The New Elements of Mathematics: Vol.: 2. : Algebra and Geometry 9027930252

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T H E NEW E L E M E N T S OF M A T H E M A T I C S

Charles S. Peirce in his middle thirties. (From the Charles S. Peirce Collection in the Houghton Library, Harvard University.)

THE NEW ELEMENTS OF MATHEMATICS by

CHARLES S. PEIRCE Edited by

CAROLYN EISELE VOLUME II ALGEBRA AND GEOMETRY

ml MOUTON PUBLISHERS THE HAGUE - PARIS

1976

HUMANITIES PRESS ATLANTIC HIGHLANDS N . J .

© Copyright 1976 Mouton & Co. B.V., Publishers, The Hague No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publishers.

I.S.B.N. 90 279 3025 2

Printed in The Netherlands by Mouton & Co., The Hague

INTRODUCTION

It is perhaps in the writing of mathematics textbooks for possible publication by Ginn and Company that Peirce reveals most consistently the influence on his overall thought of the literature in what was even then the "new mathematics," really "new" then, but being brought to the lower school level only now, seventy-five years after Peirce had tried a lone hand at making it feasible. An incomplete letter from Peirce to William E. Story reads as follows : 160 W 87th St. New York City 1896 March 22 My dear Story: I want to ask you a little question in the non-Euclidean geometry. I could get it out myself, but you can probably answer it offhand. Besides, I have some other counsel I should like from you which no man could so well give me. My question is whether generally a single circle is all that can be drawn through three points. Also, what are the properties of a set of four points through which two circles can be drawn? Can a third circle be drawn through such a set generally? The counsel I seek is this. I have spent a great deal of time writing a book called New Elements of Mathematics. Most of my labor has been to make it elementary; although I have put in a great deal of hard thinking too, — not so much mathematical as logical. It is something like Veronese's Geometry, but is (I think) far deeper logically, and certainly far simpler. Nor do I directly go in to non-Euclidean Geometry. It begins with a long logical introduction which I have written twice and am still not satisfied with it. It then takes up the theory of numbers and develops the first elements, with particular attention to logic. I then take up algebra, and develop with strict logic and in a somewhat novel way the fundamental ideas, first of rational fractions and then of continuous real quantity. My definition of continuity is some improvement upon Cantor. I only just touch upon imaginaries. But I have come to the conclusion that a Part has to be inserted here, or else appended, developing the calculus of imaginaries, the fundamentals of the theory of functions, so far as Cauchy went (without bothering with Riemann surfaces), the outlines of the calculus

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(the geometry could be stated without proof if this were put in here), and the elements of plane trigonometry. There is no essential difference between pure and applied mathematics. The mathematician does not, as such, inquire into facts. He only develops ideal hypotheses. These hypotheses are all more or less suggested by observation and all depart from or transcend, more or less, what observation fully warrants. But if the hypotheses are developed with a view to ideal interests, it is pure mathematics. If they are made crabbed and one sided in the interest of truth, it is applied mathematics. The science of Time receives a brief chapter, chiefly because it affords an opportunity for studying true continuity. I then come to Geometry. There is Physical Geometry and Mathematical Geometry. In mathematical geometry I treat successively, 1st, Geometrical Topics (topology), the doctrine of the connections of places consequent upon the intrinsic constitution of space; 2nd, Geometrical Optics, or the doctrine of the intersections and envelopes of unlimited rays of vision (I do not say of light, because the idea of propagated light involves a metrical idea, and would make the rays stop at a real Absolute surface, so that not every triad of planes would have a common point,* etc.). (* I regard the jenseits of the real absolute of hyperbolic geometry as real (according to the assumption of visual optics though imaginary in metrics.)); 3rd, Geometrical Metrics, particularly the 1st and 3rd books of Euclid. I found the problem of analyzing the topical constitution of space very puzzling; and I suppose that it is in consequence of this effect of my stupidity that I am decidedly vain of my solution of that problem. I state it in four postulates. The first two are: Postulate 1. Space is continuous. Postulate 2. Space has three dimensions. I illustrate this latter by showing that if a lump of ice sticks to the side of a bucket, and is partly immersed in water, then of the four places occupied by the wood of the bucket, the ice, the water, and the air, any pair has a common limit, a surface, any three a common limit, a line, and all four a common limit, a point-pair. The point-pair, that is, any two points, I call a simple artiad punctual locus. By a topical singularity, B, of a locus, A, I mean that S is a locus of less dimensionality than, and contained in or at the limit of A, and that from Β particles, filaments, or films can move along A along more or fewer tracks than from the majority of loci of like dimensionality with, and sufficiently near to, B. If the tracks are singularly many, I name Β a singularity by excess; if singularly few, by defect. The numerical excess I term the index of the singularity. Thus, from an ordinary point on a line, a particle can move two ways. This is true even if it be a prickly line, by which I mean a line along which the furcations are crowded infinitely close together. The extremity of a line is a topical singularity of index -1 ; a furcation of index positive. I do not regard a cusp as a topical singularity. Am I right? If a line is in its midst punctually interrupted, the census-theorem compels us to regard this as having the index of singularity -2. It is two coincident extremities. A surface may have both lines and points that are topically singular. From an ordinary line on a surface (even a file-surface, which is analogous to a prickly line) a filament may move two ways. From a bounding edge, it can only move

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one way; from a schist, or line of splitting, or nodal line, it can move more than two ways. I do not regard a cuspidal edge as a topical singularity. A schist may itself have furcations and extremities and partial extremities (that is where some of the sheets cease to be joined). The index of the last considered as a singularity of the surface is the mean of the indices on the two sides of it along the schist. The isolated topically singular points are either punctures or stitches. They may occur anywhere, even where there are other singularities. A stitch may unite two sheets, or may join a line to a surface. This sufficiently illustrates what I mean by a topical singularity. Postulate 3. Space has no topical singularity. As a corollary, space is unbounded; and because it is unbounded, it returns into itself. I warn the student against supposing that a mathematically correct inference proves anything. It may be, and by itself, always is, a petitio principii, unless we accept the postulates as "gospel truth". I divide all loci into the artiad and the perissid. An artiad locus is one which if we interrupt by a sufficiently great number of topically non-singular loci of one less dimension, all pairs, triplets, or quartettes among them having the same finite number of points in common, that number is always even, so long as the intersections are not confined to certain points. But if this test cannot be applied, then the locus is artiad if it cuts the simplest non-singular locus which it interrupts into an even number of parts. A perissid locus is one which can be interrupted by an indefinite number of loci all pairs, triplets, or quartettes containing the same number of points which may be situated anywhere in it, and this number odd. Or if this cannot be applied, it is perissid if it divides the simplest non-singular locus in which it lies into an odd number of parts. Thus, one point is perissid; a point-pair is artiad, because it cuts a ray or any simple non-singular line into two parts. A ray is perissid, according to projective geometry, because it does not cut a plane or cone into two parts. On the same doctrine, a plane is perissid, because it does not cut space into two parts; and any number of rays in it cut one another in single points. A cone is artiad. But according to the theory of functions, quaternions, B. Peirce's Geometry, hydrodynamics, and probably any branch of physics that is in harmony with the science of our time (but not according to perspective, which neglects the fact that light is propagated etc.), the plane is artiad. I adhere to the projective conception, though I do not believe it is the fact of the external universe. A limited disk, a sphere, etc. are artiad. A plane with holes in it is perissid. The Listing numbers I define a little differently from Listing, and get different numbers for space because I conceive of space as perissid while he virtually makes it artiad. When he talks of "all space," he only means the parts at finite distances. It is really limited, though he calls it unlimited, because he wrote before Riemann, and confuses the infinite with the unlimited. The Listing numbers are the Chorisis, Cyclosis, Periphraxis, and Immensity. They are respectively the number of simplest possible, topically non-singular interruptions that, being established in a locus, less the number of such interruptions that, being abolished in the locus, just suffice to prevent its holding a simplest topically non-singular, Particle (for Chorisis), Filament (for Cyclosis), Film (for Periphraxis), or Body (for Immensity), except such as can collapse by shrinking while remaining in the locus in question.

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Postulate 4. Each Listing number for space is 1. The Chorisis is 1 because space is all one piece. The Cyclosis is 1 because a straight unlimited filament cannot by any continuous change of shape shrink to nothing; but if space is interrupted by a plane that suffices to prevent it from holding such a filament. The Periphraxis is 1 because a plane film cannot by any continuous change of shape shrink to nothing; but a ray suffices to prevent space from holding such a film. The Immensity is 1, because a solid filling all space cannot by any continuous change shrink to nothing. But if space is interrupted by a point, it can no longer hold a topically non-singular, that is, unlimited body. My treatment of topology consists mainly of making clear the conceptions, exercising the imagination by numerous examples, and proving the census theorem, and explaining the topical singularities. I also demonstrate that if by a system of homaloids we mean all the points of space, together with a family of surface[s] of which there is just one containing any three points of space not contained in an innumerable multitude of those surfaces, and of which any three contain just one point in common, unless they contain an innumerable multitude; and together with all the lines of intersection of those surfaces; then there are an innumerable multitude of such systems of homaloids. I also show that the intrinsic properties of all these systems are precisely the same; so that no one can be distinguished from any other except by some extrinsic circumstance. I begin Geometrical Optics by remarking that there is a system of homaloids which are distinguished from all others by the circumstance that they look differently. Namely, all lines of possible undisturbed vision (I do not say paths of light) lie in the lines of that system. They are called rays. This gives me Postulate 5. (The Optical Property of Space.) All lines of vision lie in the lines of a system of homaloids. I now develop the elements of projective geometry, especially the theory of nets by what I call the non-metrical barycentric notation, and the theory of degrees of freedom. I use this notation. A, B, C, etc. are points, a, b, c, etc. rays, α, β, y, etc. planes. (ABC) is the plane of A, B, C. (Ab) is the plane of A and b. If a and b are coplanar (ab) is their plane, [αβγ], {ab], etc. are points. {AB}, {aß}, etc. are rays. I prove that if a and p, b and q, are copunctual, while a and q, b and ρ are coplanar, then {[ap][bq]} = {(aq)(bp)}, a useful theorem. I show that every four rays are cut by two rays. I name a one-sheeted hyperboloid, conceived as the locus of a ray cutting three fixed rays, a phimus. I define a conic as the plane section of a phimus, and thus easily get all the main propositions of conics. Another draft of the letter carries the following non-Euclidean overtone under Optical Property : I need not say that boundlessness is one thing, infinity another, nor that a point may be a perfectly real point of space although on the other side of the firmament and metrically imaginary. Thus, every pair of planes have a ray in common, and every three a point, etc., whatever system of measurement be employed. The topical properties suffice to prove that there are innumerable

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families of lines related to one another like rays. Hence, rays or straight lines, together with planes, can only be defined by means of some material thing. I chose for this thing light. The optical property of space is that light, or vision, is along lines which, continued without limit, are called rays. The rays can have but one point in common, etc. No need to state in full here. Metrical Property. The absolutely rigid body moves so that in all displacements the film occupying a certain fixed plane, called the firmament, continues to occupy that plane; but there is no point of space across which a plane film (not the firmamental film) cannot move. However, I consider this as only approximately true; for my calculations from Auwers' proper motions leave no reasonable doubt that space is really hyperbolic, or, if you please, that the proper motions have that property. My book aims to be elementary. To render it so, I have spent a great amount of labor. There are some things in it not altogether without mathematical interest, I think. But it is the logic of it which I trust will be found interesting. That part has been very maturely considered, at any rate; and this analysis of the properties of space is one of my efforts in that line to which I attach value. The five postulates adopted by Peirce offer clear evidence of his familiarity with the "Postulates of Euclidean Geometry" as summarized in a pamphlet by W. K. Clifford, a copy of which is to be found among the reprints in the Peirce Collection at the Houghton Library, Harvard University. Peirce's reference in 1896 to the hyperbolic properties of space is notable, since he entertained that idea for a time in 1891, only to give it up on further investigation. His correspondence on this matter with Simon Newcomb may be found in the aforementioned correspondence between the two men ("The Charles S. Peirce — Simon Newcomb Correspondence", Proceedings of the American Philosophical Society 101:5, October 1957, pp. 410-433). Although Peirce had opened the second letter above with the flattering statement "I look upon you as knowing all about the non-Euclidean Geometry" and turned to Story for advice, he was also exposed, as has been said, to the influence of George Bruce Halsted in the non-Euclidean area. For Halsted was spearheading in his publications on the new geometry the effort to bring to mathematicians in America an awareness of the revolution in mathematical thought. Halsted had been one of the first twenty Fellows in the then new Johns Hopkins University (1876-1878) and constituted by himself the first class of Sylvester, receiving his Ph.D. from that institution in 1879. After an instructorship in post-graduate mathematics at Princeton College, his alma mater, he was called to a professorship in mathematics at the University of Texas in 1884. The following year his Elements of Geometry

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INTRODUCTION

was published and he expressed his gratitude to Sylvester in a dedication that read "in grateful remembrance of benefit conferred throughout two informative years." His approach to the subject matter was untraditional and became even more so in the Elementary Synthetic Geometry published in 1892. It is not surprising, then, that Peirce and he were in correspondence on geometric matters in the period when Peirce was attempting to produce a geometry textbook that would reflect the "new look" that was being advocated on high levels. In a letter to his friend Judge Russell (23 September 1894) Peirce spoke of having had "some friendly dealings with him [Halsted] about his Lobatchewsky, etc." Peirce had reviewed Halsted's translation of Lobatchewsky's work in the Nation 54 (11 February 1892), and had deemed the translation "excellent," making mention at the same time of "his useful bibliography of non-Euclidean geometry" which "was already well known." The following excerpts are from the correspondence in the Peirce Collection. Halsted wrote to Peirce on 7 December 1891 : I was very much interested in your paper on "Astronomical Methods of determining the curvature of space," read before the National Academy of Sciences and if you distribute any reprints of it, I would be much obliged for a copy. Again on 15 January 1892 Halsted wrote as follows: I was much interested in your suggestion, in your letter of Dec. 13, of a modernsynthetic-geometry treatment of non-Euclidean geometry. I think I could manage the analytic treatment, Cayley, Klein, etc.; but I would be much obliged to you for a few suggestions in regard to the modernsynthetic-geometry treatment. I send you by this mail a copy of the 4th edition of my Lobatchewsky and hope in a few days to send you my translation of Bolyai. I suppose you have noticed that in the Century Dictionary definitions Hyperbolic Geometry and Elliptic Geometry (or Space) are interchanged, the definition of each being given for the other. A letter that Max Fisch dates "around 15 January 1892" was sent by Peirce to Halsted. The draft reads : When I wrote to you about a projective treatment of non-Euclidean geometry (which I also suggested in the Nation) I had the idea it was easier than it is. Namely, I thought the whole thing would readily come out by the use of Story's definition of a circle as a conic having double contact with the absolute. But it seems that does not define a circle in the sense of the locus of points equidistant from a centre. I have been intending therefore to look up another point of view. I have not had time to do so, and therefore, wishing to show my good will by answering your letter, I am obliged to do so in such feeble way as I can, thinking

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as I go along and not knowing how I am to come out. This will throw me at your mercy, and I shall have to depend on your honor to give me such credit as I am entitled to for what I may suggest. We shall naturally direct our attention to the absolute or line at infinity (beginning therefore with hyperbolic geometry, though even there the angleabsolute is imaginary). As we are to proceed projectively, let us suppose a plane projected centrally upon another, this again projected from a second centre upon a third plane, and this projected back upon the first plane so that the absolute shall coincide with its original position. Then, I take it the result is the same as shifting the original plane on itself as a rigid thing. If this is so, there is generally but one series of concentric circles whose old positions coincide with their new; and generally if any circle by three projections is brought back to coincide with itself, all concentric circles are so brought back. Let a be our first plane, with a line a upon it and on that line the four points A, B, C, D. Let our second plane be β, calling the first in the line b. And let S be the centre of projection. Then S, A, B, C, D, will be in a plane. Or say we have three planes αβγ with common lines aß = c βγ = α γα = b. And one common point O. We also have three centres of projection Χ Y Ζ with three lines XY = ζ YZ = χ ZX — y and a common plane as [...]

Another letter from Halsted to Peirce on 15 February 1892 asked for aid in showing by non-Euclidean means a fallacy in a "pseudo-proof" of Euclid's axiom. The letter ended with the statement: "I will be much obliged for a simple sentence from you in regard to it. I had the pleasure a few days ago of sending you my translation of Bolyai." (1st edition). When Halsted sent Peirce a copy of his Bolyai's Science Absolute of Space, entirely rewritten, he was President of the Academy of Science in Texas. The covering letter to Peirce, dated 17 December 1895, carried the news that he had been "so fortunate as to be able to announce important additions to the knowledge of the history of these most interesting researches and the lives of the two Bolyais." He signed, "With high regard." This time Halsted referred to the 4th edition. But beyond these contacts, there was the great influence of the famous Felix Klein which is still felt in the mathematics curriculum of the western world. Although Klein in his writings had advised against the attempt to introduce non-Euclidean geometry on an elementary level, Peirce did slip it into his textbook writings without identifying it as such. Pierce was also surely familiar with Klein's address at the opening of the Mathematical and Astronomical Congress at Chicago, Illinois, printed as "The Present State of Mathematics" and published in the Monist, IV, no. 1 (October 1893), a very influential paper. Klein's international influence on the teaching of mathematics in his own time is incalculable for he became President of the International Committee

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on the Teaching of Mathematics created in 1908 after his years of missionary work in advocating reform in the way of updating the subject matter of the curriculum. Peirce not long thereafter read a paper on "Rough Notes on Geometry, Constitution of Real Space" at a meeting of the American Mathematical Society on Saturday, 24 November 1894. It was not printed later, but one of the incomplete papers in this collection (MS. 121) could well have been that talk, since Fiske had asked Peirce for a copy and Fiske's name is written in Peirce's hand on the back of it. But it was to Story that Peirce turned for information and inspiration. During those years, Story headed the Mathematics Department at Clark University, indeed had been administrator since the founding of the institution in the fall of 1889. At the weekly mathematics conferences there, conducted by Story, non-Euclidean geometry had been the "subject of systematic discussion." Even while at the Johns Hopkins University earlier under Sylvester and in Peirce's time, Story had been working in hyper-space and non-Euclidean geometry as well as in algebraic invariants. During those years, Story published in the American Journal of Mathematics 4 (1881) the paper "On the Non-Euclidean Trigonometry," and in vol. 5 (1882), "On the Non-Euclidean Geometry," and "On the Non-Euclidean Properties of Conies." These papers and another by Halsted in volume 1 entitled "Bibliography of Hyperspace and Non-Euclidean Geometry" were surely known to Peirce whose formal interest in the subject is evidenced by his reading before the Metaphysical Club at the Johns Hopkins University in 1879 a paper on the "NonEuclidean Conception of Space" written by his pupil Miss Ladd. Story, on his part, seems to have had considerable respect for Peirce's talent. At the time of the Decennial Celebration (1889-1899) at Clark University Story was active in the organization of the Celebration ceremonies. On 27 June 1899 he invited Peirce to an entertainment in his home during the celebration the following week when he planned to have as guests Émile Picard of the University of Paris and H. B. Fine of Princeton University. Peirce not only attended the meetings but reviewed the Clark University Decennial Celebration in Science n.s. 11 (20 April 1900). Story was an editor of that volume and gave a full report therein on the professional activities of the members of his mathematics department. In describing the work of Henry Taber on the theory of matrices Story made special mention of C. S. Peirce's contribution to the multiple algebras that he called quadrates. Incidentally, when Tabor asked Peirce in 1892 for a recommendation for a mathematics appointment at Colum-

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bia University, Peirce accommodated him but indicated that he would look into the vacancy for himself. It is not surprising, then, to find Peirce telling Story about The New Elements of Mathematics mentioned above in the Story letter. It is indeed a manuscript of unusual interest in many ways (MS. 165) and Peirce's long struggle with its contents merely reflects his usual preoccupation with the challenges constantly raised by textbook presentation of that subject. Typical of Peirce's perseverance in the study of mathematics is his explanation to his wife in a letter of 1892 that the train was late at Port Jervis, "but I had in my pocket a book on the Geometry of the Circle which I am studying. It will bring me a few dollars immediately for a notice, but the chief reason for studying it is to keep up my acquaintance with every branch of mathematics, which gives me a reputation and is useful to me in other ways." There were "other ways" indeed. Peirce's philosophical and logical interest in the problem of continuity made for his deepest concentration on developments within the then infant branch of mathematics — the topology of Listing and Gauss. He studied these new ideas in great detail and came to believe that topically singular points were the only ones having identity while, in general, a point on a continuous line has no individual identity at all (MS. 159). In his theory of logical criticism "the temporal succession of ideas is continuous and not by discrete steps. The flow of time is continuous in the sense of the non-discrete" (MS. 377). Yet Peirce's purpose in writing the New Elements of Mathematics was not a philosophical one. After his enforced retirement in 1891 from service in the United States Coast and Geodetic Survey, he had turned to textbook writing in elementary mathematics as a means of economic survival. Then, as now, professional mathematicians enjoyed great financial success in such ventures and it was natural for Peirce to believe that his new and original approach to the teaching of foundations would attract wide and sympathetic support. On 15 July 1894 Charles's brother, Herbert, wrote a note to him from aboard the North German Lloyd liner Saale on the way to St. Petersburg where he had been appointed Secretary of the United States Legation. He related that "Mr. Ginn the publisher in on board and has spoken to me about your books. He says he wants a little time to consider them.... He is really interested but wants to look into the subject himself and I think is disposed to take them." The correspondence that followed depicts Charles's futile attempts to

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meet the sales-minded demands of the publisher. This effort was to result in The New Elements of Geometry (MS. 94), a response to Mr. Ginn's desire in 1894 for an updated version of Benjamin Peirce's Elementary Treatise on Plane and Solid Geometry of 1837. It is difficult to determine which of the two manuscripts (165 or 94) was completed first. Manuscript 165 is the carefully planned and conscientiously ordered work of the Story letter. Tens of pages of worksheets, and lists of definitions, scholia, glosses, theorems, and outlines for the entire work are still extant. Yet some material was repaginated for the geometry section of MS. 165 and is now missing from MS. 94. The full outline of MS. 165 suggests in a hastily written entry the existence of Chapters X and XI which now, after a separation of years, have been reunited with the main body of that manuscript. Regardless of priority, it is the story of MS. 94 that we shall relate first. On 5 April 1894 Charles had written to his brother James Mills (Jem) at Harvard University of the usual financial troubles. Consequently, I have to abandon all idea of the Arithmetic for the present; and probably that means never doing i t . . . the advantage of the geometry is that it would soon be done, as I have most of it written already. It might bring little, but might lead to further employment. The trigonometry would be a useful book and in the long run must pay tolerably well; for it would be greatly superior to Chauvenet from every point of view. The chart which I am now trying to push might, if I am lucky, bring enough at once to enable me to set to work on the Arithmetic. The Almanac about which I am negotiating with considerable prospect of success with the Times, would take all my time, but would give me a steady though very modest salary .... [P.S.] My notion about Father's geometry would be to make an introduction of three chapters. The first on the nature of mathematical reasoning and how to perform it. The second on projective geometry, especially perspective. The third on the metrical properties of space and the justification of the idea that an angle is a difference of directions. I don't know that the body of the book would require any correction. The thing would be to preserve its leading characteristic, which is to lay hold of and render definite our instinctive or natural ways of considering space, especially in reference to moving about in it. I think, that done, the work would be a masterpiece and would be much admired. Its sale would be good, too. Plimpton is anxious to have us do it. You and I should do it together, n'est-ce pas? What say you? ... I don't know that I would not reform and enlarge the part relating to plane problems and append a chapter on Descriptive Geometry, and perhaps an appendix on Modern geometry. By November 18 Charles was writing that if Jem would take hold of the Geom. and improve it in places, etc., we should make

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a fine thing of it. I have gone on the idea the Elementary Geometry must be reformed. There are many signs it will be and we might as well take opportunity by the forelock. So far, I have not ventured to use your name on the title page and certainly if you don't do anything about it nor give your name, you won't expect any considerable share of the profits. But if you will take hold why we shan't quarrel over the division, I guess. My notion is to make a book to teach most of those who use it all the geometry they are going to study. I decline to be tied down therefore to any idea. If a given thing is useful, — especially, if calculated to enlarge the mind, — and is sufficiently easy, I won't be deterred from putting it in because it is not what is called Elementary Geometry. At the same time that most of the students of the book will go no further, I wish to present the subject in a mathematical and modern spirit, and to teach them what mathematics is, in order that they may be able to judge whether they wish to pursue the study further or not. Moreover, I don't teach that geometrical "demonstrations" prove anything. Having made an initial hypothesis they serve to keep the hypothesis consistent. But whether it is true or not would have to be ascertained by special observation. Ordinary observation shows it is very near true indeed; and the astronomical observations show the errors if any are extraordinarily minute. Since writing you an account of it, I have succeeded in simplifying the hard places very much; there still exist passages which I regret the abstruseness of, and I dare say if you got hold of it, could in a few days do great things in the way of simplification. I have introduced a little algebra. I don't see why not. The students have got to learn a little algebra sometime. They had best learn it on an occasion when they can see at once its great utility. Of course I explain the thing from the very bottom. I only ask them to use letters a little, and the minus quantities, both in addition, subtraction, multiplication and division, and the distributive principle. Instead of letters, I mostly use prime numbers and not performing but only indicating the operations, I say, it is evident any other numbers would do as well since we have not done anything depending on the special values of those we have used. That it is a book, sufficiently easy, really easier than the geometries of the old school, and calculated to open the mind to mathematical ideas beyond anything elementary ever done before, is my own flattering opinion of it. Without exactly saying so, I have aimed to have the student go away with the conviction that mathematics is the art of exact generalization. Demonstration [in] Elementary Geometry of the past made the whole thing appear as nothing but the pavement over which the mathematician drives his team, with a goal in view and with a plan for reaching it. The three faculties I try to educate are Imagination, Concentration, Generalization. It will occur to you that I am taking too great liberties with father's book. Especially so, since I am in places quite diffuse, being persuaded that those who have little turn for mathematics have to have their milk watered, like other babes. But I am sure that father would not at this day have approved of the kind of logic which governs a good deal of his book. I think my work is

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in the spirit which father always radiated; although I admit that I have not been able to attain his classical simplicity. If you would take the hold of it, I believe you would do a good deal toward that. It is in some aspects like a book you always, I thought, underrated, JMP's Analytical Geometry.

This note was sent, apparently, soon after Charles had attempted to sell his manuscript to the American Book Company. For he said that "the American Book Company are favorable to the Geometry; but have not decided as yet to take it; and it is to be sent back to me for some alterations which will only take a few days. I sent it right off to them because I could hear nothing from you and because I was greatly pressed for money." However, after further reference to his personal difficulties, Peirce once again returned to the geometry theme. Now he says that "Ginn and Company said they were almost certain to take the Geometry and push it (they thought your name at least would appear on the title page) and certainly I do not think that written though it has been under a terrible load of misery anybody can deny it has a great deal of merit .... What I am coming to is, that I want your helping hand in perfecting the geometry very much ...." On an undated Friday night Charles again wrote to Jem that he was seeking the latter's advice about the Geometry (and other books). "It is a new trade to me. I am most anxious to correct my notions. I am very desirous you should take time to look over the Geometry and a piece of the Elementary Arithmetic I have here. It is a most serious work." A letter to Jem dated 2 January 1895 describes the Geometry as hanging fire, "though I was sure I was going to get it off at once." Though his personal trials had multiplied he was full of hope as he wrote: "After my Geometry, my Arithmetic of the success of which I have no doubt, and then the Philosophy." It must have been in this period that Peirce sent his brother an outline of his book explaining in an undated letter that he had not yet completed it because of illness. Great changes were necessary to make it modern. I have a preliminary Book on Logic; but don't feel sure about inserting it. Then a Book on the Fundamental Properties of Space. I. Tridimensionality. II. Continuity. III. All parts alike and that it is conceived by geometers to return into itself, etc. IV. That it has homoloidal lines and surfaces each cutting the others once, and that one system of these, namely rays and planes, are important owing to the property of light. V. That the motions of rigid bodies are such as to give certain metrical properties.

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Then a Book on Topology, chiefly the connectivity of surfaces and Euler's polyhedral formula (which I give in a very extended form, applying to polyhedral nets on surfaces of any connectivity wrapped any number of times by the net, the vertices having any number of whorls and the faces any number of circuits round them). This book is quite elementary and simple and good to begin on. Then a Book on Projective Geometry. This is as elementary as I could make it, but I arrange to make metric geometry independent of what goes before if teachers want to skip the rest. I give the elements of projective conics. I first study a "phimus" defined as the surface generated by a ray that always touches three fixed rays. This gives me the properties of conics very simply. I use a modified barycentric calculus for nets (non metric) and I also use this notation (ab) as the plane of the two rays a and b, {ABC) as the plane of the points A, B, C. (Ab) is the plane of point A and ray b. [αό] is the point of plane a and ray b. [ab] is the point of two rays a and b. [αβγ] is the point of intersection of planes α, β, γ. {aß} is the ray common to the planes a and β etc. If rays a, b both intersect both ρ and q, then {(apXbq)} = {[aqlbpl] This proves Pascal's theorem instantly, from the phimus. Metric Geometry begins with a bad chapter about the motion of a fluid and how it might be measured. Then [it] shows what the peculiarity of rigid motion is. This gives me a start in the next chapter like Euclid's, putting pons asinorum at the base. (He was evidently a student of non-Euclidean geometry.) And I soon get to direction as valid in a plane and aspect of planes round a ray. For practical postulates I assume parallels can be drawn. It is so easy practically by a ruler and a triangle and more methodical. I also assume a right angle can be made, instead of that a circle can be drawn. But I connect these with the usual postulates. Spherical geometry (which I have not got to) will be made very simple. The book has some things too advanced for children, but it will be arranged so that these can be skipped. It has a great deal that is very easy. As soon as I am through I shall send it to you. That Jem was not in a position to cooperate was evident in a letter dated 21 November 1894. It reads in part as follows: I received today your interesting account of your Geometry. It will be an epoch making book, highly original and fertile. It will be distinctly your book, and the embodiment of your ideas. It will certainly give new notions to teachers of Mathematics. How far these will stand, must remain to be seen. For myself, it will be impossible for me to take part in the book as I have already written you. I have neither time nor strength to add that to my other labors. Whatever power I may find for writing must be devoted to Quaternions and to the other subjects which I am teaching in my courses. It would be impossible for me also to cooperate in a work so distinctively thought out by another mind. But if I can make a suggestion here and there, I shall be glad to do so.

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I know nothing of the American Book Co. If they take the Geometry, you had better let them have it at once. It should appear as your book and, the sooner the better. I see no reason why the Logic and the Geometry should be in any way tied together or brought out by the same publishers. I hope you can dispose of the Logic at once, and the Geometry too. I saw Sylvester in London. He has been very ill and has aged. He probably has Angina Pectoris. He inquired for you, and spoke most affectionately of father and mother. Writing in the same vein on 16 February 1895, Peirce told Jem, Ginn wants a geometry. But a geometry with ideas in it is more than he has bargained for. A Nation's Ideals are bound to be embodied in their system of education; and a people with ideals like ours, — worshippers of Individual prosperity, — will have a system of teaching to match. Such a people rightly feels there is nothing whatever upon which they ought to be so much on their guard as the entering wedge of living thinking .... The fault of my style which I am perfectly aware of lies deeper than you think. It is aggravated by my great use of the logic of relatives which embraces everything together so easily; so that my instinct for how much people can comfortably swallow at a mouthful (never strong) is now ruined. You would have to rewrite, not merely changing wording, but ways of stating reasons. It is obvious that Edwin Ginn on his part had been writing to the Peirce brothers. In a letter dated 14 February 1895 he was gratified to learn from Jem that Charles's book met with Jem's approval and that Jem "will be willing to aid in the revision of it. We shall be very glad to see the revision of the MS., and there can hardly be a doubt but that we shall want to publish it. If it were not in some respects a matter of form we would settle the question now." Another letter from Charles to Jem on 3 March reveals that Charles had sent the Geometry to Ginn ten days earlier. Charles continued: "They positively promised me in writing an answer in 48 hours. They have not broken silence and I conclude they don't mean to take it." He spoke of his arithmetic as becoming profitable after a long time, of his Lowell lectures that the Atlantic might be induced to publish as articles, of his ability to "make the standard history of science and the possibility of adopting Marie's plan of 12 small volumes coming out singly." Then, again, there is the frustrated cry, "I see Teubner's Mitteilungen 1895 No. 1 speaks of my 'eminent vollkommenen' notation for the Logic of Relatives and gives as one raison d'être of Schroder's third volume, that my papers are so scattered. I had a great ambition to some day write a Popular Logic or a Logic for the Million — But I must be upon my guard against things I have an inclination for." One page of that letter reveals that Peirce has had "a number of

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opportunities to see that I am a good teacher for those whom the ordinary mechanical system of pedagogy does not reach. My acquaintance with logic and psychology enables me to deal well with such a one ...." Again on March 11, the troubled Charles states that he was anxious before he died "to get what I can from the geometry, and logic, and map ...." As to the geometry, I sent it to Ginn and they sent back word that "it seems to be in two parts, one a sort of mathematical philosophy, and the other more properly a text-book. Now the mathematical philosophy would meet with hardly any sale ... but Book V which you call Metrics is a proper text-book in Geometry. It is not only a good revision of your father's text-book, but much more than that. Now why cannot we make this suitable for the use of schools and colleges, publishing it by itself, and test the market with it before undertaking the other?" They had evidently merely looked at the part you looked at, which was intended merely for reference and had not examined my introductory books of Topology and Graphics. I replied saying that I fully assented to their proposal; but that at the same time I could not but think it a mistake. That nothing could be further from "mathematical philosophy" than my topology which is merely a book of the simplest definitions designed to exercise the imagination, with one single theorem, the correct form of Legendre VII.[25], which was omitted in my father's treatise. That the graphics contained merely the ordinary propositions of graphics, — the simplest of them, together with a chapter on algebra. That I thought very likely I had gone too far in Graphics, while I was led to think more likely not, considering most pupils would never have another book on mathematics. Mr. Ginn now writes : "I forwarded your letter to your brother, Mr. J. M. Peirce, this morning and I wish that you and he would largely work out this problem together, for I feel that you can yourselves do it more intelligently than it is possible for us to do it. What we want is a good revision of your father's book, with such additions as will make it even more popular than his was and cause it to meet with even a greater sale, perhaps." The "problem" to which he refers is whether it is better to prefix the two books of Topology and Graphics (I begin to feel myself the "Fundamental Properties of Space" is deadweight to be thrown away or not). But his meaning is evidently vague. He means also what modifications are to be made. Now I should be glad, very glad, to have your active collaboration. But I don't want the thing to be forever coming to the paying point. I want the money, bad. If you could sit down and read the topology and then send it to me with remarks both general and particular, why I would proceed to do what seemed necessary to making it acceptable. That instruction in geometry ought to begin with awakening the geometrical imagination, both psychology and experience show. That the first example of proof offered should be a good specimen of real mathematical reasoning and not the kind ofthing which astounds the pupil by demonstrating at length something obvious at a glance, wherein be it observed he is thoroughly in the right and the geometry a mere "game" as De Morgan called it, I equally have no doubt. For my part, I don't think it a matter of consequence

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that pupils should have the quasi-military drill of the class room about reciting this part. Let them miss it, it makes no difference; or is even better. It isn't in that direction I want their attention turned. I want to get into their vacant, wandering minds some sort of an idea of what geometry is. If I do that I have made a grand beginning, and don't care if they have not a definite idea of the proofs or do not remember the definitions. Next comes Graphics. If you could spare the time to look over this too long "Book" and give me your opinion I should be glad. I suppose I have put in too much. Still, is it not well pupils who never own another book on mathematics should have a little surplusage, to refer to, even to study later? The teacher can skip any amount. The metrics is independent of it. But the real utility of the graphics is that the students can have it as a preparatory exercise of their faculties and if they don't carry much away, it makes no difference whatever with their progress in metrics. Will you please return the graphics when you have made your remarks? The first part of the metrics I fear will have to be sacrificed. I think it is rather interesting; but it is too hard for young pupils. I am rather sorry, because I have spent much thought in trying to put this in a clear shape and it is no good except for an elementary book. But I feel it has got to go. I wish however you would read it through. After the hard part comes some easy matter perhaps worth keeping. The first five chapters of father's book have been pretty thoroughly looked over. I have not changed much; but I have changed some things. In a general way, I thought the later chapters could go, but I daresay they should be examined more carefully. Having examined the metrics will you please send me that and all the rest? Now should there not be an Appendix on Coordinates? Another on Trigonometry? Another on Brocard Geometry? I think so. Please remember that Ginn means business; but he wants a popular book. Not a baby book nor an easy one necessarily, but one that people will think is a good one to teach and a good one to keep and not sell to the next class. Remember too that it is the first and last book on mathematics (I don't count Arithmetic) for the majority. Remember that if I can once make a popular book, I shall have almost carte blanche afterwards to write all the profound ones I like. But that popular book must be made coûte que coûte. I want your generous aid, and prompt aid, about this; but I don't want you to kill yourself over it. On the whole, perhaps it will be as well to have only my name on the title page as editor, that is, if it is going to be a good book. If not, if you were with me, each could say it was all the other's fault. I asked Ginn and Co. to forward my answer to you after reading. Your loving C.S.P. What, then, had Ginn and Charles been writing to each other at this time? A letter from Charles dated only 1895 March but stamped Mar. 7 at Ginn and Co. reads:

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I assent to your proposition to publish the Metrics alone, as contained in your letter which I received this PM. But I do not approve of it. One book, that on the Fundamental Properties of Space has the character you mention of being concerned with the "philosophy of geometry." But it has always been considered as proper to insert in the elements the Postulates and Axioms. This book is the same thing in a modern form. I go on the principle it is wrong to teach as true what is known to be false. Among advanced mathematicians the axioms of geometry as self-evident truths have long been completely exploded. What takes their place is what I have drawn up under the head of The Constitution of Space. The body of the book is devoted to a clear explanation of that "Constitution." But in my Preface I expressly say that Book on the Fundamental Properties of Space should only be used for reference. The study of the book ought to begin with Topology. There is nothing of the nature of "philosophy" in my book on topology. It consists wholly of Definitions and figures designed to awaken the pupils' mathematical imagination, together with one single proposition. That proposition is substantially that of Legendre, Eléments de géometrie, livre VII, Prop. XXV which Chauvenet used, Treatise on Elementary Geometry (1st Ed. I have not Byerly's) Book VII Prop. XXXII §97. My father omits it, probably because he thought the proof unsatisfactory. It can hardly be that in 1837 he was aware it was false. My proposition is simply the truth of the matter. It has generally been considered germane to the elements, and should not be omitted. I put it first as being far the simplest, and I do not make a parade of forms of demonstration, because the pupil must learn one thing at a time, and at this stage it is most essential that demonstration be kept in the background. Otherwise he goes through life with the absurd idea that demonstration is the main element of mathematical thought, — an idea exploded by all modern mathematicians but very rife among the ungeometrical. I have had great success in teaching dull boys, especially in geometry. The reason is I am so well acquainted with psychology and logic, and I know the first thing is to excite them to really think. During that part of the instruction they seem to make little progress; but it is really by far the most important step of all they are taking. The proposition I have put first is, I am fully convinced, after experiment and reflexion, quite the most suitable to be put first of all the propositions of geometry. At any rate it must come in somewhere. There is nothing "philosophical" about it. On the contrary, it is peculiarly elementary. But by giving a separate book to it, those teachers who choose to do so can make it the last proposition by simply teaching topology after metrics. I could insert a suggestion to that effect in the preface. Let me beg you to read that over again and to ask yourself if what I say of it is not true. The other book which precedes metrics is graphics. There is certainly no "philosophy" about it. It contains merely the elementary propositions of graphics. Perhaps it goes too far. About that I was doubtful. I reached the decision I did by considering that many pupils will go no further in mathematics than this treatise and I had better insert all that will be practically useful to them and which is not too difficult. I am decidedly of [the] opinion graphics ought to be studied, because of its extreme utility, and because further exercises

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of the generalizing power and of the imagination are requisite to successful teaching of geometry before the element of demonstration is emphasized as much as it is in metrics. (By the way, you speak of my "calling" it "metrics," as if that were not the recognized term.) I am not at all unfavorable to cutting down what I have given of graphics, although my individual preference would be for retaining it. Nevertheless, although my experience of teaching is that the worst results are obtained by beginning with metrics and the best by taking topology, graphics, metrics in that order, yet, after all, I want the book to sell, and if you think the old stupid way, which makes a large percentage of the scholars dunces, is the best way to sell the book, I consent reluctantly to the excision of the parts you do not like. All the more, that I shall expand them into another book for use by the more intelligent of the teachers. Or, if you like, those books can be made appendices in large type with a recommendation in the preface to begin with them. I should incline to another appendix in smaller type on Brocard geometry, which is all the rage now. I shall be happy to draw up about a hundred problems in geometrical construction. More if desired. Now I leave you to make such decisions as you like. I dare say J.M.P. would lend his judgment if desired. I have half a mind to insert at the end a simple treatise on trigonometry as Legendre and others have done. I shall be glad to hear from you as soon as may be.

A footnote warns that "Almost any such change as you propose will require a not long introductory chapter to be inserted." By 11 March 1895 Peirce wrote directly to Ginn and Co. again asking that this letter be forwarded to J. M. P. Since Jem was a successful professor of mathematics at Harvard University, Ginn was, of course, anxious to have his name on the title page. And so Charles says, I shall be very glad if my brother's health and leisure are sufficient to enable him to collaborate in any perfectionment of my revision of my father's geometry, of which you express so favorable an opinion. I used every argument to persuade him to help me about it; and he did go so far as to promise to do so; but he found it impossible and had to give it up. I don't want him to undertake what he cannot perform promptly, because it would only be a drag upon him. If he would take hold of the thing, — although you think it so well done, — and would make such changes as he thought proper, I should think it a great advantage. He has made some examination of the MS. and has expressed his wish the thing should appear substantially as it is. But if he would even do so much as to read through my two books of Topology and Graphics, and make any positive suggestions, it would, I am sure, be very useful. You know what I think of those books. Namely, I consider the book on Topology consists of nothing but the easiest definitions with one single proposition, and that the easiest one (not puerile) in geometry. The graphics is more advanced but the

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bulk of it is in my opinion easier than and a proper and necessary introduction to Metrics. I may have carried it too far; but it is easy to omit the more difficult parts. I have constantly borne in mind that a geometry is the last book on mathematics most of the pupils will study, and therefore it ought to include as much that is useful and not too hard as possible, even if the teacher chooses to skip a good deal. My father's little treatise dates from 1837. By the time I was a schoolboy, his views about teaching geometry were so decided that he interfered with the course in my school to insist that before I was put into his book or into Legendre, I should go through a book containing substantially the same matter as my books on Topology and Graphics. I have talked much with mathematicians on the subject. I think they generally, almost unanimously, warmly agree to this. I have repeatedly been pressed to write for educational magazines etc. on the teaching of geometry. I have always postponed it until my book should be out. I shall then let drive and shall preach to all the teacher's conventions. The old way of teaching is a total failure with a considerable percentage of pupils. The modern method, — not absolutely new, for there are several treatises based on these views, — has never failed in my hands, though I have had some pupils considered quite incapable of mathematics. We are a strong party. We have modern mathematicians, modern psychologists, and modern logicians with us. We will blow the old system out of water. One firm of publishers, — I mean, a firm of powerful American Schoolbook publishers, — will have a modern treatise soon. Nevertheless, I am not obstinate if you think the additional chapters will stop the sale of the book or injure it, let us cut them out. I had that possibility constantly in view in writing the thing, and have an introductory chapter all prepared for that contingency. I don't consent to any changes in my book on the Fundamental Properties of Space. That is meant as explained in the preface for reference only. I will consent to omitting it altogether, or I will substitute for it such twaddle as may be thought more salable. I fully consent to the omission of the Topology and Graphics. I only say I am confident it will be a capital mistake. But I dare say I have gone too far in graphics. Of course, I think not; but it is a doubtful thing in my own mind. What I feel most misgivings about is the first part of the metrics. But the thing is written so that it can be omitted. There is just now a perfect rage for the Brocard geometry. I don't share it particularly; but I think it might perhaps be well to give a taste of it. I have lots of construction problems. O n e wonders if the introductory chapter referred to above contained the bitter fragment found in MS. 153 which reads: Circumstances have unfortunately compelled the editor to publish this edition, not as the author would have desired it revised, with introductory books upon topical and graphical geometry, but in the miserable shape in which it appears. A number of treatises already present the subject somewhat as it should be treated, and are meeting deserved success. The editor will at the earliest possible

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moment offer the schools his New Elements of Mathematics, which his experience in teaching leads him to hope may do away with the phenomenon of boys destined to distinguish themselves as sound thinkers unable with their utmost diligence to make any progress in the intelligent study of geometry. As is now apparent Peirce's determination to give a modern rendition of the geometry could easily have been reinforced by the materials then being published in the Bulletin of the New York Mathematical Society. For example in vol. 1 (1891-1892) one finds "The Teaching of Elementary Geometry in German Schools," a review by Alexander Ziwet of Inhalt und Methode des planimetrischen Unterrichts of Dr. Heinrich Schotten. Ziwet speaks of the sixty-year-old agitation for reform in teaching of geometry in Germany, of the publicity given it there, and the hope "that the Bulletin of the New York Mathematical Society may, in the course of time, perform a similar service towards the improvement of mathematical instruction in this country." According to Ziwet, Dr. Schotten held that "the object of mathematics teaching in the Gymnasium is not to produce mathematicians but to improve the mind, not only by training in logical thinking, but by accustoming the student to precision of language in writing and speaking, by awakening his self-activity through the solution of problems, and in the case of geometry in particular, by forming and practicing the power of mental intuition ('Anschauung')." He advocated the introduction of simple ideas in the projective geometry which must be an integral part of the whole system. The Bulletin did take up the cause — for example in publishing T. H. Safford's article on "Instruction in Mathematics in the United States" in October 1893. Peirce was thus exposed to the new radical "reform" thinking that was already in the air at that time. After writing the letter to Charles on 8 March which Charles had quoted to his brother on 11 March, Edwin Ginn wrote to Jem on 14 March, expressing again his hope that the brothers would cooperate on the geometry venture. Ginn continues: I am wondering whether it would not be well for you to consult some of your friends at Harvard who have some of the Geometry work to do, and get any suggestions you can in regard to the new book. Would it not be well for us to consult some few leading teachers in other parts of the country, so that the book may not only be suited for Harvard, but be adapted to a broader field. Would it meet your views to see other people? I feel that you will be the practical person in the issue of this book, that your brother has had less to do with teaching, has less knowledge of the real wants of the schools, and is perhaps more theoretical in many ways. I hope you will be able to give the necessary time to pruning his work and putting it in such shape as to make it meet a good degree of popularity.

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Under such pressure, Charles was willing to make almost any concession to get the material printed at last. We find him writing from his home in Milford, Pa., on 22 March in these terms: My brother sends me yours of the 18th; and both your propositions please me. The Metrics, as I have left it, is much too hard to begin with. Even my preparatory books my brother finds too hard. While I disapprove teaching geometry in that way, as do all recent writers on geometrical teaching, except that precious Society for the Improvement of the same, I have no doubt you are right that many schools will like it. However, they will not like, — they are just the people who won't like, — my father's short cuts in reasoning, — often logically indefensible, too (but they wouldn't always have heard that said). Therefore a new series of introductory chapters will be necessary; after which I will imitate Euclid by putting all the fundamental propositions (as I have done, but in too abstruse a way). Then most of my father's chapters with merely such changes as are necessary to meet the taste of bad teachers today. An appendix will contain about a hundred exercises. I will abide by any understanding my brother may reach with you concerning the royalty or copyright-price. It will be convenient for him alone to sign with you. Some agreement reached, you can announce the book at once. My work is largely done in my papers here; so that the rest won't take but a short time. Now as to the Topology and Graphics, my brother thinks parts of the former too hard. I will not have that said of it, if I have to excise the heart of it. I shall at once proceed to rewrite it, mingling with it a boy's story. It will represent conversations on imaginary practical questions in geometry with accounts of experiments. I shall call it Euclid Easy. One advantage of writing in this style is that it enables me, without offence, to hint to the teacher every detail of teaching. It will afford formal statements to be learned; and that is what teachers and pupils always conceive to be the essence of learning. At the same time, the real education will be slipped in before they know it. My wife will be glad to pay for the plates, provided you will promise to take hold and push the thing energetically and in preference to any other geometry, unless the other one, which in my opinion, will soon be out of date anyway. When it is written, I propose to get up a geometrical crusade, and I know where I can get powerful support. I want to make a hundred addresses to teachers on the subject. I will send you a specimen from which you can judge of it, and then my wife's lawyer will send you a contract. I have tried my hand at the kind of writing I propose, to the great delight of practical teachers, and I have no doubt the thing will pay, apart from its use in schools. But if my wife assumes the bulk of the expense, naturally she will have to have the bulk of the profit. Although this is not the place to discuss the purely personal circumstance in Peirce's life, let it be said that during these months the details of his personal misfortunes were harrowing and in an undated letter to Jem, obviously from this period, Charles laments, "this geometry I thought

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so easy has been my final ruin." In another note he complains of illness "owing to insufficient food and that is why the Geometry is not yet finished." Again, "I naturally became sick and my mind all a blur at a moment when my best powers were wanted to finish up my geometry." Yet on 30 August 1895 Ginn & Co. sent the last letter on the subject. It referred to a letter from Charles on the 27th and spoke of Charles supplying a list of professional men to whom the book might be sent and also of the preparation by Peirce of copy for the diagrams. One senses from the letter of 22 March that negotiations were already grinding to a halt. Yet on 28 March it was suggested that C. S. P. publish the Metrics only. In that case an announcement could be put in the "forthcoming catalogue just ready for the press." The end finally came in a letter from Jem to Charles which read : I send you a letter received today from Ginn, enclosing your MS. of "Euclid Easy." He does not appear to take much interest in it. The main idea seems to me excellent. But I can imagine that some of the decoration may appear questionable to a publisher .... I will return the remainder of your MS. What Ginn wants is obviously merely the revised form of Father's book, and any argument on the subject from a higher standpoint (and still more from a lower one, which he thinks he understands) is thrown away on him. I might undertake to pay for the plates of the first two books, Fundamental Properties of Space, and Topology, on condition of receiving a share of the royalty. But I must warn you that it is, in my judgment, idle to expect any large return from it. I do not understand why you have turned aside from the Arithmetic. The letter from Ginn to Jem is strangely dated 25 April. Ginn had sought advice on the "Euclid Easy" and found it impossible to recommend the publication of a work of this kind without seeing the whole of it. "Euclid Easy" (MS. 268) seems to have terminated the Ginn involvement in the geometry publication. Years later, as late as 4 December 1904, Charles was still writing of completing his Arithmetic and his Logic. Moreover one wonders about the extent to which MS. 137 (Topics) was involved at that time as he expressed the wishful hope: "If only I could get a month to write out my memoir on Topical Geometry people, I think, would be impressed with the desirability of having me where my work could be properly done." The following notice in the Nation of 14 July 1898 reveals the depth of Peirce's commitment to the cause of the proper basis for geometry instruction, a stance he maintained to the very end.

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It is a little strange that, after a generation of celebrity, Reye's "Geometrie der Lage" should now be translated into English for the first time. Part I. comes to us from the Macmillan Co., Prof. Holgate of Evanston being the highly competent translator. The original has long been used in some of our American universities to great advantage. In certain respects it is a more brilliant book even than the treatise of Cremona, and it covers a somewhat wider field. But its merits are too well known to need any comment from us. Later researches into continuity go to show that Topology and not Graphic forms the real foundation and generalization of geometry; and the moment is almost at hand at which Reye's book must be superseded by one which shall lay the foundations of its logic deeper still. Meantime, this well-executed translation, with a useful preface, will serve a good purpose. We shall speak more particularly of the version in noticing the following part.

As for The Elements of Mathematics (MS. 165), Peirce mentions it again in MS. 229 which is included in an appendix of this collection. It is mentioned also in MS. 517 (c. 1903?) in which Peirce tells of having devoted a year to it, of its having reached three publishers, and of its having been lost. The editor believes that this carefully designed and logically perfected presentation was well advanced before the opportunity for publication of a revision of Benjamin's Geometry was offered by Mr. Ginn. It contains the kind of logical introduction that Peirce said had been deleted from MS. 94. The editor believes that when George A. Plimpton, an associate of Ginn at Ginn & Co., indicated to Charles that he "wants us to do it," Charles rewrote the geometry part of MS. 165 into the present Books of MS. 94 and attached Benjamin's book to all as a kind of Appendix. If that was the case pages were later detached from MS. 94 and inserted with new pagination into MS. 165 as is indicated in the relevant footnotes of the present edition.

CONTENTS

Introduction to Volume 2

ν

1 Elements of Mathematics (165) Preface (1) (164) Preface (2) (94a) 1 Introduction, on Mathematics in General 2 Sequences 3 The Fundamental Operations in Algebra 4 Factors 5 Negative Numbers 6 Fractional Quantities 7 Simple Equations 8 Ratios and Proportions 9 Surds 10 Topical Geometry 11 Graphics and Perspective

1 3 4 7 34 61 75 84 91 102 131 143 165 192

2 New Elements of Geometry Based on Benjamin Peirce's Works and Teachings (94) Preface Book I Fundamental Properties of Space Chapter I Continuity Chapter II Uniformity [Chapter III] Chapter IV Homaloids Chapter V Measurement Chapter VI The Constitution of Space Chapter VII Branches of Geometry Book II Topology Chapter I Generation, Intersection, Enclosure Chapter Π Connectivity

233 235 239 241 255 259 260 264 269 270 271 273 289

XXX

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Chapter ΙΠ Polygons and Polyhedra Chapter IV Knots Chapter V Reversion and Perversion Book III Graphics Introduction Algebraical Lemmas Chapter I A Ray and a Plane Chapter Π The Fundamental Propositions of Graphics . Chapter ΙΠ Degrees of Freedom Chapter IV Projection Chapter V Notations for Graphics Chapter VI The Phimus Book IV Metrics Part I The Philosophy of Metrics Chapter I Principles of Measurement Chapter II Primary Theorems Part Π Benjamin Peirce's Presentation of the Propositions of the Elements 3 Topical Geometry (137)

296 308 316 319 321 352 357 369 385 397 419 427 429 429 445 458 475

4 Appendices A. On the Quadratic Equation (86) B. Rational Fractions (278a fragment) C. Numerical Equations (69) Method of Solving a Numerical Equation (from 1150) . . Scholium (from s-25) D. [Additional Definitions] (from 166 and 150) E. [A Gloss. Elliptic, Hyperbolic, and Parabolic Measurement] (150) F. A Geometrico-Logical Discussion (126) G. [Projective Space] (part of 114) H. [Logic of Number — Le Fevre] (229) I. [Additional Theorems] (150 and 266) J. Promptuarium of Analytical Geometry (part of 102). . . K. Pythagorean Triangles (109) L. Analysis of Time (part of 138) M. Plan of Geometry (132) N. [Non-Euclidean Representation] (105) O. An Attempt to State Systematically the Doctrine of the Census in Geometrical Topics or Topical Geometry, More

551 556 558 559 561 564 574 578 590 592 596 602 607 611 614 621

CONTENTS

Commonly known as 'Topology'; Being, a MathematicoLogical Recreation of C. S. Peirce. Following the Lead of J. B. Listing's Paper in the Göttinger Abhandlungen (from 145) P. ["Census" from Century Dictionary with Model] Q. Chapter III The Nature of Logical Inquiry (608) . . . . R. Three Problems A Problem — "A Young King" (s-59) Problem (from s-3) A Problem — The Vindication of Susan (s-58) S. Comments on Cayley's Memoir on Abstract Geometry from the Point of View of the Logic of Relatives (546) . . . . T. [Obituary. Prof. Arthur Cayley] U. [From an Address to National Academy of Sciences] (95)

xxxi

623 629 633 634 636 636 638 642 650

Key to Greek Terms

655

Key to Contents of MS. 165

659

Key to Contents of MS. 94

662

Index of Names

667

Subject Index

669

1

ELEMENTS OF MATHEMATICS (165)

PREFACE (1) (164)

This text-book contains all the mathematics which can conveniently be taught in the common schools, excepting practical arithmetic. It is so written that any one of its principal facts may be omitted in teaching. Teachers need not be informed that there is no elementary subject which has advanced so much of late years as geometry. No recent writers approve of putting beginners into metrics at the outset; but as circumstances sometimes compel the use of the old system, this work has been so arranged that the preliminary work, which saves a great deal of time in the end and often makes the difference between learning geometry agreeably and never learning it at all, but only learning to loathe it, can be passed by, if need be. To what can be taught in the schools will here be found appended further information sure to be useful to the scholar in after-life, if he has prudently preserved his school-books....

PREFACE (2) (94a)

Benjamin Peirce's Elementary Treatise on Geometry was published in 1837, and since he had been teaching the subject in Harvard College (beginning two years before his graduation) for ten years, it is safe to infer that the substance of the work had been in his mind for a long time. The whole aspect of the science has been metamorphosed since say 1830 or 1835. In the first place, the nature of the hypotheses (that is, the Axioms and Postulates, together with other propositions virtually taken for granted in the old books without explicit statement) is differently conceived. We all see, now, that geometry has two parts; the one deals with the facts about real space, the investigation of which is a physical, or perhaps a metaphysical, problem, at any rate, outside of the purview of the mathematician, who accepts the generally admitted propositions about space, without question, as his hypotheses, that is, as the ideal truth whose consequences are deduced in the second, or mathematical, part of geometry. In the second place, Listing and others have created the topical branch of geometry, which studies the connection of places. This branch deals with only a portion of the hypotheses accepted in other parts of geometry; and for that reason, as well as because of its relative simplicity, it should be studied before the others. Moreover, it is most desirable that, before the scholar comes to the difficulties which in the old system meet him at the threshold of geometry, he should have had some previous training leading up to the mental effort which he is then called upon to exert. In the third place, although in 1830 not a little had really been done by individuals toward restoring that graphical, or projective, or intersectional, branch of geometry, which formed so prominent a part of the ancient science, yet it remained, for the general world of mathematicians, a closed book. It has since become the most prominent part, not only of geometry, but of all mathematics. For more than forty years, now, geometers have

PREFACE ( 2 )

5

clearly perceived that metrical geometry is but a special problem of graphical geometry. There is reason to suspect that in the school of Euclid some instruction in this branch preceded the study of the Elements. At any rate, in its modern development, some slight treatment of it, so far, at least, as to explain the practically important principles of Linear Perspective, is a needed prop[ae]deutic to metrical geometry. In the fourth place, the whole conception of metrical geometry has been revolutionized. In 1837, the "Doctrine of Parallels" formed an urgent but unsolved problem. The earnest and persistent efforts of Legendre and of many other eminent mathematicians had been powerless to clear up its difficulties. Benjamin Peirce, in his treatise, thought to conquer them by the introduction of the idea of a difference of direction. There can be no intelligent question that this idea brings strong help to mathematical inquiry. Soon after its introduction by Peirce, it was taken up by Hamilton and by Grassmann with such effect as might have been anticipated. But Professor Peirce himself subsequently admitted, with the rest of the mathematical world, that there was no solution for the question of parallels except from the idea which the pupils of Gauss, Bolyai, Lobatchewsky, Riemann, derived through their master, from Lambert, and ultimately from the Italian Jesuit, Saccheri. The idea was that it is simply a question for observation of nature whether the sum of the angles of a triangle is less than, or more than, or possibly equal to two right angles. Subsequently, Cayley (in 1854) and Klein (more fully, in 1873) showed that metrical geometry is simply the geometry of the firmament, or absolute, or infinitely distant part of space, which constitutes a surface which is one or another quadric surface, according to the system of measurement adopted, that is, according to the way in which rigid bodies move. In the course of this inquiry, a fallacy in Euclid's 16th proposition was brought to light that had remained undetected for two thousand years. In the fifth place, Georg Cantor and others have succeeded in analyzing the conceptions of infinity and of continuity, so as to render our reasonings concerning them far more exact than they had previously been; and the fundamental researches that have largely occupied mathematicians of late years, into the theory of functions, do much to render geometrical reasonings more exhaustive and precise. All these intellectual movements ought, in the opinion of very many mathematicians, to have their effects upon the system of teaching the elements of the subject. But this is not all. Pedagogy is an art which has come in the last sixty

6

ELEMENTS OF MATHEMATICS (165)

years to be based more and more upon modern scientific psychology, and upon modern views of logic. If diligent and intelligent youths find difficulty in understanding mathematics, teachers no longer deem it becoming to flog them or to objurgate them, as they used to do, in the days when the second theorem of Euclid received the name of the "Asses' Bridge." On the contrary, they consider the fact that a considerable percentage of the best minds imagine themselves to be utterly unable to comprehend mathematical reasoning to constitute an emphatic condemnation of that old system of teaching which had such a result. Teachers now see that the difficulties of the first steps in mathematics must be divided and conquered, one by one. The three functions of the mind that are exercised in mathematics are exact reasoning, mathematical imagination, and complex generalization. The first of these, the logical part, is best acquired in the study of the theory of number, because that subject involves little other difficulty. It calls for but slight efforts of imagination and of generalization. Topology, or connective geometry, is the best field for the growth of imagination, demanding little logic and not very much generalization. Graphics, or projective geometry, carried far enough, affords good training in generalization. When some familiarity with the business of the mathematician has been acquired by such studies, metrical geometry may be taken up without fear that the student's mind will be confounded by its aggregation of various difficulties. This volume is intended to contain all the mathematics (except practical arithmetic) which is necessary for a man with a good common school education, and at the same time, to give the thoughts of the student such training as may prepare him for a study of the higher mathematics. Most of the text books of geometry have contained some algebra. Euclid's Elements is more than half devoted to that branch. Benjamin Peirce's work has a brief algebraical introduction. The present volume gives all the algebra which is indispensable to an ordinary man.

1 INTRODUCTION, ON MATHEMATICS IN GENERAL

Art. 1. Definition 1. Mathematics is the science which draws necessary conclusions. Gloss 1. "Mathematics" (pronounced mathemat'iks, as it is written in Ellis's Glossic) is a noun plural in form, but now used with a verb in the singular number. A few old-fashioned people still say "the mathematics are an important science." The Mathematickes, and the Metaphysickes Fall to them as youfindeyour stomacke serves you. ( Taming of the Shrew, I.i) Earlier writers use the singular form, mathematike. Gloss 2. Names of sciences ending in "-ics" are imitated from the neuter plurals of Greek adjectives, the word βιβλία, books, being understood. The Greek name of this science is τά μαθηματικά, or, more commonly, ή μαθηματική, where τέχνη, craft, or some such noun is understood. The adjective is formed from the noun μάθημα, lesson, from μανθάνω (μαθεΐν), learn. The root is the same that appears in the English "mind." Gloss 3. With the Alexandrine Greeks the name mathematics (ή μαθηματική) came to be applied chiefly to astronomy. Thus, Ptolemy's text-book of that science, known as the Almagest, has for its genuine title Μαθηματική σύνταξις (Mathematical Compendium). In the middle ages, by a "mathematician" was understood an astrologer, or even a sorcerer. So late as Queen Elizabeth's time, the learned Dr. Dee was practically driven out of England, because he was reputed a "mathematician." In the Roman and Medieval schools, the mathematical sciences, — that is to say, the four they studied, arithmetic, geometry, music, and astronomy, — were collectively called the Quadrivium, a fanciful name signifying literally the four roads. Memorandum 1. The definition of mathematics here given was origi-

8

ELEMENTS OF MATHEMATICS (165)

nally put forward by Professor Benjamin Peirce, in 1870. It has been received with much favor by mathematicians. The only definition which seems to have been current in the more ancient times was that of Aristotle, which was an attempt to describe the peculiar kind of abstractness and ideality of the objects of mathematical study. Its significance can only be apprehended by a person thoroughly versed in the Aristotelian philosophy. Its wording is that mathematics is the science of forms immovable but not separable from matter. This definition is laudable in recognizing that mathematics is characterized by a peculiar abstractness and ideality of its objects; but it errs in implying that metaphysics carries those qualities to a higher degree, as we shall see. The Roman school-masters, after much of the Greek geometry had been almost forgotten, defined mathematics as the "science of quantity." But what they meant was simply that the objects which mathematics studies have magnitude, which is equally true of the objects of almost all, if not of all, sciences. The phrase subsequently became very popular as a definition of mathematics in another sense, namely, that mathematics is the science of quantity itself. It must be admitted that mathematics deals very largely with quantity, and embraces the whole science of quantity; and most mathematicians feel more at home in reasoning about quantity than about anything else. Yet two of the main divisions of geometry have nothing to do with quantity, there are other branches into which it does not enter, and several in which other elements play important parts. During the nineteenth century one of those two branches of geometry, graphics, became the chief interest of geometers, while the other, topology, has also been zealously pursued. Moreover, the attention that came to be paid to non-quantitative considerations even in metrical subjects threw the popular definition in disfavor among mathematicians. One of the greatest of them, Hamilton, with the concurrence of the eminent mathematical logician, De Morgan, and of others, then proposed to define mathematics as the science of time and space, being influenced in this by the philosophy of Kant. But Hamilton's definition is, by far, the most objectionable of all that have been widely in vogue. For it implicitly denies the main characteristic of mathematics, namely that this alone among the sciences makes no researches into facts, but attends solely to ideas, without seeking to establish their truth. The doctrine of Kant that Space and Time are not real entities, contrary to Newton's inference from the phenomena of dynamics, is much discredited by

INTRODUCTION

9

modern inquiries. But even according to that doctrine the properties of space and time are truths, though they be mental truths, and are not hypotheses as the mathematician treats them. Since the mathematician treats them as hypotheses, and does not inquire into their accuracy, for him they are no more than hypotheses. His studies are not inquiries into the truth about Space and Time. Hamilton called algebra the science of Time. But the most remarkable characteristic of time, namely, that the passage from the past to the future is qualitatively different from the passage from the future to the past is not represented in algebra. The chief study of the algebraists is a two-dimensional continuum, strikingly unlike time. So the higher geometry deals largely with characters to which, confessedly, nothing in real space corresponds. It is, moreover, demonstrable that arithmetic is valid, quite irrespective of whether the objects counted are in Time or Space, or not. This remains true though it be granted that Kant is right in holding that all our positive knowledge of numbers is of numbers in Time and Space. Scholium 1. The occasions upon which the mathematician's art is called to our aid are those upon which we find ourselves imagining so complicated a state of things that we cannot clearly make out what all the consequences would be. For example, suppose we are proposing to build a suspension-bridge, which is a long heavy structure, such that equal lengths weigh nearly equal amounts, and which is hung by numerous suspenders from cables attached to towers.1 Suppose that, in the course of building such a bridge, we propose to allow a wire to hang in a festoon from tower to tower, so that it will touch one of the bridge-cables at its two highest points and also at its lowest point. Now, should we have occasion to know whether that festoon will touch the bridge-cable all along, or will be everywhere above it, or everywhere below it, or will cross it, we ask a question which only a mathematician can answer. If, then, we are not ourselves sufficiently expert mathematicians to solve the problem, we propose it to a mathematician. But as soon as the mathematician examines almost any of the questions that are propounded to him, he is pretty sure to find that the real facts are so vastly complicated, that the real problem can, practically, not be solved with exactitude. For 1 This illustration calls to mind Peirce's employment as consultant by Morison in the construction of a bridge. See Volume 3, 16a. In MS. 165 Peirce has a note to "Insert Fig. 1, Parabola and Catenary" for this passage. He failed to supply the diagrams but, as is well known, the catenary idealizes the loose festoon before attaching the roadway; the parabola after the attachment.

10

ELEMENTS OF MATHEMATICS (165)

instance, in the case supposed, there is reason to believe that the wire really consists of molecules which are not nearly in contact with one another and which are dancing about among themselves in a most intricate figure. To take account of that is beyond the mathematician's powers. Accordingly, the first thing he does is to imagine a state of things different from the real state of things, and much simpler, yet clearly not differing from it enough to affect the practical answer to the question proposed. That purely ideal state of things he proceeds to study. In the case of the bridge, the ideal wire will be continuous and homogeneous, and the ideal circumstances will differ in many other respects from the concrete reality. The mathematician now proceeds, by a method which will be described in Art. 2, to ascertain how the ideal festoon and ideal bridge-cable would be related to one another. In doing this, he not only finds out, with sensible accuracy, what will happen in the actual case, but also produces a rule by which other similar questions may be answered. He is, afterwards, generally led on by curiosity to consider what would happen under circumstances differing still more from the actual case. We see, then, that the mathematician's duty has three parts, namely, 1st, acting upon some suggestion, generally a practical one, he has to frame a supposition of an ideal state of things; 2nd, he has to study that ideal state of things, and find out what would be true in such a case; 3rd, he has to generalize upon that ideal state of things, and consider other ideal states of things differing in definite respects from the first. This description of the mathematician's duty gives the best notion of what mathematics is : it is the exact study of ideal states of things. To say that mathematics is the science which deduces necessary consequences comes to much the same thing; because no necessary consequence can be deduced except from an ideal state of things, and then only on condition that the state of things is generalized. Definition 2. A mathematical hypothesis is an ideal state of things concerning which a question is asked. Art. 2. Scholium 2. Experience shows that careful attention to the method of mathematical thought contributes not a little to success in the practice of it. It is like swimming. A person may know how swimming is done, without being able to do it; and, on the other hand, a person may swim like a South Sea Islander, without knowing how one does it. But nevertheless, a careful study of the modus operandi will almost always improve one's swimming, and will help very much in overcoming the first awk-

INTRODUCTION

11

wardness. One important part of mathematical thought, often mistaken for the whole, is the process of deducing the consequences of hypotheses. To show the way in which this is done, we may examine a little mathematical investigation to which many small boys upon their own motion apply themselves, — we mean the analysis of the game of Tit-tat-too. This game is played by two persons, playing alternately. We may call the first player X, the second O. They use a square board, or diagram, of nine equal square compartments. We may number these squares as shown in the figure. [Fig. 1] 1

7

2

3

5

6

8

9

Fig. 1

Each play consists in occupying an unoccupied square, by the player's putting his mark in it. The object aimed at is to occupy the three squares of a line, that is, one of the following triads: 1, 4, 7, 1, 2, 3, 1, 3,

2, 5, 8, 4, 5, 6, 5, 5,

3 6 9 7 8 9 9 7.

As soon as a player does this, he points at the three occupied squares, one by one, saying "tit" at the first, "tat" at the second, "too" at the third. This is called "making tit-tat-too," and wins the game. The mathematical analysis of the play consists in ascertaining the effects of the different ways of playing. It is accomplished by experiment. The first player has at the outset the choice of 9 squares, of which he can occupy any one; at the second play there are only 8 to choose from; at the third play there are only 7, etc. And thus there are only 9 χ 8 χ 7 x 6 x 5 x 4 x 3 x 2 x 1 ways of filling the board; or [362880] ways in all.2 The performance of as many of these [362880] experiments as ' Peirce wrote 725760.

12

ELEMENTS OF MATHEMATICS (165)

might be necessary to bring each imaginary game to a conclusion would be one way of conducting the mathematical investigation. It is the type of all mathematical deduction, the substance and staple of which is in every case experiment. But mathematical experiments are peculiar, in that the results of them depend upon the nature of diagrams, or "constructions," of our own creation, instead of depending upon the natures of cosmical creations. Though we must allow the performance of those [362880] experiments would be a mathematical proceeding, it would be an unskillful one. Mathematicians always abridge such lengthy operations. This they are enabled to do by the fact that such lengthy operations always involve many repetitions of essentially the same experiment. For instance, one experiment which we perform so quickly in imagination that we are apt to forget any experiment was needed to ascertain the result is as follows. Suppose that of a line of three squares one of the players has already occupied one, while the other two remain vacant; and suppose that that player now occupies one of the vacant squares. Then, unless the other player at once makes tit-tat-too, or else occupies the remaining vacant square, the former player can at once make tittat-too. This being ascertained once for all, if we bear it in mind, a vast number of experiments among the [362880] become superfluous. Such a rule, established once for all by experimentation upon diagrams, and enabling us to dispense with many repetitions of essentially the same experiment, is called by mathematicians a theorem. (A formal definition of this word will be given in Art. 11, Def. 8.) If one player has occupied two squares of a tit-tat-too line, while the third is vacant, we will call that a peril-row, because the other player must instantly fill the vacant square, or the first can make tit-tat-too, as soon as his turn comes. Herein, this little game illustrates one of the most valuable rules of life, that a difficulty promptly met is no difficulty at all. Let us suppose that both players use sufficient care instantly to fill peril-rows when they can. Then since the three squares of a tit-tat-too line, if occupied at all, must be occupied one after another, so that, at some stage, two of them must have been filled while the third remained vacant, it follows that tit-tat-too can never be made at all, except when situations are reached in which players cannot fill peril-rows. But the player whose turn it is has a right to occupy any vacant square he chooses. How, then, can it be impossible for him to fill up the one vacant square of a line? It evidently cannot be impossible for him to do that. Only, his adversary may have occupied a square which belongs at once to two

INTRODUCTION

13

different tit-tat-too lines, on each of which he has already occupied a square and on each of which the third square is vacant. In that case, though either peril-line may be blocked, both cannot. In that way only will tit-tat-too be possible, supposing both players fill up peril-lines when they can. We thus see, that tit-tat-too can only be made when the party whose turn it is to play has already occupied one square upon each of two intersecting tit-tat-too lines, while the other three squares (the one at the intersection and one other on each of the lines) are vacant. This is another theorem, productive of great economy of experimentation. N.B. (1) This theorem well illustrates the extreme care that is necessary in mathematical reasoning. An uncautious thinker might say that in order to make tit-tat-too, it is necessary to occupy the three squares of the line successively, and to give the adversary a turn after each one. Now, after two have been occupied, the adversary has it in his power to prevent the completion of the row, by himself occupying the third square. Observe that this is strictly true. Therefore, the row cannot be completed, if the adversary uses ordinary care to prevent it. Therefore, tit-tat-too cannot be made, if the adversary uses ordinary care to prevent it. All this reasoning is correct except the very last step. It is true that no one particular row can be completed, if the adversary chooses to prevent it. But it does not follow that no row can be completed if the adversary does his best to prevent it. For though any one can be blocked, it may be that it is impossible to prevent one or other or two from being completed. Such slips in reasoning are hard to avoid; but they can be avoided; and the best method of avoiding them shall now be set forth. What is true is, that, taking any row whatever, a way can be found for the first party to play, so as to prevent the second party from completing that row: but what is false is, that a way can be found for the first party to play, such that, taking any row whatever, that mode of play will prevent the second party from completing that row. The point to be observed is, that a statement of the possibility of the first play's making tit-tat-too has to deal with a number of rows and a number of ways for that party to play; from the number of rows any one, no matter what, is to be taken, while from the number of ways of playing, a suitable one is to be chosen. Now, whenever we have to deal with two collections from one of which any one that comes is to be taken, while from the other a suitable one is to be chosen, it is (or may be) an advantage, serviceable to the purposes

14

ELEMENTS OF MATHEMATICS ( 1 6 5 )

of the person who is to make that selection to postpone making it, until he sees what one of the other collection is taken. Let the reasoner, then, be always on the alert for the case of two things being said to be taken from two collections in different ways; and be careful in every such case to note which is said to be taken first. Thus, he will avoid a dangerous snare of reasoning. For example, choosing first any man whatever, a woman [is] then [to] be found (among living or dead) such that that woman was that man's mother. But it does not follow that a woman can be found such that, subsequently taking any man whatever, that woman was that man's mother. "Any man has some mother," is true; "all men have some mother," is false. "Every man loves some woman," ought to be true; "some woman is loved by every man," is another proposition. If the reasoner is the master of this principle, as soon as he finds that the completion of any one tit-tat-too row can be prevented, he will ask, "Can there, then, be simultaneously two rows that require immediate blocking? If it is not so, tit-tat-too cannot be made, against ordinary care. But if it be so, those two rows must have reached that stage at one play, since the need for blocking each is immediate. Yet one play can fill but one square. Then, the square occupied by the play that brings about the fatal situation must belong at once to both rows, that is, must be their square of intersection. That is to say, tit-tat-too can only be made by a player who has occupied the square of intersection of two peril-rows, made by such occupation. We may call such a situation, after the square of intersection is occupied, an "A situation," and before the square of intersection is occupied, an " H situation." There will be no tit-tat-too, unless, after one party has created an H situation, the other plays outside of its rows so as to allow the A situation to be completed. Now, an H situation has three vacant squares in its rows, and counting the one upon which the other party plays, an H situation leading to tit-tat-too supposes four vacant squares. Therefore, the first player cannot make tit-tat-too unless he creates an H situation at his third play at latest. For then he leaves just 4 vacant squares. The second player always leaves an odd number of vacant squares; and therefore he cannot make tit-tat-too unless he creates an H situation at his second play. For then he leaves 5 vacant squares. The number of experiments is greatly reduced by these considerations, and still further by considering that if there is an imaginary line such that all the squares occupied by either player are symmetrically placed

INTRODUCTION

15

to the right and left of it, it is unnecessary separately to consider the occupation of two vacant squares symmetrically placed on the two sides of that line. In such case[s] we may write the sign = between the numbers of the squares to show their equivalence. Let us now proceed to make a partial analysis of the play, in order to show the scholar the advantage of a systematic procedure in mathematics. We will record the supposed play by drawing a line as for a fraction and writing above it the square occupied by X and under it that occupied immediately after by O. If a play is forced, that is, is the obligatory filling of a peril-row, to prevent immediate tit-tat-too, we write a semicolon after its number. At the end, X!!! or OÜ! shall mean that X or O makes tit-tat-too; while a curved dash shall show that neither party makes it. //-situations are of different kinds. If the two occupied squares are on only one pair of intersecting lines elsewhere vacant, the situations may be called a simple H [Fig. 2]. J

X

X-

X --0--

—

Fig.2

If there are four such lines intersecting in two vacant squares, the situation may be called a double H [Fig. 3].

Fig. 3

If there are three lines intersecting in two vacant squares, the situation may be called a semidouble H [Fig. 4].

Fig. 4

16

ELEMENTS OF MATHEMATICS ( 1 6 5 )

If there are four such lines intersecting in three vacant squares, the situation may be called a semitriple H [Fig. 5]. I O ν h-· Ν I \ \11 II 1 1

XI ν O κΧI I Ν• \/ II 1 κ ν I V>I k Fig. 5

If there are five such lines intersecting in four vacant squares, it may be called a semiquadruple H [Fig. 6].

1 Η-

!/

1 • 1/

Χ-

ιs ι ι _\ χ - 1 Γ\ -ΧI Ι ι Ο χ I

—

1 1

O Fig. 6

If the two occupied squares belonging to an H are in one tit-tat-too row of which the third is vacant, so as to form a peril-row, it may be called a self-reinforced H. If then the situation be such that the adversary in blocking that peril-row does not himself create a peril-row, the other player can complete his A situation (as we will assume he always will), and the H situation may be said to be reinforced unassailably. [Unassailably reinforced H in Fig. 7(c); an unassailably reinforced semidouble H in Fig. 7 (e).] If, however, the situation is such that the adversary in blocking the peril-row of the self-reinforced H, himself creates a new peril-row, then we have to distinguish two cases. Namely, when the creator of the H blocks that new peril-row, he may or may not thereby complete an .¿-situation. If he does, the original H situation was reinforced strongly, if not, it was self-reinforced weakly. [Weakly selfreinforced single if in Fig. 7(a); strongly reinforced single H in Fig. 7(b); strongly reinforced semidouble H in Fig. 7(d); strongly reinforced semitriple H in Fig. 7(f).] If the player who creates an H, thereby also makes a peril-row, the already occupied square of which does not belong to the H, the H is said to be externally reinforced. Such reinforcement is always weak. [Externally reinforced H in Fig. 7(g).]

INTRODUCTION

0

/

X1 1 1 I 11 1

•

/

/

/

X

--X-

—

17

0

X,

O

\

\

X--

sX Ν.

Fig.7(a,b,c)

χ

9 ι

γι

!/

•• / —

/

*I

ο

1 I

-χ-

•• —

ί/

Χ

Fig. 7 (d, e)

Υ- I ι ι f I

ν

\

/

Ο

Χ

\

>
c \l Ν 1 X" 'Ν

Fig. 10

The following analysis shows the result of every possible play where X first occupies square 2, supposing that each party, at each play, makes tit-tat-too if he can do so at once, and if not prevents the other from

18

ELEMENTS OF MATHEMATICS (165)

making tit-tat-too at the next play, and, if that condition allows, makes A at once, if he can, and if not, prevents the other party from making A at the next play if he can. It is unnecessary for the pupil to go through the whole of the analysis. If he goes to case 12, it will be sufficient. In the numbering of the cases, 135, for example, does not mean one hundred and thirty-five, but the first case, and the third subdivision of that case, and the fifth subdivision of that case. This way of numbering cases, invented by Benjamin Peirce, is of great utility [Figs. 11-74]. Case 1.

0 has choice between 8 (case 11), 5 (Case 12), 1 = 3 (Case 13), 4 = 6 (Case 14), 7 = 9 (Case 15).

Case 11.

X has choice between 1 = 3 (Case 111), 7 = 9 (Case 112), 4 = 6 (Case 113), 5 (Case 114).

Caselli.

h

O has an unreinforced semidouble H, and X can only prevent the A by occupying 7 or 9. Neither of these gives X an H, and therefore there is no tit-tat-too.

Case 112.

Ζ

X has a semidouble H, and O can only prevent the A by occupying 1 (Case 1121) or 3 (Case 1122). O cannot make an H.

Case 1121.

T

Case 1122.

T

Case 113.

Case 1131.

A

T

X can only make an unreinforced single H, and A would be prevented. Consequently, there is no tit-tat-too. X can only make an unreinforced single H, and A would be prevented. Consequently, there is no tit-tat-too. X has a single H, and O can only prevent A by occupying either 1 (Case 1131), 3 (Case 1132), 7 (Case 1133), or 9 (Case 1134). O has a single Η. X can only prevent A by occupying either 7, 9, 5, or 6. The last two do not give him an H\ so that there would be no tit-tat-too. The other plays give unreinforced single Hs, and A would be prevented by O occupying 3. Consequently there is no tit-tat-too.

INTRODUCTION

19

O has a semidouble H, and X cannot occupy 1, which would give O an Α. X can only occupy either 5 (Case 11321), 6 (Case 11322), 7 (Case 11323), or 9 (Case 11324). X has a weakly reinforced single H. There is no tit-tat-too. X has no H. There is no tit-tat-too.

X has no H, and consequently there is no tittat-too. X has only an unreinforced single H, and A will be prevented. Consequently there is no tit-tat-too. X has a double composite H. If O occupies either of the pair of squares 1 and 6 or of the pair 3 and 5, X makes A by occupying the other square of the pair, and so X makes tit-tat-too. X has an A, and makes tit-tat-too.

X has an unreinforced semidouble H, and O can only prevent A by occupying 1 = 3 . He thus gets an unreinforced single H, and A is prevented; so that there is no tit-tat-too. X has choice between 7 = 9 (Case 121), 4 = 6 (Case 122), 1 = 3 (Case 123), 8 (Case 124). X has an unreinforced single H, and O, to prevent A, must occupy either 1 (Case 1211), 3 (Case 1212), 4 (Case 1213), or 6 (Case 1214). There is n o H, and consequently there will be no tit-tat-too. O has an unreinforced semidouble H, X can prevent A, but cannot get an H and consequently there is no tit-tat-too.

20

ELEMENTS OF MATHEMATICS (165)

Case 1213.

-¡«i 4

X has an unreinforced semidouble composite H. O can prevent A by occupying 3 or 9, and consequently there is no tit-tat-too. O has a weakly reinforced single H . There is no tit-tat-too.

Case 1214.

Case 122.

X has an unreinforced single I f . To prevent A, O must occupy either 1 (Case 1221) or 3 = 7 (Case 1222).

±

Case 1221.

21

y

There is n o H, and consequently will be no tit-tat-too.

Case 1222.

_

O has a weakly reinforced single H . There is no tit-tat-too.

Case 123.

i^li

There is no H, and consequently will be no tit-tat-too.

Case 124.

A

O has choice between 1 = 3 = 7 = 9 (Case 1241) and 4 = 6 (Case 1242).

Case 1241.

O has a strongly reinforced semidouble H, and makes tit-tat-too.

Case 1242.

Case 13.

_

4

O has an unassailably self-reinforced semidouble H, and makes tit-tat-too. X has choice between 3 (Case 131), 4 (Case 132), 5 (Case 133), 6 (Case 134), 7 (Case 135), 8 (Case 136), and 9 (Case 137).

T

Case 131.

X

X has an unreinforced single H, and O, to prevent A, must occupy either 4 (Case 1311), 5 (Case 1312), 7 (Case 1313), or 8 (Case 1314).

Case 1311.

_4 21

O now makes A and tit-tat-too.

Case 1312.

y

Case 1313.

—

O has a weakly reinforced simple H. There is no tit-tat-too. O has an unassailably reinforced simple H, and will make tit-tat-too.

INTRODUCTION

21

O has an unreinforced semidouble H. X to prevent A, must occupy 7 or 9. Neither way does he gain an H, and therefore there is no tit-tat-too. X has an unreinforced simple H. O, to prevent A, must occupy either 5 (Case 1321) or 6 = 8 (Case 1322). There is no H and consequently will be no tit-tat-too. O has an unreinforced simple H. To prevent A, X must occupy either 5, 8, 3, or 9. The first two do not give him an H. The others only an unreinforced simple H, and A will be prevented. Consequently, there is no tit-tat-too. O has an unreinforced simple H. To prevent A, X must occupy either 4, 7, 9, or 6. The first two give him no H, the third only an unreinforced simple H, and the fourth a weakly reinforced simple H. Consequently there is no tit-tat-too. X has an unreinforced simple H. To prevent A, O must occupy either 4 (Case 1341), 5 (Case 1342), 7 (Case 1343), or 8 (Case 1344). X has a double composite H, so that of the two pairs of squares 5 and 9, on the one hand, 3 and 8 on the other, if O occupy one of the squares X makes A on the other of the same pair; and X makes tit-tat-too. There is no tit-tat-too.

O has a strongly reinforced semidouble H, and makes tit-tat-too. O has an unreinforced semidouble H. To prevent A, X must occupy either 4 (Case 13441), 5 (Case 13442), 7 (Case 13443), or 9 (Case 13444).

22

ELEMENTS OF MATHEMATICS (165)

Case 13441.

5; 3;

Case 13442.

11 |o|x|o|

No tit-tat-too.

A-IS

There is no tit-tat-too.

Case 13443.

1

X has an unreinforced simple H. A is prevented, and there is no tit-tat-too.

Case 13444.

2.

There is no H and consequently will be no tit-tat-too.

4 ; 3;

Case 135.

1

Case 1351.

T

Case 1352.

X has an unreinforced semidouble H. To prevent A, O must occupy 5 (Case 1351) or 8 (Case 1352). There is no H and consequently will be no tit-tat-too. O has no H, but X can form only an unreinforced simple or semidouble H. A would be prevented, and consequently there will be no tit-tat-too.

Case 136.

-L

O has a strongly reinforced semidouble H and will make tit-tat-too.

Case 137.

2.

X has an unreinforced simple H. To prevent A, O must occupy either 4 (Case 1371), 5 (Case 1372), 7 (Case 1373), or 8 (Case 1374).

Case 1371.

-, 4 li

X has an unassailably reinforced semidouble composite H, and will make tit-tat-too.

Case 1372.

—

O has an unreinforced semidouble Η. X can prevent A but cannot get an H, and consequently there is no tit-tat-too.

Case 1373.

y

Casel374.

¿ase 14.

Τ

4^

X has an unreinforced semidouble composite Η. O can prevent A by occupying 5 or 6. No tit-tat-too. O has no H, and X can only get an unreinforced simple or semidouble Η and A can be prevented. Consequently, there is no tit-tat-too. X has choice between 1 (Case 141), 3 (Case 142), 5 (Case 143), 6 (Case 144), 7 (Case 145), 8 (Case 146), and 9 (Case 147).

23

INTRODUCTION

Case 141.

J,

Case 142.

1i ,»

Case 143.

i.

X has an unassailably reinforced simple Η and will make tit-tat-too.

ο

Case 144.

Case 156.

O has a strongly reinforced simple Η and will make tit-tat-too.

o

X

X has an unassailably reinforced semidouble Η and will make tit-tat-too. X has an unreinforced simple H. To prevent A, O must occupy either 1 (Case 1441), 3 (Case 1442), 7 (Case 1443), or 9 (Case 1444).

£

^

o

O has a strongly reinforced semidouble Η and makes tit-tat-too. X has an unreinforced semitriple Η. O can prevent A only by occupying 1, but gets no Η and X gets an unreinforced semidouble composite H. There is no tit-tat-too.

Case 157.

SUMMARY OF THE RESULTS s Τ

/ν

χ

X!!! (V /•w ^

OÜ! /V ^

X!!! OU! OÜ!

Οϋ! 7 Τ 4 i JL Οϋ! 4 81 Τ Τ XU! Χ!!! 5. /ν

zw

/ν

Οι»

2. 1_

(V

2 1 Χϋ! 7 5. /ν

9_ (V 6 Οϋ! 1

Χϋ! Οϋ! Οϋ!

4 4 8 9" Τ "6 8. Οϋ! 4V (V

i 1 4 5. i

! τ β τ 9" τ 6 1 7. 8. 3.

Οϋ! Χϋ! Χϋ! Χϋ! Χϋ! /ν

Χϋ! Χϋ! (V

Οϋ!

If - , T gives O the best chance; for of the 7 possible plays of X, three will give the game to O. But the best play for X is then when of O's four possible plays, one gives the game to X. For his first play, O may prefer T when of the four possible plays of X one gives the game to O. There are two first plays of O which give the game to X if he plays right. They are γ and To the former X should reply by

24

ELEMENTS OF MATHEMATICS (165)

If he plays - , - , - or - he gives the game to O. If O's first play is X can only lose by - . He should play - or - . 8 Art. 3. Scholium 2, continued. If the pupil goes back and looks over the analysis of tit-tat-too attentively, he will observe that the gist of the process consists in making experiments, or trials, systematically, and carefully observing and registering the results. Let us try and see what happens must be the motto of the mathematician. The apparatus, or prepared instrument, upon which the experiments were made was a diagram of the tit-tat-too board. In all mathematical reasoning something like this has to be prepared; and it must be something that can be observed and that we can modify. It is a likeness, or analogue, of the imaginary thing we are reasoning about, and is called an icon, or image. Two kinds of icons are chiefly used by mathematicians, namely, first, geometrical figures, drawn with lines, and, second, arrays of points or letters. In connection with our diagram, we had to use the arbitrary signs, X and O, for the marks of the two players. For such purposes, mathematicians generally use letters. Finally, we had to draw up a careful statement of the rules of the game. The results, too, being what will always happen when players play in certain ways, are a sort of rules. The theorems we used were again subsidiary rules. All those rules we expressed in words. In place of words mathematicians often use other systems of symbols, having remembered rules associated with them. Such symbols have two advantages; first, they may be made clearer and more definite than words, especially for complicated cases; and second, they can be arranged so as to make arrays, upon which experiments and observations can be made. For such symbols, mathematicians generally employ peculiar characters, or little shapes about the size of capital letters. Such are the tools with which the mathematician works. They have to be used intelligently. Here are some maxims which will aid your intelligence : 1. Keep your purpose in view. 2. Make all your procedure as regular, and systematic, and symmetrical as you can. 3. Be on the look out for regularities, symmetries, and systems. 4. Get rid of everything useless. ' Another extensive seventeen-page draft of the tit-tat-too analysis is found in MS. 1525.

INTRODUCTION

25

5. Use every means of economizing the work of thought, not in order to think less, but in order that the energy which can be dispensed with in one place may be employed in another place where it is needed more. The theorems we introduced into the analysis of tit-tat-too were examples of such economies. Art. 4. Scholium 3. Mathematical works, like others, are customarily divided into Books, the books into Chapters, and the chapters into Sections. The smallest subdivisions above paragraphs are called Articles, and are generally numbered continuously throughout the work. The formulae, or symbolically expressed rules, have to be numbered, or otherwise marked, for reference. In the present work, they will be referred to by pages, and upon each page the different formulae will be marked successively by the signs of the zodiac in their order, viz. : Τ Aries. Ö Taurus. Π Gemini.

as Cancer. Λ Leo. n j Virgo.

— Libra. Scorpio. f Sagittarius.

V^ Capricornus. as Aquarius. Κ Pisces.

There are also certain kinds of statements in mathematical works which have special names. Art. 5. Definition 3. A definition is an exact description of the kind of object to which a word or phrase is exclusively applied. Gloss 4. The word definition (in Chaucer and other old writers difinicion and diffinitiouri) was taken directly from the medieval Latin diffinitio, a corruption of the classical definitio, having the same meaning as the English word. It was an old Latin word (from de- of and finis, a bound, from root BHID, to cleave, whence English bite), adopted by Cornificius* (* the supposed author of a treatise probably written about 80 B.c., perhaps the teacher of Cicero) and by Cicero to translate the Greek όρισμός, used by Aristotle in the same sense, and derived from δρος (Corcyrean δρρος), a bound, also a term or general name. Euclid heads his lists of definitions δροι, that is, terms. Probably Aristotle's word was not familiar to him. Memorandum 2. The old logics make two kinds of definitions, the nominal, which merely explain the use of words, and the real (diffinitio quid rei) which analyze the nature, i.e. the essence, i.e. the idea of the kind of thing. The kind of thing defined is called the definitum. A real definition ought (it was said) to state the genus, or general nature, and the difference, or specific characteristic of the kind of thing. Definitions

26

ELEMENTS O F M A T H E M A T I C S

(165)

strictly mathematical are of kinds of objects of our own creation. They ought to state the purpose or idea which governs such creation. Many mathematical kinds of objects may be differently defined from different points of view. Some mathematicians prefer genetic definitions, or such as show how objects of the kinds defined may be constructed. They have the advantage of showing that the kind defined is not absurd. The old logics declare that simple ideas cannot be defined; but there is nothing in mathematics which cannot be defined, assuming the person to whom the explanation is to be made has the use and experience of his eyes, muscles, muscular sense, and ordinary speech. Axiom 1. If we know that an object is of a given kind, we may conclude that anything is true of it which is stated of that kind in its definition. This is called reasoning from definition to definition. Axiom 2. If we know that of a certain object everything is true that is stated of a given kind in its definition we may conclude that that object is of that kind. This is called reasoning from definition to definition. Art. 6. Definition 4. A division is an enumeration of the varieties of a certain kind, such that everything of that kind belongs to one or other of the named varieties. Gloss 5. Division is another old logical word, from the Latin divisio, used by Cornificius in a somewhat different sense, but by Cicero nearly in the sense of our definition. The Latin verb dividere, is formed from dis-, as under, and probably an unknown verb vïdëre, which is supposed by some to mean to sunder, by others to know. It is used to translate the διαίρεσις (from διά, as under, and αίρεΐν, to grasp) used by Plato and Aristotle in the logical sense. Art. 7. Definition 5. A postulate is a proposition forming a part of a mathematical hypothesis. Gloss 6. The word postulate (for which in Elizabethan English "petition" was used: thus Blundevile in his Logicke, 1599, speaks of "petitions, called in Latin Postulata.") comes from the Latin, postulatum used in this sense by Boethius (A.D. 500) who says it is an old word. Postulatum is the participle of postulare, to beg, frequentative from poscëre, or porcscëre, inceptive of precari from Sanskrit root PARKH, to pray, whence English pray. Postulatum is used to translate αίτημα, used by Aristotle and Euclid, from αΐτεΐν, to beg. Memorandum 3. This definition is intended to state what a postulate

INTRODUCTION

27

really amounts to, without implying that geometers have always understood it so. Ancient writers do not seem to agree as to what a postulate is. Aristotle, who lived before any of the important mathematicians whose works have come down to us, in the 10th chapter of the 1st book of his "Posterior Analytics," undertakes to explain the differences between the different kinds of geometrical first premises. When he speaks of postulates, he would seem to be referring to something different from the postulates of Euclid. Yet there are objections to that view. Perhaps, it is best to infer, as we may from many things in his writings, that he did not understand mathematical matters very well. He says distinctly that a postulate is not a hypothesis. He also says that postulates are demonstrable, δεικτά. But Aristotle sometimes uses this verbal adjective for the other; and perhaps he means δεικτέα, needing demonstration. He further says a postulate is unlike (ύπεναντίον) the opinion of the learner. Now, whatever is "unlike" a universal proposition is more than unlike; it conflicts with it. The expression, therefore, suggests that Aristotle conceives a postulate as asserting something, not universally, but only of something or other. Other writers have likewise said that a postulate asserts that something of a given description can be constructed or found. It is true that all Euclid's postulates except one are of that description. That one (certainly in no sense a postulate) is that all right angles are equal. Supposing that conception of a postulate to have been shared by Aristotle, then, when he says postulates are demonstrable, he may mean that they are demonstrable in any particular cases, by actually constructing, or finding, an object of the sort which the postulate says can be found. The five postulates of Euclid are as follows: I. Let it be granted that from any point to any point a straight line be drawn; II. And that a limited straight line be continuously produced in a straight line; III. And that with any centre and radius a circle can be drawn; IV. And that all right angles be equal to one another; V. And that if a straight line falling upon two straight lines make the interior angles on the same side less than two right angles, the two straight lines, being continued to infinity, will be concurrent on the side on which are the angles less than two right angles. Art. 8. Definition 6. An axiom is a plain truth about quantity. Gloss 7. The word axiom (in Elizabethan English "Dignities and

28

ELEMENTS OF MATHEMATICS (165)

Maximes," though axiome was occasionally used) is derived through the Latin form axioma, from the Greek άξίωμα, used in the same sense, but originally meaning a dignity, or that of which one is thought worthy, from άξιοΰν, to esteem, from άξιος, worthy, ultimately from άγειν, to reckon, draw, from the root AG, whence English act, agile, etc. The thread of ideas is this: άξιος means 'equal in value to'. Thus άξιος βοός is "worth an ox." Hence, άξιοδν means to deem worth as much as, and άξίωμα was, at first, the market value, what a thing was generally deemed worth, but later, what was generally held worth while, what was generally considered proper to be done, and finally, what was generally admitted as true. Aristotle, in the chapter referred to, speaks of "what are called the common axioms" (τά κοινά λεγόμενα άξιώματα). Eucüd calls them, "Common Notions" (κοιναί εννοιαι), and they generally went by this name in English down to Elizabeth's reign, or later. Memorandum 4. The axioms of Euclid as given in the best edition (Heiberg's) are as follows: 1. Things equal to the same thing are equal to one another; 2. And if to equals equals be added, the wholes are equal; 3. And if from equals equals be taken, the remainders are equal; 4. And if to unequals equals are added, the wholes are unequal (but Heiberg admits there is strong evidence this is spurious); 5. And things double the same thing are equal to one another; 6. And things half the same thing are equal to one another; 7. And things that fit over one another are equal to one another; 8. And the whole is greater than its parts; 9. And two straight lines do not enclose space. But the evidence is overwhelming that this is spurious, i.e. was inserted in the text by an ancient editor. Art. 9. Division 1. The contents of a mathematical treatise should be of six kinds, to wit: 1st, Analyses of facts, observed or imaginary, suggesting the hypotheses; 2nd, Formal statements of the hypotheses {postulates); 3rd, Explanations of the meanings of (A) the technical terms (definitions), (B) the peculiar symbols employed (inotations); 4th, Propositions borrowed from other branches of mathematics, or elsewhere, and accepted as indubitable without proofs (axioms); 5th, Exactly reasoned propositions, that is (see Arts. 11 and 12),

INTRODUCTION

29

(A) theorems, (B) problems; 6th, Enumerations of different cases which have to be considered (divisions); 7th, The exposition of the main part of the thought, depending on comparison, generalization, abstraction, etc., in short of the thought that governs the construction of the treatise;* (* In most mathematical writings, especially the old books, all this unmechanical part of the thought is left for the unaided sagacity of the reader to divine. In an elementary treatise such omission is fatal to the success of many students. In the present volume, this part is consigned to scholia.) 8th, Warnings against probable blunders, and notices of special stumbling-blocks (notabenes); 9th, Detailed directions for applying the knowledge gained, with recommendations of suitable methods and arrangements (A) of drawings, (B) of computations; 10th, Illustrative examples showing the meanings of statements; 11th, Immediate applications of the propositions established (corollaries); 12th, Exercises for practice; since practice, practice, practice is the one way to learn mathematics; 13th, Historical memoranda, to which the learner's sense of decency will surely bind him to attend, in gratitude to the great men the devotion of whose lives to the pursuit of truth, though it now contributes to our enjoyment of life, in most cases never received any other external reward.* (* Anecdote 1. The famous Euclid, head of the mathematical department of the University of Alexandria, was one day asked by a tyro, "What do I get by learning this?" The professor called the slave who made his disbursements and said, "Pay this gentleman three-pence, since he must gain something." It is the learned anecdotist John Stobceus who tells the story.) No other motive for remembering our benefactors of former generations need be set before American youth. But even a man devoid of the sentiment of gratitude and degraded below the lowest people living upon earth, were he one of that dreadful race which the terrible satire of Dean Swift represented as lower than horses, if such a being could be conceived to have any interest in learning anything, an acquaintance with the proceedings which have given rise to peculiar modes of thinking, is the best means [of] apprehending the significance of the thought, and affords a quite unsentimental reason for attending to [the] history of the terms and proposi-

30

ELEMENTS OF MATHEMATICS (165)

tions of mathematics. 14th, Comparisons with other knowledge in respect to (A) the substance of the hypotheses or results, (B) the method of reasoning; 15th, Glosses on the pronunciation, history, derivation, and associations of the words; 16th, Notes, anecdotes, and pleasantries to relieve the fatigue of the brain. Art. 10. Definition 7. A demonstration is a perfect proof of some truth. Gloss 8. The word demonstration, an old word in English, was formed directly from the Latin demonstratio, used by Cornificius in the same sense, and formed regularly from demonstrare, to demonstrate, compounded of de, thoroughly, and monstrare, to show, for mon-es-tar-are, from monëre, to cause to think, from the root MAN, whence English mind, man, mathematics, etc. The Latin monstrare, compared with other words meaning to show, such as exhibere, designare, estendere, indicare, significare, is found to retain the idea of causing to reflect, and not merely of causing to look. Demonstratio is used to translate the Greek άπόδειξις, the usual word for a perfect proof, which is regularly formed from άποδεικνύναι, to demonstrate, from άπό, thoroughly, and δεικνύναι, to show, prove, from the root DIK, to show, from DA, to know, whence English teach, indicate, index, didactic, etc.4 Art. 11. Definition 8. A theorem is a demonstrable statement amounting to the denial of the possibility of some general description of [a] thing. Illustrations. 1. That the remainder (if there be any) after dividing by 4 the product of a number by itself, is [0 or] 1, can be demonstrated. It is, therefore, a theorem, since it is equivalent to saying that a remainder of 2 or 3, after such a process, is impossible. 2. If space has the properties it is generally supposed to have, and if there be 4 points all equidistant from one another, then, if a fifth point is at equal distances from the other four, the latter distance is to the former as [λ/3 : 2\/2].B This can be demonstrated. It is therefore a theorem because it amounts to denying the possibility of a fifth point 4

Elsewhere Peirce states more simply : "Only I had better explain that a mathematica 1 demonstration consists in contriving a spatial diagram of the state of things in the premiss — this diagram being a usual (a tactual, in the broad sense) icon or image of the form of relationship signified in the collective premiss (I mean in the two or more premisses considered as one whole) and then in observing that the diagram is a diagram of the relation signified by the conclusion of the demonstration." 5 Peirce gives V3 : V2.

INTRODUCTION

31

equidistant from the others and either nearer or further than [ y ^ / f ] their distance from one another. Gloss 9. The word theorem is taken, perhaps through the Latin, from the Greek θεώρημα, with the same meaning, originally a spectacle, from θεωρεΐν, to behold, from θεωρός for θαρρός, a spectator. There would seem to be a root 0AF. Art. 12. Definition 9. A problem is a question how (i.e. under what circumstance) a thing of a given description, often with relations to other given things, is possible. Gloss 10. The word probleme (used by Chaucer) comes probably through the French problème, and certainly through the Latin problema, from the Greek πρόβλημα, used in a geometrical application by Plato, originally something thrown forward, regularly formed from προβάλλειν, to throw forward, to put forward, from πρό, forward, and βάλλειν, to throw, from the root GAR, whence English carbine, hyperbola, gland, parley, quell, volatile, etc. Definition 10. A problematic proposition, is a demonstrable statement that, accepting a certain hypothesis, something is possible. It is loosely called a problem. Illustration 3. That there is within a circular disk a point equidistant from every point of the border is a problematic proposition. To find that point, that is, the question how that finding is possible, is a problem. Art. 13. Definition 11. A lemma is a demonstrable statement not directly relating to the subject treated but introduced for the sake of facilitating the demonstration of a theorem or problem which it precedes. Gloss 11. The word lemma is borrowed, probably through the Latin, from the Greek λήμμα, where it sometimes has the same sense, originally a thing taken, and formed from a stem of the verb λαμβάνειν, to take, from the root RABH, take, from ARBH, when[ce] English dilemma, syllable, labor, robust, elf, etc. With Aristotle a lemma is an assumption or premise, with the Stoics a particular kind of premise, with later writers a proposition borrowed from another science; and it has had many other meanings. Art. 14. Definition 12. A corollary is a statement which an immediate application of another, just proved, shows to be true. Gloss 12. The word corollary can be accented either on the first or second syllable. (It is used by Chaucer and was derived from the Latin corollarium, which in medieval and modern Latin bears the same meaning,

32

ELEMENTS OF MATHEMATICS ( 1 6 5 )

but in the classical language it means money given to buy a garland, a gratuity. It is formed by adding the suffix -arium, usually signifying a place, to corolla, a garland, diminutive of corona, a crown or garland, from the root KAR, whence English circle, ring, cylinder, cycle, collar, etc. It first occurs in the mathematical sense in Boethius, about A.D. 500. It translates Greek πόρισμα, used in this sense among others in mathematics. Πόρισμα according to its derivation should mean something extra supplied. It is formed by attaching the suffix -ma, signifying a result, to the stem of πορίζειν, to provide, from πόρος, a resource, originally a ford, from the root PAR, to cross over, to travel, whence English, fare, far, from, fore, forth, for, fear, pirate, experience, etc.) Art. 15. Definition 13. A proposition is an assertion; especially applied in mathematics to definitions, divisions, postulates, axioms, theorems, problems, lemmas, and corollaries; and loosely applied to the whole matter under the head of the proposition, including the demonstration. Gloss 13. The word proposition, which has been in English since Wiclif's time, was taken direct from the Latin propositio, used in this sense by Cornificius, formed from propônëre, to propound, compounded of pro-, fore, and pônëre, to place, contracted from posïnëre, from old port, at, and sïnëre, to let, lay. The word propositio would seem to be meant to imitate the Greek πρόθεσις, which was used in rhetoric, and in grammar, too, after that science was invented, in the same sense. But the Greek word used in Logic and Mathematics was not πρόθεσις but πρότασις. Aristotle, for instance, uses the former in his Rhetoric, the latter in the Organon. But with him the former had the present mathematical meaning, while the latter meant a premise. The word πρότασις is formed from προτείνειν, to stretch forth, from πρό, forward, and τείνειν, to stretch, from the root TAN, stretch, whence English tense, tend, tent, tendril, tendon, tone, thin, dance, etc. Art. 16. Scholium 4. There has been, since the days of Euclid, a regular form, universally approved, for the demonstration of a theorem or lemma. The proposition is stated, first. Next, is described, with letters, an icon of the conditions of the proposition, which is called the beginning of the construction. Thirdly, the proposition is restated with reference to that construction. The construction is then completed by any additions or changes which may be necessary. Fifthly, the consequences of certain experiments upon the construction are traced out, and the proposition is thus shown to be true of it. Sixthly, this is stated to be the case; and this statement is called the proof. To that, are appended the letters

INTRODUCTION

33

Q.E.D. for quod erat demonstrandum, "which had to be shown."* (* Translating Euclid's δπερ δδεν δεϊξαι.) Gloss 14. The word construction, used in English since the first English Euclid, in 1570, is taken from the Latin constructio, little used in the geometrical sense, because the Romans were not geometers. It is regularly formed from construiré, to build, from con-, together, and struëre, to pile, from the root STAR, spread, whence English, strew, straw, star, sheet, instrument, latitude, etc. In mathematics it translates the Greek κατασκευή, used in this sense, originally, furnishing, from κατά, from top to bottom, and σκευή, equipment, from the root SKU, cover, whence English, scum, sky, house, hide, scuttle, cuticle, scutchon, etc. [scutcheon]. Gloss 15. The word conclusion, used in Middle English, is taken directly from the Latin conclusio, used in the same sense by Cornificius, formed regularly from conclùdere, to conclude, compounded of com-, completely, and -clûdëre, to close, to end, from the root SKLU, shut, whence English close, include, etc. Conclusio is used to translate Greek συμπέρασμα, used occasionally by Aristotle, and usually by his followers, in the same sense. It is formed from συμπεραίνεσθαι, to conclude, from συν, aiding, and περαίνειν, to finish, from πέρας, end, or from περάν, to pass through, from the root PAR, found in πόρισμα Gloss 12. Art. 17. Definition 14. A scholium is a remark in aid of the understanding of the course of thought, but not containing demonstrative reasoning. Gloss 16. The word scholium (plural scholia) is used in all the English Euclids. The scholia are all written by commentators. Indeed, the word was originally understood to imply such authorship. But Legendre in his Éléments de géométrie [1794] introduced "Scolies" written by himself, and succeeding authors have imitated the practice. The word is taken through modern Latin scholium (Cicero writes it in Greek letters), from the Greek σχόλιον, an interpretation by a commentator, from σχολή, a lecture, originally, spare time, from the root SAGH, hold in, refrain, whence English school, scheme, sail, hectic, epoch. Boys and girls will think it strange that school should come from the root in this sense, and will ask whether it means nothing else. It does mean, also, endure', but our word school comes from the Anglo-Saxon scólu, which was borrowed from the latin schola, meaning a school-room or lecture-room. The Greek σχολή from which it was borrowed had the same meaning; but at first it was a lecture, and before regular lectures were delivered, it was a discussion, to which the name "spare time" was given, because that was the favorite way among the Athenians of spending their spare time.

2 SEQUENCES

Art. 18. Definition 15. A correspondence is a connection established in the mind between two collections such that, considering either, every object of it is connected with the same number of objects of the other.

N.B. (2) It is not said that any object of either is connected with the same number of the other collection, since the two numbers may be different. Illustration 4. In figure 75 there is a collection of rings and a collection of dots. The lines suggest mental connections. Each dot is connected with three rings; each ring with two dots. Such a connection is called a two-to-three correspondence between the dots and rings, or a threeto-two correspondence between the rings and dots. Scholium 5. Seeking to enlarge the idea, we remark that the two collections need not be different. Thus, in figure 76, the lines suggest a threeto-three correspondence among the dots. [...]· • The lower half of this page has been cut from Peirce's manuscript, probably in a revision by himself. Consequently the beginning of Art. 19 has disappeared carrying with it axioms 3 and 4 in that article as well as Cor. 1 at the end of Art. 18 as indicated in Peirce's Table of Contents. In a listing of corollaries Peirce indicates that in this manuscript on page 40, corollary 1 appears in terms of a "5-1 1-7 correspondence."

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[Art. 19.

Axiom 3. The relation expressed by "after" is transitive. Axiom 4. The relation expressed by "after" is identical.] In that case, they would not appear as axioms. But treated as they are here, as familiar truths, they are axioms. No demonstration could make us more sure of their verity; but it would afford an insight into the manner in which different relations are connected with the relation of coming after. A full explanation of the matter would be too long. But something may be said. Figure 77 symbolizes a relationship between four objects, Α, Ε, I, 0. The wedges pointing from A to I and from E to O mean A

E

I

0 Fig. 77

that A is supposed to have a certain kind of a relation to I, like the relation of E to 0. The arrows pointing from A to O, and from E to I signify that A has not that relation to O that is signified by the wedge, nor E that relation to /. Now suppose the relation signified by the wedge to be such that the fourfold relationship signified by the diagram is necessarily false, and that to make it true it is necessary to suppose that A that has the wedge-relation to I is a different thing from that A that has not the wedge-relation to O. For example, let the wedge mean strong enough to lift. Then, the diagram represents that A is strong enough to lift I but is not strong enough to lift 0, while E is strong enough to Another sheet summarizes the axioms in the first eleven chapters and on page 40 axioms 3 and 4 should appear. Axiom ΙΠ: The relation expressed by "after "is transitive. Axiom IV: The relation expressed by "after" is identical. A Scholium 6 was also lost at this point. The part of Art. 19 still extant and herein given may well be part of that Scholium. Similarly, Peirce either mislabeled or lost in revision Corollary 10 in Art. 34. Peirce drew up elaborate summaries of the contents of MS. 165 (see the Appendix). Still extant are lists of problems, theorems, definitions, postulates, scholia, figures, exercises, glosses, illustrations, anecdotes, divisions, and notations, all of which have helped in the editing. Scholium 14 seems to have been lost to Art. 43 in Peirce's revision of page 98 of his manuscript. It concerned "the importance of algorithm for G.C.D." Problem 2 was listed on the outline as belonging to Art. 43. Yet Peirce numbered it "3" in the manuscript. The editor has attempted to adhere to Peirce's summaries where possible. As a result very few number changes have been necessary.

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lift O but is not strong enough to lift I. This is manifestly absurd. Having obtained a relation of this description, namely one which thus renders the diagram absurd (the absurdity ceasing as soon as we suppose two /4s have been confounded), the compound relation represented by a wedge and an arrow, such as the relation of A to E through I, like "able to lift something that cannot be lifted by," is a relation altogether analogous to the relation of coming after. Thus, corresponding to Axiom 3, we have the proposition that, to say that any man, A, is able to lift something that cannot be lifted by another man, E, is precisely equivalent to saying A is able to lift something that cannot be lifted by somebody who is able to lift anything that can be lifted by E; for E himself may be that somebody. Further, corresponding to Axiom 4 we have the proposition that if any man, A, can lift something that cannot be lifted by somebody that is able to lift anything that can be lifted by E, then A can lift something that cannot be lifted by anybody except a man able to lift something that cannot be lifted by E. Art. 20. Definition 16. The objects of a collection are said to be arranged in a sequence when of every pair of them one object comes after the other. Theorem 1. In any sequence there can be but one object that comes after no other. Demonstration. For let A and Β be any two objects of the same sequence. Then I say that one of the two objects, A and B, comes after some other. For, by Definition 16, of the pair of objects, A and B, one of them comes after the other. Q.E.D. Theorem 2. In any sequence there can be but one object that no other comes after. Exercise 1. The pupil will please demonstrate Theorem 2, almost exactly as Theorem 1 is demonstrated. He must first sit down and patiently study the Demonstration of Theorem 1, in the light of Scholium 4. He must really imagine that which the demonstration requires him to imagine. When he catches the idea, he has to perform a similar though slightly different process with Theorem 2. Definition 17. The last object of a sequence is one that none comes after; the first is one that comes after none. Scholium 6. In mathematics, we must be careful not to allow our thought to be bound down to particular cases. When a sequence has

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a first or last, we must never forget that it might be continued further, unless it happens that the first and last are the same. In other cases, we must think of a possible continuation of the [sequence] beyond. Definition 18. A limit (throughout mathematics, note this well, for future application) is a boundary between two extensions or multitudes. Division 2. Sequences can be distinguished into such as have both first and last objects, and those which either have no first or no last. (Throughout this book the expression "either one or the other" is to be understood as meaning what some writers express by "either one, or the other, or both.") Definition 19. A sequence which has both a first and a last is called a bounded, or doubly limited sequence. A sequence which has no first or no last is called a boundless [sequence]. Division 3. A boundless sequence may either be half-bounded (limited one way) or unlimited both ways. N.B. (3) It is necessary to dismiss the notion that that which is unlimited is necessarily larger than that which is limited. Memorandum 5. It was Georg Friedrich Bernhard Riemann (b. 1826d. 1866), one of the greatest of all mathematicians, who first made this clear in 1854.7 Illustration 5. The different instants of a second or a year form a bounded sequence; because such a lapse of time has a first and a last instant. But the instants of a second that are between the first and last instants, excluding those, form a sequence unlimited both ways. Simply taking away the limits renders the sequence unlimited. [Illustration] 6. The alphabet is arranged in a bounded sequence, the ordinal numbers in a sequence limited one way only. The proper vulgar fractions, if arranged in the order of their magnitudes, form a sequence unlimited both ways; for there is no greatest nor least irreducible proper vulgar fraction. [.Illustration] 7. If the irreducible proper vulgar fractions are arranged so that those of less denominators come before those of greater denominators and those of the same denominator, but less numerators, before those with greater numerators, they form the following sequence: 7

A space in the MS seems to indicate that Peirce intended to add to the historical material. In several places in this section Peirce uses "series" interchangeably with "sequence," contrary to modern usage. The editor has changed "series" to [sequence] where necessary.

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ELEMENTS OF MATHEMATICS (165)

Τ» 3» Τ» 4·» 4» 4» 5» 5» S' Τ» 6ι 6» 7» 7» 7» 7) 7» 7» 8» 8» 8» T> 9> 9> ?» 9» 9» 8. _1 3 7 9 ptr 9> 10» 10» ΤΪΓ» 10» CIA'·

This sequence has a first but no last, and is, therefore, limited one way. This shows how the same objects, provided they are infinitely numerous, can be arranged either in a limited or unlimited sequence. [Illustration] 8. The instants of ideal time in the future or the past form a sequence limited one way (if the present be included). The instants of all ideal time form a sequence unlimited both ways. But whether this be the case with real time is a matter of opinion. [Illustration] 9. If the whole numbers are arranged so that all the odd numbers precede all the even numbers, less[er] odd numbers coming before the greater, while less[er] even numbers [come] after the greater, they form a sequence (since of any two one comes before the other). It is a bounded sequence; for 1 will come first and 2 last. Art. 21.® Definition 20. Of objects in a sequence, that one is said to come next after another, which comes after, but not after any third that comes after, that other. More briefly, next after means after, but not after something itself after. Illustration 10. Thus, in the alphabet, we say that Ν does not come next after L; for, though it comes after L, it also comes after M, which likewise comes after L. Ν does come next after M, and M next after L. Definition 21. A sequence is said to advance by steps, if every object in it, except the last, has another next after it. A sequence recedes by steps, if every object in it, except the first, is next after some other. A sparse sequence is one which both advances and recedes by steps. A sequence is proclive at an object in it not the last, if there is no object next after that object. 8

In MS. 166, an earlier draft of Arts. 21 and 22 is to be found. It runs continuously from pp. 44-50. P. 44 of 166 is being used in place of p. 44 as given in 165. For the two pages 44 must have been accidentally interchanged at some earlier time. According to Pence's Table of Contents of this volume there should be an "Anal 1" in this section. There is an "Anal 1" on p. 47 of MS. 166 and it is being interpolated into the text as given here. Def. 22 on p. 47 of MS. 166 starts with a simple sparse sequence. "A simple sparse sequence is such a sparse sequence that if one object differs in any respect from another that comes after it, then either the former object or some object after it which the latter object comes after differs in this respect from the object that comes next after it. A sparse sequence of which this is not true, may be called a complex sparse sequence."

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A sequence is reclive at an object in it not the first, if that object does not come next after any other. A dense sequence is a sequence both proclive and reclive. Illustration 11. The alphabet is arranged in a sparse sequence. [Illustration] 12. The different instants of a minute form a sequence dense throughout. For no instant comes next after any other, but between any two there are others later than the one but earlier than the other. [.Illustration] 13. If a perfectly pointed pencil were to draw a line, the motion of the pencil would arrange the points of the line in a sequence dense throughout; for no point would come next after any other. [Illustration] 14. If we consider the sequence formed by, 1st, all the instants that begin the first to the sixtieth seconds of a minute, followed, 2nd, by all the instants without exception of the next succeeding instant, then the whole sequence is sparse throughout the first minute and dense throughout the second minute but at the instant between the two minutes it is proclive while receding by steps [Fig. 78 a]. Fig. 78a

[Illustration] 15. If we consider a sequence of which the first object is the first instant of a minute, and the last the last instant of that minute, and such that there is among the remaining objects an object next after each one of them, as in the figure (where the instants very near the last cannot be seen), that sequence is everywhere sparse except at the last object, where it is reclive [Fig. 78 b, the sequence of the Achilles]. For let A be any instant whatever; then I say that the last instant does not • · · · · · · Fig. 78b

come next after A, since by hypothesis one of the other instants comes next after it. Anecdote 2. A [sequence] exactly like this occurs in the fallacy named the Achilles. Achilles and the Tortoise run a race. A certain start is accorded to the tortoise. While Achilles runs to where the tortoise started, the tortoise runs over a certain stretch. While Achilles runs over this stretch, the tortoise advances by another; and so on, without end. This is plain; the sequence is unbounded. From this, it is ridiculously concluded that Achilles will never overtake the tortoise. This argument was seriously propounded in the fifth century before Christ, in the town of Velia, on the Campanian coast, near the present Vallo, by the philosopher Zeno;

40

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and it has made great noise in the world. Zeno thought it proved motion was an illusion. Adding to the unbounded sequence the instant when Achilles overtakes the tortoise, it becomes the bounded sequence of Fig. 78 b. N.B. (4) Illustration 9 shows a sparse bounded sequence which by transposing two parts of it becomes an unbounded sparse sequence. The same objects (being the whole numbers) can be arranged in a half-bounded sparse sequence, or by reversing in imagination the order of theArabic figures with which they are written and arranging these figures as if the tens were tenths, the hundreds were hundredths, etc. (the values not changing but only the arrangement) can be brought into a half-bounded dense sequence. By taking a number from the middle and putting it at the end, we get a bounded dense sequence; or, by removing 1 from the beginning and placing it next after all the numbers written with Is only, a doubly unbounded sequence is obtained. But we shall find that there are kinds of sequences which cannot be in one-to-one correspondence to one another. Art. 22. (Analysis 1. If we compare the natural arrangement of whole numbers with that of Illustration 9, we remark this difference between them, that if in the former cases one takes first, 1, then the number next after it, 2, then the number next after 2, and always the number next after the last taken, one will inevitably take every number. But doing the same in the other case, although one is certain to take all the odd numbers, one is not obliged by the condition stated to take any of the even ones, at all. The process of taking successive next ones does not necessarily take one through the whole sequence. Considering the numbers in their natural order, if 1 and the small numbers are distinguished in any way from the large ones there must be a number which differs in this respect from this one next after it. With the other arrangement this is not so. Thus, the first numbers are all odd and the even numbers follow after; yet in that arrangement if any number is odd, the number next after it is also odd. We may take this as the distinguishing characteristic of the two sorts of sparse sequences.) Definition 22. A complex sparse sequence is a sparse sequence composed of two partial sequences, every object of the second coming after every object of the first, while the first partial sequence has no last object nor the second partial sequence any first object. A simple sparse sequence is a sparse sequence of which any two parts,

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one coming entirely after the other, have, the first part a last object, and the last part a first object. Theorem 3. Every sparse sequence is either simple or complex. Demonstration. For let a sparse sequence consist of two parts such that every object of the second comes after every object of the first. Then, I say that if any object, M, is the last of the first part, there is an object Ν that is first of the last part; and if any object Ν is first of the last part, there is an object M that is last of the first part. First, suppose M is the last of the first part. Then, by Definition 21, there is an object next after M. Call this N. Then, Ν does not belong to the first part, since it comes after M, which is the last of the first part. N, therefore, belongs to the second part. Every object of the second part comes after M, because M is an object of the first part. N, therefore, comes after no object of the second part, because, if it did, it would come after an object that came after M, and so would not be next after M. N, therefore, is the first object of the second part. Second, in the same manner it could evidently be shown that if Ν is the object of the last part and M is the object which it is next after, M must be the last object of the first part. Hence, if the first part has a last object, the last part has a first object, and conversely. Q.E.D. Exercise 2. Develop the second part of the above demonstration. Definition 23. A character, quality, mark or predicate which cannot, in a given sequence, belong to any object having another next after it, without also belonging to that other, may be called a character which advances in that sequence. A character which cannot, in a given sequence, belong to any object coming next after another, without also belonging to that other, may be called a character which recedes in that sequence. Problem 1. In any complex sparse sequence, required to find two objects, of which that which comes before has a character which advances in the sequence, yet which the second object does not possess, while the other object has a character which recedes in the sequence, yet which [the] first does not possess. Solution. Take any object of the first and endless part of the sequence for the object that comes before, and any object of the second and beginningless part for the object that comes after. For the character which advances in the sequence, take that of belonging to the first part', for the character which recedes in the sequence, take that of belonging

42

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to the second part. Demonstration. First, the character of belonging to the first part of the sequence does advance in the sequence. For, because the sequence is, by hypothesis, sparse, every object of the first part has an object of the sequence next after it, by Definition 21. And because the first part has by hypothesis no last object, the object next after any object of that part belongs to that part. Consequently, by Definition 23, the character of belonging to the first part advances in the sequence. Second, it is evident that, for similar reasons, the character of belonging to the second part recedes in the sequence. Third, the object that comes before possesses, by hypothesis, the former character and not the latter; while the object that comes after possesses the latter character and wants the former. Thus, the choice of objects and characters answers the requirements of the problem. Exercise 3. Develop the second part of the above demonstration. Theorem 4. Conversely, in every simple sparse sequence, every character that advances in the sequence belongs to every object that comes after an object that possesses it, and every character that recedes in the sequence belongs to every object that an object that possesses it comes after. Demonstration. For imagine a simple sparse sequence; and take any two objects in it, L and R, of which (since, by Definition 16, one comes after the other) let R denote the one which comes after. Let h be any character which advances in the sequence, and which is possessed by L; and let k denote any character which recedes in the sequence and is possessed by R. Then I say that R possesses the character h, and that L possesses the character k. First, let the sequence be separated into two parts of which one shall comprise all objects that come after all those that possess the character h and come before R, while the other part shall comprise all those that do not come after all of those that at once possess the character h and come before R. Then, by Axiom 3, every object, A, of the former part comes after every object, B, of the latter, since A comes after all of certain objects, some of which, at least, Β does not come after. The two parts comprise together the whole sequence, since the latter is defined by every object of it as fulfilling a certain [condition], while the earlier is defined by no object of it fulfilling that condition. Hence, since by hypothesis the entire sequence is a simple sparse sequence, it follows, by Definition 22, that the earlier part has a last object. Let us name this last object

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of the earlier part Q. Q, then, is the last of the objects that do not come after all that at once possess h and come before R. But there is a last one of the objects that at once possess h and come before R. For separating the whole sequence into two parts, one comprising those that at once possess h and come before R together with those that are earlier than some of these and the other part comprising the rest, the latter part is entirely after the former; so that by Definition 22, there must be a last one of those that at once possess h and come before R. This is the last of those that do not come after all that possess h and are earlier than R; that is, it is Q. But, by hypothesis, A advances in the sequence. Hence by Definition 23, the object next after Q in the sparse sequence possesses h. Yet it does not both possess h and come before R, since it comes after Q which is the last of which both those things are true. Hence, it does not come before R. But Q comes before R. Consequently, the object next after Q cannot come after R; for if it did, since R comes after β, this object would not be next after Q. Thus, this object next after Q and possessing h is neither before R nor after R. But, by Definition 16, if this object and R were a pair of objects of the sequence (and R, by hypothesis is an object of the sequence), one of the two must come after the other. Hence, they are not a pair but are identical. Thus, R possesses the character h. Secondly, it is evident that by similar reasoning L possesses the character A. Thus, R possesses h and L, k. Q. E. D. Exercise 4. Develop the second part of the above demonstration. Illustration 16. In counting we use the numbers one next after another. Hence, the numbers used in counting, or the cardinal numbers, form a sparse sequence. But no count could ever be accomplished which required that in the first part of it a boundless sequence should be enumerated. Hence the system of numbers used in counting is, by Definition 22, not a complex sparse sequence ; and by Theorem [3], it is a simple sparse sequence. Hence, by Theorem 4, it is legitimate (if the premises hold good) to reason as follows. The first number, 1, has a given property; but this property is such that if any number possesses it, so does likewise the next greater number; hence, all numbers possess this property. This way of reasoning invented by the great Pierre Fermât (about 1640) is called Fermatian inference. The greatest of ancient mathematicians, Archimedes (arkimee-deez), who was killed at an advanced age, 212 B.c., thought it necessary to write a tract to prove that the number of sands on the sea-shore could be arithmetically expressed. Had he been ac-

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quainted with the Fermatian inference he need only have reasoned as follows: Any number greater by 1 than an expressible number is expressible, and 1 is expressible; hence, all numbers are expressible. This method is indispensable in reasoning about whole numbers. Anecdote 3. There was an ancient argument called the Heap, which ran as follows: If a collection is not numerous, the addition of one more will not make it numerous. Now, a collection of two things is not numerous; therefore, no finite collection is numerous. Modern pettifoggers are said to perplex witnesses by such reasoning. The reasoning is sound; but the premise that the addition of a unit to a collection we should not call numerous will make a collection we should call numerous, instead of being true, is refuted by its leading to the absurd conclusion that no finite collection is called numerous. Only, the collection which in one mood we might be disposed to call numerous, a moment later, without our being aware of the slight change of our mood, might be regarded as not deserving that epithet. In the very process of questioning our inclination changes. Art. 23. Theorem 5. No collection arranged in a sequence without a last object can be converted into a collection arranged in a sequence with a last object by merely removing one object from it without further rearrangement. Demonstration. For let R be an object of an endless sequence; and let R be removed from the sequence without deranging the other objects. Then I say the sequence is still endless. For R is not the last object, since, by hypothesis, the sequence has no last object. Consequently, by Definition 17, there are other objects after R. Of those objects all that are after others remain, by hypothesis, after them. Hence, by Definition 16, those objects form a partial sequence. That partial sequence has, before the removal, no last; since if it had, that last would be the last of the whole sequence. But, by hypothesis, the removal does not make an object come after another that it did not before come after. Therefore, after the removal, that partial sequence has no last. Therefore, after the removal, the whole sequence has no last; for if it had, that last would belong to that partial sequence, since it must come after others, which came after R before the removal. Q.E.D. Corollary 2. No collection arranged in a sequence without a first object can be converted into a collection arranged in a sequence with a first object by merely removing one object without further rearrangement. . Exercise 5. Demonstrate Corollary 2.

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Corollary 3. Hence the removal of no whole number of objects can make an unbounded sequence bounded. Exercise 6. Demonstrate Corollary 3. Lemma 1. Let a sequence, t, be composed of sequences. Namely, let its objects be all the earlier of the pairs of partial sequences resulting from cutting into two a simple sparse sequence, s. That is, it is only such parts of s as contain the first object of that sequence, s, which form objects of the sequence, t. And these sequences are arranged in t in the same order as their several last objects are arranged in s. Then, I say, the sequence, t, is a simple sparse sequence. [Fig. 79]. s t

A A A A A A

Β C D E

etc.

Β B C Β C D Β C D E

etc.

Fig. 79

Demonstration. What we mean by saying that the sequences which are partial sequences of the sequence, t, are arranged as the objects of the sequence, s, is, that the objects of t and the objects of s are in such a one-to-one correspondence, that if one object of s has another after it, then the object of t which corresponds to the former has the one which corresponds to the latter after it. Consequently, in whatever way t is separated into two parts, the second coming entirely after the first, if s is separated into two corresponding parts, the second part of s will come entirely after the first; and since, by the hypothesis and Definition 23, the first part of s has a last object, the first part of t has also a last object; and thus, by the same definition, t is a simple sparse sequence. Theorem 6. The objects of a bounded simple sparse sequence cannot be in one-to-one correspondence with those of any unbounded sequence. Demonstration. Let s be a bounded, simple, sparse sequence; let u be an endless sequence; and let ν be a beginningless sequence. Then I say no one-to-one correspondence can be established between the objects of s and those of u or of v. I shall prove that this correspondence cannot exist, by assuming that it exists, and proceeding to show that from that hypothesis an absurdity results. Suppose, then, the correspondence to exist. Let t be the sequence in one-to-one correspondence with s, in the

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manner described in Lemma 1. Then s, which is the last object of t, is in one-to-one correspondence with u. Now let the last object of s be removed, together with the object of u to which it corresponds, leaving, in place of w, a sequence, u', which, by Theorem 5, is, like u, endless. In place of s, we now have that object of t which the last object of t comes next after. We thus see that if any object of t is in one-to-one correspondence with an endless sequence, so likewise is the object of t next before it. That is, by Definition 22, the character of being in one-to-one correspondence with an endless sequence recedes in the sequence, t. It follows, then, by Lemma 1 together with Theorem 4, that if any object of t is in one-to-one correspondence with an endless sequence, so likewise is the first object of t. But the first object of t is a single object; and such an object can, by Definition 15, be in one-to-one correspondence with nothing but a single object. Now by Axiom 3, nothing comes after itself; so that a single object, which contains nothing but itself, contains nothing which comes after itself. It is therefore impossible that a single object should be in one-to-one correspondence with an endless sequence. Therefore, the last object of t, which is s, cannot be in such correspondence. It is evident that similar reasoning shows that s cannot be in one-toone correspondence with v. Hence, s can be in one-to-one correspondence neither with u nor with v. Q.E.D. Exercise 7. Develop the second part of the above proof. Corollary 4. A collection of objects cannot at one time be arranged in a bounded, simple, sparse sequence and at another time in a boundless sequence. For the identity of the objects would create a one-to-one correspondence between the two sequences. Corollary 5. If when we count a collection, taking the objects in one order, we exhaust it before we exhaust the cardinal numbers, so we shall if we count the same objects in any other order. Art. 24. Definition 24. Two collections are said to be equal to one another, if they can be put into one-to-one correspondence. A collection is said to [be] inclusible in another, if it can be put into one-to-one correspondence with a part of that other; and the latter is said to be inclusible of the former. A collection is said to be less, or fewer, than another if it is inclusible in, but not equal to, that other; and the latter is said to be greater, or more, than the former.

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Corollary 6. Every collection is equal to itself. Corollary 7. By Corollary 1, if one collection is equal to another, and that other is equal to a third, then the first is equal to the third. Axiom 5. A part of a part is a part of the whole. Corollary 8. By Axiom 5, if a collection is inclusible in another inclusible in or equal to a third, or if the first is equal to the second and the second inclusible in the third, then the first is inclusible in the third. Definition 25. An enumerable or finite collection is a collection which can be arranged in a bounded, simple, sparse sequence. An inenumerable, or infinite collection is one which cannot be so arranged. Theorem 7. An enumerable part of a collection is less than the whole. Demonstration. It is evident that, by reasoning similar to that of Lemma 1, it follows that if an enumerable collection could be put into one-toone correspondence with a collection of which it formed only a part, then the same thing would be true of a collection formed by dropping an object from the first collection; and that consequently a single object could be put into a one-to-one correspondence with a collection of which it was a part; and further, that this last is impossible. Hence, no enumerable part is equal to its whole. But it is evidently inclusible in its whole, and hence, by definition, an enumerable part is less than the whole collection. Exercise 8. Develop the above reasoning. Definition 26. A àenumerable, or indefinitely great, collection is an inenumerable collection which can be arranged in a simple, sparse sequence. Theorem 8. All (¡enumerable collections are equal. Demonstration. First, every such collection can be arranged in a simple, sparse sequence having a first object but no last. For, by Definition 26, it can be arranged in a simple, sparse sequence; and by Definitions 26 and 25, this sequence cannot have both a first and a last object. If it has a last but not a first, the reversal of the arrangement will give it a first but not a last. If the simple, sparse sequence, say w, in which the collection is first arranged, has neither beginning nor end, the new sequence may be formed by taking any object as first and next the object next after it in w, and then alternately taking the object which, in sequence w, comes next before the last but one taken and the object which, in sequence w, comes next after the last but one taken.

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For example, to so arrange the years before and after Christ begin say with A.D. 4. Then the new sequence will be A.D. 4, A.D. 5, A.D. 3, A.D. 6, A.D. 2, A.D. 7, A.D. 1, A.D. 8, B.C. 1, etc. Second, to bring any two endless, simple, sparse sequences into oneto-one correspondence, we make the first of one correspond to the first of the other, and the object next after any object R, correspond to the object next after the object to which R corresponds. In this way every one of either collection corresponds to some one of the other, and to no more. Hence, by Definition 24, the two collections are equal. Q. E. D. Exercise 9. Prove the above [sequence] of years continued will embrace any year. Exercise 10. Prove second part in full. Corollary 9. It follows that a part is not less than its whole, if both part and whole are denumerable collections. Illustration 17. Every whole number has its double, so that the doubles of whole numbers form a collection equal to the collection of whole numbers. That is, the even numbers, which are only one half the entire collection of whole numbers, are as many as all the whole numbers. [Illustration] 18. If an insurance company pays its customers on an average more than all their premiums with interest, all that is received (in premiums with interest) forms only a part of what is returned in losses. If therefore the risks are an enumerable collection, the insurance company must lose money. But if new customers come in unceasingly, the amount they pay in may always be enough to pay the losses on old risks, and leave a margin of profit. It was a long time before the companies found out this very practical truth, because they were deceived by the venerable error that a part is always less than the whole. After the great fire in New York in 1835, there was but one company that saw the wisdom of borrowing a great sum to pay losses. It became very rich in consequence of doing so and of being the only one to do so. [.Illustration] 19. More people are born every year than die. Hence, in any enumerable succession of years some men do not die. But considering the inenumerable succession the deaths are precisely as numerous as the births. Scholium 7. The above illustrations and many others illustrate the great practical advantage which may accrue from careful accuracy about matters which at first sight appear to be idle subtleties and paradoxes. Euclid makes it an axiom that the whole is greater than its part and consequently falls into fallacies in the 6th (?), 7th, 16th, 18th, 20th, 24th, 26th (?)

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Propositions of his First Book, most of which are false in some kinds of non-Euclidean geometry. Theorem 9. (Called the Fundamental Theorem of Arithmetic.) Any enumerable collection counts up to the same number in whatever order the single objects are counted. Demonstration. For if not, the smaller succession of numbers (used in one count), which form an enumerable collection, would, through the identity of the objects counted, be brought into one-to-one correspondence with the succession used in the other count, although the former succession is a part of the latter, contrary to Theorem 8. Scholium 8. This [theorem] does not deserve the title it usually bears so much as does Corollary 5; for when that is once established, this easily follows. Still, it is true, that this theorem includes that Corollary as a special case.

ALGEBRA

Art. 25. Definition 27. Algebra is generally understood to mean a branch of mathematics which studies arrays of letters and signs; in opposition to geometry which studies configurations of lines. Common algebra is that branch of mathematics by which we work out rules for performing arithmetical computations, especially when this is done by studying arrays of letters, as it usually is in modern times. But there are geometrical (commonly called graphical) ways of doing this, and some mathematicians call the study algebra, even when conducted in that way. If we call that method graphical algebra, as opposed to literal algebra, there can be no mistake in our meaning, although it is not algebra, in the ordinary sense of the word. Illustration 20. When we say that by algebra we work rules for performing computations, we do not refer to the fundamental operations merely. Thus, by means of arithmetical analysis we can ascertain at what times the two hands of a clock are in conjunction. But algebra will furnish a rule applicable, not merely when one hand moves 12 times as fast as the other, but no matter what the ratio of motions may be, so long as it is known. Gloss 17. The word algebra is pronounced al-jebraa. (So it has been accented for 200 years, at least; and so it is in Spanish, Italian, and

50

ELEMENTS OF MATHEMATICS (165)

German. But in 1623 (according to Minshen) it was accented on the second syllable in Spanish, as it ought to be. It is simply the abridged title of the Arabic treatise from which Western Europe acquired its first knowledge of this art. The full title is "Al-jebr wa'1-muqäbala," or "Redintegration and Opposition," Arabic names of algebraical operations. The author was Muhammed the son of Müsä, surnamed the Chorasmian, or Khiva-ite. He lived in Baghdad, under the great Khalif, Al-mamün, in the early part of the ninth century. When the Arabians began to gain wealth, in the palmy days of Islam, they were quite ignorant of mathematics, and unable even to keep accounts. Accordingly, Al-mamün sought out a man learned in such matters. Chorasmia, south of the Aral Sea, was already overrun by the Turks; but it had formed part of the Bactrian Kingdom, and had preserved the extraordinary knowledge of arithmetic which the Bactrians had possessed from ancient times. The learned Muhammed ben Müsä was called to Baghdad and ordered to write a commercial arithmetic and a work upon algebra; both of which were of the utmost service to the Arabians at that time, and to Europeans too when in the thirteenth century they became known in Europe. It may be added that algebra was formerly called Cossic, in English, or the Rule of Cos; and the first algebra published in England was called "The Whetstone of Wit," because the author supposed that the word cos was the Latin word so spelled, which means a whetstone. But in fact, cos was derived from the Italian, cosa, thing, the thing you want to find, the unknown quantity whose value is sought. It is the Latin caussa, a thing aimed at, a cause.) Gloss 18. The word number is used with propriety in various senses; but it will contribute to clearness to restrict it here, as much as possible, to the following meaning: Definition 28. A number is a sign, belonging to a sequence of such signs, used to show the relative places of different objects in one sequence, and to compare the places of objects in corresponding sequences. Illustration 21. The vocables Eeny meny mony mi Pasca lona bona stry, used by children in counting out, are a kind of numbers, though used for a very restricted purpose. In fact they are derived from the Romany numerals. They well illustrate the fact that numbers have no general significations; but are a mere apparatus for exhibiting relations.

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51

Definition 29. A quantity is whatever number it may be which truly answers a given question or is applied to a known object. The value of a quantity is the particular number which does truly answer the question, etc. Illustration 22. The distinction is that the identity of the quantity follows that of the question, while the identity of the value follows that of the number that answers the question. Thus, a person's age is a quantity. It is the same quantity whatever his age may turn out to be. Whoever asks how old a man is, states what the quantity is and inquires about its value. The quantity remains the same as long as the man lives, although its value changes, or, to use the technical term, varies. If two persons were born at the same time the two quantities, i.e. their ages, have the same value, or, as we say, are equal. But the quantities are not the same. Art. 26. Notation 1. In algebra, instead of writing out the description of a quantity, or setting down its name, every time we wish to mention it in our operations, we use a letter, or combination of letters, to denote it. This has two advantages, 1st, it saves time and trouble, and 2nd, it facilitates the construction of the arrays we have to study and makes the relations of their parts more clear.* (* The Greek alphabet will be found in Art. 123.) Notation 2. The sign of equality, = , means that the quantity written before it equals, or has the same value as, the quantity written after it. The whole expression is called an equation, and the expressions of the two quantities whose equality is stated are called the first and second members of the equation. Illustration 23. Thus, if a means a person's age in years, a = 15, is the statement that that person is 15 years old, to be read "a equals fifteen." Notation 3. The signs > and < are used for greater than, smaller than, a > b stating that a is greater than b. To express, "is as small as (not excluding smaller than)" and "is as large as (not excluding larger than)", various signs are used by different writers. Thus A is as small as (perhaps smaller than) B, can be written A-C.B; A g Β; Α* Β;

B>-A; B^A; Β* A.

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ELEMENTS OF MATHEMATICS (165)

The first sign is to be preferred. A statement made by means of one of the signs > < is called an inequality. A statement made by means of any sign as -< which admits identity as a possible case, and which is such that Axiom 3 holds, is called an inclusion. Memorandum 6. It is curious that though it was Italy which taught Europe arithmetic and algebra and England was not even the second most mathematical country very few of [the] algebraic signs which have outlasted the struggle for existence, are Italian, while a good half are of English origin. The sign = was brought in by Robert Record in the Whetstone of Witte, published 1557. The author could not but be familiar with the sign » in frequent use in MSS. for the Latin esse, be. Knowing of it, it is a psychological impossibility he should not have been influenced by it, whether he was conscious of being so or not. The manner in which Record gradually comes to use the sign is said, by those who have seen the book, to leave no room for doubt that it was invented in writing that work. Record said he selected the sign because nothing can be more equal than two lines so placed. Another sign oo for equality, very inferior because of its unsymmetrical shape, while equality is an equiparant relation, was introduced by Descartes in 1637. It is a modified ê, for Latin xqualis, equal. It was the usual sign with French mathematicians, for nearly a century.® It often had the shape )). The signs > and < were first used by Thomas Har[r]iot, who died in 1621. The other signs are recent; but the use of a stroke, called an obelus, to signify negation is ancient. It is mentioned by Isidorus of Seville who died A.D. 636. Art. 27. Notation 4. The sign of addition is a Greek or St. George's cross, + , read "plus" (Latin for "more" and "more than"). It denotes the sum of the quantities written before and after it. Thus 2 + 3 = 5 is to be read "2 plus 3 equals 5." If no quantity precedes the sign, the meaning is the same as if 0 were written in the vacant place. This sign is also called the positive sign. Notation 5. The sign of subtraction is —, read "minus" (Latin for "less"). It denotes the remainder after subtracting from the quantity written before it, the quantity written after it. Thus 5 — 2 = 3 is read "5 minus 2 equals 3." If no quantity is written before the sign, the vacant •

Peirce has a little note to himself in the MS to "look up this."

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53

place is to be considered filled with 0 (and the meaning of that will be explained below). This sign is called the negative sign. The sign ± is used in order to show that it is indifferent whether the following quantity is added or subtracted. Thus, the equation x 2 + 21 = lOx is satisfied byx = 5±2. Memorandum 7. The signs + and — are said to have been invented by the famous painter Lionardo da Vinci (b. 1452-d. 1519). But his manuscript called "Codice Atlantico," and preserved in Milan, to a given page of which we are referred for evidence of this, will soon be facsimiled, and then we shall see. It is most probable they originated in Italy; though Luca Paciuolo, the intimate friend of Lionardo, did not use them in his algebra, printed in 1487, while a German, Johannes Widmann, speaks of them as well known in 1489. Further than this, we are left to conjecture. One fact which bears upon the question is, that nobody would in talking Latin ever say "tres plus duo" nor even "tres duobus plus," for 3 + 2, but everybody would say simply, "duo ad tres." Nor would anybody say "quinqué duobus minus" or "quinqué minus duo" for 5 — 2, but everybody would say "duo a quinqué." On the other hand, in Italian, piu and meno (plus and minus) were so used, even before the signs had come to give support to such phrases, even in the fourteenth century. Another fact is that several of the old writers, as Recorde in his Ground of Artes (published before 1543), who use + and — nowhere else, make use of them in what they call the rule Elcatayn, that is, the double rule of false position, to mean + too much, and — too little. That is, they write + to mean that the quantity they want to find is less, and minus to mean that the quantity they want to find is more. To show precisely how the signs were used, we may take the following, from the Vulgar Arithmeticke of Hylles (London: 1600): There are 3 merchants, to wit, A. B. C. Who parting an 100 crownes do agree, That Β shall have mo crownes then A by three, And C shall have 4 crownes mo then B. Now how many crownes each Merchant must have, Is all the question and thing that I crave. Guessing that A has 33, it follows that they have together 109, which must be reduced by 9 to make 100. Guessing that A has 25, it follows that they together have 85, which has to be increased by 15. This was expressed by means of Fig. 80. Now Φ was a familiar medieval abbreviation for Latin, infra, less. The correlative word, supra, more, was written ~SP\ but the letters may have been omitted, leaving only the dash, just

54

ELEMENTS OF MATHEMATICS (165)

as the sign for a root V was formed from "radix," root, by omitting the letter. The signs might have originated in this way; but it is a mere possibility. The double sign ± was used by Newton. Definition 30. The positive and negative signs are called opposite signs. To reverse a sign is to change it from + to — or from — to + · Art. 28. Notation 6. In algebra, in order to show that a compounded expression is to be treated as one quantity, it is enclosed in parentheses ( ), in brackets [ ], or in braces { }. For the same purpose a horizontal bar, called a vinculum, may be drawn over the whole expression. Scholium 9. This notation is to algebra what punctuation is to ordinary writing. It is certainly the most important part of algebraical notation; for, not only is ordinary language utterly helpless to make clear the manner of partition of a complicated expression, but it is by rearrangements of grouping that the chief steps in reasoning are performed. If the student, as he advances in mathematics, will pay attention to the methods, which attention is essential, he can see for himself whether this is not true. But this part of notation is so unobtrusive, that its importance may easily escape remark. Memorandum 8. These signs were of slow growth. Luca Paciuolo (b. 1445-d. about 1514) in his Summa (published in successive forms from 1470 to 1494) used a ν , the initial of vniuersale after I??, the sign for a root, in order to show that the effect of the radical sign was not to stop at the first number following. Thus Rf ν 40 m 1^320 meant -y/40—\/320. The celebrated Dr. Geronimo Cardano, one of [the] strangest figures in history (b. 1501-d. 1576, commonly called, in English, Jerome Cardan), in his Practica arithmeticae universalis (1539) makes use of an L having the opposite meaning, that the effect of the root does

55

SEQUENCES

stop at the first number. Thus, he would write LI? 7 ρ ï? 14 to mean y 7 + \/14. Rafaele Bombelli of Bologna, the author of a more than excellent algebra published in 1572, had the good sense to introduce a second L turned backwards, thus J, to show where the effect of the first ceased. Thus, he would write tyHptyMJ for Thus, we find in his work (where Iψ q means y/ and I [ m e a n s ] -$0, L tyc F?q 68 ρ 2 Jm I W q 68 m 2 J J, This is readily translated into our V ( v V 6 8 + 2 - \ V 6 8 - 2 )· The Uebersichtlichkeit (as the Germans say) of the modern method of expression is far superior; and that is all-important. Only one step more was needed to get the parentheses, namely to substitute for the L something of more distinctive appearance. Simon Stevin ... did this in 1585. But as the simplest way is the way men think of last, instead of substituting, (, and, ), for Bombelli's L and J, he put the two parentheses dos-à-dos and used them so placed, together, to show where the effect of an exponent was to begin. The parenthesis, as we use it, was introduced by the French refugee in Belgium, Albert Girard, in 1627. It has been attributed to Guido Grandi (b. 1671-d. 1742). It was unusual when Rinaldini employed it in 1644. Descartes (1637) and his followers use a single brace connecting terms on one side arranged in a column which are to be algebraically added and multiplied by the number on the other side. This is admirably clear, and should be imitated. Sometimes he omits the brace. Thus he has t +r -p\>v + 2yv P* 2yi>

yv

,

—s yy. tt + 4~v

+ ÌPP Meantime the vinculum had come into extensive use. Chuquet employed it first, drawing it below the line. Thus he writes Έ?214 ρ R?2180 for V 1 4 + -\/ 1 8 0 · The celebrated François Viète (b. 1540-d. 1603) gave its present place to the vinculum; and in the seventeenth century it was not unusual to write three one over another. Thus in a work of Hudde we find t XX -

TP

fi V -

^y

+ ip • ' - q + h +

in x, + h » 0.

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Art. 29. Notation 7. The sign of multiplication is a decussate or St. Andrew's cross, χ ; that is to say, axb means the product of a and b. A dot can be used instead of the cross; or the sign may be entirely omitted and the factors simply written one after another. In this case, the parenthesis about them is always omitted. Thus ab + c means ( a x i ) + c , never e x ( i + c). In reading, omit all mention of multiplication where the sign is omitted, read · "dot" and X "times." But a(b + c) is generally read "a times b plus c" or "a into b + c." Notation 8. The long sign of division is the dividend preceding, the divisor following the sign. But the fractional form has at all times been the most usual mode of writing a quotient. There is a horizontal line called a virgula, or riga, above which is written the dividend, called the numerator, and below the divisor called the denominator. A modification of the fraction consists in writing the numerator and denominator on the same line the former preceding the latter with an oblique line /, between them like a bar sinister, but called a solidus. The colon, or sign of ratio, : , is also used by many mathematicians as a sign of division, the antecedent being arbitrarily called the dividend, the consequent, the divisor. The proportion 2 : 6 : 10 :: 3 : 9 : 15 means that one number may multiply each of the numbers 2, 6, 10 and another each of the numbers 3, 9,15 so as to make the first three products respectively equal to the last three. This has suggested that (:) be made a sign of division. Some purists object to regarding a ratio as exactly the same as a quotient; but the objection seems unmathematical and futile. Notation 9. The power of a number is expressed by an exponent or small number, or letter, above the line following the base, or quantity multiplied into itself ; as λ*. A root may be expressed by the radical sign, y/, to which no index is attached if it is the square root, but y / are written for the cube root, nth root, etc. Thus, (V2)(a/2) = 2 ^ 8 = 2. Memorandum 9. The earliest way of signifying multiplication was by writing the factors together without a sign; and this grew up without conscious invention. The early algebrists, who were all Italians, or under Italian influence, called the unknown quantity res in Latin, cosa in Italian,

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its second power, or square, census or censo (the word in Latin means worth, property, and translates Arabic mài, milul, likewise used for x 2 and having the same original meaning, and this translates Greek δύναμις, power, used for a second power from early times, probably brought by Pythagoras from Babylon and translating Sumerian ibdi, of the same meaning), its third power, cubus or cubo. Of course, when they wished to express 3x3 + 5x2 + 2x, since e is Italian for and, they would and did say "3 cubi e 5 censi e 2 cosi." Shortly after, special characters were introduced for these words/for-λ. for cos (to use the English form),3 for census, ts^ for cube; and then, using ρ and m for + and —, they wrote the above ¿5 3 p2

rt.

Subsequently when it became usual to write letters for the coefficients (or known multipliers of powers and products of the unknown quantities), such letters were, as a matter of course, simply written in place of the numbers, and thus was established, before men were aware of it, the practice of making letters written together without intermediate sign denote products. As for the dot, it was so much in use in medieval writing to separate anything which, without regard to grammatical construction, for any reason had to be separated from the rest of the writing (as for example in many MSS. after every number expressed in Arabic figures) that it should not be considered as a sign of multiplication, but only as a mark of punctuation. The sign χ was introduced by Rev. William Oughtred (b. 1574-d. 1600) in 1631. (It has been said that Har[r]iot used it. If so, he would antedate Oughtred.) It has never been in favor with mathematicians, owing to its resemblance to the letter JC. The earliest, and almost exclusive, mode of indicating a quotient is by a fraction. Fractions have been written just as at present, since Leonardo Fibonacci of Pisa (A.D. 1202) who calls the line virgula. The Arabians used the same arrangement, and probably invented it. The words numeratore, denominatore, riga are found in Paciuolo (commonly called Pacioli) in 1470. Chuquet also (1484) uses numerateur and dénominateur. The sign -i- was employed by Albert Girard to signify subtraction. Its use as a sign of division is due to Johann Heinrich Rahn the elder (b. 1622-d. 1676) in 1659, though it was probably the invention of his friend Rev. Dr. John Pell (b. 1610-d. 1685). The solidus was introduced in 1880 by Sir George Gabriel Stokes.

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It has been universally accepted. Numerical exponents, just as we use them, were introducted by Descartes (1637), and literal exponents by Newton about 1670-1704. But various approaches to them had been made. Thus Simon Stevin, in his Disme 0 123

(1585) writes the decimal 5.912 thus 5.912,10 where the numbers written above are practically exponents of -j^ ; and Rafaele Bombelli in his Algebra (1572) wrote 0/, -é>, to stand for the first four powers of the unknown quantity. Our word power (from Old French, poër, from Low Latin potere, for classical posse, contracted from potesse, compounded of potis, powerful (Root PA, feed) and esse to be), translates French potence, Italian potenza, from Latin potentia, which translates Greek δύναμις, which translates Sumerian ibdi, and was extended from meaning the second power to meaning any power. Exponent is modified from exposant, used by Fermât and others. The radical sign, in the form is found in the Coss of Christoph Rudolph (1525). Whether it is the initial Vniversale written by Cardan and others after Rf, or whether it is a part of the R, or whether it is derived from the Arabic letter jeem, the initial of jidr, root, written by the Arabians to signify the same thing, is doubtful. The word root translated Latin radix which translates Arabic jidr or jidrah, which again translates Hindoo mûla, all meaning root and used to mean the first power of the unknown quantity. Latin radix was at an earlier date used in a somewhat similar way to translate into Greek βίζα, which again probably translated the Sanskrit word. Summarizing the origin of the algebraical signs, we find about 1488. Lionardo da Vinci. 1525. Rudolph. 1557. Recorde.

+ V

About 1580. Viète. >




X+B.

Thus, the character of forming, on being added to any whole number, B, a sum greater than is formed by the fixed whole number, A, added to B, is a character which advances in the sequence of whole numbers. But if X is the number next greater than A, by Axiom 6,

FUNDAMENTAL OPERATIONS IN ALGEBRA

X+B>

71

A + B.

Hence, the character of either being no greater than a fixed whole number, A, or else giving, when added to any whole number, B, a sum greater than A + B, is a character which advances in the sequence of whole numbers. But this is manifestly true of 0; since 0 is no greater than A, no matter what number A may be; hence it is true of all whole numbers. That is if X>A then X+B>A + B. Hence, if Y>B, X+Y>X+B, and a fortiori X+Y >- X+B. If, however, Y = Β, X+ Y = X+B; and again X+ Y>- X+B. Hence, if Y >- B, X+ Y>- X+B; and iîX> A, X+Y>A + B. Q.E.D. Exercise 11. Prove that the sum of two cardinal numbers, neither being zero, is greater than either. Art. 37. Axiom 9. If, of the two factors of a product, either is increased by unity, the product is increased by the other. Axiom 10. 0 x 0 = 0. Theorem 17. The product of two whole numbers is a whole number. Demonstration. For suppose it true of two whole numbers, M and N, that M X Ν is a whole number. Then, by Axiom 9, it is true of either and the number next higher than the other. For the number next higher than M is, by Corollary 11, M + 1 ; and ( M + l)iV is, by Axiom 9, a whole number, namely, it is M χ Ν + Ν. Now, M χ Ν is, by hypothesis, a whole number. Hence, by Theorem 10, M χ Ν+N is a whole number. Now, the proposition is true for M = 0, Ν = 0. Hence, by Theorem 4 and Definition 38, it is true for all whole numbers. Theorem 18. The product of any whole number by 0 is 0. Demonstration. For suppose this true of any number M. Then, it is true of the next higher number, M'. For by Theorem 11, Μ ' = 1 + M . Now, if 0 χ M = 0 then, by Axiom 9, 0 χ (1 + M) = 0 + 0. But 0 + 0, by Axiom 7, equals 0. Hence the proposition, if true of M, is true of M '. But, by Axiom 9, it is true of 0. Hence, by Theorem 4 and Definition 38, it is true of all whole numbers. Corollary 12. It follows in the same way that Μ χ 0 = 0. Corollary 13. It follows, from Axiom 9, that 1 Χ M = M and M χ 1 = M. Art. 38. Theorem 19. If the product of two cardinal numbers is zero, one of those numbers is zero.

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Demonstration. Let Ρ and Q be any two cardinal numbers. Then, the theorem is (since by Definition 38, every whole number not 0 is greater than 0), that if Ρ and Q are both greater than 0, so also is Ρ χ β . Suppose that when Ρ has the particular value M, and Q the particular value N, M and Ν being both greater than 0, the product is greater than 0. Then, when either Ρ or β takes the value next greater, it remains true that the product is greater than 0. For, by Axiom 9, the product is greater than before, and being before greater than 0, by Axiom 5 and Definition 38, it is so still. But, when, Ρ and Q are both 1, by Corollary 13, the product is not zero. Hence, by Theorem 4 and Definition 38, the product is not zero as long as neither Ρ nor Q is zero. Q. E. D. Theorem 20. No two different whole numbers multiplied by the same number, not zero, give the same product. Demonstration. Let A and Β be two different whole numbers of which A is less than B. Let M be any multiplier greater than 0. Then the theorem is, that BM and AN are not equal. For, let A' be the number next greater than A. Then by Axiom 9, MA' = MA + M. Hence, since M'> 0, MA' being MA + Mis, by Theorem 16, greater than MA. Hence, by Theorem 4 and Definition 38, if C is any number greater than A, CM is greater than AM. Hence, since Β > A it follows that BM > AM, and therefore BM and AM are not equal. Q.E.D. Art. 39. Theorem 21. I f , offour whole numbers, the first is greater than the second, and the third at least as great as the fourth and greater than 0, then the product of the first and third is greater than that of the second and fourth. Demonstration. Let A, B, C, D be four cardinal numbers, such that A > B, C >- D, C > 0. Then, I say that A C>

B-D

First, if M is any whole number, and M ' is the number next greater, by Axiom 9, M'C = MC+C. Hence, since C> 0 by Theorem 16, M'C is greater than MC. Hence, by Theorem 4 and Definition 38, if MC>BC, then M'C>BC. Hence, the character of forming, when multiplied into C, a product greater than BC is one which advances in the series of cardinal numbers. But, by Axiom 9, if B' is the number next greater than B, B'C = BC+C, and as just seen B'C > BC. Hence, the product of C by any number greater than Β is greater than BC. Now, by hypothesis, A > B. Hence, AC> BC.

FUNDAMENTAL OPERATIONS IN ALGEBRA

73

Second, if Ν is any whole number, and N ' is the number next greater, by reasoning similar to the above, if BN >- BD, then BN' >- BD. But BD > BD. Hence, if any number, as P, be at least as great as D, BP > BD. Now, by hypothesis, C >- D. Hence, BC >- BD. Since, therefore, AC > BC and BC >- BD, it follows that AC > BD. Q.E.D. Art. 40. Theorem 22. The product of any whole number by 1 is that number. Demonstration. Let Ν be any whole number. Then, I say, that 1 · Ν = Ν. For, by Axiom 8, 1 is the number next greater than 0. Hence, by Axiom 9, 1 · Ν = 0 · Ν + Ν. But, by Theorem 18, 0 · Ν = 0. Hence, by Theorem 12, 0-N + N = N. Hence, l - N = Ν Q.E.D. Theorem 22. The product of any whole number by 1 is that number. Demonstration. For suppose this true of any number M. That is I Χ Μ = M, then it is also true of the next higher number M'. That is l x M ' = M'. For by Theorem [13], M' = 1 + M now by Axiom [9], if 1 Χ M = M then 1 X (1 +M) = 1 +M or 1 X M' = M'. 12 So that if the proposition is true of any whole number, it is true of the number next greater. Now, by Theorem [18], it is true of M = 0. That is 1 χ 0 = 0. Hence, it is true of every whole number. Corollary 13. In like manner, M X 1 = M. Art. 41. Theorem 23. {The Distributive Principle.) The multiplication of whole numbers is distributive with respect to addition. Demonstration. Let L, Μ, Ν be any three whole numbers. Then I mean, first, that L(M+N) = LM+LN, and, second, that (L + M)N = LN + MN. Suppose that the first formula holds when M — P. Then, I say it holds when Μ = Ρ', where Ρ' is the number next larger than P. For, by Theorem 13, P' = \ + P . Now, by Theorem 14, (1 + P ) + Ν = 1 + (P+N) ; so that L X (P'+N) = L X [(1 + P ) + N ] = L χ [ 1 + ( P + N ) l But by Axiom 9, the last expression equals L + HP+N) ; and by hypothesis, L(P+N) = LP + LN. Hence, L(P' + N) = L + (LP+LN) or by Theorem 14, L(P' + N) = L + (LP+LN) = (L+LP) + LN. " Peirce's demonstration reads: "For by theorem , Μ' = 1 + M, now by Axiom 8, if 1 X M = M then 1 χ(1+Λ/)' = 1+M or 1x3/' = M'." Peirce proved the theorem twice.

74

ELEMENTS OF MATHEMATICS (165)

But, by Axiom 9, L + LP = L(l + P) ; and by Theorem 13, 1 + Ρ = Ρ'. Hence, L(P' + N) = LP' + LN. Thus, if the formula holds when M is any whole number, it holds when M is the next greater whole number. 18

18 The next MS. page is missing (p. 93). However, it contained the remainder of Art. 41 with the laws of multiplication. The following two theorems were also given: Theorem 24. Multiplication is associative. Theorem 25. Multiplication is commutative. Exercises 12,13,14 asked for the demonstrations, respectively, of the second parts of Theorem 23, Theorem 24, Theorem 25.

4 FACTORS

Art. 42. Definition 39. A factor of a whole number, N, is any whole number which multiplied by a whole number gives that number, N. A part of a number, N, is a factor of it not the number, N, itself. A divisor of a number, N, is a factor of it which multiplied by a number different from that number, N, gives that number, N, as the product. Thus, the factors of 6 are 1, 2, 3, 6; its parts are 1, 2, 3; its divisors are 2, 3, 6. [Definition] 40. The greatest common divisor of two whole numbers is that divisor of both of which every other divisor is a part. (The word greatest does not refer to its being higher in the scale of numbers. This is important in algebra, where the greatest common divisor may be numerically small.) [Definition] 41. A prime number is a number with just one part. [Definition] 42. A prime factor of a whole number, N, is a prime number which is a factor of that number, N. [Definition] 43. A homogeneous number is a whole number having a single prime factor. The multiplicity of a factor is the number of times it occurs as a factor in a number of which it is a factor. A heterogeneous number is a number having more than one prime factor. [Definition] 44. A composite number is a whole number having a part that is a divisor of it. Or, it is a number that is a product of homogeneous parts. [Definition] 45. Two numbers are said to be prime to each other if they have no common divisor.

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Scholium 13. (Usage is somewhat unsettled in regard to the precise application of some of the above terms. The definitions adopted have been chosen after careful consideration upon the following principles : 1st, definitions formulated by mathematicians must be respected, especially if they are ancient; 2nd, but higher even than the authority of formal definitions, are to be placed familiar and universally admitted dicta, such as that "numbers prime to one another have no common divisor," that "a perfect number is the sum of its parts," etc. ; 3rd, still higher is the necessity of not breaking down classifications rooted in the science of arithmetic itself, such as considering 0 as a whole number; 4th, it is also most desirable to have terms which shall enable us to enunciate the main propositions of arithmetic, without introducing exceptive clauses, and in plain, uncomplicated language.) Apparently insignificant departures from the above definitions may lead to serious misunderstandings. Hence, they should be learned by heart verbatim. Corollaries 14. The factors of any whole number are its parts, together with the number itself, and are its divisors, together with those multipliers which do not affect its value. 15. All whole numbers are factors of 0; and all that are unequal to 0 are parts of it. But 0 has no divisor, but itself. 16. Every number that has parts is its own divisor. 17. 1 has no part, and consequently no divisor. It has no factor unequal to itself. 1 is not a divisor of any number. 18. 0 is not a prime number, because it has the greatest possible multitude of parts. 19. 1 is not a prime number, because it has no part at all. 20. The single part of a prime number is always 1. 21. A prime number has but one divisor, which is itself. 22. Every prime number has two factors, itself and 1. 23. 1 is not a homogeneous number because its single factor is not prime. 24. 0 is not a homogeneous number, since all prime numbers are its factors. 25. 1 is not a heterogeneous number. 26. 0 is a heterogeneous number. 27. 1 is not a composite number, because though it is a product of

FACTORS

77

l x l , those factors are neither parts, nor homogeneous. 28. 0 is not a composite number, because, in every way in which it is a product, 0 is one of the factors and this is neither a part nor homogeneous. 29. 1 is the only number prime to itself, and to every number. 30. 0 is the only heterogeneous number not composite. 31. 1 and 0 are the only numbers neither prime nor composite. 1 is the only number neither homogeneous nor heterogeneous. 32. The prime numbers together with zero are the only numbers each of which has a single divisor. They are also the only numbers having parts but never affected by being multiplied or divided by parts of them. 33. The prime numbers together with 1 are the only numbers without prime parts. They are the only numbers whose only factors are themselves and 1. 34. The prime numbers together with 1 and 0 are the only numbers not composite. Exercises 15-35. Demonstrate the above corollaries. Art. 43. Problem [2]. To find the greatest common divisor of two different cardinal numbers. Solution. If one of the numbers is zero, the other is the greatest common factor. If not, divide the greater by the smaller, the smaller by the remainder, that remainder by the last remainder, and so on until ultimately 0 is found for the remainder. The penultimate remainder is the greatest common factor. If the greatest common factor multiplied by one of the numbers gives that number, there is no common divisor. Otherwise, the greatest common factor is also the greatest common divisor. Demonstration. First, let one of the numbers be 0, the other N, different from 0. Then, by Theorem 21 and Definition 39, no factor of Ν can be greater than N. But, by Theorem 22, Ν is a factor of itself; and, by Theorem 18, it is a factor of 0. Hence, it is the greatest common factor of 0 and N. This proves the first part of the solution. In this case, the greatest common factor, N, multiplied by one of the numbers, namely, 0, gives, by Theorem 18, that number as the product. But to get that number as the product of the greatest common factor multiplied by a number, since, by hypothesis, Ν is not zero, the multiplier must, by Theorem 19, be 0. Hence, by Definition 39, that greatest common factor is not a divisor of 0, one of the two numbers. In fact, 0, by Corollary 15, has no divisor but 0; and by Theorem 18, 0 is not a divisor

78

ELEMENTS OF MATHEMATICS ( 1 6 5 )

of any number, N, different from 0. Hence 0 and Ν have no common divisor. This proves the third part of the rule in this case. Secondly, let the two numbers be A and B, both different from zero, and A > B. Let Qi be the Quotient of the division of A by Β; and let C be the remainder, so that A = Q1B

+ C.

Since A and Β have a common factor (namely, 1, at any rate is a factor of both), let X be their greatest common factor. Then, A/X and B/X are whole numbers. Then, since 61 is a whole number, so likewise, by Β JÍ

A

B Λ

Theorem 17, is Q1 ·—. Hence, by Problem 1, — — Qj - — is a whole Λ A

Β JL

number, provided Q1 ·— is not greater than —. But, by the rule of Λ

division QXB is not greater than A. But, by the associative and commuB

OB

tative principles, Q t · — = Λ Λ X^Qi-^j

For this is the same as to say that

= QtB. Now, by the commutative principle X^Qi and

by t h e associative principle, Β

=

= Q i ^ -^j ; Β

and, by the commutative principle, — X = Z —; and, by the meaning Λ A Β

B O B

of division X ·— = B. Whence, we collect that Q± ·— = Λ

were β χ · ^ greater than

JÍ.

* . Now,

JL

then, by Theorem 21, X^Çh --^j would be

greater than X·—. That is, QtB would be greater than A; which is not A

Β

Λ

Λ

the case. We are sure then, that — — Q t ·-= is a whole number. Now, by the distributive principle

Thus, X is a factor of C. It is also the greatest common factor of Β and C; for had those numbers a greater factor, it would also be a factor of A. In like manner the greatest common divisor of any two successive remainders will be equal to that of the two preceding, and therefore of A and B.

FACTORS

79

By the nature of the process of division, the remainder must be less than the divisor. Thus, each remainder will be less than the preceding remainder. Thus, the remainders subsequent to a given remainder are part of the numbers less than a given number, which is enumerable. The process goes on till a remainder zero is reached. Let R be the remainder before 0. Then X will be the greatest common factor of R and 0. But R is a factor of 0 and no greater number is a factor, of R. Hence X = R. The greatest common factor is also the greatest common divisor, in case R > 1 ; for then the number which multiplied into it gives either A or Β is less than A or Β, and is therefore a part of A or B. But when the greatest common factor is 1, there is no common divisor. Art. 44. Theorem 26. If two numbers are prime to each other, every common factor of one of them and of the product of the second with any third number is also a common factor of the first and third numbers. Demonstration. Let A and Β be two numbers prime to one another, so that, in the algorithm for their common factor, the last remainder before zero is 1. Let Κ be any third number. Then, I say that any common factor of A and BK is a factor of K. Then, every factor of A being a factor of AK, this number divides both AK and BK. Then, multiply all the dividends, divisors, and remainders of the algorithm for obtaining the greatest common factor of A and Β by K; when by the reasoning of the demonstration of Problem 3, it follows that every divisor of AK and BK is a divisor of every remainder (after multiplication by K), and consequently is a divisor of the last, before zero, which is Κ χ 1. Consequently, every divisor of A and BK is a divisor of K; and is therefore a common divisor of A and K. Q. E. D. Corollary 35. If two numbers are both prime relative to a third, so likewise is their product. For let one of the first numbers be A, and the third B. Then these being prime to one another and Κ being any second, AK and Β cannot have a common factor, unless Κ and Β have that factor in common. Hence, if Κ and Β have no common factor except 1, neither have AK and B. Corollary 36. If two numbers are relatively prime, and if the product of one by a third is divisible by the second, so likewise is that third. Corollary 37. If every one of a set of numbers A, B, C, etc. is prime to every one of another set K, L, M, N, etc., then the product of the first set, ABC etc., is prime to the product of the second set, KLM etc. This is a generalization of the last corollary but one.

80

ELEMENTS OF MATHEMATICS (165)

Corollary 38. If A is prime to K, every power of A is prime to every power of K. This follows from the last by making A=B=C=etc. and K=L=M=etc. Art. 45. Problem^ 3]. Tofindthe greatest common divisor ofseveral numbers. Solution. Find the greatest common divisor of any two, and then the greatest common divisor of the number so found and any third of the given numbers, and so on until all have been used, when the last greatest common divisor, so ascertained, will be the divisor required. Demonstration. Let A, B, C, D, etc. be the given numbers. Let Κ be the greatest common factor of A and B. Then, every divisor common to A and Β is a divisor of K. Let L be the greatest common divisor of Κ and C; then, every number which divides A, B, and C divides Κ and C, and, therefore L. Thus, the last divisor divides all the numbers A, B, C, etc. Conversely, whatever divides a greatest common divisor divides the two numbers of which it is the greatest common divisor; and thus, whatever divides the last greatest common divisor so found, divides all the numbers A, B, C, etc. Consequently, this number is the greatest common divisor of them all. Exercise 36. Find the greatest common divisor of 80290, 5111106, 6019530, 16313115. Art. 46. Definition 46. A multiple of a number is a number equal to that number, resulting from multiplying it by a whole number. [Definition] 47. We may speak of a dividend of a number, in the sense of a number that is the product of that number and of any positive whole number. [Definition] 48. We may speak of a product or container of a number, meaning the product of that number by any whole number. [Definition] 49. The least common multiple of a number of numbers, not all of the same value, is a number which is a multiple of them all and of which every other such multiple is a multiple. Exercise 37. Find a series of corollaries analogous to Corollaries 14-34. Art. 47. Problem [4]. To find the least common multiple of two unequal numbers. Solution. Divide the product of the numbers by their greatest common

FACTORS

81

divisor, when the quotient will be the least common multiple sought. Demonstration. For let A be any number, and M any whole number, not 1. Then, every multiple of A must, by Definition 46, be represented by MA', for if M were 1, the product would be equal to A, contrary to the definition. Let D be the greatest common divisor, of two unequal numbers A and B. Then, by Definition 40, A' being some part of A, and B' some part of B, A = A'D, Β = B'D. If, then, MA = MA'D is a multiple of Β = B'D, so that B'D is a divisor of MA'D, then, by Corollary 36, B' is a divisor of MA'. But A' and B' are prime to one another; otherwise, D would not be the greatest common divisor. Hence, M must be divisible by B'. Let Ν be any whole number. Then, NB'A'D will be a common multiple of A and B; and the least (in the sense intended) will be obtained, by omitting the unnecessary factor N. Thus, A'B'D is the least common multiple, but a ' B ' D = m ö =

D

D

Corollary 39. The least common multiple of more numbers than two is ascertained, by first finding the least common multiple of any two, next of that and any third, until all the numbers have been used, when the last least common multiple is the least common multiple of the given numbers. Corollary 40. If a set of numbers are all prime to one another, every common multiple of them is a multiple of their product. Art. 48. Theorem 27. Every prime number is prime to every other number except its multiples. Demonstration. For a prime number has but a single part, and therefore but a single divisor. But every number is a divisor of itself. Hence the only divisor of a prime number is itself. But the only numbers which have a given number for a divisor, except itself, are by Definition 46, its multiples. Thus, the multiples of a prime are the only other numbers having any divisor of it for a divisor of one of them. Corollary 41. If a product of several factors has a prime number as a divisor, then one of those factors has that prime number for a divisor. Art. 49. Theorem 28. (The Fundamental Theorem of Numerical Composition.) Every composite number is equal to the product of an enumerable collection of prime factors, and is not a product of any other set of prime factors.

82

ELEMENTS OF MATHEMATICS ( 1 6 5 )

Demonstration. Let M be a composite number. Then, by Definition 44, it has a part which is a divisor; that is, it is the product of two whole numbers both different from itself. M is, therefore, by Corollary 31, neither 0 nor 1. Let A and Β be two whole numbers both unequal to M, but such that AB = M. Then, neither A nor Β is 0; for OB = AO = 0 while M is not zero, and AB = M. Moreover, neither A nor Β is 1 for Al — A and M is not equal to A, while AB = M, and IB = Β and M is not equal to B, while AB = M. But every cardinal number not 0 nor 1 is greater than 1 ; and by Theorem 16, the product of two numbers, both greater than 1, is greater than either. Hence M > A and Μ > Β. But every number that is neither 0 nor 1 is either prime or composite. Hence, every composite number is a product of two smaller numbers, each either prime or composite. Now, consider the character of being greater than no number except such as are either 0, or 1, or prime, or products of enumerable collections of primes each taken as factor for an enumerable number of times. I say, that this character advances in the sequence of numbers. For suppose it to be true of P; then, it is true of 1 + P, the number next greater than P. For were it not true of 1 + P, this number must be greater than some number, neither 0, nor 1, nor prime (and therefore composite), but not a product of an enumerable collection of prime factors. But every number smaller than 1 + Ρ is either smaller than Ρ or equal to P. Now, by hypothesis, there is no number of the kind supposed that is smaller than P. Hence, the only possibility is, that Ρ itself is a composite number not a product of an enumerable collection of prime factors. But P, as a composite number, must be a product of two smaller numbers either prime or composite. If Ρ were a product of two prime numbers it would be a product of an enumerable collection of prime factors. If it is a product of a prime and a composite number, both are smaller than itself, so that if the composite number were a product of an enumerable collection of prime factors, Ρ itself would be the product of those prime factors and one more, which would make an enumerable collection. If Ρ is the product of two smaller composite factors, each of these is the product of an enumerable collection of primes, and Ρ is the product of the two collections put together. Now two enumerable collections taken together form an enumerable collection; for the number of objects in the whole is the sum of the numbers in the two parts; and the sum of two whole numbers is a whole number. Thus, we have shown that if the above character is true of Ρ it is also true of 1 + P. But it evidently is true of 0. Namely, it is true that 0 is

FACTORS

83

0; and therefore is either 0 or something else. Hence, the character is true of all numbers. It remains to be shown that a number cannot be decompounded into prime factors in two ways. Suppose this could be done. Then, there would be two different enumerable collections of primes whose products would be equal. Multiply together all the prime numbers which occur in both products, taking each as a factor as many times as it occurs in that product where it occurs the fewer number of times. Call this product Q and the remaining homogeneous factors in the first collection A"BbCc etc. and those in the second collection S'T'U" etc. Then we have QAaBbCc etc. = QSsT'Ua

etc.

Then we have AaB"Cc etc. = SST'UU etc.

That is, there would be two equal products of prime factors having no factor in common. But it has been proved, Theorem 26, that if a number is prime to a second but has a divisor in common with the product of this second by a third, then that divisor is a divisor of the third. From this it follows, that a prime number, A, cannot be a divisor of any enumerable power of another prime number, nor of any product of an enumerable collection of such powers. Hence, it is impossible that AaB"Cc etc. = STU"

etc.;

and, therefore, that there can be but one way of decomposing a number into prime factors.

5 NEGATIVE NUMBERS

Art. 50. Scholium 15. Having thus conducted the pupil through all the dry and dusty part of the doctrine of whole numbers, it is distressing to have to turn from the path, just as it is about to open into one of the most delightful regions of pure thought. But that is the condition which circumstances impose upon the teacher of elementary mathematics. It is indispensable in mathematics, when we find ourselves hampered by exceptions, owing to the narrow and arbitrary character of our hypotheses, to enlarge our hypotheses so as to get rid of their angularity. If our numbers are only to serve for counting collections, there can certainly be no number less than zero. And thus we have the irregularity that some numbers can be subtracted from others, while some cannot. Operations of such partial applicability give rise to irregularities and complexities, without end. They are to be avoided. Considering a simple sparse sequence, of which the system of whole numbers from 0 up is a particular case, there is no reason why this must have a first object. Nor is the counting of collections the only use to which the whole numbers are ordinarily put. We also use them, for indicating places in entirely unlimited sequences. As long as men only reckoned forward from a fixed epoch, the reckoning of years presented no feature essentially different from the counting of any other collection of objects. Reckoning by the vulgar era began in one or two dioceses with A.D. 532, that number being taken because it was near enough to the number of years since the Incarnation of Our Lord to be defensible as such, while it was also a "great pascal cycle" or 28 χ 19 years. The author of the invention, Dionysius Exiguus, is named in history, as he deserved to be. But nobody remembers what medieval chronicler it was who had the novel and useful idea of reckoning back in reverse order, the years before the Nativity. Dionysius made the year A.D. 1 begin with Lady Day, March 25 ; but the beginning of the year was afterward shifted to the following January 1, the feast of the Circumcision; so that the year in which the

NEGATIVE NUMBERS

85

Annunciation and Nativity were assumed to have fallen was not our year A.D. 1 but the year before. Now, it is the universal rule for all years after the first of the era that each year receives a date-number greater by one than the number of the year next before it. It was an intolerable stupidity to make a perfectly unnecessary exception to that rule. That rule requires the year next before the year 1 to be called the year 0. That is the designation which astronomers do give to it. The year before the year 0 must, by the same rule, be called 0 minus 1, or 0 less by 1, where "less," of course, merely means earlier in the sequence of years. But the zero is omitted, and astronomers make the date of that year —1, minus one. For, adhering to the mathematical principle of generalization, which forbids the creation of unnecessary exceptions, since a zero can generally be struck off from a sum and 0+N be thus reduced to +N or Ν and 0 + 0 — Ν to 0 — N, [they] give the date —N to a year which comes Ν years before the year 0. Thus, by simply carrying out its own principles, the system of whole numbers is converted into the doubly unlimited system • ·· ·™*4)

3j "~2j ~ 1 ? 0, 1 y 2) 3j 4j ·»·

Art. 51. Definition 50. The negative-positive, or datary system of integers is an unbound, simple, sparse sequence of numbers, composed of two parts one entirely higher than the other. The upper part consists of zero and the whole numbers greater than zero, which may have each the sign + prefixed, and at any rate is conceived as entitled to this positive sign, which means that they are above zero. The lower part consists of the same numbers with the negative, or minus, sign, in place of the positive sign, to signify that they are below zero; and if any positive number, +N', is next higher than another, +N, then the negative of the first, that is, the same signless number, called the modulus, tensor, or absolute value, but with the negative sign, that is —N', is next lower than the negative of the other —N. Zero may take either sign; but —0 = + 0 . Definition 51. A positive number is one "greater" than 0; a negative number is a number "less than" in a reversed order of reckoning from zero. Illustration 27. A farmer consigns to a broker a drove of 50 cattle, and draws upon him for $300. The broker sells the cattle for $5 a head, and charges a commission of 2\ percent. With how much must the farmer be credited upon the broker's books? Answer : He must be credited with — $56.25. That is, he must be debited to $56.25.

86

ELEMENTS OF MATHEMATICS ( 1 6 5 )

[Illustration] 28. A housekeeper pays a butcher $1 for a piece of beef at 25 cents a pound, and finds it weighs 3 lb. 13 oz. How much over weight is the beef? Answer: It is over weight by —3 oz. That is, it is under weight by 3 oz. [.Illustration] 29. A train on the Hudson River Railway, between Harlem and Albany, crossed the Harlem bridge at 3 P.M. and the Albany bridge at noon of the same day. How long was it in going from Harlem to Albany. Answer: It was —3 hours in going from Harlem to Albany; that is, it was 3 hours in going from Albany to Harlem. Art. 52. Definition 52. The sign of a number is the positive or negative sign which may be written before it, to show whether it comes after or before zero. Though the sign of a positive number is not generally written, the quantity is said to have that sign, whether written or not. Postulate 1. There [are] but two signs which real numbers can have. Definition 53. The absolute value, modulus, or tensor of a number is the number itself, without regard to its sign, as expressive of how far it is from zero. This value may be regarded either as signless or as positive. Postulate 2. The sign and modulus suffice to determine a real number. Definition 54. The negative of a number is the number with the same modulus but the opposite sign. Postulate 3. The negative of a number is next greater than the negative of the number next greater than that number. Postulate 4. —0 = +0. Art. 53. Memorandum 11. Negative terms were recognized by the Greek arithmetician Diophantus, early in the fourth century of our era. But it was not till the seventh (as far as we know) that negative quantities were distinctly conceived, and that was by the Hindoos. Brahmagupta calls positive numbers "possessions" and negative ones "debts." He also illustrates them by the account of time, and by measuring both ways along a line. His follower Bhascara omits negative solutions, because people did not recognize them as legitimate. The Arabians seem never to have thought of negative numbers; but the earliest practical arithmetician of Western Europe, Leonardo of Pisa, called Fibonacci, in his Abacus, written A.D. 1202, being the book which taught us the "Arabic"

NEGATIVE NUMBERS

87

figures, has two problems with negative answers, which he justifies by the idea of debt. From that time, we hear nothing more of such quantities till toward the close of the 15th century. François Chuquet, in his Triparty, written in 1484, a work whose use of the word Byllion probably indicates Italian influence, speaks of negatives most distinctly. He has a problem in which a man buys 12 apples for minus eleven pieces of money, which he defends by the supposition of a debt. So Johannes Widmann, in 1489, has a German problem about a man who bought 6 eggs and minus 2 pence, paying for them 4 pence and 1 egg. Others from time to time got the notion; but Albert Girard was the first to insist upon a thorough-going and constant application of it. Art. 54. Scholium 16. Any doubly unlimited, simple, sparse sequence can, by its definition, be separated into two parts one altogether after the other, the former having a first, the other a last object. So we may separate the integers into the positive integers together with 0, and the negative integers. If, then, we can show that a character advances in the greater part and is true of 0, and that it recedes in the lesser part and is true of — 1, we prove it to be true of all numbers. Definition 55. The addition of negative integers is to be so understood that the effect of substituting in a sum for a term not positive the number next below that term in the scale of numbers, is to make the sum the number next lower in the scale of numbers. Theorem 29. All the properties of addition which are true generally for the positive whole numbers with zero are equally true in the doubly unlimited system of integers. Demonstration. For those properties follow from Axioms 7 and 8, and with Definition 55, Axiom 7 is extended so as to apply to the lower part of the sequence just as in the upper part. Consequently, whatever can be proved by Axioms 7 and 8 to be true of every positive whole number, can, by Definition 53 and Axiom 8, be proved true of every negative integer. Exercises 38-44. Prove theorems 10 to 16 of all integers positive and negative. Art. 55. Theorem 30. The negative of the negative of a number is that number. Demonstration. For let S be the sign of a number, M its modulus.

88

ELEMENTS OF MATHEMATICS (165)

Then the negative has the same modulus, and the negative [of] the negative has the same modulus M. But the sign of the negative is not S, and the sign of the negative of the negative is not that of the negative. But there are, by Postulate 1, but two signs; so that its sign must be S. Thus the negative of the negative has the sign S and the modulus M, and these are, by Postulate 2, the only elements required to determine the number. Theorem 31. The sum of a number and its negative is zero. Demonstration. Let M be the modulus, S the sign of a number; and let S be the other sign. Then M is the modulus and S is the sign of the negative of that number. Then, since, by Postulate 1, there are but two signs, one of S and S must be positive. Suppose this to be S. If for a special value of the modulus, say Ν (+N) + i-N)

=0

I say that this will be true for + N ' the number next greater than N. For, by theorem 13, Ν ' = 1 + ( + N ) and by Postulate 3, 1 + (-ΛΓ) = ( - N ) . Hence N' + [l+(-N')]=

[l+(+N)]

+

(-N),

= 1 + [(+N) +

(-N)].

whence by Theorems 14 and 29 1 + [N' + (-N')] Whence by Theorem 13 N' + ( - N ' )

= (+N) + ( - N ) .

But, by Postulate 4, —0 = +0. Hence by Axiom 7, for Ν = 0, the equation (+JV) + (-JV) = 0 Hence, this equation holds for all values of N. [Q.E.D.] Theorem 32. The negative of the sum of two integers is the sum of their negatives. Demonstration. For by Theorems 14 and 31 the sum of the negatives added to the sum of the numbers gives 0, whence by Theorem 31 the former sum is the negative of the latter. Art. 56. Theorem 33. (The Rule of Algebraical Addition.) The sign of the sum of two numbers is the sign of the greater, if one is greater than

NEGATIVE NUMBERS

89

the other, and if not, is the sign of either. The modulus of the sum is the sum or difference of the moduli according as their signs are like or unlike. Demonstration. If the signs of the two numbers are both + , by Theorem 10, the sign of the sum is the same, and the rule of moduli is evident; and Theorem 29 extends this to the case where the signs are both — ; and the rule of moduli follows from Theorem 32. If the signs of the two numbers are different and the moduli are equal, by Theorem 31 and Postulate 4, it is indifferent which sign is taken for the sum, and the modulus of the sum is the difference of the moduli. If the positive number has the larger modulus Ν and the negative number the smaller modulus M, the sum is Ν + (—M). This sum increased by M, is by Theorems 14 and 31 equal to N; and hence, by the meaning of subtraction, Ν + (—Μ) = Ν —M. Hence, by Theorem 32, if the smaller modulus M belongs to the positive number, (—Ν) + Μ = —(Ν —M). Q.E.D. Theorem 34. [The Rule of Algebraical Subtraction.] Any remainder after subtraction equals the sum of the minuend and the negative of the subtrahend. Demonstration. Let M be any minuend, S a subtrahend, and the remainder R. Then I say R = M+(—S). For by the meaning of subtraction, M=R

+S

Adding the negative of S M + (-S)

= (R + S) + ( - S )

which by Theorems 14 and 31 equals R. Q.E.D. Art. 57. Definition 56. Let the multiplication of negative numbers be so understood, that if for a factor not positive the number next lower is substituted, the product will be diminished by the other factor. Theorem 35. [The Rule of Signs.] The effect of substituting for a factor of a product its negative is to change the product to its negative. Demonstration. First, suppose the unchanged factor, P, is positive. Then I say that P(—Q) = —(PQ). Suppose Q positive, and to have a value for which the equation holds. Then, by Theorem 29, — Q — 1 is the number next smaller than — Q and, by Theorem 32, — Q— 1 - ( β + 1 ) . But, by Definition 54, P(-Q-1) = P(-Q)-P, which, by hypothesis, equals — (PQ)—P, which, by Theorem 32, equals —[(Pg)+.P],

90

ELEMENTS OF MATHEMATICS ( 1 6 5 )

which, by Theorem 23, equals - [ P ( f i + l ) l · Then, if P(-Q) = — (PQ) holds for any positive value of Q (Ρ being positive) it holds for the next greater value. But it holds for 0 = 0 by Postulate 4 and Theorem 18. Hence it holds for all positive values of Q (P being positive). If Q is negative, let Q be its negative, which will be positive. Then the product P(—@) = PQ by Theorem 30, and as we have just shown, P(—Q) = - ( P g ) , which is -[P(- 2 + etc. — (Bx +B2+

etc.).

106

ELEMENTS OF MATHEMATICS ( 1 6 5 )

Next, add — Atx+A2x

etc.), and we get

C2x+

+ etc. — (CiX + C2x + etc.) =

D1 + D2+ etc. - (Bt + B2+ etc.). This, by the distributive principle, equals [At+A2+

etc. - (C t + C2 + etc.)]* = Di + D2 + etc. - (.Bx + B2+ etc.).

We now multiply by ^ x

+

etc.

_\Ci

+ Q +

etc>).

and so get

_ P 1 + P 2 + etc. - (Bt + B 2 + etc.) ~ A1 + A2 + etc. - (C t + C2 + etc.)'

N.B. (8). Many problems cannot be entirely expressed by equations, at least, not conveniently; although their most essential conditions can be so expressed. In such [a] case, we solve for those conditions, and then introduce the others by ordinary reasoning. Art. 68.

Exercises.

45. (From the Arithmetic of Aahmes, a papyrus written 1700 B.C. 1 8 The seventh of a heap and its whole make 19. How much is the heap? Let χ be the heap. Then, the equation is -fx + χ = 19. This gives ( l + | ) x = 19; _ 19 _ 1 9 x 7 _ 133 _ 5 1 + 4 " 7 + 1 ~ 8 ~ 8" (Aahmes gives the answer in the form 16^·^, according to the Egyptian manner of writing fractions.) 46. (From Aahmes.) Divide 100 loaves among 5 persons, so that the first three get 7 times as much as the other 2. What is the difference? Let χ be the amount each of the unfortunate 2 get. Then, 7x is the amount each of the fortunate 3 get. The equation is 2χ+3·7χ = 100. 18

Date usually given as 1650B.C. This set of problems reflects Peirce's familiarity with the literature in the history of mathematics and is an indication of the scholarly work he himself was doing in that area. Available to him at that time was Moritz Cantor's Vorlesungen über Geschichte der Mathematik in four large volumes. He also knew of Eisenlohr's translation into German of the Aahmes' Papyrus (the famous Rhind papyrus) since it was published in 1877. The now well-known Chase translation appeared in 1927 and that of Peet in 1923.

107

SIMPLE EQUATIONS

This gives 100 100 * ~ 2 + 3-7 ~ 23 ' 600 The difference is 6x = — = 26.08695652173913043478 which the present writer takes out of a table of circulating decimals. In the Egyptian system, it is 2 6 ^ 2 ^ . 47. (By Bhâskara.) Two lovers broke a string of pearls, of which £ fell to the ground, y remained on the sofa, y were caught by the girl, •j3^ were taken by her sweetheart, and 6 remained on the string. How many were there? * = i * + i * + τ* + t V * + 6 6 180 Y— — 18 1δ 1 - ¿ - W - i V " 3 0 - 5 - 6 - 6 - 3 · 48. (By Metrodorus.) Demochares has lived £ of his life as a boy, y as a youth, y as a man of military age, and 13 years as superannuated. How old is he? Let χ be his age. Then the equation is + f c + f x + 13 = χ,

\x

which gives 13 *

=

1 - i - f - f

=

13-60

13-60

60-15-12-20 "

13

6

°"

49. (From Aahmes.) There are 10 measures of corn for 10 persons. The difference between each person's share and the next is £ of a measure. Let χ be the smallest share. Then, the equation is, lOx + -g(l + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 10. This gives, Λ v 11

in _ 8l i _ 1 4J _ 35 _ lyJ — JQ — 1 " 8 0 — 80

—

7__ — _ 1 T, 1 ι 1 20 16 4 8 τ 16'

Peirce has a memorandum to correct the wording of this problem from Colebrooke. He is referring to Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bháscara translated by Henry Thomas Colebrooke, Esq. (London, 1817). It is also found in Kaye's Indian Mathematics (1915). The problem is stated there as follows: "The third part of a necklace of pearls broken in an amorous struggle fell on the ground. Its fifth part was seen resting on the couch, the sixth part was saved by the lady and the tenth part was taken up by her lover. Six pearls remained on the string. Say, of how many pearls the necklage was composed. Ans. 30." Kaye ascribes it to Sridhara. 20 A section has been torn from the page at this point.

108

ELEMENTS O F M A T H E M A T I C S

(165)

50. (By Aryabhatta, about A.D. 500.) Lovely maiden, with sparkling eyes, tell me what number is that which tripled, then increased by £ of the product, then divided by 7, its quotient diminished by y, multiplied by itself, diminished by 52, the square root extracted, 8 added, and divided by 10, gives 2. Let χ be the number. Then the equation is

10

2

·

Multiplying by 10,

Subtracting 8, V W ( 1 + !)*]" - 52 = 12. Squaring, (f) 2 - 52 = 144. Adding 52, or or

(D 2 = 196; I 2 = 196, χ 2 = 784.

That is Λ:

= ±28.

51. (Given, in substance, by Aryabhatta.) A courier went from this place η days ago, at the rate of a miles a day. Another has just started from a point h miles forward of this place, on the same road, at the rate of b miles a day. In how many days will the second courier pass the first? Also, note exceptional cases. Let t be the number of days required. Then, the equation is (t + h)a = h + tb; which gives, _ h — na If h = na, the first courier has just got to the starting-point of the second courier, and it makes no difference what the speed of the second

109

SIMPLE EQUATIONS

is, unless a — b, when they keep on together. If a — b the second courier would pass the first at the end of £ days, which means that he always will be, and if they have been travelling for an indefinite time, he always has been, behind. If b > a, na > A, / is positive, and the second courier the second courier is already ahead, and if they have been travelling long enough, passed the first some time ago. So if a > b, but na > h, the first courier is ahead, and is travelling faster, so that he has already passed the second. If a> b, h > na, the first courier is behind, but is travelling the faster, and will overtake the second. If b is negative, that is, if the second courier is travelling in the opposite direction, he certainly will pass the first, unless h ^ n a , that is, unless the first courier has already passed the second courier's starting point, in which case they met some time ago, if they have been travelling long enough. 52. (By Bhâskara, 12th Century.) One fifth of a swarm of bees lit upon a Kadamba Flower, one third upon a Silindha Flower. Three times the difference of these numbers flew to the flowers of a Kataya. There was but one bee left, which hovered about attracted by the delightful aroma of a Jasmine and a Pandamus. Tell me, charming lady, the number of bees.21 Let χ be the number. The equation is i x + i x + 3 ( i - i ) x + l = χ. This gives 1 X =

15

l-T-T-3(i-|)

=

15-5-3-3(5-3)

=

15

'

53. (By Metrodorus, a mathematician of Constantine's time.) Of four pipes, one fills a cistern in one day, the next in two days, the third in three days, the fourth in four days. How long will it take all, running together, to fill the cistern? Let xlt x2, x4 be the discharge of the four pipes in one day, in fractions of the cistern. The equations are Xi = 1, 2x2 - 1,

3Λ:3 = 1, 4X4

=

1.

The question is, how much is u

X1

+ x2 + x3 + x4

?

Again Peirce reminds himself to correct from Colebrooke.

110

ELEMENTS OF MATHEMATICS (165)

We have xt = 1, x2 = 1 111611 + + +

x3 =

x4 = 1 1 + "2+t + 4

—

12 12 + 6 + 4 + 3

=

12 25"

54. There is a number such that 20 plus the product of the number by 3 less than itself, is to one less than its double, as 17 increased by the square of one less than the number, is to one more than its double. What is the number? Let χ be the number. Then, the equation is, 20 + x(x — 3) _ 17 + (x — l) 2 χ 2 - 3x + 20 _ x 2 - 2x + 18 ,ΟΓ 2x — 1 2x + 1 2x — 1 2x + 1 ' Clearing from fractions, we get, 2x3 - Sx2 + 37*+20 = 2x3 - 5x2 + 38* - 1 8 . Add —2x3 + 5x2 — 37χ +18, and we have 38 = x. 55. (From the Bamberg Rechenbuch, 1483). Two couriers set out together. One goes 6 miles a day. The other goes 1 mile the first day, and every subsequent day 1 mile more than on the day before. In how many days have they gone the same number of miles7 Let χ be the number of days. Then 6x miles are travelled by the first. As to the second, write under his miles, the same reverse order 12345 54321

123456 654321

The sum of two, one over the other, is always χ + 1 , and there are χ such sums, making jc(x+ 1). But this counts his miles twice over. Therefore, he has gone yx(x + 1) miles. Thus the equation is |x(x + 1) = 6x. This is satisfied by χ = 0. But we do not want that solution. The χ we want is not 0. We may therefore divide by x, when the equation becomes -Kx + 1) = 6 giving χ = 11. 56. A merchant made 33[y] per cent every year upon his capital as it was at the beginning of the year, but spent yearly $1000. Yet at the end of three years he had doubled his capital. How much was it originally? Let χ be the original amount.

SIMPLE EQUATIONS

111

At the end of the first year it was f x —1000. At the end of the second year it was f ( f χ -1000) -1000. At the end of the third year it was f[f + 5z = 3, 7 x + l l j > + 1 3 z = 12. We write 1, 2, 7, A,

1, 3, 11, B,

1, 5, 13, C,

1 3 = 0, 12 M

where M = Ax+By+Cz. others, and get 1, 0, 0, 2, 1, 3, 4, 6, 7, A, B-Α, C-Α,

We subtract the first column from all the 0 1 = 0. 5 M-A

We diminish the third column by 3 times the second, and the fourth by the second, and so get 1,

0,

0,

0

2,

1,

0,

0 = 0

7, 4, -6, A, B-Α, C-3B+2A,

1 M-B

We multiply the fourth column by 6 and increase the product by the third column, and so get

128

ELEMENTS OF MATHEMATICS (165)

0, 1, 4,

1. 2, 7, A, B-Α,

0, 0, -6, C-3B+2A,

6M+C-9B

+ 2A

Hence, χ = - £ , y = £, ζ = - f 83. Solve the equations 5ί

+ ν — 13z = - 7 w—4v+ w = 0 f — 4w+7v — 4w + * = 7 m—4v+7w—4x+ =14 ν—4w -f- 7χ — 4y ζ = 21 w —4x + 7 =28 — 17/ + Λ: + l l z = 35 The student should merely go over this, and not attempt to do it himself, independently. It will be best to arrange each equation in a column, thus : 5 0 1 0 0 0 -17 0 1 -4 1 0 0 0 1 -4 1 7 -4 0 0 0 1 -4 7 -4 1 0 0 0 1 -4 7 -4 1 0 0 0 0 1 -4 1 -13 0 0 1 0 0 11 - 7 0 7 14 21 28 35

Τ U V W X Y Ζ M

We multiply the 3rd and last two rows by 5, and increase them by —1st, 13 X 1st, 7 X 1st; thus: 1 0 0 0 -17 Τ -4 1 0 0 0 V 34 - 2 0 5 0 17 5 V-T -4 1 0 W 7 -4 1 -4 1 Χ 7 -4 0 0 1 -4 1 0 Y 0 13 0 5 0 -1665Ζ+13Γ 0 42 70 105 140 56 5 Μ + 7 Γ

5 0 1 0 0 -20 0 1 0 0

0 0 0

We increase the 3rd row by 20 χ 2nd and the 4th by —2nd, thus:

129

SIMPLE EQUATIONS

0 1 1 -4 0 -46 0 0 0 1 0 0 0 13 0 42

5 0 0 0 0 0 0 0

0 1 0 6 -4 1 0 70

0 0 5 -4 7 -4 5 105

0 -17 Τ 0 0 U 0 17 5 K + 2 0 U - T 1 0 W-U -4 1 X 1 0 Y 0 -166 5Ζ+13Γ 140 56 5M+7T

We diminish the third column by the last but one, and then multiply the 7th row by - f f f . 18 -4 0 -63

5 0 0 0

0

0

1

1

0 0 0

0

0

0 0 0

6 -4 1

0 189 0 -14

0

0 0 0

70

0 0

5 -4 7 -4

1 8 9c 179·'

105

0 0 0

-17 0 17 0

Τ U 5V+20U-T W-U

1 -4 1 X 1 0 Y 0 —Β£166-Η|{5Ζ+13Γ) 140 56 5Μ+7Γ

—

We increase the 7th row by 3 times the 3rd, and multiplying the 8th by 9 diminish it by twice the 3rd. These rows thus become 000 020iW 0 - 1 2 4 ^ - iff]. And by Corollary 98, since [C] and [£>] are contained in that plane the whole of the ray {è} in which [C] and [£>] both lie is contained in that plane. Thus both [A ] and {è} are contained in one plane. Q. E. D. Theorem 50. Every plane and ray contain a common point. Demonstration. Let (a) be any plane, and {è} any ray. Then I say there is a point [ab\ which is contained both in (a) and in {b}. For let (γ) and (