Pierre Bouguer's Optical Treatise on The Gradation of Light 9781487578138

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OPTICAL TREATISE ON THE GRADATION OF LIGHT

PIERRE BOl'Gl.EI{

Pierre Bouguer's Optical Treatise on THE GRADATION OF LIGHT

Translated, with Introduction and Notes by W . E. KNOWLES MIDDLETON National Research Council of Canada, Ottawa

University of Toronto Press

Copyright, Canada, 1961, by University of Toronto Press Printed in Canada Reprinted in 2018 ISBN 978-1-4875-7898-5 (paper)

ACKNOWLEDGMENTS The translator wishes to acknowledge the courtesy of the Director of the Musee du Louvre in furnishing a photograph of the portrait of Pierre Bouguer and giving permission for its reproduction; to thank the National Research Council of Canada for secretarial assistance; and to express his gratitude to his wife for encouraging him to complete the translation and for her patient work in reading the proof. Finally, it is a pleasure, and much more than a routine matter, to thank the staff of the University of Toronto Press once again for their keenness and courtesy.

Introduction As I FINISH this translation of Bouguer's Traite, just two hundred years after its publication, I reflect on the curious way in which time has dealt with its author. While there is no doubt whatever that he laid the foundations of photometry, most optical scientists are scarcely aware of his existence, and freely attribute to others some of his most useful generalizations. A. Wolf 1 and Vasco Ronchi 2 speak highly of his work in photometry, but it is not generally realized how entirely original it was, with nothing whatever to anticipate its simple solution of a problem which was widely believed to be insoluble. Bouguer would have been a notable man if he had done nothing but his photometry, but actually he did many other things, and photometry was in a way a hobby, to which he returned in the last years of his life. Pierre Bouguer was born on February 10, 1698, at Croisic in Brittany, to Jean Bouguer, Royal Professor of Hydrography, and his wife Fran~oise, nee J osseau. Jean Bouguer was one of the leading hydrographers of his time, and an able mathematician; and if we are to believe the gracious obituary notice in the Histoire de l' Academie Royale des Sciences for 1758, "the first words that the young Bouguer heard were mathematical terms; the first objects on which his eyes rested were astronomical and hydrographical instruments." 3 Be that as it may, Pierre Bouguer was a prodigy, greatly encouraged by his father. He went to the Jesuit College at Vannes, and it is related that while he was in the cinquieme (that is to say the fifth form from the top), the headmaster heard of his exceptional talents. A conversation resulted in a request from the headmaster that the young scholar should teach him mathematics, which he did. Such a relation between an eleven-year-old boy and his teacher must, as the writer of the obituary remarks, be altogether exceptional. Two years later, we are told, when Pierre was thirteen, he seems to have objected to some inexactitude on the part of his teacher of mathematics, who was unwise enough to challenge the boy to a public 1A. Wolf, History of Science, Technology, and Philosophy in the XVIIIth Century (London, 1939), p. 167. 2Vasco Ronchi, Storia delta luce (Bologna, 1952), p. 218. 1 P. 127. This was written by Jean Paul Grandjean de Fouchy (1707-88), Secretary of the Academy.

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debate. The teacher was so thoroughly defeated that he left town and was not heard of again. The next chapter in this extraordinary story took place when Bouguer was fifteen. His father died, and to the immense astonishment of the authorities Pierre applied for his professorship. It is to the credit of eighteenth-century France that after undergoing a thorough examination, Bouguer obtained the post; and in spite of the fact that most of his students were older than himself, he seems to have been a great success as professor of hydrography. In his spare time he did a good deal of original research, including some work on ships' masts which came to the attention of Father Reyneau, a member of the Academy. This led to the "little bit of luck" which does no harm even to the great. Father Reyneau showed this work to the prominent physicist Jean Jacques d'Ortous de Mairan, who succeeded in getting the Academy to propose the subject of ships' masts for its prize essay for 1727, which Bouguer of course won easily. This piece of work brought him to the attention of the learned world; and as if to consolidate his position, he also won the 1729 prize with an essay on the best way of observing the altitudes of stars at sea, and the prize for 1731, for which the set subject was the best way of observing the variation of the compass on board ship. One may suspect the hand of de Mairan in the occurrence of three nautical subjects in five years. Meanwhile in 1729 Bouguer had published the Essai d'Optique sur la gradation de la lumiere, of which I shall write later. Here I need only remark that the originality of the subject of this book convinced the Academy (if they still needed to be convinced) of the merit of its author, who in 1730 had been transferred from Croisic to Le Havre, which is nearer Paris; and on September 5, 1731, the Academy gave Bouguer the title of Associate Geometrician, a post out of which de Maupertuis had just been promoted. In 1735 Bouguer was sent by the Academy on an expedition to Peru with Godin, de la Condamine, and Jussieu, to measure an arc of the meridian near the Equator, as a contribution to the study of the shape of the earth. On January 24 of that year he had been made a full academician, with a pension, which made up for abandoning his hydrographic post. The party sailed from La Rochelle on May 16, 1735; and in this decade of the twentieth century it is interesting to note that they "arrived at Quito about a year later." 4 Hist. Acad. R. des Sci., Paris, 1758, p. 132.

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It is evident that Bouguer was in his element on this long and arduous voyage, of which a fairly brief and very fascinating description is available. 5 Apart from the main programme of measuring the arc of the meridian, Bouguer busi ~d himself with a number of other scientific matters. He measured the dilatation of various solids by making use of the large ranges of temperature found in the Peruvian mountains; he investigated the phenomena of atmospheric refraction and the laws governing the decrease of density of the air as one ascends (referred to at length in the Traite d'Optique); devised a new pattern of ship's log; and undertook a number of other researches. He did not get back from the voyage until June, 1744. After his return, he was busy writing up the results of the expedition, but he also invented the heliometer 6 and in 1752 published a treatise on navigation, which was based on that of his father. In 1757 a treatise on ships appeared. Bouguer was the leading French theoretical authority on all things nautical, and appears to have been encouraged by Louis XV in this direction. Unfortunately he had none of the social graces, and was not at all at home in the very formal and artificial society of his time. The last ten years of his life were spoiled by a public quarrel with de la Condamine which began during the expedition and lasted until 1754. Bouguer appears to have felt that almost all the credit for the work carried on in Peru belonged to himself, and de la Condamine, not unnaturally, resisted this idea. When the quarrel died down Bouguer's health was ruined by nervous indigestion brought on by his unreasonable sensitivity to criticism of his work. 7 There is, of course, not a word of this in the official necrology, where his last illness is ascribed to his sedentary life and to his grief over the death of his brother. Nor is there any indication that he had become a free-thinker, or that he had been reconciled with the Church only on his deathbed, a fact mentioned in both the other sources to which I have referred. There appears to have been a conspiracy to play down his quarrel with de la Condamine, for most of the documents concerning it seem to have disappeared. All in all, he must have been a difficult sort of person. Nevertheless I should like to believe the story in the obituary notice that a few days before his death he got out of his bed and into a carriage, and took the manuscript of the Traite d'Optique to the printer, advising him to 1 P. Bouguer, Mbn. Acad. R. des Sci., Paris, 1744, pp. 249-97. 'Ibid., 1748, pp. 11-34. 7 F. X. de Feller, Dictionnaire Historique (Paris, 1847), I, 148. M. Prevo~t and R. d'Amat, ed., Dictionnaire de biographie fran{(lise (Paris, 1954), VI, 1298.

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hurry if he did not want it to be a posthumous work. Unfortunately this does not agree with the account given by his friend the Abbe de la Caille in his preface to the Traite. Bouguer died on August 15, 1758. It is now time for a discussion of Bouguer's work on photometry. It is very unfortunate that he was not spared to write a preface to the Traite, but he did write a preface to the Essai of 1729, and in this he tells us clearly how he came to take up the subject. 8 In 1721 de Mairan had written a memoir in which he supposed that the amount of light from the sun at two altitudes was known, and tried, though not successfully, to show how the amount at other altitudes could then be calculated. 9 Bouguer originally proposed only to find out how to make this measurement (I have already shown that he owed a good deal to de Mairan); he succeeded in doing it with the moon on November 23, 1725, by comparing its light with that of a candle. 10 As J. W. T. Walsh 11 has written, this was the real birthday of photometry. It cannot be too strongly emphasized that the measurement of light was what Americans would nowadays call a "breakthrough." Nothing like this had been done before. It is true that a cleric had published a book in 1700, purporting to describe a way of measuring light; 12 but its principles were entirely unsound. The initial achievement of Bouguer was to realize that the eye could be used not as a meter but as a null indicator; that is to say that it is able to establish the equality of brightness of two adjacent surfaces. He then made use of the law of inverse squares first clearly enunciated by Kepler. 13 After this initial success he devised other means of attenuating light in measurable ratios, and the first half of the Essai (and Book I of the Traite) describes these in detail, with numerous applications. The last part of the Essai deals with the second of Bouguer's great optical generalizations; he showed that in a medium of uniform transparency, the light remaining in a collimated beam is an exponential function of its path in the medium. The third book of the Traite, 8 P. Bouguer, Essai d'Optique sur la gradation de la lumiere (Paris, 1729), 5th page of preface (unnumbered). UJ. J. d'Ortous de Mairan, Mem. Acad. R. des Sci., Paris, 1721, pp. 8-17. 10 [Anon.) Hist. Acad. R. des Sci., Paris, 1726, pp. 11-13. 11 Photometry (2nd ed.; London: Constable & Company Ltd., 1953), p. 11, n. 1. 12 R. P. Fran~ois-Marie, Nouvelle decouverte sur la lumiere, pour la mesurer et en compter !es degres (Paris, 1700). See below, p. 47. 13Johann Kepler (1571-1630), Ad Vitellionem Paralipomena, quibus astronomiae pars optica traditur (Frankfurt, 1604), p. 10, quoted by Ernst Mach, The Principles of Physical Optics, J. S. Anderson & A. F . A. Young, trans. (London: Methuen & Co. Ltd., 1926), p. 13.

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as far as page [336], is largely but not entirely copied from the last part of the Essai. 14 The similarities and differences between the Essai and the Traite are of great interest. The latter is far from being merely a revised version of the former. In Book I, new methods of measuring light are introduced, and a new instrument, the Lucimeter. The sections on the measurement of transmission are greatly enlarged, and there is an entirely new section on the measurement of the light reflected from rough surfaces. Book II is completely new, and occupies 132 pages of the Traite. It deals entirely with reflection. The second part of the Essai starts with seven pages on the mechanism of the transmission of light through material bodies. By the time he came to write the Traite, Bouguer must have felt that this was too speculative, because he condensed it to less than a page, saying in effect that it does not matter to his theory of transmission (which is true). There follows a long passage, beginning on page [230], which is taken almost verbatim from the Essai, with occasional expansions. Finally there is the completely new and important fifth section of the third Book, which closes the Traite. To summarize: large parts of Books I and III of the Traite are derived from the Essai; Book II is new. It is well to make the point that the parts of the Traite which are entirely new were probably written in the last months of Bouguer's life. On November 12, 1757, he read to the Academy a general, rather elementary paper on photometry. 15 Near the end of this he wrote "it is better to postpone these details or to introduce them into a separate work." 16 There is little doubt that he had thought about photometry in the intervals of his many occupations; but the two hundred or so new pages, and particularly Book II, may have been written in a hurry. The existence of the lacunae in Articles VI, VII, and VIII of section two of Book I which de la Caille was unable to fill in, although he tried, 17 makes it fairly clear that the last part of Book I was written very late. Certainly parts of Book II must have been written in Bouguer's last summer, for on page [122] he refers to an observation of the moon "on March 4th, 1758, at 4 o'clock in the morning," when such a sick man ought to have been in bed. There is also internal The original pagination is indicated by figures enclosed in brackets. P. Bouguer, Mem. Acad. R. des Sci., Paris, 1757, pp. 1-23. 16"Il vaut mieux .. . renvoyer ces details a une autre fois ou !es faire entrer dans un ouvrage donne a part," ibid., p. 23. I 7See de la Caille's preface. 14

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evidence that Bouguer meant Book II to be Book III, which would indeed be more appropriate. 18 Of those parts of the Traite that are not represented in the Essai the most valuable is the last section, on the light reflected (i.e., scattered) by the air when it is illuminated by the sun. I have shown elsewhere 19 that Bouguer (and later Lambert) gave an adequate account of the photometric theory of the horizontal visual range, more than a century and a half before its re-discovery in the twentieth century. It would be interesting to know the date of Bouguer's actual discovery. It was an intellectual achievement, or perhaps we should say a display of intuition, of no mean order. Book II of the Traite, or at least the large part of it which deals with reflection from rough surfaces (including the planets), is a horse of another colour. It starts well (p. (162]) with an elegant and simple method of goniophotometry. Unfortunately Bouguer then chose to assume that any surface could be analysed into an assemblage of tiny mirrors set at all angles, and that if he could find the total area of these supposed mirrors facing in each direction, he could calculate the way in which light is reflected from any object. We may note that people have made this assumption at fairly frequent intervals during the succeeding two hundred years, with scarcely any more success. 20 In the last section of Book III, referred to above, Bouguer even treats the air molecules as little mirrors, fortunately without destroying his main result, but not without making his argument harder to understand. It is fairly certain from this that the last part of Book II I was at least put in its final form near the end of his life. It is inevitable that we should compare and contrast Bouguer and Lambert. There has unfortunately been a conscious or unconscious attempt by German authors to make a sort of national hero out of Lambert, crediting him with the exponential law of attenuation, and even with the law of inverse squares, 21 and entirely ignoring Bouguer or at least playing him down. E. Anding, 22 the translator of Lambert's great work on photometry, 23 tries hard but without much success to be quite impartial in his very able and extended Anmerkungen. 18An erratum on the unnumbered page before page [l), and the error referred to in n. 48, p. 141. 1 9W. E. K. Middleton, Isis, LI (1960), 145-9. 2osee n. 31, page 112. 21 E. Raskop, Handbuch fiir den Augenoptiker (Stuttgart: Deva-Fachverlag, 1959), p. 5. 22Lamberts Photometrie, deutsch herausgegeben von E. Anding (Leipzig, 1892), Remarks, p. 62. 23Johann Heinrich Lambert, Photometria sive de mensura et gradibus luminis, colorum, et umbrae (Augsburg, 1760).

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In the Photometria, Lambert makes several graceful references to Bouguer, beginning in the preface (leaf 2, verso). In article 315 he refers to the Essai as the Traite, and records that "last year, on his deathbed" Bouguer had finished another edition, which, however, he (Lambert) has not seen. There are also references to Bouguer in articles 360, 468, 475, 556, 557, 865, 885, 910, 1030, 1048, and 1072. But Lambert nowhere makes it clear that Bouguer actually was the discoverer of a practical method of photometry. It may be wondered whether Lambert ever actually read the Traite, even by 1774 when he published a theory of the horizontal visual range which is effectively the same as that on pages (360-3]. 24 Lambert does not here mention Bouguer's work, and we may reasonably surmise that he had not read the Traite, especially as a casual examination of it might lead one to the erroneous conclusion that only Book II is new. Even Anding, on page 61 of his Anmerkungen, shows that he has not noticed the important new parts of the third Book; so perhaps Lambert did not, either. Lambert's scientific method was very different from that of Bouguer. Bouguer was first and foremost an experimenter, in the tradition of Boyle and Hooke; and when he did apply mathematics to his experiments, he preferred geometry to analysis. He never tried to set up purely theoretical structures that were not based on any experimental data. The experiments described in the Essai and the Traite are simple and elegant, and the conclusions he draws from them (at least in Books I and III) are almost never too ambitious. Lambert, on the other hand, was an extremely able mathematician, to whom it was entirely fitting that all phenomena should at once be subjected to mathematical analysis. His instinct was to develop theory as far as possible, often on the basis of very little experiment. There is a very illuminating remark in the preface to de Saussure's great book on the hair hygrometer: "Le celebre Lambert ... ce grand geometre, considerant cet objet [i.e., the hygrometer] sous son point de vue favori, semble s'etre occupe du soin de tracer geometriquement la marche de l'hygrometre.... plutot que de l'hygrometrie proprement dite." 26 De Saussure was an even greater experimenter than Bouguer, and his opinion of Lambert is of some interest. It might not be unfair to say that Lambert was more interested in the mathematics of photometry than in photometry itself. Nevertheless he performed the essential service of developing a system of photometric units and 24 Nouveaux Memoires de l' Academie Royale des Sciences et Belles-Lettres, Berlin, 1774, pp. 74-80. 26H. B. de Saussure, Essais sur l'hygrometrie (Neuchiitel, 1783), p. IX.

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setting down the laws of photometry, or at least of radiometry, in a clear and concise form. The mention of photometry and radiometry in the same sentence leads naturally to the remark that in the year 1960 photometry is done entirely on the basis of conventions internationally agreed upon, and has very little relation to the sensations which the radiation being measured produces in the consciousness of the observer. Indeed, photometrists seldom use the eye at all, and are really doing radiometry and calling it photometry. They realize this; they know the practical reasons for it; it worries almost nobody. 26 The interesting thing is that photometry was conventional from its very inception. Bouguer clearly stated (p. [12]) that only the ratios between lights could be measured, not their absolute values; he was aware that lights of different colours are hard to decide about (p. [50]), but had no idea of making corrections of any kind because of this. And it may be suspected from his discussion of de la Hire's experiment with the great mirror and the thermometer (p. [89]) that he would have accepted physical radiometry as a procedure equivalent to photometry if the former had been technically possible at the time. It is clear that for Bouguer light was a physical something (corpuscles, he thought) entering the eye, and he assumed (p. [44] and elsewhere) that equal amounts of it produced equal sensations on adjacent parts of the retina. Having assumed this, he no longer had to think of the sensations; he was comparing amounts of light. As far as the comparison of lights of the same relative spectral radiance (and hence colour) is concerned, the photometrist of today adopts exactly this view. Twentieth-century photometry-apart from its technical resources-differs only in being much more sophisticated about the comparison of lights of different colours. It is not easy to determine the extent to which Bouguer was clear in his own mind about the relationships between what we should now call luminous intensity, illumination, and brightness. No one who has tried to make these matters clear to university students will be surprised if, at the very birth of photometry, they were somewhat obscure. It is certain that Bouguer had the idea of luminous intensity clearly in mind (see, for example, pp. [8-9]). This is the simplest of the three concepts. He also realized the necessity of distinguishing between this and brightness (eclat). At times he writes of illumination (eclairement), but usually at a point rather than of a surface. Unfortunately he uses the word intensite at one time or another for both brightness 26 lt would not be fair to fail to mention that Dr. Vasco Ronchi (Storia delle luce, p. 219) opposes the common view with great erudition and much fervour.

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(p. [186]) and illumination (p. [115]); or alternatively the word force. Illumination is also referred to simply as lumiere 27 (p. [10] and passim); but the same word on page [8] clearly means luminous intensity. In translating I have not tried to modernize this terminology in any way, fearing to give a misleading impression. Similarly, it seems best to translate the word gradation by "gradation," although the first part of the book is devoted to the measurement of light. A modern dictionary gives "gradation, gradual process." No help could be had from two of the largest "mathematical dictionaries" of the period, the Dictionnaire encyclopaedique des mathematiques of d'Alembert, Bossut, de la Lande, de Condorcet, et al. (Paris, 1789) and the Dictionnaire universel de mathematique et de physique of Saverien (Paris, 1753), as to what Bouguer intended the word to connote, at least in the title. But on page [34] he uses it in the sense of "attenuation." The Traite d'Optique was translated into Latin at Vienna within two years of its appearance. The title-page of this translation reads as follows: D. BOVGVERI ACADEMIAE SCIENTIARUM REGIAE PARISINAE & C. MEMBRI OPTICE DE DIVERSIS LVMINIS GRADIBVS DIMETIENDIS OPVS POSTHVMVM IN LATINVM CONVERSVM A

IOACHIMO RICHTENBVRG SOCIETATIS IESV MARIAE THERESIAE AVGVSTAE HONORIBVS DICATVM CVM SVB AVGVSTISSIMIS EIVSDEM AVSPICIIS IN ANTIQVISSIMA, AC CELEBERRIMA UNIVERSITATE VINDOBONENSI PHILOSOPHIAM VNIVERSAM PROPVGNARENT D. IOSEPHVS TOBENZ ET D. DANIEL TOBENZ MENSE AVGUSTO ANNO M.D. CCLXII. DIE VINDOBONAE TYPIS IOANNIS THOMAE TRATTNER, CAES. REG. ET APOST. MAI. AVL. NEC NON INCL. ORDINVM INFER. AVSTR. TYPOGR. ET BIBLIOP. 28 This word is without an accent throughout the Traite. lone "U" in scientiarum is from a different font. Note also that there is an unfilled space after "die." 27

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A brief examination of this translation reveals nothing of special interest; it seems direct and literal. The mathematics have been copied fairly well, but not checked, or at least the errors in the original have not been discovered. Of more interest is a sentence in the translator's preface (signature A2, verso), which says: "The most distinguished Lambert ... worked at these fruitful matters and indeed rivalled [Bouguer], although in his Photometria he freely confesses that he had not seen this so very celebrated posthumous work at all." Lambert does say that he has seen the Essai of 1729, which he calls the Traite. What is clear is that both works had made an impression at Vienna, and that in a good deal of the Teutonic world (though not in Berlin, where the Court was francophile) Latin was more accessible than French at the time. In 1950, a Russian translation was published. 29 I shall conclude with a few notes on the Abbe Nicolas Louis de la Caille, the friend of Bouguer who saw the Traite through the press. 30 De la Caille was born on March 15, 1713, at Rumigny and died at Paris on March 21, 1762. Like Bouguer, he was greatly encouraged by his father, who died when Nicolas was only eighteen, leaving him very inadequately provided for; but he was protected by the Duke of Bourbon. He developed a strong interest in astronomy, and in 1736 he was introduced to Cassini, and given a place at the Observatory of Paris. In 1738 he succeeded Varignon as Professor of Mathematics at the College Mazarin, and discharged his duties with great success, continuing to study astronomy in his spare time. In 1741 the Academy made him Associate Astronomer, and from then until his death he produced a long series of astronomical papers. From 1750 to 1754 he observed at the Cape of Good Hope. His death d'une fievre maligne was probably hastened by fatigue brought on by his enormous assiduity in observing. He seems to have been the model of the devoted and disinterested scientist. 29 Pierre Bouguer, Opticheskii trac/at o gradatsii sveta, trans. N. A. Tolstoi and P. P. Feofilov, ed. with a commentary by A. A. Gershun (Moscow: Akad. Nauk S.S.S.R., 1950). 30See [Grandjean de Fouchy], "Eloge de M . L'Abbe de la Caille," Hist. Acad. R. des Sci., Paris, 1762, pp. 197-212.

Contents INTRODUCTION

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PUBLISHER'S NOTE

4

EXTRACTS FROM THE REGISTERS OF THE ROYAL ACADEMY OF

7

SCIENCES BOUGUER'S CONTENTS

9

BOOK I

19

BOOK II

75

BOOK III

153

APPENDIX A:

Units of Length Used by Bouguer

245

APPENDIX B :

Mathematical Notation of the Traite

246

APPENDIX

and II

c: Problems Dealt with in Book III, Sections I 248

OPTICAL TREATISE ON THE GRADATION OF LIGHT

TRAITE

D'O PT IQ U E SUR

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GRADATION DE LALUMIERE: Ouvrage pojlhume de M. Bo u cu ER , de I' Academie Royale des Sciences, C,Cc. ET

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Pour flrvir de Suite aux Memoires de l'Acadlmie Royale des Sciences.

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OPTICAL TREATISE ON THE GRADATI ON OF LIGHT Posthumous work of Mr. PIERRE BOUGUER of the Royal Academy of Sciences, etc., and published by the Abbe de la Caille of the same Academy, etc. TO SERVE AS A SUPPLEMENT TO THE MEMOIRS OF THE ROYAL ACADEMY OF SCIENCES

AT PARIS, FROM THE PRESS OF H. L. GUERIN & L. F. DELATOUR ST. JAMES ST., AT ST. THOMAS AQUINAS

1760 WITH THE ROYAL APPROVAL AND LICENCE

[iii]1

PUBLISHER'S NOTE

Tms WORK first appeared as a duodecimo volume in 1729. 2 It was received with general applause by geometricians and physicists, and confirmed the reputation which its author had already earned by several excellent separate papers, and which he was to sustain to such a high degree. Mr. Bouguer had realized how important it was to extend his researches on light; but being obliged by circumstances to reside in a little town in Brittany, he was able to procure almost none of the things needed for success in delicate experiments. We know how much those which it is proposed to make on this subject need, among other things, complexity and precision in the instruments employed. It was thus almost impossible for him to perfect the physical part of his Essay, although this task was extremely close to his heart, because, in a way, he had made it his own. [iv] The Royal Academy of Sciences had hastened to attach to itself a person so capable of doing it honour, and Mr. Bouguer had moved from Croisic to Havre-de-Grace; but although now somewhat closer to the capital, he was scarcely in a better position to retouch his favorite work. Then the long and arduous voyage to Peru which he was induced to undertake, and the commission to give a public account of it, which the Academy gave him on his return; that given him by the Minister, which kept him working incessantly on the details of every part of the Naval Service-all this occupied him for more than twenty years. Finally, freed from the engagements which obliged him to occupy so usefully much time which he felt he could not spare, and given back to himself again, so to speak, he hastened to pick up the thread of his old ideas on the gradation of light, and passed the last two years of his life in preparing to write a new work on the same subject, but which would no longer need to be called an Essay. After making all the necessary experiments and calculations, Mr. Bouguer was busy writing them out when he was attacked by a malady The original pagination is shown in brackets throughout. Bouguer, Essai d'Optique sur la gradation de la lumiere. (Paris, 1729). This was reprinted by Gauthier-Villars of Paris in 1921 in the series "Les Maltres de la Pensee Scientifique." 1

2P.

PUBLISHER'S NOTE

5

which made him ill for several months, but could not oblige him to interrupt his labour; even when forced [v] by a clear case of dropsy to stay in bed, it was impossible to keep him from working at his book for some hours each day. In the end he had the satisfaction of seeing a fair copy of the last pages, a few days before his death. 3 The Royal Academy of Sciences, regretting the premature loss of such a distinguished Member, took up his book. He had actually read several parts of it at the meetings of that Society ; he had done me the honour of leaving me the work of publishing it. Death had not left him time to add a preface such as he alone could have done properly. Since I cannot supply the lack in the way that I should like to do, I shall be content to give here a short account of what it is necessary for the reader to know about the book. In the preface to his Essay, Mr. Bouguer brought out the fact that there was one entire part of the subject of optics which had scarcely been touched on at the period in which he wrote. This was the part which dealt with the intensity or brightness of light; and the necessity of developing this had been felt and remarked upon several times. [vi] Mr. Bouguer was induced to work on the subject by reading a memoir by Mr. de Mairan, which appeared in the volume of the Academy for the year 1721, and in which the ratio of the values of the light from the sun at the two solstices was assumed to be known. 4 When he tried to determine this ratio by experiment, Mr. Bouguer's studies led him to a rather complete theory of this part of optics. They suggested to him new ways of making a large number of other investigations in this field , and finally furnished him with the materials for his Optical Essay. Mr. Bouguer did not conceal the fact that there was cause to fear that his determinations might not be entirely accurate, no matter what care and what checks he employed; and it was to rid himself of these scruples that he later made all the new experiments that will be found in the book now being published. But a fear of still not having been able to give all his results the precision of which they are capable prevented his making up his mind on some of them. One may note in the second section of the first Book, some small lacunae that I have not been able to fill up. I had it from Mr. Bouguer himself that he had left these in the copy of his work, because he was not entirely [vii] satisfied with some numbers which he had obtained in his new experiments. He hoped in vain to be able to re-commence these in the 3Note by de la Caille: Mr. Bouguer died on August 15, 1758. •See n. 77, p. 66.

6

BOUGUER'S OPTICAL TREATISE ON LIGHT

summer of the year 1759. Nevertheless I should have liked to be able to give these numbers here, even if they were a little uncertain, but I had in my possession only the copy which I have just mentioned. I did everything possible to recover the original papers on which this was written, hoping to find these data; but I succeeded in finding only a few scraps with nothing pertinent upon them. The papers and instruments of Mr. Bouguer had been sold or dispersed during the vacation of the Academy, at a time when I was absent. I did not feel that I should therefore suppress the articles which contain these lacunae, because the experimental procedures are described there, so that physicists may follow them, and fill in the empty spaces. The reader will excuse these in a posthumous work, in which I have permitted myself no sort of amendment. Although the manuscript which I have used was not entirely correct in its algebraic expressions, not being by the hand of Mr. Bouguer, I hope that the reader will have reason to be satisfied with the accuracy of this publication. I have gone over everything with care, [viii] and have made an equally close check of all the calculations. I owed this duty to the friendship with which the author honoured me, and to the confidence which he placed in me, especially on the subject of this work.

Extracts from the Registers of the Royal Academy of Sciences,]uly 26th, 1758, and March 31st, 1759 MR. CLAIRAUT AND THE ABBE DE LA CAILLE were appointed to examine a book written by Mr. Bouguer and entitled: Optical Treatise on the gradation of light. On the strength of their report, the Academy declared the book worth publishing under the title of Supplement to the Transactions of the Academy. In testimony whereof I have signed this certificate. At Paris, on the 27th of June 1760. (signed) GRANDJEAN DE FOUCHY Perpetual Secretary of the Royal Academy of Sciences

ROYAL LICENCE LOUIS, BY THE GRACE OF Goo, KING OF FRANCE AND NAVARRE: To our beloved and trusty Counsellors, Gentlemen keeping our Courts of Parliament, Ordinary Request Masters of our Household, Great Council, Provost of Paris, Bailiffs, Seneschals, their civil lieutenants and our other justicers concerned: GREETINGS! Our beloved Members of the Royal Academy of Sciences, in our dear city of Paris, had representations made to us to the effect that they needed our Letters of privilege for the printing of their works: For these reasons and wishing to please the Deponents we allow them by this Privilege to entrust any Printer they wish with the publication of all daily Observations or Researches, or annual Reports on all that will have taken place during the meetings of the aforesaid Royal Academy of Sciences, Books, Briefs, or Treatises prepared by members of the Academy, and generally all that the Academy will want to publish. However the aforesaid works will have first to be examined and declared worth publishing. The Deponents will decide together or separately on the number of volumes, sizes, margins, types. They will have as many printings made as they will want and will be allowed to have their books sold and distributed in all our Kingdom for a period of twenty consecutive years starting on the day when this Privilege is signed. No other books than the abovedescribed ones for the Academy shall be published under these terms. Nobody, whatever his quality or rank, will be permitted to introduce foreign editions anywhere under our jurisdiction. On the other hand no bookseller or printer shall publish, distribute or sell the aforesaid works, wholly or in part, make translations of them or publish extracts from them, on any account, without a clear and written authorization from the aforesaid Deponents or from those who are appointed by them, under penalty of confiscating the counterfeited copies, of fining each offender in the amount of three thousand pounds of which

8

BOUGUER'S OPTICAL TREATISE ON LIGHT

one-third for Us, one-third for the Hotel-Dieu of Paris and one-third for the aforesaid Deponents or any one appointed by them, and of paying all costs and damages; on condition that this Privilege shall be recorded at full length on the Register of the Corporation of Booksellers and Printers in Paris within three months of the date of issue of this Privilege, that the aforesaid works will be printed in our Kingdom and nowhere else, on good paper and with beautiful type in compliance with publishing Regulations; and that before offering them for sale the manuscripts or printed documents having been used as proofs for the printing of the aforesaid works will be handed over to our very dear and trustworthy Knight, Sir Daguesseau, Chancellor of France, Commander of our Orders, who will then have two copies placed in our public Library, one in the library of our Chiteau of the Louvre and one in the library of our aforesaid very dear and trustworthy Knight, Sir Daguesseau, Chancellor of France, failing which this Privilege shall be cancelled. We require that the full contents of this Letter shall be to the full and peaceful benefit of the aforesaid Deponents and their executors, without tolerating their enduring any trouble or hindrance. We wish the full copy of this Letter to appear either at the beginning or at the end of the aforesaid works. All copies shall be duly checked and certified "true copies" by one of our beloved and trustworthy Counsellors and Secretaries. We, first of all, command our Gentleman-Usher or Sergeant to put this Letter of Privilege into effect by taking all necessary steps, without asking for any other authorization and notwithstanding hue and cry, Norman Charter and letters to the contrary. Such is our pleasure. Issued in Paris on the nineteenth day of the month of March in the year of grace one thousand seven hundred and fifty and the thirty-fifth year of our Reign. By the King in his Council.

Signed, MOL. Recorded on Register XII of the Royal Union of Booksellers and Printers in Paris, No. 430, Folio 309, according to Article 4 of the 1723 Regulation which forbids people of all ranks, other than Booksellers and Printers, to sell, distribute or advertise any books for sale whether they claim to be the authors or otherwise. Eight copies of each published book will be supplied to the above-named Union as is prescribed by Article 108 of the same Regulation. At Paris, on the fifth of June, 1750. Signed, LE GRAS, Syndic.

CONTENTS [iii]

PUBLISHER'S NOTE BOOK ONE VARIOUS METHODS OF MEASURING LIGHT, WITH SEVERAL APPLICA· TIONS OF THESE

[l]

19

Means of finding the ratio between the intensities of two different lights

[3]

20

ARTICLE I. Use of candles or lamps to measure light Use of lenses and burning glasses to measure light ART. u. Details of the precautions which must be taken in comparing two lights ART. III. The means of deriving from a single lamp or candle, in several experiments, the lights which are to be compared Second arrangement of the candle when it is desired to render the light extremely oblique To determine how much light is weakened in passing through a transparent body ART. IV. Second way of deriving the lights to be compared from a single flame or candle ART. v. Solution of most of the preceding problems by the introduction of some portion of the daylight into a dark room ART. VI. Construction of an instrument suitable for measuring the intensity of light, which enjoys some of the properties of a dark room ART. vu. The second way of using lenses or telescope objectives in the measurement of light Description of another instrument for measuring light Reflections on the heliometer and on the inventor of this instrument ART. vm. In using the instrument represented in Figs. 10 and 11 the intensity of the light is directly measured and not its total amount ART. IX. Comparison of the preceding means of measuring light with those which have been proposed by different authors On the method of Mr. Huyghens On that of Father Fran~ois-Marie, Capucin On that of Mr. Celsius

[3] [6]

20 21

[8]

22

[16]

27

[19]

29

[21]

31

[22]

32

[27]

35

[31)

38

[35] [35)

40 40

(37]

42

(39]

43

[43] [45) [46] [48]

45 46 47 47

SECTION ONE

10

BOUGUER'S OPTICAL TREATISE ON LIGHT

SECTION Two Applications of the preceding means of measuring light to the solution of several optical problems

ART. I. Observations made to determine what intensity the light must have in order that it may cause a feebler one to disappear ART. II. Experiments on reflection by mirrors ART. 111. Observations on the transparency of glass and plate glass ART. IV. Experiments on the transparency of sea water ART. v. Observations made on the light reflected by rough surfaces ART. VI. Observations on the quantity of light reflected to us by each part of the sky or atmosphere ART. VII. Observations made to discover how much more illumination one large portion of the sky gives than another ART. VIII. To find how much stronger the light of the sun is than that sent to us by a large portion of the visible sky, or how much stronger it is than the shadows of objects exposed to broad daylight ART. IX. To find how much the light of celestial bodies increases or decreases with the changes in their altitudes above the horizon ART. x. Use of the preceding observations to discover the transparency of a certain thickness of air ART. XI. Observations made to determine how much stronger the light of the sun is than that of the full moon Why the rays of the moon collected at the focus of a large burning glass give no indication of heat ART. XII. Observations made to determine by how much the parts of the sun near its centre are more luminous than those which are near the edges of this body

(51]

50

(51] (56]

50 53

(58] (61]

54 55

(65]

58

(70]

60

(73]

62

(75]

62

(79]

65

(82]

66

(85]

68

(88]

69

(90]

70

(97)

75

[98)

76

BOOK TWO RESEARCHES ON THE QUANTITY OF LIGHT REFLECTED BY SURFACES BOTH POLISHED AND ROUGH SECTION ONE Remarks on the reflection caused by perfectly polished surfaces which extinguish no rays

ART. I. On the virtual focus in reflection by plane surfaces and on (98) the intensity of the light which they reflect (99) ART. II. On the virtual focus in reflection from curved lines ART. III. On double virtual foci by reflection from curved surfaces (100) Thoughts on the uncertainty of certain astronomical (103] observations

76 76

78

79

11

CONTENTS

On the intensity of light reflected by an infinitely polished convex spherical surface ART. v. Showing that the light reflected in every direction by the globe is of exactly the same intensity when we receive it at a very great distance and when the luminous point is also very far away from the globe ART. vr. On the quantity of light reflected by surfaces on which perfectly polished small hemispheres take the place of wrinkles or roughness ART. vu. On the light reflected by a globe the surface of which is covered by an infinite number of perfectly polished small hemispheres ART. VIII. On the light which would be sent to us by the planets, and particularly the moon, on the two preceding hypotheses, namely (1) supposing their surfaces were perfectly polished, (2) that they were covered with an infinite number of perfectly polished small hemispheres ART. IV.

Two Researches on the quantity of light effectively reflected from physical bodies of which the surfaces are polished ART. I. Showing that in reflection there is always a part of the light which is as if extinguished ART. II. Showing that the number of rays reflected at different inclinations has not a constant ratio with the number of incident rays Difficulties in making these experiments on the surface of quicksilver ART. m. On the different intensities of reflection produced by the surface of water according to the different angles of incidence of the light ART. IV. Construction of a table which indicates the quantities of light reflected by water at all the different inclinations Table of the amounts of light reflected by the surface of water ART. v. Construction of a table of the quantities of light reflected by the plate glass of which mirrors are made Table of the reflections produced by plate glass ART. VI. Comparison of the light reflected by the surfaces of water and of mercury when these two liquids are contained in the same vessel ART. vu. On the quantity of light reflected at the smaller inclinations by the interior surface of transparent bodies when the light presents itself to emerge from them ART. vm. On the quantity of light reflected by the internal surface at the largest angles of incidence ART. IX. On the light absorbed or extinguished by the surfaces of transparent bodies independently of reflection

[105)

81

(107)

82

(111)

84

[117)

88

(118)

88

(123)

91

(123)

91

(125)

92

(127)

93

(131)

95

(134)

97

(136)

97

[136) (137)

98 98

[138)

99

[144)

102

(150)

106

[156)

109

SECTION

12

BOUGUER'S OPTICAL TREATISE ON LIGHT

SECTION THREE

On the reflection of light by mat or rough surfaces Showing that small polished hemispheres will not suffice for the calculation of the intensity of the light reflected by mat surfaces ART. II. A means of observing the various intensities of light reflected by surfaces looked at from the direction in which they are illuminated Table of the brightness of the surface of frosted silver, fine plaster, and Dutch paper, or of the intensity of light reflected by these surfaces ART. III. On the distribution of the small asperities, deduced from the preceding experiments Table of the ratios of the asperities of surfaces of frosted silver, plaster, and Dutch paper ART. IV. On the expression of the small asperities by the ordinates of a curved line which we shall call the numerator of asperities ART. v. On the intensity of the light or the colour of rough surfaces when they are illuminated and seen in two absolutely different directions: important theorem ART. VI. Corollaries of the preceding theorem The position of the eye being given, to find the direction in which the luminous body must be put in order that the surface shall be as bright as possible Showing that a surface may preserve exactly the same intensity of light whatever situation is given to it in relation to the incident and visual rays ART. VII. The situation of a luminous body being given with reference to a mat surface, to find the direction in which the eye must be placed in order that the intensity of the reflected light may be as great as possible The angle made by the incident rays with the visual rays being given, to find the situation which a mat surface must have in order that its brightness or the intensity of its colour may be at a maximum ART. vm. On the intensity of the light or of the colour of a rough surface when the plane which passes through its centre, the luminous body, and the eye is not perpendicular to this surface ART. IX. On the numerator of asperities in the planets ART. x. To infer from the numerator of asperities, once determined, how much more or less brightness the different points of a planet should have in all its different situations with reference to the sun ART. XI. Showing that on the surface of the planet there are always an infinite number of points which are equally bright Some circumstances in which all the points of the planet which are equally illuminated are situated on elliptical arcs

(161)

112

[161)

112

[162]

113

[165]

114

[165]

115

[168]

116

[169]

116

[174) [177)

120 121

[178]

122

[179)

123

[183)

125

(184)

126

[186) (189)

128 129

[192)

130

(195]

132

[196]

133

ART. I.

13

CONTENTS

SECTION FOUR On the absolute magnitude of the small faces of the same inclination in rough surfaces, and on the total quantity of light that they reflect (198]

ART. I. In rough surfaces, to determine the ratio which exists between their total extent and that of the little faces at each inclination ART. 11. Application of the preceding solution to plates of frosted silver and to surfaces of plaster, and the precise determination of the quantities of light absorbed by these two surfaces Table of values of a4z/2J zsds, or the ordinates CK of Fig. 29 raised to the second power, one-quarter of which expresses the quantity of light reflected by plaster and by Dutch paper, supposing that no rays are extinguished ART. III. Application of the general solution of Article I to the numerator of asperities which appears to apply best to several planets ART. IV. On the totalamountoflightthattheplanetsreflecttous in their different phases ART. v. To find what part of the light which they receive from the sun the planets will send to us when they are in opposition, supposing that no rays are extinguished Showing that of 300,000 rays of the sun at least 172,000 are extinguished on the surface of the moon ART. VI. First elucidation. Do the little faces of the asperities send back as parallel the parallel rays which they receive, or do they reflect them in the manner of convex surfaces? ART. VII. Second elucidation. We enquire what becomes of the rays which are reflected from one little asperity to the other

135

(199]

135

(205]

138

(208]

141

(210]

142

(212]

143

(218]

146

(222]

149

(222]

149

(226]

151

(229]

153

On the law according to which light is diminished in passing through dijferent thicknesses of a diaphanous body (230]

154

BOOK THREE RESEARCHES ON THE TRANSPARENCY AND OPACITY OF BODIES SECTION ONE

ART. I. Showing that light does not decrease according to the terms of an arithmetic progression when going through homogeneous and equally thick layers of a diaphanous body ART. II. Showing that when the thickness of the medium increases by equal quantities the light diminishes according to the terms of a geometric progression ART. III. On the use of the logarithmic curve to represent the weakening of light On the curve which will be called the gradulucic in what follows

(230]

154

[231]

155

(233]

156

(235]

157

14

BOUGUER'S OPTICAL TREATISE ON LIGHT

ART. IV. On what ratio we should make the specific transparency (237) of media depend ART. v. Showing that all that has just been established can also be (241) applied to the bodies which we call opaque ART. VI. Showing that if, when light goes perpendicularly through a diaphanous body, one is obliged to take note of the alterations which it suffers in meeting the two surfaces, one may entirely (243] neglect the effect of second and third reflections, etc.

159 160

161

SECTION Two Method of calculating the intensities of light passing through different thicknesses of transparent bodies when the rays are sensibly parallel (247]

General principle for the calculation of the intensities of light PROBLEM I. Having found by the methods explained in the First Book what part of the light traverses a certain thickness of a body everywhere equally transparent, to find what part of the light passes through all other thicknesses of the same body PROB. u. Knowing the diminution suffered by light in passing through a certain thickness of a transparent body, to find the thickness which the light must penetrate in the same body in order to suffer any other desired diminution Showing that at a depth of about 311 ft. in sea water, the light of the sun becomes equal to that of the full moon seen at the surface of the earth PROB. III. Knowing by the methods explained in the First Book the diminution suffered by light in passing through a certain thickness of different diaphanous bodies, to find the specific transparency of these bodies Showing that air is about 4,575 times as transparent as sea water PROB. IV. To find the subtangent of the logarithmic curve which belongs to each transparent body PROB. v. Knowing by experiment the diminution suffered by light in passing through a certain thickness of a transparent body, to determine the thickness which must be given to this body to make it become opaque Showing that at a depth of 679 feet sea water has lost its transparency

164

(248]

165

(250]

166

(255)

168

(256]

169

(257)

169

(258)

170

(258)

170

(261]

172

(264)

173

SECTION THREE Method of calculating the intensities of light when the luminous body is not at an infinite distance (265)

174

First Proposition To compare in a general way the different intensities which light has at different distances from a luminous body when this light (272) passes through a body equally transparent throughout (272] Preparatory discussion

178 178

15

CONTENTS

Knowing the distances m and n to the luminous body, to find the ratio pjq of the intensities p and q of the light SOLUT. II. To find at what distances (m and n) we must place ourselves from the luminous body in order that these two distances may be in a certain given ratio (b to c), and that the intensities of the light will also be in a given ratio (p to q) SOLUT. III. To find at what distances m and n from a luminous body one must place oneself when the difference n - m of these distances or the thicknesses of the diaphanous body is known and we desire that the intensities p and q of the light may be in a given ratio SOLUT. IV. Knowing the intensity of the light at a certain distance from a luminous body, to find at what other distance we must place ourselves so that the light may appear stronger or weaker exactly in a certain ratio Construction of the problem by the intersection of two loci In the particular case in which the rays do not suffer any refraction, to find in a simple manner an infinite number of points where the two lights have the required ratio SOLUTION I.

Second Proposition To establish a prescribed ratio between the light received directly from a luminous body and that received at the same distance after it has passed through a transparent medium Explanations SOLUTION 1. The thickness u of a diaphanous body being given, to find at what distance x the luminous body A must be placed in order that the direct light received from it and that which passes through the diaphanous body may be in the ratio p to q SOLUT. 11. The distance x or Ap at which we wish to place ourselves from the luminous body A being given, to find the thickness u or Np which the diaphanous body should have in order that the ratio p to q may exist between the direct light and that which is transmitted by the diaphanous body Third Proposition Two luminous bodies A and B being situated at a certain distance apart in a medium everywhere equally transparent, to find the ratio between their lights when these are received at given points E, e, etc., and, on the other hand, to find at what points E, e, etc., the lights must be received when we wish that there should be a certain given relation between them Preparatory discussion SOLUTION I. To find the ratio of the lights when that of the distances to the two luminous bodies is known Showing that when the two luminous bodies are situated at the two foci of a hyperbola and when the observer is on the circumference of that curve, the ratio between the two lights is free from all logarithms

[275)

180

[277]

181

[279]

183

[281] [283]

184 186

[286]

188

[288) [288)

189 189

[291)

191

[292)

191

[293) [293)

192 192

(295]

194

(296)

194

16

BOUGUER'S OPTICAL TREATISE ON LIGHT

SOLUT. u. To find the ratio of the distances to two luminous bodies when we know the relation between the two lights To determine all the points, E, e, etc., where the intensities (p and u) of the light from the two luminous bodies A and B are in a constant ratio (of c toe) Remarks on the different characteristics of the curve CEe (Figs. 37 and 39) Showing that the curved line CEe (Figs. 37 and 39) has an asymptote only because the medium is not perfectly transparent

(298]

195

(299]

196

(302]

198

(305]

200

On the diminution suffered by light in passing through bodies which have not everywhere the same density (306]

201

SECTION FOUR

Showing that it is always easy to construct the curve of the different densities of a medium when its gradulucic is known Showing that in general we may employ logarithms for drawing the gradulucic when we know the various densities of the medium On the transparency of the atmosphere On the densities of the atmosphere at different heights above sea level Rule for finding the heights of the highest mountains by means of the barometer Remarks on the observations made on the peak of Tenerife On the various masses of air which the rays of the stars have to traverse in order to reach us Table of the masses of air contained in the atmosphere and the intensities which the light from the stars has after having passed through these masses On the various intensities which the light of the stars has at the surface according to whether they are more or less elevated above the horizon Showing that in passing vertically through the atmosphere the light of the stars does not lose one-fifth part of its intensity

(311]

204

(312] (315]

204 206

(315]

206

(320] (322]

210 211

(323]

212

(332]

218

(333]

219

(335]

220

Essay on the quantities of light reflected by the inner parts of transparent bodies, on the various intensities of the aerial colours, etc. (337]

222

SECTION FIVE

I. On the reflection produced by the inner parts of diaphanous bodies when these bodies are everywhere equally illuminated and equally dense in all their parts u. On the intensity of the light from the object, and of that which is reflected by the interior parts of the medium, when this light passes through the medium in order to illuminate them III. Observations on the light reflected by the internal parts of crystal glass IV. Conjectures on the transparency of sea water

(338]

222

(341]

225

(344] (347]

226 228

17

CONTENTS

Explanation of the rays of light that are seen in the sea when the sun is very high v. On the light reflected by the interior parts of the medium when it is not of the same density throughout and is illuminated with different amounts of light VI. Various applications of the formulae of the preceding article to the aerial colours vu. On the aerial colours near the zenith when the sun is on the horizon, and near the horizon when the sun is at the zenith Two different ways in which we observe very distant terrestrial objects Table of the intensities of the aerial colours of terrestrial objects, according to their distance from the observer Showing that it appears that terrestrial objects cannot be seen at a greater distance than 45 leagues of 20 to a degree VIII. Remarks on the different ways of determining the portion of the light reflected by the particles of air Why the shadows at morning and evening are of a very remarkable blue colour

[348)

229

[350) 230 [354] 232

[358] 235 [360]

236

[363]

238

[364]

239

[367] 240

BOOK ONE Various Means of Measuring Light, with Several Applications o__f These WHEN SEVERAL CANDLES are situated almost in line, and we receive on a surface large enough and distant enough the shadow of an opaque object placed [2]1 between the candles and the surface, we notice a sensible difference between the light given by one or two candles and that produced by all the others. We may easily distinguish thirty or forty different degrees of light; the first is found next to the shadow, the second is formed by the rays 2 of the first two candles, and near the fourth or fifth we may distinctly see the tenth, the twentieth, or the thirtieth, with all the intermediate degrees. In this way we obtain a kind of scale, which furnishes most of the values we are likely to need when we undertake to measure light; and we may conclude from this that we shall succeed up to a certain point. We may see very distinctly the difference between 30 and 31 degrees, and even between 39 and 40. It will thus be possible to carry the precision of our determinations as far as that; and we may even propose to do better, as will be seen in what follows. 1Numbers in brackets indicate the paging of the original. (Unless otherwise identi· fied, footnotes throughout the text are those of the translator.) 2Rayons. Bouguer, of course, lived during the vogue of the corpuscular theory of light. It is clear from a passage in the Essai d'Optique (sect. 2, chap. 1) that he thought of "rays" in this way.

[3]

SECTION ONE

Means of Finding the Ratio between the Intensities3 of Two Different Lights ARTICLE I

Use of candles or lamps to measure light 4 as the first of all the ways of measuring light that natural comparison which is offered us by the united brilliance of several candles or lamps. It is true that we shall not succeed in determining the absolute intensity of light; we shall always discover the intensity only in a relative manner; but is it not exactly the same when in applied mathematics we inter-compare quantities which are capable of increase or decrease? Have we any better idea of the magnitude of the fathom 5 or the foot? And do we take the trouble to get to the bottom of the nature of extension in the operations of practical geometry? Here we shall consider light only as it is radiated by luminous bodies, or as it falls on the objects around us. Acting to a greater or lesser extent on our eyes, it affects us more or less strongly; but we shall neglect everything metaphysical [4] in the sensation and, limiting ourselves to the discovery of the relations between its external and accidental causes, think only of its measurement. My readers will see at once that when we take the light of several candles as a unit or common measure, we ought to be extremely careful to choose them all of the same size; we must also place them at the same distance from the surface which they illuminate, and see to it that all their rays fall on it with the same inclination. Now these two last conditions are often very difficult or even impossible to fulfil; WE MAY CONSIDER

3 Forces. The reason for a note on this purely verbal matter is that the word force sometimes means "intensity" in the modern photometrist's sense, i.e., candlepower, but in other places denotes other quantities which are frequently called "intensity" by other physicists, but not nowadays in photometry; for example, flux per unit area. In this section it has the first meaning. However, we shall have no compunction in translating it by "intensity" throughout, to avoid a feeling of anachronism. 4This heading is not in the original in this place; it is taken from the Table of Contents. 6 Toise. The old measure of length in France. (See Appendix A.)

VARIOUS MEANS

OF

MEASURING LIGHT

21

besides the circumstance that we have not always at our disposal several exactly similar candles or lamps, each one necessarily takes up a certain amount of room, so that if we place a certain number side by side, they will never cast an equal illumination on the surface exposed to their light. We may also perhaps regard it as one more objection to their use, that they always give a light which increases in jumps, or by an entire unit each time we increase their number. But all this difficulty is avoided if we make use, not of several candles, but of the decrease or increase in the light from one only, according to whether we take it at a greater or lesser distance [5] from the luminous body; and we have in this way an infinity of gradations, which may be as small as we wish. The property of the rays to extend in straight lines causes them to spread apart in the same proportion that they go away from the source. If we receive them at a distance ten times as great, they will occupy a space ten times as wide and ten times as high. The same quantity of light will thus be distributed over an area one hundred times as great, and each point will be illuminated one hundred times less, or, which comes to the same thing, the light will be one hundred times as feeble. To explain this in a more general manner: the rays always form a pyramid of which the luminous body is the apex, and which has the surface on which the light is received as a base. The more one retreats from the light, the greater the surface over which the same rays spread out. This area or surface increases precisely in the same proportion as the square of the distance to the luminous body; however, as it receives over its entire extent only the same number of rays, it is evident that it receives proportionally less at each point, and that the force or intensity is smaller in precisely the same ratio that the square of the distance to the lamp is greater. 6

[6]

Use of lenses and burning-glasses to measure light

THESE REMARKS may apply equally well to the foci of convex or concave lenses, as well as those of mirrors. After the rays have crossed in such a focus, or have taken directions similar to those they would have had if they had left it, they proceed to get farther and farther apart, and occupy regions which, always getting larger and larger, are proportional to the squares of the distances to the point where the rays crossed, which then takes the place of the luminous body. Thus 6This law was first stated by Kepler, Paralipomena ad Vitellionem (1604), p. 10. See E. Mach, The Principles of Physical Optics, J. S. Anderson & A. F . A. Young, trans. (London: Methuen & Co. Ltd., 1926), p. 13.

22

BOUGUER'S OPTICAL TREATISE ON LIGHT

the use of lenses and burning-glasses provides us with one more easy way to change the light in any desired ratio. To determine this ratio, we need only measure the exact distance to the focus, or, which comes to the same thing, we shall measure the bases of the pyramids formed by the rays in spreading apart. We take the diameters of these bases and raise them to the second power, and these last quantities in the inverse order express the intensities of the light. We must, however, avoid using distances either too great, or too small. If we used them too large, the light would be attenuated not only by divergence but also by meeting the coarse parts [7] of the air which could, in a long path, intercept many rays. It is true that things would have to be taken quite far to make this trouble evident, at least when the weather is sufficiently clear; for a thickness of 200 fathoms of air scarcely absorbs or intercepts more than a hundredth part of the light which passes through it,7 as we shall prove later.8 On the other hand it is necessary to be very careful not to use, in these observations, distances which are too small. This is in order that we may be able to consider the luminous body as a point, or to neglect the dimensions of the focus of the mirror or burning-glass without significant error. Our ordinary table candles have a flame of which the diameter is most frequently half an inch, 9 and the height 1½ to 2 inches; and if, with the idea of making the light stronger, we go away only an inch or two, we run the risk of error in choosing a point in the flame from which to measure the distance. It is not the same if we go back 1 or 2 feet 9 or even more, or if when using lenses of very short focu~, such as an inch or two, we place ourselves at least 10 or 12 inches away. But since we already have different means of increasing or decreasing the intensity of the light, we have only to employ one if another is not available, or to combine them, when one alone is not sufficient. [8]

ARTICLE II

Details of the precautions which must be taken in comparing two lights WE SHALL CLARIFY most of the things which have just been indicated by setting out in greater detail the precautions which we should take, 7This is about twice the value given by modern theory for pure air at ordinary pressure and temperature, with no particles larger than the molecules of the atmospheric gases. Bouguer's value corresponds to a visual range of about 160 km. or less. (For theory, see the translator's Vision Through the Atmosphere (Toronto: University of Toronto Press, 1952). 8See page 66. 9 Pouce; pied. (See Appendix A.)

VARIOUS MEANS OF MEASURING LIGHT

23

if we were to determine how many times as much light is radiated by a flame of a certain size, as by a candle. We should first cause the rays of these two luminous bodies to fall with the same obliquity on two different parts of a surface, which we should remove to a greater or lesser distance from the flame or from the candle, until the two parts of the surface appear equally illuminated. It will then remain only to measure the two distances, and their squares will express the ratio between the absolute intensities of the two luminous bodies. 10 If, for example, we have to approach three times as near to the candle, this is an indication that it radiates nine times fewer rays than the flame, since in order to make the two lights perfectly equal, we are obliged to increase that of the candle nine times, by using a distance three times as small. Or perhaps one of the distances [9] is not exactly one-third of the other ; let it be 6 feet while the other is 15. The squares of these two numbers, 225 and 36, then show us that the flame produces more illumination than the candle in the ratio 225:36, or that its light is 6¼ times as strong.

To receive the two lights that we wish to compare, we may use a simple piece of cardboard ECD (Fig. 1), which is folded at CH in order that we may expose each half of its surface more perpendicularly to each source of light. All this piece of cardboard will be black; but it will have two holes of exactly the same size, say 3 or 4 lines 11 in diameter, over which is placed oiled paper, or two exactly similar pieces of glass, made equally rough or white with emery or sandstone. Oiled paper and ground glass sometimes have a greater transparency than necessary, and I have more often used two pieces of plain paper, 101 t does not appear that Bouguer arrived at any arrangement like the photometric bench so ubiquitous today. 11 Lignes. (See Appendix A.)

24

BOUGUER'S OPTICAL TREATISE ON LIGHT

very fine and white, taken from adjoining parts of the same sheet. I let the two lights fall on these two pieces of paper, and estimated their equality by looking from the back, or sometimes from the front. A second piece of cardboard FC served as a diaphragm, preventing the two lights being confused before illuminating the two little (10) surfaces P and Q. This diaphragm was black on both faces, and joined the first piece of cardboard exactly on the ridge formed by the fold. The same instrument is represented by ECD in Figure 2. 12

~ \

\ Ff!J-2,· \

\

-4r -~i

r4 :\\~';." i \ \ 41\ ii\

l \ \

~p

N

/·

~~

If we suppose that we are not at liberty to change the distances of the two luminous bodies from this instrument enough to render their two lights equal, we need only make use of convex or meniscus lenses to reduce them to equality. It is essential that these lenses be exactly the same, so that we need not fear a lack of transparency in one or the other. The lamp A and the candle B 13 (Fig. 2) being placed at the same distance from the observer or from the instrument ECD, the 12The sort of photometric field thus formed could provide only a very moderate precision. Later (p. 28) Bouguer suggests that it is better if the surfaces P and Q are juxtaposed. 13So in the original. The engraver has evidently reversed them.

VARIOUS MEANS OF MEASURING LIGHT

25

first of these sources ought to radiate much more light than the second, but if we move the lens K away from the observer (provided we do not carry it more than half-way to the lamp) we shall be able to diminish the light from the lamp in a very great ratio, and easily make it equal at P to that of the candle B which, passing through the lens L, finally falls on Q. The two lights form the cones RMS and TNX; but we estimate their intensity only by observing the luminous points P and Q, or just the parts P and Q which are covered with white paper, and we are always very careful to ensure that (11] these points are found exactly on the axes of the luminous cones, and that they form only a very small part of the bases RS and TX. In this way each light is altered in two different ways. They are enfeebled more or less, according to the inverse ratio of the squares of the two distances AK and BL, in reaching the lenses Kand L. Supposing that BL were 10 feet, and AL only 9, the light of the candle would be proportionately more reduced than that of the flame, and the two reductions would be in the ratio 100 to 81. But each light receives a further attenuation which can add a great deal to the first. The rays of the flame, after crossing or being united at the focus M of the lens K, spread apart again and are distributed in all the space RS, while the rays of the candle receive a less considerable diminution in filling only the space TX. We shall find the ratio of these two latter reductions by taking the squares of MP and NQ; but if we wish to avoid various difficulties, and nevertheless neglect nothing essential, we have (as we have already stated) only to take the diameters RS and TX of the lighted spaces, instead of these distances. Although the edges [12] of these spaces fall on the black opaque parts of the instrument, they will always be visible enough for easy measurement of these diameters RS and TX, after we have made the two luminous points P and Q equally bright. If the first diameter is 40 lines, I multiply the square of this, 1,600, by that of AK, 81, and I have 129,600. I perform the same operation for the other light. I multiply the square of BL, 100, by that of the diameter TX of the luminous cone which has N as its apex, and if this diameter is 16 lines, the product will come to 25,600. It is well to remark here that, since we are dealing here only with ratios, it is not absolutely necessary to use the same units to express all these dimensions. Having measured AK in feet, 14 we are obliged to employ the same unit for BL. Next we have expressed in lines the diameter RS of the luminous cone having M as its summit, and we also use lines for the other cone. These operations give 129,600 and 14

Here avec le pied de Roi, reflecting the ancient origin of this unit.

26

BOUGUER'S OPTICAL TREATISE ON LIGHT

25,600 for the changes we have had to make in the two lights in order to make them perfectly equal; the first has had to be diminished 51/16 times as much as the second.1 6This is therefore an indication that the first, considered absolutely, is 51/16 times as intense as the second, on the assumptions which we have made. [13] When we advance the lenses Kand L towards the middle point of the distance from the luminous bodies to the instrument ECD, we increase the light which they receive; but these same lenses then disperse it more by letting it fall at a greater distance from their own foci M and N, and a more considerable reduction results from the combination of these two changes. It is not exactly the middle point that produces the greatest diminution, but a point separated from this by the quantity KM (or LN), which becomes a little greater as we bring the lens nearer the luminous body, as all readers familiar with optics 16 know. In general, the first diminution of the two lights occurs in the ratio of AK 2 to BL2, and the second in the ratio of MP 2 to NQ2, or of the squares of the diameters RS and TX of the two cones of light.17 Thus the double alteration follows the ratio AK 2 X RS2 : BL2 X TX 2 , and since we succeed by this means in rendering the two illuminations received at P and Q equally bright, it follows necessarily that the first source of light radiates more than the second in the same ratio or that the flame A sends out more light than the candle B, in the ratio AK 2 X RS 2 to BL2 X TX 2 • [14] It will always be easy to put this method into practice, and to make it exact it will generally be sufficient to take a certain number of precautions, which it is proper to observe at all times. One should keep out all stray light with the greatest possible care, and the observation will succeed still better if the room is made absolutely dark by hanging it in black. The piece of cardboard CF forms an essential part of the instrument ECD; but I usually place a second piece at F, to prevent the light from the two sources falling obliquely on the two faces of CF. One should always arrange these cards or diaphragms so that they shade one another, in order to avoid all reflections. The two lenses K and L should also be set in two large surfaces or pieces of black cardboard, which will also increase the darkness. 16 In the original "20 1/16 times." The good Abbe de la Caille seems to have failed to find this error. Cinq does sound like vingt. 16L'Optique. This means "geometrical optics," here and elsewhere. 17Since the divergences of the two beams are in general not the same, both these formulations can scarcely be true. On considerations of energy it is easy to see that the second, which Bouguer uses, is the correct one.

VARIOUS MEANS OF MEASURING LIGHT

27

Finally I have noticed that another point, already mentioned, is of no less importance; namely to give exactly the same shape and size to the two parts P and Q which serve as images or pictures, in order to make all circumstances absolutely uniform for the observer who examines them. These two illuminated images cannot be too close together. It is absolutely necessary that one should be able to see them both in the same glance, and if possible it is even well [15] to arrange them in such a way that their edges touch. 18 I have almost always looked at these images through a hole made in a final diaphragm which I use at some distance from the eye, so that it hides all other objects. The observation will be a little more troublesome and at the same time less exact (because less direct) whenever we undertake to compare two lights which do not appear at the same time, or which it would be too difficult to bring together. We must then use a third light as a common measure, and we shall compare it with the others in succession. A candle or a lamp will do for this auxiliary source, but it must be the same in the two comparisons, and it is always well to repeat the experiments so that we may take the mean of the different results. We may sometimes do even better, in order to avoid the accidental changes which a luminous body may suffer over a certain time. Instead of using only one candle or lamp as a common measure, we may use several, say four or five burning together; their united light will be more uniform. 19 In any event, we cannot strive too much to make the observation more direct whenever this is possible; [16] and other procedures that we are about to propose may serve that purpose.

ARTICLE III A means of deriving from a single lamp or candle, in several experiments, the lights which are to be compared

IN THE RESEARCHES with which we are concerned, we often have to determine in what proportion a polished surface, for example a mirror, reflects the light that it receives; or to find the exact degree of transparency of a diaphanous body having a certain thickness. 18The importance of this is, of course, recognized in modern practice. Even a relatively narrow division greatly increases the difficulty of matching the two lights. (See for example Y. Le Grand, Rev. d'Optique XII (1933), 145-59.) 19 1n modern photometry this "substitution method" is the rule rather than the exception, as it reduces or eliminates many instrumental errors. The relative stability of modern sources of light makes all the difference.

28

BOUGUER'S OPTICAL TREATISE ON LIGHT

To solve these problems, we might perhaps be tempted to choose two candles of exactly the same size, or two perfectly similar lamps. We should cause the light from one of these to be reflected, or to pass through the transparent body, and then compare it with the light of the other candle; but it is very easy to give the experiment another form which will give much greater assurance of success. Suppose first that we are dealing with reflection. When we have installed the mirror or reflecting surface in a vertical position (Fig. 3),

we imagine its plane [17] produced to C; and having taken two screens of exactly the same colour or equally white, we place them at equal distances in front of and behind the mirror. We locate these screens exactly parallel to one another, or else set them at equal angles to the straight line joining them. They will be illuminated by the lamp or candle P placed on the same straight line ED. We then seek the point A from where we may see the screen D by reflection in the mirror, at the same time that we see directly, and in the same glance, the other screen E which we may have placed a little above its true position, or a little to one side. It is necessary that the two screens, seen thus by reflection and directly, should appear to form one surface only, or should seem to touch. 20 We must also make their apparent whiteness, 20The engraver has perversely rendered this impossible by bevelling the edges of the mirror.

VARIOUS MEANS OF MEASURING LIGHT

29

or the colours of their light, perfectly equal by moving the candle P forwards or backwards along the straight line ED; after doing this we have only to measure the two distances EP and DP, and the squares of these distances will show the ratio according to which the reflection by the mirror causes the intensity of the light to decrease. Evidently, if the mirror reflected without alteration all the rays that it received, we should have to place the candle P precisely at the mid-point C of the distance between the two screens, so that they might appear [18] equally illuminated, since everything else would be symmetrical. But since much of the intensity of the light is always lost by reflection, we can restore this loss only by bringing the candle P much nearer to the screen D which is seen by reflection. This screen is then more strongly illuminated than the other, in the same ratio that the square of PE is greater than that of PD; and thus this same ratio should express the reduction of the light by reflection at the mirror, since the greater proximity of the candle to the screen D does nothing but re-establish the equality in the tone of the two colours which the reflection had destroyed. Perhaps we need not say that it is always important to shade the eye, but we should not fail to note that the mirror must also be shaded, and receive no other light than that coming from the screen D; for otherwise part of the light spread upon its surface would be added to that from the screen. It is also well to interchange the two screens and repeat the experiment, in order to have no fear of inequality in their colours, between which there might be some slight difference. 21 Furthermore, the procedure has to be changed a good deal if we wish to observe rays of light which form rather small angles with the surface of the mirror, for we should then have to put [19] the two screens D and E at a very small distance from one another, and besides the fact that it might become difficult to put the candle between them, we should be likely to commit errors of too great a magnitude in measuring the two distances.

Second arrangement of the candle, when we wish to render the light extremely oblique It is better in this case to place the candle very close to the mirror; we may put it above the mirror or a little to one side, arranging things by means of various diaphragms so that it neither illuminates the 21This has a wonderfully modern sound. Some equivalent procedure would certainly form part of an experiment of this kind nowadays.

30

BOUGUER'S OPTICAL TREATISE ON LIGHT

mirror nor bothers the eye. It will illuminate the two screens D and E nearly perpendicularly (Fig. 4): the screen D which is seen by reflection in the mirror, and the screen E which is seen directly in the same field of view. But as the brightness of the colour of the first will be enfeebled by reflection, we shall have to move the screen E noticeably farther away to render the tone of the two colours 22 or lights exactly the same. After we have succeeded in doing this, it remains only to measure the distances PE and PD from the candle to the two screens, and we shall have in the squares of these quantities the ratio that we wished to discover. 23

" :

.. _..---~ ~

A

If we were obliged to make the distance PE 60 inches, while PD is 50, the screen D would be more strongly [20] illuminated than the other in the ratio 3600:2500, and the excess of illumination that the screen D would then receive would only serve to replace the loss by reflection. 24 As the screen E stands at a noticeably greater distance from the observer than the other, it appears smaller, and therefore it must be made larger, or else the size of the other must be reduced by diaphragms, until they appear perfectly equal, according to our previous recommendations. This is the only additional care that we are obliged to take because of the greater distance of the screen E, for we need have no fear that its brightness will appear less when it is observed from a greater 22Le ton des deux couleurs. Throughout the work Bouguer uses ton de la couleur as synonymous with vivacite (brightness). Couleur frequently does duty for lumiere. We do not feel that we should attempt to modernize Bouguer's photometric notation. 23These arrangements are still among the best ways of measuring the absolute reflectance of a mirror, possibly with the addition of some optical device to give a better photometric field. A block of magnesium carbonate would be sawn in half and scraped to make two screens of identical high reflectance. The arrangement of Fig. 3 can give very good precision, especially if we can rotate ( or at least reverse) the lamp. 24 Giving a reflectance of about 70 per cent, very near the accepted modern figure for a mercury-backed mirror.

VARIOUS MEANS OF MEASURING LIGHT

31

distance. If for example the observer is three times as far away, he will receive one-ninth as many rays 25 from each point of the object; but on the other hand the object will form a smaller image in the eye, and, the points of this image being closer together in exactly the same ratio, the brightness of the image, or the intensity of its light or its colour will still be the same, 26 For the same reason, two walls which are parallel but at different distances appear equally illuminated by the sun, or equally white, provided that the distances are not great enough to permit the air to impress on one of these objects some slight tint of its colour. If we are ten [21] times as distant from one wall as from the other, we shall receive only a hundredth as much light from each point of the more distant surface; but the same area of the retina corresponds to one hundred times as many of these same points, and consequently the action of the light on the same area at the back of the eye will be equal in both instances. Therefore in our experiment it is sufficient to measure with care the two distances PD and PE, which alone determine the brightness of the two screens, and it is quite unnecessary for the observer to pay aFiy attention to his distance from them. To determine how much light is weakened in passing through a transparent body

IN NEARLY the same way we may find the amount which light is attenuated in passing through a diaphanous body. In Figure 5 the

--•·-

..

•

transparent object, in the form of a rectangular parallelipiped, is indicated at B, and above it we have placed the candle P, which illuminates the screens D and E almost perpendicularly. The first of these screens is seen through the transparent body, the other directly, See n. 2, p. 19. is, of course, a very important law. It begins to break down when the object becomes so distant that it subtends only a few minutes of arc at the eye, for reawns most probably connected with the structure of the nervous system. 26

26This

BOUGUER'S OPTICAL TREATISE ON LIGHT

32

but in the same field of view. The latter would appear more strongly illuminated than the former if it were not taken to a greater distance from the candle. By some slight adjustments we may ensure (22] that the two colours appear equally bright, or that the two objects appear to have the same degree of whiteness, and then we take the squares of the two distances to the candle; if one is double or triple the other, the screen D will be four or nine times as strongly illuminated as the other, which will show that the transparent body weakens the light which traverses it four, or nine, times.

ARTICLE IV Second way of deriving the lights to be compared from a single flame or candle WE MAY ALSO succeed in the preceding experiments by following a somewhat different procedure. We wish to discover to what extent the plane mirror B (Fig. 6) weakens the light which it receives at the angle of incidence PBC; 21 we let this light fall on the screen D, and arrange things so that the screen E, placed at a greater or smaller distance, shall be directly illuminated to the same degree. All readers who know

P

p ;i ·--.. . --•.•

c..........

············r ·-::::......... \

_,,.,/· . . ...., ............ .. -, ______ ~/ .F~. 6'. ·---...

.J.

. A

~-~=:: :.-: .~

•-JD.. ----------- ·------,_, ______''·fE

the first principles of catoptrics will be aware that the rays which fall on the first screen will be directed, after reflection from the mirror, as if they had come from the point p, which is situated as far to the side of and behind the mirror as the candle is in front of it. If the reflection had destroyed none of the light, and if (23) we had moved one of the screens until both were equally illuminated, we should have found an exact equality between the distance pD, or the sum PB BD of the

+

27 Nowadays the angle of incidence is measured from the normal to the surface. In the eighteenth century no rule seems to have become established. (See n. 28, p. 34).

VARIOUS MEANS OF MEASURING LIGHT

33

incident and reflected rays, and the distance PE. But in reality we must always carry the screen E a little farther away because of the imperfect reflection by the mirror, which absorbs a certain number of rays. Therefore we have only to take the squares of the distances pD and PE to obtain the ratio in which the light is weakened by reflection. In this experiment we must take a special precaution. This is to make the rays PE and BD nearly parallel, so that we may observe the two screens almost perpendicularly from some point A at an adequate distance, while they are illuminated almost normally, or else both with the same inclination. As to the greater or smaller distance from the observer to the screens, we have already seen that this makes no difference in the intensity of the light. It will not be difficult to determine the precise degree of transparency of a diaphanous body bounded by two plane parallel faces. We cause the light from a candle to fall on the screen D (Fig. 7) after it has

4.,.....

r~-~~~'::_=;;:1;··:;&7:

·-------::=ce=.jE

traversed this body, which is at B ; and we [24) adjust the distance of the other screen E until it is found to be equally illuminated by direct light. We must, however, consider the fact that the rays which go through the transparent body change their direction; on emerging they are directed as if they had diverged from a point p, which is like a sort of focus, and is separated from P by a distance which always depends on the thickness and the nature of the transparent medium. Thus two changes always occur in the light in its passage through the transparent body; a part of the rays is destroyed or intercepted because the medium is not absolutely clear, and it is this change that we wish to find . On the other hand the intensity of the light is increased because the light is directed as if the luminous body had been brought up top. Now it follows from this that we have only to measure the two distances PE and pD, and take their squares, in order to have the simple ratio according to which the light is weakened by the lack of transparency of the body B. But it remains to determine the point p, or the virtual focus from

34

BOUGUER'S OPTICAL TREATISE ON LIGHT

which the light seems to diverge after it has gone through the diaphanous body. To do this we must examine the route followed by the rays in their whole trajectory, as shown in Figure 8. The ray PABC, which crosses the two parallel faces [25] of the body KL perpendicularly, suffers no refraction. The others, PD and PG, bend so as to approach the perpendicular to the surface when they enter the transparent body, and take directions DE and GH as if they had come from the point Q, which is situated behind the luminous body in such a way

that there is a constant ratio between QA and PA for all rays near the perpendicular. This ratio is the same as that between the sines of the angles of incidence and of refraction, for these angles are measured relative to the perpendicular to the refracting surface, 28 and there is rigorously the same ratio between the lines QD and PD, which are the secants of the complements of these angles, and which do not differ here, as to their length, from the lines QA and PA. 29 Now the rays suffer a second change in direction on reaching E and H; they bend away from the perpendicular and follow the directions EF and HI, exactly parallel to their initial paths PD and PG, but situated as if they had come from the point p. Thus there is the same ratio between DE and Pp as between QD and QP. But instead of QD and QP we may put the sine of the angle of incidence and its excess over the sine of the angle of refraction, and similarly we may substitute the thickness AB for DE. Thus we shall find Pp by the following proportion: the sine of the angle of incidence is to its excess over the sine of the angle of refraction, as AB is to Pp. We are also [26] free to use, instead of these sines, the numbers which express their constant ratio. This ratio is approximately 3:2 for glass 30 and 4:3 for water. We have then this proportion: AB is to Pp as 3 is to 1 for the first of these transparent substances, or as 4 is to 1 for the second. That is to say, the virtual focus pis distant from the luminous body 28See

n. 27, and notice the difference in usage. We have translated this passage literally. Bouguer means that to a first approximation QA = QD, and PD = PA; but that it is QD and PD which are actually in the correct ratio. 3°Crystal. Glass, such as that used for mirrors. 29

VARIOUS MEANS OF MEASURING LIGHT

35

one-third of the thickness of the diaphanous body, if it is glass, and one-quarter of that thickness if it is water. 31 The virtual focus or point p having been determined, it is entirely clear that we must pay attention to it in the experiment represented by Figure 7. The rays from the luminous body fall on the screen D as if they had diverged from the point p, or as if the luminous source had been carried there, and thus we must pay attention to this in the alteration of the light. But it remains for us to show that we need not consider this in the experiment of Figure 5, discussed in the previous Article. If in imagination we compare each point of the screen D to the luminous body of Figure 8, it is true that if we place the eye at C (still in Fig. 8) we shall receive a greater number of rays from each of these points. They will in fact be directed as if [27] they had come from the point p. But we must also consider that all the screen will behave as if brought closer. It is at P, and we shall see it as if it were at p. Thus it will appear under a greater angle; and if we take a certain number of its points, they will occupy a greater area at the back of the eye, which will make a perfect compensation in the intensity or strength of the light. Each point will, so to speak, be more luminous, but these points will appear more widely separated in the same proportion. 32 We thus see that we need pay no attention to the virtual focus in Figure 5, when we look at the object through the transparent body; but it must receive careful attention in Figure 7, where the rays which illuminate the object produce the same effect as if they came from the point p, without any compensation occurring.

ARTICLE V Solution of most of the preceding problems by the introduction of some portion of the daylight into a dark room

I HAVE LOOKED for some other ways of avoiding the accidental irregularities of the auxiliary sources that one is obliged to use in 31 1n modern terms, this very useful proposition might read: "the light-source is apparently advanced by a distance x = (n - 1) t/n, where n is the index of refraction of the transparent substance and t its thickness." This theorem is constantly used in the practice of photometry (see J. W. T. Walsh, Photometry (2nd ed.; London: Constable & Company Ltd., 1953), p. 24). 32This seems to us to be a loose argument; but it is evident that Bouguer intended the word point to mean "a small but finite area," both here and elsewhere. The conclusion is of course quite correct.

36

BOUGUER'S OPTICAL TREATISE ON LIGHT

measuring light. I introduced some rays of daylight into a [28] dark room through two different apertures. I caused them to be reflected , or made them pass through some transparent substance, and afterwards compared them with the direct light that I obtained from the second opening, the size of which I measured with care, like that of the first. I always chose a clear day, and arranged things so that the two openings corresponded to a part of the sky nearly opposite the sun. Figure 9 shows us the interior of the dark room; the two openings are indicated by P and Q. I was able to alter their position a little, to move them up or down, and also to change their size, making all these adjustments by means of simple pieces of cardboard, which I made to slide over each other. When the two openings were of their maximum size, they formed squares with sides 7 or 8 inches long.

To investigate how much a mirror, or the surface of a liquid, diminishes light by reflecting it, I placed the mirror, or else the vessel containing the liquid, horizontally at 0. This reflected the light that it received from the opening P on to the frame GH at R. I then enlarged or diminished the other opening Q until the light which it furnished and which fell at S, appeared as intense as the other ; and in the [29] size of the two openings I had the ratio in which the light was enfeebled by its reflection at the mirror or the surface of the liquid placed at 0. In this observation it is essential that the two rays PO and QS, which I ordinarily make 7 or 8 feet long, should be exactly parallel, so that the two beams of light may come from two parts of the sky equally elevated above the horizon, making their intensities precisely the same. Thus the second opening Q must necessarily be a little higher than the other, P, if it is desired to have the two images Rand S

VARIOUS MEANS OF MEASURING LIGHT

37

side by side, and quite near one another. It is no less important that the frame GH should then be placed exactly vertical, so that the reflected ray OR and the direct ray QS may strike it with precisely the same inclination; otherwise the two lights, although rendered equal, will not illuminate equally. This arrangement also serves to fulfil an important condition: the direct ray QS will be found to have exactly the same length as the sum of the incident and reflected rays PO and OR, which is necessary in order that the quantities of light introduced into the room may be sensibly proportional to the size of the two openings. As to the frame GH, we may cover it, if desired, with vellum [30] or white paper, and limit the images Rand S by diaphragms; or on the contrary we may use a sheet of black cardboard, pierced with two holes. We shall give these a diameter of 3 or 4 lines, and cover them with pieces of white paper. We may easily see the absolute necessity of having a curtain which extends from the frame as far as the two openings, in order to separate the two shafts or pyramids of light throughout almost their entire trajectory; and it will also be seen that it is no less essential to hang another obstacle above the reflecting surface 0, in order to prevent the image R being illuminated otherwise than by reflection. We recommend the use of the black screen pierced with two holes, only because by removing the pieces of white paper which close these, one may bring up his eye and assure himself that all the diaphragms or curtains which are sometimes needed do not intercept some part of the two main beams. When it is desired only to compare the properties of two surfaces as far as reflection is concerned, we may make a large number of observations easily and rapidly by a slight change in the arrangement just described. We have only to place in a horizontal position, behind the frame, and side by side, the two surfaces that we desire to examine, and look at [31] the two luminous points Rand Sas in a mirror. To these we shall furnish more or less direct light by changing the ratio of the two openings, which we shall now take care to place at the same height. It is not necessary, in these new experiments, that the frame should be set vertical; indeed, it will be better to slope it backwards so that it may be more strongly illuminated, and so that the two luminous points R and S, the images of which must appear equally bright, may be seen better by reflection in the two surfaces. When the comparison has been made for a certain angle of incidence, we may make it for another by placing the same luminous points R and S higher or lower.

38

BOUGUER'S OPTICAL TREATISE ON LIGHT

ARTICLE VI Construction of an instrument, suitable for measuring the intensity of light, which enjoys some of the properties of a dark room MOREOVER, we can obtain all the advantages of a dark room by constructing a very simple instrument or machine which contains the chief properties of this in miniature. Two tubes, blackened inside, and also provided with some diaphragms placed mainly near the middle of their length, are joined at their lower extremities by means of a hinge at BC [32] (Fig. 10). These tubes have at the bottom two holes

3 or 4 lines in diameter, Rand S, closed by two pieces of very thin and very white paper. The two other extremities each have a circular opening an inch or so in diameter, but perfectly equal, and these give free entry to the light. One of these tubes is formed of two parts, one sliding in the other, and as we lengthen or shorten it we produce exactly the same effect as if the diameter of its opening D had been made smaller or larger in the same ratio. For then this opening corresponds to a smaller or larger region of the sky. 33 With this instrument, do you wish to find out how much more or less light is sent to us by a certain region of the sky than by another which is at a greater or lesser elevation, or more or less distant from the sun? You need only arrange the two tubes to form a certain angle; you will direct them towards the two parts of the sky, and then 33 This means of applying the inverse square law is still very useful for special problems.

VARIOUS MEANS OF MEASURING LIGHT

39

introduce their lower ends into a larger tube 7 or 8 inches long, into which you can look. If you adapt this well enough to the shape of the forehead and the rest of the face below the eyes, you will succeed in seeing absolutely no light but that of the two luminous points Rand S. It is often necessary that the observer should have a helper who will [33] take care of pointing the instrument, and who will also lengthen or shorten the tube BD, until the two images Rand S appear equally bright. If the tube AB is 12 inches long, and it is necessary to make the other 15 inches, this is as if we had decreased the diameter of the opening D in the ratio 12 to 15, or 4 to 5. Therefore this opening will furnish a weaker illumination in the ratio 16 to 25, and this will indicate that the part of the sky which corresponds to the opening D is really more luminous than the other in the ratio 25 to 16. The same instrument will also serve to measure the various degrees of intensity in the light sent to us by terrestrial objects, when each of these is of a uniform colour, and presents a sufficiently large surface. The more brightly an object is illuminated, or the more light it emits, the more we must lengthen the tube BD which is pointed towards it. In lengthening this tube we make its opening correspond to a smaller part of the object, and the luminous point Sis less illuminated by it. When the two luminous points, or the two images 34 R and S, are of the same intensity, we have nothing more to do but to take the squares of the lengths of the two tubes; and putting them in the reverse order, we shall have the ratio of the intensities of the light from the objects that we wish to compare. [34] But if the observation is to succeed, the objects must reflect enough light to be visible on the two pieces of paper or ground glass which cover the holes Rand S. The objects must also be large enough; they must subtend an angle considerably larger than that subtended by either opening A or D, at the other extremity of its tube. In fact, if each object appeared only under an angle of 2 or 3 minutes, and if by lengthening the tube BD one could not reduce the angular size of its opening to less than 7 or 8 minutes, the object would occupy only the middle of it, so to speak. The two luminous points or small images R and S would then always receive the same number of rays in spite of the lengthening of the tube, and we should not succeed in producing the necessary attenuation of the two lights which we must use to measure them. In this case, to avoid giving an enormous size to the instrument, or narrowing the opening D in the tube DB, which would HJmages, here and in the next paragraph.

this word.

It is not clear why Bouguer liked to use

40

BOUGUER'S OPTICAL TREATISE ON LIGHT

plunge us into other difficulties, we can scarcely avoid having recourse to a new expedient which we are now going to propose. (35]

ARTICLE VII

A second way of using lenses, or telescope objectives, in the measurement of light

INSTEAD OF RECEIVING the light at a greater or lesser distance from the focus of a lens through which it is made to pass, in order to augment or diminish its intensity, as we explained in the first and second articles, we have only to receive it exactly at the focus, and change its intensity by giving the lens a larger or smaller opening. For this we shall purposely choose lenses of a very slight convexity; we could even use the objectives of our longest astronomical telescopes, 35 which would make the images larger, so that it will be easier to choose and distinguish the parts of which the brightnesses or intensities 36 are to be compared. D F

Fi'q.:11. --

EA

Figure 11 represents the instrument. Two identical objectives AE and D F are adapted to two tubes, and their foci are located at the lower extremities, at 6 or 7 feet from the lenses, or at 10 or 12 feet. Each of the lower ends is closed, except for a small hole about 3 or 4 lines in diameter, (36] which is covered with a piece of very white 35Astronomical telescopes were made extremely long in order to reduce the effect of various aberrations. The achromatic (crown and flint) doublet was invented in 1729 by Hall, but was not generally known until the London optician Dollond made such objectives available about 1758, the year of Bouguer's death. See H. C. King, The History of the Telescope (London: Charles Griffin & Co. Ltd., 1955). ,.L'eclat ou la force.

VARIOUS MEANS OF MEASURING LIGHT

41

paper, or of ground glass. The instrument will then be complete, and it remains only to direct the two telescopes towards the objects you would observe. These objects will be displayed on the lower ends of the tubes, but you will find only the parts which correspond to the two pieces of paper. To distinguish them better you will take away all stray light by inserting the ends of the two telescopes in a wider tube to which the eye is applied, as we recommended when we were dealing with the instrument represented in Figure 10. If the two objects that you observe are not equally luminous, but if one is much brighter than the other, they will be reduced to equality by diminishing the aperture of one of the two objectives. The actual extent of the two lenses will then mark the ratio which exists between the two different amounts of light. But it must be noted that in this kind of observation there is only one legitimate way of reducing the apertures of the objectives. Since we wish the size of the surface of the lens to express the number of rays which get through it, we must not cover the central parts more than the edges; the former being thicker, are less transparent, the latter more so. We have to cover them all proportionally, and for that we must use (37] diaphragms that have exactly the form of sectors. 37 In Figure 12 we have represented one of these objectives, partly

Fy.12.

covered; it is surrounded by a circular ring or diaphragm MNO, on which we have drawn divisions related to the whole circumference. Other diaphragms, cut in the form of sectors, are placed between the circular diaphragm and the lens itself; and by sliding these over one another, we can render any desired part of the objective inoperative. In the state shown by our figure, the light passes through only the space FCEH, which is exactly four-twelfths of the whole extent of the lens. It is consequently diminished in the same ratio. 37The modern photometrist would not be worried by the lack of transparency of the glass, but might well investigate the effect of spherical aberration.

42

BOUGUER'S OPTICAL TREATISE ON LIGHT

If all the objects which it is proposed to observe should be almost in the same direction, we could replace the two long tubes by one only. The objectives would both remain at the top, side by side, and one eyepiece 38 would be enough at the bottom. Such an instrument would then not differ from that which I put into the hands of astronomers in 1748, giving it the name of Heliometer or Astrometer 39 ; an instrument which the Royal Society of London reproduced in 1753 almost in the same form, with the assurance that it had been proposed to them several years earlier. 40 I still congratulate myself less for having in this way learned about a clever English optician, 41 than for having been the means [38] of at last doing him justice; for it appears that his discovery was neglected as if it had not been well enough explained, or as if people had not realized all its value. Apparently it would have remained utterly unknown if my proposal for a heliometer had not rescured it from oblivion. 42 Almost exactly the same thing happened in this affair as in the last century with regard to the micrometer. It was thought of in England; 43 but everything which had been done having been entirely ignored by the public at the time, it could be regarded as absolutely useless or even non-existent. Thus the world really owed this invention to Messrs. Auzout and Picard, and to them alone. 44 In the same way I believe it to be obliged to me for the heliometer, as I borrowed nothing from our scientific neighbours, who had themselves completely lost sight of it, or even despised the almost similar idea which had been communicated to them. I only thought of this second instrument for the precise measurement of celestial intervals which are a little larger than those one may measure with the first; but you will see that I am giving it another employment which may become of considerable importance. We may leave the heliometer 38This is the first we have heard of an eyepiece (oculaire) . Probably no actual eyelens was intended. 3 9Mem. Acad. R . des Sci., Paris, 1748, pp. 11-34. • 0Servington Savery, Phil. Trans., 48 (1754), 167-78. It is stated at the head of this paper that it was actually read on October 27, 1743, and a letter from James Short vouching for this and quoting the minutes of the Society appears on pages 165-6. Bouguer's paper brought this to Short's attention. ••Presumably Savery. 42 It is hard to reconcile all these facts with the tremendous energy shown by the Council of the Royal Society in preparing to observe the transits of Venus, 1761 and 1769. See H. Woolf, The Transits of Venus (Princeton, Univ. Press, 1959), chap. III. 43 By William Gascoigne, certainly as early as 1641. King (History of the Telescope, p. 96) thinks it may have been in 1639. Gascoigne's work was brought to the light of day by Richard Townley (Phil. Trans., 1 (1667), 457-8), after he had read of Auzout's work. 44 Adrien Auzout, Traite du micrometre (Paris, 1667). It is uncertain how much credit is to be given to Auzout and how much to the Abbe Picard.

VARIOUS MEANS OF MEASURING LIGHT

43

only its two objectives, and we shall then see the two images more easily in the same glance, and avoid all the difficulties found in trying to observe these images through equally thick parts of an ocular. 46

[39]

ARTICLE VIII

In using the instruments represented in Figures 10 and 11, the intensity of the light is directly measured, and not its total amount 46 IT IS ALSO EASY to see that we do not measure with this instrument the absolute amounts of light, but only intensities, or the number of rays in relation to the surface of the luminous body. It is the same when we use the instrument shown in Figure 10. We must always distinguish carefully between these two things, the absolute quantity of light and its intensity. The absolute quantity depends as much on the way in which the rays lie side by side, or are dense or compressed, as on the size of the surface of the luminous body which lights us; while the intensity expresses only the way in which the rays are more or less compressed, without regard to their total number. If when we look at a luminous body it seems very bright, the intensity of its light will be very great; and yet it might happen that, all things considered, this luminous body produced very little illumination, because it had little [40] surface area. The contrary is equally possible; the intensity of the light may be found to be very small, but nevertheless the luminous body may give out a great deal of light, if it has a very large surface. This is related to what we said 47 about the wall illuminated by the sun: when one recedes from it, it appears to have the same brightness, 48 or in other words the intensity of its light is always the same, although this wall might reflect to us a greater or lesser total amount of light. It keeps exactly the same degree of whiteness,49 because the number of rays which enter the eye continues to be proportional to the size of the image traced on the retina. In order to make things clearer, may we be permitted to add that 46 Presumably the diaphragm with the paper-covered holes was substituted• Bouguer does not say. 46 ln modern terms, luminance and not flux. 47See p. 31. 48Sa couleur paroit egalemenl forte. See n. 22. 49 Blancheur. Clearly this means "apparent whiteness," and is synonymous with couleur (n. 48). Nowadays we seldom think of whiteness in this way, at least in scientific contexts.

44

BOUGUER'S OPTICAL TREATISE ON LIGHT

in the observation represented by Figure 2 we compare absolute amounts of light and not intensities. For we receive rays from all parts of the surface of the luminous bodies at the same time; we do not inquire whether the surface of the flame appears more or less bright; we observe all the light we receive from it. In the present instance we assume another point of view. We consider only a given part of the surface of the luminous body, and the expression we find for the intensity of its light is in relation to the apparent extent of this part. Instead [41] of drawing conclusions about the total quantity of light, then, we immediately look for the brightness. We examine portions of two luminous objects or two objects having an equal apparent extent and find out whether one sends out more or less light than the other. It is evident that the results will be different according to the two senses in which we solve the problem, but besides the fact that the two solutions each have their usefulness, it is always easy to pass from one to the other. When we know the brightness of the light we have only to multiply it by the surface of the luminous body, if we are always at the same distance from it, and we shall have the total quantity of light. If, on the contrary, we have this latter quantity, it need only be divided by the area of the surface of the luminous body and we shall have the brightness or the degree of intensity with which each part of our retina is affected. Finally, if two objects or luminous bodies are at very different distances we must change the length of the telescopes a little so that the two images will be delineated equally distinctly on the bottom of the two tubes, and then the intensity or brightness of the lights will not be expressed exactly by the apertures of the objectives. The distance from the object to the objective does not affect the intensity of the light, provided that we have nothing to fear from the lack of transparency of the air. If [42] the object is much farther away it offers a much larger portion of itself to illuminate the bottom of the telescope under the same angle, and there is an exact compensation between the real size of this part and the least number of rays which each of its points sends towards the objective. The number of rays admitted into the telescope which come to us from parts of the object apparently equal in size will then always be the same in spite of the greater or lesser distance of the object. But this same quantity of light which comes through the objective and which is proportional to the size of its aperture 50 must then lose more or less of its intensity according to whether it spreads out for a greater or lesser distance and covers a &0

Jts area, of course.

VARIOUS MEANS OF MEASURING LIGHT

45

greater or smaller surface at the focus. The more the light is weakened by this second cause the less we have to have recourse to the other, that is, to diminish the aperture of the objective in order to render the two images equally bright. Thus, everything considered, the intensities of the light which we observe are proportional to the reciprocal of the apertures of the objectives and to the squares of the lengths of the telescopes. 51

(43]

ARTICLE IX

Comparison of the preceding means of measuring light with those which have been proposed by various authors WE SHALL FINISH the enumeration which we have had to make of the different means which can be employed to measure light by comparing them with the methods proposed by certain authors. It will have been seen that we have made it a rule always to reduce the two lights to a perfect equality by making the weaker one increase or by diminishing the stronger. We have sometimes made them suffer these changes immediately before receiving them, and at other times we have increased or diminished their intensities as they left the luminous bodies, so to speak; but the order which we have followed in this has not brought any difference into the result. All my readers know that it does not matter what arrangement we give to numbers or fractions when we multiply them together; their product is always the same. We believe that this indispensable rule, to bring everything to equality in this comparison, cannot be replaced by any other, for this ratio is the only one about which one is sure not to make very large mistakes (44]. In general our senses have not been given to us to enable us to determine with precision how much the different causes of our sensations differ in intensity. They have been disposed by an infinite wisdom simply to notice the different relations which these have with us. The difference between them may be very great while we scarcely feel it; it no longer interests us; there is nothing in it which threatens us; and it is no less possible that a very small difference in the cause may correspond to a very different sensation, because this small increase in the agent sometimes puts it in a condition to alter our 61 No reader who is familiar with photometry will feel that Bouguer was labouring the point in this article. A really unambiguous photometric notation is scarcely available even yet.

46

BOUGUER'S OPTICAL TREATISE ON LIGHT

constitution, and so it is important for our own safety that we should be aware of it promptly. But it is not the same when two causes act on our senses feebly. When they act at the same time on the same organ, and when the impressions we get from them appear to us to be equally intense, and all the other circumstances are absolutely the same, we cannot doubt that they must then be very nearly equal. We shall have proof of this as far as light is concerned in the first of the experiments which will be reported in the following section. Thus we have been right to insist on all the precautions which contribute to make everything equal [45] on both sides in the comparisons that we are obliged to make. It is not enough that we should see with one glance the lights of which we wish to discover the ratio and which we make equal. We must look at them with the same obliquity. It is also necessary, as we have expressly recommended, that they should be circumscribed by diaphragms, or otherwise, so that they form areas of the same size and of the same apparent shape, and we must bring them so close to one another that they seem to touch in painting themselves on the retina. If in that state, and when we examine them in a place hung with black, or sufficiently dark, they appear to us equally intense, and if at the same time we have taken the precaution to observe them through an opening made in another diaphragm which cuts off our view of all other objects, it is certain that we have only to take into account all the changes which we have made them suffer to reduce them to equality, in order that we may know, exactly enough, what were their original intensities. It is only because they have omitted one or more of these essential details that various scientists have proposed very insufficient and defective methods of measuring light. Mr. Huyghens reports in his treatise Cosmotheoros, 52 Book II, p. 136, [46] that, to compare the light of the sun with that of the star Sirius, he looked at the first of these heavenly bodies through a long tube which had only a very small hole at the top, and that he had made the two lights equally bright. But apart from the fact that this clever mathematician may not have made all the necessary distinctions between the total quantity of light and its intensity, it is only too certain that we can only judge directly the strength of two sensations when they affect us at the same instant. How can we assure ourselves otherwise that an organ as delicate as the eye is always precisely in the same state, that it is not more sensitive 52 Christian Huyghens (1629-95), Cosmotheoros sive de terris coelestibus earumque ornatu conjec/urae (The Hague, 1698).

VARIOUS MEANS OF MEASURING LIGHT

47

to a slight impression at one time than at another? And how can one remember the intensity of the first sensation when one is actually affected by the second and when an interval of several hours or even days has gone by between the two? To succeed in this determination he would have had to have recourse to an auxiliary light which he could make use of in the two observations, and which would serve as a common term of the comparison. We are obliged to make practically the same strictures on the methods described by Father Fran~ois-Marie, a Capucin of Paris, in a little book which he published in 1700 under the title Nouvelles decouvertes sur /,a lumiere. 63 [47] When I published in 1729 my work entitled Essai d'Optique, I was not acquainted with this work and I could not speak of it at that time. This good cleric believed that by placing several pieces of glass of the same thickness one upon the other, or by causing the rays to suffer several successive reflections by mirrors, he had a scale which diminished by exactly equal steps, or which followed the terms of a decreasing arithmetic progression; and when he wished to evaluate the intensity of a light he looked to see how many pieces of glass or mirrors he had to use to make it disappear entirely. Father Fran~ois-Marie was deceived about the equality of his degrees of light, as several other people have been, who have fallen into the same error; but what made his expedients still more defective was the bad use he made of them. His results must depend more or less on the transparency of his pieces of glass, and not only this, but on the differing state of his eyes, which would be more or less sensitive at one time than at another. When his sight was a little fatigued all lights would ordinarily appear to him stronger. 54 He would then need a greater number of pieces of glass to weaken them to the same extent. Each observer would in this way attribute a different degree of the scale to the light which he was measuring. People would not be able to agree when observing at different times or [48] in different countries, and the measurements would never give exact ratios. The method proposed in 1735 by Mr. Celsius, the famous Swedish astronomer, in the History of the Royal Academy of Sciences, 66 was HR. P. Franc;ois-Marie, Nouvelles decouvertes sur la lumiere pour la mesurer & en compter les degres (Paris, 1700). This charming little book is interesting because of the great pains its author takes to convince his own conscience and his superiors that it is not impious to try to measure light, the gift of God. 54The logic of this is hard to understand, since the observations are to be made at threshold. But the reproducibility of such measurements is poor, as Bouguer says. MHist. Acad. R . des Sci., Paris, 1735, pp. 5-8. Celsius' idea was reported to the Academy by de Mairan, and seems to have been received with acclaim. Bouguer's very sound arguments against it are those which would occur to a man in his fifties.

48

BOUGUER'S OPTICAL TREATISE ON LIGHT

no better, and seems to have been worth even less. He made it depend on the distinctness with which we discover the smallest objects at different distances, according to whether they are more or less illuminated, and he paid no attention to the fact that it is still more difficult to submit this distinctness to a precise law than to measure the intensity of light. He claimed that to see in an equally distinct manner some small object twice as far away as another it is necessary that it should be illuminated 256 times as much, in conformity with the eighth power of the distance; but it is certain that if someone with very short sight easily reads small letters at 4 or 5 inches in a dimly lit place there is not enough light in the world to make him identify them at 14 or 15 inches, unless his eyes are of some very peculiar construction. Our distinct vision really has very narrow limits. We see well enough all the objects which are included in a certain range of distance, but if they are taken farther from us, or brought considerably closer, the greatest [49] or least light is not sufficient for us to be able to perceive the smallest parts, because the lack of distinctness does not then arise from the small number of rays, but from their particular directions, which cause them not to come together sufficiently exactly on the retina. The principle indicated by Mr. Celsius could therefore only agree with experience by the most extreme chance. If it were good for this astronomer it could not be so for observers whose eyes were made differently from his. It would, furthermore, be found to fail even for him whenever the distances were considerably smaller or larger than the first ones which he used to set up his pretended rule. We have nothing of this sort to fear in the methods we are employing. As we are considering only the amount of light or its brightness, it does not matter whether the observer has long or short sight, good sight or bad. If the rays cross before having reached the retina or if they come together farther back, nevertheless they act on the back of the eye. There is nothing lost, and the total effect is always the same as regards the intensity of the impression. We see no other obstacle to the application of our rules except when the lights which we wish to compare are of [50] different colours. The problem is then susceptible to various solutions, and indeed it has various meanings. One might ask to what extent one of the lights is more suited to communicate heat, and it would then be necessary to have recourse to experiments quite different from those with which this treatise is concerned. If, on the contrary, it is a question of the distinctness with which lights of different colours permit us to see objects, we can have recourse to our methods, and it will suffice to pay attention to making the distance

VARIOUS MEANS OF MEASURING LIGHT

49

from the eye to the objects exactly the same in order not to fall into the same difficulty as Mr. Celsius. A comparison of two lights of different colours in the way that we prescribe is chiefly embarrassing in case it is necessary to do it with more care, that is to say, when the two intensities closely approach equality; but there is a point where one of two lights will certainly appear stronger than the other, and another point where this light will appear more feeble. We have then only to take the mean between these two limits. 56 Finally, the rules which we propose are extremely simple and are founded only on wellknown principles, in spite of all the precautions which render their application a little more difficult. Nevertheless we shall not employ others in order to arrive at curious pieces of knowledge and to discover different things which people have attempted without success to find, or which it is not yet advisable to look for. 66The comparison of lights of different colours is now known to be subject to large personal differences, and to be very sensitive to the conditions of observation. The procedure described by Bouguer is, however, still useful.

SECTION TWO Applications of the Preceding Means of Measuring Light to the Solution of Several Optical Problems [51]

ARTICLE I

Observations made to determine what intensity a light must have in order that it may cause a feebler one to disappear WE SHALL PUT at the head of all our observations those which have taught us how much intensity a light must have so that its presence may render the effect of another much more feeble light absolutely invisible. 57 All our organs, the most delicate as well as the coarsest, are subject to very similar limitations. In the same way that a loud noise prevents us hearing another feebler one, we do not see, in the presence of a strong light, another of which the intensity is much less, if the two strike our retina in the same place. Having placed a candle at a distance of 1 foot from a very white surface, I placed beside the [52] candle a ruler of a certain width, and I then placed another candle of the same size as the first at various distances until I ceased to distinguish the shadow of the ruler caused by the second candle. The shadow was very visible when I had carried the second candle only to 4 or 5 feet from the surface. All the space occupied by this shadow was still illuminated by the first candle, but at each side of this space the light was augmented by a sixteenth or a twenty-fifth part by the rays from the other candle, and this increase was quite visible. It was still faintly visible when I carried the second candle to 6 or 7 feet, and finally it disappeared or, to explain it in another way, all the surface appeared to me to have an absolutely uniform whiteness, when I put this candle at a distance of about 8 feet. Thus the difference between the two lights ceased to be visible only when 67The modern point of view is directed to the incremental threshold, exactly the converse of Bouguer's concept.

VARIOUS MEANS OF MEASURING LIGHT

51

the small part added was about sixty-four times as weak as the first. 58 I could have derived the two lights which I was comparing from the same luminous body with equal ease, but I repeated the experiment several times, taking care to substitute one candle for the other to see if they gave equal illumination. 59 Observers who make this experiment can easily vary it in a great number of different ways that will be perfectly equivalent, but they [53] will doubtless find slightly different ratios according to the different constitution of their eyes. I have not noticed that the ratio is changed by great brightness of the lights provided that one can easily sustain their brilliance. This does not mean that it is not a good idea, when one has lights of this kind to compare, always to commence by reducing them to a moderate intensity by subjecting them to equal or, more correctly, proportional changes. In any case, each observer can easily determine the precise degree of the delicacy of his own sight, distinguish the parts of the retina which are most sensitive, 60 and judge of the precision which he can hope for in his observations. I do not fear an error of more than a sixtieth part in any comparison of light, and another person will perhaps obtain an eightieth part. But we must not believe that all experiments will be as successful as this. Sometimes they will be very far from it, and that is what I have to fear, I freely confess, for several of my own. Not only must we always realize the great difference between very simple experiments and those which, being complicated, demand several separate and successive comparisons, but in order to obtain a certain precision it may be necessary to arrange things so as to have only one observation to make, and thus to be willing to sacrifice much time and sometimes to incur quite considerable expense. We should without doubt [54] try to obtain the precision referred to above. It is good to have this always in view, but it will be very rarely obtained . It will often happen that we shall not intercept all stray light sufficiently well. Some slight reflection will be mixed with the rays whose intensity we are comparing; or we may not bring them close enough together; or they may be too different in colour, and perhaps also the back of the observer's 68 ln

view of the diffuse edges of the shadow, this figure still seems reasonable. One of many demonstrations of Bouguer's attention to detail. 60This passage raises an interesting question: was Bouguer aware that very faint lights are more easily visible in parafoveal vision? The numbers quoted in the next sentence are very much against the supposition that he had compared lights in the range of brightnesses for which the fovea is less sensitive. 69

52

BOUGUER'S OPTICAL TREATISE ON LIGHT

eyes, quite apart from the point where the optic nerve is inserted, will not have everywhere the same sensitivity. 61 To go back to the ratio which makes one light disappear in the presence of another, it is easy to see how many applications may be made of this. All our readers have noticed that one can only see with the greatest difficulty what happens in a dark room when one looks into it from a brightly illuminated place. A window on which the sun is shining sometimes does not allow us to see an object even outdoors when it is in the shade. We are at the edge of a basin full of water and frequently we cannot distinguish the bottom of it. This also arises from the fact that the too feeble light which it sends to us is effaced by the light of day and by that which the water itself reflects to our eyes. In case the impression that these last sources of light make on our eyes is only fifteen or twenty times as strong as that which is made by the light coming from the bottom of the basin, [55J it will not prevent us seeing through it. We see it through the other light as through a gauze or a net, but we entirely cease to distinguish it if the impression caused by all the external light is eighty or a hundred times stronger than the other. It is also sometimes the same when our eyes are not struck by the light at the moment, but when they preserve the impression of it. Our retina has some resemblance to those phosphors which are penetrated, so to speak, by the light to which they are exposed. If the light is very weak its action must be repeated for the effect to become noticeable. On the other hand, the rays which strike the back of the eye communicate to it a vibration which may subsist for a long time after we have passed into a dark place, and our eyes are then in the same state as if they were still affected by a certain amount of light. Now if this impression is much stronger than that made on us by an object in the shade we shall not discover this at all, and we shall only begin to see it when the vibration of the retina, having become weaker, ceases to surpass sixty or eighty times the intensity of the feeble gleam of the object. Certain places in a darkened room are more strongly lighted than others; there is often an infinity of different nuances in the shadows, and it is [56J very certain that shadows serve to render other parts more prominent; but if the impression which remains to us from the light outside is still too strong, not only shall we not discover the dark parts of objects, but we shall not even distinguish those which are the most strongly illuminated. To speak correctly, we shall see only 61 Much more recent photometrists have been known to disregard this last source of error.

VARIOUS MEANS OF MEASURING LIGHT

53

the dominant light which still exists in our eyes, but of which the effect continually diminishes, apparently according to the terms of a geometric progression. 62

ARTICLE II Experiments on reflection by mirrors

I HAVE MADE a great number of observations on the quantity of light that mirrors reflect in proportion to that which they receive, but I shall content myself by reporting one or two here by way of example before I go further into this subject. Having chosen a flawless piece of glass one line in thickness, I had it silvered especially for this purpose, and I found, using the arrangement which is represented in Article III of the preceding section, that the reflection weakened the light in the ratio 1764 to 2809 or 628 to 1000 when the angles of incidence and reflection were 15°. (57] Having put the tablet D (Fig. 4) at a distance of 42 inches from the candle, I was obliged to remove the other tablet, E , to 53 inches in order that the two might have the same whiteness when I looked at them directly and by reflection. Thus the light reflected by the mirror was to the light which it had received as the square of 42 is to the square of 53. The second light being expressed by 1/ 2809, which is unity divided by the square of 53, the other will be expressed by 1/ 1764, which is unity divided by the square of 42. The reflection appeared to me a little less bright on a metal mirror which had received a very fine polish. The more distant tablet being 53 inches from the candle, the other tablet, D, only had to be moved to 40 inches, and thus this mirror reduced the light from 1000 degrees to 561 by reflection under an angle of incidence of 15 degrees. When I made the angles of incidence and reflection much smaller the reflection became sensibly stronger. It appeared to me that under the angle of 3° the glass mirror sent back about 700 rays out of 1000 which it had received, and that the weakening was a little less still on the metal mirror. I confirmed these two experiments by putting two perfectly similar mirrors in series, on which two (58] successive reflections took place. The effect being then as if doubled, it was easier to determine it exactly, but the smallness of the angle gave rise to new difficulties 62 A natural assumption for Bouguer, but far from the truth. See for example Y. Le Grand, Light, Colour, and Vision (New York: J. Wiley & Sons, 1957), p. 239.

BOUGUER ' S OPTICAL TREATISE ON LIGHT

which had to be surmounted. The images became multiple even in the metal mirror, which then appeared to have waves in spite of the perfection of its polish. This obliged me to put the tablets only at very small distances.

ARTICLE III Observations on the transparency of glass and of plate glass Tms IS ONE of the first experiments that I made before publishing my Essai d'Optique. I wished to discover how many times light diminished in passing through 16 pieces of the ordinary glass of which window panes are made. These were 9½ lines thick altogether. During the night I had a torch and a candle carried to a sufficiently great distance and in different directions, and letting their light enter a sort of box through two different holes, I received the light separately on the back of the box with the same obliquity. I was able to examine their intensities because I had made a third opening through which to look. I first had the torch and the candle moved [59] nearer or farther away until they seemed to give equal illuminations, but having then made the light of the torch pass through my 16 pieces of glass it became so feeble that in order to re-establish equality I was obliged to have the candle carried to a distance 15½ times as large, a distance which caused the light to diminish 240¼ times. It is then clear that the 16 pieces of glass weakened the light 240 times, since that of the torch suffered by their interposition precisely the same change as that of the candle, and this must have been 240 times as weak, since I received it at a distance 15½ times as great. I next made the same experiment in a slightly different way. I remarked that the light of the moon was equal to that of a tallow candle distant 27 feet from me; but after having made it pass through my 16 pieces of glass it was now only equal to that of the same candle received from about 430 feet. 63 Now the light of the candle was about 254 times less strong at the second distance than at the first, for 729 and 184,900 are the squares of 27 and 430, and the second of these squares is about 254 times the first. Thus the light became 254 times 6 3This comparison was evidently done with a field brightness of about 10-4 cd/m 2 , or a little more. This corresponds to a contrast threshold of about 10 per cent (see H. R . Blackwell, J. Opt. Soc. Am., 36 (1946), 624), and the agreement with the other test was at least as good as one might expect. The difference of colour between the moon and the candle would not be very important at this brightness.

VARIOUS MEANS OF MEASURING LIGHT

55

as weak, according to this second test, in going through our 16 pieces of glass, while according to the first it diminished only 240 times. One might take [60] the mean between these two experiments and say that the light was diminished about 247 times. The glass that I was using being of the commonest quality, I have lately been curious to examine the transparency of the plate glass of which our mirrors and the lenses of our telescopes are made. This material is often slightly coloured, and this cannot help spoiling its transparency a little. Six pieces, which formed a thickness of 11 ½lines taken together, only weakened light in the ratio 3 to 10. This I have verified by the procedure which is represented in Figure 5, and I also examined it twice in a dark room where I made the daylight enter through two openings. One of these beams of light passed through the 6 pieces of plate glass, and it was necessary to set the ratio of the two apertures at 331 to 100 in order to make this beam as strong as the other which I received directly. On another occasion it seemed to me that the light was weakened not in the ratio of 30 to 100 but 27 to 100, and I am persuaded that the latter of these determinations is the more certain. One may find more transparent plate glass. I have seen a piece which was 3 inches thick and which hardly weakened light by half. However I have since realized that this advantage came principally [61] from the fact that this mass really formed only one piece and was not distributed in several sheets. 64

ARTICLE IV Experiments on the transparency of sea water PROPOSING to examine the transparency of sea water I had a canal made of planks, closed at the two ends by two pieces of glass placed perpendicular to its length. This canal was about half a foot wide and 115 inches long, or 9 feet 7 inches, which was the entire length of the planks. Then during the night I had a lighted torch carried to a very great distance and made its light pass through the canal, and I had a candle placed near me in order to determine at what distance this light would appear equal to that of the torch. To judge better of the equality of the lights, I received them beside each other on the same 64 The loss due to the reflections at 12 air-glass surfaces (assuming normal incidence) would be about 40 per cent.

56

BOUGUER'S OPTICAL TREATISE ON LIGHT

surface with the same obliquity, and I arranged things so that no stray light should be mixed with them. I also consulted the eyes of all the people who were present. 65 As soon as I found by having the candle moved nearer or farther away that it had to be placed 9 feet distant, I had the canal filled with sea water (62] but this water, as you might imagine, diminished the light of the torch considerably; to such an extent that it only became equal to that of the candle again when the latter was received from a distance of 16 feet. It follows from this that the 115 inches of sea water made the light about three times as weak, for the square of 16 being about three times that of 9, this is an indication that the candle whose light served me as a standard gave three times as much illumination in the first case as in the second. I repeated the same experiment several times, but in taking the mean between all the tests I obtained the result that the light did not diminish quite as much, but only in the ratio of 5 to 14. I verified these same experiments by another method which ought to be much more exact because it only needs one torch and so it does not matter whether the intensity of the light suffers some change while the observation is being made. The torch having been carried to a great distance, I made the light pass through the canal, which was empty. This passage made it diminish a little because of the two pieces of glass which were at the ends of the canal, but the diminution was much more considerable when I had the canal filled with sea water, and it was then necessary (63] to find in what ratio this new diminution occurred, since it was produced only by the lack of transparency of the water. To this end I compared the light of the torch which passed through the canal in the two cases with the light of the same torch which I caused to pass through a convex telescope lens, and which I weakened more or less according to whether I received it at a greater or lesser distance behind the focus; that is to say that instead of taking the light of a candle as a standard as before, I took that of the torch itself, which I caused to pass through a convex lens. You will easily see that I had to put this lens beside the canal, and that in this way the two lights which came from the torch, and passed through the canal and through the convex lens, fell side by side and nearly perpendicularly on the surface that I exposed to them, so that it was on this surface that I had to examine their equality. The canal being empty, the light which passed through was equal to that which 66

1n intellectual circles scientific experiments were frequently demonstrated to

guests.

VARIOUS MEANS OF MEASURING LIGHT

57

passed through the convex lens and which I received behind the focus at a distance of 8 inches; but when the canal was filled with water the light which passed through was found to be equal only to that which passed through the convex lens and which I received at 13 inches from the focus. Thus in order to know the diminution (64] which a thickness of 9 feet 7 inches of sea water produced, we have only to see how much the light which passes through the convex lens is weaker when it is received at 13 inches behind the focus than when it is received only at 8 inches, and for that we have only to take the squares of the two distances 13 and 8 inches. This was done, and it is found in this way that the diminution occurs in the ratio of 64 to 169, a ratio which is not much different from that of 5 to 14. The experiments which I have just reported were made at Croisic with a certain amount of care. They agreed fairly well among themselves; nevertheless I confess that I have no reason to be satisfied with them and that I think them to be very defective. I had occasion to recognize this in the torrid zone when I remarked that there one may sometimes discover the bottom of the sea in places where it is 100 or 120 feet deep. For this it is necessary that the bottom should be of white sand and that the sun should be high in the sky. The light of this luminary is weakened by its passage of 100 or 120 feet and one must also consider an equally long path for the return of the rays to the eye. This double path nevertheless leaves the light with an intensity still quite perceptible and we must infer from this that sea water, when it is very pure, is much more transparent than I found. The sea is never as limpid on our European coasts as it is (65] far from land in the parts of the torrid zone where the flux and reflux is not considerable and where the sea is also deep enough to prevent the agitation of the surface being communicated to the bottom. I had the water at Croisic taken from the harbour itself. I let it settle and passed it through a cloth, but it would have been well to filter it with more precautions. As it is necessary that the water should have a rather large thickness, which obliges one to use a large quantity, so that its defect of transparency may be perceptible, the experiment becomes long and difficult, and perhaps one would have great difficulty in arriving at a more exact determination than that which I base, although conjecturally, on the rough observation made in the tropics. I suspect that the thickness of 10 feet of sea water weakens light at the very most in the ratio of 3 to 5 or perhaps even of 3½ to 5. I shall deduce reasons for this later.

58

BOUGUER'S OPTICAL TREATISE ON

LIGHT

ARTICLE V Observations made on the light reflected by rough surfaces

WE HAVE SPOKEN of the reflection produced by mirrors. Rough surfaces also reflect a considerable amount of light which often is sufficient to (66] illuminate us perceptibly. I have examined this effect chiefly when it was produced by plaster and by paper. I exposed a circular surface of plaster 1 inch in diameter to the light of a candle from which it was distant exactly 9 inches. This disk of plaster was as white as possible, and was struck by the light at an angle of 75°. 66 The light made an angle of the same number of degrees with the disk in proceeding to fall at 3 inches distance on the point P of the instrument represented by Figure 1. In this state the light reflected by the plaster was very perceptible. I had taken care to keep out stray light by a multiplicity of diaphragms, but as the nearness of the candle made the thing difficult and even the coarse parts of the air perhaps reflected some light, I measured that received at the point Pin the two cases, that is to say, when I took away the disk of plaster and when I once again exposed it to the light of the candle. For this measurement I used a mirror which reflected the rays of the same candle on to the other point, Q, of the instrument in Figure 1, and I merely moved this mirror nearer or farther away until the two points P and Q appeared to be equally (67] illuminated . I had made the two points or little boards opaque. They were illuminated from the front by rays sent out by the disk and reflected by the mirror, and I looked at them also from the front, placing myself near the mirror. When I took away the disk of plaster the point Q received, as I have already indicated, a rather strong illumination. I had to put the mirror at 60 inches from the candle and it was then distant 54½ inches from point Q, so that the rays travelled 114½ inches in all to go from the candle to point Q; but when I put back the disk of plaster in its place the point P was found to be illuminated much more strongly, and to re-establish equality between the two lights I only had to put the mirror 36 inches away from the candle and 30½ from the point Q, which gave 66½ inches for the length of the incident and reflected rays together. In this second case the light reflected by the mirror was about three times as strong as in the first. The two illuminations were, as we know, in the inverse ratio of the squares of the distances, or the paths The angle between the surface and the ray. See n. 27.

G6

VARIOUS MEANS OF MEASURING LIGHT

59

travelled by the rays. Thus if we express the first by 1/13,110¼ or by 76/1,000,000, which is unity divided by the square of 114½ inches, the second will be expressed by 1/4,422¼ or by 227 /1,000,000, which is [68) unity divided by the square of 66½ inches. Now this last was equal to that which the disk of plaster sent to the point P, added to the stray light, and therefore we must subtract 76/1,000,000, which is the expression of the stray light, and we shall obtain the fraction. 151/1,000,000 for the light reflected by the surface of the plaster alone However, a very considerable reduction must be made in this, since the mirror which furnished us our comparison light did not reflect all the light which it received, but extinguished a large part of it. It reduced the light from 100 degrees to 55 degrees under large angles of incidence 67 or when the rays made a large angle with the surface. Thus the light reflected by the disk of plaster itself should be expressed not by 151/1,000,000 but by 8,305/100,000,000, or by 1/12,041. It now only remains to draw attention to the fact that all these expressions are nothing more than the inverse ratios of the squares of distances measured in inches. The light reflected by the plaster is 1/12,041 while the light which the disk received is 1/81 since it is 9 inches distant from the candle. Now it follows from this that the plaster reflects under these conditions the 149th part of the light which it receives, since 1/12,041 is 149 times less than 1/81. 68 A different result will be found if we change the circumstances of the experiment considerably. The rays which fell on our disk made an angle of 75° with its surface, and they were sent back, so to speak, at an angle of reflection of the same number of degrees. Angles nearer a right angle and unequal would not produce an extreme difference in the ratio, and nothing would be perceptibly changed if the surface of the disk were increased a little, or if it were diminished no matter in what ratio. It would receive more or less light and would also send back more or less, always proportionately. The greatest changes ought to come from the various distances from the disk that one puts oneself to receive this light. Is the distance two or three times as great? Then the reflected light should be four times or nine times as weak, and so on. 69 An experiment that I made on some very fine paper gave me practic67See n. 66. 68Something improbable seems to be hidden in this. If we calculate this through, making the (rather small) corrections for the size of the disk, it would seem that the reflectance of Bouguer's plaster was only about 27 per cent. 89The modern reader will of course find this discussion far from satisfactory, but it is worth noting that the subject of goniophotometry is still a little untidy.

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BOUGUER'S OPTICAL TREATISE ON LIGHT

ally the same result, and there is every reason to believe that things are about the same for most bodies of which the surface is rough and which are very white. Each of their parts sends back at about 3 inches distance the 150th part of the light which strikes it. The light reflected ten or twenty times farther is one hundred times or four hundred times less still. It is only the [70] fifteen-thousandth or sixty-thousandth part of the light which the small surface receives, but on the other hand we can, if we wish, make the surface much larger, and in order to evaluate the total effect, which can then become quite considerable, we must evidently look for the sum or the integral of all the little elements of light, taking into account their obliquity. ARTICLE VI Observations on the quantity of light reflected to us by each part of the sky or of the atmosphere IF WE ARE often illuminated only by that secondary light sent to us by opaque bodies, we are ordinarily much more so by that sent to i.:s by the entire mass of air which forms the atmosphere. We are obligated to the great transparency of this medium for the daylight which we enjoy, but the Author of nature has arranged everything with such wisdom that the slight degree of opacity of the air, or its slight defect in transparency, is almost as useful to us as its transparency itself. These beginnings of opacity are the reason why the air reflects sunlight to us; and without this property we should be plunged into the most profound shadow whenever we did not see [71] that luminary and when it did not directly illuminate the objects around us. It is true that each part of the sky sends down very few rays; but as we receive some from all the visible sky a very large amount of light results, which is comparable with the strongest that we know. The intensity of this secondary light is very different according to the various portions of the sky that we observe. It is sensibly stronger at a small height above the horizon than at a greater height. The difference is still more marked in places more or less separated from the sun, and we should add that the ratio between these diverse intensities is subject to change as the sun gets higher or lower. The atmosphere is very bright up to a distance of 3° or 4° from the sun. Beyond this the differences in the light are sometimes scarcely visible and one must be extremely attentive to see them. The eye in passing continuously from a point to a neighbouring point ordinarily perceives

VARIOUS MEANS OF MEASURING LIGHT

61

the same colour of light. It does not distinguish any appreciable increase or decrease and we are disposed by this apparent uniformity to think that points at some distances from each other which we do not compare directly, or which we do not see with the same glance, are also precisely of the same tint. 70 But we have only to [72] use the Lucimeter 71 which we have represented in Figure 11. We may bring together, so to speak, the most distant points of the sky, and find ourselves in a condition to judge the degree of their light, which would be almost impossible by any other means. I have found several times, when the air was very clear and when the sun was at an altitude of about 25°, that at 8° or 9° distance from the luminary the light of the sky was four times as strong as at 31 ° or 32°, for I was obliged to make the tube directed towards the first point twice as long as the one pointing towards the second. I have sometimes formed an angle of 60° between the two tubes of the Lucimeter, placing the instrument in such a way that the middle of the angle was pointing 90° from the sun, measured on the horizon. I have had to make the tube which was pointed more nearly towards the sun ... 72 inches long while the other was 24 inches in length. Thus two points of the sky distant 60° from one another and both situated in the vertical plane which was distant 90° from the sun did not enjoy an equal light. The light which the first sent to me was to that of the second as the square of 24 is to that of .... These two lights were to one another approximately as . .. is to .. .. The apparent height of the sun was . . . degrees and the two points of the sky were elevated by . . .. [73] But what appeared to me very remarkable is that when the sun is only 15° or 20° high, and if we always observe parts of the sky situated on the same almacantar or at the same height above the horizon, the sky light, after having diminished up to a certain distance from the luminary on each side, then proceeds to become greater up to the point opposite the sun. Thus on this line parallel to the horizon there are two parts the least luminous, or two minima which are, I believe, about 110° or 120° distant from the sun, and between these two minima equally distant from the sun is found, opposite the sun, a maximum or most luminous place. 73 1°Teinte. See n. 48. 71The term lucimetre was first suggested by Fran!;ois-Marie (see n. 53) as an alternative for photometre; it does not seem to have come into use. It is, of course, a hybrid. 72The manuscript was incomplete here and there, as de la Caille points out in his preface. 73The correct explanation of this had to wait for more than a hundred years, when Lord Rayleigh did his theory of scattering by gas molecules.

62

BOUGUER'S OPTICAL TREATISE ON LIGHT ARTICLE VII

Observations made to discover how much more illumination one large portion of the sky gives than another IMAGINING two vertical planes perpendicular to one another which divide all the apparent surface of the sky into four equal parts, I also wished to know how much more illumination the first quarter, beginning at the sun and extending to 90° distance, gave than the second quarter, beginning at 90° distance from the sun and ending [74] at 180°. For this I placed two boards at a certain distance from one another. I put them in an exactly vertical position and perpendicular to a large wall, to which they were attached. They were facing opposite directions and each was illuminated only by a quarter of the apparent surface of the sky, for I waited for the instant when the sun began to pass behind the wall. Then I pointed the two tubes of my Lucimeter towards the boards. The device pointed upwards at an angle of approximately . . . 0 and made with the boards equal angles which were ... 0 • 74 Things being in this state I was obliged to make the tube which was directed towards the more brightly illuminated board . . . inches long while the other tube was only 24 inches in length. Now it follows from this that the first quarter of the apparent surface of the sky diffuses more light than the second in the ratio of ... to ... , so that if we take 1,000 for the light of the second we shall have ... for that of the first. The sun was then ... 0 in altitude. But if the sun had been higher or lower, the ratio which I had just discovered would have been a little different; it would have approached closer to unity the higher the sun. ARTICLE VIII

[75]

To find how much stronger the light of the sun is than that sent to us by a large portion of the visible sky, or how much stronger it is than the shadows of bodies exposed to broad daylight THE SECONDARY LIGHT sent to us by the atmosphere might be the subject of an infinite number of other experiments, just as it is the cause of an infinity of phenomena very worthy of attention. Everybody has remarked that the shadows of bodies are less strong the 74See

n. 72.

VARIOUS MEANS OF MEASURING LIGHT

63

farther away they are cast. This does not refer to the penumbra which results from the luminous body having a certain size, so that all its rays do not proceed from a single point. The reduction of intensity of which we speak depends on an entirely different cause. It results from the fact that the shadow received farther from the opaque body is illuminated by a greater part of the sky, which must destroy something of the darkness of the shadow. This difference is distinctly seen if we examine the shadow of a very high building. If it is perfectly isolated, the intensity of its shadow is less strong at its foot 75 and [76) very near one of its faces, because it is partly destroyed by the light sent out by at least half of the apparent sky. Let us consider points in the shadow more distant from the building. These will be illuminated not only by the first half of the sky but also by a large part of the other half, that which is behind the building; to which perhaps should also be added even the light of the sky reflected by the walls. Finally, if the building is surmounted by a turret or a vane, the extremity of the shadow will be illuminated by almost all the sky, and one can make the mistake of confounding it with the penumbra. It will be seen that it is always quite easy to determine the size of that part of the sky which, illuminating each part of the shadow, makes it less dark or diminishes its intensity. The shadow which we examine is nothing else but a more feeble light. It is the ordinary light of day which fills the entire half of the visible atmosphere, or some other part, and it suffices to compare it once with the direct light of the sun to be able to establish the relative intensity which it should have in all other cases. 76 I again had recourse to my Lucimeter to make this comparison. I pointed one of its tubes towards the back of a wall lit by the sun, where I had placed a very white board. At the same time [77) I pointed the other tube towards another board which I had placed in the shadow and in a re-entrant corner where it could only be illuminated by a quarter of the apparent surface of the sky. The two tubes were equally inclined to the two boards. The inclination was ... 0 and the two boards were placed exactly vertical. I waited till the first was directly exposed to the sun and that luminary had then ... 0 of altitude. 76This is obscure. It is evident from a passage four sentences below that by the intensity of a shadow Bouguer means its darkness. We may ask, "less strong than what?" 78 lt is difficult to believe that Bouguer really meant this. The remainder of this Article is incomplete and we can only assume that if he had lived it would have been greatly different. There is no reference to the varying turbidity of the sky.

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BOUGUER'S OPTICAL TREATISE ON LIGHT

To make the two lights equally intense at the bottom of the instrument I was then obliged to make the tube which was directed towards the place illuminated by the sun ... inches long, while the other was only 24 inches long. Thus the sun, in the circumstances which I have indicated, gives more illumination than a quarter of the apparent surface of the sky in the ratio of ... to .... The quarter of the sky of which I observed the light was enclosed between two vertical planes perpendicular to one another. This quarter began at a distance of 90° from the sun and ended at 180°. But we should not fail to consider that the board which was lit by the sun was also illuminated at the same time by that quarter of the apparent surface of the sky which began at the sun itself. I had therefore observed the sum of the two illuminations from the sun and from the first quarter of the sky, and this sum was stronger (78] than the light from the second quarter of the sky in the ratio that we have just indicated. Taking 1,000 to express the latter, we should have ... for the sum of the two others. But we have already realized in the preceding article that when the light of the second quarter of the sky is expressed by 1,000 that of the first quarter is represented by . . . . It follows from this that we have only to subtract ... from ... , which expresses the sum of the two illuminations from the sun and the first quarter of the sky, and we have arrived at ... for the expression of the light from the luminary itself. Moreover we are supposing that this light and that from the first quarter of the sky would produce the same effect as if they had had the same direction, and we are also neglecting some other considerations which we shall at least indicate below, and to which one may pay attention if one wishes. Finally, if the number ... designates the light of the sun alone, ... expresses the light of the first quarter of the sky, and ... that of the second. But it suffices to admit this calculation to be able to resolve these questions in an infinity of other cases. A wall is, for example, exposed directly to the sun and this luminary has the same altitude as before. Its light will still be expressed by ... but that of the first quarter of the sky will be given by ... , and that of the entire half of the sky by .... Thus [79] the wall will be illuminated with the light expressed by ... while the other side of the wall, which will be in the shade but illuminated by all the other half of the sky, will receive an illumination represented by .... The side of the wall exposed to the sun will then be about . . . times more strongly illuminated than the other.

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ARTICLE IX To find how much the light of celestial bodies increases or decreases with the changes in their altitudes above the horizon THIS OBSERVATION cannot be made with equal facility on all the heavenly bodies. The light of the stars is too weak, and that of the sun is, on the contrary, too strong, to allow them to be compared conveniently with the different sources of light which we have here below. That is why one can hardly make this observation except on the moon. But we have also to take this planet when it is almost in opposition with the sun; its phase then changes only very slightly, and one may be sure that practically all the change in its light will come from its different altitudes above the horizon. After that the operation will be of the greatest simplicity. My chief observations of the moon were made when it had an apparent altitude of 66° 11' and 19° 16'. [80] On the twenty-third of November, 1725, it had the second of these altitudes towards 10:30 in the evening, and its light, received in the box of which I spoke in Article III, appeared to me to be equal to that of four candles which were placed 50 feet from me. The next day at about 3 o'clock in the morning, as the moon was still a little way from the meridian and at an altitude of 66° 11', I examined its light a second time, and it was then equal to that of my four candles placed 41 feet away. After this it was easy for me to find the ratio which I was looking for, for the squares of two distances 41 and 50 feet are 1,681 and 2,500, and since these squares express the intensity of the light from the candles in the two observations, they also express the intensities of the light from the moon, which were equal to them; that is to say that the different intensities with which the moon illuminates us when it is at 19° 16' and 66° 11' apparent altitude are proportional to the two numbers 1,681 and 2,500 or, what comes to the same thing, one of these intensities is just about two-thirds of the other. By observing the moon in the same way at every degree of altitude one could make a table which would show the intensity of the light at all altitudes, and this table would serve for all the other celestial bodies, since [81] their rays must suffer the same attenuation in coming through the atmosphere. I determined to observe the moon when it was at altitudes of 66° 11' and 19° 16', only because the sun has these same apparent altitudes at Croisic at midday on the days of the summer and winter solstices. Thus I learned how much more

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the sun illuminates us in one season than in the other. I have also examined the moon several times when it was about to set, its lower edge appearing to touch the surface of the sea. I found that it then gave about 2,000 times less light than when it had an altitude of 66° 11'. The same thing could then also happen to the sun, but this ratio is subject to very great variations. This doubtless arises from the fact that the lower part of the atmosphere is almost always unequally charged with vapours, and these vapours produce different effects; effects more appreciable on the light of celestial bodies which are rising or setting. We see also by the same reasoning that astronomical refractions, which are ordinarily the same at large elevations, are subject to great irregularities when the stars are very low.

[82]

ARTICLE X

Use of the preceding observations to di scover the transparency of a certain thickness of air

WE MAY REGARD it as one of the principal uses of these last observations to put us in a position to obtain a rather detailed knowledge of the transparency of the air. Mr. de Mairan had already made the ingenious remark, in the Memoirs of the Royal Academy of Sciences for 1721, 77 that supposing one could measure the ratio between the intensities of the light of a heavenly body at two different elevations it would not be impossible to derive from this the transparency or opacity of the atmosphere. This learned academician did not indicate any means of measuring light. He even doubted that the thing was possible.78 It was also necessary to discover the true law followed by light in its attenuation, when it passes through greater or lesser thicknesses of the same medium. This law had not been sufficiently well examined up to that time. I commenced by making some observations on the light that we received from the moon, and in looking into this question with more care I thought of a way of resolving this problem 77Jean Jacques d'Ortous de Mairan, 1678-1771, a physicist of some prominence, wrote on many subjects but seemed to have a leaning towards geophysics. The reference is to pp. 8-17. 78 De Mairan says that he had thought it impossible (in 1719, see Mem . Acad. R. des Sci., pp. 104-35) but that he now (ibid., 1721, p. 9) felt that it would require " ..... observations fort dijficiles, mais de la possibilite desquelles j e ne crois pas que nous soyons en droit de desesperer."

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effectively. I shall give an idea of this solution here in advance, [83] expecting to return to the same subject further on. Our experiments show us that when a celestial body is at 19° 16' apparent elevation its light is only two-thirds of what it is when the body is 66° 11' high. This difference in intensity only comes from the fact that the light has a much shorter path to travel in the atmosphere when the star is very high than when it is very near the horizon. It is true that in the first case the light already makes a rather large journey and that it must meet a great many parts of the air which are able to intercept it, but it traverses a considerably longer path in the second case, and it is the extent to which the second path is longer than the first which is the cause of the light being more feeble. We must then look for the two different quantities of air which the rays from the stars have to penetrate at the two elevations 66° 11' and 19° 16'. We can easily find them, for we have several systems which represent as exactly as is necessary the diverse condensations and dilatations of the air. 79 Now suppose the quantities of air which the rays have to traverse are equivalent to thicknesses of 4,275 fathoms and of 11,744 fathoms of our dense air [84] here at the surface. 80 We should conclude that it is in traversing a space equivalent to 7,469 fathoms of our air that the light loses about a third of its intensity, or that it decreases in the ratio of 1,681 to 2,500. 81 In fact the light should be equally weakened after having passed through equal thicknesses in one path or the other; that is to say, it should be precisely the same after having made the first trajectory and after having made the first 4,275 fathoms of the second. Thus the entire attenuation that we perceive when the star is near the horizon comes only from the last 7,469 fathoms of the second trajectory or from its excess over the first. If it were quite certain that the light was 2,000 times as weak when the star rises or sets than it is at 66° 11' of elevation, we could determine in the same way the degree of transparency of another thickness of air and we should only have to make new observations to determine an infinity of others. But it is much more worthwhile to examine in a 79This refers to the laws connecting barometric pressure and altitude. See below, p. 206. 80 Bouguer's note: See the table of masses of air towards the end of this work, Book III, Sect. V, p. 332. 81 Applying modern theory to this observation (see W. E. K. Middleton, Vision Through the Atmosphere, Toronto: University of Toronto Press, 1952), it turns out that the air with which Bouguer worked would have permitted a visual range of about 110 km. This is very clear air.

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general way, as we shall do below, in what proportion light diminishes when it passes through different thicknesses of transparent bodies. This proportion will not be difficult to discover. Then relieved by this theory, which will be applicable [85] to all media, from making new experiments without end, it will suffice that we should have one only, on the exactness of which we can count, to draw from it all the inferences and the knowledge we shall need. Thus it will suffice to know, for example, that light decreases in the ratio of 4 to 5 in going through a thickness of 10 feet of sea water, or that it diminishes in the ratio of 1,681 to 2,500 in traversing 7,469 fathoms of dense air, in order to be in a position to discover how much it is weakened in penetrating every other thickness of these same media.

ARTICLE XI Observations made to determine how much stronger the light of the sun is than that of the full moon AFTER HAVING repeated most of the preceding observations I wished to see how many times more the sun illuminates us than the moon. According to our method it was necessary to compare the light of these two bodies with that of a candle or a torch which served as a common measure, but since the light of the sun is extremely strong it was necessary to make it suffer extremely great attenuations and it was necessary that these attenuations should always proceed [86] by known degrees. For that I used a concave telescope lens which made the rays very divergent, and I had only to go a little closer to it or a little farther away to make the intensity of the light vary quickly and in whatever proportion I wished. I made one of these experiments on the twenty-second of September, 1725, the day of the full moon. Having closed all the windows of a room, and the sun being at an elevation of 31 °, I made its light enter through a hole which had a diameter of one line, against which I had applied the concave glass. Then receiving the light at a distance of 5 or 6 feet, at a point where the divergence of the rays was 108 lines, and where the light was in consequence weakened 11,664 times (since instead of occupying an area one line in diameter it occupied one which had a diameter of 108 lines and which was 11,664 times as great), it appeared to me exactly equal to the light of a candle situated at a distance of 16 inches. After that it only remained

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to make a similar observation on the moon during the night. I still had to use the same concave glass in order that its lack of transparency might cause a similar diminution in one observation as in another. I waited for the time when the moon had an elevation of 31°, but on receiving the light from near the lens and when the divergence of the rays was only 8 lines, it had so little intensity that I was obliged to have the candle put at a distance of 50 feet to make the two lights equal. [87] Now to find the result of these two observations I have but to consider that the light of the moon was weakened only 64 times by the concave glass, and that if we had made it diminish 11,664 times, as we did that of the sun, it would then have been necessary to place the candle not 50 feet away but 675. Now since the light of the sun when it is diminished 11,664 times is equal to that of a candle placed 16 inches away, as we have come to know by the first observation, and since the light of the moon, diminished the same number of times, is only equal to that of the same candle carried to a distance 675 feet or 8,100 inches, as we conclude from the second observation, it follows that the light of the sun is to that of the moon as 65,610,000 (which is the square of 8,100) is to 256 (which is the square of 16) . Thus it appears that the light of the sun is about 256,289 times as great as that of the moon. 82 I am reporting here only the third of the tests which I made; for at the full moon of July, 1725, I had found that the sun gives us 284,089 times as much light as the moon; at that of August, 331,776 times; and in another test, 302,500 times. I think we may conclude from all this that the sun gives us about 300,000 times as much light as the moon, but the difficulties in determining such a ratio are so great that I dare not regard it as too exact. However, it [88] is still worth remarking that the moon was at nearly its mean distance from the earth when I was making my observations. The different distances of the sun should not change the ratio of the two lights appreciably, because this body cannot move farther from the earth, without moving away from the moon at the same time, and almost in the same ratio. It follows from this that if its direct light is diminished, its reflected light, which comes to us from the moon, should diminish in the same way. But as to the diverse distances of the moon, they should produce a change, and a considerable change; for in relation to us that planet should be 62 According to the Smithsonian Physical Tables (9th ed. ; Washington, 1954), p. 730, the difference is 14.3 stellar magnitudes, a ratio of about 525,000 to 1.

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like a torch of which the flame has always the same intensity, but of which the light necessarily changes when it is received at appreciably unequal distances. The difference can be about one quarter, since the greatest and least distances from here to the moon are about in the ratio 8 to 7, and the squares of these two numbers are about as 4 to 3. 83 However it may be with these observations, they agree as well as is possible with what is known already of the intensity of moonlight. Experiments had been made showing that this light, collected in the focus of the largest concave mirrors, produced no sensible heat and did not even act on the thermometer of [89] Mr. Amontons, which is very sensitive. Mr. de la Hire the younger exposed the concave mirror of the observatory, which is 35 inches in diameter, to the rays of the full moon when it passed the meridian during the month of October, 1705, and he reassembled these rays into a space 306 times as small. 84 This was to increase the intensity of the moonlight 306 times at most, but even though, by using more perfect mirrors, such as clever experimenters have since made, one might succeed (though it has not yet been done) in augmenting the moonlight 1,000 times, it would still be an extremely long way from being equal to the light which comes to us directly from the sun. It would still only be the 300th part and it would not be surprising if in that state it produced absolutely no heat that could be felt. As to the light of the sun, it must be exceedingly great, for we find that it is here 300,000 times as strong as that of the moon and we are about 400 times as distant from the sun, which weakens the light 160,000 times. [90]

ARTICLE XII

Observations made to determine by how much the parts of the sun near its centre are more luminous than those which are near the edges of this body I will end this second section by reporting some attempts I made to discover whether each part of the surface of the sun radiates absolutely the same amount of light. When we look at this body through a long 83Ten

to eight would be nearer the modern value. P. de la Hire, Mem . Acad. R . des Sci., 1705, pp. 346-7. However, Lynn Thorndyke (History of Magic and Experimental Science (New York, 1958), VIII, 342) says that Geminiano Montanari wrote in his Astrologia convinta (1685) that a very delicate and sensitive thermometer showed a rise in temperature when the rays of the moon were turned on it by a big burning-glass. 84G.

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telescope which magnifies objects greatly, 86 we must pay a great deal of attention in order to notice any difference in the intensity of its light. It looks like a flat surface of which the brightness is, so to speak, everywhere the same. But the same thing may happen then as when we compare parts of the sky at a considerable distance from the sun. The light sent to us by these different parts of the atmosphere appears sensibly the same, since the change from one to the other takes place gradually or by insensible degrees; while if we succeed by some expedient in bringing certain of these parts together by suppressing, so to speak, all the intermediate space which separates them, we immediately realize that there is a considerable difference [91] in their light. Perhaps we must also use a similar expedient to make sure whether the sun is equally bright over all its surface when it is observed with a long telescope. To solve this problem I used the Heliometer, of which I spoke in Article VI I of the preceding section. The two objectives of the instrument furnished me two images of the sun which touched. These had the same intensity of light, since the two glasses were exactly equal and of the same aperture; but the surface of one of the two lenses being divided into twelve parts, I had to cover 3¼ of them, or change the surface of the two objectives in the ratio of 12 to 8¾, or 48 to 35, in order that the centre of the second image should not appear more luminous than a portion of the same size taken from the other image at three-quarters of the radius. Thus the light of the sun is not the same over all the surface of its disk. If one compares the centre and a place distant from it three-quarters of the semi-diameter the quantities of rays that one receives are in the ratio of 48 to 35. 86 This observation has its difficulties; the principal one is that of assuring oneself of the exact situation of the two parts of the disk of which the brightness is being compared. It is necessary that one of these points should be noticeably [92] removed from the centre to produce a greater difference in their light, but on the other hand it is very inconvenient to make this point too near the edge. I have observed it at 41 ° 25' distance from the edge, but I freely confess that I should have repeated this observation more times than I have done, although I have verified it three or four times. It is quite certain that the sun is less luminous in those parts of its disk which are most BliTelescopes were made very long to reduce the effect of aberrations. This was before the invention of the achromatic doublet. MFrom the data given by R. V. Karandikar (J. Opt. Soc. Am., 45 (1955), 483-8) this would be about 0.83, or 48 to 40.

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distant from the centre. Now it follows from this that not only does each luminous point taken separately on the surface of the sun not emit an equal quantity of light in every direction, but also that it emits less in more oblique directions. When we consider the points more nearly approaching the edges, this diminution is even greater than that undergone by the sines of the angles formed by the surface of the sun and the emerging rays. In fact, if each luminous point emitted an equal quantity of light in every direction the disk would become continually more luminous at a greater distance from the centre and it would finish with a circle of infinite brightness. Let us represent the sun by the globe ADB, Figure 13, and let the observer be situated on the extension of the

IC.

semi-diameter CD. This [93] observer will receive rays coming out perpendicularly from the sun only from the point D, and he will be illuminated from all the other points E only by oblique rays. We should also remark that he will relate all the parts Dd, Ee, etc., of the spherical surface to the corresponding parts Cc, Ff, etc., of the plane AB, which is perpendicular to the visual rays, and it is necessary that these latter parts should be equal in order that the first will appear to be so, or that they should correspond to equal spaces on the retina. In this case these parts Dd, Ee, etc., will be the greater the more obliquely

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they are seen. They will increase as we approach the edge of the disk in the same ratio that the sines EF diminish. They will thus attain a greater number of luminous points and the disk would necessarily be infinitely bright at the edge if each separate luminous point furnished the same quantity of light in all directions, since the last parts Ee are infinitely greater than the one in the middle, Dd. 87 This hypothesis being completely contrary to the observations, so that we are obliged to reject it, each luminous point must necessarily radiate less light the smaller the angles which the emerging rays make with the surface of the sun. Suppose for an instant that it is the sines of [94] these angles which express the quantities of light. We shall receive from the point D the quantity of light expressed by the total sine DC, 88 and that which comes to us from the point E will be expressed by the sine EF. Then the disk would be equally bright everywhere. For if a given part Ee is larger because of its inclination, and if it contains for this reason, a larger number of luminous points, on the other hand each of these points will send us less light in precisely the same ratio. Thus, under this second hypothesis, which might be found true in some cases, each of the parts Ee of an equal apparent extent ought to send us an equal amount of light, and all the surface of the body would appear of a uniform brightness. But, as we have seen, the visible parts of the sun's disk which are most distant from the centre are less luminous than the others. We are assured of this for the point E which corresponds to three-quarters of the semi-diameter CA and is 41 ~ 25' distant from the apparent edge A. It is difficult for us to mark with precision the interval included between the two points being compared, and this has an influence on the ratio between the two amounts of light. But we cannot doubt that the weakening in question is real. It is thus indubitable that the light radiated obliquely [95] by each particular point of the sun diminishes still more rapidly than the sines of the angles which the rays make with the surface of the sun where they emerge. Thus instead of taking the sines CD, EF, etc., for the quantities of light which each point D, E, etc., sends towards us, we must take some measure which diminishes more quickly, such as the lines DC, EG, etc., intercepted between the circumference of the circle and a certain curved line AGCHB. This This part of the subject was handled better in every way by J. H. Lambert. The trigonometric functions had still to be liberated from their geometrical derivation so that they could be used freely and easily as numbers. Nowadays we should merely assume at once that DC was unity. 87

88

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curved line, which we can know only by multiplying the observations, necessarily has two branches which form an angle of retrogression at C, and end at A and B. We also know, though less surely, that if we express by the total sine the light which the point D sends to us perpendicularly, that which comes to us obliquely from the point Eat 41 ° 25' from the apparent edge of the disk, is not expressed by the sine EF but by a line EG, smaller in the ratio of about 35 to 48. Thus GF is about a quarter of EF when AF is about a quarter of the semi-diameter AC. Instead of indicating the quantities of light which come to us from different points, we might be equally glad to have an expression for the different quantities of light which the same point sends out in all the different directions. If these quantities were exactly proportional to (96) the sines of the angles which the rays form with the surface of the luminous body at their points of exit, then for the point D we should only have to draw the circle DLK. DK, the prolongation of CD, is the diameter of this, and if we draw the chords DK, DL, etc., they will mark the ratios of the quantities of light projected into each direction DK, DL, etc. But the light radiated obliquely is still weaker, and for the direction DL which makes an angle of 41 ° 25' with the surface of the luminous body, it is expressed by DM, which is less than DL by about one-quarter. The number of rays sent towards all the different directions is not then expressed by the chords DK, DL, of the circle DKL, but by those of a kind of oval DMKM, narrower than the circle by about one-quarter. 89 89 Bouguer dealt with the sun as if it were a solid body with a definite surface; we now know that it has a complicated atmosphere which both emits and absorbs light .

BOOK TWO

Researches on the Quantity of Light Reflected by Surfaces Both Polished and Rough IN THIS SECOND BOOK we propose to give a more extended account of our observations on the reflection produced by mirrors and other surfaces. We shall continue to refer this reflection to that which would be given by perfectly polished mathematical surfaces which did not extinguish or weaken a single ray, and we shall commence with some remarks on the properties which these latter surfaces must have in this connection. We shall only say extremely simple things, but on these it is nevertheless absolutely necessary to insist, since we shall remove some difficulties which, without being very considerable in themselves, have nevertheless hindered the greatest physicists or mathematicians who have dealt with optics.

[98]

SECTION ONE

Remarks on the Reflection Caused by Perfectly Polished Surfaces Which Extinguish No Rays ARTICLE I On the virtual focus in reflection by plane surfaces, and on the intensity of the light which they reflect ALL OUR READERS know (and we have already obliged them to pay particular attention to this) that when light is reflected by a perfectly plane surface it takes precisely the same direction as if the luminous body were as much behind the surface as it is really in front of it. The rays of the luminous body P in Figure 6, after having struck the mirror B, take the direction BD as if they came from a virtual focus p. But what is peculiar to the plane mirror, and the real subject of the present remark, is that the point p does not change its position at all even when the reflection takes place from another point than B, or if one puts one's eye at any other place than D. It will [99] perhaps also be superfluous to add that the light has the same intensity at the beginning B of the reflected ray as at the end of the direct ray, since the reflection does not change this at all. If then we express by 1/ PB 2 the intensity of the light in the place where it strikes the plane we shall have 1/ PD 2 or 1/(PB + BD) 2 for the intensity at D, since the light follows the inverse ratio of the squares of the distances.

ARTICLE II On the virtual focus by reflection from curved lines IT IS NOT the same with curved surfaces, where the apparent position of the image is always subject to change with a change of the point where the light is incident. In the interests of simplicity we shall at first confine ourselves to the examination of what happens in the

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circle. If the rays LD and Ld leaving the luminous body L (Fig. 14) strike the circumference of the circle AIB in two points infinitely close together, D and d, and are reflected along the lines DM and dm, these reflected rays, when produced, will cut in G and will appear to come from this point, which will in consequence be the apparent position of the image of the luminous point L . If after this the position of the eye is changed, the apparent position of G will change also, being always found on the curved line OGI which the geometers know [100] by the name of caustic by reflection. This is formed by the succes-

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sive intersections of the reflected rays; it has them all for tangents. Our readers already know that in order to find the distance DG from the circle on the prolongation of these rays, we have only to make the following proportion: DL + ¼DF, the sum of the lengths of the incident ray LD and one quarter of the chord DF which is the prolongation of the reflected ray DM, is to one quarter of this same chord as the incident ray LD is to DG. To say this in a word, we always have DG = (¼DF X LD)/(LD + ¼DF).

ARTICLE III On double virtual foci by reflection from curved surfaces BuT A REMARK which does not appear to have been sufficiently insisted upon is that as regards convex or concave mirrors we never have surfaces which can be considered absolutely as simple curved lines. They have always really two caustics and at the same time two different images. It is the more essential that we should pay attention to this, because these two images sometimes have everything necessary to be seen equally distinctly. This has nothing in [101] common with those which are repeated by our ordinary mirrors because of their double surfaces, that of the glass and that of the quicksilver. The surfaces which are being considered here are curved but absolutely simple. Instead of the circle AIB let us then consider a perfectly spherical surface. It will still be true that the incident rays infinitely near one another which fall at D and at d will be reflected as if they came from the point G, but it will be no less certain that infinitely neighbouring rays which fall to one side of the point D as at o will not come to the eye as if they came from the point G, but as if they came from a point E in which the reflected rays cut the axis BA of the globe which passes through the luminous body. In fact the incident rays Lo and the reflected rays oµ are in a different plane from the rays LD and DM and these two planes have CL for their common section. It is therefore necessary to distinguish two different virtual foci of the reflected rays according to whether the incident rays fall above or below Dor whether they fall to one side. 1

tThis is not the place to enter into the long history of geometrical optics. The effect mentioned by Bouguer is one of the several aberrations produced by reflection from a spherical surface.

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If the aperture in our uvea were only an extremely narrow vertical slit, as we see it in some animals, we should receive only the rays which have the point G for virtual focus, or which [102] appear to come from there. We should receive them because together they form, as it were, a lamina MGm, of which the width is in the vertical direction. If on the contrary our uvea had the form of a horizontal slit we should only be affected by those rays which appear to come from the point E and which form a plane MEµ of which the width is in the horizontal direction. But since light enters our eyes through a perfectly round hole the two images can act on our retina at the same time and if, in spite of this, we almost always discover only one of them it is principally because our eyes are not adapted to discovering with the same distinctness the smallest objects at all distances, and because only one of the two images is at a suitable distance for our vision. In this case our eyes have only to choose between two images, but they sometimes succeed in sorting out the most suitable one among a much greater number. When we observe the stars with a long telescope it may form seven different images in a row, so to speak, on the axis of the telescope and placed at different distances from the objective according to the different refrangibility of the seven primary colours. The red image is the closest to the observer, the orange image a little farther away, the yellow still a little farther, etc. If the constitution of the atmosphere [103] remained the same, and if the change in the height of the star were not considerable, we should always discover the same image, that which is at an appropriate distance from our eye and of which the colour is at the same time the most suitable to make an impression on our retina. But if the air suddenly becomes charged with vapours, or if we take the star at two very different elevations, which may weaken certain rays more than others, we may, as I first observed in Peru, discover another image which will sometimes lead us into considerable errors, possibly more than 20" of arc, in the situation of the star which is being observed. 2 This remark is the more important in that it indicates one of the difficulties which prevent us from carrying the precision as far in the determination of the right 2 Bouguer's footnote: See the book De la figure de la terre determinee par les observations f aites au Perou, page 207 and following. Translator's note: The reference in Bouguer's footnote is to P. Bouguer, La Figure de la terre determinee par les observations de la Condamine et Bouguer (Paris, 1749). The point seems to be that the parallax resulting from the longitudinal chromatic aberration of the objective makes the coincidence of the star and the cross-hairs very sensitive to eye position. Bouguer did not believe in the use of an eye ring (ibid., p. 213).

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ascension of stars as in that of their declinations. Since attention has not been paid to this change of the focal length of telescopes, it is only too certain that among the great number of observations that we have there must be many which have not all the accuracy on which we now pride ourselves. In order to avoid this source of error absolutely we must employ precautions which have not perhaps been sufficiently well known, but this is not the place to explain them. An object simply plunged into water [104) furnishes a special example of the doubling of images. If BAb (Fig. 15) is a portion of the surface

of water, and if the object is at 0, we can perceive its image in two different places, either at G on the caustic by refraction or at Eon the vertical straight line AO which passes through the object, and which is not less a caustic than the other line. The first image is visible by means of rays such as ODM, Odm, which direct their path higher or lower one than another in coming towards us, while the second image, that which we see as in E, is formed by the rays ODM, 0~µ, 3 which are arranged one beside the other. We shall discover one or the other and it is also quite possible that of the two only one will be formed in the eye, and this will be composed of their common part, while the parts which spill over reciprocally can only produce the bad effect of rendering vision less distinct. This is the solution of a difficulty which greatly occupied the Reverend Father Tacquet, Barrow, and Smith, as well as several other authors, and which Mr. Newton himself 31n

the text, Odm; but Figure 15 requires Ooµ.

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regarded as forming a very spiny problem, even if it were not really insoluble. 4 (105]

ARTICLE IV

On the intensity of light reflected by an infinitely polished convex spherical surface WHAT WE HAVE just said is not at all foreign to our subject, but we shall return to something which is still more pertinent. We must, as we have just seen, distinguish two different groups of planes of rays in the bundle reflected by each visible part of the surface of the globe in Figure 14. These planes take the form of very narrow fans and they have their breadth in the vertical direction or in the horizontal according to whether they come from the point G or from the point E. It is also evident that the light diminishes in each plane, not as the square of the distance to the virtual focus but as the distance itself, since the widths of the fan increase simply as the distances. But what must be carefully noted is that the multiplicity of planes of rays which leave one of the foci is expressed by the number of rays contained in the planes which come from the other. Thus we must multiply together the two numbers of rays contained in the vertical and horizontal planes in order to have the total quantity of rays. To explain this in another (106] way: the bundle of rays increases in size as it becomes more distant from the globe, and if its thickness is represented by the angle MGm formed by the rays which are directed one above the other, its width is represented by the angle MEµ formed by the rays which proceed side by side. The totality of rays thus continually occupies greater and greater spaces, of which the extent depends equally on the distances from the two virtual foci G and E. From this it follows that we must multiply these distances by one another in order to have the inverse ratio according to which the light is weakened as it advances. Thus in the present case it is not 4Bouguer's footnote: See the Scholium of proposition VIII of the Lessons on Optics of Mr. Newton. Translator's note: The complete reference is to the Scholium of Prop. VIII of Section III of the Lectiones Opticae of Newton. He refers in this scholium to Lecture V, Article 13 of Barrow's Lectiones Opticae (London, 1669). Bouguer's reference to Smith is to Robert Smith (1689-1768), A Complete Sys/em of Opticks (Cambridge, 1728). Father Andreas Tacquet (1612-60) wrote a Catoptrica tribus libris exposita, published posthumously in 1669. The phenomenon would be described nowadays as the astigmatism of a plane refracting surface, but is not felt to be a paradox.

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weakened in the inverse ratio of the square of one of the distances, but in the inverse ratio of MG X ME. Quite close to the globe the intensity of the reflected light is exactly the same as that of the incident light. This latter intensity is smaller, the greater the square of the distance to the luminous point L, and we can express it by 1/ LD 2 • This same expression will do for the light when it is first reflected, since the rays make exactly the same angle with the surface of the globe in their reflection as in their incidence, and since nothing at this place alters the thickness of the bundle. But as the light goes away from the surface which reflects it, it [107] is distributed in areas µm which, as we have just seen, are greater the more distant they are from the virtual foci E and G, and the light must consequently be more feeble in the same ratio. To find its force or intensity at M we have then only to make the following proportion: the product of the distances ME and MG is to that of DE and DG as the intensity of the light on arriving at Dis to its intensity or force at M. Supposing then that we take I/LD 2 for the intensity of the incident light near the point D; we shall have (DE X DG)/(LD 2 X ME X MG) for the intensity of the reflected light received perpendicularly at M .

ARTICLE V Showing that the light reflected in every direction by the globe is of exactly the same intensity when we receive it at a very great distance and when the luminous point is also very Jar away from the globe WHENEVER the luminous point L is very far away from the globe the distance DG from the virtual focus G to the point D where the reflection takes place becomes exactly one-quarter of the chord DF. For in the expression (¼DF X LD)/(LD + ¼DF) for this distance DG, the term LD in the denominator becomes infinitely great in relation to the other, [108] which it makes negligible, and the expression reduces to ¼D F. This condition occurs whenever the incident rays come from such a great distance that they may be regarded as parallel to one another. In this case we may always take DG as one quarter of the chord D F, which is the prolongation of the reflected ray, and this point will be the virtual focus of the rays which fall on Dd. As regards the other focus, the greatest or the least distance of the luminous body does not make it leave the line CL. It is always exactly at the inter-

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section E of the reflected ray and the diameter which is parallel to the incident rays. We may also make another remark which is just as important. When the luminous point is at an infinite distance, the product of DE and DG, which is one of the terms of the proportion of which we made use in order to find the intensity of the reflected light, is always constant and is exactly one-quarter the magnitude of the square of the semi-diameter of the globe. In fact, the rays LD being parallel to CA, the angle DCA, which is then equal to the angle LDK, is also equal to the angle KDM and to the angle CDE. Thus the triangle CED is isosceles and similar to the triangle FCD, and the product of DE and DF will always be equal to the square of the semi-diameter CD. The product of DE and DG will always be one-quarter [109] of this, or equal to the square of half the semi-diameter. But if this product is constant, that of the distances ME and MG multiplied together will be no less constant if we suppose that the reflected light is always received at the same distance, which is great enough to enable one to regard the diameter of the globe as negligible in comparison. It follows from this that the intensity of the light suffers an equal diminution in every direction by reflection from the globe. If we admit the two suppositions that the luminous body is infinitely far away and that the reflected light is received at a distance which can be taken as infinitely great but always the same, the direct light falling on the globe has everywhere the same intensity, since all its rays are considered parallel, and since the distance of each point of the globe to the luminous body may be considered the same. Furthermore, this light is then subject to an equal diminution in every direction, since it is weakened in the ratio of the two products MG X ME and DG X DE, which are then constant. In this way we see the truth of this new and curious theorem which will be extremely useful to us in what follows, that when the luminous body is infinitely distant and when it illuminates a globe the surface of which is infinitely polished and which absorbs no light, this globe will reflect in every direction a light of the same intensity, provided [110] that we receive it from a great distance. We must except only the place where the shadow of the globe falls, but this shadow is only a single point in comparison with the immensity of the whole spherical surface which receives the reflected light. We are insisting on this property in the interests of greater clarity. The reflected light which is distributed equally into such great spaces is all furnished by only one-half of the globe, and if it were possible

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that it could be furnished by the entire globe, the ratio between the two lights would be expressed by the inverse of the surfaces or the squares of their semi-diameters. The reflected light would be as much weaker as the square of the distance which it reaches is greater than the square of the radius of the globe. But it is only one-half of the globe which receives the direct light and reflects it, and we must again remark that the quantity of the direct light is not expressed by the surface of the globe, because of the obliquity of almost all the parts of the surface which are, so to speak, flying away. It is expressed only by the area of the plane of one of the great circles, which is four times as small as the entire surface of the globe. It is thus, to speak properly, an amount of light expressed by one-quarter of the surface of the globe which spreads at a distance on to an entire spherical surface. Now it follows from this that its diminution cannot follow the ratio of the squares of the semi-diameters but [111] should correspond to one of these squares and to one-quarter of the other. This is the physical explanation, so to speak, of a part of the theorem which we have proved geometrically.

ARTICLE VI On the quantity of light reflected by surfaces on which perfectly polished small hemispheres take the place of wrinkles or roughness KNOWING THE INTENSITY of the light reflected by globes of which the surfaces are infinitely polished, it is easy for us to see what phenomena would be furnished us by surfaces which were covered with an infinity of tiny eminences formed in the shape of hemispheres, but of which the individual surfaces were always perfectly polished. If instead of the illuminated half of the globe of Figure 14 we consider a plane equal to its base, or in the plane of one of the great circles of the globe, and which would be entirely covered with small hemispheres of greater or lesser size, it does not matter in what ratio, this plane, reflecting light into an infinity of directions, will still radiate exactly the same number of rays as the globe, provided that it is observed from the same very great [112] distance, and that the luminous body is also at a distance which can be considered infinite. Figure 16 represents this circular plane, entirely covered with small hemispheres of various sizes which take the place of roughness and wrinkles. I ts diameter is the same size as the axis of the globe, and we

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I

suppose that by multiplying the number and varying the size of the little hemispheres with which the plane is filled, no empty space is left. These conditions being admitted, the two lights reflected towards each direction will be equally intense. 6 This is because the great number of small hemispheres compensates perfectly for their small size. It is true that each small hemisphere sends out less light the smaller it is, the number of rays which it reflects being proportional to the square of half its semi-diameter, but on the other hand the number of hemispheres will be greater, precisely in the same ratio. If the diameter of each little hemisphere of the plane of Figure 16 is one thousand times as small as the diameter of the globe of Figure 14, each little hemisphere will reflect into each direction a light which will be one million times as weak as that sent out by the globe, but for the same reason, that the diameter of each small hemisphere is one thousand times less than that of the globe, there will be one million hemispheres in Figure 16, which will contribute to strengthen the total light [113] and make it equal to that of the globe. On every other supposition the compensation will not be made less exactly. Thus it is evident that the light reflected in a certain direction will still be exactly the same whether it is sent out by a very large number of small hemispheres or by a smaller number of larger hemispheres, or by one only, as in Figure 14, provided that they occupy exactly the same base or cover the same plane P IQI (Fig. 16). Nevertheless it is necessary for the truth of this theorem that in oblique reflections the little hemispheres should not interfere with one another by intercepting each other's light, as otherwise they would not be in the same situation as the half globe of Figure 14. To fulfil this condition we might suppose that the hemispheres have different 6 I.e., the light reflected by the one large hemisphere and that reflected by the surface shown in Figure 16. This will not be true in directions near the general surface.

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sizes according to a certain rule. We should still have some other restrictions to put in, but we believe we can do without them, while freeing ourselves from the majority of the physical inconveniences which present themselves. We shall only examine this matter now in an abstract manner which will, however, have its applications in what follows. If we wish to compare the intensity of the light reflected by the surface IP IQ (Fig. 16), clothed with all its little hemispheres, with that of the light reflected by the [114] same surface when it is perfectly polished, the comparison will be quite easy provided that the diameter of the surface is small enough in relation to the distance from the luminous body and to the distance at which the reflected light is received. It will be remembered that if we express the intensity of the light by 1/LC2 when it arrives at C it will be expressed at O by 1/(LC CO) 2 after reflection on the mathematical plane. As regards the same surface covered with small hemispheres, we still simply make the mental substitution of one hemisphere in the place of all the small ones, and as 1/LC2 continues to express the intensity of the light at the distance LC from the luminous body, we have only to diminish it in the same ratio that the square ¼/C 2 of half the semi-diameter of the large hemisphere which we imagined, is less than the square of CO. That is to say, the intensity of the light at O will be ¼IC 2/(LC2 X CO 2). Thus the quantities of light reflected by the perfectly smooth plane and by the plane covered with little hemispheres will be as 1/(LC + CO) 2 is to ¼JC2/(LC2 + CO 2). In addition the reader will doubtless remark that the mathematical plane reflects light only into a certain region, leaving everything else in the shade without a single ray, and that if we increase its extent the intensity of its light does not increase. The plane covered with little hemispheres, on the contrary, [115] illuminates every direction equally, and if its extent is augmented the light which it sends out will really become greater because each of its parts contributes to reflecting light into the same directions. This is also indicated by the general expressions for the two terms of the ratio. It will be more difficult to find the intensity of the light reflected by the surface covered with small hemispheres if the diameter of this surface is considerable in relation to the distances CL and CO. One could divide the surface into infinitely small parts by mutually perpendicular co-ordinates. Each part will be a small rectangle which will be equal to a small circle of which one will find the diameter. This

+

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small circle could hold a hemisphere equivalent in itself to all the other smaller ones which are contained in the little elementary rectangle. Then after the distance from each of these small rectangles to the luminous body had been found, and after we had taken unity divided by the square of this distance for the expression of the intensity of the light when it came from the small rectangle, we should only have to diminish this in the same ratio that the square of the distance to the point O is greater than the square of half the semi-diameter of the small circle which is equal to the small elementary rectangle, and we should obtain the effect of it. Finally, as this effect will itself be an element of the light reflected towards O and an infinitesimal of [116] the second order, we have only to integrate twice, which can be done by using only the quadratures of the circle and the hyperbola, and we shall have the total intensity of the light. But we return to the supposition that the surface IP IQ is small enough; and we shall make a remark which will perhaps be found to be rather important, although it has much connection with things said already. If in this particular case the quantity of the light reflected in each direction is exactly equal to that sent out by the globe of Figure 14, the phenomena nevertheless become quite different when we look directly at the globe or at the plane covered with little hemispheres. On the globe we shall see, it is evident, only one small luminous area, which will be the actual image of the luminous body L, while the plane of Figure 16 will appear entirely illuminated in consequence of the effect of each small hemisphere, and we shall see a uniform light spread over all its surface, supposing that the hemispheres with which it is covered are small enough so that we cannot discern separately the tiny luminous points furnished by each of them. Another difference as regards the plane is that while the total quantity of its light will be the same everywhere, the more or less oblique position of the eye will nevertheless cause a change in its appearance. The [117] plane will appear more or less narrow. It will occupy more or less space on the retina, and the image, always painted with the same number of rays, will acquire more or less brightness in the eye. The intensity of the light will increase in the same ratio as the image becomes narrower or less extended . It will increase in the same ratio that the sine of the obliquity of the visual rays in regard to the plane becomes smaller. 6 6The effects of multiple reflection and shadowing make the phenomenon much more complicated than this would indicate. This article and the two following, it is sad to say, really lead nowhere.

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ARTICLE VII On the light reflected by a globe the surface of which is covered by an infinite number of perfectly polished small hemispheres

Now LET us suppose that it is not the plane of Figure 16 which is covered with an infinite multitude of small hemispheres, but the surface of the globe of Figure 14 itself. The number of small hemispheres which will cover half the globe will be, as is well known, double the number of those which are necessary to cover the plane of Figure 14. Thus, each small hemisphere producing its effect, the light reflected by the globe, when it is received in the direction of the luminous body itself, will be exactly double what it was before. Its intensity will then be found by this proportion: the square of the distance at which (118] the reflected light is received is to half the square of the radius of the globe as the intensity of the direct light failing on the globe is to that of the reflected light. If, instead of receiving this reflected light in the direction of the luminous body, we receive it in some other direction, it must necessarily be less, since we cease to be illuminated by the same number of small hemispheres. If, for example, we are situated in the direction which makes a right angle with CL we shall receive half as much light as if we were situated very near CL.

ARTICLE VIII On the light which would be sent to us by the planets, and particularly the moon, on the two preceding hypotheses, namely (1) supposing their surfaces were perfectly polished, (2) that they were covered with an infinite number of perfectly polished little hemispheres THE FIRST HYPOTHESIS contradicts all the observations if we try to apply it to celestial bodies which reflect the light of the sun to us. If their surfaces were perfectly polished, all their disks would appear completely dark except for one point, which would be the image of the sun. Besides this, this image seen in the planet would always give us the same (119] quantity of light. When the moon is in quadrature, or even in the neighbourhood of its conjunctions, it would illuminate us just as much as when it is in opposition. This light, although it would appear to come to us from only one point of the planet, would still be very considerable. We know the

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ratio between the diameters of the planets and their distances from the earth. By applying the proportion which we found in the preceding article, this is all we need to know in order to be able to discover how much weaker the light they reflect to us would be than that which they themselves receive from the sun, if their surfaces were perfectly polished. We learn by the proportion indicated above that for the moon the ratio in question would be expressed by unity compared to about 190,800. As we are a little closer to the sun than the moon is when it is in opposition, it is true that the light which we receive from the first of these two bodies is a little greater than that which is received by the moon. But this is only about one two-hundredth part. Thus the light reflected by the surface of the little planet, supposed perfectly polished, would be a little less still in relation to the light which comes directly from the sun. It would be the 191,700th or 191,800th part of it. [120] But let us pass to the second hypothesis. Let us suppose that the surface of the planet is entirely covered with small wrinkles or bumps which are absolutely equivalent to small hemispheres, each with a perfectly polished surface. Let us further suppose that these small hemispheres produce such a bright reflection that they absorb absolutely no rays. Then, each little hemisphere furnishing a tiny image of the sun, every part of the planet exposed to our sight and illuminated by the sun would appear luminous. Then the moon and all the other planets would appear to us exactly in the way we see them. The full moon would give twice as much light as the moon in quadrature. Its light would always be exactly proportional, not to the apparent size of its phases, but to the real size of the illuminated part which was turned towards us. Another very remarkable peculiarity is that, other things being equal, the edges of the disk of the planet would appear brighter than the parts which are nearer the centre, for the reason that the parts at the edges which appear to us to be equal are really larger, and contain a larger number of wrinkles or elevations which present directly towards us part of their tiny surfaces, each of them sending us light. [121] This is what we actually remark in the primary or secondary planets properly so-called. The light from their edges is stronger than that from their centres, and they differ essentially in this from the sun, of which the light, as we have observed, decreases as one considers points in its disk more distant from the middle. 7 1 Bouguer was in error about the planets, which are actually darker at the edges. See H. C. van de Hulst, in The Atmospheres of the Earth and Planets (Chicago: University of Chicago Press, 1949), p. 98.

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Finally, if we continue to admit the second hypothesis, the light of the moon in opposition would, as we have seen, be twice as strong as if the surface of the planet were perfectly spherical, without any roughness. Thus instead of sending us an amount of light which would be 1/191,700 or 1/191,800 of that which we receive from the sun, it would send us back 1/95,900. However, the direct observation of which we gave an account towards the end of the preceding book gave a result only one-third as large, since we found that the light of the moon is only the 300,000th part of that of the sun. But it may be that the moon is not covered with wrinkles or elevations which are equivalent to small hemispheres, and besides this it is certain that these asperities, or all the parts of the surface of the planet, are not equally well adapted to reflect light, and that some of them absorb a very considerable quantity of it. [122] In fact we see in the moon spots five or six times as dark as others, which reflect five or six times less light in proportion to their extent. I found this ratio on March 4, 1758, at 4 o'clock in the morning, between the darkness of Grimaldi and that of the middle of the Mare Humorum. It is true that Grimaldi, which had always appeared to me to be the darkest of all the spots, does not occupy a large part of the disk of the planet, but the other dark parts are much more extended and it is not astonishing that, taken together, they should cause a decrease in the light of about two-thirds of the total. There is nothing in this which we cannot also notice down here in terrestrial objects, of which some reflect more than two-thirds of the rays that they receive, others only half, and others incomparably less. We shall not fail to examine this subject again; and at least in regard to terrestrial objects, which we can more easily submit to our experiments, we do not despair of succeeding in finding out whether the weakening of the light which they reflect is principally caused by the extinction of the rays, or whether it results from the fact that the small elevations or wrinkles are not equivalent to little hemispheres.

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SECTION TWO

Researches on the Quantity of Light Effectively Reflected from Physical Objects of Which the Surfaces are Polished ARTICLE I In reflection there is always a part of the light which is as if extinguished IT MAY BE DOUBTED whether there is in nature any object which reflects all the light it receives and which does not absorb a considerable part of it. If the body is coloured it sends back chiefly rays of the same colour and weakens almost all the others, but even a body which is illuminated only by a light of the same colour as itself must still extinguish part of it. If it is struck by a very strong light it is heated more or less noticeably, and it is difficult to see how it could acquire this heat unless a certain number of the rays dissipate part of their force against it by penetrating it more or less, and by exciting an agitation of their own in the little molecules of which it is formed. This agitation or shaking is (124) nothing else than a communicated motion, which cannot help answering to some loss of motion in the light itself. However this is accomplished, the intensity or the number of rays must necessarily diminish. 8 In fact, even if the surface of mercury, and mirrors to which art has succeeded in giving the brightest polish, produce a strong reflection, nevertheless these surfaces do not reflect all the light which strikes them. When I brought into the dark room represented in Figure 9, daylight which I took at 11½ 0 of elevation, and made it fall on the surface of quicksilver, I had to establish a ratio of 6,400 to 4,826 between the two openings P and Q, on one occasion, in order to render this light equal to the direct light. Thus of 1,000 rays the mercury 8The existence of infrared radiation was, of course, quite unknown to Bouguer; but nevertheless his argument is sound. This paragraph is interesting from the standpoint of the history of theories of heat.

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reflects 754 under an angle of incidence of 11½ 0 • 9 On another occasion I was obliged to set the ratio between the two openings at 6,400 to 4,497, which gave me a result slightly different from the preceding. Of 1,000 rays which struck the surface of the quicksilver only 703 were reflected. Nearly the same thing has occurred to me when making experiments on metal mirrors. It follows that at least a quarter of the light is lost by reflection, and [125] apparently there are no bodies which extinguish or deaden light less than this. 10

ARTICLE II Showing tlzat the number of rays reflected at dijferent inclinations Jzas not a constant ratio to the number of incident rays BuT WHAT STILL more clearly distinguishes the physical surfaces with which we are now dealing from the mathematical surfaces which we considered in the preceding section is that in reality reflection is not equally strong under all angles of incidence. In general it is stronger under small angles and weaker under greater ones. The difference is very great when rays strike the surface of very transparent bodies with different degrees of obliquity, but it is almost as great for certain opaque bodies and I have never seen it absolutely lacking for any. The body which has yielded me the greatest inequality is black marble. I examined the reflection which it produces according to the various procedures which correspond to Figures 3, 4, and 6, and which I explained in Articles III and IV of the first section of the preceding book. First of all I was astonished to see that under an angle of incidence of 3° 35' this piece of marble, although it was only of a [126] very mediocre appearance, nevertheless furnished me a reflection approaching that given by quicksilver. Of 1,000 rays which struck it, 600 were thrown back. But under an angle of incidence of 15° it only reflected 156 out of 1,000 rays; under an angle of 30° it sent back only 51; and under an angle of incidence of 80° it reflected only 23. Everything being disposed as in Figure 3, and the interval DE between the two boards D and E being 600 parts, I was obliged to put the candle at a distance of only 79 parts from the first board.

9Bouguer's note : In what follows the angle of incidence of rays will always be taken as that which they form with the surface and not with the perpendicular to it. 10 Bouguer's measurements of specular reflection, here and elsewhere, often seem low, but not always (see Book I, n. 24).

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Similar experiences made on metal mirrors gave me very much less noticeable differences. The greatest was found to be scarcely oneeighth or one-ninth part, but always in the same direction. As regards the reflection from quicksilver, I could observe it directly only by means of the dark room into which I introduced daylight, and I was at liberty to bring in this light only down to a certain elevation because of the arrangement of my lodgings. For an angle of incidence of 21 ° I once found that out of 1,000 rays, 637 were reflected; and at another time 666. After that I compared the reflection which it gave under all the other angles with those given by metal mirrors. I placed the mercury and the mirror side by side [127] behind the frame GH in the dark room of Figure 9, and I employed the second of the methods which I explained in Article V of the first section of the preceding book. It appeared to me that the quicksilver surpassed the metal mirror, principally under the smallest angles of incidence, but very little, and the difference was hardly perceptible at large angles. I made mention above of the difficulties which presented themselves to me in the experiments that I made on metal mirrors. I experienced even greater ones in the observations that I made on quicksilver. Wishing to satisfy myself fully, and dissipate all my doubts, I looked at everything several times and paid the matter more attention, but as I gave it more time I almost always remarked that the reflection became weaker. The place where I was observing was dark and well closed up, and it was warm weather, and I finally recognized that the continual weakening which I noticed in the reflection resulted from the facility with which the mercury became tarnished, merely by contact with the air, which was probably somewhat humid. Besides this, in spite of every precaution one might take, imperceptible particles of fluff are always flying about in the apartment, which, falling on the surface of the liquid, do not go into it because of their very small specific gravity. These little floating bodies then throw [128] their shadows a long way and so intercept a great number of the reflected rays whenever the angles of incidence are very small. This accident does not take place to the same extent when one makes experiments on other liquids. They are less heavy and the little pieces of fluff plunge into them almost entirely. There is still another source of error in the experiments made on mercury, when the vessel which contains it is not sufficiently large. We know that the surface of this liquid is not perfectly plane, especially if it is of small extent; it is convex, and its convexity must weaken the intensity of its reflected light, since rays which should be sent

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back parallel become divergent. We use a larger vessel to avoid this trouble; but we sometimes fall into quite the contrary difficulty, which one might perhaps not think of-the convexity of the liquid changes into a concavity. This happens when the vessel is circular and has a diameter of about 20 lines. 11 When the vessel is full, the edges of the mercury present a large convexity, turned outwards. As the particles of the liquid in this situation are on the point of falling, only their mutual adhesion prevents it; they remain attached to each other (129) as far as the centre of the surface, forming a sort of chain which holds them. But is it not evident that the last parts must yield a little because of the effort which they contribute in sustaining the first parts, which because of their weight tend to be thrown outwards? The inner parts thus advance a little towards the edges, leaving a kind of empty space towards the middle of the surface, or a slight concavity. I have actually observed this. For the better understanding of this effect, on which we are insisting because of its singularity, we may imagine a series of drops of mercury on a horizontal plane. These, touching each other at their sides, form a circular crown, in the middle of which there will be an empty space. If one distributes additional quicksilver in different parts of the crown, it will acquire a new width, and the empty space in the middle will shrink little by little. We have only to continue to add new mercury until we have enough so that finally the drops may touch at the centre, making the empty space in the middle disappear, and nevertheless the quicksilver will at first have less depth than at the edges. It follows from all this that the vessels must have an appreciably greater width, say 4 or 5 inches in diameter. This is the only way we can have perfectly plane surfaces, supposing, however, that (as I suspect) there are not then formed several concentric crowns (130) decreasing in height as they are more distant from the edges, or nearer the centre. The dimensions of stationary circular waves of this kind, and even their number if there are really more than one of them, must depend on the width of the vessels, and on the ratio between the specific gravity of the liquid and the cohesion of its particles. Although it is difficult to observe the reflection produced by liquids, chiefly because their surfaces are always horizontal, I have nevertheless carefully examined the reflections produced by water. I chose that of 11 This statement has no support from the theory of surface tension, and one can only suppose that Bouguer was the victim of some optical illusion. The explanation that follows is rather fanciful.

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Arcueil. I thought that water might serve for a standard of comparison, and would furnish me with more certain results, since there is without doubt less difference in this respect between various pure waters than between different kinds of surfaces which depend on the particular nature of the bodies to which they belong, and also sometimes on the perfection of their polish. (131]

ARTICLE III

On the different intensities of reflection produced by the surface of water, according to the dijferent angles of incidence of the light

To THE REFLECTION produced by the surface of water from Arcueil, I have applied almost all the various means of measuring light that I have devised. At different times I used daylight, which I introduced into a dark room through two different openings. I also compared the reflected light with that sent back by metal mirrors at the same angle, or by the surface of a triangular glass prism which I laid beside the water's surface. It was then only a question of examining the reflection by these latter surfaces separately. Having determined by these or other means the quantities of light reflected by water, and knowing already those reflected by mercury, I undertook a final verification by comparing these two reflections with one another directly, several times. When I let daylight fall on the water surface in the dark room at an angle of incidence of 13½ 0 , I had to set the ratio of the two openings, from which I derived the direct and (132] reflected light, at 11,644 to 54,756, so that the two lights should become equal. Thus of 1,000 rays which fell on the surface of the water it reflected only 213, or 244 according to another observation. I did not content myself with introducing the light into the dark room at this angle only; but I think I should avoid giving too many details. I will only add that for an angle of incidence of 25°, the size of the opening which was destined to furnish the reflected light being still expressed by the number 54,756, the other had to be made 5,274, showing that of 1,000 rays only 97 were then reflected. 12 2'fhe values calculated from Fresnel's equation R· = 1 s~i_n2c-'('-ci_-----'r) + 1 t_a_n2~('-ci_-__,_r) 2 tan 2 (i + r) 2 sin2 (i + r) ' are 0.245 and 0.086 at the two angles mentioned by Bouguer. This is quite good agreement. 1

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From these observations we already see how greatly the light reflected by water is diminished at large angles of incidence, or when the rays form a large angle with the reflecting surface. We still continue to take inclinations or angles of incidence from the surface, and not from the perpendicular. But the difference becomes much greater still when we compare the smallest inclinations with those which approach a right angle. I have sometimes been undecided as to whether the reflection at the surface of water is not stronger at very small angles of incidence than that from the surface of quicksilver. Everything considered, it is not quite as strong, although I think it would be very difficult to state the exact difference. [133] I will end by saying that the light reflected by water at the smallest inclinations is just about three-quarters of the direct light. 13 There is no one who has not felt the effect of this strong reflection when walking in clear calm weather on the edge of a pond opposite the sun. The reflected light is sometimes a third, a half, or even a greater part of the direct light received from the luminary, and this addition cannot fail to be very evident. As the sun approaches the horizon the direct light goes on diminishing in a manner which we shall try to specify below, while at the same time the reflection becomes proportionately stronger. Thus there is a certain elevation of the sun which makes its total heat, the united action of the direct and reflected light, as strong as possible. That is the subject of a problem, and I am satisfied to have indicated all the principles of its solution, simply leaving some calculations to be made in order to finish it. The elevation of the sun which gives this maximum is 12° or 13°. 14 If on the contrary we consider the reflection produced by water at large angles of incidence, we find it extremely feeble. I have assured myself that for almost perpendicular incidence this reflected light is scarcely 1/37 of [134] that reflected by quicksilver; and it results from all my observations that it is then only a sixtieth part or rather a fifty-fifth, of the direct light. 15 At an angle of 50° the light reflected by the surface of water is about 1/32 of that reflected by mercury. Since it goes on increasing as the angle becomes smaller, it is already twice as strong in proportion at an angle of 39°, for it is then 1/16 of that reflected by quicksilver. 13At

grazing incidence, all the light is actually reflected. calculation on the basis of the best available data for the very clearest air suggests a maximum at about 6° solar elevation. But Bouguer seems not to have realized that a very moderate amount of turbidity would quite remove this maximum. 16At normal incidence it is (1.33 - 1)2/(1.33 1)2 = 1/49. 14A

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ARTICLE IV Construction of a table which indicates the quantities of light reflected by water at all different inclinations

IT WOULD BE useless to try to construct a table of the number of rays reflected by water at all the different angles of incidence by employing absolutely nothing but observations. We should have to make an extremely large number of experiments; we should still be reduced to taking the mean between different results, and in spite of this the numbers of the table would not follow each other well and we should remark irregularities of too great a magnitude. It is much more worthwhile to base the work on a certain number of chosen observations, and to infer from them all the other numbers of (135] the table in a way which will at the same time agree as well as possible with all the other experiments. Expressing the quantity of direct light by unity, I have taken for the reflected light an indeterminate expression A + Bz 3 + Cz 6 , in which z indicates the versed cosine of the angle of incidence, or the excess of unity, taken for the total sine, over the right sine of the angle of incidence. I needed only three observations in order to determine the coefficients A, B, and C. The angle of incidence 90° makes z = 0, and the first term A should then itself indicate the quantity of the reflected rays, since the others disappear. This quantity, as we have seen, is the fifty-fifth part of the direct light, and this gives us the fraction 1/55 for the value of A. I then subjected the two other terms: (1) to observations made at an angle of incidence of 25°, which showed me that of 1,000 rays there were only 97 reflected; and (2) to those which showed me that at an TABLE 1 Table of the Amounts of Light Reflected by the Surface of Water (The number of direct rays is expressed by 1,000) Angles of incidence (Degrees)

½

1 1½ 2 2½ 5 7½

Number of rays reflected

Angles of incidence (Degrees)

Number of rays reflected

Angles of incidence (Degrees)

Number of rays reflected

721 692 669 639 614 501 409

10 12½ 15 17½ 20 25 30

333 271 211 178 145 97 65

40 50 60 70 80 90

34 22 19

18 18

18

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angle of incidence infinitely small, so to speak, the reflected light is three-quarters of the incident or direct light. The indeterminate expression is in this way changed to 1/55 (1/3) z3 (2/5) z6, which thus indicates the intensity of the reflection for all angles of incidence, in proportion to the direct light, which is always taken as unity. This formula is very simple and formed of very few terms. Nevertheless it agrees [136] rather exactly with all those of my observations which I trust the most. 16

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ARTICLE V Construction of a table of the quantities of light reflected by the plate glass of which mirrors are made THE SAME METHOD served me for the plate glass of which we make mirrors and telescope lenses; but besides finding that the reflection did not follow the same law as for water, I remarked very large differences between one sample of glass and another, which doubtless resulted from their different degree of polish and perhaps as much or [137] more from the different substances which have been used in their composition. In general it appears, at least at large angles of incidence, that a strong reflection greatly depends on the opacity of the different media. 17 At perpendicular incidence several pieces of plate glass TABLE 2 Table of the Reflections Produced by Plate Glass

(The number of direct rays is expressed by 1,000) Angles of incidence (Degrees)

Number of rays reflected

Angles of incidence (Degrees)

Number of rays reflected

Angles of incidence (Degrees)

Number of rays reflected

2½ 5

584 543 474 412 356

15 20 25 30 40

299 222 157 112 57

50 60 70 80 90

34 27 25 25 25

7½

10 12½

16 Because of the low value found by Bouguer at grazing incidence the Table (1) differs widely from the modern values up to about 10°. The rest of the Table agrees within 10 per cent. 11 These results are not up to Bouguer's usual standard. The reflection depends (apart from the effect of imperfect polish) on the relative index of refraction of the two media (e.g. glass and air). He should have obtained about 4 per cent at perpendicular incidence, or 1/25. At grazing incidence the reflectance is unity. His observations at intermediate angles may easily have been vitiated by polarization by reflection, a phenomenon which was unrecognized at the time.

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reflected 1/40 of the direct light, and in several others I observed that it was 1/31. However this may be, if the direct light is still indicated by unity, the formula 1/40 + (7 /10)z 3 - (1/40)z 6 expresses sufficiently well for these surfaces the progression of these reflections for all the different inclinations. Here unity still takes the place of the total sine and z is still the versed cosine of the angle of incidence.

[138]

ARTICLE VI

Comparison of the light reflected by the surfaces of water and of mercury when these two liquids are contained in the same vessel THE DIFFERENT LA ws of reflection at different surfaces give rise to several singular phenomena. We shall pause for a few moments to consider one of these which we prefer because of the occasion which it presents for giving some useful illustrations. It has some analogy with that which we spoke of in the first book concerning a basin full of water, of which the bottom is visible under a certain inclination and lost to view under another. If, on to mercury contained in a vessel, we pour a certain quantity of water, a kind of mirror is formed, of which the water will be the glass, so to speak. We shall see two images. Objects will be painted 18 on the surface of the mercury and on that of the water, and at small angles of incidence these images will be so much the same in intensity that one may deceive oneself by taking one for the other. At large angles of incidence the image depicted on the surface of the water will disappear. This will happen when [139] its intensity is less than the sixtieth or the eightieth part of that of the other lights which strike the eye, as we have observed in the first book. Then we shall only see the image traced on the mercury. As the eye is lowered the image painted on the water gains in intensity and the other, on the contrary, loses, for the reason that the greater the amount of light reflected from the surface of the water the less is left to enter into this medium and to trace the other image on the quicksilver. Finally, if one places one's eye still lower, this latter image becomes incomparably weaker than the first. Thus one may render whichever of the two one wishes stronger or weaker. Two entirely different cases will be distinguished, and it will be found that the mean angle of incidence which separates them is about 10°. Something similar Throughout this article Bouguer uses the verb peindre to express this idea.

18

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happens with our ordinary mirrors, but the angle of incidence which makes the separation of the two cases by making the two images equally strong is ordinarily 13° or 14° in several mirrors. It is not the same in every case because of the differences between the glasses. 19 The angle of incidence which gives the same degree of intensity to the two images, on the water and on the mercury in our experiment, causes about one-third of the light to be reflected at the first surface of the water, as may be seen by consulting the first of the small tables above. Of 1,000 rays which leave the object and which go [140] to meet the first surface at an angle of 10°, only about 333 contribute to the formation of the upper image. There are thus 667 which go into the water and which trace the second image after striking the surface of the mercury. On the mercury below, a second reflection takes place, which causes a new weakening of the light. The 667 rays will be reduced to about 500. The diminution caused by the defect of transparency of the water counts for nothing, not only in the first part of the trajectory of the rays, but also in the second return part, if the thickness of the water is only a few lines or a few inches, for this medium is so diaphanous that when it has little depth it intercepts no perceptible light. The image painted on the mercury then sends us back 500 rays, but this light, striking the surface of the water from the inside or from below, again suffers a new reflection which makes a considerable portion of the rays go down again into the water, and only those which are not reflected come to us. This part of the 500 rays which will be sent back towards the interior of the water will be from 160 to 170 rays. Now the remaining part which goes through the surface and comes to us will give to the image painted on the mercury the same brightness as the other, which we see on the surface of the water, and which sends back 333 rays to us. All these things will become still clearer [141] if we glance at Figure 17 in which Eis the object, AB represents the surface of the mercury, and CD that of the water. We have already seen that the rays reflected by the latter surface, coming towards the eye 0, are directed as if they came from the point K which is as far below CD as the object E is above it. These rays are reflected between F and H at a point which we have not marked in order to preserve the distinctness of our figure, but they always come as if they had left the point K. This point is the virtual focus of the rays, not only those which are directed towards the upper and lower edges of the aperture of the iris, but also of those which are thrown towards the right or towards the left. Thus the reflected rays incident at 10° form at O a light of which 333/K0 2 UAJso the silvering, etc.

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is the intensity, just as we have 1000/EF2 for its intensity when it strikes the surface of the water at F and when it has not yet suffered any decomposition or separation. No difficulty can be found in this. As to the image which is painted on the mercury at G, there are several more things to consider. The rays EFGHO, which come to the eye, are reflected at G and are bent twice by refraction at F and at H. Besides this they have two virtual foci, according to whether they follow paths one beside the other, or one more or less above or below the other, [142) as we remarked concerning Figure 15. The first rays appear to come from point L, the point of intersection of their last direction HO produced and of the vertical EL from the object E. Indeed it is clear that as all these rays are situated in different vertical planes which have the perpendicular EL for their common section, their prolongations must also all cut in the same line and strike the eye as if they really came from there. But the other rays, such as Efgho, which are directed a little higher or a little lower than one another, without departing from the same vertical plane, have for a virtual focus the point M which my readers who are practised in the study of caustics will have no embarrassment in determining. To mark this point in a very simple manner we erect the perpendicular HN to GH, prolong it as far as the vertical line from the point G, and after that draw NP horizontally as far as the prolongation of GH. Then taking m and n to express the constant ratio which exists between the sines in refraction, we have only this proportion to make: mn is to 2m 2 - 2n 2 as HP is to a fourth term (2m 2 - 2n 2 )HP/mn. This last term will be equal to the interval

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between the two virtual foci L and M. The ratio of m to n is very close to 4 to 3 for water, and we shall therefore have 7 /6HP for the interval LM, (143] so that it will always suffice to augment HP by a sixth to get this interval. But since there are two virtual foci for the image traced on the mercury, it will be the same for this image as for that of the object of Figure 15 which is plunged in water. The intensity of its light will depend on the product of the two distances from the eye to the foci L and M, and the resulting image will always be a little confused. 20 Rigorously, the same thing should take place in regard to all our ordinary mirrors. The image painted on the surface of their backing, although almost always brighter, ought never to be as clear as that which is traced on the first surface of the glass. It will not be the same if we make the thickness JG of the water very small in relation to the height of the object. HP will become almost zero, and so will LM, and the two virtual foci L and M will be confounded with the virtual focus K, which belongs to the upper surface of the water. Thus we see that the two images will then be altered in their brightness or intensity only by the decomposition produced by the reflections. The image furnished by the surface of the water is subject to only one decrease of intensity, which reduces it from 1000 to 333, because of the angle of incidence of 10° in the particular case with which we are here principally concerned. (144] But, we repeat, the image which is painted on the mercury always receives three successive diminutions by the separations. The first is by the initial reflection at F, where the light which enters into the water becomes reduced to 667. In the second place it suffers another dimunition at G on the surface of the quicksilver which, absorbing part of the light, reduces to 500 that which is reflected along GH. Finally it is weakened a last time at H by the part which is separated by being reflected along HR as it meets the interior surface of the water; and only the remainder, 333, comes to bring the second image to the eye.

ARTICLE VII On the quantity of light reflected at the smaller inclinations by the interior surface of transparent bodies when the light comes out of them

IT REMAINS to examine this last reflection, which still belongs to polished surfaces, but which is suffered by light not on entering a 2

osee Book II, n. 4.

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medium but when, on the contrary, it comes out, arriving at one of the surfaces from the inside. This reflection offers us the most singular phenomena. From the smallest inclinations [145] up to those of a certain number of degrees, the greatest part of the ray is reflected, perhaps to as great an extent as by metal mirrors or by the surface of quicksilver, while the other part, which does not come out, is extinguished or absorbed, so that the surface of transparent bodies is found to be opaque from the inside. If we cause the inclination or the angle of incidence to increase by only a few degrees above a certain limit we notice that the strong reflection ceases suddenly, a large part of the rays passes out, and few are weakened or extinguished. Finally, as the angle of incidence increases still further, the quantity of reflected light becomes less and less, and when the angle of incidence approaches 90° almost all the rays come out of the diaphanous body, its surface losing almost all the property of absorbing light which it had before, and taking on almost as much transparency as it had in the other direction, or when it was struck from outside. The first of these observations has been made by all of our readers who have made some optical use of triangular glass prisms. In consequence of this property these prisms have even been sometimes substituted in catadioptric telescopes in the place of plane metal mirrors, thus profiting by the strong reflection which they produce in the circumstances indicated. [146] If a beam of light strikes them from the interior at an angle of incidence of 10°, 20° or 30°, their surface produces the same effect, or very nearly so, as the surface of quicksilver, extinguishing about a quarter or a third of the rays and reflecting two-thirds of three-quarters. 21 This property is maintained practically at the same intensity right up to an angle of incidence of 49° 49'. 22 But if the angle of incidence is augmented by only a degree the quantity of internally reflected light suddenly diminishes and a very great part of the rays passes out, so that the surface suddenly becomes transparent. All transparent bodies have this same property and differ among themselves only in the greater or lesser angle of incidence where the strong reflection ends, as well as the extinction of the other part of the light which is not reflected. For water this angle is about 41 ° 32', and in each medium it depends so much on the constant ratio which is found in refraction between the sines that this law alone suffices to 21 Reflection under these conditions is total. It is difficult to guess why Bouguer thought there was some extinction. 22 Bouguer's footnote: Taking 31 and 20 for the ratio of sines in refraction for rays of mean refrangibility.

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leave nothing to be desired in the explanation of the phenomenon, at least as far as the adventitious opacity of the surface is concerned. When in Figure 15 rays such as OD [147] leave the point O so as to come out of the water through the surface AB, they are broken at D in order to take the direction DM, and if we mentally prolong the broken ray until it meets the vertical OA in E, the constant ratio which is characteristic of the refraction would be between the lines DO and DE, which are the secants of the angles ADO and ADE, while AD takes the place of the total sine. It is this ratio whose discovery is attributed by Huyghens to Snellius, a ratio which, as Mr. Descartes first made known, necessarily holds also between the sines of the complementary angles. 23 This same ratio of DO to ED is about as 4 to 3, or more exactly, as 529 to 396, for rays of mean refrangibility when they emerge from water, and it will be the same thing whether the light leaves a point 0 deeper in the water or, on the contrary, farther up. But in the second case there is a boundary to consider, for if the point O is too high, or if the angle ODA is smaller than 41° 32', or if OD does not surpass the line DA by a quarter of its own length (which comes to the same thing), the law of refraction will not be observed. The secant DE will become imaginary, in the language of geometricians, because it must always be three-quarters of OD, and so would become smaller than the total sine DA, which would imply a contradiction. [148] Now the laws of nature are inviolable to such an extent that the refraction no longer takes place; it changes to reflection. 24 The light then rebounds only downwards, the surface acting as if it were totally devoid of transparency, and at the same time a part of the light is extinguished . But the physical explanations of refraction which have been given up to the present time do not extend to this second peculiarity. 25 If one wishes to assure oneself very easily of what we are saying concerning the strong reflection produced by the inside surface of water when the angle of incidence is less than 41½ 0 , he has only to put some mercury and some water in a cylindrical phial of transparent glass ABCD (Fig. 18), and it will be even better if the vessel has the 23This might well be a fair statement of the respective contributions of Snellius and Descartes in this disputed matter. Bouguer was no chauvinist. But it seems that the law was actually discovered by Thomas Harriott in 1602. See J. Lohne, Centaurus, 6 (1959), 113-21. 24 An almost perfect expression of the scientific outlook in the Age of Reason. 26For the modern theory, see, for example, R. W. Wood, Physical Optics (3rd. ed. ; New York: The Macmillan Co., 1934), p. 416. For some reason some light seems to have disappeared whenever Bouguer made experiments of this kind. See Book II , n. 21.

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form of a rectangular parallelipiped. EF is the surface of the mercury and GH that of the water. A vertical board is placed at I, at the mean height of the two surfaces EF and GH. It is illuminated by a candle which is situated at exactly the same height; and if the eye is placed at O exactly opposite the board I, we shall see that the two images which are painted on the surface of the mercury at L and on the interior side of the surface of the water at K, have to all appearance the same intensity provided that the angles of incidence do not quite reach 41½ 0 • [149) It is true that I think I have noticed, even contrary to my expectations, that a piece of crystal placed directly on the surface of mercury weakens the intensity of the reflection a little. Water should naturally produce the same effect but, since that caused by the crystal is almost inappreciable, one is still better justified in neglecting the second. If one wished to make this experiment more perfect, it would not be difficult to make the interval which separates the two images disappear, by having them reflected a second time outside on two small mirrors of the same quality which were put one beside the other. Then, seeing the two images absolutely with the same glance, the comparison which would be made between them would be more certain. But if all the different angles of incidence from zero to 41½ 0 for water, and to 49° 49' for crystal glass, give reflections as strong as those from mercury, and extinguish the rest of the light, we have only to increase the angle of incidence a little and the reflected light is found to be considerably diminished and at the same time there will be very little light extinguished. The experiment represented by Figure 17 furnishes us the proof of this. For the particular angle of incidence EFC of 10° for water, the angle GFH is about 42° 31', as is the angle GHF, which is the angle of incidence of the ray GH with [150) reference to the upper surface CD on its inside; but although the angle of incidence is only increased by a degree, the surface regains the greater part of its transparency. Of 500 rays which come to strike it

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along GH only 160 or 170 are reflected 26 along HR and 333 emerge, taking the direction HO. We are quite sure that this last part which escapes is 333 in relation to the total quantity 500 which approaches along GH. But it is certainly possible that the part reflected towards the interior of the water along HR might be a little less than we say, and it must necessarily be so if there is any considerable part extinguished.

ARTICLE VIII On the quantity of light reflected by the internal surface under the largest angles of incidence AT THIS TIME we pass to the examination of what happens under the largest angles of incidence, but we shall occupy ourselves only with plate glass, on which we thought we might make experiments which would more possibly be of utility. I will confess freely that this research gave me a great deal of trouble, and that I used up much more time on it than I thought I should. The difficulties which [151] I found in this matter resulted from the fact that the reflected light, the intensity of which I wished to determine, only reaches us after having received various alterations; and from the fact that it is not always easy in practice to disentangle, in the altered light which we observe, the particular effect of the reflection produced only by the internal surface of the transparent body. I thought I should diversify the observations in order to have enough data to be able to employ mathematical analysis. I compared, for example, the direct light from an object: (1) with its image painted on the first surface, (2) with another object more or less strongly illuminated than the first, which I looked at through the same glass, and (3) with the image painted on its internal surface, but very much altered when I observed it. The calculation was easy. The problem was among the number of those which are called simple. Designating by a the intensity of the direct light, by b the light reflected by the first surface of the crystal, by e the intensity of the light after it has passed through the same medium, by c the intensity of the internal image as it is furnished by observation after all the alterations which it has suffered, and by z the unknown intensity of this same image, disengaged from all which is foreign to it, we find 27 z = (a - b) X c/e 2•

Actually, by Fresnel's equation, about 15. is obscure. It is impossible to obtain this formula if one takes Bouguer's notation as it is (very clearly) stated. I obtain z = (a - b)c/e. 26

27 This

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But this determination is always too uncertain [152) because the number of observations that we need is too great. These, never being exact enough in view of the smallness of the quantity we wish to discover , can throw us into considerable errors when we combine them. We must not even be surprised if we find the unknown in the wrong direction, or if we are deceived not only in its quantity but also in whether it is positive or negative. Besides, in this solution we have supposed that all the light is reflected or transmitted and that no part of it is extinguished by the surfaces, but if only one of these same surfaces absorbs some rays, as we have seen that the last surface does, we should have to introduce a new unknown into the problem, which would then become more difficult. We should then be obliged to have recourse to some other observation different from those which we have just employed, and the quantities which we have to discover would be still more inevitably lost among the errors to which our data would be subject. Finally, by thinking about it, I imagined a means which I believe to be sufficiently exact and which should naturally have presented itself to me much sooner. This means is almost as direct as if we were able to take the eye quite into the interior of the transparent body to observe the image painted on the second surface before [153] it had been altered. It consists in comparing experiments made on pieces of glass one of which has twice the thickness of the other, taking great precautions to have their material perfectly homogeneous. For this purpose I had them taken from a larger piece. We chose the thickest glass and took two samples side by side, of which one was exactly twice as thick as the other. The samples on which I made the experiments which I am going to report were extracted from a piece of glass which was very white and which had not that slight tint of black which I have noticed in other glasses which were nevertheless reputed to be very fine. This piece was five lines thick. We gave to the two samples a width of eight lines and four lines. It is in this direction that the light made its passage. I first examined to what extent their exterior faces weakened the light by reflection, and I found that it was 36 times at an angle of incidence of 75°, that is to say that of 1,000 rays only 28 were reflected. In place of the mirror which is represented in Figure 3, I next placed the two pieces of glass one above the other in the direction BC, and I disposed the two boards D and E in such a way that the angle of incidence was 75°. I had reduced these same boards to little surfaces or objects which formed squares [154) of which the sides were only a line and a half or

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two lines long. Finally, instead of comparing one of the images with the other object seen directly, or without the interposition of any transparent object, I compared the internal image painted in the sample of glass which was only four lines thick, with the other object seen through the eight lines of thickness of the other sample. Anyone who pays the matter the slightest attention will agree that I made all the circumstances absolutely the same in regard to the two objects, and that we should find no other difference in their appearance than that which was produced by the internal reflection of which I wished to discover the intensity, and which took place in the thinner sample. The reflections produced on entry into the two samples, taking place with the same obliquity of 75°, should weaken the light equally. It was the same thing again at the exit. Besides this, the paths of the two lights were of exactly the same length in the two samples, for in the sample twice as thin, which furnished me the internal image by reflection, there was a double path because of the return of the light in the sample; but this double path was equivalent to the single path in a straight line which the light made in the other sample while it passed through a double thickness. It is true that all the alterations did not take place in the same order, but as each one can be expressed by a ratio [155] it does not matter in what order they occur. Their results should always be the same. Thus, everything being exactly equal on each side except only for the internal reflection which took place in the thinner sample, we may free ourselves from all the alterations which we can regard as foreign, and I found myself precisely in the same situation as if I had immediately compared the internal image which is in question with the other object seen directly without the interposition of the thicker sample of crystal. Now as you will easily guess, I was obliged to bring the candle very near the object which I saw by reflection. The distance from one object to the other was 100 inches and I was obliged to put the candle at a distance of only 16 inches from the first object, which made the distance to the other 84 inches. It follows from this that at an angle of incidence of 75° the internal reflection weakens the light 27 or 28 times, and as the reflection which is produced externally with the same inclination weakens the light 36 times, we recognize that the internal reflection is a little stronger than the other. Repeating this experiment on the same pieces of glass and on others I sometimes found the two reflections equally strong, but (156] ordinarily the internal reflection appeared to me to be brighter. The image internally reflected was always a little redder than the other

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object seen directly through the other piece of glass. This was inconvenient, but another inconvenience still more considerable was found in the imperfection of the glasses which I used, for in the direction that I made the light go through them they had threads which more or less spoiled the transparency. These apparent threads were really different layers of the substance which resulted from the way in which it had been cast. Glasses are much more transparent when one looks in the direction of their real thickness, but I had not two of them of which one was twice as thick as the other, and which were certainly of the same substance. 28

ARTICLE IX On the light absorbed or extinguished by the surfaces of transparent bodies independently of reflection WE HAVE SEEN above that all the surfaces of transparent bodies absorb or extinguish a certain quantity of light on their inside surfaces under angles of incidence below a certain limit. Suspecting that they conserve something [157] of this property at the largest angles of incidence, which give them practically all their transparency, I was curious to examine this. It was again necessary to disentagle the effect of this extinction of light from all the other effects with which it is always confounded. To this end I placed beside a piece of glass A (Fig. 19) four other pieces B, which together were of exactly the same length as the first. They were disposed somewhat obliquely in reference to the boards C and D . I wished that the first angle of incidence should be 75°. I also left some space between the four pieces of glass B in order to avoid any unwanted reflection. As the light passed through these four pieces one after the other, the piece A should weaken it equally, if one leaves out the reflections. The light had to make passages of the same length, but in the case of the four pieces B it suffered six extra reflections in entering and leaving the three last pieces. Observations made this difference very noticeable. I had to bring the board C nearer the candle Lin order to make the two images or objects equally bright. It appeared that the diminution produced by these six reflections was in the ratio of 243,049 to 360,000. 28This well-conceived experiment was seriously handicapped by the materials at Bouguer's disposal, and it seems possible that the polish of his samples was not of the best. The internal and external reflections are theoretically equal in magnitude.

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t i

I

!

I

This diminution, as will be seen, (158] belongs to the two surfaces of three pieces of glass and it is certain that each piece has contributed to it equally or proportionally. It has weakened the light in the same ratio. Thus each piece must have weakened the light in the ratio of 877 to 1,000, 29 which, being repeated three times, gives that of 243,049 to 360,000. To discover the simple ratio we could, strictly, have been contented to put two pieces of glass at B whose lengths together equalled that of A. We employed four pieces so that we might better submit the alteration of the light to observation, for in subdividing it as we are obliged to do, we also subdivide the error, if there is one. But this simple diminution suffered by the light in going through two surfaces of glass is appreciably stronger than is allowed by the two reflections, the external and internal together. Thus there is necessarily some other hidden cause of alteration. The external reflection produces a diminution of 1/36th part, for our pieces of crystal had been taken from the same plate glass as those with which we have been principally concerned in the preceding article, and besides this they were given the same rather perfect polish. The external reflection, then, reduced the light from 1,000 degrees to 972. The other 29Supposing the index of refraction of the glass was 1.55, the two reflections together should have reduced the light to about 91 per cent. A dense lead glass might give Bouguer's result.

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reflection, which took place on the interior of the other surface, produced a diminution of [159] a 27th or a 28th part, that is to say, it should reduce the 972 degrees to 936. Thus we have 936 degrees for the result of the two reflections combined, but if we compare them with the 877 degrees furnished by the direct observation of the intensity of the light after it had suffered the total loss caused by its passage through the two surfaces, we conclude that 59 rays were absorbed and so about I/16th of the light was extinguished, because it would only be reduced by the two reflections from 1,000 degrees to 936 and not to 877. We can scarcely avoid attributing this effect to the second surface, which continues to act as if it was still somewhat deprived of its transparency. At all the small angles of incidence from zero up to 49 degrees 49 minutes it extinguishes ½or ¼of the rays. We have seen it, then, suddenly absorb only a much smaller part, and we now find that it still preserves a little of this property for almost perpendicular incidence. I have repeated the same observation on other pieces of glass and, if I have not found the same results, a difference in the same sense has always occurred which shows me that there is a true extinction of the light. It has often appeared that it is about the 24th part, or even equal to the internal reflection. I cannot say whether the difference between these [160] determinations should be attributed only to differences in the pieces of glass or also to the inevitable errors of observation; perhaps both have contributed to them. However this may be, I have come to prefer the last results. Thus while I took a 36th part for the weakening caused by the external reflection on glass at large angles of incidence and a 27th or 28th part for the internal reflection suffered by the light which is turned back into the transparent body when it seeks to emerge, I took a similar fraction for the extinction of the rays which takes place at the same time. Finally, there is an infinite number of cases in which it is not so much a question of separating these three different alterations as of having their joint effect. The three together weaken light by about a tenth part in our pieces of ordinary plate glass which have received a sufficiently good polish. 30 But it cannot be too often repeated that we must not affirm anything too positively when we are obliged to rely on observations which demand such delicate, or rather over-nice, attention, especially with regard to the third alteration, the investigation of which is more complicated. 30 This is a good round number, but surely one-eighth would have better expressed the results given in detail above.

[161)

SECTION THREE On the Reflection of Light

by Mat or Rough

Surfacel 1

ARTICLE I WE BELIEVE we have paid enough attention to the reflection of light from polished surfaces and that we should now extend our researches to mat or rough surfaces such as those of an infinite number of bodies which surround us. These latter surfaces have an infinity of tiny wrinkles or roughnesses which, presenting numbers of their small faces on all sides, reflect light in every direction. The rays which are lost to one spectator serve for another who is placed in another position, and it is not impossible that the intensity of the light or the brightness of the surface may not appear absolutely the same to one and to the other. The little hemispheres with which we imagined in the first section that certain surfaces were covered, give us a sufficiently exact idea of the matter, provided that we do not confine ourselves to the supposition of little hemispheres but have recourse to other figures which may be determined [162] by circumstances or by observations. If we can succeed in finding out the distribution of the little faces which the roughnesses present in each direction, we can place ourselves, so to speak, at the source of the majority of these phenomena; we shall be in a position to draw an infinite number of conclusions on the subject of the light reflected by ordinary bodies; and this will be less a hypothesis that we have imagined than the actual reality of the thing. 31 Bouguer's treatment of this subject, which forms such a large part of the second book, is the least satisfactory part of the whole work. He was not the last to make the assumption that a rough surface could be analyzed into little mirrors, nor the last to fail to get any very useful results by doing so (see V. G. W. Harrison, Definition and Measurement of Gloss (Leatherhead, Surrey: The Printing and Allied Trades Research Assn., 1945)). Our footnotes in this part of the work will be infrequent, partly in consequence of the rather limited success (to put it mildly) of his treatment. But he did establish the subject of goniophotometry.

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ARTICLE II

A means of observing the various intensities of light reflected by surfaces looked at from the direction in which they are illuminated IT APPEARED to us that we could succeed in making this observation by using two surfaces exactly similar and of the same colour which we cause to vary, one by giving it different obliquities, and the other by varying its distance, in order to be able to relate the effect produced only by the obliquity of the first to that produced by the known distance of the second. The latter surface is represented by DC in Figure 20. I place it at different distances from the candle or lamp L, exposing [163] it perpendicularly to the light, while I leave the surface AB always at the same distance, giving it more or less oblique positions. The more I diminish the angle of incidence on the latter surface, the

-----t..

,,;a,;~~L"',:·-··-·-··--····-·-•-~lit

Ft7. 20.

(!

more it loses the brightness of its colour or of its light, and I easily succeed in evaluating the precise amount of the change by moving the other surface D, that which receives the light perpendicularly, farther away, until the two intensities appear perfectly equal when I place my eye at some point 0. As is evident, I find in the square of the distance LF the expression for the intensity of the light from each surface. In a word, if I am obliged to move the second one three or four times as far away from the candle, it is an indication that the obliquity of the first, AB, decreases the intensity or the tone of its colour nine or sixteen times. I applied this method to silver plates which Mr. Germain, the King's goldsmith, had had frosted with the most extreme delicacy, and then whitened in an acid solution. 32 Their whiteness was much 82 Dans l'eau seconde. lam indebted to Dr. L. Ragey, Director of the Conservatoire National des Arts et Metiers, Paris, for the information that l'eau seconde, sometimes called l'eau verte, is a solution of copper nitrate produced in the recovery of silver from an alloy of silver and gold. Apparently it had enough excess nitric acid to act a little on the frosted silver. It is interesting that the translator of the Traite into Latin renders this passage by "tum vero per aquam illam fortem, quam secundam vocant (& cujus vis dissolvendi jam multum usu hebetata est)." He was evidently concerned, as I was, to save the passage from being obscure.

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greater than that of the finest paper. When I put the surface AB 60 inches away from the candle and made the angle of incidence 75°, the tone of its whiteness was weakened so much that I was obliged to have the other plate carried 7 inches farther, or 67 inches away from the candle. When I made the angle [164] of incidence 60° I had to have the other plate carried 75 inches from the luminous body. An obliquity of 45° in the plate AB brought about a still larger change; I was obliged to have the other plate put at a distance of 89 inches. Finally the angles of incidence of 30° and 15° were found to be equivalent to distances of 105¾ inches and of 131. Similar experiments were made on several other surfaces; on that of polished marble, of plaster, of Dutch paper, etc., but I shall content myself with adding here to the description of the preceding those which I made on the plaster, which had been chosen of the best quality and of the greatest whiteness. I put an interval of 24 inches between the candle and the first surface AB, and when I made the angle of incidence 75°, the other surface CD, which received the light perpendicularly and served me as a standard of comparison, had to be taken to 27½ inches from the candle. An angle of incidence of 60° obliged me to make the distance from the other surface to the candle 30 inches, and the other angles of incidence of 45°, 30°, and 15° corresponded to distances of 33, 40½, and 54½ inches. Expressing by one thousand the brightness of the surface AB or the intensity of the reflected light when it was illuminated and looked at perpendicularly, the [165] brightness of the light at the other angles of inclination is found to be expressed by the numbers which I have inserted in the following little Table. TABLE 3 Inclinations of incident and visual rays to the surface

Brightness of the surface, or intensity of the reflected light Frosted silver

Plaster

Dutch paper

90 75 60

1,000 802 640

1,000 762 640

1,000 971 743

45 30 15

455 319 209

529 352 194

507 332 203

(Degrees)

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ARTICLE III On the distribution of the small asperities deduced from the preceding experiments IT WILL BE SEEN that these surfaces do not present the same number of little faces in every direction and that they have much fewer which send the light back at small inclinations than at large ones. It is just about the same in all the surfaces which I have examined. The distribution is not absolutely the same but there is a great similarity between them all, and I have [166] found none which have their small wrinkles or asperities equivalent to little hemispheres. In addition it must be remarked that if it suffices to take the inverse ratio of the squares of the distances to the second surface, in order to know according to what law the brightness of the first or the intensity of the light which it reflects changes for different obliquities, we must still pay some further attention in order to discover the distribution of the small elements or, what amounts to the same thing, to determine the number of rays which they reflect. In this second part of the research we must necessarily take into account the inclination of the visual rays with reference to the surface, for its brightness or the tone of its colour must be more or less increased in the experiments according to whether the surface is looked at more or less obliquely. In fact when the surface is observed under a smaller angle of inclination it is reduced to a smaller space on the retina, and the rays reflected by the little faces are then found to be pressed more closely together. We have found, for example, for the plaster, that the change suffered by the intensity of the light when the surface is struck perpendicularly and then with an angle of incidence of 30°, corresponds to distances of 24 inches and of 40½. The squares of these [167] two numbers taken inversely seem thus to show us that there are more of the little elements of roughness which send back light in a direction perpendicular to the surface than in the other direction in the ratio of 1,640 to 576, but the number of little asperities of which the small faces are oblique is two times smaller still, and is really only 288, in comparison to 1,640, which expresses the number of the others. We have given the reason for this. When the number of small faces at the inclination of 30° appeared to be 576, it was as if doubled by the picture, only half as extensive, painted on the back of the eye, and painted with the same number of rays, which really doubled its intensity.

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Thus, to obtain the number of little faces, it is necessary to join or compound the direct ratio of the sines of the angles formed by the visual rays and the surface AB with the inverse ratio of the squares of the distances from the second surface CD to the candle, or, what comes to the same thing, we must continually divide the sines of the obliquities of one by the squares of the distances of the other. The total sine being designated by unity, we have then 1/576 for the small asperities which present their little faces in the direction perpendicular to the large surface, while the number of little faces which correspond to the angle of incidence of 30° will be represented by ½/1,680 or 1/3,360. [168] In conformity with this rule, and taking 1,000 for the number of the little faces which present themselves in a direction perpendicular to the large surface, or which are parallel to this same surface, I have found the quantities noted in the following Table for the plates of frosted silver, the surface of plaster, and the most beautiful Dutch paper which has ever been moulded, which I looked at in the direction of its little grooves. The inclinations contained here in the first column are the complements of those of the other table, since the little faces are perpendicular to the incident and visual rays which were then being considered, but the inclinations are still measured from the large surface just the same. TABLE 4 Inclinations of the little faces with respect to the large surface (Degrees)

[169]

Distribution of the little asperities; or of their Ii ttle faces For Frosted Silver

For Plaster

For Dutch Paper

0 15 30

1,000 777 554

1,000 736 554

1,000 937 545

45 60 75

333 161 53

374 176 50

358 166 52

ARTICLE IV

On the expression of the small asperities by the ordinates of a curved line IT WILL BE SEEN that we succeed in counting, so to speak, the little faces which each object presents towards every direction, and that we

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distinguish them from one another by the different obliquities that we give to the surfaces, although each one in particular may be imperceptible. The methods of interpolation well known to geometricians make it easy to extend the small Table if one wishes to do so, and to make it contain all the different situations of the little faces. But instead of indicating the ratios by numbers it is just as simple and as natural to express them by the ordinates of a curved line. It is the only means of making visible at a glance the details of an infinite number of facts and of making it possible to deduce easily all their different consequences which, however, must still be admitted only with the widest reservations which are almost always necessary in matters of physics. If at the centre of the mat or rough surface AB of Figure 21 we erect the perpendicular CD, and [170] suppose it to have a length of 1,000 parts and to express the sum or the number of the small faces

which the rough surface presents perpendicularly to this direction, we have only to mark on all the other radii of the semi-circle EDF the corresponding numbers of the little Table, paying attention to the fact that the degrees of the semi-circle here denote the inclinations of the rays, while in the table we have marked the inclinations of the little faces themselves, which are the complement of these. Suppose now that we are dealing with the surface of plaster. We mark on the radius C 60 from C to G 554 parts, then from C to H 176, and doing the same thing on all the other radii we locate a sufficiently great number of points to be able to trace the curve CHDH, which will furnish us the scale that measures the little faces turned towards every direction. The faces of the same inclination are uniformly spread over all the rough surface, since we suppose it to be of the same colour and equally

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BOUGUER'S OPTICAL TREATISE ON LIGHT

mat everywhere. But in the mind we bring together all the faces which are equally inclined and imagine them as if they were at the centre of the surface. We express them all at once by the lines CD, CG, which are perpendicular to them, and these lines are the ordinates 33 of our curve, to which we may give the name numerator of asperities. [171 This curve has almost always the form of a kind of oval less wide then high, and of which the lower end ordinarily terminates in a point. Two arcs of a circle of about 114° very nearly represent it for silver plates, provided that we make the angle at the top formed by the two arcs disappear, and round it off a little. For the plaster this oval is in reality appreciably sharper at the top than at the bottom, and its width is only about 11/20 of the length of its axis CD. For the Dutch paper its width is about two-thirds of its major axis. If our curved line, the numerator of asperities, had exactly the form of a circle ChDh, rough surfaces would have a very singular property even when they were illuminated extremely obliquely. Provided that they were observed by the spectator with the same obliquity and from the same side, they would always appear of the same tone of colour or light. 34 It is true that if the angle of incidence were changed to make it 30°, for example, after it had been 90°, the ordinate Ch which would express the number of little faces struck perpendicularly would be only half as large as CD, according to the ratio of the sines of the angles of incidence, because of the properties of the circle ChDh, of which the chords CD, Ch, etc., are proportional to the sines of the angles ECD, ECh. But there would be [172] an exact compensation since, in conformity with what we have already remarked several times, the visual ray hC making an angle of 30° with the surface AB, this surface would appear twice as narrow; and a quantity of rays of light only half as large falling on a space half as large in the retina, would make an equally bright impression on each point. It would be exactly the same for all the other inclinations. They would change absolutely nothing in the intensity of the colour or the light. It follows from this that the circumference of the circle ChDg, considered as a numerator of asperities, serves to separate two extremely different cases in the reflection of light produced by rough surfaces. Whenever the numerator is narrower than the circle, the intensity of its light or colour diminishes when, after it has been illuminated at 330rdonnees,

here and elsewhere. We should call such a line a radius vector. This is, of course, an expression of the law which Lambert (Photometria, p. 41) stated explicitly. 34

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a right angle, it is illuminated at an acute angle of incidence and also observed in just about the same direction. If, on the contrary, the numerator comes outside the circle ChDg; if, for example, it takes the form of the semi-circle EDF, as would happen if the roughnesses were equivalent to small hemispheres, the rough surface would become more and more luminous as it was illuminated and looked at from [173] a smaller angle of inclination ECH. It should even be remarked that in the particular case of small hemispherical asperities the surface would finally have an infinite intensity of light, since it would always throw out the same quantity of light and at the same time would be reduced to an extremely small space in the eye. 35 We must however consider that we are here concerned principally with oblique reflection compared to perpendicular reflection, for it is quite possible that on two different hypotheses for the numerator, between which the circle ChDh serves as a dividing line, two different inclinations might give exactly the same intensity to the light. For this it suffices that the two ordinates CG, CH of the numerator on one hypothesis should be exactly proportional to the corresponding ordinates of the numerator of asperities on the other hypothesis, which is outside ChDh. Then the little surfaces exposed perpendicularly to the light, and consequently also the quantity of reflected rays, would diminish in the same ratio as the two images of the surface AB in the eye, and the intensity of the light would be as much augmented by this cause as it would be diminished by the other, on the two hypotheses. But in general I do not believe that it is among our terrestrial objects down here that we ought to look for numerators of asperities which come outside [174] the circle ChDh. I see this only in the planets properly so-called, the parts of which appear to me to be constituted a little differently to those of the bodies surrounding us, as we shall show in more detail below. Some of the planets have mountains twice as high as ours, and their surfaces thus present more small inclined faces in proportion. That is to say that the number or the sum of these small faces does not diminish in as great a ratio as the ordinates or chords Cg, Ch of the circle CgDh. We are quite sure of this because we see in a very distinct manner that several of these planets, such as the moon and Venus, have, other things being equal, the edges of their disks more luminous than the parts near their centres. 36 See Book II, n.6. Book II, n.7.

35

36See

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ARTICLE V On the intensity of the light or the colour of rough surfaces when they are illuminated and seen in two absolutely dijferent directions: important theorem become a little more difficult and the applications of our curved line demand a little more geometry when the surfaces are illuminated along a certain direction and observed at the same [175) time by a spectator situated on another line. However I give here a very general and simple theorem in which it suffices to suppose that the luminous body and the eye are at a great enough distance from the surface.

THE RESEARCHES

If in this case we bisect by the ordinate CE (Fig. 22) the angle LCO formed by the incident rays LC from the luminous body and the visual rays CO which meet at the centre C of the surface AB; and if we pass the

G L

perpendicular PM to the visual rays, through the extremity E of the ordinate, this line, cutting the axis of the numerator CGEH at P, will give us CP, which expresses the brightness of the light on the surface or the tone of its colour. A demonstration of this theorem presents no difficulty, and we shall reduce it to a simple explanation. First it is very clearly evident that the ordinate CE, which bisects the angle LCO, and which for this

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reason we shall sometimes name the mean ordinate, expresses the area of the small faces which are in a proper situation to send light towards the observer, but the number of rays which falls on these little faces does not depend uniquely on their size. It depends also on the sine of the angle of incidence, for the smaller this sine the less the width [176] presented by these little surfaces to the light, and the fewer rays they receive. Now the ordinate CE being perpendicular to the little faces with which we are concerned here, the angle ECL or ECO is the complement of the angle of incidence, and CEM is then equal to the angle of incidence itself. From this it follows that CE, which expresses the area of the small faces, is reduced to CM, or receives light only in proportion to CM. Therefore CM represents the absolute or total amount of light which is thrown towards O; but we are concerned with the intensity of this light or the tone of colour which is assumed by the surface AB, and this intensity, as we have seen, is inversely proportional to the sine of the angle which the visual rays form with the surface. Thus there are, so to speak, two different angles of incidence to consider in the present research, that formed by the rays from the luminous body when they strike the little faces, and that of the visual rays with the surface AB itself. This latter is equal to the angle CP M, and since the light appears stronger or the rays become more closely packed together, to the same extent that the sine of this angle is smaller or that the image occupies less space in the back of the eye, we must increase CM in the same ratio that the sine of the angle CP Mis smaller than the total sine. We shall thus have CP for the intensity of the light, or for the brightness with which the surface AB appears to be illuminated to the observer situated at 0. [177] We may deduce some rather curious consequences of this theorem. We shall be content to indicate the main ones.

ARTICLE VI Corollaries of the preceding theorem

l. IT IS EVIDENT that for each position of the spectator we can put the luminous body in two different places which will produce precisely the same effect as far as the intensity of the light from the surface or the tone of its colour is concerned. The straight line PM cuts the numerator of asperities at two points E and E', and each of these two points has

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all the properties of the other, provided that the ordinate which ends in them bisects the angle formed by the incident rays and by the visual rays CO. Thus we have only to make an angle equal to OCE' on the other side of CE' and we shall have the second direction which the rays incident from the luminous body must have in order that the surface may appear as luminous as at first. 2. IF THE OBSERVER consents to change his position we shall find an infinite number of other situations for the luminous body which will be absolutely equivalent. We have only to draw an infinite number of straight lines PM' from the point P, intersecting the curve, and if we drop perpendiculars CO' on (178] these straight lines from the point C they will mark for us the situation of the visual rays. For each one of these situations there will always be two positions of the luminous body which will be absolutely equivalent, the straight lines CE" and CE"' serving as mean ordinates.

The position of the eye being given, to find the direction in which the luminous body must be put in order that the surface may be as bright as possible 3. WE SEE ALSO that when the position of the eye is given it is very easy to find the direction in which the luminous body must be situated in order that it may produce the greatest possible effect. We have only to place the perpendicular em to the visual ray CO in such a position that it will be tangent to our curve. This will give the maximum Cp for the intensity of the light, and we shall again obtain the direction of the incident rays by carrying the angle OCe to the other side of Ce. 4. IT IS EASY to see that it is almost never immaterial to interchange the eye and the luminous body, and that to increase the brightness of the surface AB the direction which makes the smaller angle with this surface is always to be preferred for the position of the eye. If the luminous body [179] is put along CO and the eye along CL, it is quite true that it will be the same little faces which reflect the light, and that its absolute quantity will still be the same. But the surface being viewed less obliquely in the second case, it will appear wider, and the rays will be less close together. In conformity with our general theorem, we should then also be obliged to drop the perpendicular Eµ from the point Eon to CL (which would have become the visual ray), and we should have Cir and not CP for the intensity of the light. It is not more difficult to verify this fourth corollary by experiment than the preceding ones. I have actually verified them all, and the

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experiment has always appeared to me to succeed as well as I could hope, not by giving me precisely or rigorously the same results as the theory, but by never failing to confirm it up to a certain point. If there should be a maximum, one is in reality found. If two lights should be of the same intensity, or if one should be much stronger or weaker than the other, the observation has agreed. The only difference is that it has sometimes been necessary to change the directions a little.

Showing that a surface may preserve exactly the same intensity of light, whatever situation is given to it in relation to the incident and visual rays 5. Now, WE REMARKED above that the [180] numerator of asperities for the plates of frosted silver approached closely to two arcs of a circle, each 114° long. This agreement gives rise to a peculiarity which is worthy of attention. If the angle formed by the incident rays of light with the visual rays is 66° we may give to the silver plate any situation we wish and the impression of its whiteness will not suffer any change. This property is valid in general for all numerators of this kind, and this is like a new theorem, of which we have thought we should take notice. Whenever the angle made by the incident rays with the visual rays is equal to 180° minus the extent of the arcs of the circle which form the numerator, the situation of the rough surface which receives the light does not influence the intensity of its colour in any way. We shall easily be convinced of this if we cast our eyes on Figure 23, where the curved line CHDH, which expresses the little faces of the rough surface by its ordinates, is formed by two equal, circular arcs. If we suppose that CE is the mean ordinate, and that after having drawn the straight line DEM through the point E we drop the perpendicular CO from the point C to that line, we have only to make the angle ECL equal to the angle ECO, as we have just seen, and we shall have, in the two lines CL and CO, the directions which the incident and visual rays must have in order that the surface may appear to be illuminated with [181] a light of which the intensity will be expressed by CD. This is the same intensity as when the surface is illuminated perpendicularly and viewed perpendicularly also. But because of the properties of the circle, the angle CEM is measured by half of the arc DEC and the angle ECM is consequently equal to 90° minus½ DEC. Thus the angle MCL included between the incident and visual rays is equal to 180° minus DEC. Now exactly the same thing will be found if we take any other chord Ce for mean ordinate. The angle Cem will

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BOUGUER'S OPTICAL TREATISE ON LIGHT

always be equal to half the arc DeC. The angle mCe will be its complement; and the angle /Co 37 formed by the incident ray lC and the visual ray Co, and which is double the angle mCe, will then be 180° minus half the entire arc DC, while the intensity of the colour of the surface AB will continue to be expressed by the same line CD.

0

I have not failed to verify this theorem by experiment, as I did the preceding ones, and I have been able to do this easily because of the silver plates with which I was provided, which were perfectly similar and equally mat. I left one of these in a certain position after having arranged things so that the rays of light which it received made an angle of 66° with the visual rays. The surface was equally inclined to one and the other, or, which comes to the same thing, the axis CD bisected [182] the angle between these rays, or served as mean ordinate. The other plate was then placed next to the first, and its inclination was greatly varied in relation to the rays. In all its different positions it appeared to me to be as bright as the other. A difference was only found when the plate was too oblique in reference to the incident or the visual rays. The little asperities which are only too noticeable on these plates then interfered with one another and I knew in advance that these particular cases must be left out. I could extend these researches considerably by giving the solution of several curious little problems, but it is sufficient to have exhibited the principle, and we must beware of getting gradually away from our 37The

symbol l does not appear in the figure.

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subject by throwing ourselves into purely geometrical discussions. 38 We shall content ourselves with adding to the solution of a problem, of which we have already solved the inverse, that of another which might sometimes be useful. (183]

ARTICLE VII The situation of a luminous body being given with reference to a mat surface, to find the direction in which the eye must be placed in order that the intensity of the reflected light may be as great as possible To RESOLVE this problem we have only to erect perpendiculars EF (Fig. 24) to the curve from various points E of the numerator of asperities and to draw to these same points the ordinates EC. After

this we erect the perpendicular GH from the point G, which is onequarter of EC from E, and produce it until it meets the line EF. Next, taking the point Has the centre of a circle which passes through 38 More clearly than some of his contemporaries, Bouguer recognized the danger of being led into extravagantly clever displays of geometry or analysis. The 18th-century attitude towards this learned occupation was satirized by Voltaire in Candide and by Swift in Gulliver's Travels. Since this part of the work was probably written in the last years of Bouguer's life, it is interesting to recall that Candide appeared in 1759. The somewhat tedious geometry in these pages would not have been superfluous if Bouguer's fundamental postulate (of the little mirrors) had proved more fruitful.

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the point E, this circle will cut the axis CD in two points I and i. Through these points and the point E, we draw straight lines such as EM, and, dropping perpendiculars such as CM to them from the point C, we then transfer EM to EN, perpendicularly to the incident ray of light LC, of which the situation is given. We thus mark the point N. In the same way we look for an infinite number of these points, the locus of which forms a curve nNn. It now only remains to determine the point where this curve cuts the incident ray LC. One may quite easily deduce from this the situation which must be given to the visual ray in order that the intensity [184) of the light from the surface should be a maximum or a minimum. From the points of intersection N, we have only to erect a perpendicular NE to the incident ray. It will meet the numerator of asperities in E. This point will indicate the mean ordinate CE, and we have then only to make the angle ECO equal to the angle LCE to have the direction which must be given to the visual ray. It will not be difficult for readers slightly versed in transcendent geometry to find the demonstration of this, as well as of that of the following problem:

The angle made by the incident rays with the visual rays being given, to find the situation which a mat surface must have in order that its brightness or the intensity of its colour may be a maximum THE SURFACE AB (Fig. 25) is exposed to the light of the luminous body. The angle made by the incident rays and the visual rays is given, and it is desired that the intensity of the colour of this surface should be a maximum or a minimum. We shall make the angle BCR equal to half the angle made by the incident rays with the visual rays. From each point E, e, of the numerator of asperities we draw ordinates EC, eC, and so on, and at the same time we draw the tangents, EH, eh, D

Ff!J.~.5.

A

B

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and so on, to the curve. Erecting now at the mid-point of each ordinate perpendiculars GH, gh, and so on, until they meet (185] the corresponding tangent, we shall have an infinite number of points H, H', h, etc. The locus of all these points will sometimes be a straight line. It will be so, for example, when the numerator of asperities has two circular arcs for its two halves, as it practically has for the two plates of frosted silver. But in general it will be a curved line which joins all the points H, H', h, etc., and its intersection H' with the straight line CR will furnish us the solution of the problem proposed. We have now only to draw from the point of intersection H' the tangent H' E' to the numerator of asperities and we shall have the point E' through which we must lead the mean ordinate CE'. Thus nothing remains but to arrange the surface AB in such a way that CE' really bisects the angle formed by the incident and visual rays, and then the surface will have the brightest colour or the least bright colour possible. (186]

ARTICLE VIII

On the intensity of the light or of the colour of rough surfaces when the plane which passes through their centres, the luminous body, and the eye, is not perpendicular to these surfaces WE NOW CEASE to make the plane which passes through the eye and the luminous body perpendicular to the mat surface, and for a few moments we are going to consider our subject in a more general manner which will be applicable in a much greater number of cases than all the particular researches which we might undertake. We are still supposing that the distances from the luminous body L (Fig. 26) and

128

BOUGUER'S OPTICAL TREATISE ON LIGHT

from the eye Oto the plane AB are extremely large in relation to the diameter of this plane. We then imagine a spherical surface which passes through the luminous body and the eye, and of which C is the centre. The eye is in the plane FZG which is perpendicular to the surface AB, but the luminous body Lis not in the same plane. It is in another, IZC, which makes with the first a given angle, the magnitude of which is expressed by the arc FI; and the luminous body is elevated above the surface AB by a certain number of degrees denoted by IL. All these quantities are known, and if we further imagine on the spherical surface [187] an arc of the great circle LO it will be easy for us to determine its size. If we use spherical trigonometry for this we shall simply have to solve the triangle LZO in which we know the sides ZL and ZO as well as the included angl~. We shall find the arc LO, and we shall also look for the angle ZOL, of which we shall have need in what follows. This being done, we easily see that in the present research we cannot confine ourselves to considering the numerator of asperities as a simple curve CHDH, but that we must revolve it around its axis DC in order to obtain a conoidal or spheroidal surface. The mean ordinate which expresses the number or sizes of the little faces from which, in the present circumstances, we receive a useful reflection, is found in the numerator which has been carried to DEC, and the mean direction of which, being produced to Q, bisects the arc LO, or the angle made by the incident ray of light LC with the visual ray CO. Thus we shall know the angle made by this mean ordinate with one ray or the other. We have only to find the side ZQ of the spherical triangle ZOQ, of which we know ZO, and of which we have already found QO and the angle 0. We shall have [188] the angle ZCQ made by the direction or mean ordinate CE with the axis CD of the numerator CED, and the knowledge which we have of this curved line will give us the length of the mean ordinate. From the point E, we drop the perpendicular EM onto the visual ray, and look for CM. A right triangle is to be solved which is in the plane LCO and of which we have already found the angles and the hypotenuse CE. The side CM will represent, as before, the small faces of the asperities, reduced because of the obliquity with which they receive the light. Finally, we erect a perpendicular MP to the visual ray from point Min the plane FZG, and produce it as far as the axis CD of the numerator. This is to obtain CP, which will be as much greater in relation to CM as the smallness of the sine of the angle OCG formed by the visual ray and the surface AB augments the intensity

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129

of the light from this same surface. It is evident that CP gives us the intensity which we proposed to discover. The same method may serve when the rough surface is illuminated at the same time by several luminous bodies, as it is, for example, by all the points of the visible sky when it is lighted by daylight. There are then an infinite number of little faces of different inclinations which contribute to the reflection [189]. We shall thus find an infinite number of different values of CP, and to obtain the total effect we must have recourse to the ordinary methods of finding the integrals or sums of elementary quantities. It will still be nearly the same problem if the luminous body and the eye are not at a distance which can be regarded as infinite. But then the problem will be much more complicated, since besides other difficulties, we must pay attention to the changes in the angle OCG, which will become larger or smaller for different points on the surface AB, almost in the same way as we noticed in Article VI of the first section.

ARTICLE IX On the numerator of asperities in the planets

THE EXPERIMENTS that we have indicated in Article II of this section were directed precisely as if we had wished to discover the light sent to us by each point of a planet when it is in opposition. Each part of the planet is then struck more or less obliquely by the rays of the sun according to whether it is more or less close to the edge of the disk of the planet, and we also see it with the same obliquity. The circumstances are then absolutely the same as when we make the observations represented by [190] Figure 20. We examined on silver plates and on other surfaces how much the tone of their whiteness was subject to change at different obliquities. If the moon were made of plaster and there were no marks on it, the little table in Article II would indicate the different intensities of its light at all the various points. Suppose that the planet were represented by the globe of Figure 13. The point E which is distant 30° from the edge A of the apparent disk would only illuminate us with an intensity expressed by 352 while the point D would illuminate us with an intensity of 1,000. But things happen very differently in these heavenly bodies. Their edges are more luminous than the parts in the centre, 39 the reverse of what happens with the aesee Book II, n. 7.

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130

sun, and this has already led us to say that the numerator of their asperities is a kind of oval which is broader than the circle. One might investigate the brightness of different points of the planet's disk by direct observation, and put oneself in a position to draw the numerator of its asperities. The ordinates of this curved line not being proportional to those of the circle ChDh of Figure 21, neither are they proportional to the sines of the angles A Cg, A Ch; and if we denote these sines by S we must necessarily add at least some constant quantity b in order to have the ordinates of the numerator of the planets. Instead of [191] the quantity b by itself we might add several other terms, multiplying them by some powers or functions of S. But I believe that we shall always have a sufficiently general expression if we stop at S + bsm;am. 40 It will serve for the planets properly socalled; if one wishes one may extend it to the sun by taking S - bsm/am, which will indicate how much light is sent out by this luminary from its different points, in proportion to the sine S of the angle made by its rays with the surface where they emerge. We shall then only have to compare this expression with the intensities of light found by observations at three different points in order to determine not only the constant b in relation to the total sine a, but also the exponent m. If we limited ourselves to the particular expression S - bs 2/a 2 for the sun, taking its uncertainty away from m, since we have not enough observations on this body, and supposing that we could regard as exact those of which we spoke in Article XII of the second section of the first book, we could put in the place of s the total sine and threequarters of this successively, and we should then have the same ratio between a - b and ¾a - (9/16)b as between 48 and 35. This would give us 1/24 a for the required value of b.

[192]

ARTICLE X

To infer from the numerator of asperities, once determined, how much more or less brightness the different points of a planet should have in all its different situations with reference to the sun THE NUMERATOR of asperities being traced, we can then easily submit to calculation all the phenomena to which the light of the planet is subject, even when it is not in opposition. Let us represent this planet by the globe of Figure 27. It is lit by the sun L, which must be imagined at a practically infinite distance, while the observer is also at a very Thus in the original. S and s appear to be identical.

40

RESEARCHES ON REFLECTED LIGHT

131

0~

great distance in the direction EO. The situation of the point C whose brightness we wish to evaluate is exactly known. We know how much it is elevated above the plane GMK which passes through the centre of the planet, through that of the sun, and through the eye of the observer. Its elevation corresponds to the arc QC, of which CR is the sine, and the point Q is distant by the whole arc QM from the visual ray OM, which is directed to the centre of the planet, and which is perpendicular to its apparent disk GFKI. The line EP exactly bisects the angle LEO formed by the incident rays from the sun and our visual rays, and (193] this angle is considered to be the same for every point on the surface of the planet. Now if the curve CHD is the numerator of asperities of the planet, the mean line CH which bisects this angle 41 is exactly parallel to EP, and the angle BCH is equal to the angle ECR since the sides of one are perpendicular to those of the other. Now as the ordinate CH is proportional to the sine of the angle BCH plus a certain quantity, it will also be proportional to ES plus a quantity which we shall indicate by b X (ES/ EKr in conformity with what we said a moment ago, that is to say that we shall have ES + b(ES/ EKr for the sum of all the little asperities or faces which contribute to the brilliance of the neighbourhood of the point C of the planet. We should multiply this sum by the sine of the angle of incidence of the direct rays if this angle was not the same for all the points on 41

Cet angle. What angle is not clear.

132

BOUGUER'S OPTICAL TREATISE ON LIGHT

the surface of the planet. 42 The angle OEL being constant, and EP being parallel to CH, the incident rays make the same angle with CH as with EP, and it is the complement of this angle which we always consider as the angle of incidence and which is here always the same. We can then neglect this multiplication, which would not change anything in the ratios which we wish to discover, but we [194] should pay great attention to the angle formed by the visual rays and the small part of the surface of the planet around the point C. The smaller the sine of this angle or the more obliquely the neighbourhood of the point C is looked at, the greater the intensity of its light should appear, since the little surface, without suffering any diminution in the light which strikes it, will occupy less space on the retina, as we have already repeated so many times. From the point R, we drop the perpendicular RT on to the visual ray OE, which is directed to the centre of the planet, and from the point C we draw the line CT, which will be perpendicular to the same visual ray, and is the sine of the angle CEO. Now as CE is perpendicular to the surface of the planet at C, and TC is perpendicular to the visual ray, the angle ECT will be equal to the angle formed by the visual rays with the surface at C, so that we shall have ET for the sine of this angle. Thus it remains only to divide by ET the sum of the little asperities in the neighbourhood of the point C, ES + b(ES/EK)m, and we shall have [ES+ b (ES/EK)m]/ET for the brightness of the neighbourhood of the point C. No difficulty will present itself in finding the particular values which enter into this general expression. We shall easily find ES and ET by employing [195] either plane or spherical trigonometry. After this we have only to make the same calculation for any other point and we shall see by how much the intensity of the light is stronger or weaker at one point than at another.

ARTICLE XI Showing that on the surface of the planet there are always an infinite number of points which are equally bright IN ADDITION it is evident that there are an infinite number of points on the surface of the planet which have exactly the same brightness 42 1.e., the angle of incidence on the supposed little mirrors which reflect light to 0, not the angle of incidence on the surface of the planet.

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133

and that this must happen wherever there is the same ratio between ET and ES+ b (ES/EKr as between the corresponding quantities which belong to another point different to C. If in the plane GMK we draw a parallel sr to SR and a parallel tr to TR, and if [Es+ b (Es/EKrl/Et is equal to [ES + b (ES/EKrl/ET it is quite clear that we have only to erect at the point r, the intersection of sr and tr, a perpendicular to the plane GMK as far as the surface of the planet, and it will indicate another point which will have precisely the same brightness as the first point C. [196] We have thus the means of finding in a direct manner, or a priori, so to speak, although depending on some preliminary observations, an infinite number of points which will have the same brightness as the point C. To ES, which is known, we shall add the interval SP = b (ES/EKr. Having then drawn sr parallel to SR we also add to Es the quantity Sp = b (Es/EK)m; and if we draw a line pt parallel to PT, making the same ratio between Ep and Et as between EP and ET, we shall have the point t, from which it is necessary to draw tr parallel to TR in order to have the point of intersection r which will fall exactly below the new point c, on the surface of the planet, of which the brightness is the same as that of the first point C. 43 Some circumstances in which all the points of the planet which are equally luminous are situated on elliptical arcs WHEN THE ORDINATES CH of the numerator of asperities are simply proportional to the sines of the angles BCH plus a constant quantity b, the short lines SP, Sp, which are added to the sines ES, Es, are equal, and it will not then be difficult to assure oneself that all the points R, r, etc., will then be located exactly on a straight line. Thus all the perpendiculars RC, re, etc., which are erected on the plane GMK, will be precisely coplanar, and all the [197] points on the surface of the planet which have the same brightness will be situated on the circumference of a small circle; but this small circle seen from far off and projected on the plane of the disk GFK will appear to be an arc of an ellipse. In a word, all the points of the planet which are equally luminous will form portions of this curved line such as 121 in Figure 28, and the more distant these arcs are from the centre, the more the brightness goes on increasing. This also agrees quite well with observation, provided that we except the extremities of these curved lines, towards which the small roughnesses cover up or hide one another. 43 Figure

27 is not drawn in such a way as to make this clear.

134

BOUGUER'S OPTICAL TREATISE ON LIGHT

When the planet is in opposition all these ellipses are converted into perfectly concentric circles which increase in brightness as we approach the edges of the disk. But for this it is necessary that the little faces of the asperities should not be simply proportional to the cosine S of the_ir inclination with respect to the main surface, but that these cosines should be augmented by a certain quantity which, while possibly the same for all, increases some of them more in proportion than others.

(198]

SECTION FOUR

On the Absolute Magnitude o{ the Small Faces of the Same Inclination in Rough Sutfaces, and on the Total Quantity of Light That They Reflect UP TO THE PRESENT we have only compared the small faces of the asperities with one another as to their magnitude, according to whether they were more or less inclined. But should it not be possible to know more about them? Could not one make an absolute determination, or discover what part of the large surface is formed by all those of the same inclination? This proportion should be finite, for if each little face is extremely small, others having the same inclination will be found in every part of the large surface, and their prodigious numbers ought to compensate for their extreme smallness. We have already seen that as far as reflection is concerned, the effect of hemispherical asperities is quite comparable to that produced by an entire surface when it is perfectly polished. Roughnesses of every other shape certainly have analogous properties, and we should not confine ourselves to finding out simple ratios if it is possible for us to succeed in determining absolute values.

(199]

ARTICLE I

In rough surfaces, to determine the ratio which exists between their total extent and that of the little faces at each inclination Tms PROBLEM is not as difficult as it might appear at first. The more little faces there are of a certain inclination the fewer there must be of others, for since each one occupies a certain amount of room on the large surface each one must tend to exclude all the others. Knowing, then, by our observations, the law followed by the little faces in their distribution, we have only to divide up the extent of the large surface according to this law and we shall have the part which is occupied by

136

BOUGUER'S OPTICAL TREATISE ON LIGHT

these little faces of each inclination. In this examination we must be careful not to lose sight of an essential point, namely that the little faces of the same size occupy more or less space according to whether they are more or less inclined. A little face which is parallel to the large surface encompasses a space as large as itself, while the others cover much less. We then see two different ways of solving this problem: one which is perhaps [200] more elegant considering it from a geometrical standpoint, the other perhaps agreeing more closely with physics or with the nature of the things themselves. For all the little asperities we may mentally substitute a large conoid which will represent them and which will cover all the large surface, and in this form the question reduces itself to a search for the figure of the asperities by the ratio which exists between the small faces of different inclinations. We have followed this method in regard to surfaces of which the roughnesses present little faces equally in all directions. The asperities have then the form of little hemispheres, and we have imagined a large one which should be equivalent to all. But there is every reason to believe that all the little asperities are not similar a nd that some of them do not present little faces at all inclinations. Some present more of them , others fewer, and the irregularity only disappears because of their very large number in each appreciable part of the large surface. However this may be, we shall represent the extent of the faces of each particular inclination by the surface of a circular zone, or rather a spherical one, which we shall consider as if it were by itself, or isolated . We shall give a spherical form to this zone, since each little face always adds its effect to that of others which have a very slightly different situation. Furthermore we shall continue to suppose in these researches that the body reflects [201] light equally all round at each inclination, which is exactly true in an infinite number of cases. At the same time we shall also pretend, but only for the clearness of our Figure, that these zones are placed stepwise one above the other in a discontinuous manner. Then let CHDT (Fig. 29) be the numerator of asperities of the surface AB. From our experiments we know the relation which exists between the ordinates CH, CD of this curve. We know the form of this curve or, if you wish, the nature of it, but at the moment we do not wish to settle its exact size or to discover its absolute dimensions. We shall begin by changing this curve into another curve EKC, of which the corresponding ordinates are as the square roots of the ordinates of the first one. Those of EKC will serve as the radii of the

RESEARCHES ON REFLECTED LIGHT

137

zones of which we have just spoken, and the similar portions of these different zones being proportional to the squares of their radii, these portions will really be proportional to the ordinates of the numerator of asperities CHD, as they ought to be in order to represent the sum of all the little faces at each inclination. (202] In this section we can scarcely avoid having recourse to algebraic analysis, just as we have been obliged to do in some of the articles of the preceding section. All our readers must know that geometry and algebra are, as it were, the logic of true physics, and those among them who are not sufficiently well versed in these advanced sciences have only to pass over our calculations and be contented to see the results. 44 We denote by a the radius AC of the surface AB, which it is always permissible to make circular, and by c its circumference. We obtain ½ac for its extent. We also let s be the sine of the variable angle FCA which is the complement of the inclination of the little faces; z will express the corresponding ordinate CH of the numerator which has a relation with s that is known through our observations, and of which the greatest value is equal to the total sine a; u will denote the ordinates CK of the second determinator, 45 which will be proportional to the 44 l t is quite evident that Bouguer felt that he was writing for a general, educated public, rather than for specialists. Perhaps his concern for his readers is a reflection of the general, good-natured politeness of French society at this period. It is interesting to contrast his method of exposition with that of Lambert which was much more condensed and professional. 46La seconde determinatrice. This is our first encounter with this curve, which deserves a note only because it returns to confuse an otherwise excellent and useful argument in the third book.

138

BOUGUER 1S OPTICAL TREATISE ON LIGHT

square roots of the ordinates of the first. Finally, r will indicate the length of the axis EC of the second curve. It is clear that all the difficulty is reduced to finding the absolute length of this axis, or its ratio to AC or DC, since everything else will follow. The ordinates z of the first determinator CHDT being as the squares u 2 of those of the [203] second, we have u 2 = r 2z/a. Thus we shall have KC or u = ryz/ya. This ordinate serves as the radius of the little arc Kk which, making a revolution around the axis CE, forms the zone by which all the little faces of the same inclination are represented. We shall obtain the area of this zone by multiplying KL, its vertical height, by the circumference of the circle of which KC is the radius. As regards KL, we have it by this proportion, FC = a: FG = ds::KC = ryz/ya:KL = (rdsyz)/aya;and thecircumference, of which KC is the radius, being KC X c/a or (cryz)/aya we shall have cr 2zds/a 3 for the surface of the zone. It is evident, on the other hand, that this zone occupies on the surface AB an area which is less, the greater the angle the zone makes with this surface. The small arc Kk projects on to Mm, which is as much smaller in relation to Kk as FP = s is in relation to the total sine FC = a. We shall then have cr 2zsds/a 4 for the entire projection of the zone, or for the part of the surface AB which serves it as a base. But since we know the relation which exists between each ordinate of the numerator of asperities CHD and the sine FP = s, we can reduce the projection cr 2zsds/a 4 to one variable, which permits us to integrate it or to obtain the value of (cr 2/a 4) f zsds, the sum of the bases of all the little faces or asperities. Now this sum should be equal [204] to the area½ ac of the entire surface AB which serves as a base to all the little faces or asperities with which it is covered. We shall then obtain the equation (cr 2/a 4 ) f zsds = ½ac, from which we derive r 2 = a 5/2J zsds; and that is all we wished to discover. Having found the value of r or of the axis CE of the second determinator of the asperities, we shall find the squares of its ordinates u or KC in an equally well-known manner, obtaining u 2 = a 4z/2f zsds. Thus we shall know which particular sphere each little face with a certain inclination should be compared to, and instead of having simple ratios as above, we shall henceforth submit to an exact calculation the absolute quantity of the reflected light, provided that no part of it is absorbed. We may even discover this latter part too, by comparing our calculations with direct observation. Moreover, the surface AB being a circle of which a is the semidiameter, if we increase it to make it two or three times as large, or

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139

even change its shape, we have still only to make the squares r2 or u 2 also two or three times as large, since it is not the values of rand u themselves but those of their squares which are proportional to the sum of the little faces at each inclination. Finally, the reflected light, if [205] it is not subject to any extinction, will be diminished, as we know, by distance, in the same ratio as the square of the distance at which it is received is greater than one-quarter of the squares r 2 or u 2 , since our little zones are nothing else but portions of spheres. 46

ARTICLE II Application of the preceding solution to plates of frosted silver and to surfaces of plaster, and precise determination of the quantity of light absorbed by these latter surfaces

THESE TWO SURFACES are not extremely different in the law followed by the reflection which they produce, and one may almost always neglect differences which exist mainly between the lower parts of the numerators of asperities, for these lower parts correspond to the little faces which make very large angles with the surface AB (Fig. 29), and because of this position they occupy much less area. However, we have felt we should not confuse these two kinds of surface, and so we have wished to make the calculations for them absolutely separately. It will be remembered that the numerator of asperities for the frosted silver is not [206] much different from two arcs of a circle each of 114° in extent, and having DC for a common chord. This important observation makes it possible for us to integrate exactly our general formulae for these plates, while in the majority of other cases we can only succeed by approximation. From the point C we draw the tangent CQ to one of the arcs DHC, DTC. We denote the sine QS bye, and the cosine CS or D V by f. We also drop the perpendicular FR on to QC, and we denote by x this perpendicular or sine, which is variable. We learn from trigonometry that the sine FP of the sum of the two arcs AQ and FQ, is equal to [Jx + e (a 2 - x 2 ) ½]!a. This expression is that for FP, which we had called sin our general solution. On the other hand, as the chords HG = z are always proportional to the sines FR = x in this particular investigation, where it is a question of the reflection caused by plates of mat silver or by the surface of plaster, we shall have this proportion: DV = f: DC= a:: FR = x : 48

Bouguer's note: See Article V of the first section of this book.

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BOUGUER'S OPTICAL TREATISE ON LIGHT

HC = z = (ax)/J. Thus we are in a condition to make all the necessary substitutions in our general formulae r 2 = a 5/2J zsds and u 2 = a 4z/2f zsds. We shall introduce ax/f in the place of z; [fx + e (a 2 - x 2) ½l/a in the place of s; and [fdx (a 2 - x 2) ½ - exdx]/ a (a 2 - x 2) ½in the place of ds; and we shall have 47 ,2

= __________a_0_ _ _ _ _ _ _ __

2f{[f(a;(:

2 ~ ) : 2) 1

ex][ x 2

+

7 x)½]}dx (a 2

-

2

which [207] is changed by integration into the equation

r

2

~a6J

=--------~----~-------~ (l + + e2)x 3

2efx2(a 2

2

-

x 2) 1

a 2ef(a 2

-

x 2) 1

-

a 3ef ·

It remains only to put DV = f for the greatest value of x and VC = e for that of (a 2 - x 2)½; and we shall finally obtain

r

2

3 8 -a 2

2 2 + 212 = ae e

+!

4

2 3a 3 3 , and r = 2 -ae a- 2e , or r = a

V

3a 2a- 2e

Thus, considering CKE, the second determinator of asperities for frosted silver, we see that its axis CE, or the radius corresponding to the little faces which are parallel to the surface AB, is appreciably greater than the radius CA or CD for this same surface. It is not equal to them, as would be the case if the little roughnesses were equivalent to little hemispheres, because the oblique little faces are here smaller or less in number, which is to the advantage of the other faces, since all the space which is not occupied by one must necessarily be taken up by others. As the arcs DHC and DTC each measure 114°, the angle DCQ is 57° and the angle A CQ 33°. Of these e and j are the sine and the cosine, and we find CE to be 1,815 parts in proportion to 1,000 for CD. If we are concerned with the reflection which takes place perpendicularly to the surface AB we may again use the proportion on which we have insisted so strongly. Taking the square of one-half of CE, we compare it to the square of the distance at which the reflected light is received, and (208] we shall find to what extent this light is weakened by reflection, supposing that none of it is absorbed. With regard to plaster, it has seemed to me to be more certain not 471n its original notation this equation is guaranteed to mislead a modern reader. It may have misled de la Caille since the last term of the next equation seems superfluous. It does not seem worthwhile to pursue this further.

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141

to attribute to the numerator of its asperities a fictitious geometrical curve, but to borrow the values of z, or the ratio between its little invisible faces, from the table in Article III of the preceding section, and to introduce these values directly into our formulae r 2 = a 5/ 2 zsds and u 2 = a 4z/2 zsds. I then obtained these integrals by approximation, and did this not only for plaster but also for the Dutch paper. The following table contains the small number of results which it has seemed sufficient to calculate:

f

f

TABLE 5 Table of values of a4z/2 f zsds or the ordinates CK of Figure 29 raised to the second power, one-quarter of which expresses the quantity of light reflected by plaster and by Dutch paper, supposing that no rays are extinguished Angles made by the small faces with the large surface (Degrees)

Values of CK 2 or r 2 or u 2 For plaster

For Dutch paper

00

15 30

24875 18308 13781

21765 20396 11862

45 60 75

9303 4378 1244

7792 3613 1132

[209] These values are the squares of the semi-diameters of spherical zones which are perfectly equivalent to the little faces presented by the asperities of the surfaces of plaster and of Dutch paper. Thus we have only to make a simple proportion, in conformity with what we have established in the first section of this third 48 book, in order to obtain the quantities of light reflected by these surfaces to a distance D. This distance will be expressed in units of which the semi-diameter of the circular surface AB contains 100, and we shall then say that D 2 is to the intensity of the light which the surface receives, as onequarter of the value of CK 2 indicated in the little table is to the intensity of the light reflected to the distance D. We must nevertheless still make a reduction for the angle of incidence if the rays do not fall perpendicularly on the little faces which produce the reflection. When we conform to this rule in order to find how much light is sent back to a distance of 3 inches in a perpendicular direction by a disk 48So in the original. There is further internal evidence that Bouguer may at some time have intended that the third book should have been the second, and vice versa. It should be noted that much of the matter of the first and third books appears in the Essai of 1729, but that the second book is new.

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BOUGUER'S OPTICAL TREATISE ON LIGHT

of plaster 1 inch in diameter, while it is illuminated in the same direction, we find that this is about 1/58, while it would only be 1/144 if the little asperities were equivalent to small hemispheres. It is true that if the light is not incident at a right angle, but at an angle of 75°, we must, (210] as we have just said, diminish this 1/58 a little, following the ratio of the sines of the angles of incidence. But the difference will not be very considerable. Everything considered, the plaster could send back approximately 1/60 of the light which it receives, but this quantity is very different to that which is furnished us by experiment, since we have observed that only 1/149 or 1/150 of the light received is really reflected. It follows from this that the plaster, in spite of its extreme whiteness, absorbs or extinguishes many rays. Of 10,000 which strike it, there ought to be 166 or 167 reflected; and there are really only 67. Out of 167 rays we find, then, that 100 are extinguished, or about three-fifths. 49

ARTICLE III Application of the general solution of Article I to the numerator of asperities which appears to apply best to several planets

that the numerator of asperities for several of the planets cannot be as narrow as the numerator of asperities of our plaster, and that for these celestial bodies we must even give a greater width than that of a perfect circle to this curve. If we continue to denote the total sine by a and the cosines of the inclinations of the little faces or the sines of the angles (211] A CH, A CG (Fig. 21) by s, we shall take the quantities s + b (s/ar for the ordinates z of the numerator, as we have already done. Then making the necessary substitutions in our general formulae r 2 = a 6/2 f zsds and u 2 = a 4z/2f zsds, which give us the squares of the axis CE = r of the second determinator of asperities CKEN (Fig. 29), and its variable ordinates CK = u, we shall have

WE HAVE RECOGNIZED

and which determine the spheres whose surfaces are suitable to represent all the little asperities of the visible portions of the planet. See Book I, n. 68.

49

143

RESEARCHES ON REFLECTED LIGHT

Supposing, as there is every reason to believe, at least until we have a greater number of observations, that we may regard as absolutely constant the quantity added to the sines s of the inclinations of the little faces, the exponent m will become zero, and the ordinates z of the numerator being proportional to s + b, we shall then have for the second determinator 3 r2 = -3a -2a + 3b

[212)

and

ARTICLE IV

On the total amount of light that the planets reflect to us in their different phases

A GREAT NUMBER of researches about the light of the planets suggest themselves. In the preceding section we have already examined how much more luminous these celestial bodies are at some points of their disks than at others, when they have no markings. We might still ask, what is the ratio which exists between their total light in their different phases? We shall try to answer this second question. Figure 27 represents the planet illuminated by the sun in the direction LE, while the observer is situated on EO at a distance which can also be regarded as infinite. We shall not repeat what we have already said in order to explain this figure, and we shall not go into details on the manner of finding the values of ES and of ET. We let a be the radius of the globe, y the sine CR of the arc CQ 50 of which the point C is elevated or depressed with reference to the plane GMK, t the sine QZ of the angle made by QE with the line EP which bisects the angle formed by the incident rays of light with the visual [213) rays; and finally by g we denote the sine of half this last angle. We shall then have and ET= [(a 2

-

g2)½ (a 2

-

t 2 )½ (a 2

-

y 2 )½ - gt(a 2

-

y2)½]/a 2

The first of these quantities is the sine of the complement of the angle of inclination of the little faces which serve to reflect light in the neighbourhood of C in the present case, and the other quantity ET 60DQ in the original.

144

BOUGUER'S OPTICAL TREATISE ON LIGHT

is the sine of the angle ECT, which is equal to that made by the visual rays, or better, the reflected rays, with the surface of the planet in the neighbourhood of the same point, since CT is perpendicular to these rays, just as EC is to the surface of the globe. If we next imagine beside the quarter-circle FQ another quarter-circle Fq infinitely near the first one, we shall have adt/(a 2 - t 2)½ for the value of the small arc Qq, and in conformity with the doctrine of Archimedes we must multiply this little arc by CX = dy in order to obtain the small trapezium on the surface of the planet which, being included between the two quarters of a circle FQ and Fq, has the small arc CX 51 for inclined height. We shall then have adydt/(a 2 - t 2)½ for this small trapezium and it remains to discover how many little faces it contains, which contribute to the reflection in proportion to their size, or, which comes to the same thing, we must look for the radius of the small sphere which is comparable to it. (214] Rigorously we could content ourselves with multiplying the little spherical surface adydt/(a 2 - t 2)½ which corresponds to CX 61 , by ES b (ES/ EKr, or by (a2 _ y2)½ (a2 _ 12)½/a b (a2 _ y2)½m (a2 _ 12)½m/a2m,

+

+

which, being the sine ES of the angle BCH augmented by a certain quantity b (ES/EKr, expresses the sum of the small faces of the same inclination, of which BCH is the complement. As far as the ratio between the total quantities of light is concerned, this is the same thing as multiplying by a quarter of the square u 2 of the ordinates of the second determinator, since these squares are proportional to the ordinates z of the numerator. We then obtain dydt (a 2 - y 2)½ + [bdydt (a 2 - y 2)½m (a 2 - t 2)½m-½J/a 2m- 1, for the sum of the small faces which reflect the light in an effective manner in the present case. But we must reduce them in the same ratio that the sine of the angle of incidence of the direct light is smaller than the total sine, that is to say, the value which we have just found should be divided by the total sine a and multiplied by the sine of the angle of incidence which is (a 2 - g 2) ½, since, CH being parallel to PE and perpendicular to the small faces, the angle of incidence is the complement of the angle NEP, of which g is the sine. In consequence we shall obtain [dtdy (a 2 - y 2 )½ (a 2 - g 2)½]/a + [bdtdy (a 2 - y 2)½m (a 2 - t 2)