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THE 
 
 LIFE 
 
 OF 
 
 JOHN DOLLOND, F.R.S. 
 
 INVENTOR 
 
 OF THE 
 
 ^c!)romatit Celesrope. 
 
 With a copious Appendix of all the Papers referred to. 
 
 BY 
 
 JOHN KEJLJLY, ILIL.B. 
 
 RECTOR OF COPFORD, IN THE COUNTY OF ESSEX ; , 
 
 Author of the Triglott Celtic Dictionary, and a Translator of the 
 Bible into the Manks Gaelic. 
 
 THIRD EDITION WITH ADDITIONS. 
 
 PRINTED BY W. M. THISELTON, 37, GOODGE STRKE'[ 
 
 1808. 
 
LOAN STACK 
 

 CONTENTS 
 
 Page 
 TAeXj/eo/JohnDollond, F.R.S 5 
 
 u4 Letter from Mr. John Dollond to Mr. James Short, F.R.S. con- 
 cerning an Improvement of Refracting Telescopes . . . . 17 
 
 Letters relating to a Theorem of M. Euler, of the Royal Academy of ' 
 Sciences at Berlin, and F.R.S. for correcting the Aberrations in 
 the Object-Glasses of Refracting Telescopes •. ^1 
 
 ~/i Description of a Contrivance Jbr Measuring Small Angles, bi/ 
 Mr. John Dollond; communicated by Mr. J. Short, F.R.S. . 33 
 
 An Explanation of an Instrument for Measuring Small Angles, the 
 
 first Account of which was read before the Royal Society, May 10, 
 
 1753. By Mr. John Dollond. In a Letter to James Short, M. A. 
 
 and F.R.S 36 
 
 433 
 
11 CONTENTS. 
 
 / 
 
 Page 
 An jiccount of some Experiments concerning the different Refrangibility 
 
 of Light. By Mr. John Dollond. With a Letter from James 
 
 Short, M.A. F.R.S. Acad. Reg. Suec. Soc. 50 
 
 Some Account of the Discovery, made by the late Mr. John Dollond^ 
 F.R.S. which led to the grand Improvement o/" Refracting Teles- 
 copes, in order to correct some Misrepresentations, in Foreign 
 Publications, of that Discovery: with an Attempt to account for the 
 Mistake in an Experiment made by Sir Isaac Newton ; on ivhich 
 Experiment, the Improvement of the Refracting Telescope intirely 
 depended, By Peter Dollond, Member of the American Philoso- 
 phical Society at Philadelphia 6l 
 
 An Attempt to explain a Difficulty in the Theory of Vision, depending 
 on the different Refrangibility of Light. By the Rev. Nevil Maske- 
 lyne, D.D. F.R.S. and Astronomer Royal 78 
 
 An Account of an Improvement made by Mr. Peter Dollond in his New 
 Telescopes. In a Letter to James Short, M.A. F.R.S. tvith a 
 Letter of Mr. Short^o the i2e?;. Thomas Birch, D.D. fe.R.S. 8 8 
 
 A Letter from Mr. Peter Dollond, to Nevil Maskelyne, F.R.S. and 
 Astronomer Royal; describing some Additions and Alterations made 
 to Hadley's Quadrant, to render it more serviceable at- Sea . 92 
 
 Remarks on the Hadley's Quadrant, tending principally to remove the 
 Difficulties which have hitherto attended the Use of the Back-^observa- 
 
CONTENTS. Ill 
 
 Page 
 tion, and to obviate the Errors that might arise from a JVant of 
 
 Parallelism in the two Surfaces of the Index-Glass. By Nevil 
 
 Maskelyne, F. R. S. Astronomer Royal 9^ 
 
 An Account of an Apparatus applied to the Equatorial Instrument for 
 
 , correcting the Errors arising from the Refraction in Altitude. By 
 
 Mr. Peter Dollond, Optician; communicated to the Royal Society 
 
 by the Astronomer Royal .119 
 
 A 'I 
 
THE 
 
 LIFE 
 
 OF 
 
 JOHN DOLLOND, F.R.S. 
 
 In modem times the attention of men has been employed rather in 
 improving what they know than in attempting to make new disco- 
 veries. When a man, therefore, has been fortunate enough, by 
 extraordinary research, or by a strong effort of genius, to surprise 
 the world with a new invention, a lively interest is immediately excited 
 in every mind to trace the steps, investigate the means, and collect 
 
6 ' LIFE OF JOHN DOLLONp, F.R.S. 
 
 every incident which led to the result: — and to the honour of human 
 nature be it said, while curiosity exerts itself in this manner on the 
 invention, the inventor is not less the object of regard and consi- 
 deration: we wish to learn the history, the life, the character of the 
 man, and, as far as it is possible, to be acquainted with him. The 
 subject of the following memoir is entitled to this introduction, and 
 the public will receive with satisfaction the following account of the 
 inventor of the achromatic telescope. 
 
 John Dollond, fellow of the Royal Society, was born in Spitalfields, 
 I J Oh on the tenth day of June in the year 1706: his parents were French 
 
 protestants, and at the time of the revocation of the ^dict of Nantz, 
 which happened in the year l685, resided in Normandy; but in what 
 particular part of it is not at present precisely known : M. de Lalande 
 does not believe the name to be of French origin : but however this may 
 be, the family were compelled soon after this period to seek refuge in 
 England, in order to avoid persecution and to preserve their religion. 
 
 The fate of this family was not a solitary case; fifty thousand per- 
 sons pursued the same measures, and we may date from this period 
 the rise of several arts and manufactures, which have become highly 
 beneficial to this country. An establishment was given to these 
 refugees, by the wise policy of our government, in Spitalfields, and 
 particular encouragement granted to the silk manufactory. 
 
 The first years of Mr. Dollond's life were employed at the loom; 
 but, being of a very studious and philosophic turn of mind, his leisure 
 hours were engaged in mathematical pursuits; and though by the 
 death of his father, which h ippened in his infancy, his education 
 gave way to the necessities of \ is family, yet at the age of fifteen. 
 
LIFE OP JOHN DOLLONDj F.R.S. 7 
 
 before he had an opportunity of seeing works of science or elementary 
 treatises, he amused himself by constructing sun-dials, drawing geo- 
 metrical schemes, and solving problems. 
 '^* 'An early marriage and an increasing family afforded him little 
 opportunity of pursuing his favourite studies: but such are the powers 
 of the human mind when called into action, that difficulties, which 
 appear to the casual observer insurmountable, yield and retire before 
 perseverance and genius: even under the pressure of a close applica- 
 tion to business for the support of his family, he found time, by 
 abridging the hours of his rest, to extend his mathematical knowledge, 
 and made a considerable proficiency in optics and astronomy, to which 
 he now principally devoted his attention, having in the earlier stages 
 of his life prepared himself for the higher parts of those subjects by 
 a perfect knowledge of algebra and geometry. 
 
 Soon after this, without abating from the ardour of his other lite- 
 rary pursuits, or relaxing from the labours of his profession, he began 
 to study anatomy, and likewise to read divinity; and finding the 
 knowledge of Latin and Greek indispensably necessary towards at- 
 taining those ends, he applied himself diligently, and was soon able 
 to translate the Greek Testament into Latin ; and as he admired the 
 power and the wisdom of the Creator in the mechanism of the 
 human frame, so he adored his goodness displayed in his revealed 
 word. 
 
 It might from hence be concluded that his sabbath was devoted to 
 retired reading and philosophical objects; but he was not content with 
 private devotion, as he was always an advocate for social worship, and 
 with his family regularly attended the public service of the French 
 
8 LIFE OF JOHN DOLLOND, P.R.S. 
 
 protestant church, and occasionally heard Benson and Lardner, whom 
 he respected as men and admired as preachers. In his appearance he 
 was grave, and the strong lines of his face were marked with deep 
 thought and reflection; but in his intercourse with his family and 
 friends, he was cheerful and affectionate; and his language and senti- 
 ments are distinctly recollected as always making a strong impression 
 on the minds of those with whom he conversed. His memory was 
 extraordinarily retentive, and, amidst the variety of his reading, he 
 could recollect and quote the most important passages of every book 
 which he had at any time perused. 
 
 He designed his eldest son, Peter Dollond, for the same business 
 with himself; and for several years they carried on their manufactures 
 together in Spitalfields ; but the employment neither suited the ex- 
 pectations nor disposition of the son, who, having received much 
 information upon mathematical and philosophical subjects from the 
 instruction of his father, and observing the great value which was set 
 upon his father's knowledge in the theory of optics by professional 
 men, determined to apply that knowledge to the benefit of himself 
 and his family; and accordingly, under the directions of his father, 
 commenced optician. Success, though under the most unfavourable 
 circumstances, attended every effort; and in the year 1752 John Dol- 
 lond, embracing the opportunity of pursuing a profession congenial 
 with his mind, and without neglecting the rules of prudence towards 
 his family, joined his son, and in consequence of his theoretical 
 knowledge, soon became a proficient in the practical parts of optics. 
 
 His first attention was directed to improve the combination of the 
 eye-glasses of refracting telescopes; and having succeeded in his sys- 
 
LIFE OP JOHN DOLLOND, P.K.S. 9 
 
 tem of four eye-glasses, he proceeded one step further, and produced 
 telescopes furnished with five eye-glasses, which considerably surpassed 
 the former; and of which he gave a particular account in a paper 
 presented to the Royal Society, and which was read on the 1st of 
 March 1753, and printed in the Philosophical Transactions, vol. 
 xlviii. page 108. — See appendix, pp. 17 — 20. 
 
 Soon after this he made a very useful improvement in Mr. Savery's 
 micrometer: for instead of employing two entire object-glasses, as 
 Mr. Savery and M. Bouguer had done, he used only one glass cut 
 into two equal parts, one of them sliding or moving laterally by the 
 other. This was considered to be a great improvement, as the mi- 
 crometer could now be applied to the reflecting telescope with much 
 advantage, and which Mr. James Short immediately did. An account 
 of the same was given to the Royal Society, in a paper which was 
 afterwards printed in the Philosophical Transactions, vol. xlviii. page 
 178; and in another paper, part ii. page 551. — See appendix, pp. 
 33 — 49.* 
 
 Mr. Dollond's celebrity in optics became now universal; and the 
 friendship and protection of the most eminent men of science flattered 
 and encouraged his pursuits. To enumerate the persons, both at 
 home and abroad, who distinguished him by their correspondence or 
 cultivated his acquaintance, however honourable to his memory, 
 would only be an empty praise. We cannot, however, forbear men- 
 
 * This kind of micrometer was afterwards applied by Mr. P. DoUond to the achro- 
 matic telescope.— /See Appendix^ pp. 88 — ^91. 
 
 B 
 
10 LIFE OP JOHN DOLLOND, F.R.S. . 
 
 tigning the names of a few persons, who held the highest place in 
 his esteem as men of worth and learning: — Mr. Thomas Simpson, 
 master of the Royal Academy at Woolwich; Mr. Harris, assay- 
 master at the Tower, who was at that time engaged in writing and 
 publishing his Treatise on Optics; the Rev. Dr. Bradley, then astro- 
 nomer royal; the Rev. William Ludlam, of St. John's college, Cam- 
 bridge; Mr. John Canton, a most ingenious man, and celebrated 
 not less for his knowledge in natural philosophy, than for his neat 
 and accurate manner of making philosophical experiments. To this 
 catalogue of the philosophical names of those days, we must add that 
 of the present astronomer royal, the Rev. Dr. Maskelyne, whose 
 labours have so eminently benefited the science of astronomy. 
 
 Surrounded by these enlightened men, in a state of mind prepared 
 for the severest investigation of philosophic truths, and in circum- 
 stances favourable to liberal inquiry, Mr. Dollond engaged in the 
 discussion of a subject, which at that time not only interested this 
 country, but all Europe. Sir Isaac Newton had declared, in his Trea- 
 tise on Optics, page 112, " That all refracting substances diverged 
 the prismatic colours in a constant proportion to their mean re- 
 fraction;" and drew this conclusion, " that refraction could not be 
 produced without colour;" and consequently, " that no improvement 
 could he expected in the refracting telescope.'" No one doubted the 
 accuracy with which Sir Isaac Newton had made the experiment; yet 
 some men, particularly M. Euler and others, were of opinion that 
 the conclusion which Newton had drawn from it went too far, and 
 maintained that in very small angles refraction might be obtained 
 without colour. Mr. Dollond was not of that opinion, but defended 
 
LIFE OP JOHN DOLLOND, F.R.S. 11 
 
 Newton's doctrine with much learning and ingenuity, as may be seen 
 by a reference to the letters which passed between Euler and Dollond 
 upon that occasion, and which were published in the Philosophical 
 Transactions, vol. xlviii. page 287 — see Appendix, pp. 21 — 32; and 
 contended that, " If the result of the experiment had been as de- 
 scribed by Sir Isaac Newton, there could not be refraction without 
 colour." 
 
 A mind constituted like Mr. Dollond's could not remain satisfied 
 with arguing in this manner from an experiment made by another, 
 but determined to try it himself, and accordingly, in the year 1757, 
 began the examination ; and, to use his own words, with, " a resolute 
 perseverance," continued during that year, and a great part of the 
 next, to bestow his whole mind on the subject, until in the month 
 of June 1758 he found, after a complete course of experiment, the 
 result to be very different from that which he expected, and from that 
 which Sir Isaac Newton had related. He discovered " the difference 
 in the dispersion of the colours of light, when the mean rays are equally 
 refracted hy different mediums.''^ The discovery was complete, and he 
 immediately drew from it this practical conclusion, '^ That thQ object- 
 glasses of refracting telescopes were capable of being made without 
 the images formed by them being affected by the different refrangi- 
 bility of the rays of light." His account of this experiment, and of 
 others connected with it, was given to the Royal Society, and printed 
 in their Transactions, vol. I. page 743 — see Appendix, pp, 50 — 60; 
 and he was presented in the same year, by that learned body, with 
 Sir Godfrey Copley's medal, as a reward of his merit, and a memorial 
 of the discovery, though not at that time a member of the society. 
 
 B 2 
 
12 LIFE OF JOHN DOLLOND, P.R.S, 
 
 This discovery no way affected the points in dispute between Euler 
 and Dollond, respecting the doctrine advanced by Sir Isaac Newton.* 
 A new principle was in a manner found out, which had no part in 
 their former reasonings, and it was reserved for the accuracy of Dol- 
 lond to have the honour of making a discovery which had eluded the 
 observation of the immortal Newton. -{- 
 
 This new principle being now established, he was soon able to con- 
 struct object-glasses, in which the different refrangibility of the rays 
 of light was corrected, and the name of achromatic given to them by 
 the late Dr. Bevis, on account of their being free from the prismatic 
 colours. Dr. Hutton, in his Mathematical Dictionary, has said that 
 this name was given to them by M. de Lalande; but that is a mistake. 
 
 As usually happens on such occasions, no sooner was the achro- 
 matic telescope made public, than the rivalship of foreigners, and 
 the jealousy of philosophers at home, led them to doubt of its 
 reality; and Euler himself, in his paper read before the Academy of 
 Sciences at Berlin, in the year 17^4, says, " I am not ashamed 
 frankly to avow, that the first accounts, which were published of it, 
 appeared so suspicious, and even so contrary to the best established 
 principles, that I could not prevail upon myself to give credit to them ;" 
 
 * See note at bottom of pages 79 — 80, for Priestley's remarks, &c, 
 t The cause of this difference of the results of the 8th experiment of the 2nd part 
 of the first book of Newton's Optics, as related by himself, and as it was found when 
 tried by DoUond in the years 1757 and 1758, is fully and ingeniously accounted for by 
 Mr. Peter DoUond in a paper read at the Royal Society on the 21st of May 1789, and 
 afterwards published for J. Johnson in St. Paul's Church Yard-— 5ee Appendix, pp.QX—^ 
 yy ; also in Hutton's Dictionary— Article, Chromatic. 
 
WPE OP JOHN DOLLOND, P.R.&. 13 
 
 and he adds, " I should never have submitted to the proofs which 
 Mr. Dollond produced to support this strange phaenomenon, if M. 
 Clairaut, who must at first have been equally surprized at it, had not 
 most positively assured me that Dollond's experiments were but too 
 well founded." And when the fact could no longer be disputed,, they 
 endeavoured to find a prior inventor, to whom it might be ascribed, 
 and several conjecturers were honoured with the title of discoverers. 
 
 Mr. Dollond's improvement in refracting telescopes was of the 
 greatest advantage in astronomy, as they have been applied to fixed 
 instruments; by which the motions of the heavenly bodies are de- 
 termined to a much greater exactness than by the means of the old 
 telescope. Navigation has also been much benefited by applying 
 achromatic telescopes to the " Hadley's sextant:" and from the im- 
 proved state of the lunar tables, and of that instrument, the longitude 
 at sea may now be determined by good observers to a great degree of 
 accuracy; and their universal adoption by the navy and army, as well 
 as by the public in general, is the best proof of the great utility of 
 the discovery. 
 
 In the beginning of the year 1761, Mr. Dollond was elected fellow 
 of the Royal Society, and appointed optician to his majesty, but did 
 not live to enjoy those honours long; for on the 30th of November, 
 in the same year, as he was reading a new publication of M. Clairaut, 
 on the theory of the moon, and on which he had been intently en- 
 gaged for several hours, he was seized with apoplexy, which rendered 
 him immediately speechless, and occasioned his death in a few hours 
 afterwards. Besides Mr. Peter Dollond, whom we had occasion to 
 mention in this memoir, his family, at his death, consisted of three 
 
14 LIFE OP JOHN DOLLONDj F.R.S, 
 
 daughters and a son, who, possessing the name of his father, and 
 we may add, a portion of the family abilities, carried on the optical 
 business in partnership with his elder brother. 
 
 Since the last edition of this Life, we have to mention the death 
 of Mr. John Dollond, the partner of, his elder brother Mr. Peter 
 DoUond, which has occasioned the latter to take into partnership his 
 nephew, the son of his eldest sister, Mr. George Huggins, who has, 
 by the King's permission, taken the name of Dollond. 
 
 THE END OP THE LIFE. 
 
THE APPENDIX. 
 
■ii 
 
APPENDIX. 
 
 A Letter from Mr, John DoUond to Mr. James 
 Short, F.R S. concerning an Improvement of 
 Refracting Telescopes. 
 
 Read March 1, 1753. 
 
 SIR, 
 
 JLT is well known, that the perfection of refracting 
 telescopes is very much limited by the aberration of the rays of light 
 from the geometrical focus; which arises from two very different causes; 
 that is, from different degrees of refrangibility of light, and from the 
 figure of the sphere, which is not of a proper curvature for collecting 
 the rays in a single point. The object-glass is chiefly affected by the 
 first of these; nor has there been yet any method discovered for recti- 
 fying that aberration so, as in the least to remove the indistinctness of 
 the image arising from it. We are therefore reduced to the necessity 
 of contracting their apertures, which renders it impossible to magnify 
 much without very long glasses. 
 
 c 
 
18 A Letter from Mr, John Dollond to Mr. James Short, 
 
 But the case is widely different with regard to the eye-glasses; for, 
 though they are very much affected by both the aberrations before- 
 mentioned, yet, by a proper combination of several together, their 
 errors may be in a great measure corrected. If any one, for instance, 
 would have the visual angle of a telescope to contain 20 degrees, the 
 extreme pencils of the field must be bent or refracted in an angle of 
 10 degrees; which, if it be performed by one eye-glass, will cause 
 an aberration from the figure, in proportion to the cube of that angle: 
 but if two glasses are so proportioned and situated, as that the re- 
 fraction may be equally divided between them, they will each of them 
 produce a refraction equal to half the required angle: and therefore 
 the aberration being in proportion to the cube of half the angle taken 
 twice over, will be but a fourth part of that, which is in proportion 
 to the cube of the whole angle; because twice the cube of one is but 
 J of the cube of two; so the aberration from the figure, where two 
 eye-glasses are rightly proportioned, is but a fourth of what must 
 unavoidably be, where the whole is performed by a single eye-glass. 
 By the same way of reasoning, when the refraction is divided between 
 three glasses, the aberration will be found to be but the ninth part of 
 what would be produced from a single glass; because three times the 
 cube of one is but one ninth of the cube of 3. Whence it appears, 
 that, by increasing the number of eye-glasses, the indistinctness, 
 which is observed near the borders of the field of a telescope, may 
 be very m;ich diminished, though not intirely taken away. 
 
 The method of correcting the errors arising from the different 
 refrangibility of light is of a different consideration from the former; 
 for, whereas the errors from the figure can only be diminished in a 
 
concerning an Improvement of Refracting Telescopes, 19 
 
 certain proportion to the number of glasses, in this they may be 
 intirely corrected, by the addition of only one glass; as we find in 
 the astronomical telescope, that two eye-glasses, rightly proportioned, 
 will cause the edges of objects to appear free from colours quite to 
 the borders of the field. Also in the day telescope, where no more 
 than two eye-glasses are absolutely necessary for erecting the object, 
 we find, by the addition of a third rightly situated, that the colours, 
 which would otherwise confuse the image, are intirely removed: — I say 
 intirely removed; but this is to be understood with some limitation; 
 for though the different colours, which the extreme pencils must 
 necessarily be divided into by the edges of the eye-glasses, may in this 
 manner be brought to the eye in a direction parallel to each other, so 
 as, by the humours thereof, to be converged to a point in the retina; 
 yet, if the glasses exceed a certain length, the colours may be spread 
 too wide to be capable of being admitted through the pupil or aperture 
 of the eye; which is the reason, that, in long telescopes, constructed 
 in the common manner, with three eye-glasses, the field is always 
 very much contracted. 
 
 These considerations^ Sir, first set me on contriving, how to en- 
 large the field by increasing the number of eye-glasses, without any 
 hinderance to the distinctness or brightness of the image: and though 
 others had been about the same work before, yet observing, that the 
 five-glass telescopes, sold in the shops, would admit of farther im- 
 provement, I endeavoured to construct one with the same number of 
 glasses in a better manner; which so far answered my expectations, as 
 to be allowed by such persons, as are the best judges, to be a consi- 
 derable improvement on the former. 
 
 c 2 
 
20 A Letter from Mr. John Dollond to Mr. James Short, &c. 
 
 Encouraged by this success, I resolved to try, if possibly I might 
 gain some farther enlargement of the field by the addition of another 
 glass; and by placing and proportioning the glasses in such a manner, 
 as to correct the aberrations as much as possible, without any 
 detriment to the distinctness, I have obtained as large a field, as is 
 convenient or necessary, and that even in the longest telescopes, 
 which can be made. 
 
 These telescopes with six glasses having been well received, and 
 some of them being gone to foreign parts, it seems a proper time to 
 settle the account of its origin; which is one of the motives, that has 
 induced me to trouble you with this short sketch of the considerations, 
 that gradually led me to its construction ; and I am emboldened. Sir, 
 to write thus much, from the many favours I have already received at 
 your hands, as well as from a sense of your being a proper person to 
 judge in such cases. And though I am sensible, that you are not 
 unacquainted with the theory contained in this letter, yet forasmuch 
 as the subject has never been fully treated by any author, I shall en- 
 deavour, as soon as may be, to draw up a more particular explanation 
 of the aberrations of light by refraction; but shall add no more at 
 present, only beg leave to take this opportunity of subscribing 
 myself 
 
 Your much obliged 
 
 and most humble servant^ 
 
 John Dollond. 
 
 Vine Court, 
 February 21, 1753. 
 
21 
 
 Letters relating to a Theorem of Mr. Euler, of 
 the Royal Academy of Sciences at Berlin, and 
 F.R.S. for correcting the Aberrations in the 
 Object-Glasses of Refracting Telescopes. 
 
 No. 1. 
 A Ze«er/rom Mr. James Short, F.R.S. to Peter Duval, Esq. F.R.S. 
 
 Read April 9, 1752. 
 
 DEAR SIR, 
 
 1 HERE is published, in the Memoirs of the Royal 
 Academy at Berlin, for the year 1747, a theorem by Mr. Euler, in 
 which he shews a method of making object-glasses of telescopes, in 
 such a manner, as not to be affected by the aberrations arising from 
 the different refrangibility of the rays of light: these object-glasses 
 consisting of two meniscus lenses, with water between them. 
 
 Mr. John Dollond, who is an excellent analyst and optician, has 
 examined the said theorem, and has discovered a mistake in it, which 
 arises by assuming an hypothesis contrary to the established principles 
 of optics; and, in consequence of this, Mr. Dollond has sent me 
 
11 Letters relating to a Theorem, 8$c. 
 
 the inclosed letter, which contains the discovery of the said mistake, 
 and a demonstration of it. 
 
 In order to act in the most candid manner with Mr. Euler, I have 
 proposed to Mr. DoUond to write to him, shewing him the mistake, 
 and desiring to know his reasons for that hypothesis; and therefore I 
 desire, that this letter of Mr. Dollond's to me may be kept amongst 
 the Society's papers, till Mr. Euler has had a sufficient time to 
 answer Mr, Dollond's letter to him. -i-s-'., «|'>'^ 
 
 I am, SIR, 
 
 Your most humble servant, 
 
 James Shoit. 
 
 Surrey Street, 
 April 9, 1752. 
 
 .nnr^- n 
 
of Mr, Euler, of the Royal Academy of Berlin, 23 
 
 No. 2. 
 
 A Letter from Mr. John Dollond to James Short, A.M. F.R.S. 
 concerning a Mistake in M. Euler's Theorem for correctirig the 
 Aberrations in the Object-Glasses of Refracting Telescopes. 
 
 Read November 23, 1752. 
 
 SIR, 
 
 1 HE famous experiments of the prism, first tried 
 by Sir Isaac Newton, sufficiently convinced that great man, that the 
 perfection of telescopes was impeded by the different refrangibility of 
 the rays of light, and not by the spherical figure of the glasses, as 
 the common notion had been till that time; which put the philoso- 
 pher upon grinding concave metals, in order to come at that by 
 reflection, which he despaired of obtaining by refraction. For, that 
 he was satisfied of the impossibility of correcting that aberration by a 
 multiplicity of refractions, appears by his own words, in his Treatise 
 of Light and Colours, Book I. Part 2. Prop. 3. " I found more- 
 " over, that when light goes out of air through several contiguous 
 " mediums, as through water and glass, as often as by contrary 
 
24 Letters relating to a Theorem^ ^c. 
 
 " refractions it is so corrected, that it emergeth in lines parallel to 
 " those in which it was incident, continues ever after to be white. 
 " But if the emergent rays be inclined to the incident, the whiteness 
 " of the emerging light will by degrees, in passing on from the place 
 " of emergence, become tinged in its edges with colours." 
 
 It is therefore, Sir, somewhat strange, that any body now-a-days 
 should attempt to do that, which so long ago has been demonstrated 
 impossible. But, as so great a mathematician as Mr. Euler has lately 
 published a theorem * for making object-glasses, that should be free 
 from the aberration arising from the different refrangibility of light, 
 the subject deserves a particular consideration. I have therefore care- 
 fully examined every step of his algebraic reasoning, which I have 
 found strictly true in every part. But a certain hypothesis in page 
 285, appears to be destitute of support either from reason or experi- 
 ment, though it be there laid down as the foundation of the whole 
 fabriek. This gentleman puts m : 1 for the ratio of refraction out of 
 air into glass of the mean refrangible rays, and M'. 1 for that of the 
 least refrangible. Also for the ratio of refraction out of air into 
 water of the mean refrangible rays he puts nil, and for the least 
 refrangible N'.l, As to the numbers, he makes w=-|-i., JW=xi=^ 
 and n=4-; which so far answer well enough to experiments. But 
 the difficulty consists in finding the value of N in a true proportion 
 to the rest. 
 
 Here the author introduces ' the supposition above-mentioned ; 
 
 * Vide Memoirs of the Royal Academy of Berlin for the Year 1747' 
 
of Mr, Euler, of the Royal Academy of Berlin. 25 
 
 which is, that m is the same power of M, as n is of N; and therefore 
 puts n=:m^, and N=M^. Whereas, by all the experiments that have 
 hitherto been made, the proportion will come out thus, m—l: 
 n—1: :m—M:n—N. 
 
 The letters fixed upon by Mr. Euler to represent the radii of the 
 four refracting surfaces of his compound object-glass, ^refg h and k, 
 and the distance of the object he expresses by a; then will the focal 
 
 distance be= J ^ . Now, says he, 
 
 w(7— t) +^(7— 7-tt— t; — t— / "TT 
 
 it is evident, that the different refrangibility of the rays would make 
 no alteration, either in the place of the image, or in its magnitude, 
 if it were possible to determine the radii of the four surfaces, so as to 
 
 have n(t-i)+m(^-t+i-i) = ^(7-i)+M7-t+i-i-)- And 
 this. Sir, I shall readily grant. But when the surfaces are thus 
 proportioned, the sum of the refractions will be=0; that is to say, 
 the emergent rays will be parallel to the incident. For, if w(-^— x) + 
 
 • Mi— 7+i-i) =iV(t-i)+i^i'(:^-t+i-i). thenn^N{^^^)^ 
 w_3f (i— -i:-j-L«.x)=0. Also if n—Nim—M: :w— 1 :7ra— 1, then 
 
 w-l(t-i)+w-l(:^-t+i-i)=0; or otherwise ?2(t~i)+w,(;^-t 
 Hhi— i")— /+i=0; which reduces the denominator of the fraction 
 expressing the focal distance to -^. Whence the focal distance will be 
 = a; or, in other words, the image will be the object itself. And as, 
 in this case, there will be no refraction, it will be easy to conceive 
 how there should be no aberration. 
 
 And now. Sir, I think I have demonstrated, that Mr. Euler's 
 theorem is intirely founded upon a new law of refraction of his awn ; 
 but that, according to the laws discovered by experiment, the 
 
26 
 
 Letters relating to a Theorem, 8^c. 
 
 aberration arising from the different refrangibility of light at the 
 object-glass cannot be corrected by any number of refractions 
 whatsoever. 
 
 London, 
 March 11, 3752, 
 
 I am, SIR, 
 
 Your most obedient humble servant, 
 
 John DoUond. 
 
of Mr, Euler, of the Royal Academy of Berlin. ^ 
 
 ^ No. 3. 
 
 Mr. Euler's Letter to Mr. James Short, F.R.S. 
 Read Jvdy 8, 1753. 
 
 MONSIEUR, 
 
 V OUS m'avez fait un tres sensible plaisir, ent g.yant 
 dispose M. Dollond de remettre la proposition de ses objections 
 contre mes verres objectifs, jusqu' k ce que j'y aurois repondu, et je 
 vous en suis infininient oblige. Je prend done la liberte de vous 
 addresser ma reponse k lui, en vous priant, apres I'avoir daignee de 
 votre examen, de la vouloir bien lui remettre: et en cas que vous 
 jugiez cette matiere digne de I'attention de la Societe Royale, je vous 
 prierois de lui communiquer les preuves detaillees de ma theorie, que 
 j'ai exposee dans cette lettre. Cependant j'espere, que M. Dollond en 
 sera satisfait, puisque je tombe d'accord avec lui du pen de succes, 
 qu'on sauroit se promettre de mes objectifs, en les travaillant selon la 
 maniere ordinaire. 
 
 J'ai I'honneur d'etre, avec la plus parfaite consideration, 
 
 MONSIEUR, 
 
 Votre tres humble, et 
 
 tres obeissant serviteur, 
 
 L. Euler. 
 
 Berlin, 
 Juin IQ, iy5'J>. 
 
 D '2 
 
28 Letters relating to a Theorem, 8^e, 
 
 No. 4. 
 
 ^ Monsieur Monsieur Dollond. 
 Read July 8, 1753. 
 
 MONSIEUR, 
 
 JjiTANT tres sensible a I'honneur qu6 vous me faites, 
 au sujet des verres objectifs, que j'avois propose, j'ai celiii de vous 
 marquer d'abord ingenument, que j'ai rencontre aussi ici le plus 
 grands obstacles dans I'execution de ce dessein, vu qu'il s'agit de 
 quatre faces, qui doivent etre travaillee exactement selon les propor- 
 tions que j'avois trouvees : cependant ayant fait les- experiences sur 
 quelquesuns, qui parurent le mieux reussi^ nous avons trouv^, que 
 I'intervalle entre les deux foyers des rayons rouges et violets etoit 
 beaucoup plus petit, qu'il ne seroit d'un verre simple de la meme 
 distance focale. Neantmoins je dois avoiier, qu'un tel verre, quand 
 meme il bien seroit parfaitement execute sur mes principes, auroit 
 d'autres defauts, qui le mettroient au dessous meme des verres ordi- 
 naires : c'est qu'un tel verre n'admet qu'un tres petite ouverture en 
 consequence des grandes courbures, qu'on doit donner aux faces in- 
 terieures: desorte que lorsqu'on donne une ouverture ordinaire, 
 I'image devient tres confus. 
 
of Mr. Elder J of the Royal Academy of Berlin^ ^9 
 
 Ainsi puisque vous vous etes donne la peine, Monsieur, d*executer 
 de tels verres, en en faisant des experiences*, je vous prie de bien 
 distinguer les defauts. qui peuvent naitre de* la diverse refrangibilit6 
 des rayons, de ceux, qui viennent d'une trop grande ouverture: pour 
 cet efFet vous n'aurez qu'a laisser une tres petite ouverture. 
 
 Or si ma theorie etoit juste, dont j'aurai bientot I'honneur de 
 parler, il seroit moyen de remedier k ce defaut; il faudroit renoncer a 
 la figure spherique qu'on donne ordinairement aux faces des verres, et 
 tacher de leur donner une autre figure, et j'ai remarque que la figure 
 d'une parabole leur procureroit Tavantage, qu'ils admettroieiit une 
 ouverture tres considerable. Notre savant M. Lieberkuhn s'est appli- 
 que ^ travailler des verres dont la courbure des faces decroit depuis le 
 milieu vers le bords, et il s'en est aperqu de tres grands avantages. 
 Par ces raisons je crois, que ma theorie ne soufFre encore rien de ce 
 cotd. 
 
 Pour la theorie, je conviens avec vous, monsieur, que posant la 
 raport de refraction d'un milieu dans un autre quelconque pour les 
 rayons moyens comme w ^ 1, et pour les rayons rouges comme il/ a 1, 
 la raison de m — M ^ m — 1 sera toujours si ^ peu pres constant, 
 qu'elle satisfera a toutes les experiences, comme la grand Newton a 
 remarque. Cette raison ne difFere non plus de ma theorie que 
 presque imperceptiblement: car puisque je soutiens que ilf=w*, et 
 que m difFere ordinairement fort peu de I'unite, soit m=^\-\-.:; et 
 
 * Mr. Dollond, in his letter to Mr. Euler, here referred to, does not say that he had 
 made any trials himself, but only he had vmderstood that such had been naade by othersj 
 without success. 
 
30 Letters relating to a Theorem^ S^c. 
 
 puisque M=w*=l+a I m ^ peu pres, et I (i+w)=/7W=rw, aussi fort 
 i peu pres, j'aurai 7w~il:f=l+w-rl-acd= (l~*) w, et m-l = 6<j, done 
 
 la raison —11^ — sera=l— a, ou fort a pea pres constante, Del^ ie 
 w— I ^ •* 
 
 conclude que les experiences d'ou le grand Newton a tire son raport, 
 ne sauroient etre contraires k ma theorie. 
 
 En second lieu, je conviens aussi que si la raison —^^^^^^^ — = Const. 
 
 m — I 
 
 fetoit juste k la rigueur, il n'y auroit plus moyen de remedier au 
 defaut qui resulte de la diverse refrangibilite des rayons, de quelque 
 maniere qu'on disposeroit divers milieux transparens, et que I'intervalle 
 entre les divers foyers tiendroit toujours un raport constant a la dis- 
 tance focale entiere du verre. Mais c'est precisement cette consi- 
 deration, qui me fourriit le plus fort argument: I'oeil me paroit une 
 telle machine dioptrique parfaite, qui ne se ressent en aucune maniere 
 de la diverse refrangibilite des rayons: quelque petite que soit sa 
 distance focale, la sensibilite est si grande, que les divers foyers, s'il y 
 en avoit, ne manqueroient pas de troubler tres considerablement la 
 vision. Or il est bien certain, qu'un oeil bien constitue ne sent point 
 I'eiFet de la diverse refrangibilite. 
 
 La structure merveilleux de Toeil, et les diverses humeurs^ dont il 
 est compose, me confirme infiniment dans ce sentiment. Car s'il 
 s'agissoit seulement de produire une representation sur le fond de 
 Toeil, une seulehumeur auroit ete suffisante; et le Createur n'y auroit 
 pas surement employe plusieurs. Dela je conclud, qu'il est possible 
 d' aneantir I'effet de la diverse refrangibilite des rayons par une juste 
 
of Mr. Euler, of the Royal Academy of Berlin, 3 1 
 
 arrangement de plusieurs milieux transparens, done puisqiie cela ne 
 
 seroit pas possible, si la formule— ^^^^ — = Const, etoit vraye a la ri- 
 
 m — 1 
 
 gueur, j'en tire la consequence qu'elle n'est pas parfaitement conforme k 
 
 la nature. 
 
 Mais voila une preuve directe de ma these: je conqois diverse 
 milieuz transparens, A, B, C, D, E, S^c. qui different entr'eux egale- 
 ment par raport a leur densite optique: desorte que la raison de 
 refraction de chacun dans le suivant soit le meme. Soit done dans le 
 passage du premier dans le second la raison de refraction pour les 
 rayons rouges=r: 1, et pour les violets=i;: 1 ; qui sera la meme dans 
 le passage du second dans le troisieme, de celuicy dans le quatrieme, 
 du quatrieme dans le cinquieme, et ainsi de suite. Del^ il est clair, 
 que dans le passage du premier dans le troisieme sera= r^ : 1 pour les 
 rayons rouges, et=i;2: i pour les violets: de meme dans le passage du 
 premier dans le quatrieme les raisons seront r^:l eX v^:\. 
 
 Done si dans le passage dans un milieu quelconque la raison de 
 refraction des rayons rouges est=r'*:l, celle des rayons violets sera 
 =t;":l; tout cela est parfaitement conforme aux principes du grand 
 Newton. Posons r»=/?, ett;«=:^ desorte que i2 ;] , et ^: 1 expri- 
 ment les raisons de refraction des rayons rouges et violets dans un 
 passage quelconque: et ayant nlr=lR etn lv=l Fnousauronsl R: 
 
 lrz=l V-.lvy ou— — =— -. Ou bien mettes t;=r«. et a cause de / v:=. 
 I V I V 
 
 I Tt 1 
 
 « / r, on aura— -=^, oulF=:ccl R, et partant F=zR». 
 
32 Letters relating to a Theorem^ S^c. 
 
 Voila done le fondement du principe, que j'ai employe dans ma 
 piece, qui me paroit encore inebranlable: cependant j'en soumets la 
 decision a I'illustre Societe Royale, et a votre jugement en particulier, 
 ayant I'honneur d'etre avec la plus parfaite consideration, 
 
 MONSIEUR, 
 
 Votre tres humble 
 
 et tr^s obeissant serviteur, 
 
 L, Euler. 
 
 Berlin, 
 .Tuin 15, 1752. 
 
33 
 
 A Description of a Contrivance Jor measuring 
 small Angles, hy Mr, John DoUond; commu- 
 nicated by Mr, J. Short, F.R S. 
 
 Read May 10, 1753. 
 
 JLiET an object-glass of any convenient focal length (being truly 
 ground and well centred) be divided into two equal parts or segments, 
 by cutting it straight through the center; and let a piece of machinery 
 be so contrived, as to hold these two segments in the same position to 
 each other, as they stood in before they were cut asunder; and to be 
 capable at the same time of drawing them to different distances from 
 that position, in the manner as is represented in the figure. 
 
 Each of these segments will form a distinct image of any object 
 to which they are directed; differing in nothing from that, which 
 might have been made by the whole glass before it was cut, except in 
 brightness. And while these segments are held in their original po- 
 sition, the images will coincide, and become one single image as at 
 first; but, in proportion as they are drawn off from that situation, the 
 images will separate more or less, according to the distance they are 
 drawn to. By this means the images of two different objects, or of 
 different parts of the same object, not very far from each other, may 
 
 E 
 
34 A Description of a Contrivance for Measuring small Angles, 
 
 be brought to a contact or coincidence at the focus: and this coinci- 
 dence may be viewed to a very great nicety with a proper eye-glass. 
 
 The measure of the angle subtended by the two objects, whose 
 images are thus brought to a coincidence, depends upon three things: 
 first, a careful observation of the coincidence of the images: — 
 secondly, an exact measure of the distance, which the glasses are drawn 
 but to from that situation, which makes the image single: — and, 
 lastly, a true knowledge of the focal distance of the glass. How the 
 angle is to be . found from these measures,, and how it may likewise 
 be come at, by viewing two land-objects at a convenient distance, 
 will be shewn hereafter in the explanation of the figure. It is easy 
 to understand, in the meantime, that the angle will be measured with 
 more accuracy, in proportion to the length of the glass, which is used 
 for that purpose ; but the difficulty of managing long telescopes is no 
 less apparent. Therefore the most practicable method of using this 
 micrometer to advantage, is, to apply the divided object-glass to the 
 object end of a reflecting telescope: for, as the apertures of these 
 sort of telescopes are large in proportion to their lengths, they will 
 admit of very long glasses; nor will the measures be any way affected 
 by the metals or glasses, which the reflector is composed of: and the 
 angles will be found in the same manner, as though the images were 
 viewed with a single eye-glass, in the manner of a common refracting 
 astronomical telescope; but with this advantage, that, as the images 
 will be exhibited larger and distincter by the reflecting telescope; 
 and as every part thereof will be much more manageable than a long 
 refracting telescope; so the contact or coincidence of the images will 
 be more accurately pbserved. 
 
by Mr. John Dollond, 35 
 
 It would be however unnecessary now, as well as impro- 
 per, to say much about the advantages of this method 
 above those which have hitherto been put in practice; 
 because, as a machine is now' making for this purpose, A H B 
 the experiments, which will shortly be tried, will be more 
 convincing, as well as more intelligible, than any thing 
 that might be offered at present. 
 
 Explanation of the Figure. 
 
 The two semicircles represent the two segments of the 
 object-glass, whose centers C and D are drawn off to the 
 distance C D, and the points A and B are two objects, or 
 different parts of the same object; therefore the lines 
 ACG and BDG represent two rays that pass through 
 the centres or poles of the segments, and are therefore 
 not at all refracted, but go straight through to G, where they 
 intersect; and G being the respective focus to the distance 
 of the objects from the glass, the two images will coincide 
 at that point. It appears from the figure, that ABiCD: : 
 GH'. GE; and from a common proportion in optics, GH: 
 GE::HE: EF. Therefore, AB.CD-.iHEi EF; F being 
 the focus of parallel rays; and consequently the angles 
 AEB and CFD are equal. That is, the angle subtended 
 by the distance of the centres of the segments from the 
 distance of the focus of parallel rays is equal to the angle 
 subtended by the distance between the objects A and B 
 from the end of the telescope. 
 
 E 2 
 
3d 
 
 An Explanation of an Instrument Jhr measuring 
 small Angles, the Jirst Account of which was 
 read before the Royal Society, May 10, 1753. 
 Sy Mr, John DoUond. In a Letter to James 
 Short, M.A. and F.R.S. 
 
 Read April 25, 1754. 
 
 SIR, 
 
 1 HE account which I gave you, some time ago, of 
 a new micrometer, was contained in as few words as possible; being 
 rather desirous, that experiments might be made, before I said much 
 concerning it: — ^but since your many repeated experiments have con- 
 firmed what was expected from it, I have endeavoured to draw up a 
 more full account of this instrument, with demonstrations of the 
 principles which it is fopnded upon, which I here send you enclosed, 
 and which you may lay before the Royal Society, if you think proper. 
 
 I am, SIR, 
 
 Your most obedient, humble servant, 
 
 John DoUond. 
 
 Denmark Court, 
 April 4^ 1754. 
 
37 
 
 Explanation of an Instrument for Measuring Small Angles, S^c. 
 
 JSeFORE I enter upon particulars relating to this micrometer, it 
 will be proper to make a few preparatory observations on the nature of 
 spherical glasses, so far as may be necessary to render the following 
 explanation more easily understood. 
 
 Observation I. — It is a property of all convex spherical glasses to 
 refract the rays of light, which are transmitted through them, in such 
 a manner, as to collect all those that proceed diverging from any one 
 point of a luminous object, to some other point; whose distance from 
 the glass depends chiefly on its convexity, and the distance of the 
 object from it. 
 
 Observation II. — ^The point, where the rays are thus collected, may 
 be considered as the image of that point, from which they diverge. 
 For if we conceive several radiant points thus emitting rays, which, 
 by the refractive quality of the glass, are made to converge to as many 
 other points, it will be an easy matter to understand, how every part 
 of the object will be truly represented. As this property of spherical 
 glasses is explained and demonstrated by all the writers on optics, it 
 being the very foundation of the science^ the bare mention of it is 
 sufficient for the present purpose. 
 
 Observation III. — It will be necessary, however, to observe farther, 
 that the lines connecting every point in the object, with its corres- 
 ponding ones in the image, do all intersect in a certain point Qf the 
 axis or line passing through the poles of the glass, wker^e its two 
 
38 Explanation of an Instrument for Measuring Small Angles, ^c. 
 
 surfaces are parallel, and may be properly called its centre: whence it 
 appears, that the angles subtended by the object and its image from 
 that point, must be equal: and therefore their diameters will be in the 
 same ratio, as their distances from that point. 
 
 Observation IV. — As the formation of the image by the glass depends 
 entirely on the property above-mentioned, that is to say, its collecting 
 all the light, that is incident on it, from the several points of the 
 object into as many other points at its focus ; it follows, that any seg- 
 ment of such a glass will also form an image equal, and every way 
 similaf, to that exhibited by the whole glass; with this difference only, 
 that it will be so much darker, as the area of the segment is less than 
 that of the whole glass. 
 
 Observation V. — ^The axis of a spherical glass in a line connecting 
 the centres of the spheres, to which the two surfaces are ground; and 
 wherever this line passes through the glass, there ^the surfaces are 
 ■parallel. But if it happens, that this line does not go. through the 
 substance of the glass, such a glass is said to have no internal centre; 
 but it is conceived to be in its plane produced, till it meets the axis: 
 and this imaginary point, though external to the glass, is as truly its 
 centre, and is as fixed in its position to it, as if it were actually within 
 its substance. 
 
 Observation VI. — If a spherical glass, having its centre or pole near 
 its middle or centre of its circumference^ should be divided by a 
 straight line through the middle; the centre will be in one of the seg- 
 ments only. For how exact. soever a person may be supposed to be 
 in cutting it through the centre; yet 'tis hard to conceive, how a ma- 
 thematical point should be divided in two: therefore the centre will 
 
by Mr. John Dollond. 39 
 
 be internal to one of the segments, and external to the other. But 
 if a small matter be ground away from the straight edge of each seg- 
 ment, both their centres will become external ; and so they will more 
 easily be brought to a coincidence. 
 
 Observation VII. — If these two segments should be held together, 
 so as to make their centres coincide; the images, which they give of 
 any object, will likewise coincide, and become a single one. This 
 will be the case, when their straight edges are joined to make the glass, 
 as it were, whole again: but let the centres be any-how separated, 
 their images will also separate, and each segment give a separate and 
 distinct image of any object, to which they may be exposed. 
 
 Observation VIII. — ^Though the centres of the segments may be 
 drawn from their coincidence, by removing the segments in any 
 direction whatever; yet the most convenient way for this purpose is, 
 to slide their straight edges oiie along the other, till they 
 are removed, as the figure in the margin represents 
 them: for thus they may be moved without suffering 
 any false light to come in between them. And by this 
 way of removing them, the distance between their cen- 
 tres may be very conveniently measured; viz. by having a Vernier's 
 division, commonly, though falsely, called a Nonnius's, fixed to the 
 brass-work, that holds one segment, so as to slide along a scale on 
 the plate, to which the other part of the glass is fitted. 
 
 Observation IX. — As the images of the same object are separated 
 by the motion of the segments, so those of different objects, or dif- 
 ferent parts of the same object, may be made to coincide. Suppose 
 the sun, moon, or any planet, to be the object; the two images 
 
40 Explanation of an Instrument for Measuring Small Angles , S^c. 
 
 thereof may, by this contrivance, be removed, till their opposite 
 edges are in contact: in vvhich case, the distance between the centres 
 of the two images will be equal to the diameter of either; and so of 
 any other object whatever. 
 
 Observation X. — This divided glass may be used, as a micrometer, 
 three different ways. In the first place, it may be fixed at the end of 
 a tube, of a suitable length to its focal distance, as an object-glass; 
 the other end of the tube having an eye-glass fitted as usual in astro- 
 nomical telescopes. Secondly, it may be applied to the end of a tube 
 much shorter than its focal distance, by having another convex glass 
 within the tube, to shorten the focal distance of that, which is cut 
 in two. Lastly, it may be applied to the open end of a reflecting 
 telescope; either of the Newtonian, Gregorian, or Cassegrain con- 
 struction. And though this last method is much the best, and most 
 convenient, of the three; yet, as the first is the most natural, as well 
 as tlie easiest to be understood, it will be proper to explain it fully, 
 and to demonstrate the principles, on which this micrometer' is con- 
 structed, by supposing it made use of in the first way: — which being 
 done, the application of it to other methods will be readily under- 
 stood. 
 
 , Having thus, by the foregoing observations, given a general idea of 
 the nature and effects of this divided object-glass, I shall proceed to 
 demonstrate the principles, from whence the measures of the angles 
 are to be obtained by this instrument; which will be done by the 
 following propositions. 
 
ly Mr, John Dollond. 
 
 A\ 
 
 PROPOSITION I. 
 
 Suppose a divided object-glass fixed at the end of a tube, j^ JX B 
 
 according to the first method^ and the tube directed to 
 the object intended to be measured; and suppose, like- 
 wise, the segments removed from their original position, 
 in the manner directed under Observation VIII. till the 
 opposite edges of the tivo images are seen in contact at 
 the focus of the eye-glass: then, I say, the angle sub- 
 tended^ by the distance between the centres of the seg- 
 ments, from the focus of the eye-glass, where the edges 
 are seen in contact, is equal to the angle subtended by 
 the diameter of the object from that same point. 
 
 DEMONSTRATION. 
 
 Let the line A B represent the diameter of the object 
 to be measured; and the points CD the centres of the 
 two glass segments: also G the focus where the images 
 of the extremities of the object are coincident. It is 
 evident, from Observation III. that AG and B G are 
 straight lines, that pass through the centres of the seg- 
 ments, and connect the extreme points of the object 
 with their corresponding points in the images; and 
 therefore, as the diameter of the object and the distance 
 between the centres of the segments are both inscribed 
 between these two lines, they must needs subtend the 
 
 F 
 
42 Explanation of an Instrument for Measuring Small Angles, 8^c. 
 
 same angle from the point where those lines meet; which is at G. 
 Q. E. D. 
 
 The focal distance C G, or D G, is variable, according to the dis- 
 tance of the object from the glass: so that it decreases as the distance 
 of the object from the glass increases; and when the object is so far 
 off, that the focal length of the glass bears no proportion to its dis- 
 tance; then will it be least of all, as C F or D F; and the point 
 F is called the focus of parallel rays. Any other focus, as G, being 
 the focus of a near object, is called a respective focus; as it respects a 
 particular distance: but the focus of parallel rays respects all objects 
 that are at a very great distance; such as is that of all the heavenly 
 bodies. 
 
 PROPOSITION II. 
 
 The distance H^ of the object from the glass is to E F, the focal dis- 
 tance of parallel rays, as the distance H G of the object from its 
 image is to E G, the distance of the image from the glass: that is, 
 HE : EF :: HG : EG. 
 
 The ' demonstration of this proposition may be gathered from any 
 treatise of dioptrics; it being a general rule for finding the respective 
 focus to any given distance, when the focus of parallel rays is known. 
 
 PROPOSITION III. 
 
 The angle subtended by the diameter of the object, from the glass, is 
 equal to that subtended, by the opening of the centres of the segments, 
 from the focus of parallel rays. That is, the angle A E B equal to 
 the angle C F D. 
 
by Mr, John Dollond. 43 
 
 DEMONSTRATION. 
 
 It appears, by inspection of the figure, that ABi CD: :HG : EG. 
 
 And by the last proposition HEiEF: :HG:EG. 
 
 Then, as the two last terms of these two analogies are alike; the 
 two first terms of one will be in the same proportion as the two first 
 terms of the other; which gives the following proportion: AB: CD: : 
 HE : EF. Whence the truth of the proposition is evident. 
 
 From this proposition it appears, that the angle subtended by the 
 diameter of the object from the glass, is found without any regard to 
 the distance of the object, or to the distance of the respective focus, 
 where the image is seen; as the measure depends intirely upon the 
 focus of parallel rays and the opening of the segments. We may 
 likewise, from hence, derive a rule for the quantity of the angle, 
 without considering the length of the glass. Let an object, whose 
 diameter is known, be set up at some known distance; the angle it 
 will subtend from the glass may then be found by trigonometry: then 
 let it be measured by this micrometer, and the distance, between the 
 centres of the segments, found on the scale already mentioned, will 
 be the constant measure of the same angle, in all other cases: because 
 the distance of the object makes no alteration in the measure of the 
 angle, as has been demonstrated: and thus having obtained the dis- 
 tance between the centres of the segments, which answers to any one 
 angle, all other distances may be computed by the rule of three. 
 
 All that has been hitherto said relates to the first method of using 
 this micrometer; that is, by fitting it to the end of a tube suited to 
 its focal length, and by viewing the images with a proper eye-glass, in 
 the manner of an astronomical telescope. But the length of the 
 
 F 2 
 
44 Explanation of an Instrument Jbr Measuring Small Angles, ^c. 
 
 tube, in this way, would be very troublesome; and therefore it will be 
 proper to consider other methods, foran easier management. I shall, 
 therefore, proceed to the second method, mentioned in Observation X. 
 which is, by using another object glass to shorten the focus of that 
 which serves for the micrometer. To facilitate the understanding of 
 this method; it will be necessary to premise the following observation. 
 
 Observation XI. — Rays of light, which are brought to such con- 
 vergency as to form the image of an object, proceed, after that, 
 diverging, in the manner they did when they issued from the object 
 before they were transmitted through the glass; and therefore they 
 may be again collected by another spherical glass, so as to form a 
 second representation of the same object; which may again be repeated 
 by a third glass, 8^c. So that the first image may be considered as an 
 object to the second glass, and the second image will be an object to 
 the third, and so on. Though these images may be very different, 
 in respect to their magnitudes, yet they will be all similar; being true 
 representations of the same object: this will hold good, though the 
 second glass should be put so near the first as to receive the rays be- 
 fore the image is formed: for as the rays are tending to meet at a 
 certain distance, the second will receive them in that degree of con- 
 vergency, and, by an additional refraction, bring them to a nearer 
 focus; but the image will still be similar to that which would have been 
 made by the first glass, if the second had not been there. 
 
 Upon this principle all refracting telescopes are made; some of 
 which are a combination of four, five, or six glasses. The first glass 
 forms an image of the object; the second repeats the image, which 
 it receives from the first; and so on, till the last glass brings a true 
 
iy Mr, John I>ollond. ' 45 
 
 representation of the object to the eye. The same may be said of re- 
 flecting telescopes: for a spherical mirror acts in the same manner, in 
 that respect, as a spherical glass. 
 
 Now let this be applied to the subject in hand. Suppose the focal 
 distance of the divided object-glass to be about forty feet; and suppose 
 the segments to be opened wide enough to bring the opposite edges 
 of an object in contact: then let. another object-glass, uncut, be fixed 
 within the tube, of a proper degree of convexity, to shorten the focus 
 of the other as much as may be required; suppose to twelve feet: by 
 what has been just now observed, this glass will represent the two 
 images in the same form which would have been exhibited by the 
 divided glass, if this other glass had not been there. For though the 
 images are not yet formed, when the second glass receives the rays: 
 yet, as those rays are converging towards it, the second glass must 
 represent those images in the same position, and form, as the ten- 
 dency of the rays requires. For while the segments are fixed in their 
 position to each other, their images will also be fixed in their position; 
 and let them be repeated ever so many times, by refraction through 
 spherical glasses, or by reflection from spherical mirrors, they can 
 suffer no alteration in their position to one another. By this means, 
 the telescope may be shortened, at pleasure, though the scale for the 
 measure of the angles will remain the same. The only inconvenience, 
 which the shortness of the telescope introduces, is a want of sufficient 
 distinctness; which will so far hinder the exactness of the observation, 
 as the contact of the edges cannot be so accurately determined, as 
 they might be with longer telescopes. 
 
 This difficulty is intirely removed by fixing the divided glass at the 
 
4.Q Explanation of an Instrument for Measuring Small Angles , 8^c. 
 
 end of a reflecting telescope: for the reflections and refractions, which 
 the rays nmust undergo in passing through the telescope, will no way 
 alter the position of the images, which the rays, that have passed 
 through the segments, are tending to: for, as has been already ob- 
 served, a number of reflections and refractions may repeat the images, 
 and alter their magnitudes; but can make no alteration in their 
 proportions. 
 
 Therefore this way of fixing the divided glass to a reflecting teles- 
 cope, which was the third method proposed, is, by far, the best; as 
 such telescopes of moderate and manageable lengths, when well made, 
 are capable of magnifying considerably, and shewing objects to great 
 advantage. This micrometer being applicable to the reflecting teles- 
 cope, with so much certainty, is no inconsiderable advantage: for 
 any one will easily understand, that, to measure the diameter of a 
 planet exactly, . it is necessary, that the planet be magnified, and 
 shown distinctly, which could not be obtained, in the common way, 
 without very great lengths; such as rendered it very diflicult, , not to 
 say impracticable, to take exact measures. Besides, the common mi- 
 crometer is limited, in this respect, upon another account; viz. because 
 the diameter of the planet cannot be measured, without having the 
 whole planet within the field of the telescope, which confines the mag- 
 nifying power within very narrow bounds; whereas, by this method, 
 nothing more is required, than to see the contact of the edges, which 
 allows the magnifying power to be increased at pleasure. 
 
 In the common micrometer, the object is to be taken between two 
 wires, so that the contact of its edges with those wires cannot be 
 observed at one view; and the least motion of the telescope, whilst 
 
hy Mr, John Dollond, Af 
 
 the observer is turning his eye from one wire to the other, must oblige 
 him to repeat the observation; whereas, by this method, the contact 
 of the edges of the images is not at all affected by the motion of the 
 telescope. Whence the comparison of this micrometer with the 
 common sort, in this respect, stands thus: the one requires great 
 steadiness in the telescope, but yet it is applicable to none, but such 
 as are very difficult to keep steady; the other does not require such 
 steadiness, though it is applicable to short telescopes, which are easily 
 managed, v . 
 
 These advantages not only add to the certainty of the observation, 
 but assist vastly in the expedition ; for an observer may make twenty > 
 observations, in this way, where he could scarcely, with much 'fatigue, 
 be sure of one with the common micrometer. Expedition in making 
 observations, must be allowed a very great advantage, in this climate, 
 where the uncertainty of the weather renders astronomical observations 
 so precarious, that no opportunities, even the most transient, should 
 be let slip. An instance of this was given to the Royal Society, in an 
 account of the eclipse of the sun last October. 
 
 As the motion of the telescope gives the observer no great incon- 
 venience, in this method; neither does the motion of the object at all 
 disturb his observation (I mean such a motion, as that of the heavens 
 is.) This gives him leave to take the diameter of a planet, in any 
 direction; or the distance between two stars or planets, let their 
 situation be how it will ; in which respect the common micrometer is 
 absolutely defective; as it can give no angles, but such as are per- 
 pendicular to the line of their motion ; though the diameters of the 
 planets, in other directions, are very much wanted; it being highly 
 
48 Explanation of an Instrument for Measuring Small Angles, S^c. 
 
 probable, from the laws of motion, and what we see in Jupiter, that 
 such planets, as revolve round their axes, have their polar diameters 
 shorter than their equatorial ones. 
 
 The distances of Jupiter's satellites from one another, or from Ju- 
 piter's body, cannot be measured, with any certainty, in the common 
 way, as their position is always very far from being at right angles 
 with the line of their motion: neither can the moon's diameter, which 
 must be taken from horn to horn, scarce ever be obtained that way, 
 because it very rarely happens, that the diameter, to be measured, lies 
 at right angles to the line of her motion. The same may be said of 
 the distance between two stars. But this micrometer gives angles, in 
 every direction, with equal ease and certainty; the observation being 
 also finished in an instant, without any trouble or fatigue to the ob- 
 server. For as there are no wires made use of, this way, in the field 
 of the telescope, so the observer has no concern about any illumination. 
 The largeness of the scale deserves also to be taken notice of, as it 
 may, in this micrometer, be increased almost at pleasure, according 
 as the smallness of the object requires. Another inconvenience 
 attending the common micrometer is, the variation of the scale, ac- 
 cording to the distance of the object. As the telescope must be 
 lengthened, or drawn out farther, for short distances; the scale, which 
 depends upon that length, is thereby increased; which renders the 
 measure of the angle very undertain: whereas, in this micrometer, the 
 scale is the same at all distances; so that the angle may be measured 
 with the utmost certainty, without any regard to the distance of the 
 object. 
 
hy Mr. John Dollond. 49 
 
 , Upon the whole, it may be concluded, that this micrometer is a 
 complete instrument in its kind; having many advantages above the 
 common sort, without any of their disadvantages: and there is no 
 doubt, but, when brought into practice, it will tend much to >the 
 advancement of astronomy. 
 
 G 
 
50 
 
 An Account of some Experiments concerning the 
 different Refrangibility of Lighit. By Mr. John 
 DoUond. With a Letter Jrom James Short, 
 M.A, F.R.S. Acad. Reg. Suec. Soc. 
 
 To the Rev. Dr. Birch, Secret. R. S. 
 Read June 8, 1758. 
 
 DEAR SIR, 
 
 X HAVE received the enclosed paper from Mr. 
 Dollond, which he desires may be laid before the Royal Society. It 
 contains the theory of correcting the errors arising from the different 
 refrangibility of the rays of light in the object-glasses of refracting 
 telescopes; and I have found, upon examination, that telescopes made 
 according to this theory are intirely free from colours, and are as dis- 
 tinct as reflecting telescopes. 
 
 I am, DEAR SIR, 
 
 Your most obedient humble servant, 
 
 James Short. 
 
 Surrey Street, 
 
 June 8, 1758. ' ' 
 
51 
 
 Experiments concerning the different Refrangihility of Light, 8^c. 
 
 XT is well known, that a ray of light, refracted by passing through 
 mediums of different densities, is at the same time proportionally 
 divided or spread into a number of parts, commonly called homogeneal 
 rays, each of a different colour; and that these, after refraction, pro- 
 ceed diverging; a proof, that they are differently refracted, and that 
 light consists of parts that differ in degrees of refrangibility. 
 
 Every ray of light passing from a rarer into a denser medium, is 
 refracted towards the perpendicular; but from a denser into a rarer 
 one, from the perpendicular; and the sines of the angles of incidence 
 and refraction are in a given ratio. But light consisting of parts, which 
 are differently refrangible, each part of an original or compound ray 
 has a ratio peculiar to itself; and therefore the more a heterogeneous 
 ray is refracted, the more will the colours diverge, since the ratios of 
 the sines of the homogeneal rays, are constant; and equal refractions 
 produce equal divergencies. 
 
 That this is the case when light is refracted 'by one given medium 
 only, as suppose any particular sort of glass, is out of all dispute, being 
 indeed self-evident; but that the divergency of the colours will be the 
 same under equal refractions, whatsoever mediums the light may be 
 refracted by, though generally supposed, does not appear quite so 
 clearly. 
 
 However, as no medium is known, which will refract light without 
 diverging the colours, and as difference of refrangibility seems thence 
 
 G 2 
 
52 Experiments concerning the different RefrangiUlity of Light, 
 
 to be a property inherent in light itself, opticians have, upon that 
 consideration, concluded, that equal refractions must produce equal 
 divergencies in every sort of medium: whence it should also follow, 
 that equal and contrary refractions must not only destroy each other, 
 but that the divergency of the colours from one refraction would 
 likewise be corrected by the other; and there could be no possibility 
 of producing any such thing as refraction, which would not be affected 
 by the different refrangibility of light; or, in other words, that how- 
 ever a ray of light might be refracted backwards and forwards by 
 different mediums, as water, glass, 8^c. provided it was so done, that 
 the emergent ray should be parallel to the incident one, it would ever 
 aft6r be white; and conversely, if it should come out inclined to the 
 incident, it would diverge, and ever after be coloured. From whence 
 it was natural to infer, that all spherical object-glasses of telescopes 
 must be equally affected by the different refrangibility of light, in 
 proportion to their apertures, whatever material they may be formed 
 of. 
 
 But it seems worthy of consideration, that notwithstanding this 
 notion has been generally adopted as an incontestable truth, yet it 
 does not seem to have been hitherto so confirmed by evident experi- 
 ments, as the nature of so important a matter justly demands; and 
 this it was that determined me to attempt putting the thing to issue 
 by the following experiment. 
 
 I cemented together two plates of parallel glass at their edges, so 
 as to form a prismatic or wedge-like vessel, when stopped at the ends 
 or bases ; and its edge being turned downwards, I placed therein a 
 glass prism with one of its edges upwards, and filled up the vacancy 
 
hy Mr. John Bollond, 53 
 
 with clear water: thus the refraction of the prism was contrived to be 
 contrary to that of the water, so that a ray of light transmitted 
 through both these refracting mediums would be refracted by the 
 difference only between the two refractions. Wherefore, as I found 
 the water to refract more or less than the glass prism, I diminished 
 or increased the angle between the glass plates, till I found the two 
 contrary refractions to be equal ; which I discovered by viewing an 
 object through this double prism; which, when it appeared neither 
 raised nor depressed, I was satisfied, that the refractions were equal, 
 and that the emergent rays were parallel to the incident. 
 
 Now, according to the prevailing opinion, the object should have 
 appeared through this double prism quite of its natural colour ; for if 
 the difference of refrangibility had been equal in the two equal re^ 
 fractions, they would have rectified each other : but the experiment 
 fully proved the fallacy of this received opinion, by showing the di- 
 vergency of the light by the prism to be almost double of that by the 
 water; for the object, though not at all refracted, was yet as much 
 infected with prismatic colours, as if it had been seen through a glass 
 wedge only, whose refracting angle was near 30 degrees. 
 
 N,B. This experiment will be readily perceived to be the same as 
 that which Sir Isaac Newton mentions* : but how it comes to 
 differ so very remarkably in the result, I shall not take upon me 
 to account for J but will only add, that I used all possible pre- 
 
 * Book I. Part ii. Prop. 3. Experiment viii. of his*"Optics. 
 
54 Experiments concerning the different Refrangihility of Light, 
 
 ■ caution and care in the process, and that I keep the apparatus by 
 nie to evince the truth of what I write, whenever I may be pro- 
 perly required so to do. 
 
 I plainly saw then, that if the refracting angle of the water-vessel 
 : could have admitted of a sufficient increase, the divergency of the 
 coloured rays would have been greatly diminished, or entirely recti- 
 fied ; and there would have been a very great refraction without colour, 
 as now I had a great discolouring without refraction : but the incon- 
 veniency of so large an angle, as that of the vessel must have been, 
 to bring the light to an equal divergency yvith that of the glass prism, 
 whose angle was about 6o degrees, made it necessary to try some ex- 
 periments of the same kind, by smaller angles. 
 
 I ground a wedge of common plate glass to an angle of somewhat 
 less than 9 degrees, which refracted the mean rays about 5 degrees. 
 I then made a wedge-like vessel, as in the former experiment, and 
 filling it with water, managed it so, that it refracted equally with the 
 glass wedge ; or, in other words, the difference of their refractjons 
 was nothing, and objects viewed through them appeared neither 
 raised nor depressed. This was done with an intent to observe the 
 same thing over again in these small angles, which I had seen in the 
 prism : and it appeared indeed the same in proportion, or as near as 
 I could judge; for notwithstanding the refractions were here also 
 equal, yet the divergency of the colours by the glass was vastly 
 greater than that by the water ; for objects seen by these two refrac- 
 tions were very much discoloured. Now this was a demonstration, 
 that the divergency of the light, by the different refrangibility, w^s 
 
hy Mr. John Dollond, ' 55 
 
 far from being equal in these two refractions. I also saw, from the 
 position of the colours, that the excess of divergency was in the 
 glass ; so that I increased the angle of the water-wedge, by different 
 trials, till the divergency of the light by the water was equal to that 
 by the glass ; that is, till the object, though considerably refracted^ 
 by the excess of the refraction of the. water, appeared nevertheless 
 quite free from any colours proceeding from the different refrangibility 
 of light ; and, as near as I could then measure, the refraction by the 
 water was about -f of that by the glass. Indeed I was not very exact 
 in taking the measures, because my business was not at that time about 
 the proportions, so much as to show, that the divergency of the 
 colours, by different substances, was by no means in proportion to the 
 refractions ; and that there was a possibility of refraction without any 
 divergency of the light at all. 
 
 Having, about the beginning of the year 1757, tried these experi- 
 ments, I soon after set about grinding telescopic object-glasses upon 
 the new principles of refractions^ which I had gathered from them ; 
 which object-glasses were compounded of two spherical glasses with 
 water between them. These glasses I had the satisfaction to find, as 
 I had expected, free from the errors arising from the different refran- 
 gibility of light: for the refractions, by which the rays were brought 
 to a focus, were everywhere the differences between two contrary re- 
 fractions, in the same manner, and in the same proportions, as in the 
 experiment with the wedges. 
 
 However, the images formed at the foci of these object-glasses were 
 still very far from being so distinct as might have been expected from 
 the removal of wso great a disturbance ; and yet it- was not very diffi- 
 
56 Experiments concerning the different Refrangihility of Light, 
 
 cult to guess at the reason, when I considered, that the radii of the 
 spherical surfaces of those glasses were required to be so short, in 
 order to make the refractions in the required proportions, that they 
 must produce aberrations, or errors, in the image, as great, or 
 greater, than those from the different refrangibility of light. And 
 therefore, seeing no method of getting over that difficulty, I gave 
 up all hopes of succeeding in that way. 
 
 And yet, as these experiments clearly proved, that different sub- 
 stances diverged the light very differently, in proportion to the refrac- 
 tion ; I began to suspect, that such a variety might possibly be found in 
 different sorts of glass, especially as experience had already shown, 
 that some made much better object-glasses, in the usual way, than 
 others : and as no satisfactory cause had as yet been assigned for such 
 difference, there was great reason to presume, that it might be owing 
 to the different divergency of the light by their refractions. 
 
 Wherefore, the next business to be undertaken, was to grind 
 wedges of different kinds of glass, and apply them together, so that 
 the refractions might be made in contrary directions, in order to dis- 
 cover, as in the foregoing experiments, whether the refraction and di- 
 vergency of the colours would vanish together. But a considerable 
 time elapsed before I could set about that work; for though I was de- 
 termined to try it at my leisure, for satisfying my own curiosity, yet I 
 did not expect to meet with a difference sufficient to give room for 
 any great improvement of telescopes ; so that it was not till the latter 
 end of the year that I undertook it, when my first trials convinced 
 me, that this business really deserved my utmost attention and appli- 
 cation. 
 
by Mr. John Dollond. 57 
 
 I discovered a difference, far beyond my hopes, in the refractive 
 qualities of different kinds of glass, with respect to their divergency 
 of colours. The yellow or straw-coloured foreign sort, commonly 
 called Venice glass, and the English crown glass, are very nearly alike 
 in that respect, - though in general the crown glass seems to diverge 
 the light rather the least of the two. The common plate glass made 
 in England diverges more ; and the white crystal or flint English glass, 
 asi it is called, most of all. 
 
 It was not now my business to examine into the particular qualities 
 of every kind of glass that I could come at, much less to amuse my- 
 self with conjectures about the cause, but to fix upon such two sorts 
 whose difference was the greatest; which I soon found to be the 
 crown, and the white fiint or crystal. I therefore ground a wedge of 
 white flint of about 25 degrees, iand another of crown of about 29 de- 
 grees, which refracted nearly alike ; but their divergency of the co- 
 lours was very different. I then ground several others of crown to 
 different angles, till I got one, which was equal, with respect to the 
 divergency oi the light, to that in the white flint : for when they 
 were put together, so as to refract in contrary directions, the re- 
 fracted light was intirely free from colour. Then measuring the re- 
 fractions of each wedge, I found that of the white glass to be to that 
 of the crown nearly as 2 to 3 ; and this proportion would hold very 
 nearly in all small angles. Wherefore any two wedges made in this 
 proportion, and applied together, so as to refract in a contrary di- 
 rection, would refract the light without any difference of refrangi- 
 bility. ■ ^ 
 
 H 
 
58. Experiments concerning the different Refrangihility of Light, 
 
 To make therefore two spherical glasses, that shall refract the 
 light in contrary directions, it is easy to understand, that one must 
 be concave, and the other convex ; and as the rays are to converge 
 to a real focus, the excess of refraction must evidently be in the con- 
 vex ; and as the convex is to refract most, it appears from the experi- 
 ment, that it must be made with crown glass, and the concave with 
 white flint or]ass. 
 
 And further, as the refractions of spherical glasses are in an 
 inverse ratio of their focal distances ; it follows, that the focal dis- 
 tances of the two glasses should be inversely as the ratios of the 
 refractions of the wedges : for being thus proportioned, every ray of 
 light that passes through this combined glass, at whatever distance 
 it may pass from its axis, will constantly be refracted, by the dif- 
 ference between two contrary refractions, in the proportion required ; 
 , and therefore the different refrangihility of the light will be intirely 
 removed. 
 
 Having thus got rid of the principal cause of the imperfection 
 of refracting telescopes, there seemed to be nothing more to do, 
 but to go to work upon this principle : but I had not made many 
 attempts, before I found, that the removal of one impediment had 
 introduced another equally detrimental (the same as I had before 
 found in two glasses with water between them) : for the two glasses, 
 that were to be combined together, were the segments of very 
 deep spheres ; and therefore the aberrations from the spherical sur- 
 faces became very considerable, and greatly disturbed the distinctness, 
 of the image. Though this appeared at first a very great difficulty, 
 yet I was not long without hopes of a remedy : for considering 
 
by Mr, John Dollond, 59 
 
 the surfaces of spherical glasses admit of great variations, though 
 the focal distance be limited, and that by these variations their 
 aberrations may be made more or less, almost at pleasure, I plainly 
 saw the possibility of making the aberrations of any two glasses 
 equal ; and as in this case the refractions of the two glasses were 
 contrary to each other, their aberrations, being equal, would intirely 
 vanish. 
 
 And thus, at last, I obtained a perfect theory for making object- 
 glasses, to the apertures of which I could scarcely conceive any limits : 
 for if the practice could come up to the theory, they must certainly 
 admit of very extensive ones, and of course bear very great magni- 
 fying powers. 
 
 But the difficulties attending the practice are very considerable. 
 In the first place, the focal distances, as well as the particular surfaces, 
 must be very nicely proportioned to the densities or tefracting powers 
 of the glasses ; which are very apt to vary in the same sort of glass 
 made at different times. Secondly, the centres of the two glasses 
 must be placed truly on the common axis of the telescope, other- 
 wise the desired effect will be in a great measure destroyed. Add to 
 these, that there are four surfaces to be wrought perfectly spherical ; 
 and any person, but moderately practised in optical operations, 
 will allow, that there must be the greatest accuracy throughout the 
 whole work. 
 
 Notwithstanding so many difficulties, as I have enumerated, I have, 
 after numerous trials, and a resolute perseverance, brought the matter 
 at last to such an issue, that I can construct refracting telescopes, with 
 
 H 2 
 
6o Experiments concerning the different Refrangihility, 8§c. 
 
 such apertures and magnifying powers, under limited lengths, as, in 
 the opinion of the best and undeniable judges, who have expe- 
 rienced them, far exceed any thing that has been hitherto produced, 
 as representing objects with great distinctness, and in their true 
 colours. 
 
 John Dollond. 
 
^^P' 
 
 ■% 
 
 
m 
 
 * <.# 
 
 %» 
 
 /^ //'/rnv/;/ //V ./ '/'//r'///AW// /l-n/// /t/i nr/z/i/i 11/ J'lr/'/i/i/iif fir J.//,'/i/u/ir /i.. I. 
 
6i 
 
 Some Account of the JDiscovay, made hy the late 
 3Ir, John Dollond, F.R S. which led to the 
 grmul Improvem^ent of Refracting Telescopes, 
 in Order to correct some Misrepresentations, in 
 Foreign Publications, of that Discovery : with 
 an Attenvpt to account for the Mistake in an 
 Experiment made hy Sir Isaac Newton; on 
 which Exp)erhnent, the Improvejnent of the Re- 
 fracting Telescope intirely dejyended. By Peter 
 Dollond, Member of the American Philosophi- 
 cal Society at Philadelphia. 
 
 ADVERTISEMENT, 
 
 IVlY intention in writing the following paper was, to correct several 
 false representations, relating to the invention of the achromatic 
 refracting telescope, and to secure to my late father, Mr. John 
 Dollond, as well as to this country, the honour of so valuable a dis- 
 covery. 
 
62 Advertisement hy Mr. Peter Dollond, 
 
 With this view, the paper was presented to the Royal Society, by 
 the Rev. Dr. Maskelyne, Astronomer Royal, in expectation of its 
 being published in the Philosophical Transactions, It was read at a 
 meeting of the Society on the 21st of May, 1789; but afterwards, 
 contrary to my expectation, it was resolved, in a council of the So- 
 ciety, that the paper should not be printed in their Transactions; I 
 therefore take this method of submitting it to the public; as I hum- 
 bly conceive, it relates to a subject of a sufficient degree of importance 
 to claim their attention. 
 
 Peter DoUond, 
 
 St, Paul's Church-yard, 
 Sept. 1, 1789. 
 
63 
 
 Some Account of the Improvement in Rejracting Telescopes, $s^. 
 
 The correction of any inaccuracies or false representations in the 
 history of science is certainly of some consequence to the public, and 
 deserves the attention of the Royal Society ; particularly so, when 
 such false representation tends to deprive any one of that praise, to 
 which he may be justly entitled, by having contributed tdwards the 
 advancement of science ; even though it may be in things of little 
 moment. Then certainly it must be much more so, when it relates 
 to matters of great importance ; such as was the discovery which 
 brought forward the grand improvement of the refracting telescope. 
 
 I was led to these reflections, by having seen some accounts of that 
 discovery in different publications, which were related in a manner 
 that lessened the merit of my late father John Dollond, and gave it 
 to others, who never thought themselves in any manner entitled to 
 claim with him, or ever appeared to be inclined so to do. Their own 
 characters ^yere too exalted in science to need any additional merit of 
 any discovery, to which they had not an undoubted right. 
 
 The celebrated M. Euler, of Berlin, and M. Klinginstierna, pro- 
 fessor of mathematics at Upsal, in Sweden, are the persons alluded 
 to. These gentlemen have been mentioned by different foreign 
 authors, who have had occasion to give accounts of the improve- 
 ment of the refracting telescope, as being the discoverers of the 
 PRINCIPLE on which that improvement was founded ; and nothing 
 has been left to Dollond, but the credit of being the first who put the . 
 
64 Some Account of the Discovery made by Mr. John Dollond, 
 
 same into practice ; whereby he has been deprived of the honour 
 which is justly due to his memory, for having made so useful a dis- 
 covery. 
 
 In order to set this matter in a proper light, I shall mention so 
 much from Sir Isaac Newton's Optics, as is necessary for the purpose ; 
 and then endeavour to prove, that what was attempted by Euler and 
 Klinginstiema was not done from any knowledge of the principle on 
 which the improvement was founded ; but that Dollond was actually 
 the discoverer of that principle, as well as the person who first put the 
 same in practice- 
 When Newton had made his great discovery of the different re- 
 frangibility of light, he fully explained that to be the cause of the 
 imperfection of refracting telescopes, and that it was not occasioned 
 by the spherical figures of the glasses, as has been the generally 
 received opinion. But as mathematicians had made many attempts to 
 correct the errors arising from spherical figures, by giving the glasses 
 figures from the conic sections, he took that opportunity of mention- 
 ing an ingenious thought of his own, of composing the object-glasses 
 of two glasses with water between them ; by which means he says, 
 that the spherical figures of the glasses might have been corrected, 
 and telescopes brought to a sufficient perfection, had it not been for 
 the different refrangibility of the several sorts of rays. — Neivtoris Op- 
 tics, 3d. Edition, p. QO. 
 
 Newton having completed the principal experiments relating to the 
 different refrangibility of light, and having determined the proportions 
 of the sines of incidence to the sines of refractions in the different 
 coloured rays, as given by his glass prisms, proceeds to try the eighth 
 
relating to Refracting Telescopes. 65 
 
 experiment of the second part of the first book of his Optics, to dis- 
 cover their proportions in different refracting mediums. This expe- 
 riment he tried, by placing a prism of glass in a prismatic vessel of 
 water. Refracting the light through these different mediums, he 
 found that light, as often as by contrary refractions it is so corrected 
 that it emergeth in lines parallel to those in which it was incident, 
 continues ever afterto be white* but if the emergent rays be inclined 
 to the incident, the light will become coloured. — Newton s Optics^ 
 p. 112. 
 
 The conclusion drawn from this experiment was, that the di- 
 vergency of the different coloured rays was constantly in a given 
 proportion to the mean refraction in all sorts of refracting mediums. 
 This was the principle established by the Newtonian experiment, and 
 was doubted by no one, until the beginning of the year 1757; when 
 Dollond tried the same experiment as above related, and found the 
 result to be very different; for the light after being refracted in con- 
 trary directions through the glass and water prisms, if the emergent 
 rays were parallel to the incident rays, they were found to be con- 
 siderably coloured ; from whence it followed, that the dissipation of 
 the different coloured rays was not in the same proportion to the mean 
 refraction in water as in glass. And further experiments proved, that 
 there was also a very considerable difference of the same nature to be 
 found in different kinds of glass. — See this appendix, p. 50. 
 
 This was the new principle, which brought forward the improve- 
 ment of refracting telescopes; a principle so contrary to the 
 generally received opinion, that Euler had much difficulty to prevail 
 on himself to believe what was told him by his friends on that subject; 
 
66 Some Account of the Discovery made hy Mr. John Dollond, 
 
 as appears by his own papers published in the Memoirs of the Royal 
 Academy at Berlin. For he first supposes the goodness of Dollond's 
 telescopes to be owing to the greenish colour of the crown-glass, 
 which did not transmit all the red rays; he afterwards endeavours to 
 account for it from the construction of the eye-glasses; and at last 
 declares it to be very extraordinary, that the English optician should 
 have made such an improvement, by reasoning, as it were, in a man- 
 ner quite contrary to the nature of things ; for so indeed the new 
 discovered principle appeared to him. Notwithstanding these decla- 
 rations of Euler, which were published in the year 17^^ M. De la 
 Lande, in the second volume of his Astronomy, p. 837, published 
 in the year 17^4, says, that Euler, in 1747, endeavoured to correct 
 the different refrangibility of light, by a method which Newton 
 pointed out for correcting the errors of the spherical figures of the 
 glasses; which was, by two lenses with water between them, as re- 
 cited above: and he says, that DoUond tried to confute Euler, who 
 had demonstrated an error in Newton's theory of colours; but the 
 dispute having given occasion to Dollond to examine the thing more 
 narrowly, he afterwards acknowledged the error of Newton, and in 
 the year 1759 he found out a method of making achromatic telescopes 
 that succeeded very well. 
 
 Now this account of De la Lande's is by no means the true state 
 of the facts, as appears by the Letters which passed between Euler 
 and Dollond, see the former part of this Appendix, pp. 21 — 32 j for 
 though Euler argues from his hypothesis, that the result of Newton! s 
 experiment could not be exactly as he relates it, yet he does not pre- 
 tend to controvert any of Newton's laws of refraction, as being con- 
 
^ relating to Refracting Telescopes. Qy 
 
 trary to experiment, but believed, that the divergency of the different 
 coloured rays differed scarce sensibly from bearing a given proportion 
 to the mean refraction, in all sorts of refracting mediums; by which 
 it appears, that the error afterwards discovered by Dollond was not 
 even suspected by Euler; therefore that part of De la Lande's account 
 cannot be true; for Dollond could not be said to acknowledge an error, 
 supposed to be discovered by Euler in Newton's theory of coteurs, 
 by having actually discovered one himself of a different nature. The 
 true state of the fact is, that in 1747 Euler endeavoured to correct 
 the errors arising from the different refrangibility of light in object- 
 glasses, by a method which was not founded on any experiment, 
 but on an hypothesis, which did not appear to be on a true principle, 
 so that the attempts which were made to put this method in practice 
 did not succeed : this was certainly the case; for after M. Clairaut 
 had examined the controversy between Euler and Dollond, he pro- 
 nounced, that what Euler had done appeared to be more ingenious 
 than useful. 
 
 Euler indeed says, that the structure of the eye gave him the 
 greatest reason to suppose, that the different refrangibility might 
 be corrected by several refractions through different kinds of mediums; 
 for which purpose he thought the eye to be so constructed. But 
 this reasoning had no weight with Dollond, as he perceived and 
 mentioned to his friends, that the refractions of the eye, at the several 
 surfaces and humours, are all made the same way, and consequently, 
 for want of contrary refractions, the colours produced by the first 
 refractions could not be taken away. How this can subsist with the 
 perfection of our vision, has been ingeniously explained by the 
 
 I 2 
 
68 Some Account of the Discovery made hy Mr. John Dollond, 
 
 Astronomer Royal, in an account which he proposes to lay before the 
 Society. — See p. 78 of this Appendix. 
 
 Klinginstierna has likewise been considered as a party concerned 
 in the improvement of the refracting telescope ; though De la Lande 
 does not mention his name, yet some others do. This has been oc- 
 casioned by his having, in the year 1755, considered the controversy 
 between Euler and Dollond, and having formed a theorem of his own, 
 by which he was also induced to believe, that the result of Newton*s 
 experiment could not be as he had related it ; except when the angles 
 of the refracting mediums were small. This he communicated to 
 Dollond, in a letter to his friend Mr. Mallet, who was then in 
 England. As this theorem has never been published in English, I shall 
 give a copy of it here, as taken by my father from Klinginstierna's 
 letter to Mallet, that mathematicians may judge of the truth of the 
 deductions. 
 
 " Remarks on the Law of Refraction of Rays of Light of different 
 Kinds, through different Mediums. See Newton's Optics., Book I. Part 
 ii. Prop. 3. Eocper. viii. 
 
relating to Refracting Telescopes. 6g 
 
 " Upon any right line, as TH, let there be drawn two arches TIH, 
 TGH, and let a right line TIG be drawn intersecting the arches 
 in I and G. Join IH and GH ; let F E K be a transparent wedge, 
 having its acute angle FEK equal to the angle IHG; let the two 
 faces of this wedge be contiguous to two different transparent me- 
 diums ; and let the ratio of refraction out of the medium, that 
 joins the surface EF into the wedge, be as the ratio of TH to TI, 
 and the ratio of refraction out of the wedge into the medium join- 
 ing the surface EK as the ratio of TG to TH. 
 
 " Now if AB represents a ray of light entering into the wedge, 
 and the angle AB a is made equal to HIG, then will the angle CB 
 a be equal to the angle THI. Ang. BC Z'=ang. THG and DC />== 
 HGL; so that the incident ray AB will be parallel to the emergent 
 ray CD, which has been twice refracted. 
 
 " Now if the incident ray is compounded of divers simple rays, 
 each of which, after two refractions, should emerge parallel to the 
 common incident ray, the refractions of each will be represented by 
 so many right lines Tig- joining Hf, H^, the same as before. 
 
 " According to Newton's law of refraction quoted above TH— TI 
 should be in a constant ratio to TH— TG; that is, if an arch of a 
 circle is described on the centre T with the interval T H, meeting 
 the lines TIG, Tig in L and /; then by that supposition LI should 
 be to LG as li to Ig. But these proportions will not hold, unless L 
 and / were in an arch described on the chord TH, but they are in an 
 arch whose centre is T. 
 
 " Therefore Newton's law of refraction does not seem to follow 
 
70 Some Account of the Discovery made bij Mr. John Dollond, 
 
 clearly from his 8th experiment, which our wedge with two conti- 
 guous mediums refers to. 
 
 " If we should suppose such a law of refraction as we find necessary 
 to bring out every simple ray parallel to the incident, after two re- 
 fractions through this wedge FEK, it can be demonstrated, that the 
 same law will not have the same effect in another wedge of a different 
 angle, but for every different angle there will be a different law re- 
 quired. 
 
 " Whence it seems to follow, that there must be some mistake in 
 this experiment of Newton's, which he himself gives as an universal 
 one, for it does not seem likely that the law which really obtains in 
 nature should depend upon a greater or less angle of a wedge. 
 
 " Nevertheless it must be observed, that the less the refractions are, 
 the nigher will the Newtonian law be to that which is required for 
 producing a perfect parallellism of the emergent rays to the common 
 incident ray ; for in this case Llto LG will be very nearly in a given 
 ratio. It does not seem that the aberration of the rays in object- 
 glasses, proceeding from the different refrangibility, can be corrected 
 by any refractions, which is what Mr. Newton plainly insists upon. 
 However, this whole affair deserves to be more accurately examined 
 by experiments." 
 
 I shall here only remark, that it appears by this copy of a, letter 
 from Klinginstierna, that the supposed error in the result of Newton's 
 experiment, which he thus labours to demonstrate, is the same as 
 before attempted to be ascertained by Euler, and not the error 
 which was afterwards discovered by Dollond. 
 
relating to Refracting Telescopes, 71 
 
 The account given by De la Lande, of the improvement of the re- 
 fracting telescope, was copied by most foreign writers on the subject, 
 with little variation, except in giving sometimes a little more of the 
 honour to Euler, and also making mention of Klinginstierna. 
 
 But in the Eulogy on l^uler, written and published by M. N. Fuss, 
 professor of mathematics at St. Petersburg, in the year 1783, p. 41 
 and 42, he gives the whole of the discovery to Euler, except " that 
 Dollond is allowed to have found out two sorts of glass, which 
 crowned at last, in 1757? the happy conjecture of Euler, by the in- 
 vention of achromatic telescopes, which formed a new epoch in 
 astronomy and dioptrics." As this account is the most curious of any 
 I have found, I shall here give it at length, and contrast it with what 
 Euler says himself on the subject, in a paper read before the Royal 
 Academy of Sciences at Berlin, in 17 64, and published in the 
 volume of their Memoirs for 17 66, p. II9. 
 
 Mr. Fuss says, '^ The examination of the Newtonian theory had 
 given Euler an opportunity of investigating the different refrangibility 
 of light, and the bad effects which the dispersion of the colours pro- 
 duced in refracting telescopes, which had been almost intirely aban- 
 doned upon account of this defect. The consideration of the 
 wonderful structure of the eye made him suppose, that a certain 
 combination of different transparent bodies could remedy this* incon- 
 venience. He proposed for this purpose, in the year 1747, object- 
 glasses composed of two glasses, the cavity between which could he 
 filled with water. 
 
 " His opinion was attacked by the famous English artist, Dollond, 
 who opposed to him the authority of Newton: M. Euler soon shewed 
 
ft Some Account of the Discovery made by Mr, John DoUond, 
 
 h'ltn the error of his principles. Some experiments made upon me- 
 aiiSGuses, the cavities of which were filled with different liquids, con- 
 firmed him in his opinion: and Mr. Dollond, who had in the mean 
 time discovered two sorts o£ glass, which were proper for examining 
 it further, crowned at last, in 1757, the happy Oonjecture of M. 
 Euler, by the invention of achromatic telescopes, which formed an 
 epoch in astronomy and dioptrics. 
 
 " The success of Mr. Dollond, who availed himself, with so much 
 advantage, of a discovery, which he had at first attacked as contrary 
 to experiment, induced M. Euler to extend his researches further 
 upon the subject of dioptric instruments, &c." 
 
 I shall now subjoin a translation of M. Euler's paper, read at the 
 Academy of Sciences at Berlin in the year 1764. 
 
 *' Although I have already frequently discussed this subject^ I see 
 myself again obliged to resume it, in consequence of the astonishing 
 discoveries which have been lately made upon the nature of glass, and 
 its different kinds. I am not ashamed frankly to avow, that the first 
 accounts, which were published of it, appeared so suspicious, and 
 even so contrary to the best established principles, that I could not 
 prevail upon myself to give credit to them. That there should be two 
 kinds of glass, in which the refraction of the mean rays is nearly 
 equal, whilst that of the extremes differs most enormously, appeared 
 to me to shock good sense; and perhaps I should never have submitted 
 to the proofs, which Mr. Dollond produced to support this strange 
 phenomenon, if Mr. Clairaut, who must at first have been equally 
 surprized at it, had not most positively assured me, that Dollond's 
 experiments were but too well founded. But at length the experi- 
 
relating to Refracting Telescopes, 73 
 
 ments made at Petersburg by M. Zeiher have efFectually succeeded ia 
 removing my prejudice; that ingenious philosopher having incontesta- 
 bly proved that it is the lead;, which is used in some compositions of 
 glass, that produces in it that strange quality of augmenting the dis- 
 persion of the extreme rays, without changing sensibly the refraction 
 of the mean ; and by increasing the quantity of lead in the composi- 
 tion of glass, he has been enabled to make glass, which produces a 
 much greater dispersion of the rays than the flint-glass of Dollond. 
 
 " Now I must intirely renounce this principle, whidi until now 
 has appeared so well-founded, that the dispersion of the extreme rays 
 depends solely upon the refraction of the mean rays ; and I am obli- 
 ged to acknowledge, that the dispersion depends principally upon the 
 quality of the glass, without the mean refraction thereof being sensi- 
 bly affected thereby." 
 
 As it appears by the above paper, that Euler was at last so con- 
 vinced of the truth of the discovery made by Dollond, as to renounce 
 his favourite hypothesis, it must be inferred, that the account given 
 of this matter by Fuss is very far from being the true state of the 
 facts, and indeed so much so, as to be very inexcusable, even in an 
 eulogy. 
 
 There is another publication of a later date, which I shall take the 
 liberty of mentioning, " Extracts of the Observations made at the 
 Royal Observatory at Paris, in the year l^JQ^ > by Count Cassiniy In 
 page 106 he gives an account of the improvement of the refracting 
 telescope, by way of prelude to his describing a method proposed by 
 M. I'Abbe Rochon, of putting fluids, and also a kind of mastic, be- 
 tween the glasses of achromatic object-glasses, as a good method of 
 
 K 
 
74 Some Account of the Discovery, made by Mr, John Dollondj 
 
 mending bad glasses, or, as he calls it, a method to correct 
 the non-sphericity of the glasses ; which he mention^ as being similar 
 to that ingenious idea proposed by Newton, for correcting the errors 
 of the sphqrical figures of object-glasses. The account he gives of 
 the improvement of the refracting telescope is as follows. He says, 
 " It was the celebrated Euler who first proposed to correct the errors 
 arising from the different refrangibility, by using different refracting 
 mediums, s'- h as water and glass. The late Mr. Dollond having 
 availed himself of and realized this ingenious idea, has a just right to 
 partake of the glory." — By these publications it seems, that Dollond, 
 who explained the fallacy of Euler's hypothesis, who afterwards dis- 
 covered the true principle, on which the different refrangibility of 
 light could be corrected, and he, who put the same in practice, so 
 much to the benefit of science, is only to be allowed to partake of 
 the glory, and that with Euler, who never himself thought he had 
 the least right to claim any part of the discovery with Dollond, as 
 most fully appears by the paper above recited from the Berlin 
 Memoirs. 
 
 I can account for these false representations no other way than by 
 supposing, that those who wrote them have not taken sufficient pains 
 to inform themselves of the true history of the discovery; for I 
 would not wish to attribute what they have said to any partiality ; and 
 I am induced to hope, that when the state of facts which I have here 
 adduced shall be candidly considered, that they will retract their de- 
 clarations, and acknowledge, that Dollond was the sole discoverer of 
 the principle which led to the improvement of refracting telescopes. 
 
 I now come to a more agreeable part of this paper, which is, to 
 endeavour to reconcile the different results of the 8th experiment of 
 
relating to Refracting Telescopes » 75 
 
 the second part of the 1st Book of Newton's Optics^ as related by 
 himself, and as it was found by Dollond, when he tried the same 
 experiment, in the year 1757. Newton says, that light, as often as 
 by contrary refractions it is so corrected, that it emergeth in lines 
 parallel to the incident, continues ever after to be white. Now DoUond 
 says when he tried the same experiment, and made the mean refracti- 
 on of the water equal to that of the glass prison, so that the light 
 emerged in lines parallel to the incident, he foi..^J the divergency of 
 the light by the glass prism to be nearly double to what it was by the 
 water prism. The light appeared to be so evidently coloured, that it 
 was directly said by some persons, that if Newton had actually tried 
 the experiment, he must have perceived it to have been so. Yet who 
 could for a moment doubt the veracity of such a character ? — therefore 
 different conjectures were made by different persons. Mr. Murdoch 
 in particular gave a paper to the Royal Society in defence of Newton; 
 but it was such as very little tended to clear up the matter. Philo- 
 sophical Transactions^ vol: liii. p. 192. — Some have supposed that 
 Newton made use of water strongly impregnated with Saccharum 
 Saturni, because he mentions sometimes using such water, to increase 
 the refraction, when he used water prisms instead of glass prisms. 
 Newton s Optics, p. 62. — ^And others have supposed, that he tried the 
 experiment with so strong a persuasion in his own mind, that the di- 
 vergency of the colours was always in the same proportion to the mean 
 refraction, in all sorts of refracting mediums, that he did not attend 
 so much to that experiment as he ought to have done, or as he usually 
 did. None of these suppositions having appeared at all satisfactory, I 
 hav£ therefore endeavoured to find out the true cause of the difference, 
 
 K 2 
 
76 Some Account of the Discovery made hy Mr. John Dollondy 
 
 i , 
 
 and thereby shew, how the experiment may be made to agree with 
 Newton's description of it, and to get rid of those doubts, which 
 have hitherto remained to be cleared up. 
 
 It is well known, that in Newton's time the English were not the 
 most famous for making optical instruments: — telescopes, opera-glasses, 
 &c. were imported from Italy in great numbers, and particularly from 
 Venice ; where was manufactured a kind of glass which was much more 
 proper for optical purposes than any made in England at that time. The 
 glass made at Venice was nearly of the same refractive quality as our 
 crown glass, but of a much better colour, being sufficiently clear and 
 transparent for the purpose of prisms. It is probable that Newton's 
 prisms were made with this kind of glass ; and it appears to be the more 
 so, because he mentions the specific gravity of common glass to be 
 to water as 2.58 to 1. Newton s Opt. p. 247, which nearly answers 
 to the specific gravity we find the Venetian glass generally to have. 
 Having a very thick plate of this kind of glass, which was presented 
 to me about twenty-five years ago by the late professor Allemand, of 
 Leyden, and which he jthen informed me had been made many years, 
 I cut a piece from this plate of glass to form a prism, which I conceived 
 would be similar to those made use of by Newton himself. I 
 have tried the Newtonian experiment with this prism, and find it 
 answers so nearly to what Newton relates, that the difference which 
 remains may very easily be supposed to arise from any little dif- 
 ference, which may and does often happen in the same kind of glass 
 made at the same place at different times. Now the glass prism 
 made use of by Dollond to try the same experiment was made of 
 English flint glass, the specific gravity of which I have never known 
 
relating to Refracting Telescopes. 77 
 
 to be less than 3, 12. This difference in the densities of the prisms 
 ^ijsed by Newton and Dollond was sufficient to cause all the dif- 
 fer^ice which appeared to the two experimenters in trying the same 
 experiment. 
 
 From this it appears, that Newton was accurate in this experiment 
 as in all other?, and that his not having discovered that, \vhich was 
 discovered by Dollond so many years afterwards, was owing intirely to 
 accident; for if his prism had been made of glass of a greater or less 
 density, he would certainly have then made the discovery, and refract- 
 ing telescopes would not have remained so long in their original im- 
 perfect state. 
 
78 
 
 jin Attempt to explain a Difficultij in the Theory 
 of' Vision, depending on the different Refrangi-^ 
 hility of Light. By the Rev, Nevil Maskelyne^ 
 D.D. F.R S. and Astronomer Royal. 
 
 Read June 1 8, 1789. 
 
 1 HE ideas of sight are so striking and beautiful, that we are apt to 
 consider them as perfectly distinct. The celebrated Euler, taking this 
 for granted, has supposed, in the Memoirs of the Royal Academy of 
 Sciences at Berlin for 1747, that the several humours of the human 
 eye were contrived in such a manner as to prevent the latitude of 
 focus arising from the different refrangibility of light, and considers 
 this as a new reason for admiring the structure of the «ye; for that 
 a single transparent medium, of a proper figure, would have been 
 sufficient to represent images of outward objects in an imperfect man- 
 ner; but to make the organ of sight absolutely complete, it was 
 necessary it should be composed of several transparent mediums. 
 
An Attempt to explain a Difficulty in the Theory of Vision ^ ^c. 79 
 
 properly figured, and fitted together agreeable to the rules of the 
 sublimest geometry, in order to obviate the effect of the different re- 
 frangibility of light in disturbing the distinctness of the image ; and 
 hence he concludes, that it is possible to dispose four refracting sur- 
 faces, in such a manner as to bring all sorts of rays to one focus, at 
 whatever distance the object be placed. He then assumes a certain 
 hypothesis of refraction of the differently refrangible rays, and builds 
 thereon an ingenious theory of an achromatic object-glass, composed 
 of two meniscus glasses vi^ith v^^ater between them, with the help of an 
 analytical calculation, simple and elegant, as his usually are. 
 
 He has not, however, demonstrated the necessary existence of his 
 hypothesis, his arguments for which are more metaphysical than 
 geometrical ; and, as it was founded on no experiments, so those made 
 since have shewn its fallacy, and that it does not obtain in nature. 
 Moreover, which is rather extraordinary, it does not account, accord- 
 ing to his own ideas, for the very phenomenon which first suggested • 
 it to him, namely, the great distinctness of the human vision, as was 
 observed to me, many years ago, by the late Mr. John Dollond, F.R.S. . 
 to whom we are so much obliged for the invention of the achromatic 
 telescope,'* for the refractions at the several humours of the eye. 
 
 * As a misstatement of this fact has been made by both Paley and Priestley, we shall 
 quote their own words for the satisfaction of the reader, — " At last it came into the 
 mind of a sagacious optician, to inquire how this matter was managed in the eye; in 
 which there was exactly the same difficulty to contend with, as in the telescope. His 
 , observation taught him, that, in the eye, the evil was cured by combining together 
 lenses composed of different substances, i. e. of substances which possessed different 
 
60 An Attempt to explain a Difficulty in the Theory of Vision^ 
 
 being all made one way, the colours produced by the first refraction 
 will be increased at the two subsequent ones instead of being corrected, 
 whether we make use of Newton's or Euler's law of refraction of the 
 differently refrangible rays. 
 
 Thus Euler produced an hypothetical principle, neither fit for ren- 
 dering a telescope achromatic, nor to account for the distinctness of 
 the human vision; and the difficulty of reconciling that distinctness 
 with the principle of the different refrangibility of light discovered by 
 Sir Isaac Newton remains in full force. 
 
 In order to go to the bottom of this difficulty, as the best probable 
 means of obviating it, I have calculated the refractions of the mean, 
 most, and least refrangible rays at the several humours of the eye, and 
 thence inferred the diffusion of the rays, proceeding from a point in 
 an object, at their falling upon the retina, and the external angle 
 which such coloured image of a point upon the retina corresponds to. 
 
 4 
 refracting powers. Onr arHst borrowed from thence his hint ; and produced a correction 
 
 of the defect, by imitating, in glasses made from different materials, the effects of the 
 
 different humours through which the rays of light pass before they reach the bottom of 
 
 the eye." — See Paley, p. 23. " M. Euler did not pretend to controvert the experiments 
 
 of Newton ; but he said that they were not contrary to his hypothesis, but in so small a 
 
 degree as might be neglected, and asserted that, if they were admitted in all their ' 
 
 extent, it would be impossible to correct the difference of refrangibility occasioned by the 
 
 transmission of the rays from one medium into another of different density ; a correction 
 
 which, he thought,-was very possible, since he supposed it to be actually effected in the 
 
 structure of the eye, which he thought was made to consist of different mediurqs for 
 
 that very purpose. To this kind of reasoning Mr. DoUond made no reply ; but by 
 
 appealing to the experiments of Newton, and the great circumspection with which it 
 
 was known that he conducted all his inquiries." — See Priestley, p. 458. 
 
by the Rev. Dr. Maskelyne. 81 
 
 I took the dimensions of the eye from M. Petit, as related by T)v. 
 Jurin; and, the specific gravities of the aqueous and vitreous humours 
 having been found tp'be nearly the same with that of water, and the 
 refraction of the vitreous humour of an ox's eye having been found by 
 Mr. Hawksbee to be the same as that of water, and the ratio of re- 
 fraction out of air into the crystalline humour of an ox's eye having 
 been found by the same accurate experimenter to be as 1 to ,68327, I 
 took the refraction of the mean refrangible rays out of air into the 
 aqueous or vitreous humour, the same as into water, as 1 to ,74853, 
 or 1,33595 to i ; and out of air into the crystalline humour as 1 to 
 ,68327, or 1,46355 to 1. Hence I find, according to Sir Isaac New- 
 ton's two theorems, related at Part II. of Book I. of Optics, p. 113^ 
 that the ratio of refraction of the most, mean, and least refrangible 
 rays at the cornea should be as 1 to ,74512, ,74853 and ,75197; at 
 the fore-surface of the crystalline as 1 to ,91 173, ,91282, and ,91392; 
 and at the hinder-surface of the crystalline as 1 to 1,09681, 1,09550, 
 and 1,09420. 
 
 Now, taking with Dr. Jurin 15 inches for the distance at which 
 the generality of eyes in their mean state see with most distinctness, 
 I find the rays from a point of an object so situate will be collected 
 into three several foci, viz. the most, mean, and least refrangible 
 rays at the respective distances behind the crystalline, ,5930, ,6034, 
 and ,6141 of an inch, the focus of the most refrangible rays being 
 ,0211 inch short of the focus of the least refrangible ones. 
 
 Moreover, assuming the diameter of the pencil of rays at the cor- 
 nea, proceeding from the object at 15 inches distance, to be -i-th of 
 an inch in a strong light, which is a large allowance for it, the semi- 
 
82 An Attempt to explain a Diffi,culty in the Theory oj Vision^ 
 
 angle of the pencil of mean refrangible rays at their concourse upon 
 the retina will be 7° 12-', whose tangent to the radius unity, or ,1264 
 multiplied into ,0211 inch, the interval of the foci of the extreme re- 
 frangible rays, gives ,002667 inch for the diffusion of the different 
 coloured rays, or the diameter of the indistinct circle upon the retina. 
 Now, I find, that the diameter of the image of an object upon the 
 retina is to the object as ,6055 inch to the distance of the object from 
 the centre of curvature of the cornea; or the size of the image is the 
 same as would be formed by a very thin convex lens, whose focal dis- 
 tance is, 6055 inch, and consequently a line in an object which subtends 
 an angle of l' at the centre of the cornea will be represented on the 
 retina by a line of -^gi^-^th inch. Hence the diameter of the indistinct 
 circle on the retina before found, ,002667 will answer to an external 
 angle of ,002667 X5 67 8'= 15' 8'', or every point in an object should 
 appear to subtend an angle of about 15', on account of the different 
 refrangibility of the rays of light. 
 
 I shall now endeavour to shew that this angle of ocular aberration 
 is compatible with the distinctness of our vision. This aberration is 
 of the same kind as that which we experience in the common re- 
 fracting telescope. Now, by computation from the tabular apertures 
 and magnifying powers of such telescopes, it is certain that they admit 
 of an angular indistinctness at the eye or no less than 57'; therefore 
 the ocular aberration is near four times less than in a common refracting 
 telescope, and consequently the real indistinctness, being as the square 
 of the angular aberration, will be 14 or 15 times less in the eye than 
 in a common refracting telescope, which may be easily allowed to be 
 imperceptiWe. 
 
by the Rev. Dr, Maskelyne. 83 
 
 Moreover, Sir Isaac Newton has observed, with respect to the 
 like difficulty of accounting for the distinctness with which refracting 
 telescopes represent objects, that the erring rays are not scattered 
 uniformly over the circle of dissipation in the focus of the object- 
 glass, but collected infinitely more densely in the centre than in any 
 other part of the circle, and in the way from the centre to the cir- 
 cumference grow continually rarer and rarer, so as at the circum- 
 ference to become infinitely rare ; and by reason of their rarity are 
 not strong enough to be visible, unless in the centre and very near it. 
 
 He farther observes, that the most luminous of the prismatic 
 colours are the yellow and orange, which ^ affect the sense more 
 strongly than all the rest together; and next to these in strength are 
 the red and green; and that the blue, indigo, and violet, compared 
 with these, are much darker and fainter, and compared with 
 the other stronger colours, little to be regarded; and that therefore 
 the images of the objects are to be placed not in the focus of the 
 mean refrangible rays, which are in the confine of green and blue, 
 but in the middle of the orange and yellow, there where the colour is 
 most luminous, that which is in the brightest yellow, that yellow which 
 inclines more to orange than to green. 
 
 From all these considerations, and by an elaborate calculation, he 
 infers, that though the whole breadth of the image of a lucid point 
 be -jiyth of the diameter of the aperture of the object-glass, yet the 
 sensible image of the same is scarce broader than a circle whose dia- 
 meter is -a-foth part of the diameter of the aperture of the object- 
 glass of a good telescope; and hence he accounts for the apparent 
 diameters of the fixed stars as observed with telescopes by astrono- 
 mers, although in reality they are but points. 
 
84 An Attempt to explain a Difficulty in the Theory of Vision, 
 
 The like reasoning is applicable to the circle of dissipation on the 
 retina of the human eye; and therefore we may lessen the angular 
 aberration, before computed at 15', in the ratio of 250 to 55, which 
 will reduce it to 3' 18'". 
 
 This reduced angle of aberration may perhaps be double the appa- 
 rent diameter of the brightest fixed stars to an eye disposed for seeing 
 most distinctly by parallel rays; or, if short-sighted, assisted by a pro- 
 per concave lens; which may be thought a sufficient approximation in 
 an explication grounded on a dissipation of rays, to which a precise 
 limit cannot be assigned, on account of the continual increase of 
 density from the circumference, to the centre. Certainly some such 
 angle of aberration is necessary to account for the stars appearing 
 under any sensible angle to such an eye; and if we were, without rea. 
 son, to suppose the images on the retina to be perfect, we should be 
 put to a much greater difficulty to account for the fixed stars appearing 
 otherwise than as points, than we have now been to account for the 
 actual distinctness of our sight. 
 
 The less apparent diameter of the smaller fixed stars agrees also 
 with this theory; for the less luminous the circle of dissipation is, the 
 nearer we must look towards its centre to find rays sufficiently dense 
 to move the sense. From Sir Isaac Newton's geometrical account of 
 the relative density of the rays in the circle of dissipation, given in his 
 system of the world, it may be inferred, that the apparent diameters 
 of the fixed stars, as depending on this cause, are nearly as their 
 whole quantity of light. 
 
 In farther elucidation of this subject let me add my own experi- 
 ment. When I look at the brighter fixed stars, at considerable 
 
by ike Rev. Dr, Maskelyne. * 85 
 
 elevations, through a concave glass fitted, as I am short-sighted, to 
 shew them with most distinctness, they appear to me without scintil- 
 lation, and as a small round circle of fire of a sensible magnitude. If 
 I look at them without the concave glass, or with one not suited to 
 my eye, they appear to cast out rays of a determinate figure, not ex- 
 actly the same in both eyes, somewhat like branches of trees (which 
 doubtless arise from something in the construction of the eye) and to 
 scintillate a little, if the air be not very clear. To see day objects 
 with most distinctness, I require a less concave lens by one degree 
 than for seeing the stars best by night, the cause of which seems to 
 be, that the bottom of the eye being illuminated by the day objects, 
 and thereby rendered a light ground, obscures the fainter colours blue 
 indigo and violet in the circle of dissipation, and therefore the best 
 image of the object will be found in the focus of the bright yellow 
 rays, and not in that of the mean refrangible ones, or the dark green, 
 agreeable to Newton's remark, and consequently nearer the retina of 
 a short-sighted person ; but the parts of the retina surrounding the 
 circle of dissipation of a star being in the dark, the fainter colours, 
 blue, indigo, and violet, will have some share in forming the image, 
 and consequently the focus will be shorter.- 
 
 The apparent diameter of the stars here accounted for is different 
 from that explained by Dr. Jurin, in his Essay on Distinct and In- 
 distinct Vision, arising from the natural constitution of the generality 
 of eyes to see objects most distinct at moderate distances, and few 
 being capable of altering their conformation enough to see distant 
 objects, and among them the celestial ones, with equal distinctness. 
 
60 An Att^pt to explain a Difficully in the Theory of Vision, 
 
 But the cause of error, which I have pointed out, will affect all eye?, 
 even those which are adapted to distant objects. 
 
 If this attempt to shew the compatibility of the actual distinctness 
 of our sight with the different rcfrangibility of light shall be admitted 
 as just and convincing, we shall have fresh reason to admire the 
 wisdom of the creator in so adapting the aperture of the pupil and the 
 different rcfrangibility of light to each other, as to render the picture 
 of objects upon the retina relatively, though not absolutely, perfect, 
 and fitted for every useful purpose ; " where," to borrow the words of 
 our religious and oratorical philosopher Derham, " all the glories of 
 the heavens and earth are brought and exquisitely pictured." 
 
 Nor does it appear, that any material advantage would have been 
 obtained, if the image of objects on the retina had been made ab- 
 solutely perfect, unless the acuteness of the optic nerve should have 
 been increased at the same time; as the minimum visihile depends no 
 less on that circumstance than the other. But that the sensibility of 
 the optic nerve could not have been much increased beyond what it is, 
 without great inconvenience to us, may be easily conceived, if we only 
 consider the forcible impression made on our eye by a bright sky, or 
 even the day objects illuminated by a strong sun. Hence we may 
 conclude, that such an alteration would have rendered our sight pain- 
 ful instead of pleasant, and noxious instead of useful. We might 
 indeed have been enabled to see more in the starry heavens with the 
 naked eye, but it must have been at the expence of our daily labours 
 and occupations, the immediate and necessary employment of man. 
 I shall only mention farther, and obviate an objection to the dif- 
 
hy the Rev. Dr, Maskelyne, BT 
 
 fusion of the rays upon the retina by the different refrangibility of 
 light. It may be said, that the ocular aberration, being a separate 
 cause from any effect of the telescope, should subsist equally when we 
 observe a star through a telescope as wheh we look at it with the 
 naked eye; and that therefore the fixed stars could not appear so 
 small as they have been found to do through the best telescopes, and 
 particularly by Dr. Herschel with his excellent ones. To this I answer, 
 that the ocular aberration, which is proportional to the diameter of 
 the pupil when we use the naked eye, is proportional to the diameter 
 of the pencil of rays at the eye when we look through a telescope, 
 which being many times less than that of the pupil itself, the ocular 
 aberration will be diminished in proportion, and become insensible. 
 
86 
 
 jiii Account of an Improvement made by Mr. Peter 
 DoUond in his New Telescopes. In a Letter 
 ^o James Short, M.A. F.R. S. with a Letter of 
 Mr. Short's to the Rev. Thomas Birch, D.D. 
 S^ecref. R. S. ' 
 
 DEAR SIR, 
 
 1 HAVE sent you inclosed, a letter which I received 
 this morning from Mr. Dollond, concerning an improvement which 
 he has made in his new telescopes. He, some months ago, sent me 
 a telescope, in this new way, of 3 J feet focal length, with an aperture 
 of 3f Inches; I examined it, and I approved of it; I have tried it 
 with a magnifying power of 150 times, and I found the image distinct, 
 bright, and free from colours. 
 
 You may, if you please, lay Mr. Dollond's letter before the Royal 
 Society. 
 
 I am, DEAR SIR, 
 
 Your most obedient and humble servant, 
 
 James Short. 
 
 Surrey Street, 
 February 7, 1765. - 
 
89 
 
 Mr. DolloncTs Letter to Mr. Short. 
 
 Read February 7, 1765. 
 
 SIR, ,, 
 
 1 TAKE the liberty of sending you the following 
 short account of an improvement I have lately made in the compound 
 object glasses of refracting telescopes. 
 
 The dissipation of the rays of light may be perfectly corrected in 
 object glasses, by combining mediums of different refractive qualities; 
 and the errors or aberrations of the spherical surfaces may be corrected 
 by the contrary refractions of two lenses, made of the different me- 
 diums; yet as the excess of refraction is in the convex len&, and 
 though the surfaces of the concave lens may be so proportioned as to 
 aberrate exactly equal to the convex lens, near the axis; yet as the 
 refractions of the two lenses are not equal, the equality of the aber- 
 rations cannot be continued to any great distance from the axis. 
 
 In the year 1758, when my father had constructed some object 
 glasses for telescopes in this manner, viz. with one convex lens of 
 crown glass, and one concave lens of white flint glass; he attempted 
 
 M 
 
£0 An Accounl of an Improvement made by Mr. Peter Dollond, 
 
 to make short object glasses to be used with concave eye glasses, in 
 the same manner; but as the field of view, in using a concave eye 
 glass depends on the aperture of the object glass, the limits of the 
 aperture were found to be too small: this led my father to consider 
 that if the refraction of the crown glass (in which the excess was) 
 should be divided by means of having two lenses made of crown 
 glass instead of one, the aberration would thereby be decreased, and 
 the apertures might then be larger : this was tried with success in 
 those object glasses, when concave eye glasses were used, and these 
 have been ever since made in this manner : some trials were likewise 
 made, at the same time, to enlarge the apertures of longer object 
 glasses, where convex eye glasses were used, by the same method ; 
 but these not succeeding, in the same manner, the method of making 
 them with one lens of crown glass, and one of white flint glass, was 
 continued. 
 
 As I could not see any good reason why the method, which was 
 practised with so much success, when concave eye glasses were used, 
 should not do with convex ones ; I determined to try some further 
 experiments in that way. After a few trials, I found it might be 
 done; and in a short time I finished an object glass of 5 feet focal 
 length, with an aperture of 3:| inches, composed of two convex lenses 
 of crown glass, and one concave of white flint glass. 
 
 Thinking that the apertures might be yet admitted larger; I at- 
 tempted to make one of 3 J feet focal length, with the same aperture 
 of 3f inches, which I have now completed, and am ready to show 
 the same to the Royal Society, if desired. 
 
 The difliculty of procuring good glass of so large a diameter, and 
 
in his New Telescopes. gi 
 
 of the thickness required, added to the great exactness of the sur- 
 faces, in order to correct the aberration in such large apertures, has 
 prevented me from attempting to extend them any farther in that 
 lerigth. 
 
 I am, SIR, 
 
 Your most obedient, 
 
 and most humble servant, 
 
 Peter DoUond. 
 
 M 2 
 
m 
 
 A Letter J^rom Mr. Peter DoUond, to Nevil Mas- 
 kelyne, F.R.S. &^ Astronomer Royal; describing 
 some Additions cmd Alterations made to Hadley's 
 Quadrant, to render it more serviceable at Sea. 
 
 Read March 29, 1772. 
 
 REVEREND SIR, 
 
 ■\ 
 
 xHE particular attention which you have always 
 
 shown to any improvement tending to the advantage of astronomy 
 
 or navigation, makes me take the liberty to trouble you with an ac- 
 
 * count of some additions and alterations which I have lately made 
 
 to the Hadley's quadrant. 
 
 The general use of this instrument at sea is so well known, that no 
 mention need be made of the importance of any improvements in 
 the construction, that may render the observations more exact, and 
 occasion more frequent opportunities of making them. 
 
Description of some j^dditionSj §c. to Hadley's Quadrant. 93 
 
 The glasses of the Hadley's quadrant should have their two surfaces 
 perfect planes, and perfectly parallel to each other. From several 
 years practice in grinding these glasses, I have found out methods of 
 making them to great exactness; but the advantage, that should arise 
 from the goodness of the glasses, has often times been defeated by 
 the index glass being bent by the brass frame that contains it: to 
 prevent this, I have contrived the frame, so that the glass lies on 
 three points, and the part that presses against the front of the glass 
 has also three points exactly opposite to the former. These points 
 are made to confine the glass by three screws at the back, that act 
 exactly opposite to the points between which the glass is placed* 
 This little contrivance may be of some use; but the principal im- 
 provements are in the methods of adjusting the glasses, particularly 
 for the back observation. 
 
 The method hitherto practised for adjusting that part of the instru- 
 ment, by means of the opposite horizons at sea, has been attended 
 with so many difficulties that it has scarcely ever been used ; for so lit- 
 tle dependance could be placed on the observations taken this way, 
 that the best Hadley's sextants made for the purposes of observing 
 the distances of the moon from the sun or fixed stars, have been 
 always made without the horizon glass for the back observation ; for 
 want of which, many valuable observations of the sun and moon have 
 been lost, when their distance has exceeded 120 degrees. 
 
 To make the adjustment of the back observation easy and exact, I 
 have applied an index to the back horizon glass, by which it may be 
 moved into a parallel position to the index glass, in order to give it 
 the two adjustments, in the same manner as the fore horizon glass is 
 
94 Description of &ome Additions f ^c. (o Hadley's Quadrant, 
 
 adjusted. Then, by moving the index to which the back horizon 
 glass is fixed, exactly 90 degrees (which is known by the divisions 
 made for that purpose) the glass will be thereby set at right angles 
 to the index glass, and consequently will be properly adjusted for use, 
 and the observations may be made with the same accuracy by this, 
 as by the fore observation. 
 
 To adjust the horizon glasses in the perpendicular position to the 
 plane of the instrument, I have contrived to move each of them by 
 a single screw, that goes through the frame of the quadrant, and is 
 turned by means of a milled head at the back, which may be done 
 by the observer while he is looking at the object. 
 
 To these improvements. Sir, I have added your method of placing 
 darkening glasses behind the horizon glasses, which you have been 
 «o ki^d as to give me liberty to apply to my instruments. These 
 glasses, which serve for darkening the object seen by direct vision, 
 in adjusting the instrument by the sun or moon, I have placed in 
 such a manner as to be turned behind the fore horizon glass, or be- 
 hind the back horizon glass, that they may be used with either; there 
 are three of these glasses of different degrees of darkness; the 
 lightest or palest I do imagine will be of use in taking the sun's alti- 
 tude when ihe horizon appears glaring, which I believe often happens 
 by the reflection of the sea. 
 
 If these additions and alterations should be thought to be real im- 
 provements, which I cannot doubt. Sir, if they are honoured with 
 your approbation, I hope they may serve in conjunction with those 
 improvements you have made yourself in respect to the obviating any 
 possible errors in the parallelism of the planes of the index glass, and 
 
to render it more serviceable at Sea. 95 
 
 in regard to the adjustment of the telescope parallel to the plane of 
 the quadrant, to extend the use of this most valuable nautical instru- 
 ment, and to add to the exactness of the celestial observations taken 
 with it to determine the longitude at sea. But of these particulars I 
 need say no more, since you are, v^^ithout doubt, in every respect, 
 the properest person to give an account of them. 
 
 . I am, SIR, 
 
 Your most obedient, 
 
 humble servant, 
 
 Peter DoUond. 
 
 London, 
 Februaiy 25, 1772. 
 
96 
 
 RemarJcs on the Hadley's Quadrant, tending prin- 
 dpally to remove the Difficulties which have 
 hitherto attended the Use of the Back-observation, 
 and to obviate the Errors that might arise J^rom 
 a Want of Parallelism in the two Surfaces of 
 the Index-Glass. By Nevil Maskelyne, F.R.S. 
 Astronomer Royal. 
 
 Read May 28, 1772. 
 
 X H£ back-observation with Hadley's quadrant being founded on the 
 same principles, and in theory, equally perfect with the fore observa- 
 tion, and being at the same time necessary to extend the use of the 
 instrument up to 180 degrees (it being impracticable to measure 
 angles with any convenience beyond 120 degrees with the fore- 
 observation) it may seem surprizing that it hath not been brought 
 equally into general use, more especially since the method of finding 
 the longitude by observations of the moon, has been practised at sea 
 for some years past; since this method v^^ould receive considerable 
 
Remarks on the Hadleys Quadrant^ 8^e. 97 
 
 advantage from the use- of the back-observation in taking distances 
 of the sun and. nnoon between the first and last quarter, could such 
 observations be as much depended upon as the fore-observation. The 
 causes of this seem to have been principally these two, the difficulty 
 of adjusting the back-horizon-glass, and the want of a method of 
 directing, the sight parallel to the plane of the quadrant. The back- 
 horizon-glass, like the fore-one, requires two adjustments : — the first, 
 or common one, disposes it at right angles to the index glass, when 
 the index stands at (O) upon the arch; which is usually performed by 
 setting (0) of the index of the arch of the quadrant by double the 
 dip of the horizon of the sea, and then holding the quadrant vertical 
 with the arch downwards, and turning the back-horizon-glass about, 
 by means of its lever or perpetual screw, till the reflected back-horizon 
 appears to coincide with the fore-horizon seen directly. But this 
 operation is so difficult in practice with the back -horizon-glass wholly 
 silvered, except a small transparent slit in the middle, as it has been 
 usually made, that few (if any) persons have ever received proper sa- 
 tisfaction from it. If the back-horizon-glass was silvered in every 
 respect like the fore-horizon-glass (which it ought to be) the upper 
 part being left unsilvered, and a telescope was applied to it, perhaps 
 this adjustment might be rendered somewhat easier and more exact; 
 but it could not even thus be made so exact as the adjustment of the 
 fore-horizon-glass may, by making use of the sun's limbs. 
 
 The second adjustment of the back-horizon-glass, in the common 
 construction of the quadrant, is still more troublesome, since it cannot 
 be executed without setting the index 90 degrees off the arch, in or- 
 der to place the index-glass parallel to the back-horizoR-gla33 ; when 
 
 N 
 
98 Remarks on the Hadlefifs QuadrOntf 
 
 this adjustment may be performed in the same manner as the corres- 
 ponding adjustment of the fore-horizon-glass. But the bending of 
 the index, that follows the setting it oiF the arch, is a very disagree- 
 able circumstance, having a tendency, especially on board of ship, to 
 expose both the index and centre work to damage ; and may even, 
 without extraordinary precautions taken by the instrument maker in 
 placing the plane of the index-glass exactly according to the length 
 of the index, disturb its perpendicularity to the plane of the quadrant : 
 on these accounts it would be much better if this adjustment of the 
 back-horizon-glass could be performed, like those of the fore-horizon- 
 glass, with the index remaining upon the arch of the quadrant. 
 Fortunately, this desideratum has been lately effected by an ingenious 
 contrivance invented by Mr. Dollond, which he has given an account 
 of in a letter addressed to me*, which I have presented to this So- 
 ciety, by means of an additional index applied to the back- 
 horizon-glass; whereby both the adjustments may be made by the 
 same observations and with nearly the same exactness as those of the 
 fore- horizon-glass: — for a further knowledge of which see the account 
 itself. , ^ 
 
 Besides the difficulty of adjusting the back-horizon -glass, the 
 want of a method of directing the line of sight parallel to the plane 
 of the quadrant has proved also a considerable obstacle to the use of 
 the back-observation : this will easily appear from the following pro- 
 position, that the error of the angle measured arising from any small 
 deviation of the visual ray from a parallelism to the plane of the quad- 
 
 * See page 92 for the Letter alluded to. 
 
by the Rev. Dr. Maskelyne. 99 
 
 rant, is to twice an arch equal to the verse-sine of the deviation, as 
 the tangent of half the angle measured by the quadrant is to radius, 
 very nearly. Thus a deviation of 1° in the line of sight, will pro- 
 duce an error of about l' in measuring an angle of 90°, whether by 
 the fore or back-observation ; but the same deviation will produce an 
 error of 4' in measuring an angle of 150°, of & in taking an angle 
 of 160°, and 12' in taking an angle of 170°. Hence a pretty 
 exact adjustment of the line of sight, or axis of the telescope, 
 is requisite in measuring large angles, such as those are taken by the 
 back-observation : and therefore a director of the sight ought by no 
 means to be omitted in the construction of the instrument (as it 
 commonly has been since Mr. Hadley's time, though recommended 
 by him), except a telescope be made use of, which, if rightly placed, 
 answers the same purpose better, especially in observing the distance 
 of the moon from the sun between the first and last quarter. The 
 director of the sight may be placed exact enough by construction ; 
 but the telescope cannot, and Mr. Hadley, not having been aware 
 ef the importance of an exact position of it, has accordingly given 
 no directions for the placing of it. I shall therefore endeavour to 
 supply this defect in the following remarks. 
 
 In the first place, I would by all means recommend an adjusting 
 piece to be applied to the telescope, whereby its axis may be brought 
 parallel to the plane of the quadrant: in the next place, the back- 
 horizon-glass ought to be silvered in the same manner as the fore- 
 horizon-glass : and thirdly, two thick silver wires should be placed 
 within the eye-tube in the focus of the eye-glass parallel to one 
 another, and to the plane of the quadrant. If tHey were put at such 
 
 N 2 
 
100 Remarks on the Hadleifs Quadrant y 
 
 a distance as to divide the diameter of the field of view into three 
 equal parts, it might be as convenient as any other interval. In this 
 manner wires were placed in the telescope by Mr. Hadley, as appears 
 by his accountof the instrument in Philosophical Transactions, No. 420. 
 These wires are to be adjusted parallel to the plane of the quadrant, by 
 turning the eye-tube round about which contains the wires, till they 
 appear parallel to the plane of the quadrant. The axis of the teles- 
 cope, by which is meant the line joining the centre of the object-glass 
 and the middle point between the two wires, is to be adjusted parallel 
 to the plane of the quadrant by either of the two following methods. 
 
 Method I. — When the distance of the moon from the sun is 
 greater than QO degrees, by giving a sweep with the quadrant and 
 moving the index, bring the nearest limbs to touch one another at 
 the wire nearest the plane of the quadrant. Then, the index re- 
 maining unmoved, make the like observation at the wire farthest from 
 the plane of the quadrant; and note whether the nearest limbs are in 
 contact as they were at the other wire: if they are, the axis of the 
 telescope is parallel to the plane of the quadrant: but if they are not, 
 it is inclined to the same, and must be corrected as follows. If the 
 nearest limbs of the sun and moon seem to lap over one another at 
 the wire farthest from the plane of the quadrant, the object end of 
 the telescope is inclined from the plane of the quadrant, and must be 
 altered by the adjustment made for that purpose : but, if the nearest 
 limbs of the sun and moon do not come to touch one another at the 
 wire farthest from the plane of the quadrant, the object end of the 
 telescope is inclined towards the plane of the quadrant, and must be 
 altered by the adjustment accordingly. Let these operations be re- 
 
by the Rev. Dr. Maskelyne. 101 
 
 peated until the observation is the same at both the parallel wires, 
 and the axis of the telescope will be adjusted parallel to the plane 
 of the quadrant. In like manner, the axis of the telescope may be 
 also adjusted parallel to the plane of the quadrant for the fore-observa- 
 tion. 
 
 Method II. — Set the index to (o) and hold the plane of the quadrant 
 parallel to the horizon of the sea, with the divided arch upwards, the 
 two wires being parallel to, and including both the direct fore-horizon, 
 and the reflected back-horizon, between them. Raise or lower the 
 plane of the quadrant until the direct and reflected horizons coincide 
 together: if the coincidence happens in the middle between the two 
 wires, or rather, to be more exact, above the middle by such a part of 
 the field of view as answers to the number of minutes in the de- 
 pression of the horizon .(which may be easily estimated if the angular 
 interval of the wires be first found by experiment, in manner here- 
 after mentioned) the axis of the telescope is parallel to the plane of 
 the quadrant ; but if it does not, the line of sight is inclined to the 
 plane of the quadrant, and must be corrected as follows. If the direct 
 and reflected horizons, when they coincide, appear higher above the 
 middle between the wires, than what the quantity of the depression 
 of the horizon amounts to, the object end of the telescope is inclined 
 from the plane of the quadrant, and must be altered by the adjust- 
 ment made for that purpose; but if the two horizqns appear to 
 coincide in a lower part of the field of the telescope, the object end 
 of the telescope is inclined towards the plane of the quadrant, and 
 must be altered by the adjustment accordingly. Repeat these ope- 
 rations till the two horizons appear to coincide above the middle between 
 
102 Remarks on the Hadleys Quadrant, 
 
 the two wires, by the quantity of the depression of the horizon, and 
 the axis of the telescope will be adjusted parallel to the plane of the quad- 
 rant. In order to find the angular interval between the wires, hold the 
 quadrant perpendicular to the horizon, as in observing altitudes ; and 
 turn about the eye-tube with the wires until they are parallel to, and in- 
 clude the direct fore-horizon and reflected back-horizon between them. 
 Move the index from (o) along the divided arch, at the same time 
 taising or lowering the telescope by the motion of the quadrant until 
 the direct horizon appears to coincide with the upper wire, and the 
 reflected back-horizon with the lower wire ; the number of degrees 
 and minutes shown upon the arch, increased by double the depression 
 of the horizon, will be the angular interval of the wires ; its propor- 
 tion to the depression of the horizon will be therefore known ; and 
 hence the space in the field of the telescope answering to the depres- 
 sion of the horizon, may be easily estimated near enough for 
 adjusting the axis of the telescope in the manner before mentioned. 
 The first of the two methods here given for adjusting the position of 
 the telescope will probably be found most convenient ; and the greater 
 Xhe distance of the sun and moon is, the more nearly may the adjust- 
 ment be made, because the same deviation of the axis of the 
 telescope will cause a greater error. 
 
 The telescope should be fixed by the instrument-maker so as to 
 command a full field of view when the instrument is placed at 90® 
 if the instrument be an octant, or 120° if it be a sextant ; because 
 the index-glass then stands more oblique with respect to the incident 
 and reflected rays, and consequently the field of view of the telescope, 
 as far as it depends upon the index-glass, will be more contracted 
 
by the Rev. Dr. Maskelyne, 103 
 
 tlian in any other position of the index: but if there is a fait field of 
 •view in this case, there necessarily must be so in every other position 
 of the index. 
 
 The two parallel wires will be very useful on many occasions, as 
 well in the fore as the back-observation. In taking the altitude of 
 the sun, moon, or star, direct the sight towards the part of the 
 horizon underneath, or opposite to the object, according as you 
 intend to observe by the fore or back- observation, a^nd hold the 
 quadrant that the wires may constantly appear perpendicular to the 
 horizon, and move the index till you see the object come down 
 towards tlie horizon in the fore-observation, or up to it in the 
 back-observation,, and turn the instrument in order to bring the 
 object between the wires ; then move the index till the sun 
 or moon's limb, or the star touch the horizon. The nearer 
 the object is brought to an imaginary line in the middle between 
 the wires (it is indifferent what part of the line it is brought 
 to) and the truer the wires are kept perpendicular to the horizon, 
 the more exact will the observation be. In the fore-observation, the 
 object appears in its real position ; but in the back-observation, the 
 object being brought through the zenith to the horizon, the real 
 upper-limb will appear the lowest ; and the contrary. Either limb 
 of the sun may be used in either observation ; but it will be most 
 convenient in general to make the sun appear against the sky, and not 
 against the sea ; and then the objects appearing inverted through the 
 telescope, the sun will appear lowest, and the horizon highest. The 
 observed altitude is to be corrected for dip, refraction, and sun's 
 semi -diameter, as usual. 
 
104 Remarks on the Hadleys Quadrant^ 
 
 In taking the distance of the nearest limbs of the sun and moon, 
 whether by the fore or back-observation, having 'first set the index 
 to the distance nearly, by the help of the Nautical Almanac, and 
 brought the moon to appear anywhere on or near the diameter of the 
 field of view of the telescope, which bisects the interval between the 
 wi^es, give a sweep to the quadrant, and the sun and moon will pass 
 by one another ; if in this motion the nearest limbs, at their nearest 
 approach, just come to touch one another, without lapping over, on 
 or near any part of the diameter of the field of the. telescope which 
 bisects the interval between the wires, the index is rightly set ; but if 
 the nearest limbs either do not come to meet, or lap over one another, 
 alter the index, and repeat the observation till the nearest limbs come 
 to touch one another properly. This method of observing will be 
 found much more easy and expeditious than without the wires, since 
 in that case it would be necessary to make the limbs touch very near 
 the centre of the telescope, but here it is only necessary to make 
 them do so anywhere on or near the diameter of the field of the 
 telescope which bisects the interval between the two wires. 
 
 The same method may be used in taking the moon's distance from 
 a fixed star. 
 
 It may not be amiss here to make some remarks on the rules that 
 have been usually given for observing the sun's altitude, both with the 
 fore and back-observation, which have all been defective, and to point 
 out the proper directions to be followed, when a telescope is not used 
 w'ith two parallel wires to direct the quadrant perpendicular to the 
 horizon, and to shew the principles on which these directions are 
 founded. 
 
hy the Rev. Dr. Maskelyne, 105 
 
 Observers are commonly told, that in making the fore-observation 
 they should move the index to bring the sun down to the part of the 
 horizon directly beneath them, and turn the quadrant about upon the 
 axis of vision ; and when the sun touches the horizon at the lowest 
 part of the arch described by them, the quadrant will shew the 
 altitude above the visible horizon. I allow that this rule would be 
 true, if a person could by sight certainly know the part of the hori- 
 zon exactly beneath the sun ; but, as this is impossible, the precept 
 is incomplete. Moreover, in taking the sun's altitude in or near the 
 zenith, this rule intirely fails, and the best observers advise to hold 
 the quadrant vertical, and turn one's self about upon the heel, stop- 
 ping when the sun glides along the horizon without cutting it : and it 
 is certain that this is a good rule in this case, and capable with care of 
 answering the intended purpose. We have thus t^yo rules for the 
 same thing, which is a proof that neither of them is an universal one, 
 or sufficient in all cases alone. 
 
 In taking the back-observation, observers have been advised either 
 to turn the quadrant about upon the axis of vision, or, holding the 
 quadrant upright, to turn themselves about upon the heel, indifferently. 
 The true state of the case is this; that, in taking the sun's altitude, 
 whether by the fore or back-observation, these two methods must be 
 combined together; that is to say, the observer must turn the quad- 
 rant about upon the axis of vision, and at the same time turn himself 
 about upon his heel, so as to keep the sun always in that part of the 
 horizon-glass which is at the same distance as the eye from the plane 
 of the quadrant: for, unless the caution of observing the objects in 
 the proper part of the horizon-glass be attended to, it is evident the 
 
106 Remarks on the Hadley's Quadrant, 
 
 angles measured cannot be true ones. In this way the reflected sun^ill 
 describe an arch of a parallel circle round the true sun, whose convex 
 side will be downwards in the fore-observation, and upwards in the 
 back-observation, and consequently, when, by moving the index, the 
 lowest point of th6 arch in the fore-observation, or the uppermost 
 point of the arch in the back-observation, is made to touch the hori- 
 zon, the quadrant will stand in a vertical plane, and the altitude above 
 the visible horizon will be properly observed. 
 
 The reason of these operations may be thus explained: — the image 
 of the sun being always kept in the axis of vision, the index will al- 
 ways show on the quadrant the distance between the sun and any 
 object seen directly which its image appears to touch; therefore, as 
 long as the index remains unmoved, the image of the sun will de- 
 scribe an arch everywhere equidistant from the sun in the heavens, 
 and consequently a parallel circle about the sun, as a pole ; such a 
 translation of the sun's image can only be produced by the quadrant 
 being turned about upon a line drawn from the eye to the sun, as an 
 axis ; a motion of rotation upon this line may be resolved into two, 
 one upon the axis of vision, and the other upon a line on the qua- 
 drant perpendicular to the axis of vision ; and consequently a proper 
 combination of these two motions will keep the image of the sun 
 constantly in the axis of vision, and cause both jointly to run over a 
 parallel circle about the sun in the heavens ; but when the quadrant is 
 vertical, a line thereon perpendicular to the axis of vision becomes a 
 vertical axis ; and, as a small motion of the quadrant is all that is 
 wanted, it will never differ much in practice from a vertical axis; 
 therefore the observer, by properly combining and proportioning two 
 
by the Rev. Dr. Maskeli/ne. 107 
 
 motions, one of the quadrant upon the axis of vision, and the other 
 of himself upon his heel, keeping himself upright (which gives the 
 quadrant a motion upon a vertical axis) will cause the image of the 
 sun to describe a small arch of a parallel circle about the sun in the 
 heavens, without departing considerably from the axis of vision. 
 
 If it should be asked, why the observer should be directed to perform 
 two motions rather than the single one equivalent to them on a line 
 drawn from the eye to the sun as an axis, I answer, that we are not capa- 
 ble, while Iboking^ towards the horizon, of judging how to turn the 
 quadrant about upon the elevated line going to the sun as an axis, by 
 any other means than by combining the two motions above-mentioned, 
 SK> as to keep the sun's image always in the proper part of the horizon- 
 glass. When the sun is near the horizon, the line going from the 
 eye to the sun will not be far removed fromt the axis of vision; and 
 consequently the principal motion of the quadrant will he performed 
 on the axis of vision, and the part of the motion made on the vertical 
 ams will be but small. On the contrary, when the s^in. is near the 
 zenith, the line going to the; sun is not far. removed from a vertical 
 line, and consequently the principal motion oi the quadrant will be 
 performed on. a.vertipal axis, by the observer's turning himself about, 
 and the part of the motion made on the axis of vision will be but 
 Sflfiall. In intermediate altitudes of the sun, the motions of the qua- 
 drant QU: the. ax,is of vision and on a vertical axis will be qiore ^ually 
 4ivided. Hence appears the reason of the method used by the best 
 observers in taking the sun's altitude when, near the zenith by hold- 
 ing the quadrant vertical and turning about upon the heel, and the 
 
 q % 
 
108 Remarks on the Hadleifs Quadrant, 
 
 defects of the rules that have been commonly given for observing 
 altitudes in other cases. 
 
 As it may conduce to the setting this matter in a still clearer light, 
 I shall here describe in order the several motions that will be given to 
 the reflected image, by turning the quadrant about upon the axis of 
 vision, a vertical axis, or the line drawn from the eye to the sun, 
 successively. 
 
 I. If the quadrant is turned about upon the axis of vision, the 
 same being directed to the point of the horizon exactly beneath 
 or opposite the sun, the image of the sun will move from right 
 to left, or from left to right, across the horizon-glass, the same 
 way as the arch of the quadrant is carried, both in the fore and 
 back-observations, with a velocity which is to the angular velocity 
 of the quadrant as the sine of the sun's altitude to the radius, 
 describing an arch convex downwards in both cases; and when 
 the motion of the sun in this arch is parallel to the horizon, the 
 quadrant is held truly perpendicular to the horizon, and conse- 
 quently in a proper position for taking the sun's altitude. But, 
 if the axis of vision be directed to, and turned round a point in 
 the horizon beside the vertical circle passing through the sun, the 
 sun*s image, when its motion is parallel to the horizon, will be 
 neither in the axis of vision nor the sun's vertical, but between 
 both; at the same time, the plane of the quadrant will not be 
 vertical, and [^the altitude found by bringing the sun's image to 
 touch the horizon will not be the true altitude. 
 II. If the quadrant be held perpendicular to the horizon, and turned 
 
by the Rev. Dr. Maskelyne. lo^ 
 
 about upon a vertical axis, or one nearly so, the sun will describe 
 an arch convex downwards in the fore-observation, and upwards 
 in the back-observation, the motion of the sun being the same 
 way as the axis of vision is carried in both cases, and being to 
 the angular motion of the quadrant, as the verse-sine of the sun s 
 altitude to the radius in the fore-observation, but as the verse- 
 sine of the supplement of the sun's altitude to 180° to the radius 
 in the back-observation. The sun therefore will move slower 
 than the axis of vision in the fore-observation, and consequently 
 will be left behind, with respect to the axis of vision, or seem to 
 move backwards; and the sun will move quicker than the axis of 
 vision in the back-observation, or will seem to get before it. 
 When the motion of the sun in this arch is parallel to the horizon, 
 the plane of the quadrant coincides with the vertical circle passing 
 through the sun, and consequently the quadrant is in a proper 
 
 • position for taking the sun's altitude. But if the quadrant be 
 held a little deviating from the perpendicular position to the hori- 
 zon, and turned about upon an axis, either vertical or only 
 nearly so, the arch described by the sun apparently will cut the 
 horizon, but will never move parallel to it, and consequently the 
 quadrant will not be brought into a proper position for observing 
 the sun's altitude. 
 
 III. If the quadrant be turned on the line going to the sun as an 
 axis, the reflected sun will be kept constantly in the axis of vision, 
 and will describe an arch of a parallel circle about the real sun, 
 with a velocity which is to the angular motion of the quadrant, 
 as the sine of the suji's altitude is to the radius; and when the 
 
1 10 Remarks on the Hadleijs Quadrant , 
 
 motion of the reflected sun is parallel to the horizon, the quad- 
 rant is vertical. 
 
 Hence naturally arise the three methods of taking an altitude, 
 which have been mentioned before. In the first, the axis of vision is 
 supposed; always directed to one and the same part of the horizon, 
 aamely, that which is in the sun's vertical. In the second, the ob- 
 Sjerver is required to hold the quadrant truly vertical, and to turn 
 himself upon a vertical axis ; but it is evident neither of these motions 
 can be accurately performed. In the third method, the observer is 
 only required to move both himself and the quadrant, so as to keep 
 the sun always in or near the axis of vision, which may be performed 
 very well, because the axis of vision is a visible and certain direction 
 for it. One exception, however, should be made tO; this general rule, 
 namely, in taking the sun*s altitude when very lowj by the back- 
 observation: in which case ijt will be best to use the second method, 
 OX, else, to hold the quadrant, perpendicular by judgment; which will 
 l)e:much facilitated by using a telescope containing wires; in ijts focus 
 parallel to the plane: of; the quadrant, as described in p. 103 of this 
 ^ppmdisQ.:. for, in this- case,^ thq perpendicular position, of the qua- 
 4rant cannot be; attained 5q near by thq method of; turning the 
 quadrant on a line going to the sun as an axis,, as it can by any other 
 
 ^aethod^ 
 
 Xt remains tptfTeatof the; errors which may arise from a defect of 
 parallelism in thor two sarfaces; of the indexrglags, and to: pqintiQiit the 
 means of obviating themi in the; celestial; observations. It? ig well 
 k«o>vj?i. that, if 3j pe»pilt oC parallel ^a^s^ fells upoi^i, a^ glass wtoe two 
 
% the iRev, Dr. Maskelyne. 1 1 1 
 
 surfaces are inclined to one another, and some of the rays are reflected 
 at the fore-surface, and others passing into the glass and suffering a 
 reflection at the back-surface and two refractions at the fore-surfac6 
 emerge again from the glass, these latter rays will not be parallel t6 
 those reflected at the fore-surface, as they would have been if the 
 surfaces of the glass had been parallel, but will be inclined to the 
 same. I find that the angle of their mutual inclination, which may 
 be called the deviation of the rays reflected from the back-surface, 
 will be to double the inclination of the surfaces of the glass (which is 
 here supposed to be but small), as the tangent of the angle of inci- 
 dence Out of air into glass, is to the tangent of the angle of refraction. 
 Hence, in rays falling near the perpendicular, the deviation will be 
 about three times the inclination of the surfaces ; and if the angles 
 of incidence be 50°, 6o°, 70°, 80° or 85°, the deviations of the re- 
 flected rays will be about 4, 5, 7, 13, or 26 times the inclination of 
 the surfaces, respectively. Had the deviation been the same at all 
 incidences of the rays on the index-glass, no error would have been 
 produced in the observation ; because the course of the ray would 
 have been equally affected in the adjustment of the instrument, as in 
 the observation. But, from what has been just laid down, this is far 
 from being the case, the deviation increasing according to the obli- 
 quity with which the rays fall upon the index-glass ; so that in very 
 oblique incidences of the rays, such as happen in measuring a large 
 angle by the fore-observation or a small angle by the back-observation, 
 the least defect in the parallelism of the planes of the two surfaces oi 
 the index glass may produce a sensible error in the observation. 
 What is here said only takes place in the fullest extent, if the 
 
112 ■" Remarks on the Hadler/s Quadrant, 
 
 thickest or thinnest edge of the index -glass, or, to express the same 
 thing in other words, the common section of the planes of the 
 surfaces of the index-glass stands perpendicular to the plane of the 
 quadrant; but, if the common section of the planes is inclined to 
 the plane of the quadrant, the error arising from the defect of the 
 parallelism of the surfaces will be lessened in the proportion of the 
 sine of the inclination to the radius ; so that at last, when the 
 common section becomes parallel to the plane of the quadrant, the 
 error .entirely vanishes. For this reason : Mr. Hadley very properly 
 directed the thickest and thinnest edges of the index-glass to be placed 
 parallel to the plane of the quadrant. But as it may well be ques- 
 tioned whether this care is always taken by the instrument-maker, and 
 it cannot be supposed that the glasses can be ground perfect parallel 
 planes, it would certainly be an advantage acquired to the instrument, 
 could the error arising from a want of parallelism of the planes be 
 removed in whatever position the common section of the planes 
 should be placed with respect to the plane of the quadrant. This will 
 be effected for celestial observations, if the upper part of the index- 
 glass be left unsilvered on the back, and made rough and blacked, the 
 lower part of the glass being silvered as usual, which must be covered 
 whenever any celestial observations ar© made. Then, if the teles- 
 cope be sufficiently raised above the plane of the quadrant, it is 
 evident that the observations will be made by the rays reflected from 
 the fore-surface of the upper part of the index-glass, and consequently, 
 if the quadrant be adjusted by making use of the same part of the 
 index-glass, the observations will be true, whether the two surfaces of 
 the index-glass be parallel planes or not. The sun or moon may 
 
hy the Rev, Dr. Mashelyne, ,113 
 
 be thus observed by reflection from the unsilvered parts of the index- 
 glass and horizon -glass, so that a paler darkening glass will suffice, 
 and they will appear much distincter than from an index-glass wholly 
 silvered with a deep darkening glass ; for although the surfaces of a 
 glass may be parallel, yet there always arises some little confusion from 
 the double reflection. Neither will the moon appear too weak by two 
 unsilvered reflections, even when her crescent is very small,- except 
 she should be hazy or clouded; and then the light may be increased 
 by lowering the telescope so as to take in part of the silvered reflection ^ 
 of the index-glass, which in this case must be uncovered: the same is 
 also to be understood with respect to the sun, should his light be too 
 much weakened by haziness or thin clouds. The horizon-glasses 
 should be adjusted, or the error of adjustment found by the sun or 
 moon; the first will be in general the best object for the purpose; 
 and, as the sun or moon seen directly through the unsilvered part of 
 the horizon-glass will be much brighter than the image of the same 
 seen by two unsilvered reflections, it must be weakened by a darken- 
 ing glass placed beyond the horizon-glass, the reflected image being 
 farther weakened, if necessary, by a paler darkening glass placed in 
 the usual manner between the index-glass and the horizon-glass. 
 
 If a quadrant was designed principally for taking the distance of 
 the moon from the sun and fixed stars, and v/as not wanted for ob- 
 serving terrestrial angles, it would be the best way to have none of 
 the glasses silvered, but to leave the horizon-glasses intirely trans- 
 parent, and to put a red glass for an index-glass of the same matter 
 with the darkening glasses, which would reflect light irom the fore- 
 surface only. ' • 
 
 p 
 
314 ReTTiarks on the Hadleifs Quadrant y 
 
 The sun's altitude might also be observed with this instrument, 
 either by the fore or back-observation; and the altitude of the moon 
 might be taken vi^ith it in the night. But the altitudes of stars could 
 not be observed with it, nor the moon's altitude in the day time, which 
 would however be no great inconvenience, as these observations might 
 be well enough supplied by common quadrants. , 
 
 The following rules for the size of the glasses and the silvering 
 them, and the height of the telescope may be of use. The index 
 glass and two horizon-glasses should be all of equal height, and even 
 with one another in height both at top and bottom. The telescope 
 should be moveable parallel to itself nearer to or farther from the 
 plane of the quadrant, and the range of its motion should be such 
 that its axis when at the lowest station should point about -tV^h of an 
 inch lower than the top of the silvering of the horizon-glasses, and 
 when at the highest station should point to the height of the middle 
 of the unsilvered part of the index-glass. The height of the glasses, 
 and the quantity of parts silvered and parts unsilvered, should vary 
 according to the aperture of the object-glass, as in the following ta- 
 ble ; where the first column of figures shews the dimensions in parts 
 of an inch answering to an aperture of the object-glass of T2_.ths of 
 an inch in diameter; the second column what answer to an aperture 
 of the object-glass of T^ths of an inch in diameter; and the third, 
 tvhat are suitable to an aperture of the object-glass of ^^g-ths of an 
 inch in diameter. 
 
hy th& Rev, Dr. Maskelyne. 
 
 115 
 
 Diameter of aperttu'c of obj ect-glasss • • • • 
 
 Height of glasses 
 
 Height of silvered part of index glass 
 
 Height of unsilvered part of di tto 
 
 Height of silvered part of horizon-glasses 
 Height of unsilvered part of ditto 
 
 Pafts of an Inch. 
 
 ,30 0,40 
 
 ,90 1,13 
 ,50 0,63 
 ,40 0,50 
 ,25 0,33 
 ,6510,80 
 
 0,50 
 
 1,37 
 0,77 
 0,60 
 0,42 
 0,95 
 
 If the telescope has a common object-glass, the first aperture of 
 Wths of an inch will be most convenient; but if it has an achromatic 
 object-glassj one of the other apertures of v^ths or -i^ths of an inch, 
 will be most proper. The field of view of the telescope should be 5 
 or (5 degrees, and the objects should be rendered as distinct as possible 
 throughout the whole field, by applying two eye-glasses to the teles- 
 cope. The breadth of the glasses should be determined as usual, 
 according to the obliquity with which the rays fall on them and the 
 aperture of the object-glass. 
 
 I shall conclude this paper with some easy rules for finding the 
 apparent angular distance between any two near land objects by the 
 Hadley's quadrant. 
 
 To firid the angular distance between two near objects by the fore- 
 observation. Adjust the fore-horizon-glass by the object intended to 
 be taken as the direct-object; and the angle measured by the fore- 
 observation on the arch of the quadrant between this object and any 
 other object seen by reflection will be the true angle between them 
 as seen from the centre of the index-glass. But, if the quadrant be 
 already well adjusted by a distant object, and you do not chuse to alter 
 it by adjusting it by a near one, move the index, and bring the image 
 
 p 2 
 
Il6 Remarks on the Hadlmfs Quadrant, 
 
 of the near direct object to coincide with the same seen directly, 
 and the number of minutes by which (o) of the index stands to the 
 right hand of (o) of the quadrant upon the arch of the excess is the 
 correction, which added to the angle measured by the arch of the 
 quadrant between this direct object and any other object seen by re- 
 flection will give the true angular distance between them reduced to 
 the centre of the index. 
 
 To Jind the angular, distance between two near objects by the back- 
 observation. 
 
 It is supposed that the horizon-glass is truly adjusted ; if it is riot, 
 let it be so. Observe the distance of the objects by the back-observa- 
 tion, and take the supplement of the degrees and minutes standing 
 upon the arch to 180 degrees, which call the instrumental angular 
 distance of the objects; this is to be corrected as follows. Keep the 
 centre of the quadrant or index-glass in the same place as it had in 
 the foregoing observation, and observe the distance between the near 
 object, which has been just taken as the direct object, and some dis- 
 tant object, twice; by making both objects to be the direct and 
 reflected ones alternately, holding the divided arch upwards in one 
 case and downwards in the other, still preserving the place of the cen- 
 tre of the quadrant. The difference of these two observations will 
 be the correction, which added to the instrumental angular distance, 
 found as above in the first observation between the first object and 
 any other object seen by reflection, will give the true angular distance 
 between them reduced to the centre of the index glass. 
 
by the Rev, Dr. Mashetyne, 117 
 
 I" 
 
 But if you should happen to be in a place where you cannot com- 
 mand a convenient distant object, the following method may be used. 
 
 The back- horizon-glass being adjusted, find the instrumental an- 
 gular distance between the objects; this is to be corrected by means 
 of the following operations. Set up a mark at any convenient dis- 
 tance opposite or nearly so to the object which has been taken as the 
 direct object; and looking at the direct object move the index of the 
 quadrant, and bring the image of the mark to coincide with the di- 
 rect object, and read off the degrees and minutes standing on the 
 arch of the quadrant, which substract from 180 degrees, if (O) of 
 the index falls upon the quadrantal arch; but add to 180 degrees, if 
 it falls upon the arch of excess; and you will have the instrumental 
 angular distance of the object and mark. Invert the plane of the 
 quadrant, taking care at the same time not to change the place of its 
 centre, and looking at the same direct object as before, move the in- 
 dex of the quadrant, and bring the image of the mark to coincide 
 again with the direct object, and read off the degrees and minutes 
 standing on the arch, and thence also find the instrumental angular 
 distance of the object and mark. Take the sum of this and the for- 
 mer instrumental angular distance; half of its difference from 36o° 
 will be the correction, which added to the instrumental angular dis- 
 tance first found between the same direct object and the other object 
 seen by reflection will give the true angular distance between them 
 reduced to the centre of the index-glass. 
 
 It is to be observed, that if the mark be set up at the same distance 
 from the quadrant as the direct object is, there will be no occasion to 
 invert the plane of the quadrant, but the observer need only make 
 
11§ Remarks on the Hadlei/s Quadrant^ i^c. 
 
 the image of the mark coincide with tlae direct object, then turn him-' 
 self half round, and, now taking the mark for the direct object cause 
 the image of the former direct object to coincide with the mark, the 
 divided arch of the quadrant being kept upwards, and the place of 
 the centre of the quadrant remaining also the same; in both cases: 
 half the difference of the sum of the two instrumental angles from 
 360° will be the correction of the adjustment as before. 
 
 Should only one of the objects be near, and the other remote (that 
 is to say, half a mile distant or more) let the distant object be taken 
 for the direct one, and the near object for the reflected one; and the 
 true distance of the objects as seen from the centre of the index-glass 
 will be obtained without requiring any correction, whether it be the 
 back or fore-observation that is made use of; only observing, as usual, 
 to take the supplement of what is shown upon the arch to 180° in 
 the back-observation. 
 
m 
 
 An Account of an Apparatus applied to the 
 Equatorial Instrument for correcting the Errors 
 arising from the Refraction in Altitude* By 
 Mr, Peter DoUond, Optician; commvmcated to 
 the Royal Society hy the Astronmner Royal, 
 
 Read March 4, 1779. 
 
 X HE refraction of the atmosphere occasions the stars or planets to 
 appear higher above the horizon than they really are; therefore, a 
 correction for this refraction should be made in a vertical direction to 
 the horizon. 
 
 The equatorial instrument is so constructed, that thd correction 
 cannot be made by the arches or circles which compose it, when the 
 star, &c. is in any other vertical arch except that of the meridian ; 
 because the declination arch is never in a vertical position but when 
 the telescope is in the plane of the meridian. 
 
120 jiccount of an Apparatus applied (o the Equatorial Instruinent, 
 
 To correct this error, a method of moving the eye-tube which 
 contains the wires of the telescope in a vertical direction to the hori- 
 zon has been practised; but as the eye -tube is obliged to be turned 
 round in order to move it in that direction, in the different oblique 
 positions of the instrument, the wires are thereby put out of their 
 proper situation in every other ^position of the instrument, except 
 when it is in the plane of the meridian; for the equatorial wire 
 should always be parallel to the equator, that the star in passing over 
 the field of the telescope may move along with it, otherwise one can- 
 not judge whether the telescope be set to the proper declination, 
 except at the instant the star is brought to the intersection of the 
 wires, which is only a momentary observation. 
 
 The method I have now put in practice for correcting the refraction 
 of the atmosphere is, by applying two lenses before the object-glass of 
 the telescope; one of them convex, and the other concave; both 
 ground on spheres of the same radius, which in those I have 
 made is thirty feet. The convex lens is round, of the same diameter 
 as the object-glass of the telescope, and fixed into a brass frame or 
 apparatus, which fits on to the end of the telescope. The concave 
 lens is of the same width, but nearly two inches longer than it is 
 wide, and is fixed in an oblong frame, which is made to slide on the 
 frame that the other lens is fixed into, and close to it. These two lenses 
 being wrought on spheres of the same radius, , the refraction of the 
 one will be exactly destroyed by that of the other, and the focal 
 length of the object-glass, will not be altered by their being applied 
 before it: and if the centres of these two lenses coincide with each 
 
by Mr. Peter Dollond. 121 
 
 other, and also with that of the object-glass, the image of any object 
 formed in the telescope will not be moved' or suffer any change in its 
 position. But if one of the lenses be moved on the other, in the di- 
 rection of a vertical arch^ so as to separate its centre from that of 
 the other lens, it will occasion a refraction, and the image will change 
 its altitude in the telescope. The Quantity of fhf» rpfrartion will be 
 always in proportion to the motion of the lens, so that by a scale of 
 equal parts applied to the brass frame, the lens maybe spt to occasion 
 a refraction equ^l to the refraction of the atmosphere in any altitude. 
 If the concave lens be moved downwards, that is, towards the horizon, 
 its refraction will then be in a contrary direction to that of the at- 
 mosphere,- and the star will appear in the telescope as if no refraction 
 had taken place. 
 
 There is a small circular spirit level fixed on one side of the apparatus, 
 which serves to set it in such a position, that the centres of the two 
 lenses may be in the plane of a vertical arch. This level is also used 
 for adjusting a small quadrant, which is fixed to it, and divided into 
 degrees, to shew the elevation of the telescope when directed to the 
 star; then the quantity of refraction answering to that altitude may 
 be found by the common tables, and the concave lens set accordingly, 
 by means of the scale at the side, which is divided into half minutes, 
 and, if required, by using a nonius, may be divided into seconds. 
 
 It must be observed, that when a star or planet is but a few degrees 
 above the horizon, the refraction of the atmosphere occasions it to be 
 considerably coloured. The refraction of the lens acting in a contrary 
 direction would exactly correct that colour, if the dissipation of the 
 
 a 
 
122 jiccount of an jipparatiLS applied to the Equatorial Instrument ^ 
 
 r^ys of light were the same in glass as in air; but as it is greater in 
 ^lass than in air, the colours occasioned by the refraction of the at-, 
 mosphere will be rather more than corrected by those occasioned by 
 •the refraction of the lens. 
 
 The following is a drawing of the refraction apparatus, which may 
 serve to give a more clear idea of it. 
 
by Mr. Peter Dollond. 123 
 
 EXPLANATION OF THE PLATE. 
 
 AA. The circular brass tube, which fits on to the end of the telescope, 
 BB. The oblong concave lens in its frame, which slides over the fixed 
 
 convex lens. 
 c. The circular spirit level, which shews when the oblong lens is in 
 
 a vertical arch. 
 
 D. The quadrant to which the spirit level is fixed, for shewing the 
 
 angular elevation of the telescope. 
 
 E. The milled head fixed to a pinion, by which the whole apparatus is 
 
 turned round on the end of the telescope, in order to set the 
 oblong lens in a vertical arch. • ' 
 
 F. Another pinion for setting the quadrant to the angular elevation 
 
 of the telescope. By means of these two pinions the air bub- 
 ble must be brought to the middle of the level. 
 aa. Is the scale, with divisions answering to minutes and half minutes 
 of the refraction occasioned by the concave lens. 
 
 W. M. Thiselton, Printer, Goodge Street, London. 
 
m