, 600. ⇣ º ⇣Æ”:: 11, –A 11, 264, 264, 578. 578. º 559, . –A Æ” ⇣ ⇣ ⇣ º373, 388, 394, 475, 504, J”6,:99, [570], 583, 596, 599. JJ⇣. ªª :358, 114, 129, 145, 166, 189, 225, QQ327, Y…KØ. :⇣A:Æ6, 99, 114, 129, 145, 166, ⌦⌦Æ” º 104, 104, 429, 429, 511, –A : 11, 578. . ⇣ ⇣ ⇣ PY 295. 559, º 225, 327, 358, . . .599. .166, 189, 225, 327, 358,264, …K A⇣Ø:Æ:J”6,. [291], : 99, [570], 583, 596, YØ. 189, 114, 129, 145, 514, 515,373, 516,388, 517,394, 520,475, . . . .504, ºº PY 295. 559, YØ⇣Ø: : 6,.[291], 99, 114, 129, 145, 166, 189, 225,Q327, 373,29,388, 475,38, 504, 25, 28, 30, 394, 31, 36, º Q..ªª @@ : 358, ⇣ .[291], ⇣ Æ” ⇣ 559, P@Y : 226, 228, 229, 245, 246. PY Ø : 295. 226,114, 228,129, 145, 166, 189, 225, 327, 41, 358, 43, . 373, . . . 388, 394, 475, 504, YØ :⇣ 6,: 99, P@Y 226, 295. 228, 229, 245, 246. 559, PY Ø⇣Æ”: .:[291],

’ÊÑØ⇣

⇣ Ø⇣ È“Ç

’Ê⌦ÇÆ⇣ ⇣K

QÂîØ⇣

Q⌦íÆ⇣ ⇣K ˙ÊîØ@⇣

⇣ Q¢Ø⇣ÜÒØ ·”

◆ Ø⇣ @Q¢

…J⌦ Ø⇣ ✏ ⇣ Ü⇣ …Ø@ –ÒØ⇣

⇣ –AÆ”

º

⇣:(v.) PY 295. –YØ⇣Æ” : (II) [532]; 102, 396, 407. P@Y :[291], 226, 228, 229,(V) 245, 246. ⇣ ⇣ –YØÆ”:(v.) : (II) [532]; 102, 396, 407. PY [291], 295. P@Y : 226, 228, 229,(V) 245, 246.

⇣⇣ H H.. AAJª Jª ⌘Jª

(v.)(ver.), : (I) 9.574, QH⌦... ª⇣JJ@ ’ÀhÒ: 570 – 623. Q128 ⌘ ª @ QH⌦.JªA⇣Jª Translation hÒÀ (v.) : (I) 9. hÒÀ :: (I) 9. ’À :⌦À570 (ver.), 623. . Å⌧ :(v.) 195, 526, [568], 586. hÒÀ (v.) (I) 9.574, hÒÀ (v.) : (I) 9.Arabic QQH. ⌘ª⌘JªA⇣Jª 128 Translation @ ’À : 570 (ver.), 574, 623. glossary 227 hÒÀ : (I) 9. [568], QH⌦⌦. JªA⇣Jª Å⌧⌦À ::(v.) 195, 526, 195, 526, [568], 586. 586. ⇣ ⇣ . ⌘ ˙ Ê” : 520, 520, 564. 128 Translation H A Jª Q ⌦.ª⌘Jª@ hÒÀ⌦⌦ÀÀ::(v.) : (I) 9. [568], 586. – Å⌧ 195, 526, 586. Å⌧ Å⌧ À : 195, 195, 526, 526, [568], [568], 586. 128 Translation ⌦ ⇣ ⌘ ⌘ hÒÀ (v.) : (I) 9. : 4, 5, 6, 6. H A Jª : (I) – ª @@ 130 Translation QQ⌘⌦⌦. ªJª ⌘Jª Å⌧⌘⌦À : 195, 526, [568], 586. Å⌧⌦À :(v.) 195, 526, [568], 586. 128 Translation 130 Translation ⌘ ⇣ ˙Å⌧ Ê”⌦À: ::520, 564. ––– 195, 526, [568], QQΩÀ⌘⌦ªJªYª 195,520, [527], [569],586. 587. @ : 71,…73,J” 76, 82,◆ ⌘84, 88, 126,Translation 128 130 Translation QQΩÀ⌦⌘⌘Jª 282, 294, . . . ; ( @Q⌦Jª ) 279. ˙Ê”⇣ ⌦À::: 520, Å⌧ 195, 520, 526, 564. [568], 586. 520, 520, 564. – ª @ Yª 128 Translation ⌘ QΩÀ⌘ªYª – ⇣ ⌘ @ ⇣ 130 Translation Ê” 520, 520, 520, 564. 564. Q Qª∫@ ⇣K : 7, 8, 62, ⇣J” ::: 520, ˙˙…˙Ê” – 111, 111, 484,Translation ⌘J” :69,264, Ê” 520, 520, 564. P@128 ΩÀ⌘ 561, Yª 612. »A [275], [287], [300], 378, 408. 128 Translation ⌘ ⌘ ⇣J” :: 520, Q ª @ : 7, 8, 62, 69, 111,130 – 111, 483,520, 611.564. QP@ΩÀQªYª ˙…Ê” 521, 521, 565. @ ⇣⇣K Translation ∫ ⌘ 128 Translation ⌘ ⇣ P@ Q ∫ K ⇣ 112, 115, 520, 201, 216, ΩÀ Yª : 77, …˙J”Ê” 128 Translation 264, 520, [275],564. [287], [300], 378, 408. ⌘J”J” :: 520, ……»A ˙◆ Ê”243, : 520, 564. 130 A“ª 13, 94, 101, 146, 168, 210,Translation 130 Translation QÂѪ ⇣ËQ”✏ (. .... .⇣Ë◆Q”✏. ) : 454, 456, 468, P@ΩÀQ220, ∫Yª⇣K 234, ⌘ ⇣⌘J”J”211, »A : 264, [275], [287], [300], 378, 408. ˙ Ê” : 520, 520, 564. 471. : 264, [275], [287], [300], 378, 408. 212, 213, . . . . … A“ª ΩÀ Yª 130 Translation ◆ ✏⌘⌘ ⇣◆ ✏ ΩÀQ∫Yª⇣K⇣ : [573]. ⇣ P@QÂѪ QÂѪ 264, [275], [287], [300], 378, 408. Ë»A . ËQ”)[275], : [275], 454,[287], 456, 468, 471. [300], 378, …Q”J”⌘J”J”(.::: .264, [264], [287], [300], P@A“ª ≠ ✏QQJ∫∫⇣J∫” ⇣KK …⌘J” 274, 285, 356, . . . . »A »A 264, [275], [287], [300], 378, 408. 408. ◆ ◆ P@ ⇣ ⇣ ⌘ 130 Translation …ø ✏ ✏ 378, 409. QÂѪ : [527], [528]. ⇣ Ë Q” (. . . Ë Q” ) : 454, 456, 468, 471. … J” (. . . ) : 454, 456, 468, 471. A“ª : 264, [275], [287], [300], 378, 408. ≠✏✏ JJ∫” ⇣K ·∫” (v.) : (IV) 423, 542,Òm⇣◆»A ◆⇣◆:⇣Ë◆✏Q”✏: (I))(IV) ⇢⌘' (.(v.) ⇣Ë◆✏Q”✏545, 547, 554. P@ Q ∫ ⇣ 190. ◆ ⇣ QÂѪ …ø . . 454, 456, 468, 471. ⇣ ·∫” (v.) 423, 542, 545, 547, 554. 455, 457, 469, 472. ⇣ Ë Q” (. ... ËË:Q” ))[275], ::: 190. 454, 456, 471. »A J” :....264, [287], [300], 378, 408. A“ª ⇢ ' 130 Translation ✏ ✏ P@ Q ∫ K : 572. …ø Òm (v.) (I) ‡Òª (v.) : 1, 3, 7, 14, 32, 62, 63, 66, 73, 76, . ≠ J J∫” Ë Q” (. Q” 454, 456, 468, 468, 471. : 14, 91, 93, 100, 104, 105, QÂѪ ✏⇣ ✏QÂѪ ◆ ◆ ⌘J”199, ⇣ËQ”✏⇢⌘J”' (.: (v.) ⇣ ✏:::)[275], ·∫” (IV) 423, 542, 545, 547, 554. A“≠ »A [287], [300], 378, 119, . .14, . . 32, A“ª (v.) (IV) 423, 542, 545, 547,408. 554. »A : 7,264, [275], [287], [300], 378, 408. …ø ø107, .264, .. Ë:Q” :(IV) 454, 456, 468, 471. J J∫” (v.) 424, 543, 546, ‡Òª (v.) : 1, 3, 62, 63, 66, 73, 76, . . .(I) Òm (v.) 190. ◆ ◆ ✏ ✏ ↵ ↵ ⇣ ⇣ ⇢ ' ⇣ ✏ ✏ ✏ QÂѪ Òm (s.) : 107, 190, 512, 516, 588. ·∫” (v.) : (IV) 423, 542, 545, 547, 554. · ” : 16, 21. A“ª 555. (v.) :: . (IV) 423, 542, 545, 547, ËQ”76, )(IV) : 454, ≠ ” :3,16, 21. A“✏✏ ø øJJ∫”:(v.) 285,:·1, [291]. : 32, 526, ‡Aæ” ·∫” (v.) 423,456, 542,468, 545,471. 547, 554. 554. ‡Òª 7,QÂѪ 14, 62,527. 63, 66, 73,·∫” .....◆Ë:Q” ⇢↵548, ' (.(v.) A“…ø (s.) 107, 190, 512, 516, 588. ◆ Òm (I) 190. …ø ⇣ ⇣ ✏ ✏ ✏ ⇣ A“ª · ” : 16, 21. ◆ ◆ …ø Ë Q” (. . . Ë Q” ) : 454, 456, 468, 471. : 16, 21. ≠ 16, (v.) :. (IV) 423, 542, 545, 547, 554. ‡Òª (v.) :⇣390. 62, 63, 66, 73,·∫” 76, . . . 21. ⇣ËQ”✏ )32,: 454, ’ª A“ª ‡Aæ” A“✏  øJ:J∫” 320, ✏1, 3,(. 7,. . 14, ⇢↵'': (s.) Òm : 107, 190, 512, 516, 588. Ë Q” 456, 468, 471. ↵ ⇣ ✏ ⇣ ⇢ ✏ · ” : 16, 21. ↵ 190. ‡ ≠ ···∫” :: (v.) (IV) 423, 542, 545, 547, 554. (v.) : 1, 3, 7, 14, 32, 62, 63, 66, 73,Òm .(v.) . :21. .:(I) …ø ÈJ⇣76, K16, : . 602. ⌦””kA 16, 21. A“‡Aæ” ·‡Òª ’ªKAø ø✏✏J⇣J∫” A“ª …ø 408, [446], : 101, 131, [169], A“ª ⇢ ' ‡ ’ª ÈJ kA K : 602. ⇢ ' Òm (s.) : 107, 190, 512, 516, 588. A“  ø ⌦ ↵ 190. JJJ∫” ⇢”':(v.) ‡ (v.) :.(I) (IV) 423, 542, 545, 547, 554. ⇣J∫” Òm·∫” (v.) (I) 190. ‡Òª (v.) 508, : 1, 3, 7, 14, 32, 591. 62, 63, 66, 73,Òm . . .::21. ·⇣76, 16, ‡Aæ” A“KAø  ø[479], [532], [584], ≠ ·≠ ’ª ↵ ✏ ⇣ (v.) :·∫” 602. ⇢⌦76, ''K:(v.) :K(s.) :.:16, 21.190. ::21. 21. ‡Òª 1, 3, (v.) 7, 14, 62,423, 63, 66, 73,ÒmÄA .16, .547, .(I) ✏: 32, ⇢”kA ·ÈJ545, 16, (IV) 542, 554. ‡Aæ” Òm 107, 190, 512, 516, 588. ‡ ‡‡ ⇣ A“  ø ≠J ª qÇ : 4. : 109. · KAø ≠ J J∫” ⌦ ’ª A“  ø : 285, 291. ≠ J J∫” ⇣ ↵ :.:: .4. ⇢”⇢kA '':K (s.) ‡Òª : 1, K3,: :7, 16, 14, 32, 62, 63, 66, 73,Òm .602. . 21. ÈJ76, ’ª (v.) ÄA ::K16, 16, ·qÇ 21. 21.31, 21. 190, ‡Aæ” ·‡Òª (s.) 190, 512, 512, 516, 516, 588. 588. ‡ 32, 62,32, 63, 66, 73,Òm⌦76, . .16, .::: 107, .107, ≠J (v.):ÄA : 1,1,3,3,7, 7,14,14, (v.) (I) 190. ’ªKAø⌦ª (v.) ÄA ::: 4. 16, 21. 190, 512, 516, 588. ‡ qÇ ‡Aæ” ÄA 21. ✏JKAø ⇢kA'KKK::(s.) 63, 66, 76, . . . . ·È⇣‡Òª ↵ :3,3,73,16, ÄA :. .16, 16, 21. ÈJ⇣⇣⇣⌦76, K(v.) 602. ≠J (v.) ::·1, 1, 7, ’ª 14, 32, 62, 62, 63, 66, 66, 73, 73,Òm 76, .: :..107, ⌦ÆJ62, ⌦ª (v.) ’ª áÇ (v.) 19.190, 513, 517, ⇢ ' ” 21. ‡Òª 7, 14, 32, 63, . . . : 320, 390. Òm : (I) 190. (s.) :: (II) 107, ⇣ÄA ‡Aæ” : [290], [570], [571]. ‡ áÇ (II) 19. ✏JKAø qÇ 4. ·≠J K : :(v.) 16, 21. ⇣ ÈJ kA K 602. ª È⇣‡Aæ” ÆJ ⌦ ⌦ ⇢589. ' K(s.) ⌦ ÈJÒm⇣⌦kA : 602. : 107, 190, 512, 516, 588. ÄA KK::(v.) : 16,: 21. ·È⇣✏JKAø ⇣áÇ ≠J ≠J ÆJ⌦⌦⌃Jª⌦⌦ª∫: ⇣K 95, 103. ⇣ ÈJ kA 602. ‡Aæ” 603. (II) 19. ⌦ ⌦ qÇ : 4. ‡Aæ” áÇ ⇢ ' A“ª An ancient Greek treatise on magic 129 Òm (s.) : 107, 190, 512, 516, 588. ⇣ ⇣·≠J ÄA K : :(v.) 16, 21.squares áÇ ÆJ⌦⌦⌦⌃Jªª⌦∫:⇣K 245. ·È✏J⌦KAø KAø : 4. (II) ‡ 19. ≠J qÇ : 4. ⇣ (v.) : qÇ ÈJ⇣⌦kAK: : 4.602. ⇣✏JÆJ⌦ª : ⇣ 110. ⌦KAø » ≠J ⌃J ∫ K áÇ ··È≠J KAø ⇣⇣ ✏ :(v.) qÇ 4. : (II) 19. áÇ ⇣≠J ✏ÈJÆJ⌦⌦ª⌃Jª⌦∫⇣K : 624. ≠J⇣J∫” : 109. á Ç ⌧” : 602. 363, 377, 379. ≠J ÈJ kA K : ⇣ ✏ ⌦ ⇣ » áÇ ááÇ Ç⌧”(v.) : 363, 377, 379. ⇣áÇ :: 66, (II) 19. ⇣È⌦✏JÆJ⌦ªª⌦ ⇣ ÄAK : : 16, 21. (v.) (II) 19. (s.) : 293, 449, 481, 482, qÇ : 4. ∫K ⌦ ⇣ ⌦ B≠J :⌦⌦⌦⌃Jª7. » ✏ ≠J ⇣ ⌧” : 363, 377, 379. á Ç ≠J 497, ⇣ 487, áÇ (v.)496, : (II) 19.500, [571]. ⇣K » ‡Òª (v.) : 1, 3, 7, 14, È⇣È⇣✏J⌦✏JÆJ:ÆJ⌦⌦⌦⌃Jªª7. 32, 62, ÜÒÇ ⌧” : 63, 419.66, 73, 76, . . . . ∫ » qÇ : 4. ⇣ B≠J ⌦ ª ⌦⌦ ⇣ ÜÒÇ 419.377, 379. á⇣⇣⇣ Ç✏ ⌧”⌧”: :363, ∫11. K525. áÇ » áÇ áÇ (v.) : (II) 19. ⌦ÆJ⌦⌃Jªª:⌦7. ⇣ÈÈ⇣B✏J✏J⌦ÆÀÆJ °≠J : 7, ⇣ ⌧” :: 419. ⇣áÇ ⇣ ⌦B :⌦⌦⌃J7. ÜÒÇ 420. ✏ ≠J ∫ K ⇣K » á≠í ⇣Ç⌧”(v.) 363, 321, 377, 379. 11. °≠JÆÀ⌦⌃J:⌦⌦:∫11. :: 321, áÇ : (II) 19.324, 459, 512, 529, [530]. ⇣ ⇣ ✏ ≠í : 321, 321, 324, [278], 321,379. 321,459, 324,512, 460, 529, [530]. : 419.377, ⇣⇣ÇÇ✏ ⌧”⌧”⌧”:: 363, » ááÜÒÇ BÈ⇣≠J :ÆÀ7.:: 11. ⇣ 363, 377, 379. 11. ¢ ° ⇣ áÇ ⌃J ∫ K 530, [531]. ≠J K ⌦⌦⌃J7.⌦⌦:∫11. ⇣ Ç✏513, »» á≠í 321, 377, 321, 379. 324, 459, 512, 529, [530]. ⌧” ::⇣ :363, È⇣¢:::ÆÀÆÀ570 : 11. ⇣ ÜÒÇ ⌧” ’ÀB° (ver.), 574, 623. [571], 575, 624. [529]. ≠J í ⌧ K :: 419. [528]. ⌦ áÇ ⇣ ⇣⇣⇣ ⌦í⌧”⌧:K::321, ≠J [528]. ≠í 321, 324, 459, 512, 529, [530]. °BÈ⇣¢ÆÀ: ÆÀ:7.:11.11. 419. » ÜÒÇ ÜÒÇ 419.377, 379. á Ç✏ ⌧”⌧”:⇣: 363, » ⇣B°BÈ¢ÆÀ::ÆÀ7.:7.:11. ⇣ ⌦í⌧”⌧K:: 419. ≠J [528]. hÒÀ (v.)11.: (I) 9. ÜÒÇ ⇣ Ç✏ ⌧”(v.) ≠í : 321, 324, 459, 512, 529, [530]. Q ¢ : (I)321, ⇣È°¢ÆÀÆÀ:: 11. á : 363, 377, 379. ⇣: K321, Q≠í ¢⇣ ⌦í(v.) (I)321, 324, 459, 512, 529, [530]. ≠J ⌧ : : [528]. BB⇣ :: 7. 7. 11. ≠í ÜÒÇ⌧”: :321, 419.321, 324, 459, 512, 529, [530]. °È°¢ÆÀÆÀÆÀ::: 11. 11. 11. ¢ ⇣

⌦ QQ—⌦¢ :: 16. ⇣⇣ØP ⇣⇣ : 8. 444. È ¢ÆK:: 63, 63, 444. È 8. I ⇣ ⇣ØP :: 8. ⌦Å . j.´ (v.) : () 4. ⇣K11. È ØP ≠J í ⌧ : [528]. Q — ¢ : 16. ⌦ : 63, 444. 228 Arabic glossary 63, 444. 444. QQ⌦⌦⌦¢¢ :: 63, (I) 208, 209, 210, 211, 212, 212, 213, 214, Å Æ K : 11. ✏ (v.) ——¢¢ :: 16. ¯¯P (v.) :: (III) 54, 600, 601. Y´ (I) 321, 16. Që ⇣ P (v.)214, (III) 54,: 491, 491, 600,324. 601. : 63, 444. ⌦ Æ K (v.) : (I) 208, 209, 210, 211, 212, 212, 213, ¯ P (v.) : (III) 54, 491, 600, 601. (v.) : (III) 54, 492, 601, Q—¢¢ÆK::(v.) : (I) 9, 320, 332, 368, 390, 412, 607, 614, 630. (v.) : (I) 9, 320, 332, 368, Å 11. 16. Q——⌦¢¢390, 16. 444. 63, 16. ⇣ ::: (v.) 602. 413, 608, 615, 631. ë 213, 214, ✏ : 105, 105, 106??, 565. ⇣Ë Y´ Å Æ K : 11.: (I) 208, 209, 210, 211, 212, 212, °É Q‡Aí ⇣ Å 11. 444. 445. 63, °É : 45, 100, 103, 104, 112, —⌦¢⇣ÆKÆ⇣::K 16. È¢É@  ⇣°É : 19, 229, 246, 255, 264, [273], 286, 214, 295, 298, ⇣ 213, ë Æ K (v.) : (I) 208, 209, 210, 211, 212,[273], 212, È¢É@  È¢É@  Å : 11. 119, 167, 302, 427, ... . ⇣  Æ⇣KK:⇣:: 16. 11. —Å ¢ Å Æ 11. ⇣ È¢É@ È¢É@  ‡Aí : 19, 229, 246, 255, 264, [273], 286, 295, 298, ë ÆÆ⇣KKÆK(v.) : (I) 208, 209, 210, 211, 212,[273], 212, 213, 22214, ), 27, 32, 33, 53, XY´ °É@ ::: 22 (((XY´ ë (v.) : (I) 208, 209, 210, 211, 212, 212, 213, 214, ⇣ — ¢ : 16. °É@ 22 XY´ Å : 11. ≠ì (v.) : XY´ (I) 407, 407, [478], [478], 507, 507, 606. 606. : 11. ⇣ (v.) : (I) … Æ K (v.) : (I) 18. ⇣ 18. °É@ : 22 ( ‡Aí Æ K : 19, 229, 246, 255, 264, [273], [273], 286, 295, 298, 56, 63, 80, 84,[478], .[478], . . ; 507, (cen132 Translation ⇣ ≠ì (v.) :63, (I) 407, 606. ë Æ K (v.) : (I) 208, 209, 210, 211, 212, 212, 213, 214, ≠ì (v.) : (I) 407, 507, 606. ⇣ ⇣ ë⇣ÆK ⇣:(v.) (v.) (I) 208, 208,209, 209,210, 210,211, 211, 212, 212, 212, 212, 213, 214, Å 11.:: (I) È¢É@ (v.) ë 208, 209, 210, 211, 213, 214, ⇣ ≠ì (v.) : (I) (I) 407, [478], 507, 507, 606. 606. tral part) 96,407, 167.[478], ≠ì : …132 Æ K (v.) : (I) 18. È¢É@  ‡Aí : 212, 19, 229, [273], [273], (v.)295, : (I)298, 407, [478], 507, 606. ⇣ 286, Translation ⇣ KÆÆ⇣:KK(v.) 212, 213, 246, 214, 255, . . . . 264, 134 Translation È¢É@  ⇣ ‡Aí :11. 19, 229, 246, 255, 264, [273], [273], 286, 295, 298, Å ë Æ : (I) 208, 209, 210, 211, 212, 212, 213, 214, …ì (v.) : (VIII) ⇣ È¢É@  : 147, (VIII) ˙ÊÓ (VIII) [23], 79, 112, 124, 131, 132,(v.) 145, 155, ⇣⇣(v.) : 115, 121, 223, 225, 248, …‡Aí Æ112, K⇣EÆ⇣(v.) : : (I) 18. 246, Translation …ì (v.) : (VIII) KK(v.) ::: 19, 229, 255, 264, [273], [273], 286, 295, 298, …ì (v.) : (VIII) 19, 229, [264], 132 Translation [23], 79,132 124, 131, 132, 145, 147, 155, ‡Aí Æ 19, 229, 246, 255, 264, [273], [273], 286, 295, 298, ë Æ K : (I) 208, 209, 210, 211, 212, 212, 213, 214, 426, 428, 429, 431, . . . . 507, ≠ì (v.) : (I) 407, 606. ‡Aí Æ K : 19, 229, 246, 255, 264, [273], [273], 286, 295, 298, 132 Translation …ì (v.) (VIII)155, [478], ⇣⇣[273], Translation Translation 286, 298,124,134 ˙ÊÓ E (v.) (v.)::[273], (VIII) [23], 295, 79, 112, 131,134 132,(v.) 145, 147, …ì (v.) :: :(VIII) (VIII) ≠ì (v.) (I) 407, [478], 507, 606. …132 Æ K (I) 18. : ⇣ ⇣ K(v.) Translation …‡Aí KÆEKÆ(v.) : :(I) 18. ≠ì (v.):214, :()(I) 407, ⇣ë⇣ÆAÓ338, (I) 408, [478], [479],507, 508,606. (I) 208, 209, 210, 211, 212,[273], 212, 213, : 347, 19, 229, 246, 255, 264, [273], 286, 298, 357. (v.) 22. (v.):295, :147, 407, [478], 507, 606. ÈK : 102, 519. ©≠ì ì (v.) ()(I) 22. 134(v.) Translation ⌦ ˙ÊÓ E (v.) : (VIII) [23], 79, 112, 124, 131, 132, 145, 155, ⇣ © ì : () 22. …132 Æ K (v.) : (I) 18. 607. © ì (v.) : () 22. ⇣ …⇣ Æ⇣ÆKK Æ(v.) (v.) (I) 18. (v.) (I) 18.246, 255, 264, Translation K : 19, 229, [273], [273], 286, 295, 298, …‡Aí ::: (I) 18. …ì (v.) : (VIII) ì (v.) :: () () 22. 22. 134(v.) Translation ÈK⌦ AÓ⇣ E :⇣(v.) 102, 519. [23], 79, 112, 124, 131,©© ì …ì (v.) (VIII) ˙ÊÓ : (VIII) 132, 145, 147, (v.) ::: () 22.155, (v.) (VIII) 198, 200, 201, ˙ÊÓ E (v.) : (VIII) [23], 79, 112, 124, 131, 132, 145, 147, 155, …ì (v.):295, : 16, (VIII) [23],255, 112,[273], [273], K(v.) : 19, 229, 246, 264, 286, 298, …‡Aí ÆKK :Æ(v.) : :197, (I)(VIII) 18. 134 Translation (s.) [23]. 233, 234, 609, 611, 616. ®Ò 12, 231, 236,79, 299, 313, 328, 389. (v.) (VIII) ©…ì ì (s.) : : 16, [23]. ÈK⇣˙ÊÓ EE :(v.) 102, 519. © ì (s.) : 16, [23]. ˙ÊÓ : (VIII) [23], 79, 112, 124, 131, 132, 145, 147, 155, ⌦…ÆAÓ⇣124, 131, 132, 132, 145, . . . . © ì (s.) : 16, [23]. (v.)::: (I) (VIII) [23], 79, 79, 112, 112, 124, 124, 131, 131, ©132, 132, 145, 147, 147, 155, KE (v.) 18. [23], ˙ÊÓ (v.) (VIII) 145, 155, ì (s.) (v.) () 134 Translation (v.)::::16, (I)22. 22. (s.) 16, [23]. ®Ò ©ì [23]. (v.) : () 22. ÈK⇣ÈK⇣⌦ AÓAÓ⇣ KEE::: 12, 102,197, 519. 520.231, 236, 299, 313, 328, 389. (s.) : 16, [23]. 102, 519. 134 © ì (v.) : () 22. …⌦ ÆK (v.) 18. [23], 79, 112, 124, 131,©132, (v.):: (I) (VIII) 147, 155, ìÒ”(v.) :145, 39, 132, 147, 230, 230, [568], [568],Translation 630. ì : () 22. ⇣ÈK˙ÊÓ :(s.) 39, 132, 147, 630. 16, [23]. Ë 197, 231, 236, 299, 313, ®Ò 197, 231, 236, 299, 313, 328, 389. ⇣⇣ÈK⌦ AÓAÓKE::: 12, 102, 519. © ìÒ” : 39, 132, 147, 230, [568], 630. © ìÒ” : 39, 132, 147, 230, [568], 630. Translation 134 Translation 102, 519. [23], 79, 112, 124, 131,134 134 Translation E :(v.) : (VIII) 132, 147, 155, ⌦AÓ328, ÈK˙ÊÓ 102, 519. ì (s.) : 16, [23].  ⌦®Ò © ìÒ” ::145, 39, 132, 147, 230, [568], 630. 389. 39, 132, 147, 230, [568], 630. ©ìÒ” ì (s.) : 16, 12, 197, 231, 236, 299, 313, 328, 39, 132,[23]. 147, 230, 230, [568], [569],Translation Ë 389. ©134 : 39, 132, 147, 630. 134 ⇣ AÓKKE:::(v.) Translation ®Ò 12, 236, 389. © ì (s.) : 16, [23]. ˙ÊÓ :197, (VIII) [23], 79,299, 112,313, 124,328, 131, 132, 145, 147, 155, ⇣ ÈK 102, 519.231, ⌦ á Ø (v.) : (III) 462; (VIII)  389. ©⇣ Ø ì (s.):: (III) 16, [23]. (v.) 462; (VIII) 458, 520, 521, 540, 631. ⇣ Ø(v.) ºA 460.197, 231, Ë ®Ò 12, 236, 299, 313, 328, á : (III) 521, 540, Ë ⇣®ÒAÓJÎKKKE::::: 12, á (v.) : (III)462; 462;(VIII) (VIII)458, 520, 134 Translation ®Ò 12, 197, 231, 236, 299, 313, 328, 389. ⇣ ÈK 102, 519. 197, 389. ©©⇣ ìÒ” ìÒ” : 39, 132,462; 147,(VIII) 230, [568], 630.  I : (I) 231, 222, 236, [223].299, 313, 328, á Ø (v.) : (III) (v.) : (III) 463. (VIII) 459, .ºA⇣⌦ kJÎ. : (v.)  (v.) (III) 462; (VIII) 458, 520, 521, 540, 540, :˙228, 39, 132, 147, 230, [568], 630. ˙áØØ: ✏(v.) ::: (I) 228, 574, 580. 460. ˙ØË (v.) (I) 580. 461. ˙574, Ø: 462; (v.) :(VIII) (I) 228, 574, 580. Ø (v.) (I) 228, 574, 580. (v.) (III) 458, 520, 521, ⇣K@(I) ©⇣⇣ ìÒ” ìÒ” : :522, 39, 132, 147, 230, [568], 630. ÈK⌦ AÓkK.E:: 12, 102,197, 519. 521, 541, 548, 548, 551, 630. ®Ò 231, 236, 299, 313, 328, 389. ˙ Ø (v.) (I) 228, 574, 580. Ë ˙ Ø (v.) : 228, 574, 580. ÜA Æ (s.) : 543. I (v.) : (I) 222, [223]. ✏ © : 39, 132, 147, 230, [568], ✏⇣(s.) (s.) :: 543. ⇣K@Æ551, ⇣Ø Z˘Î :: (V) 522, 525, 524, [527], 526, 530,  543??, 559, 560, 586. . JÎ :(v.) ˙554, Ø:556, : (I) 228, 574, 580. (v.) (V) 9, 9, 523, ˙Æ551, (v.) (I) 228, 574, 580. ºA 460. ÜA⇣ÜA 543. 554, 557, 558, 566. K@ (s.) :(v.) 543. Ë ✏ ⇣ ®Ò K : 12, 197, 231, 236, 299, 313, 328, 389. Ë ✏ á Ø (v.) : (III) 462; (VIII) ⇣ I k  (v.) : (I) 222, [223]. @ : [207]. Ë ⇣ 531, 552, 557, 560, 561, 587. ÜA Æ K@ (s.) : 543. I. . JÎ k. : (v.) (v.) : (I)9,222, [223]. (v.) :462; (I):560, 228, 574, 580. (s.) :˙ØØ ˙ÆØ (v.) (I)Ø:559, 228, 574, (s.) (s.) 469.580. ÜA K@469. (s.) :::(s.) 543. 469. á⇣Ø⇣:Ø (v.) :469. (III) (VIII) Z˘Î : :(V) 522,[223]. 524, 526, 530,Ø543??, 551, 586. (s.) 544. (s.) :556, 543. ºA 460. I (I) 222, .IkJÎk..K@(v.) ⇣ ®Ò : :12, 197, 231, 236, 299, 313, 328, 389. á Ø (v.) : (III) 462; (VIII) ºA 460. Ë Øá⇣:˙Ø(s.) 424, [571]; á⇣ ØÒÀ@ XY´ 1,580. 18. [207]. :::(III) (VIII) Ø (v.) (I) 574, ::(v.) 424, [571]; 1, 18. (v.) (I) 228, 574, 580. ⇣Ø ⇣228, (v.) : (I)9,222, . JÎ. (v.) (s.) ::462; 469. :˙424, 469. (s.) 469. Ø 543??, (s.) 469. Z˘Î : (V) 522,[223]. 524, 526, 530, 556, 559, 560, 586. á Ø :551, 424, [571]; á ØÒÀ@ XY´ 1,)1,18. ✏ á Ø : [571]; á ØÒÀ@ XY´ 18. 425, [572]; ( 1, ºA : 460. ⇣ ⇣ ⇣ ºA. JÎ 460. ✏⇣(K@::[273], ÜA Æ560, (s.) :˙Õ@ 543. Ë (v.) I kJÎk526, :460. [207]. ºA Yg .. @:@:(v.) : :(I) 4,222, 6, 9,[223]. 191, 250, [257], [261], [318], 408, 530, 599. ⇣Ø˙ØØ:18. áÜA 424, [571]; á ØÒÀ@ XY´ 1, 18. ⇣ ⇣ Ø (s.) : 469. . (v.) : ( ) 2, 5; ( ˙ Ø ) 224, 225, Ø (s.) : 469. I (v.) (I) ˙ Ø (v.) (I) 228, 574, 580. Ø ˙Õ@ ) 2, 5; ( ˙ Ø ) 224, 225, Ø (v.) : ( ˙Õ@ ) 2, 5; ( 224, 225, Ø (v.) (I) 228, 574, 580. : [207]. Ø (v.) : ( ˙Õ@ ) 2, 5; ( ˙ Ø ) 224, (v.) (I) á 424, [571]; á ØÒÀ@ XY´ 1, 18. 9, 522, 524, 530, 543??, 551, 556, 559, 586. .I Æ K@ (s.) : 543. 424,556, [571]; Z˘Î : (V) 9, 522, 524, 526, 530, 543??, 559, 560, 586.1, 18.˙Ø) 225, ⇣ Æ✏⇣:✏⇣K@551, k. @(v.) : [207]. .Yg ⇣ ÜA (s.) : 543. Z˘Î (v.) : (V) 9, 522, 524, 526, 530, 543??, 551, 556, 559, 560, 586. ˙ Ø (v.) : (I) 228, 574, 580. Ë (I) 228, ºA á⇣ Ø@˙⇣ÆØ ÒK@”[273], ::(v.) 14, 457, 499. (s.) 469. 469. : (I) 4, 6, 9, 191, 250, [257], [261], 408, ÜA (s.) 543. Ø:::: 457, (v.) (500. ˙Õ@5;574, ) (2, 5; (599. ˙Ø93, ) 225, 224, 225, :(v.) 14, 457, 14, 458, See p. Ø ([318], ˙Õ@ ):: 499. 2, ˙530, Ø580. ) 224, .k. (v.) I.JÎ @:(v.) :460. : [207]. [207]. á Ø@ Ò ” :” :14, 499. á Ø@ Ò : 14, 457, 499. : (V) 9, 522, 524, 526, 530, 543??, 551, 556, 559, 560, 586. (I) 222, [223].Z˘Î ⇣ Ø (v.) ( ˙Õ@ ) 2, 5; ( ˙ Ø ) 224, 225, Ø (v.) : ( ˙Õ@ ) 2, 5; ( ˙ Ø ) 224, 225, ⇣Yg ⇣ Z˘Î.JÎk. :(v.) (V) 9, 6, 522, 524, 526, 530, 543??, 551, 556, 559, 560, 586. ˙179, Ø (v.) : 499. (I) 228, 574, ˙127, Ø (v.) : ()(s.) (I) 228, 574, 580. ºA :(v.) á⇣⇣Ø :”[273], [571]; á408, ØÒÀ@ XY´ Z˘Î :::73, 9, 522, 524, 559, 560, 586. (I) 4, 9, 191, 250, [257], [261], [318], 408, 530, ÈÍk 72, 75, 76, 78, 79,526, 80, 530, .[257], . . . 543??, 180, 181, ⇣574, (v.) (() ˙Õ@ ): (2, 5; (599. ˙1,Ø580. )18. 224, 225, Ø@ØØn. :424, 14, (v.) :556, 420, 630. (v.) (457, ˙Õ@ 2, 5; ˙ Ø )420, 224, 225, Ø :): 469. Ø125, (v.) () 420, 630. Ø (v.) () 630. ØÒ: 551, (s.) ::(I) 469. 420, 630. 126. I :460. [207]. Yg (v.) ::(V) (I) 4, 6, 9, 191, 250, [261], [273], [318], 530, 599. ˙ (v.) : 228, 580. á Ø@ Ò ” : 14, 457, 499. (v.) (I) 4, 6, 9, 191, 250, .Yg.. . @(v.) : 424, [571]; á ØÒÀ@ XY´ 1, 18. (v.) : (I) 228, 574, 580. :(v.) 457, 499. (v.) : (I) 4, 6, 9, 191, 250, [257], [261], [273], [318], 408, 530, ˙14, ØØ556, (v.) (I) 228, 574,599. Ø (I) 228, 574, 580. ⇣⇣˙Ø127, ⇣ [257], ✏⇣Ø:Ø:551, ⇣⇣ØÒÀ@ á 424, [571]; á ØÒÀ@ XY´ 1, 18. (s.) :):: :469. (s.) ::Ø 469. ºA JÎ : 72, 460. (v.) (181, ˙Õ@ ) (2,˙XY´ 5; (1, ˙Ø580. )18. 224, 225, (v.) : ( ˙Õ@ 2, 5; Ø )630. 224, 225, Z˘Î (v.) : (V) 9, 522, 524, 526, 530, 543??, 559, 560, 586. : 535, 535. [261], [273], [318], 409, (v.) () 420, ÈÍk : 73, 75, 76, 78, 79, 80, . . . . 125, 179, 180, ✏ (v.) : () 420, 630. 536, 536. Ø 424, [571]; á ✏ á Æ J” : 535, 535. ⇣ ⇣ Yg..  (v.) : (I) 4, 6, 9, 191, 250, [257], [261], [273], [318], 408, 530, 599. ˙ Ø (v.) : (I) 228, 574, 580. ⇣ á Æ J” : 535, 535. á Æ J” : 535, 535. ✏Ò(v.) ⇣⇣ 531, Ø (s.) :420, 469. (s.) 469. 600. Z˘Î :73, (V) 9, 522, 524, 526, 530, 543??, 551, 559, 560, Ø(v.) (v.) :2, () 630. á⇣⇣ØØ ”:[273], :˙Õ 14, 457, ˙Õ (v.) ::420, Ø”(v.) (v.) :24, ()420, 630. Ø (v.) (499. ˙Õ@ 2, 5; ( 599. ˙Ø30, ) 225, 224, (v.) :⇣✏Ø (I) 24, 27, 28, 29, 30, 225, 36, . . . . ÈÍk 72, 79, 80, .. 125, 127, 179, 180, 181, ˙Õ :)420, Ø (v.) :(I) ([318], ˙Õ@ ): (v.) 5; ((I) ˙530, Ø581. )586. 224, Ø ::::556, () 630. Èk 247. :25, (I) (v.) :⇣J” 630. ÆØ@Ø@127, :() 535, 535. ˙ (I) 228, 574, 580. . ...:::::(v.) Yg (v.) :73, (I)75, 4, 76, 6, 191, [261], 408, ⇣ÈÍk ZA (s.) 469. ÈÍk 72, 75, 76,9,78, 78, 79,250, 80, .[257], ....˙Õ .Ø 125, 179, 180, 181, (v.) (I) 228, 575, á Æ J” 535, 535. Ò : 14, 457, 499. (s.) : 469. : 535, 535. ⇣ Ø (s.) : 469. 72, 73, 75, 76, 78, 79, 80, . . . . 125, 127, 179, 180, 181, Ø (s.) : 469. 72, 73, 75, 76, 78, 79, 80, . (v.) : (V) 9, 522, 524, 526, 530, 543??, Ø556, ((I) ˙Õ@5;630. ) (2,˙Ø5; ( ˙Ø) 225, 224, 225, :: :(v.) ((I) ˙Õ@559, ):: :499. 2, )586. 224, ” (v.) : (v.) 14, 457, Ø (v.) () 420, 630. () 420, Z˘Î 560, ⇣ .107, ˙Õ (v.) Èk . :: 72, 247. áá⇣Ø˙Õ Ø@Ø@127, Ò✏ÒØØ ”551, :(v.) 14, 457, 499. ÈÍk 73,126, 75, 127, 76, 78, 179, 180, 181, 125, . . . 79, . 80, . . . . 125, ZA (s.) :: 469. (s.) 470. Ø:Ø97. (v.) ((I) ˙Õ@5;630. ) (2,˙Ø5; ( ˙Ø) 225, 224, 225, ⇣˙Õ Ø✏⇣J”Ø97. ((I) ˙Õ@ ):::2, 224, ✏  : 247. ⇣Ë Yg ˙Õ : :(v.) 16, (v.) :469. (v.) 420, ˙Õ@. 125, : ˙Õ@ 16, ˙Õ@ : ()16, 97.)630. (v.) ::(v.) () ˙Õ@ 16,420, 97. á Æ 535, 535. Èk 521, 553. ⇣ ⇣ ⇣ ZA Ø (s.) . ÈÍk : 72, 73, 75, 76, 78, 79, 80, . . . 127, 179, 180, 181, © Ø (v.) : ( ˙Õ@ ) 2, 5; ( ˙ Ø ) 224, 225, Èk  : 247. Æ:(v.) J”✏✏⇣ (I) 535, (v.) ((v.) ))::2, 5; (); ˙Õáá408, : :Ø ((˙Õ@ )420, () ˙224, Ø)˙225, 225, (v.) Ø ((535. ˙Õ@ 2, 5;630. ()2,˙(2, ) 224, 224, (v.) : :(I) (I) 2, (Ø)5; ⇣˙Õ@ Èk : 247. ˙Õ@ Ø224, (v.) :Ø ˙Õ@ 2, 5; Ø5; (I) 4, 6, 9, 191, [257], [261], [273], ˙Õ [318], 530, 599. (v.) () Ø::(v.) (v.) () 420, ✏.. .250, ˙Õ (v.) :5):((I) ⇣Ë Yg ˙Õ (v.) : :(v.) (I) 24, 24, 25,˙630. 27,( 28, 29, 30, 30, 36 Æ J” 535, 535. ⇣ : 16, 97. ˙Õ@ : 16, 97. ⇣ ⇣ : 521, 522, 554. J”Ø(v.) : 535, 535. Èk 247. 553. ©áØÆ225, :Ø(::˙Õ@ )420, 2, 5; ( ˙Ø593, )630. 224, 225, 225, [526], 576, 595. (I) 421, 631. (v.) :: 97. () 420, (v.) (I)16, 631. (v.) () ✏✏. :: 521, ⇣ËÈ⇣ØP ⇣Yg ˙Õ@ :(v.) ˙Õ@ : 16, 97. ˙Õ (v.) :421, (I) 630. ⇣ ˙Õ (v.) : (I) ⇣ 553. 8. ©ØØØ(v.) : :Ø ˙Õ@ )420, 2,()5; ( ˙Ø)630. 224, 225, ✏.  : 247. 8. (v.) (I) 421, 631. ⇣ËÈk ≠ (v.) 630. ËYg Yg 521, 553. (v.) 420, (v.) ()420, 630. (v.) :::(:() () 420, 630. 553. ˙Õ ::: (I) ˙Õ (v.) (I) (v.) (I) : 16, 97. (v.) : (v.) (I) ˙Õ@ : ˙Õ@ 16, 97. ⇣ÈØP ⇣✏ :: 521, ˙Õ@ : 16, 97. ˙Õ@ : 16, 97. ⇣ (v.) Ë Yg : 8. 521, 553. Ø ()◆ 420, ◆ ◆≠˙Õ ◆˙Õ : (I)630. (v.) :: (v.) (I) ⇣¯È⇣ØP ⇣⇣✏P:(v.) :133, 16, 97. ˙Õ@ : ˙Õ@ 16,149, 97. : 144, 133, 144, 149, 340. : 133, 340. : 144, 133, 144, 340. 149, 340. ⇣ : 149, ≠˙Õ Ø (v.) : () 420, 630. 553. 54, 491, 600, 601. ˙Õ (v.) : (I) ⇣ÈËÈØP ⇣Yg ˙Õ (v.) : (I) ØP :: 8.521, 8. : (III) (v.) : (I) ◆ ◆ (v.) : (I) ˙Õ@ : 16, 97. ˙Õ@ : 16, 97. 8. 133, 149, 144, 340. 149, 340. : 133,: 144, ⇣P :(v.) ¯È⇣ØP ◆16, ◆ (v.) 8. : (III) 54, 491, 600, 601. ˙Õ : (I) : 16, 97. ˙Õ@ : ˙Õ@ 97. 133,: 149, 144, 340. 149, 340. : 133,: : 24. 144, : 24. : 24. 24. ⇣⇣

≠Ø ⇣ ˙Õ@ÒK : 16, ˙ÕØ : (I) 24, 24, 25, 27, 28, 29, 30, 30, 36, . . . . ⇣97. ⇣ (v.) (v.) : () 420, 630. ≠ Ø (v.) :29, () 420, 24, 24, 25, ≠ 27, 28, 30, 630. 30, 36, .28, . . 29, . glossary 229 ˙ÕÒ⇣K(v.) 30, 30, 36, . . . . ˙Õ@ : 16,: (I) 97.24, 24, 25, 27, Arabic ¯ ˙Õ@Ò⇣K(v.) : 16, 97. ˙Õ : (I) 24, 24, 25, 27, 28, 29, 30, 30, 36, . . . . ˙Õ (v.) : (I) 24, 24, 25, 27, 28, 29, 30, 30, 36, . . .glossary . Greek ˙Õ@Ò⇣K(v.) : 16,: (I) 97. 24, 24, 25, 27, 28, ¯ 29, 30, 30, 36, . . . . ◆ ⇣ËQÂÑ 144, 340. ¯ ἀνισόπλευρος : (page) 105n. ⇣ 149, ⌦ : 133, ˙Õ@ 97. ◆⇣ ÒÒK⇣K:: 16, ˙Õ@ 16, 97. ¯ ἀντικείμενοι : 25n. ËQÂÑ⌦ : 133, 144, 149, 340. ◆ ἀρτιάκις ἄρτιος : 89n. ¯ PAÇ⌦ : 24.⇣ËQÂÑ : 133, 144, 149, 340. ⌦ ¯¯ ἀρτιάκις περισσός : 89n. ◆ ⇣ËQÂÑ PAÇ 24. 144, 149, 340. ⌦ :: 133, ἀρτιοπέριττος : 89n. Q⌦Ç⌦ : 7.⇣◆PAÇ : 24. : ⌦ ◆ ⌦ : 133, ⇣ËQÂÑ ËQÂÑ 144, 149,Greek 340. ἰσόπλευρος : 105n. An ancient 135 : 133, 144, 149, 340. treatise on magic squares ⌦ QPAÇ 24. 7. ⌦Ç⌦ :: 7. An ancient Greek treatise on magic squares 135 ὁμώνυμος : 93n. Q⌦Ç⌦ @ :⌦: :48, ¯QÂÑ⌦ , QÂÑ 49, 50, 52, 53, 60, 68, 68, 70, 71, 75, 82, 84, . . . . 48, 49, 50, 52, 53, 60, 68, 7. ὅρος : 225. PAÇ : 24. ⌦ ⇣È◆ J÷fl⌦ : 141. PAÇ : 24. 68, 70, 71, 75, . . . . ⌦ 71, 75, 82, 84, . . . . ◆Q⌦Ç⌦ ⌦: ,7.QÂÑ⌦ @ : 48, 49, 50, 52, 53, 60, 68, ὁ68,ὑπό70,: 225. ⇣ȯQÂÑ fl ⌦ J÷ : 141. : 107n. , QÂÑ⌦ @ : 48, 49, 50, 52, 53, 60, 68, πάντη 68, 70, 71, 75, 82, 84, . . . . Q¯QÂÑ 7. 40, ˙Ê÷fl⌦ , ·÷ 42, 51, 54, 55, 60, 71, 76, 28, 40, 42, 51, 54, 55, 60, Q⌦⌦ÇflÇ⌦@ ⌦:⌦::⌦:28, περισσάρτιος : 89n. ,7.QÂÑ : 48, ⌦ @28, fl⌦ ,⌦·÷76, fl⌦@ :78, 82, 84, . .51, . .52, ˙¯QÂÑ Ê÷71, 40,49, 42,50, 54,53, 55,60, 60,68, 71,68, 76,70, 71, 75, 82, 84, . . . . τετράγωνον : 105n. ¯QÂÑ , QÂÑ @ : 48, 49, 50, 52, 53, 60, 68, 68, 70, 71, 75, 82, 84, . . . . : 19n. ¯QÂÑ⌦⌦ , QÂÑ⌦⌦ @ : 48, 49, 50, 52, 53, 60, 68, τινές 68, 70, 71, 75, 82, 84, . . . . ὑπερβαίνειν : 117n.

Bibliography B¯ uzj¯an¯ı, Ab¯ u’l-Waf¯ a’ : see Sesiano. Y. Eche [‘¯ ash] : Les bibliothèques arabes publiques et semi-publiques en Mésopotamie, en Syrie et en Egypte au Moyen Age. Damas 1967. P. Fermat : Varia opera mathematica. Toulouse 1679. : Œuvres complètes, ed. P. Tannery & Ch. Henry (4 vols). Paris 1891–1912. Fihrist : see Ibn al-Nad¯ım. B. Frénicle de Bessy : “Table generale des quarrez de quatre”, in Divers ouvrages de mathematique et de physique (Paris 1693), pp. 484–507. G. M. Ghidini : “I termitidi devastatori delle cellulose”, Bollettino del R. Istituto di patologia del libro III/2 (1941), pp. 62–69. Ibn ab¯ı Us.aybi‘a : ‘Uy¯ un al-anb¯ a’ f¯ı .tabaq¯ at al-at.ibb¯ a’, ed. A. Müller. Cairo 1882 (2 vol.). Ibn al-Nad¯ım : Kit¯ ab al-fihrist, ed. with notes by G. Flügel, J. Roediger and A. Müller (2 vol.). Leipzig 1871–1872. : The Fihrist of al-Nad¯ım, tr. B. Dodge (2 vol.). New York 1970. am¯ a’, ed. J. Lippert (u. A. Ibn al-Qift.¯ı : Ibn al-Qift.¯ıs Ta’r¯ıkh al-h.uk¯ Müller). Leipzig 1903. W. Kutsch : T ¯ abit b. Qurra’s arabische Übersetzung der ᾿Αριθμητικὴ Εἰσαγωγή des Nikomachos von Gerasa. Beirut 1959. M. Moschopoulos : see Tannery, Sesiano. Nicomachos : Nicomachi Geraseni Pythagorei Introductionis arithmeticae libri II , ed. R. Hoche. Leipzig 1866. See also Kutsch. J. Sesiano : “Herstellungsverfahren magischer Quadrate aus islamischer Zeit (II)”, Sudhoffs Archiv 65 (1981), pp. 251–265. : “Herstellungsverfahren magischer Quadrate aus islamischer Zeit (II0 )”, Sudhoffs Archiv 71 (1987), pp. 78–89. : Un traité médiéval sur les carrés magiques. Lausanne 1996. : “Le traité d’Ab¯ ul’-Waf¯ a’ sur les carrés magiques”, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 12 (1998), pp. 121–244.

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: “Les carrés magiques de Manuel Moschopoulos”, Archive for history of exact sciences 53 (1998), pp. 377–397. : “Une compilation arabe du XIIe siècle sur quelques propriétés des nombres naturels”, SCIAMVS 4 (2003), pp. 137–189. : Les carrés magiques dans les pays islamiques. Lausanne 2004. : “Magic squares for daily life”, in Studies in the history of the exact sciences in honour of David Pingree [= Islamic philosophy, theology and science, vol. LIV], Leiden 2004, pp. 715–734. : burg 2014.

. Saint Peters-

: Magic squaresNin the tenth century, two Arabic treatises by Ant.¯ ak¯ı xı.ki Cham 2017. and B¯ uzj¯ an¯ N ancient times : Magic squares, their history and construction from to AD 1600 . Cham 2019. N k T. E. Snyder : Our enemy, the termite. Ithaca/New York 1935. N M. Stifel : Arithmetica integra. Nuremberg 1544. X N =Moschopoulos xki P. Tannery : “Le traité de Manuel sur les carrés magii ques, texte grec et traduction”, Annuaire de l’Association pour l’encouragement des études grecques en France N k 20= (1886), (N, k) pp. 88–118. Reprinted in Tannery’s Mémoires scientifiques, IV, pp. 27–60. k : Mémoires scientifiques (17 vols). Paris/Toulouse 1912–1950. k=2 1 1

2 2

3 3

4 1

5 2

6 3

... ...

6 6

7 7

8 1

9 2

10 3

... ...

14 7

15 8

16 2

17 3

18 4

19 5

... ...

21 7

22 8

23 9

24 3

2 2

... ...

... ...

31 16

32 2

33 3

... ...

47 17

48 3

... ...

63 18

64 4

... ...

79 19

80 5

7 4

8 2

9 1

10 2

11 3

12 3

13 2

14 3

15 4

16 1

17 2

18 2

19 3

20 2

... ...

k=3 1 1

2 2

... ...

27 1

28 2

k=4 1 1

15 15

16 1

81 1

82 2

... ...

... ...

Index Ab¯ u K¯amil : 10. Ab¯ u’l-Waf¯a’ : see B¯ uzj¯ an¯ı. al-Ant.¯ak¯ı : 4, 9–13, 165, 202n, 212. Archimedes : 10. al-Asfiz¯ar¯ı : 12. al-‘Asqal¯an¯ı : 11. Bayt al-h.ikma : 163. Boëtius : 10. bordered magic squares : see magic squares. al-B¯ uzj¯an¯ı, Ab¯ u’l-Waf¯ a’ : 4, 12, 124, 129, 165–169, 175n, 188, 213. chess (moves) : 88–91, 204. complements : 2 & 19n (def.), 25n. composite magic squares : see construction methods, magic squares. construction methods : 3 (def.) ♦ ordinary magic squares: 4, 5, 16–19, 164–165 ♦ bordered squares: (odd orders) 4, 13–14, 18–27, 165–173 ◦ (principles) 169–173 ◦ (with separation by parity) 13–14, 30–87, 174– 203  (evenly-even orders) 14, 88–91 (4 × 4), 94–99, 204–205 (4 × 4), 209–212 ◦ (principles) 210–212  (evenly-odd orders) 14, 90–95, 205– 209 ◦ (principles) 206–209 ♦ composite squares: 3, 104–123, 213–219  (odd orders) 110–113, 213 ◦ (example) 213  (evenly-even orders) 14, 104–107, 214–216 ◦ (examples) 124–133, 136–137, 140–143, 146– 153  (evenly-odd orders) 14, 106–111, 214, 215–219 ◦ (examples) 134–135, 138–139, 144–145, 154–161. cross, central : 14, 109nn, 114–123, 154–161, 216–219. Diophantos : 10. A. Dürer : 7. early readers (interpolations) : 15, 17 & 17n, 19 & 19n, 21 & 21nn, 25 & 25n, 35, 39, 41 & 41n, 43 & 43n, 45 & 45nn, 47, 51 & 51nnn, 53 & 53n, 55 & 55n, 57, 59 & 59n, 69, 71 & 71n, 73 & 73n, 75 & 75n, 77

234

Index

& 77n, 89 & 89n, 91 & 91nn, 93 & 93nn, 95 & 95n, 97 & 97n, 99 & 99nn, 107 & 107nnn, 111, 113 & 113n, 115 & 115n. equalize, equalization : (meaning) 49 (§ 18), 183  (equalization rules) 48–55, 119, 183–187, 189–196, 198–201. Euclid : 10–11, 16–17, 89n, 163. L. Euler : 8. P. Fermat : 7–8. Fihrist : 17n, 163. B. Frénicle de Bessy : 3n. ‘general construction methods’ : see construction methods. gloss : see early readers. hidden numbers : 10–11. al-H . ij¯az¯ı : 16–17, 163. Ibn ab¯ı Us.aybi‘a : 12–13.

Ibn al-Haytham : 5, 12, 165n. Ibn al-Nad¯ım : 17n, 163–164. Ibn al-Qift.¯ı : 12–13. Ish.¯aq ibn H . unayn : 16–17, 163. al-Khayy¯am : 12. al-Kh¯azin¯ı : 12–14, 16n, 173, 212. al-Khw¯arizm¯ı : 10. ‘large’ number : 2 & 19n (def.). magic squares : (ordinary) 1 (def.), 16–19, 164–165, 174 ◦ (bordered) 1–2 (def.), 164–165 ◦ (composite) 3 (def.) ◦ (pandiagonal) 8 & 205 (def.) ◦ (number of possibilities) 3, 8 ◦ (with non-consecutive numbers) 5 ◦ (magic use) 6–7 ◦ (orders) 3 (def.) ♦ bordered squares of odd orders: 13, 18–29, 165–173 ◦ (examples) 26–29  (with separation by parity) 13–14, 30–87, 174–203 ◦ (order n = 5) 40–41, 54–57, 180, 188 ◦ (order n = 4t+1) 40–41, 42–45, 56–71, 180–182, 189–197 ◦ (order n = 4t+3) 40–41, 46–49, 70–77, 180, 182–183, 197–203 ◦ (examples) 71n, 76–87 ♦ bordered squares of even orders: (evenly-even orders) 14, 89n, 94– 99, 204, 209–212, 219 ◦ (examples) 102–103  (evenly-odd orders) 14, 89n, 90–95, 204, 205–209, 219 ◦ (examples) 100–101  (‘evenlyevenly-odd’) 89n, 104–106 & 107, 204 ♦ composite squares: 14, 104– 161, 213–219  (with equal subsquares) 104–107, 112–113, 214–215

Index

235

◦ (examples) 104–105, 124–137, 213–215  (with unequal parts) ◦ 104–105, 108–109, 112–115, 214, 215–216 ◦ (examples) 104–105, 138– 153, 216  (with a cross in the middle) 108–109, 114–123, 216–219 ◦ (examples) 154–161, 218–219 ♦ squares of small orders: (n = 2) 1–2, 88–89, 125, 204, 213  (n = 3) 1, 3, 11n, 16–17, 18–19, 33n, 163, 167, 175  (n = 4) 1, 3, 16–17, 88–91, 163, 204–205  (n = 5) 2, 20–23, 41, 43n, 54–57, 167–168, 188  (n = 6) 3, 16–17, 93, 163, 206, 209, 218n  (n = 8) 3, 89n, 94–95, 99n, 103, 204, 209–210, 212, 215 ♦ history : (antiquity) 4, 8, 163, 165, 176, see also: treatise ◦ (tenth century) 4, see also: Ant.¯ ak¯ı, B¯ uzj¯ an¯ı ◦ (eleventh and tweltfth centuries) 4–6, 165, 174, 219 ◦ (India, China, Byzantium) 8 ◦ (Europe) 7–8. See also: construction methods, magic sum, natural square, order, sum due. magic sum : 1–2 (def.). al-M¯ah¯an¯ı : 16–17, 164. manuscript Ankara Saip I, 5311 : 9–11, 13–15, 27n, 55n, 89n, 102 & 103n. manuscript London BL Delhi Arabic 110 : 11–15, 26, 28, 100, 102, 125, 173, 212, 214, 215. ‘median’ (number) : 2 & 19n (def.). M. Moschopoulos : 8. al-Mufad.d.al ibn Th¯ abit ibn Qurra : 12, 16n, 30n, 124, 152, 154, 155. See also: treatise. natural square : 5 & 17n (def.), 19n, 164–165. al-Nawbakht¯ı : 16–17, 164. ‘neutral placings’ : 51 & 97 (def.), 53n (ref.), 68–71, 74–75, 120–123, 187, 206–207. Nicomachos : 9–11, 16–17, 89n, 163. numerals : 5, 27n. order : (def.) 1  (categories of) 3, 88–89, 204  (divisibility) 3, 213, 215. ordinary magic squares : see magic squares. pandiagonal magic squares : see magic squares. ‘planetary’ squares : 6–7, 163. ‘small’ number : 2 & 19n (def.). M. Stifel : 8.

236

Index

‘sum due’ : 2 & 44–47 (def.). Th¯abit ibn Qurra : 9, 12, 163. treatise translated by al-Mufad.d.al : 4, 10, 12–14, 16–161, 71 n. 99, 109 n. 163, 163–164  (reading public) 14  (corruptedness) 43 n. 58, 45 n. 61, 47n. 64, 49 n. 66, 51 n. 71, 59 n. 90, 69 (§ 31), 89 nn. 114–116, 93 n. 125, 105 n. 148, 111 n. 166, 123 n. 194, 199. See also: early readers, manuscript. wafq : 7, 16, 228. al-Zanj¯an¯ı : 6.

boethius Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaften

Begründet von Joseph Ehrenfried Hofmann, Friedrich Klemm und Bernhard Sticker. Fortgeführt von Menso Folkerts. Herausgegeben von Richard L. Kremer und Friedrich Steinle.

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The recently discovered Arabic translation of an anonymous ancient treatise describing advanced methods for constructing magic squares has improved our knowledge of Greek mathematics considerably. The early tenth-century translator reports that he found two manuscripts of this treatise, for the greater part damaged by termites. However, since the preserved parts of

ISBN 978-3-515-12852-0

9

7835 1 5 1 28520

one made up for the destroyed parts of the other, he could manage to complete his work. This translation was then the starting point for studies on magic squares in Arabic. It appears thus that there existed Greek studies on the subject of magic squares, and at a remarkable stage of advancement. But this being the only text preserved, we are left in ignorance of the earlier history.

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