*-' 
 
 1HRI 
 
 HB 
 
 flBB^B 
 
\A\ 5NAN V 
 
 OF THI: 
 
 UNIVERSITY OF CALIFORNIA. 
 
 OK 
 
 Mrs. SARAH P. WALSWORTH. 
 
 Received October, (894. 
 Accessions No* tf'JM. CL/v No. 
 
INSTITUTES 
 
 NATURAL PI 
 
 im&-. irtJ 
 
 THEORETICAL AND PRACTICAL. 
 
 HV WILLIAM ENFIELD, LL. D. 
 
 < jS - .. ] 
 
 
 iOMB CORI'.El 
 
 it'ftiL 
 
 CHANGE IN THE ORDER O 
 
 AND THE ADDITIO 
 
 I 
 
 VN APPEND!^ TO THE AS 
 
 SELECTED FBO.M 
 
 ;.a 
 
 \\ IWt's PKACTlCAIi A 
 
 UEL WE13RKR, A. M. A. A. S. 
 
 Late President of HarvarJ College. 
 
 THIRD AMERICAN EDITION, WITH IMPROVEMENTS. 
 
 Oranis philosophic difiicultas in eo versari videtur, ut a phaenomenis motuura investigemus vires nature, deindt 
 ub bis viribus demoiistremus phienoineiui relujua< Newton. 
 
 BOSTON : 
 
 PUBLISHED BY CUMMINGS AND HILLIARD, AT THE BOSTON BOOKSTORE, 
 NO. 1 CORNHILL. 
 
 Unirenity Press...HiIIiard and Metcalf. 
 1820. 
 
"tli \-fnr ot tin iiu.. (it-iidfji 
 -.: of a book, the ri-'bi w he. 
 
10 THT 
 
 11EV. JOSEPH 
 
 I OR O. _AWS, FELLOW OF THE nOYAL I 
 
 IS TE8TIMO.ST 01 iBSPJiCT 
 
 A CHAKACTEF EMINENTLY 
 
 (. DMPREHENSIVE AND ENLARGED VIEWS OT 1 SCIENCE, 
 ASSIDUOUS AND SUCCESS; JL RESE/ *^ : HES INTO N 
 AN ARDENT ^E 01 TRUTK, 
 T, ;>- - HE gj. 
 
 Ti OF 
 
 AN ACTIVE SPIRIT OF PHILANTHROPY, 
 THIS WORK 
 
 IS INSCRIBED 
 
 BY HIS AFFECTIONATE FRIEND, 
 
 AND OBEDIENT SERVANT, 
 
 WILLIAM ENFIELD. 
 
PREFACE. 
 
 JN OTHING can be an adequate apology fop obtruding upon the 
 world a new Elementary Work, in a branch of Science already well 
 understood, except the plea of utility. It is wholly upon this ground, 
 that 1 venture to submit the following Treatise to the public in- 
 spection. 
 
 The difficulty which I met with, in providing my Classes* with a 
 Text-book in Natural Philosophy, neither, on the one hand, materially 
 deficient in Mathematical Demonstration, nor, on the other, too copious, 
 or too abstruse, for the purpose of elementary instruction, first sug- 
 gested the idea of this work. And the apprehension that others may 
 have met with the same difficulty, induces me to make it public, in 
 hopes that it may be of some use to those who wish'to study, or to 
 teach, this science systematically. 
 
 To that class of readers who are satisfied with general views, this work 
 will be of little service. Sketches of philosophy, sufficiently compre- 
 hensive to answer their purpose, will easily be found. But the knowl- 
 edge, which is gathered up in this cursory manner, must unavoidably 
 be superficial, and will, in many particulars, be confused and inaccu- 
 rate. What Cicero, says of philosophy in general, is particularly true 
 of Natural Philosophy : Difficile est enim in philosophia pane a esse ei 
 nota, cui non sinl aut pleraque, aut omma.f It may be laid down as 
 an universal maxim, that there is no easy method of obtaining excel- 
 
 * In the Warrington Academy. { Tuscul. Quatst. II. 1. 
 
vi PREFACE. 
 
 JBB 
 
 lence. The small portion of learning, or science, which is to he ac- 
 quired hy the help of facilitating expedients, has heen justly compar- 
 ed to a temporary edifice built for a day.* It is as unreasonable to 
 hope to acquire knowledge without undergoing the labour by which it 
 is usually gained, as it would be to expect that an acorn will become 
 an oak, without passing through the ordinary process of vegetation. 
 All the knowledge of Natural Philosophy which can be acquired by 
 
 * 
 
 cursory reading, without the assistance of mathematical learning, 
 must consist in an acquaintance with leading facts and general conclu- 
 sions. To understand the manner in which the laws of nature have 
 been inferred from these facts, and to be able with certainty and pre- 
 cision to apply these laws to the explanation of particular phenomena, 
 necessarily requires a previous knowledge of the elements of Geome- 
 try, Trigonometry, the Conic Sections, and Algebra. A mechanic, 
 who should set about making a machine without the requisite tools, 
 would not act more absurdly, than a student who should attempt to 
 understand the science of Natural Philosophy without these helps. 
 A preceptor, who professes to teach this science in the easy and amus- 
 ing method of experiment alone, is an architect without his rule, 
 plumb-line, and compasses. 
 
 Facts are, it is true, the materials of science ; and much praise is 
 unquestionably due to those who have increased the public store, by 
 new experiments accurately made, and faithfully related. But it is 
 not in the mere knowledge, nor even in the discovery of facts, that phi- 
 losophy consists. One who proceeds thus far, is an experimentalist ; 
 but he alone, who, by examining the nature, and observing the relation 
 of facts, arrives at general truths, is a philosopher. A moderate 
 share of industry may suffice for the former : patient attention, deep 
 
 Knox on Liberal Education, 9. 
 
PREFACE. vii 
 
 reflection, and acute penetration, are necessary in the latter. It is 
 therefore no wonder, that amongst many experimentalists there should 
 be few philosophers. 
 
 The hardy perseverance, and the vigorous exertions, which are 
 necessary to form this character, are so contrary to that effeminacy 
 and frivolity which distinguish the present age, that, if it were not for 
 the provision which is made in our universities, and other seminaries, 
 for the propagation of sound learning of every kind, there would be 
 some reason to apprehend, that all the more abstruse and difficult 
 branches of science would be excluded from the modern system of 
 education, and consequently would fall into disesteem and neglect. 
 
 It is by no means the intention of this treatise to encourage the in- 
 dolent spirit of tlie times, by opening a bye-path to the Temple of 
 Philosophy. The known and beaten road is the safest and the best. 
 It has been with a view of assisting the student in his progress, that I 
 have attempted to arrange the leading truths of Natural Philosophy 
 in a perspicuous method, and to demonstrate them with conciseness ; 
 adding a brief description of experiments, adapted to illustrate and 
 confirm the propositions to which they are respectively subjoined. 
 
 Being more desirous to be useful than to appear original, I have 
 freely selected from a variety of authors such materials as suited my 
 design. Those who are conversant with this class of >vriters will per- 
 ceive that, amongst many others, 1 have made use of the works of 
 NEWTON, Keil, Winston, Gravesande, Cotes, Smith, Helslium, Rowning, 
 and lastly, Rulherforlh, whose arrangement I have in part adopted. 
 
 With respect to any inaccuracies or mistakes which may have es- 
 caped my attention, 1 must rely on that candour, which those who are 
 best acquainted with the extent and difficulty of this undertaking will 
 be most inclined to exercise. 
 
ADVERTISEMENT 
 
 TO THE SECOND LONDON EDITION, 
 
 BY THE EDITOR. 
 
 J.N laying before the public a new edition of " The Institutes of 
 Natural Philosophy, by the late Dr. Enfold? the Editor feels it incum- 
 bent on him to assure the reader, that he has endeavoured, as far as 
 was consistent with an elementary book, to avail himself of those ad- 
 vantages which the publication of new discoveries, and new works in 
 science, has afforded him ; and although the limits of an advertise- 
 ment will not allow him to particularize all the additions that will be 
 found interwoven with the various parts of the volume, yet it may be 
 expected that, in this place, some notice should be taken of the most 
 material of them ; and it is presumed that the following account will 
 be deemed sufficient for the purpose. 
 
 In the first book, the propositions on the divisibility of matter, and 
 the attraction of cohesion, are more fully discussed, and a very useful 
 corollary is drawn from that on the attraction of capillary tubes. To 
 the first and third propositions of the second book, considerable addi- 
 tions are subjoined ; and in the second chapter is inserted a new prop- 
 osition, from which, in conjunction -with others, are deduced many co- 
 rollaries and scholiums, connected with the remaining parts of the 
 book.* Several examples are also given in the two first sections of the 
 fifth chapter, which will be found useful to the young student, as illus- 
 trative of the theory of falling bodies. 
 
 In the third book is given, independently of the additions noticed 
 in the margin,! an important proposition on the specific gravities of 
 
 See Prop. A, (p. 13.) and Cors. Schols. &c. to Prop. 14, 17, 24, 26, 28, 30, 31, 36, 44, 46, 49, 52, 
 .53, 54, 57, und 58. 
 
 f See additions to Prop. 3, 6, 12, 13, 18, A, (p. 89) 50, and 55. 
 
ADVERTISEMENT. ^ ix 
 
 bodies, with which are connected examples, and a table of the compar- 
 ative weights of many of the most useful substances in nature. De- 
 scriptions, accompanied with figures, are likewise given of the Pyrom- 
 eter, Air-Pump, Barometer, with its application to the measuring of 
 altitudes, 6jc. Fahrenheit's Thermometer, with a Table of Heat ; dif- 
 ferent kinds of Hygrometers, the Steam-Engine, and the Hydrometer. 
 
 The principal additions to the book of Optics will be found con- 
 nected with the propositions mentioned below ;* in the course of 
 which are introduced Mr. Delaval's Theory of Colours ; brief ac- 
 counts of Dr. Blair's achromatic Lenses, and Dr. Herschel's grand 
 Telescope. 
 
 On the subject of Astronomy, are arranged under the different arti- 
 cles several useful Tables, and the important discoveries of the illus- 
 trious Dr. Herschel, which have been carefully selected from the last 
 twenty volumes of the Philosophical Transactions. The reference in 
 the margin,! will direct the reader to those propositions to which the 
 most material additions are subjoined. 
 
 Some valuable treatises, on Magnetism and Electricity, particularly 
 those of Mr. Cavallo, having appeared since the, original publication 
 of this volume, it was thought necessary very considerably to enlarge 
 this part of the work ; and it is hoped that the principal discoveries 
 in these brandies of science will now be found under their respective 
 heads. 
 
 By the suggestion of a friend, on whose judgment the public has 
 long placed great confidence, it has been deemed proper that the first 
 principles of chemistry should form a part of the present volume ;$ and 
 
 See Prop. 5, 13, 32, A, (p.!6r) 42, B, (p. 175) 61, 62, 66, 68, 69, 76, D, (p. 199) 94, 122, 128, 144, andi!45. 
 
 t See Def. 1, 12. Prop. 8, 16, 20, 32, 35, 39, 51, 57, 72, 78, 79, 83, 109, 116, 117, 118, 119, 120, 123, 136, 
 167, 168, 177, 179, A, B, (p. 323) 182, 183. 
 
 + It has been deemed best to omit the Introduction to Chemistry. As an elementary treatise, it has been 
 found defective, and as far as our information extends, it has not been generally used in those seminaries, 
 where these Institutes are taught, the Chemical Professors generally recommending their respective favourite 
 authors. 
 
 b 
 
ADVERTISEMENT. 
 
 although we have chiefly confined ourselves to the interesting discove- 
 ries of the philosophers, Black, Priestley, and Lavoisier, on Heat and the 
 Fictitious Airs, it is nevertheless presumed, that enough has been said 
 on these subjects to render the doctrines and introductory practice to 
 modern chemistry perfectly intelligible to any person who may be de- 
 sirous of farther prosecuting the study of this amusing and useful 
 science. 
 
 The reader ought to be apprized, that besides additions to the old 
 plates, two new ones are now given : one, as already noticed, accom- 
 panying descriptions of several pneumatic and hydraulic machines, 
 and the other containing figures relating to subjects in magnetism, 
 electricity, and chemistry. 
 
 It is hoped that the augmentations to the volume, although they 
 compose about one third of the whole work, will be found such as 
 ought, at this period, to be comprehended in an elementary book of 
 science ; and that the speculations of Dr. Herschel, towards the end 
 of the astronomical part, will not be considered as an exception : they 
 are at least the speculations of a great mind, and capable of exciting, 
 in every well-disposed heart, emotions of interest and exquisite pleas- 
 ure, inasmuch as they lead to the grandest and most sublime notions 
 of the great Author of the universe. 
 
 The editor will only add, that in the additions to this work, he has 
 uniformly aimed at conciseness ; and he will consider his exertions well 
 rewarded, if it be found, by a candid and discerning public, that he 
 has not sacrificed perspicuity to brevity, and that he has not omitted, 
 within his prescribed limits, any material article that might serve to 
 render the original work, in its present enlarged state, generally ac- 
 ceptable and useful. 
 
 May 14, 1799. 
 
ADVERTISEMENT 
 
 TO THE FIRST AMERICAN EDITION. 
 
 THE principal object in undertaking this American Edition of Dr. Enfield's Institutes 
 of Natural Philosophy was to supply our Colleges with & book, which is held in so high es- 
 timation for the use of Students in recitations to their Instructors. In some of these semi- 
 naries of learning Ferguson's Astronomy is recited, and in one or more of them after the 
 exercises in Enfield's Philosophy, exclusive of the Astronomical part. It was the opinion 
 of the Rev. Dr. Willard, President of Harvard College, and Professor Webber, that it 
 would be a valuable improvement in this part of Collcgial Instruction to substitute En- 
 field's Astronomy for Ferguson's, provided some additions were made, which should be 
 equally or more important than certain Articles in the latter, particularly those relating to 
 the Calculation and Projection of Solar and Lunar Eclipses, including the necessary Ta- 
 bles. On account of some particular circumstances, measures were not taken to make 
 the proposed additions till the printing was far advanced, and assurances given that it 
 would be finished within a short period. Hence it was extremely difficult to execute the 
 plan. Professor Webber, however, consented to attempt what then seemed to be practi - 
 cable, and what, it was hoped, might in some good degree answer the purpose. And there 
 appears to be a propriety in giving a particular account of the alterations, that have been 
 made. 
 
 The errors, which occurred, are corrected. And it was thought expedient to retain the 
 Introduction to the First Principles of Chemistry, but to annex it as an Appendix.* 
 
 A change is made in the order of the branches by inserting Magnetism and Electricity 
 between Pneumatics and Optics, instead of placing them after Astronomy ; as it was 
 thought a more natural and useful arrangement for a regular course of instruction, the 
 propriety of closing with Astronomy being particularly obvious. 
 
 In the Astronomical part, the alterations are comprised in a few particulars, the sub- 
 stance of which is almost entirely taken from Ferguson's Astronomy, as well as the figures 
 to which they refer, f 
 
 An Appendix of about 80 pages is subjoined to the Astronomical part, and constitutes 
 the most distinguishing and important peculiarity of this edition. It contains the most 
 useful Solar and Lunar Tables, and a Table of Logistical Logarithms, together witli their 
 explanation, and twenty-two Problems, illustrating their use and application, and exem- 
 plifying the Projection of Eclipses. ' These are selected from a work of Mr. Alexander 
 
 See note, page ix. 
 
 f The particular additions are the explanations of the circles of perpetual apparition and occultation un- 
 der Prop, XX. Scholium 2 to Prop LX Scholium to Prop. I.XXXIL Scholium to Prop. CXIV. Expla- 
 nation of the figures <.{ orbits of Satellites under Scholium to Prop. CXVI. Scholium to Prop. CXLIII 
 Cor. to Prop. CLXXIH. Cor. to Prop. ('LXXV A small alteration is made in the demonstration of Prop. 
 CX1X, The Scholium to Prop. CXIX. s.nA Scholium ,3 to Prop. CXX, are omitted. The figures, together 
 with four projections of eclipses, which are also added with the Appendix to the Astronomical part fill 
 two plates. 
 
xii ADVERTISEMENT. 
 
 Ewing, teacher of Mathematics at Edinburgh, entitled " Practical Astronomy," and pub- 
 lished in 1797. With respect to the Projection of Eclipses, there is a small alteration in 
 ' making two distinct Problems for the Lunar and Solar Projections, instead of placing 
 
 them under the respective Problems for calculating those Eclipses. And two notes are 
 added to the Problem for projecting a Solar Eclipse. The former contains the necessary 
 directions for a different mode of projection ; and the latter the Rev. President Willard's 
 method of finding the point on the sun's limb, where a Solar Eclipse begins ; the knowl- 
 edge of which point is of great importance to observers. 
 
 In the w hole execution of the work it has been the unfeigned endeavour of the editors to 
 merit the approbation of the public. 
 
 Boston, January, 1802. 
 
 ADVERTISEMENT 
 
 TO THE THIRD AMERICAN EDITION. 
 
 THE following work has been, we believe, for some years, a classic in every College iu 
 New England, but considerable complaints having lately been made of its incorrectness 
 and deficiency, it was expected that some other system would be brought forward to sup- 
 ply its place. None however has appeared, and in the mean time the former editions 
 have been taken up, and copies were continually called for, which could not he furnished, so 
 that the public necessity seemed absolutely to require another edition to be im media; ely put 
 to the press. Such corrections, however, have been made as the time permitted, and they 
 will be found much more considerable, both in number and importance, than those in any 
 former edition. The principal alterations occur in Book II, Props. 52, 57, and 80 ; B. Ill, 
 P. 16 and 51; B. IV, P. 4 j B. VI, P. 13, 21, 22, 23, 55,56, 61. and 147; B. VII, P. 7, 36, 
 39, 45, 60, 70 ; Prob. at the end of Chap. IV, P. 80, 115,136; Table at the end of Part I. P. 1 67, 
 and 170. In the Appendix, Explanation of Tables I, II, and VI, and the examples under 
 Problems 8 and 1 5.* Less considerable alterations occur in every part of the work, too nu- 
 merous to be particularized. These corrections have been made by the direction of JAMES- 
 DEAN, A. M. A. A. S. of Windsor, Vt. It is not pretended that the work even now is 
 rendered perfect. The articles of Electricity and Magnetism, especially, are much be- 
 hind the present state of science, and Physical Astronomy, also, is too concise and obscure ; 
 but even as it has hitherto been, it seems to be the almost unanimous opinion ofinstructere, 
 that it is better adapted to the state of science in this country than any other work extant ; 
 and the publishers can venture to assure the public that the present edition is very much 
 improved. 
 
 Boston, January, \ 820. 
 
 * See Errata. 
 
CONTENTS. 
 
 BOOK I. 
 
 OF MATTER. - - 1 
 
 BOOK IT. 
 
 OF MECHANICS, OR THE DOCTRINE OF MOTION. 
 
 CHAP. I. Of the General Laws of Motion, - 10 
 
 CHAP. II. Of the Comparison of uniform Motions, - 13 
 
 CHAP. III. Of the Composition and Resolution of Forces, 15 
 CHAP. IV. Of Motion as communicated by Percussion in Non-Elastic 
 
 and Elastic Bodies, - - - - 18 
 CHAP. V. Of Motion as produced by the Attraction of Gravitation, 23 
 . 1. Of the Laws of Gravitation in Bodies falling without Obstruction, ibid. 
 . 2. Of the Laws of Gravitation in Bodies jailing down inclined Planes, 26 
 . 3. Of the Pendulum and Cycloid, - 31 
 . 4. Of the Centre of Gravity, - - 39 
 CHAP. VI. Of Motion as directed by Certain Instruments called Mechan- 
 ical Powers, - - - ' . 44 
 CHAP. VII. Of Motion as produced by the United Forces of Projection 
 
 and Gravitation, - 51 
 
 .1. Of Projectiles, - - ibid. 
 
 . 2. Of Central Forces, - - 57 
 
 BOOK III. 
 
 OF HYDROSTATICS AND PNEUMATICS. 
 PART I. 
 
 OF HYDROSTATICS. 
 
 CHAP. T. Of the Weight and Pressure of Fluids, - ?'! 
 
 CHAP II. Of the. Motion of Fluids, - 79 
 
 . 1. Of Fluids passing through the Bottom or Side of a Vessel ibid. 
 
 . 2. Of Rivers, - .... 83 
 
 CHAP. HI. Of the Resistance of Fluids, 86 
 
 CHAP. IV. Of the Specific Gravities of Bodies, 89 
 
 Table of Specific Gravities, 95 
 
xiv CONTENTS. 
 
 PART. II. 
 
 OF PNEUMATICS. 
 
 CHAP. I. Of the Weight and Pressure of the Air, 96 
 
 CHAP. II. Of the Elasticity of the Air, " 99 
 
 Of the Syphon, . 106 
 
 Of the Syringe, Common Pump, and Forcing Pump, 107 
 
 Of the Condenser and Air-Pump, - - 108 
 
 Of the Barometer, - - . . . 109 
 
 Of the Thermometer, . - . Ill 
 
 Table of Heat, - . 112 
 
 Of the Hygrometer, - - ibid. 
 
 OJ the Steam Engine, - 118 
 
 Of the Hydrometer, ... - 114 
 
 BOOK IV. 
 
 OF MAGNETISM. - - 115 
 
 BOOK V. 
 
 OF ELECTRICITY, 127 
 
 BOOK VI. 
 
 OF OPTICS, OR THE LAWS OF LIGHT AND VISION. 
 
 CHAP. I. Of Light, .... - 147 
 
 CHAP. II. Of Refraction, - . -150 
 
 . 1. Of the Laws of Refraction, . ibid. 
 
 . 2. Of Images produced by Refraction, - - . - 167 
 
 CHAP. III. Of Reflection, 172 
 
 . 1. Of the Laws of Reflection, .... ibid. 
 
 . 2. Of Images produced by Reflection, . . 180 
 
 CHAP. IV. Of Vision, 183 
 
 . 1. Of the Laws of Vision, ..... ibid. 
 
 . 2. Of Vision as affected by Refraction, - - 190 
 
 . 3. Of Vision as affected by Reflection, - . . 196 
 
 CHAP. V. Of Colours, .... 202 
 
 . 1. Of the different Refrangibility of Light, - ibid. 
 
 . 2. Of the Rainbow, ...... 207 
 
 CHAP. VI. Of the Optical Instruments, - . . 215 
 
 . 1. Of Telescopes, - .... ibid. 
 
 . 2. Of Microscopes, ..... ggo 
 
 . 3. Of the Magic Lantern, 228 
 
 . 4. Of the Camera Obscura, ... S23 
 
CONTENTS. 
 
 BOOK VII. 
 
 OF ASTRONOMY. 
 PART I. 
 
 FTHE MOTIONS OF TUB HEAVENLY BODIES. 
 
 CHAP. I. Of the Solar System in General, - 
 
 CHAP. II. Of the Earth, - - - . 
 
 . 1. Of the globular Form of the Earth, and its diurnal Motion about 
 
 its Axis, and of the Appearances which arise from these, ibid. 
 
 . 2. Of the Annual Motion of the Earth round the Sun, - SJ30 
 
 . 8. Of Twilight, . . -247 
 
 4. Of the Equation of Time, 
 
 CHAP. III. Of the inferior Planets, Mercury and Venus, - - 
 
 CHAP. IV. (sf the superior Planets, Mars, Jupiter, Saturn, and the 
 
 Herschel, - 259 
 
 CHAP. V. Of the Moon, ..... sfig 
 
 . 1. Of the Variations in the Appearance of the Moon, - ibid. 
 
 . 2. Of Eclipses, . . g68 
 
 CHAP. VI. Of the Satellites of Jupiter, Saturn, and the Herschel, 277 
 
 CHAP. VII. Of Comets, - ... 281 
 
 CHAP. VI 11. Of the Sun, - .... 284 
 
 CHAP. IX. Of the Parallaxes, Distances, and Magnitudes of the 
 
 Heavenly Bodies, . - 285 
 
 PART II. 
 
 OF THE CAUSES Or THE CELESTIAL MOTIONS AND OF OTHER PHENOMENA. 
 
 CHAP. I. Of the Cause of the Revolutions of the Heavenly Bodies in their 
 
 Orbits, - 296 
 
 CHAP. II. Of the Lunar Irregularities, - . . 300 
 
 CHAP. III. Of the Spheroidical Form of the Earth, . 307 
 
 CHAP. IV. Of the Precession of the Equinoxes, - 309 
 
 CHAP. V. Of the Tides, - 311 
 
 PART III. 
 
 OF THE FIXED STARS. - 314 
 
 APPENDIX TO THE ASTRONOMY, 
 
 Containing Solar and Lunar Tables, with their explanation and use, and the Projection of Eclipses, selected 
 
 from " Swing's Practical Astronomy." 
 
 Explanation of the Tables, - - 329 
 
 Problems, shewing the use and application of the Tables, and Projection 
 
 of Edipses, . . 336 
 
 Solar and Lunar Tables, .... 861 
 
ERRATA. 
 
 Page 48, for Prop. LXII. read Prop. LVII. 
 
 68, line 28, for APN read A;;N. 
 
 82, 23, for AE read ED, and for ED read EB. 
 
 99, 8, for BL read BL". 
 
 329, 19, for 54| read 43J. 
 
 336, 26, 28, 33, and 36, for 71 7' read 71 7' 25". 
 
 28, for 4h. 44' 28" read 4h. 44' 29f ". 
 
 29, for Ih. 22' 40" read Ih. 22' 38|". 
 
 33, for + 4 44 28 read -f 4 44 29|. 
 
 34, for 28 25 read 28 26|. 
 
 38, for 81 45' read 81 45' 25" ; and for 5h. 27' read 5h.27f. 
 
 39, for Ih. 3* 40" read Ih. 3' 381". 
 
 349, 5, 13, and 27, for north" read ascending. 
 6, 13, and 2", for " south" read descending. 
 
 N. B. A correspondent, whose kindness has enabled us to correct many other errors, which appeared in 
 former editions of this work, furnished us several corrections for the example under Prob. VIII. page 
 341, and for the example under Prob. XV. page 345 ; but we reget that they were mislaid, till it was too late 
 for their insertion: 
 
BOOK I. 
 
 OF MATTER. 
 
 ^t 
 
 DEFINITION I. MATTER is an extended, solid, inactive, and moveable 
 
 substance. 
 
 4 
 
 SCHOLIUM. Extension and solidity are discovered to be properties of matter by tbe senses. 
 Both by the sight and touch we perceive material substances to have length, breadth and thick- 
 ness, that is, to be extended : and from the resistance which they make to the touch, we acquire 
 the idea, and infer the property of solidity. It is unnecessary here to inquire, whether solidity 
 necessarily supposes impenetrability. Natural Philosophy, being employed in investigating the 
 laws of nature by experiment and observation, and in explaining the phenomena of nature by 
 these laws, has no concern with metaphysical speculations, which are generally little more than 
 unsuccessful efforts to extend the boundaries of human knowledge beyond the reach of the human 
 faculties. 
 
 DBF. II. A body is any portion of matter. 
 
 COROLLARY. All bodies have some figure ; for, being portions of matter, they are finite, 
 and therefore bounded by lines either straight or curved. 
 
 PROPOSITION I. 
 
 Matter may be, and mere extension is, infinitely divisible, or capable of 
 being divided beyond any supposed division. 
 
 1. Any particle of matter, placed upon a plane surface, has an upper and a lower side, or a 
 part which touches, and another which does not touch the plane, and is therefore divisible. 
 
 2. Let CO, MD, be two parallel right lines, to which let AB be drawn perpendicular. In the Plate 1. 
 line MD, on one side of the perpendicular AB, take, at equal distances, the points E, F, G, H. F 'E' l ' 
 On the other side of AB, in the line CO, take any point C, and join CE, CF, &c. Each of the 
 
 lines CE, CF, Sec. will cut off a portion from AB : But whatever number of lines be drawn 
 
 1 
 
S OF MATTER. BOOK I. 
 
 in the same manner from C to MD produced, there will still remain a portion of AB not 
 cut off; because no line can be drawn from the point C to the line MD, which shall coincide 
 with CO : The line AB is therefore infinitely divisible. 
 
 Plate 1 *' ^ e t the right lines AC and GH be drawn perpendicular to the right line BF. In AC, 
 Fij*. 2. produced at pleasure, take any points C, C, &c. from which, as centres, with the distances 
 CA, CA, &c. describe arcs of circles KAL, NAO, &c. touching BF in the point A, and 
 cutting HG. The farther the central point is taken from A, the greater will be the circles, 
 and the nearer will the arcs approach to the line BF ; but (El. III. Pr. 16. Cor.) the arcs, 
 touching BF in A, cannot toucli it in any other point. The line HG is therefore infinitely 
 divisible. 
 
 From this proposition the following theorems are derived by Dr. Keill, 
 in his fifth lecture. 
 
 THEOREM i. Any quantity of matter, however small, and any finite space, however large, 
 being given, (as for example, a cube circumscribed about the orb of Saturn,) i is possible for 
 the small quantity of matter to be diffused throughout the whole space, and to fill it so that 
 there shall be no pore or interstice in it, whose diameter shall exceed a given finite line. 
 
 COR. Hence there may be given a body, whose matter, being reduced into a space 
 absolutely full, that space may be any given part of its former magnitude. 
 
 THEOREM n. There may be two bodies of equal bulk, whose quantities of matter, being 
 unequal, in any proportion ; yet the sum of their pores, or quantity of void space in each of 
 the two bodies shall be to each other nearly in a ratio of equality. Example. Suppose 1000 
 cubic inches of gold when reduced into a space absolutely full, to be equal to one cubic inch : 
 then 1000 cubic inches of water, which is 19 times lighter than gold, will, when reduced, contain 
 J-jth part of an inch of matter. Consequently the void spaces in the gold, will be to those in 
 water as 999 to 99'Jif , or nearly in the ratio of equality. 
 
 In the present state of knowledge, it is impossible to determine how far the division of matter 
 can be actually carried, or whether there be any indivisible atoms by the arrangement and 
 combination of which all sensible bodies are formed. We are, however, furnished both by art and 
 nature with many astonishing instances of minute division. If a pound of silver and a grain 
 of gold be melted together, the gold will be equally diffused through the whole silver ; and if a 
 grain of the mass, containing only the 5761th part of a grain of gold, be dissolved in aqua fortis, 
 the gold will subside. 
 
 A grain of gold may be spread by the gold-beater into a leaf containing 50 square inches, 
 awl this leaf may be divided into 500,000 parts : and by a microscope, magnifying the diameter 
 of an object 10 times, and its area 100 times, the 100th part of each of these, that is, the 50 
 millionth part of a grain of gold will be .visible. 
 
 The natural divisions of matter are still more wonderful. In odoriferous bodies a surprising 
 subtilty of parts is perceived : several bodies scarcely lose any sensible part of their weight in 
 a great length of time, and yet continually fill a very large space with odoriferous parti- 
 cles. Dr. Keill has computed the magnitude of a particle of assafcetida to be only the 
 
BOOK I. OF MATTER. 3 
 
 38 
 
 l,000,OUO,000,000,000,OUO th P art f a cubic inch A S a ' 1 "' Mr ' Leewenho <" k informs us that 
 there are more animals in the milt of a cod-fish than there are men on the whole earth, and a 
 single grain of sand is larger than four millions of these animals. Moreover, a particle of the 
 blood of one of these animalcula has been found, by calculation, to be as much smaller than a 
 globe of the -j^th of an inch in diameter, as that globe is smaller than the whole earth. 
 Nevertheless, if these particles be compared with the particles of light, they will be found to 
 exceed them in bulk as much as mountains do single grains of sand. 
 
 These instances may serve to shew the amazing fineness of the parts of bodies, which are 
 nevertheless compounded. Gold, when reduced to the finest leaf, still retains those properties 
 which arise from the modifications of its parts. Microscopic animalcula are, without doubt, 
 organized bodies, and the globules of their blood are possessed of specific qualities. Even the 
 rays of light are compounded of an indefinite variety of particles, which, when separated, have 
 the power of exciting the ideas of colours. 
 
 DBF. III. That force by which the parts of the same body, or of different 
 bodies, on their contact, or near approach, are united to or tend towards each 
 other, is called the attraction of cohesion. 
 
 PROP. II. The attraction of cohesion appears in solid bodies. 
 
 EXP. 1. Observe the different degrees of cohesion in different kinds of wood, suspending 
 weights, from pieces of equal diameter, placed vertically or horizontally, till they break. 
 
 2. Measure the different degrees of cohesion in silk, thread, horse-hair, &c. by weights 
 suspended from cords of each, placed vertically or horizontally. 
 
 The result of sundry experiments, made by professor Musscheubroek, to shew the cohesive 
 power of different solids, may be seen in the following table. In estimating the absolute 
 cehesion of solid bodies, he applied weights to separate them according to their length : the 
 pieces of wood which he used were parallelopipedons, each side of which was ^%-ths of an inch, 
 and the metal wires made use of were T Lth of a Rhinland inch in diameter, and they were 
 drawn asunder by the following weights : 
 
 lb. Ib. 
 
 Fir 600 Copper 299- 
 
 Elm - 950 Brass .... 360 
 
 Alder - - 1000 Gold 500 
 
 Oak - 1150 Iron .... 450 
 
 Beech - 1250 Silver ... 370 
 
 Ash - 1250 Tin .... 49^ 
 
 Lead 29* 
 
 From the experiments of Guy ton Morveau, the following are the utmost weights, which wires 
 of 0.787 of an English line in diameter can support without breaking. 
 
4 OF MATTER. BOOK I. 
 
 lb. Av. 
 
 A wire of Iron supports - - 549.25 
 
 Copper - - 302.278 
 
 Platinum - 274.32 
 
 Silver ... . 187.137 
 
 Gold - 150,753 
 
 Zinc - - 109.54 
 
 Tin - . 34.63 
 
 Lead - ... 27.621 
 
 PROP. III. The attraction of cohesion takes place between particles of 
 the same fluid. 
 
 Exp. 1. A drop of water, at the end of a small cylinder of wood, will hang in a spherical 
 form. The drop is spherical, because each particle exerts an equal power in every direction, 
 drawing other particles towards it on every side as far as its power extends. 
 - 2. Two globules of mercury, on meeting, unite, which union can arise only from their strong 
 attraction. Drops of water will do the same. 
 
 PROP. IV. The attraction of cohesion takes place between two solid 
 bodies of the same kind ; and the more perfect the contact, the greater is the 
 attracting force. 
 
 EXP. 1. Two glass bubbles, floating near each other on water, rush together. 
 
 2. A glass bubble floating on water in a glass vessel, moves towards the side of the vessel. 
 
 3. Two circular pieces of cork, placed upon water, and brought near each other, will be 
 attracted. 
 
 4. Two plates of glass laid together, though perfectly dry, will cohere. 
 
 5. Two leaden balls, having each a flat surface of a quarter of an inch in diameter, scraped 
 smooth, on being forcibly put together, will cohere so strongly as sometimes to require a weight 
 of nearly 1001 b. to separate them. 
 
 6. Two polished plates of brass, smeared with oil, will cohere strongly. 
 
 M. Musschenbroek found that the adhesion of polished planes, about two inches in diameter ? 
 heated in boiling water, and smeared with grease, required the following weights to separate 
 them : 
 
 Cold Grease. Hot Grease. 
 
 Planes of Glass - ISOlb. SOOlb. 
 
 Brass - 150 800 
 
 Copper - 200 850 
 
 Marble - - - 225 600 
 
 Silver - - 150 250 
 
 Iron - - 300 950 
 
 PROP. V. The attraction of cohesion takes place between solids and 
 fluids. 
 
BOOK I. OF MATTER. 
 
 Exp. 1. A plate of glass, or metal, will retain drops of water, or mercury, when inverted. 
 
 2. If a plate of glass be in part immersed in a vessel of water, the water which lies contiguous 
 to the glass will rise above the level. 
 
 3. Water rises above its level between two parallel plates of glass at a small distance from 
 each other, and in a glass tube having a fine bore, called a capillary tube. 
 
 4. The fluid will rise between parallel plates, and in capillary tubes, in vacuo. Hence it 
 appears, that the ascent of fluids in capillary tubes is not owing to the pressure of the air. 
 
 5. Human blood will rise to a great height in a tube having an exceedingly fine bore. 
 
 6. Water will ascend in the cavities of sponge, sugar, and other porous bodies. 
 
 7. If a drop of oil be poured upon a plate of glass laid horizontally, and another plate of glass 
 be so placed as to meet the first plate at one edge, and be at such a distance from it at the 
 other, as just to touch the drop of oil; this drop, because its touching surface is continually 
 enlarging, will move, with increasing velocity, towards that edge. If the planes be lifted up on 
 the side where they meet, the motion will be retarded, stopped or reversed, according to the 
 degree of elevation. 
 
 8. The same phenomenon takes places in a tube of unequal bore. 
 
 9. A circular piece of ice, two inches and a half in diameter, exactly balanced and brought 
 to touch the surface of some mercury, will be so strongly attracted by the mercury, as to require 
 more than nine pennyweights, in the opposite scale, to restore the equilibrium. 
 
 10. A piece of wood having a smooth and plane surface, suspended from a beam and balanced, 
 on touching a surface of water, will be attracted; and it will require an additional weight in 
 the opposite scale to separate them. 
 
 SCHOLIUM 1. As it is by the attraction of cohesion that the parts of a body are kept 
 together ; so when a body is broken, this attraction is bnly overcome. Hence the reason of 
 soldering of metals ; gluing of wood, &c. Hence also may be explained why some bodies are 
 hard, others soft, and others fluid, which properties may result from the different figures of the 
 particles, and the greater or lesser degree of attraction consequent thereupon. Elasticity may 
 arise from the particles of a body, when disturbed, not being drawn out of each other's attraction ; 
 as soon, therefore, as the force upon it ceases to act, they restore themselves to their former 
 position. 
 
 SCHOLIUM 2. Solids are dissolved in menstruums from the particles of the solid being more 
 attracted to the fluid than to themselves. Precipitation arises from a like cause ; for if to the 
 solution of any solid in a fluid, some other solid or fluid be added, the particles of which are 
 attracted by the fluid with a greater force than those of the solid which was dissolved, the solid 
 falls to the bottom in a fine powder. Thus silver dissolved in aqua fortis is precipitated by 
 copper. 
 
 PROP. VI. The heights to which a fluid arises between parallel plates 
 of glass are inversely as the distances of the plates. 
 
 The absolute attractive force of the plates will always remain the same, whatever be the 
 distance of the plates. The same weight of fluid must, therefore, at different distances of the 
 
OF MATTER. BOOK 1. 
 
 plates, be supported. But the quantity of fluid supported can only continue the same, when 
 the height of the column supported is reciprocally as its base ; that is, when as much as the 
 height is increased the base is diminished, and the reverse. Now, the length of the base 
 remaining unvaried, the base can only be made greater or less, by increasing or diminishing 
 the distance between the plates. Therefore, the force, and the quantity of fluid supported, 
 remaining the same, the height will be greater as the distance of the plates is less, and the 
 reverse. 
 
 Let H, B, D, express the height, base and distance, when the plates are at any given distance, 
 and h, b, d, express the same when they are brought nearer : from what has been shewn, 
 II : h : : b : B ; but b : B : : d : D j therefore H : h : : d : D. 
 
 Ext. Let two parallel plates of glass be immersed, at different distances from each other, in 
 a vessel of coloured water. 
 
 PROP. VII. The suspension of the fluid, in capillary tubes, is owing to 
 the attraction of the ring of glass contiguous to the upper surface of the fluid.* 
 
 Every ring of glass below the surface attracts the water above it as much downwards, as it 
 attracts the water below it upwards, and consequently can contribute nothing towards the 
 support of the column : and the action of the lowest ring upon all the fluid in the tube, within 
 its surface of attraction, must either concur with the force of gravity to bring the fluid 
 downwards, or, acting upon it at right angles, can have no effect in suspending it within the 
 tube. The fluid therefore can only be supported by the ring of glass contiguous to its upper 
 surface, which, attracting upwards, opposes the action of gravitation by which the fluid endea- 
 vours to descend. This reasoning may be applied to the fluid raised between parallel plates 
 of glass. 
 
 EXP. Let a capillary tube be composed of two parts, the bore of one of which is wider than 
 that of the other : immerse its wider orifice in water, till it is filled to any height less than the 
 length of the wider part ; the fluid will only rise to the height to which it would rise if the tube 
 were throughout of the same bore with the wider part : but immerse the tube till the fluid enters 
 the smaller part, and the whole column will be suspended, provided its length do not exceed 
 that of the column which a tube of the smaller bore is capable of supporting. 
 
 Hence it is manifest, that the water is sustained by the attraction of the narrower part of the 
 tube, for the wider part could not sustain so long a column : it is also manifest, that it is 
 sustained by the ring contiguous to the upper surface ; for if it were sustained by the ring at the 
 lower surface, no reason could be assigned why this should now support the greater column in 
 both parts of the tube, when it was before only able to sustain a column which filled a part of 
 the wider tube. 
 
 Next, let the tube be inverted, and the water be raised into the lower extremity of the wider 
 part; when the suspended column is of greater length than that which a tube of the same bore 
 
 * This proposition has been disputed. Dr. Hamilton, in his second lecture, supposes that the suspension 
 arises from the attraction of the annulus lying just within the lower orifice of the tube. But Mr. Parkinson 
 rejects both suppositions, and concludes, that the fluid is sustained by the immediate attraction of the glass- 
 See Parkinson's Hydrostaticks, p. 39. 
 
BOOK I. OF MATTER. 
 
 with the wider part is capable of sustaining, it will immediately sink : whence it is manifest, 
 that the suspension of the column in this case depends upon the attraction of the wider part of 
 the tube ; for the narrower part could sustain a larger column : and also, that it is sustained by 
 the ring contiguous to the upper surface; for if it were sustained by the ring at the lower 
 surface, it has been seen that this ring could support a much longer column. 
 
 SCHOL. The reason why the narrower or wider ring sustains a column of the same length in 
 the unequal tube above described, as in a tube throughout of the same diameter as the upper 
 ring, is that the moving forces of the columns are in both cases the same ; as will be more fully 
 shewn hereafter. Book III. Pr. iv. Cor. 
 
 PROP. VIII. In capillary tubes, the heights to which the fluid rises are 
 inversely as the diameter of the bores. 
 
 The fluid being suspended (Prop. VII.) by the ring of glass contiguous to the upper surface, 
 and the distance to which the attracting force of glass reaches being unvaried ; the attracting 
 force which sustains the fluid will be as the number of attracting particles, that is, as the 
 circumference, or diameter of the ring, or of the tube. Let Q, q, then, represent the quantities 
 of fluid to be raised in two tubes of different bores ; D, d, the diameters of their bores ; and 
 H, h, the heights to which fluids rise in the tubes ; because Q, q, represent two cylinders of the 
 fluid, from the properties of the circle and cylinder (El. XII. 2. n, and 14.) Q : y : : DDK : ddh ; 
 and from the nature of this attraction, which is as the diameters of the tubes, Q : q : : D : d ; 
 therefore DDH : ddh : : D : d ; and consequently D : d : : h : H. 
 
 COH. From this proposition it appears, that in any glass capillary tube, the height to which 
 it will elevate water, and keep it sustained, multiplied into the diameter of the tube, is a given 
 quantity ; this is found by experiment to be .053 part of an inch : by means of this value the 
 diameter of a capillary tube being given, the height to which it will elevate water will be known, 
 for it will be equal to .053, divided by the diameter; thus suppose the dianwter is 5 \ of an inch, 
 the height to which the water will be elevated = .053 X 20 = 1.06. 
 
 EXP. Let two tubes of different bores be immersed in a vessel of coloured water ; it will be 
 found, that the water will rise as much higher in the smaller tube, as the diameter of its bore is 
 less than that of the larger tube. 
 
 PROP. IX. Between two glass plates, meeting on one side, and kept 
 open at a sin ill distance on the other, water will rise unequally ; and its 
 upper surface will form a curve, in which the heights of the several points 
 above the surface of the fluid will be to one another reciprocally, as their per- 
 pendicular distances from the line in which the plates meet. 
 
 Let AE be the surface of the fluid ; AF the line in which the plates meet ; HL the curve Plate 1. 
 formed by the surface of the raised fluid ; GB, 1C, KD, LE, perpendicular to AE, expressing Fl &- 3< . 
 the heights of the respective points G, I, K, L, in the curve, above the surface of the fluid, and 
 
8 OF MATTER. ^ BOOK I. 
 
 AB, AC, AD, AE, perpendiculars to AF, expressing the distances of the same points from the 
 line in which the plates meet: these heights and distances are reciprocally proportional. For let 
 the lines GB, 1C, KD, LE, represent pillars of fluid of an equal but exceedingly small breadth ; 
 those portions of the glass plates, which, by their attraction, support these pillars being equal, 
 will sustain equal quantities of fluid ; that is, the pillars will be equal. But the pillars may be 
 considered as parallelepipeds, which (El. XI. 34.) are equal when their bases and altitudes are 
 reciprocally proportional. And the bases, being equal in breadth, are as their lengths, that is, as 
 the intervals between the plates : and since the intervals continually increase as the distance 
 from the line in which the plates meet increases, these intervals, at the points 6, C, D, E, are 
 as their distances AB, AC, AD, AE. from the line AF. Since, then, the heights of the pillars 
 are reciprocally as the intervals, the heights GB, 1C, &c. are reciprocally as the distances 
 AB, AC, &c. This is the property of an hyperbola, whose assymptotes are AE and AF. 
 
 Exp. Let coloured water rise between two glass plates (their inner surfaces being first 
 moistened) meeting on one side according to the proposition. 
 
 PROP. X. Some bodies appear to possess a power the reverse of the 
 attraction of cohesion, called repulsion. 
 
 Exp. i. If apiece of iron be laid upon mercury, the surface of the mercury near the iron 
 will be depressed. 
 
 2. A fine needle laid upon water will swim. 
 
 5. Two circular plates of tinfoil being placed upon water, and pressed down by a small 
 additional weight upon their surface, repelling the water, will have a cavity round them : 
 but when they are brought near each other, they will rush together ; the re-action of the water 
 on the outer side of the plates being greater than the re-action on the inner side, where the two 
 cavities produced by repulsion are united. 
 
 4. Mercury, poured into a recurved glass tube, having the bore on one side exceedingly fine, 
 and on the other large, will not rise so high in the narrow, as in the wide bore : water will rise 
 higher. 
 
 5. Melted glass, dropped into water, forms globules with a stem, (called Prince Rupert's 
 drops) which on breaking the stem will burst with great violence, and fall into powder. 
 
 PROP. XI. All bodies on or near the surface of the earth tend towards 
 its centre, by the attraction of gravitation. 
 
 A stone, or other heavy body, let fall, will move towards the earth till it meet with some other 
 body to obstruct its course. And bodies move in lines perpendicular to the surface, because the 
 point to which they ultimately tend is the centre of the earth, and the line of direction produced 
 coincides with the radius, and is at right angles with the surface, which is nearly spherical. 
 Some bodies ascend, because they are acted upon by a force greater than the attraction of 
 gravitation, and in a contrary direction. Vapours, smoke, &c. do not descend, because they 
 are lighter than the air, and supported by it. 
 
BOOK I. OF MATTER. 9 
 
 EXP. 1. Smoke or steam will descend in an exhausted receiver. 
 
 2. Any boiling fluid being placed in a scale and balanced, the balance will be destroyed by 
 evaporation. 
 
 SCHOL. 1. When we speak of attracting powers, we do not attempt to explain their nature, 
 or assign their causes. Having derived general principles, or laws of nature, from phenomena, 
 we only give a name to these principles, in order to explain other appearances by them. 
 
 SCHOL. 2. The tendency of all bodies towards the earth really results from their tendency 
 towards the several parts of the earth. For by an experiment made by Dr. Maskelyne upon 
 the side of the mountain Schehallien, he found the attraction of that mountain sufficient to draw 
 the plumb-line sensibly from the perpendicular. See Phil. Trans. Vol. LXV. or Sir J. Pringle's 
 Discourses. 
 
BOOK H. 
 
 OF MECHANICS, 
 
 OB 
 
 THE DOCTRINE OF MOTION. 
 
 CHAP. I. 
 
 Of the General Laws of Motion. 
 
 PROPOSITION I. 
 
 J^ VERY body will continue in its state of rest, or of uniform motion in a 
 right line, until it is compelled, by some force, to change its state. 
 
 Any body at rest on the surface of the earth will always continue so, if no external force be 
 impressed upon it to give it motion, and if the obstacle which hinders the attraction of gravitation 
 from carrying it towards the centre be not removed. A body being put into motion by some 
 external impulse, if all external obstructions were removed, and the attraction of gravitation, 
 suspended, would move on for ever in a right line ; for there would be no cause to diminish the 
 motion, or to alter its direction. This cannot be fully established by experiment, because it is 
 impossible entirely to remove all obstructions ; but, since the less obstruction remains the 
 longer motion continues, it may be reasonably inferred, that if all obstacles could be removed, 
 motion once communicated to any body would never cease. 
 
 Exp. 1. A body at rest requires some degree of force to put it in motion : and when in 
 motion, it will continue to move longer on a smooth surface than on a rough one; instances of 
 which occur in the use of friction rollers ; in the exercise of skating, &c. 
 
 2. If a stone be whirled round in a string, on being set at liberty it will continue to move 
 with the force which it has acquired. 
 
 3. If a vessel containing a quantity of water be moved along upon a horizontal plane, the 
 water, resisting the motion of the vessel, will at first rise up in the direction contrary to that in 
 which the moving force acts : when the motion of the vessel is communicated to the water, it 
 will persevere in this state ; and if the vessel be suddenly stopped, resisting the change from, 
 motion to rest, it will rise up on the opposite side. In like manner, if a horse which was 
 standing still, suddenly starts forwards, the rider will be in danger of being thrown backwards; 
 if the horse stops suddenly, the rider will be thrown forwards. 
 
CHAP. I. OF THE LAWS OF MOTION. 11 
 
 SOHOL. This proposition suggests two methods of distinguishing between absolute and 
 apparent motions. (1.) Absolute motion, or change of absolute motion, mav sometimes be 
 distinguished from apparent, by considering the causes which produce them. When two bodies 
 are absolutely at rest, they are relatively so; and the appearance is the same when they are 
 moving at the same rate, and in the same direction : a relative motion, therefore, can only arise 
 from an absolute motion in one or both of the bodies, which (oy the Prop.) cannot be produced 
 but by force impressed. Hence, then, if we know that such a cause exists, and acts upon one 
 of the bodies, and not upon the other, we may conclude that the relative motion arises from a 
 change in the state of rest, or absolute motion of the former; and that with respect to the 
 latter, the effect is merely apparent. Thus when a person on board a ship observes the coast 
 receding from him, he knows the appearance arises from the motion of the ship upon which the 
 wind or tide is acting. 
 
 (2.) Absolute motion may sometimes be distinguished from apparent hy the effects produced, 
 A body in absolute motion endeavours to proceed in the line of its direction : if the motion be 
 only apparent, there is no such tendency. It is in consequence of the tendency to persevere 
 in a rectilinear motion, that a body, revolving in a circle, constantly endeavours to recede from 
 the centre. This effort is called a centrifugal force ; and as it rises from absolute motion only, 
 whenever it is observed, we are convinced that the motion is real. 
 
 Exp. Let a bucket, partly filled with water, be suspended by a string, and turned round 
 till the string is considerably twisted ; theik let the string untwist itself. At first the water 
 remains- at rest, but as it acquires the motion of the bucket, the surface grows concave to the 
 centre, and the water ascends up the sides,, thus endeavouring to recede from the axis of motion, 
 and this effect increases till the water and bucket are relatively at rest. When this is the case, 
 let the bucket be suddenly stopped and the absolute motion of the water will be gradually 
 diminished by the friction of the vessel ; and at length, when it is again at rest, the surface 
 becomes plane. Thus the centrifugal force does not depend upon the relative, but upon the 
 absolute motion, with which it begins, increases, decreases and disappears. 
 
 PROP. II. The change of motion produced in any body is proportional 
 to the force impressed, and in the direction of that force. 
 
 Effects are proportional to their adequate causes. If, therefore, a given force will produce a 
 given motion, a double force will produce the double of that motion. If a new force be impressed 
 upon a body in motion, in the direction in which it moves, its motion will be increased propor- 
 tionally to the new force impressed : If this force acts in a direction contrary to that in which 
 the body moves, it will lose a proportional part of its motion : If the direction of this force be 
 oblique to the direction of the moving body, it will give it a new direction. 
 
 EXP. Let one clay ball, suspended by strings, strike another clay ball suspended in the 
 same manner, at rest or in motion, it will communicate a degree of motion greater or less in 
 proportion to the force of the striking body : In the opposite direction, motion will be destroyed 
 in the same proportion. 
 
OF MECHANICS. BOOK II. 
 
 COK. Since the effect produced by two bodies upon each other, depends upon their relative 
 velocity, it will always be the same whilst this remains unaltered, whatever be their absolute 
 motions. 
 
 PROP. III. To every action of one body upon another, there is an equal 
 and contrary re- action : Or, the mutual actions of bodies on each other are 
 equal and in contrary directions, and are always to be estimated in the same 
 right line. 
 
 Whatever quantity of motion any body communicates to another, or whatever degree of 
 resistance it takes away from it, the acting body receives the same quantity of motion, or loses 
 the same degree of resistance in the contrary direction : the resistance of the body acted upon 
 producing the same effect upon the acting body, as would have been produced by an active force 
 equal to, and in the direction of, that resistance. 
 
 Con. 1. Hence it appears, that one body acting upon another, loses as much motion as it 
 communicates ; and that the sum of the motions of any two bodies in the same line of direction 
 cannot be changed by their mutual action. 
 
 Con. 2. This proposition will explain the manner in which a bird, by the stroke of its wings, 
 is able to support the weight of its body. For if the force with which it strikes the air below it, 
 is equal to this weight, then the re-action of the air upwards is likewise equal to it ; and the bird 
 being acted upon by two equal forces, in contrary directions, will rest between them. If the 
 force of the stroke is greater than its weight, the bird will rise with the difference of these two 
 forces : And if the stroke be less than its weight, then it will sink with the difference. 
 
 EXP. Let a clay ball in motion strike another equal to it at rest : The striking body will lose 
 half its quantity of motion, which will be communicated to the other body. 
 
 SOHOL. These three laws of motion may be illustrated by experiments, but their best confir- 
 mation arises from hence, that all the particular conclusions drawn from them agree with 
 universal experience. They were assumed by Sir Isaac Newton as the fundamental principles 
 of mechanics ; and the theory of all motions deduced from them, as principles, being found to 
 agree, in all cases, with experiments and observations, the laws themselves are considered as 
 mathematically true. 
 
 CHAP. II. 
 
 Of the Comparison of uniform Motions. 
 
 PROP. IV. The quantities of matter in bodies are in the compound ratio 
 of their magnitudes and densities. 
 
 If the magnitudes of two bodies be given, the quantities of matter will be as the densities : 
 If their densities be given, the matter will be as the magnitudes : therefore the matter is 
 
CHAP. II. COMPARISON OF MOTIONS. IS 
 
 universally in the compound ratio of the magnitudes and densities. For example ; If A and B 
 be two balls equal in r: -gnitude, the quantity of matter in A will be to that in B, as the density 
 of A is to that of B : if both be of the same density, their quantities of matter will be as their 
 magnitudes. 
 
 PROP. V. The velocities, with which bodies move, are directly as the 
 spaces they describe, and inversely aa the times in which they describe these 
 
 spaces. 
 
 * 
 
 It is manifest, that the degree of velocity increases as the space a body passes over in a given 
 time increases, and as the time in which it passes over a given space decreases ; and the reverse. 
 For example ; If one ball A move over six feet, and another ball B over three feet in the same 
 time, A has double the velocity of B : but if the ball A passes over six feet in two seconds of 
 time, and the ball B passes over six feet in one second, the velocity of B is double of that of A. 
 
 PROP. VI. The spaces which bodies describe are in the compound ratio 
 of their times and velocities. 
 
 It is evident, that the longer time any body continues to move, and the greater velocity it 
 moves with, the greater space it will pass through ; and the reverse. If, for example, the body 
 A moves for one second, and the body B moves for two seconds, both moving with the same 
 velocity ; A will move through half as much space as B : If A moves with two degrees of velocity, 
 and B with one degree of velocity ; A will, in the same time, pass over twice as much space 
 asB. 
 
 PROP. VII. The times in which bodies move are directly as the spaces, 
 and inversely as the velocities. 
 
 The greater space any body passes through, and the less degree of velocity it moves with, 
 the greater must be the portion of time taken up in the motion ; and the reverse. For example ; 
 If the ball A moves with the same velocity with the ball B, but passes over double the space, 
 A will move twice as long as B ; If A moves over the same space with B, and with half the 
 velocity, it must, in this case also, move twice as long as B. 
 
 PROP. A. If bodies be acted upon by different constant forces, the ve- 
 locities communicated will vary in a ratio compounded of the forces and times. 
 
 Let F, V, T, represent force, velocity and time, and be supposed variable ; it is evident that 
 the velocity will be increased and diminished in the same ratio with both force and time, and 
 these being independent of each other, V will be as F x T. 
 
 COR. If, therefore, F be compared with any other known force /capable of generating a 
 velocity equal to v in the time t, then V : v : : F x T :/x t. 
 
OF MECHANICS. BOOK II. 
 
 PROP. VIII. The power required to move a body at rest is as the 
 quantity of matter to be moved. 
 
 Each particle of matter in any body resisting motion, a force must be exerted upon each 
 particle to overcome this resistance ; if, therefore, two bodies containing different quantities of 
 matter are to be moved, the greater body will require the greater force. 
 
 DBF. I. The momentum of any body is its quantity of motion. 
 
 PROP. IX. In moving bodies, if the quantities of matter be equal, the 
 momenta will be as the velocities. 
 
 It is manifest, that if the body A be equal to the body B, but A has twice the velocity of B, 
 A has twice as much motion as B. 
 
 PROP. X. The velocities of two bodies being equal, their momenta will 
 be as their quantities of matter. 
 
 The bodies A and B moving with equal velocities, since every portion of matter in A has as 
 much motion as an equal portion of B, it is evident, that if A has twice the quantity of matter in 
 B, it must have twice as much motion. 
 
 
 
 PROP. XI. The momenta, of moving bodies are in the compound ratio of 
 their quantities of matter and velocities. 
 
 The greater quantity of matter there is in any body, and the greater velocity it moves with, 
 the greater will evidently be its quantity of motion ; and the reverse. If, for example, the 
 body A be double of the body B, and moves with twice its velocity, the momentum of A will be 
 quadruple of that of B: For it will have twice the momentum of B from its double velocity, 
 and also twice the momentum of B from its double quautity of matter. 
 
 COR. Hence, if in two bodies the product of the quantities of matter and velocities are equal, 
 their momenta are equal. 
 
 PROP. XII. The velocities of moving bodies are as their momenta 
 directly, and their quantities of matter inversely. 
 
 The greater momentum any body has, and the less quantity of matter it contains, the greater 
 must be its velocity. For example ; If the body A is half of'B, and their momenta are equal, 
 A will move with twice the velocity of B ; and if A and B are equal, and the momentum of A 
 is double of that of B, its velocity will also he double. 
 
 PROP. XIII. The force, or power of overcoming resistance, in any moving 
 body, is as its momentum. 
 
CHAP II. 
 
 COMPARISON OF MOTIONS. 
 
 15 
 
 Since a body having a certain degree of motion is able to overcome a certain degree of 
 resistance, it is manifest, that with an increased momentum, it will be able to overcome a greater 
 resistance. 
 
 COR. Hence the momentum of any body is measured by its power of overcoming resistance. 
 
 SCHOL. Let Q, q, denote the quantities of matter in any two bodies, D, d, their densities, 
 and B, I, their bulk or magnitude, V, t>, their velocities, T, (, the times of their motion, S, s, the 
 spaces over which they pass, P, p, the moving powers, M, m, their momenta, andF,/, their 
 force. The preceding propositions may be thus expressed : 
 
 PROP. IV. 
 
 Q 
 
 q :: 
 
 BD :bd 
 
 
 
 
 
 S s 
 
 
 V. 
 
 V 
 
 v :: 
 
 
 
 
 
 
 T ' 7 
 
 
 VI. 
 
 s 
 
 s : : 
 
 TV : tv 
 
 
 VII. 
 
 T 
 
 t :: 
 
 S s 
 
 
 
 
 
 V ' v 
 
 
 A. 
 
 V 
 
 v : : 
 
 FxT : /X 
 
 t 
 
 VIII. 
 
 p 
 
 p :: 
 
 Q 
 
 1 
 
 
 IX. 
 
 M 
 
 m : : 
 
 y 
 
 v 
 
 if Q=^. 
 
 X. 
 
 M 
 
 m :: 
 
 Q 
 
 1 
 
 ifV=v. 
 
 XI. 
 
 M 
 
 m : : 
 
 QV 
 
 qv 
 
 
 XII. 
 
 V 
 
 v : : 
 
 M 
 
 m 
 
 
 
 
 Q 
 
 7 
 
 
 XIII. 
 
 J * 
 
 M 
 
 m 
 
 
 CHAP. III. 
 
 Of the Composition and Resolution of Forces. 
 
 DBF. A. Equable motion is either simple or compound. Simple motion 
 is that which is produced by the action, or impressed force of one cause. 
 Compound motion is that which is produced by two or more conspiring 
 powers, i. e. by powers whose directions are neither opposite nor coincident. 
 
 PROP. XIV. A body acted upon by two forces united, will describe the 
 diagonal of a parallelogram, in the same time in which it would have 
 described its sides by the separate action of these forces. 
 
 Plate 1. 
 
 If in a given time, a body, by the single force M impressed upon it at a point A, would be Fig. 4.' 
 carried from A to B; and by another single force N impressed upon it at the same point, would 
 be carried from A to C ; complete the parallelogram ABDC ;, and with both forces united, the 
 body will be carried in the same time through the diagonal of the parallelogram from A to D. 
 
16 OF MECHANICS. BOOK II. 
 
 For since the force N acts in the direction of the right line AC parallel to BD, this force (by 
 Prop. II.) has no effect upon the velocity with which the body approaches towards the line BD 
 by the action of the force M. The^body will therefore arrive at the line BD in the same time, 
 whether the force N is impressed upon it or not ; and at the end of that time will be found 
 somewhere in the line BD. For the same reason at the end of the same time it will be found 
 somewhere in the line CD ; therefore it must be found at the point D, the intersection of these 
 two lines. And (by Prop. I.) it will move in a right line from A to D. 
 
 EXP. Two equal leaden weights, suspended at the end of a triangular frame of wood to give 
 them a steady motion, and let fall at the same instant from equal heights, striking a ball sus- 
 pended by a cord at the point in which their lines of direction meet, will carry it forwards in 
 the diagonal of the parallelogram of those lines produced. 
 
 COR. 1. Hence, the velocity produced by the joint action of two forces is to that with which 
 the body moves by the action of each force singly, as the diagonal of the parallelogram to either 
 side ; for the diagonal is described in the same time with either side. 
 
 COR. 2. If two sides of a triangle represent the directions and quantities of two forces, 
 the third side will represent the direction and quantity of a force equivalent to both acting 
 jointly: For the third side may be considered as the diagonal of a parallelogram. 
 
 COR. 3. A body may be moved through the same line by different pairs of forces. In plate 1. 
 fig. 4. AD is the diagonal both to the parallelogram ABDC, and to the parallelogram AEDF ; 
 and consequently expresses a force equal to AB, AC, and to AF, AE. 
 
 COR. 4. Hence we learn why any heavy body let fall perpendicularly from the top of a 
 mast, when a ship is under full sail, will fall to the bottom, in the same manner as if it had been 
 at rest. ^ 
 
 SCHOLIUM. This proposition may be farther illustrated. If two men sit upon the opposite 
 sides of a boat in full sail, and toss a ball to one another, they will catch the ball in their turn, 
 just as they would have done if the boat had been at rest. The ball is here acted upon by two 
 forces : (1.) it partakes of the motion of the boat, which is common to the ball, the boat and the 
 men : (2.) the other force is that with which the man throws it across the boat. By these two 
 forces together, the ball will describe the diagonal of a parallelogram, one of whose sides is the 
 line that the boat has described whilst the ball is flying across; and the other side is a line 
 drawn from one man to the other. 
 
 PROP. XV. The velocity produced by two joint forces, when they act 
 in the same direction, will be as the sum of the forces, and when they act in 
 opposite directions, will be as their difference ; and the velocity will be the 
 greater the nearer they approach to the same direction, and the reverse. 
 
 Plate 1. j n (|, e parallelograms ABCD, in which AB, AC, express the direction and quantity of two 
 joint forces, the side AB being placed at different angles with AC, it is manifest, that as AB 
 approaches towards AC, the diagonal increases, till at length it becomes equal to AC + CD, 
 that is, to AC -f AB, and the velocity is as the sum of the forces, since they act in, the same 
 direction. 
 
CHAP. III. COMPOSITION OF FORCES. 17 
 
 In the parallelograms ABDC, as AB recedes from CD, the diagonal decreases, till at length p l ate * 
 it vanishes with the angle, and the two sides AB, AC, constitute one right line, the parts of '"' 
 which, AB, AC, representing forces acting in opposite directions, if the forces be equal, they 
 will destroy each other; if unequal, the velocity will be as their difference. 
 
 PROP. XVI. Any single force or motion may be resolved into two forces 
 or motions ; and the directions of these may be infinitely varied : also any 
 two forces may be compounded into single forces. 
 
 A body moving in the line AD, maybe considered as receiving its direction and velocity Mate 1. 
 from two forces acting jointly in the directions AB, AC, or from two other forces expressed by l %' 
 AF, AE : For (Prop. XIV. Cor. 3) each pair would produce the same effect. In like manner 
 the direction and quantities of the forces will be diversified with every change of the sides of 
 the parallelogram, the diagonal remaining the same. 
 
 It is also manifest, that any two joint forces may be compounded into one, being expressed 
 by the sides of a parallelogram, or its diagonal. 
 
 PROP. XVII. If a body is acted upon by three forces, which are pro- 
 portional to, and in the direction of, the three sides .of a triangle, the body 
 will be kept at rest. 
 
 Let a body placed at D be acted upon by three forces AD, GD, FD, proportional to, and in Plate 1. 
 the direction of, the three sides of the triangle GED : complete the parallelogram GEFD ; and >s ' ' ,} 
 make AD eqjwl to, and in the direction of, the diagonal ED. 
 
 If the body at D be acted upon by the forces AD, ED, equal and in opposite directions, it 
 will be kept at rest. But the force ED (Prop. XVI.) is equivalent to the two forces DG, DF, 
 that is, DG, GE ; therefore the body acted upon by the three forces AD, DG, DF, that is by 
 three forces proportional to, and in the direction of, the sides of the triangle GED, will be at 
 rest. 
 
 EXP. Let three weights in the proportions of 3, 4, 5, be suspended from cords, which pass 
 over pullies and meet in a point ; if the directions of the cords be parallel to the sides of a 
 triangle (drawn in a plane parallel to the plane of the cords) whose sides are to each other as 
 the weights, a ball at the point in which the cords meet will be kept at rest. 
 
 COR. The body will be at rest if the three forces are proportional to the three sides of a 
 triangle drawn perpendicular to the direction of the forces ; for such a triangle is similar to the 
 former. Draw Ag, Cd, and Be, perpendicular to the sides GE, GD, DE, forming a triangle 
 ged, which is equiangular to GED ; hence, the sides about their equal angles being proportional, 
 the forces which are proportional to the lines GE, GD, and DE, are also proportional to ge, gd, 
 and de. 
 
 SCHOLIUM. A boy's kite, as it rests in the air, is an instance of a body resting whilst three 
 forces act upon it. For the kite is acted upon by the wind ; by its own weight ; and by the 
 string that holds it. 
 
18 OF MECHANICS. BOOK II. 
 
 PROP. XVIII. The force of oblique percussion is to that of direct or 
 perpendicular action, as the sine of the angle of incidence to radius. 
 
 Plate l. Let a body strike upon the plane AD, at the point D, in the direction Bl) : the line BD 
 
 Vis ' S ' expressing the force of direct impulse may be resolved into two others, BC, BA, the one 
 
 parallel, the other perpendicular to the plane. Of these, the force BC, parallel to the plane, 
 
 cannot affect it. The whole force upon the plane may therefore be expressed by BA. But BA 
 
 is to BD as the sine of the angle of incidence BDA is to radius. 
 
 SOHOL. If the surface to be struck be a curve, let AD be made tangent to the curve at D, 
 and the proof will be the same. 
 
 PROP. XIX. The force of oblique action produced by percussion is to 
 that of direct action, as the cosine of the angle, comprehended between the 
 direction of the force and that in which the body is to be moved, to radius. 
 
 Plate 1. L e t FD represent a force acting upon a body at D, and impelling it towards E ; but let DM 
 be the only way in which it is possible for the body to move. The force FD may be resolved 
 into two forces FG, FH, or GD ; of which only the force GD impels it towards M. And, FD 
 being radius, GD is the cosine of the angle FDG, or MDE, comprehended between the direction 
 of the force, and that in which the body is to be moved. 
 
 CHAP. IV. 
 
 Of Motion, as communicated by Percussion in Non-Elastic and Elastic Bodies. 
 
 DBF. II. Bodies are non-elastic, which, when one strikes another, do not 
 rebound, but move together after the stroke. 
 
 COR. Hence their velocities after the stroke are equal. 
 
 DBF. III. Bodies are elastic, which have a certain spring, by which their 
 parts, upon being pressed inwards by percussion, return to their former state, 
 throwing off the striking body with some degree of force ; when the elasticity 
 is perfect, the body restores itself with a force equal to that with which it is 
 compressed. 
 
 EXP. The existence of this property is visible in a ball of wool, cotton, or sponge compressed. 
 
 PROP. XX. When one non-elastic body in motion, strikes upon another 
 at rest, or moving with less velocity in the same direction, the sum of their 
 momenta remains the same after the stroke as before. 
 
 For (Prop. III. Cor. 1.) as much motion as the striking body communicates, so much it loses ; 
 
CHAP. IV. OF ELASTIC BODIES. 19 
 
 whence, if the motions of the bodies are in the same direction, whatever is added to the motion 
 of the preceding body will be subducted from that which follows, and the sum will remain the 
 same. 
 
 PROP. XXI. When two non-elastic bodies, moving in an opposite 
 direction, strike upon each other, the sum of their momenta, after the stroke, 
 will be equal to the difference of their momenta before the stroke. 
 
 For (from Prop. III. Cor. 1.) that body which had the least motion will destroy a quantity 
 equal to its own in the other ; after which they will move together with the remainder, that is, 
 the difference. 
 
 EXP. Let two cylinders filled with clay, A, B, be of equal weight, and suspended by cords 
 from equal heights ; let two other cylinders of the same kind, C, D, but in weight as 2 to 1, be 
 suspended from the same height. The heights from which they are let fall, in the arc formed 
 bv the motion of the cylinder (from the nature of the pendulum, afterwards to be explained) will 
 be the measure of their velocity; and (by Prop. XI.) their momenta will be as their velocities 
 multiplied into their quantities of matter; whence the cases of the two preceding propositions 
 may be established by the following experiments. N. B. Quantity of matter is expressed by q, 
 velocity by v, and momentum by m. 
 
 No. 1. Prop. XX. Case 1. Let the cylinder A fall from the height of 18 inches, upon the 
 cylinder B at rest. The momentum of A before the stroke (by Prop. XI.) is 18 ; for the quan- 
 tity of matter is 1, and the velocity 18; whence ql xcl8=ml8. After the stroke, the 
 quantity of matter being (Def. II.) 2, and the velocity of each cylinder 9, the momentum will 
 be 18 ; qZ x v9 = m 18. 
 
 No. 2. Case 2. Let A Jail from 18 inches, and B from 9, in the same direction ; their 
 momenta before the stroke are 18 4- 9 = B7 ; after the stroke, the quantity of matter will be 2, 
 and the velocity 1S-J ; whence v ISf x q 2 = m 27. 
 
 No. 3. Prop. XXI. Case 1. Let the equal cylinders A and B fall in opposite directions, 
 from the height of 12 inches; the momenta being equal and opposite, the motion of both will be 
 destroyed. 
 
 No. 4. Case 2. Let A fall from the height of 12 inches, and meet B falling in the opposite 
 direction from 6 inches ; their velocity after the stroke being 3, and quantity of matter 2, the 
 momentum will be6;<72xw3=m6. 
 
 No. 5. Prop. XX. Case 5. Let the cylinder C, double of the cylinder D, fall from 12 inches 
 on D at rest. Before the stroke, the quantity of matter in C is 2 and its velocity is 12 ; whence 
 its momentum is 24 ; q 2 X v 12 = in 24. After the stroke, the velocity will be 8, and quantity 
 of matter 3 ; whence q 3 X v 8 = m 24. , 
 
 No. 6. Case 4. Let C fall from 12 inches, and D from 6 inches in the same direction. 
 Before the stroke, the velocity of C is 12, and quantity of matter 2; whence its momentum is 
 24; q 2 x v 12 = m 24; and tiie velocity of D is 6, and its quantity of matter 1 ; whence 
 5! x v6 = wi6; therefore the whole momentum is 30. After the stroke, the velocity of the 
 whole is 10, and the quantity of matter 3 ; whence q 3 X v 10 = in 30. 
 
 No. 7. Prop. XXI. Case 3. Let C fall from 6 inches, and D from 12, in opposite directions, 
 the quantity of matter in C being 2, and its velocity 6 ; and the quantity of matter in D being 
 
20 OF*MECHAN1CS. BOOK II. 
 
 1, and its velocity 12, their momenta will be equal, and being opposite, will destroy each other. 
 C q 2 X v 6 = in 12 ; D q 1 X v 12 = ro 12. 
 
 No. 8. Case 4. Let C fall from 3 inches, and D from 12, in opposite directions : Before 
 the stroke, the momentum of C is 6; qZxv 3 = m 6, and the momentum of D is 12; 
 ql x v 12 = m 12 ; whence the difference of their momenta is 0. After the stroke, the velocity 
 is 2, and quantity of matter 3 ; whence the momentum is G ; q 3 x v 2 = m6. 
 
 PROP. XXII. When one elastic body strikes upon another of the same 
 kind, the one loses, and the other gains, twice as much momentum, as if the 
 bodies had been void of elasticity. 
 
 For, since (by Def. III.) perfectly elastic bodies, on percussion, restore themselves with a 
 force equal to that with which they are compressed, whatever momentum is gained by one bodj, 
 or lost by the other, on percussion, from the law of re-action, the same must be gained, or lost, 
 from the power of elasticity. 
 
 COR. I. Hence the momentum of elastic bftdies after percussion may be found, by doubling 
 the momentum which would have remained, had the bodies been non-elastic. 
 
 2. If one of the bodies, considered as non-elastic, would lose more than half its momentum, 
 as elastic, it loses more than all, that is, acquires a negative momentum in a contrary direction. 
 
 EXP. The following experiments may be made with ivory balls suspended from strings ; 
 they correspond with the preceding experiments on non-elastic bodies. 
 
 Let A and B be equal balls ; and let C be a ball double of the ball D. 
 
 No. 1. A, falling from 18 inches on B at rest, has 18 degrees of momentum before the 
 stroke ; therefore, after the stroke, supposing the balls non-elastic, the same momentum 
 belonging to the two equal balls together, each has 9 degrees of momentum, and A has lost and 
 B gained 9. This being doubled, A, as elastic, will lose 18, and B will gain 18 degrees of 
 momentum : whence A will be at rest, and B will move with 18 degrees of momentum. 
 
 No. 2. A, falling from 18 inches, and B from 9 in the same direction ; as non-elastic, after 
 the stroke, each has 13^ momentum, or A has lost 4|, and B gained 4|. As elastic, after the 
 stroke, A loses 9, B gains 9 ; therefore A rises to 9 inches, B to 18. 
 
 No. 3. A and B, falling in opposite directions from 12 inches, as non-elastic, would lose all 
 their momentum ; as elastic, each loses 24 degrees of momentum ; that is, gains 12 in the 
 contrary direction. 
 
 No. 4. A, falling from 12 inches, and B in the opposite direction from 6, as non-elastic, the. 
 momentum of each, after the stroke, will be in the direction of A ; whence A loses 9, and B 
 loses 9, moving 3 degrees in the contrary direction. As elastic, A loses 18, or has 6 in the 
 contrary direction, and B loses 18, or gains 12 in the contrary direction. 
 
 No. 5. C, double of D, falling from 12 inches on D at rest, the momentum of C, before the 
 stroke, being 24, and of D nothing ; as non-elastic, C, after the stroke, having its momentum 
 16, and moving with the velocity 8, will have lost 4 degrees of velocity, and 8 of momentum : 
 and D will have gained 8 of each. As elastic, therefore, C will lose 8 degrees of velocity, or 
 (Prop. XI.) 16 of momentum, and D will gain 16 of each ; that is, C will move with 4 degree* 
 of velocity, and D with 16 
 
CHAP. IV. OF ELASTIC BODIES. 21 
 
 No. 6. C, falling from 12 inches, and D from 6 in the same direction, before the stroke, the 
 velocity of C is 12, and its momentum 24 ; and the momentum of D 6. After the stroke, as 
 non-elastic, the momentum of C is 20, because q 2 x v 10 = m 20 ; and the momentum of D is 
 10, because q 1 X v 10 = m, 10 ; therefore C has lost 4 degrees of momentum, or 2 degrees of 
 velocity, and D gained 4 of each. If, therefore, the gain or loss be doubled on account of the 
 elasticity of C and D, C will lose 8 degrees of momentum, or 4 of velocity, ar.d 13 will gain 8 
 of each; that is, C will move with 8 degrees of velocity, and D with 14. 
 
 No. 7. C, falling from 6 inches, and D from 12, in opposite directions, their momenta being 
 equal, would destroy each other without elasticity: Therefore, being elastic, each will acquire 
 the momentum of 12 in opposite directions ; that is, D will return to 12, and C to 6. 
 
 No. 8. C, falling from 3 inches, and D from 12 in opposite directions; since the momentum 
 of C, before the stroke, is 6, and of D 12, as non-elastic bodies they would, after the stroke, 
 move in the direction of D, with the velocity of 2 ; whence C would move in the direction 
 contrary to its first motion with 4 degrees of momentum, and lose 10 ; and D would lose 10 : 
 Therefore, being elastic, C will lose 20 degrees of momentum, and also D 20 ; whence C will 
 move in the contrary direction with 14 degrees of momentum ; that is, will return to 7 ; and D 
 will return to 8. 
 
 COR. 1. If the sum of two conspiring momenta, or the difference of two contrary momenta, 
 be divided by the sum of the quantities of matter in both the moving bodies, the quotient will 
 give the common velocity after the stroke. 
 
 SCHOL. Let A and B be two spherical bodies, moving with their centres in the same line: 
 and let their velocities be a and b. The momentum of A, before the stroke is Act, and that of 
 B is B6 ; their sum or their difference, is Aa + B6, or Aa B6. Therefore (by Prop. XX. and 
 XXI.) the momentum, after the stroke, is expressed by Aa db B6, and, their common velocity by 
 
 AaB6 . AAartABfr 
 
 Hence the momentum of A, after jthe stroke, is - - ; and that of B is 
 
 ~~ 
 
 A + B 
 
 Next, suppole the bodies perfectly elastic. Subtract the momentum of A considered as uon- 
 
 clastic, after the* stroke, , from its momentum, before the stroke, Aa; and the 
 
 remainder, T- - will express the momentum in that case lost by A, and gained by B. 
 
 A -p -D 
 
 Subtract this remainder, T~T~R ' ^ rom ^ e momentum of A, as non-elastic, after the stroke, 
 
 AAoAB& ,-, r ABoiBBJi 
 
 - -- - ; and add the same remainder to the momentum of B, after the stroke, -- : 
 A -}- li A -}- B 
 
 AAa 2 AB6 ABa ... 
 
 the difference, - - - ^ - , will express the momentum of A, after the stroke, and 
 A -|- JtS 
 
 2ABo BB6=F AB6 
 
 the sum - - - ^- - will express the momentum or B, after the stroke, supposing 
 
 . .. . Aa2B& Ba , 
 
 them perfectly elastic. And - - -- jr - , and - ^ - , will express their respec- 
 
 ^ive velocities. 
 
 
S3 OF MECHANICS. BOOK II. 
 
 COR. 2. If there be any number of elastic, equal, and spherical bodies, whose centres are 
 placed in the same line, and the first body strikes upon the second in the direction of that line, 
 all the bodies will be at rest except the last, which will move oft' with the velocity of the first. 
 
 EXP. Several equal ivory balls, being so suspended as to have their centres in a right line, 
 if the first be let fall upon the second, the last will fly oft', to the height from which the first 
 fell. 
 
 COR. 3. When the striking ball is less than the quiescent, there will be an increase of 
 momentum. 
 
 EXP. Let the ball 1) fall from 12 inches upon C, double of D, at rest. If they were non- 
 elastic, they would proceed together, and their velocity, being the same, C, after [the stroke, 
 would have double the momentum of D; that is, C would have 8 degrees, and D 4; whence 
 D would have communicated more than half its momentum to C. The effect being doubled by 
 the elasticity of the bodies, U communicates to C 16 degrees of momentum, and loses as much 
 itself, or returns with 4 degrees of momentum in the contrary direction : while C moves for- 
 wards with 4 degrees more momentum than D had at the first. Thus the whole sum of 
 momentum is increased from 12 to 20 degrees ; but as much as the momentum is increased in 
 the direction in which D first moved, so much is given to D in the contrary direction. In 
 this manner may momentum be continually increased by a series of bodies. 
 
 COR. 4. If a non-elastic body strikes upon an immoveable obstacle, it will lose all its 
 motion ; an elastic body will return with a force equal to the stroke. 
 
 EXP. Let a leaden ball, and an ivory ball, strike upon any fixed plane. 
 
 CHAP. V. 
 
 Of Motion, as produced by the Attraction of Gravitation. 
 
 SECT. I. 
 
 Of the Laws of Gravitation in Bodies falling without Obstruction. 
 
 PROP. XXIII. The motion of a body, falling freely by the attraction 
 of gravitation, is uniformly accelerated, or its velocity increases equally in 
 equal times. 
 
 A new impression being made upon the falling body, at every instant, by the continued 
 action of the attraction of gravity, and the effect of the former (by Prop. I.) still remaining, the 
 velocity must continually increase. Suppose a single impulse of gravitation, in one instant, to 
 give it one degree of velocity ; if, after this, the force of gravitation were entirely suspended, 
 the body would continue to move with that degree of velocity, without being accelerated or re- 
 tarded. But, because the attraction of gravitation contiaues, it produces as great a velocity in 
 
CHAP V. OF FALLING BODIES. 33 
 
 the second instant as in the first; which being added to the first, makes the velocity in the 
 second instant double of what it was in the first. In like manner, in the third instant, it will be 
 tripled ; quadrupled in the fourth ; and in every instant, one degree of velocity will be added 
 to that which the body had before ; that is, the motion will be uniformly accelerated.* 
 
 COH. The velocities of falling bodies are as the times in which they are acquired. 
 
 PROP. XXIV. The force of the attraction of gravitation acting upon 
 any body is as its quantity of matter. 
 
 For each particle of matter in any body being acted upon by gravitation, the greater 
 number of particles are contained in any body, the greater force must be exerted upon it ; that is. 
 the force increases as the quantity of matter increases. 
 
 EXP. Let two unequal balls, suspended by threads of the same length, be let fall at the same 
 time from points equally distant from the lowest points of the arcs in which they move : The 
 vibrations of each will be performed in equal times, and consequently their velocities will be 
 equal ; whence the momenta (Prop. XI.) will be as the quantities of matter ; but (by Prop. XIII.) 
 the force producing motion, is as the quantity of motion, or momentum, produced : Therefore 
 the force of gravitation is as the quantity of matter ; that is, as much greater force is exerted 
 upon the larger body than upon the less, as its quantity of matter is greater than that of the less. 
 
 COR. 1. The weight of any body is as its quantity of matter; for weight is the degree of 
 force with which any body is acted upon by gravitation. 
 
 COR. 2. If the attraction of gravitation were increased in any ratio, the weight of a given 
 body would be increased in the same ratio. Substituting, therefore, W, Q, F, for the weight, 
 quantity of matter, and force of gravity, respectively, and supposing them to be variable ; W 
 will be as Q x F. 
 
 PROP. XXV. The velocities of bodies falling from the same height, 
 without resistance, are equal. 
 
 If two bodies of different quantities of matter fall from the same height, the attracting force 
 which acts upon the greater body, will (Prop. XXIV.) exceed that which acts upon the less, as 
 much as the greater body exceeds the less in quantity of matter; whence they must move with 
 equal velocities. 
 
 EXP. A guinea, and a feather, or other light body, in the exhausted receiver of an air-pump 
 will fall through the same space in the same time. 
 
 PROP. XXVI. The spaces described by falling bodies are as the 
 squares of the times from the beginning of the fall, and also as the squares 
 
 * All bodies descending- in vacn*jfer gravity, whether great or small, dense or rare, are found to fall 
 through 16.1 feet in one second, and to SKquire a velocity in falling which would carry them uniformly through 
 32.2 feet in the next second, and an increase of velocity, equal to this, is found to be added to every succeed- 
 ing second of time. 
 
84 OF MECHANICS. BOOK II. 
 
 of the last acquired velocities ; or in the ratio compounded of the times and 
 velocities. 
 
 Plate. 1. j n t) )c triangle ABC, let AB express the time in which a body is falling, and BC the velocity 
 which it has acquired at the end of the fall ; let AF, AD, be parts of the time AB ; and through 
 F, D, draw FG, DE, parallel to BC. 
 
 Because the triangles ABC, ADE, are similar, AB is to AD as BC to DE; but AB and AD, 
 express times of descent, and BC expresses the velocity acquired in the time AB ; therefore, 
 since (Prop. XXIII. Cor.) the velocities are as the times, DE expresses the velocity acquired 
 in the time AD. In like manner GF, a'ay other right line parallel to BC, expresses the velocity 
 acquired in the time AF. Therefore the sum of the lines which maybe supposed drawn parallel 
 to CB in the triangle ADE ; that is, the whole triangie ADE, will represent the sum of the 
 several velocities with which the falling body moves in the time AD. For the same reason, the 
 triangle ABC will represent the sum of the velocities with which the falling body moves in the 
 time AB. Since therefore it is manifest, that the space which a body passes through in any 
 moment of time is as the velocity with which it moves at that moment ; and consequently, that 
 the spaces through which it passes in any times whatsoever, are as the sums of the velocities 
 with which it moves in the several moments of those times; the spaces passed through in the 
 times AD, AB, are to each other.as the triangles ADE, ABC. But the triangle ADE (El. VI. 
 19.) is to the triangle ABC in the duplicate ratio of the homologous sides AD, AB, and also of 
 DE,BC; that is, the spaces are as the squares of the times, and also as the squares of the last 
 acquired velocities ; consequently the spaces described are in the compound ratio of the times 
 and the velocities. 
 
 EXP. Let there- be two pendulums, one of which vibrates twice as fast as the other, a ball 
 let fall from such a height above the ball of the shorter pendulum as to reach it in one vibration } 
 must fall from four times this height, to reach the longer pendulum in one of its vibrations. 
 
 COR. 1. Hence, if the forces are variable, the spaces described are as the forces and squares 
 of the times ; or as the squares of the velocities directly, and forces inversely. For by the Prop, 
 (calling S, V and T, the space, velocity and time) S is as T x V and (by Prop. A. p. 13.) V is as 
 
 F X T.-. S is as F x T 2 ; also, S is as ^- ; for T is as ^ .-. S is as Y X ^ or as 1! 
 
 r r . r r 
 
 COB. 2. The times in which bodies fall from unequal heights, and their last acquired 
 velocities, are as the square roots, or in the subduplicate ratio of their heights. Since TT is as 
 S, T will be as >/S ; and since VV is as S, V will be as >/ 
 
 COR. 3. If the time of the fall of a body be divided into equal parts, the spaces through 
 which it falls in each of these parts, taken separately, will be as the odd numbers 1, 3, 5, &c. 
 The spaces being as the squares of the times or velocities, if the times be as the numbers 1, 2, 
 5, 4, the spaces will be as 1, 4, 9, 16; whence, in the first time the space will be as 1, in the 
 second time, the space passed over will be as S, in the third, as 5, &c. 
 
 SCHOI/. Since S is as T 2 , and as in the first second of time a body freely descending by 
 the force of gravity falls through 16.1 feet, we easily find the space described in any given 
 number of seconds; for S = 16.1 x T 2 . Thus in 5" a body will fall through 402 feet; for 
 16.1 x 25 = 402. Again, the spaces fallen through in the 1st, 2d, 3d, seconds, are 16.1; 
 16.1 x 3 ; 16.1 x S respectively. 
 
CHAP. V. OF FALLING BODIES. So 
 
 PROP. XXVII. The space which a body passes over in any given 
 time from the beginning of the fall, is half that which it would pass over in 
 the same time, moving with the last acquired velocity. 
 
 For the triangle ABC (by Prop. XXVI.) expresses the space passed over in the time AB S!* te , J' 
 when the motion is uniformly accelerated ; the last acquired velocity is expressed by BC ; and 
 the rectangle of AB, BC, rightly expresses the space passed through in the time AB with the 
 equable velocity BC ; since therefore the triangle ABC is half of the rectangle AB, BC, the 
 preposition is manifest. 
 
 PROP. XXVIII. The motion of a body thrown upwards is uniformly 
 retarded by gravitation : the time of its rise will be equal to that in which a 
 body falling freely acquires the same degree of velocity with which it is 
 thrown up ; and the height to which it rises will be as the square of the time, 
 or first velocity. 
 
 The same force which accelerates a falling body, acting in an opposite direction upon one 
 thrown upwards, must retard it: and, since the action of gravitation is uniform, in whatever 
 time it generates any velocity in a falling body, it must in the same time destroy the same 
 velocity in a rising body : through whatever space the falling body must pass to acquire any 
 velocity, the rising body must pass through the same to lose it ; whatever ratio the spaces bear 
 to the velocities and times in one case, must take place in the other: the effect of gravitation in 
 rising bodies being in all respects the reverse of its effect upon falling bodies. 
 
 SCHOL. As the force of gravity near the surface of the earth is constant, and known by 
 experiment, and as the spaces described by falling bodies vary as the squares of the times, (T 2 ) 
 or as the squares of the velocities (V 2 ); hence every thing relating to the descent of bodies, 
 when accelerated by the force of gravity ; and to their ascent, when they are retarded by that 
 force, may be deduced from the foregoing propositions. 
 
 (I.) When a body falls by the force of gravity, the velocity acquired in any time, as T", is 
 such as would carry it uniformly over 2 F T in 1" ; where F = 16.1 feet. 
 
 EXAM. The velocity acquired by a falling body in 6" = 32.2 x 6, or such as would carry it 
 uniformly through 193 feet in 1". 
 
 V 2 
 (2.) The space fallen through to acquire the velocity V is . For S : F : : V* : 2F|* or 
 
 EXAM. If a body fall from rest till it acquire a velocity of 20 feet per second, the space 
 
 sot* 
 
 fallen through is - ' = 6.2 feet. 
 
 S\. I i 
 
 V* 
 
 From these three expressions, V =2FT ; S = and S = FT 3 (Cor. 1. Prop. XXVI.) 
 
 any one of the quantities S, T, V, being given, the other two may be found. 
 
 4 
 
26 OF MECHANICS. BOOK II. 
 
 EXAM. 1. To find the time in which a body will fall 400 feet ; and the velocity acquired. 
 
 Since S = FT* v T = | = -1^2 = 5" nearly, and V being equal 2 FT = 32.2 x 5" = 
 ^ F ^ 16. 1 
 
 161 feet = velocity acquired. 
 
 EXAM. 2. If a body be projected perpendicularly downwards, with a velocity of 20 feet pet- 
 second, to find the space desci%ed in 4". 
 
 The space described in 4'' by the first velocity is 4 x 20, and the space fallen through by the 
 action of gravity is 16.1 x 4 2 , therefore the whole space described is 337.6 feet. 
 
 EXAM. 3. To what height will a body rise in 3", which is projected perpendicularly upwards 
 with a velocity of 100 feet per second ? 
 
 The space described in 3" by the first velocity is 300 feet, and the space through which the 
 body would fall by gravity in 3" is 16.1 x 3 2 = 144.9 feet, therefore the height required is 
 300 144.9 = 155.1 feet. 
 
 SECT. II. 
 
 Of the Laws of Gravitation in Bodies falling down inclined Planes. 
 
 DBF. IV. An inclined plane is a plane which makes an acute or obtuse 
 angle with the plane of the horizon. 
 
 PROP. XXIX. The motion of a body, descending down an inclined 
 plane is uniformly accelerated. 
 
 In every part of the same plane, the accelerating force has the same ratio to the force of 
 gravitation acting freely in a perpendicular direction, and is therefore (El. V. 9.) equally 
 exerted in every instant of the descent; whence, (as was shewn concerning bodies falling freely, 
 Prop. XXIII ) the motion must be uniformly accelerated. 
 
 COR. Hence, whatever has been demonstrated concerning the perpendicular descent of bo- 
 dies, is equally applicable to their descent down inclined planes, the motion in both cases being 
 uniformly accelerated by the same power of gravitation. 
 
 PROP. XXX. The force, with which a body descends by the attrac- 
 tion of gravitation down an inclined plane, is to that with which it would 
 descend freely, as the elevation of the plane to its length ; or as the sine of 
 the angle of inclination to radius. 
 
 Plate 1. ' je * ^ ke *' ie ' en gth f an inclined plane, and AC its elevation, or perpendicular height. If 
 
 Fig. 11. the force of gravitation with which any body descends perpendicularly be expressed by AC, 
 
 and this force be resolved into two forces, AD. DC, by drawing CD perpendicular to AB ; 
 
 because the force CD is destroyed by the reaction of the plane, the body descends down the 
 
 inclined plane only with the force AD. And (El. VI. 8. Cor.) AD is to AC, as AC is to AB ; 
 
 that is, the force of gravitation down the inclined plane is to the same force acting freely, as the 
 
CHAP. V. 
 
 OF INCLINED PLANES. 
 
 elevation of the plane is to its length, or as the sine of the angle of inclination ABC is to the 
 radius AB. 
 
 COR. 1. Hence, the force necessary to sustain a body, on an inclined plane, is to the absolute 
 weight of a body, as the elevation of the plane to its length, for the force requisite to sustain a 
 body must be equal to that with which it endeavours to descend ; which has been shewn to be 
 to that with which it would descend freely, as the elevation of the plane to its length. 
 
 COR. 2. If H be the height of an inclined plane, L its length, and the force of gravity be 
 represented by unity ; the accelerating force on the inclined plane is represented by =-. For by 
 
 the Prop, the accelerating force is to the force of gravity (1) as IT is to L .-. the accelerating 
 
 H 
 force = -T-. 
 
 TT 
 
 COR. 3. Hence -j- varies as the sine of the angle of inclination. 
 
 COR. 4. If a body fall down an inclined plane, the velocity V generated in T" is such as 
 
 TT 
 
 would carry it uniformly over ^ x 2 F Tfeet in 1", where, as before, F is equal 16.1. 
 
 TT 
 
 For (by Prop. A. p. 13.) the velocity varies as the force and time, (i. e.) as j- X T, and the 
 
 |T 
 
 velocity generated by the force of gravity in one second is 2 F, therefore V = -p x 2 F T. 
 
 Ex. If L : H : : 2 : 1, a body falling down the plane will, at the end of 4", acquire a velocity 
 of | x S2.2 x 4 = 64.4 feet per second. 
 
 COR. 5. The space fallen through in T" from a state of rest, is ^ x FT, for (Prop. XXVI.) 
 the spaces described vary as the squares of the times. 
 
 Ex. 1. If H = , the space through which a body falls in 5" is x 16.1 X 25 = 201-J feet. 
 
 Ex. 2. To find the time in which a body will descend 40 feet down this plane. Since 
 
 , therefore T- 
 
 40 x 2 
 X lt>. 
 
 = 2.2 seconds. 
 
 COR. 6. The space through which a body must fall, from a state of rest, to acquire a velocity 
 
 T "Y'S V" 2 
 
 V, is =y x . For (Cor. 1. Prop. XXVI.) S is as , therefore thespace through which the body 
 H 4F V 
 
 falls by the force of gravity, is to the space through which it falls down the. plane, as the square 
 of the velocity directly, and as the force inversely in the former case, is to the same in the latter ; 
 and if F (16.1) be the space fallen through by gravity, 2 F is the velocity acquired in 1"; 
 L L V* 
 
 hence F : S : : 
 
 : jj X V* and s = JJ x 4F* 
 
 Ex. 1. If L = 2 H, and a body fall from a state of rest till it had acquired a velocity of 40 
 
 40 2 
 
 feet per second, the space described is X -j-j == 50 feet nearly. 
 
S8 OF MECHANICS. BOOK II. 
 
 Ex. 2. If a body fall 40 feet from a state of rest down this plane, to find the velocity 
 
 rr 
 
 * acquired. V 2 = 4FS X 7- = 64.4 X 40 X i = 1288, and V = 35.8 feet per second. 
 
 lj 
 
 PROP. XXXI. The space described in auy given time by a body 
 descending down an inclined plane, is to the space through which it would 
 fall perpendicularly in the same time, as the elevation of the plane to its 
 length. 
 
 Plate i. Let AC represent the force with which a body would fall perpendicularly ; CD being drawn 
 
 from C perpendicular to AB ; AD, as was shewn (Prop. XXX.) will represent the force with 
 which the body descends down the inclined plane AB. And, since the spaces through which 
 bodies fall in any given time must be as 'the forces which move them, the space through which 
 the body will fall down the inclined plane AB, is to that through which it will fall perpendicu- 
 larly in the same time, as the force AD, to the force AC. But AD is to AC (El. VI. 8. Cor ) 
 as AC the elevation to AB the length of the plane ; therefore the space through which the 
 body will fall in a given time down the inclined plane AB, will be to the space through which 
 it would fall perpendicularly in the same time, as the elevation of the plane to Its length. 
 
 COR. I. A body would fall down the inclined plane from A to D, in the same time in which 
 it would fall perpendicularly from A to C. For, the spaces passed through in any given time 
 are as AC to AB, that is, (El. VI. 8. Cor.) as AD to AC ; consequently, if AC is the space 
 passed through in any given time by the body falling freely, AD will be the space passed through 
 in the same time, down the inclined plane AB. 
 
 COR. 2. Having the space through which a body falls in a perpendicular direction, we can 
 easily find the space which a body will describe in the same time, on planes differently inclined, 
 by letting fall perpendiculars, as CD on those planes respectively. 
 
 PROP. XXXII. The velocity, acquired in any given time by a body 
 descending down an inclined plane, is to the velocity acquired in the same 
 time by a body falling freely, as the elevation of the plane to the length. 
 
 In an uniformly accelerated motion, the velocities produced in equal times are as the forces 
 which produce them ; but (by Prop. XXX.) the force with which a body descends down an 
 inclined plane, is to that of its perpendicular descent, as the height of the plane to its length j 
 therefore the velocities produced in equal times are in the same ratio. 
 
 PROP. XXXIII. The time, in which a body moves down an inclined 
 plane, is to that in which it would fall perpendicularly from the same height, 
 as the length of the plane to its elevation. 
 
 Plate 1. The square of time in which AB is passed over, is to the square of the time in which AD is 
 
 Fig. 11. passe( i over (compare Prop. -XX VI. with Prop. XXIX. Cor.) as AB to AD ; that is, since AB, 
 
 AC, AD (El. VI. 8. Cor.) are continued proportionals, as the square of AB to the square of AC. 
 
 Therefore the times themselves are as the lines AB, AC ; that is, as the length of the plane to its 
 
 elevation. 
 
CHAP. V. OF INCLINED PLANES. S9 
 
 COR. Hence, if several inclined planes have equal altitudes, the times in which those planes ^| at e * 
 are described by bodies falling down them, are as the lengths of the planes. For the time of '^ 
 the descent down AC is to the time of the fall down AB, as AC to AB ; and the time of the fall 
 down AB is to the time of the descent down AG, as AB to AG ; therefore (El. V. 11.) the time 
 of the descent from A to C is to the time of descent from A to G, as AC to AG ; that is, the 
 times are as the lengths of the planes. 
 
 PROP. XXXIV. A body acquires the same velocity in falling down an 
 inclined plane, which it would acquire by falling freely through the perpen- 
 dicular elevation of the plane. 
 
 The square of the velocity which a body acquires by falling to D, is (by Prop. XXVI. com- Plate 1. 
 pared with Prop. XXIX. Cor.) to the square of the velocity it acquires by falling to B, as the Flg< 1] 
 space AD is to the space AB, that is (El. VI. 8. Cor.) as the square of AD is to the square of 
 AC ; and consequently the velocity at D is to the velocity at B, as AD is to AC. But, because 
 AD and AC (Prop. XXXI. Cor.) are passed over in the same time, the velocity acquired at D 
 is (by Prop. XXXII.) to that which is acquired at C, as AD to AC. Since then the velocity at 
 D has the same ratio to the velocities at B, and at C, namely, the ratio of AD to AC, the 
 velocities at B and C (El. V. 9.) are equal. 
 
 COB. 1. Hence the velocities acquired by bodies falling down planes differently inclined are Plate I. 
 equal, where the heights of the planes are equal. The velocities acquired in falling from A to Fl - 12- 
 C, and from A to G, are each equal to the velocity acquired in falling from A; to B, and therefore 
 equal to one another. 
 
 COK. 2. Hence if bodies descend upon inclined planes, whose heights are different, the 
 velocities will be as the square roots of. their heights. For (Fig. 8 and 9) the velocity in D is 
 equal to that in A, and the velocity in D is equal to that in G. Therefore the velocity in D 
 (Fig. 8.) is to that in D (Fig. 9.) as \/~AB is to \S~VO (by Cor. Prop. XXVI.) 
 
 PROP. XXXV. A body falls perpendicularly through the diameter, 
 and obliquely through any chord of a circle, in the same time. 
 
 In the circle ADB, let AB be a diameter, and AD any chord ; draw BC a tangent to the P i &te i 
 circle at B ; produce AD to C, and join DB. Because ADB (El. HI. 31.) is a right angle, a Fig. 12. 
 body (by Prop. XXXI. Cor.) will fall from A to D on the inclined plane in the same time in 
 which it will fall from A to B perpendicularly. In like manner let the chord AE be produced 
 to G ; and because AEB is a right angle, a body will fall from A to E in the inclined plane in 
 the same time in which it would fall from' A to B. 
 
 COR. 1. Hence all the chords of a circle are described in equal times. 
 COR. 2. Hence also the velocities, and accelerating forces, will be as the lengths of the 
 chords. 
 
 PROP. XXXVI. If a body descends along several contiguous planes, 
 the velocity which it acquires by the whole descent, provided it lost no motion 
 in going from one to another, is the same which it would acquire if it fell 
 
30 OF MECHANICS. BOOK II. 
 
 from the same perpendicular height along one continued plane ; and this 
 velocity will be the same with that which would be acquired by the perpen- 
 dicular fall from the elevation of the planes. 
 
 p ate * Let AB, BC, CD, be several contiguous planes ; through the points A and D, draw HE, DP, 
 
 "'* parallel to the horizon, and produce the contiguous planes CB, CD, to G and E. By Prop. 
 XXXIV. Cor. the same velocity is acquired at the point B, whether the body descends from 
 A to B, or from G to B. Therefore, the line BC being the same in both cases, the velocity 
 acquired at C must be the same, whether the body descends through AB, BC, or along GC. 
 In like manner, it will have the same velocity at D, whether it falls through AB, BC, CD, or 
 along ED ; that is, (by Prop. XXXIV.) its velocity will be equal to the velocity acquired by the 
 perpendicular fall from H to D. 
 
 COR. Hence, if a body descends along any arc of a circle, or any other curve, the velocity 
 acquired at the end of the descent is equal to the velocity acquired by falling down the perpen- 
 dicular height of the arc ; for such a curve may be considered as consisting of indefinitely small 
 right lines, representing contiguous inclined planes. 
 
 SCHOL. The velocity of a body, passing from one inclined plane to another, is diminished 
 in the ratio of radius, to the cosine of the angle between the directions of the planes. Let BC, 
 or But (Fig. 20.) represent the velocity acquired at B, and resolve BC into Bn and Cn, by 
 letting fall the perpendicular Cn; mn will be the velocity lost, therefore the Telocity at B is to 
 the velocity diminished by passing from AB to BD as BC to Bre, or as radius to the cosine of 
 the angle between the directions of the planes. 
 
 PROP. XXXVII. If two bodies fall down two or more planes equally 
 inclined, and proportional, the times of falling down these planes will be as 
 the square roots of their lengths. 
 
 Plate 1. Let the inclined planes be AB, BC, DE, EF ; let AG, DH, be lines drawn parallel to the 
 
 Fig. 14. horizon ; let AB, DE, be equally inclined to the plane of the horizon, and also BC, EF ; let AB 
 
 be to DE as AG to DH and as BC to EF, and draw GB, HE. 
 
 Because ABG, DEH, are similar triangles, AB is to DE (El. VI. 4.) as P,G to EH, and v/AB 
 
 to v/DE as v/BG to v/EH; also AB is to DE as BG + BC is to HE + EF, and v/AB to 
 
 v/DE as VBG+BC, or ,/GC is to \/ HE + EF, or 
 
 And since (by construction) AB is to DE as BC to EF, AB is to DE as AB -f BC is to 
 
 DE + EF, and ,/AB to x/DE, as \/AB + BC to VDE~+EF~. But AB, DE, being planes 
 equally inclined, the accelerating force of gravitation will be the same upon each, and the bodies 
 descending upon them may be considered as falling down different parts of the same plane. 
 Hence, (Prop. XXVI. Cor. 2. and XXIX. Cor.) the time of descent along AB is to that along 
 DE, as v/AB to v/DE ; and the time of descent along GC is to that along HF, as v/GC to 
 v/HF ; that is, as v/AB to v/DE. Again the time of descent along GB is to that along HE as 
 v/BG to v/EH ; that is, as v/AB to </DY,. Since, therefore, the time of descent along the 
 whole plane GC is to that along the whole plane HF, as v/AB to v/DE, and that of the part 
 GB is also to that of the part HE, as v/AB to v/DE, the time of descent along the remainder 
 
CHAP. V. OF PENDULUMS. 31 
 
 BC is to that along the remainder EF (El. V. 19) as ,/AB to ,/DE. Consequently, the time 
 of descent down BA + BC is to that down DE -f EF, as v/AB to ,/UE ; that is, as 
 VAB + BC to \/UE + KF. 
 
 COR. Hence, if bodies descend through arcs of circles, the times of describing similar arcs 
 will be as the square roots of the arcs. For such similar arcs may be considered as composed 
 of an equal number of proportional sides, or planes, having the same inclination to each other, 
 and their elevations equal : whence, by this proposition, the times of descent will be as the 
 square roots of the lengths of the arcs. 
 
 PROP. XXXVIII. If a body be thrown up along an inclined plane, or 
 the arc of a curve, it will, in the same time, rise to the same height, from 
 which, with equal force, it would have descended ; and any velocity will be lost 
 in the same time in which it would, in descending, have been acquired. 
 
 For the same force of gravitation has, in every respect, the same efficacy to retard the 
 motion of bodies ascending, as to accelerate them descending on an inclined plane or curve. 
 
 SECT. III. 
 
 Of the Pendulum and Cycloid. 
 
 DBF. V. A pendulum is a heavy body, hanging by a cord or wire, and 
 moveable with it upon a centre. 
 
 PROP. XXXIX. The vibrations of a pendulum are produced by the 
 force of gravitation. 
 
 Let the ball A, suspended from the centre B by the cord BA, be drawn up to C and let fall r ? ate * 
 from thence : it wiTl descend by the force of gravitation to A, from whence (being prevented '**' 
 from falling farther by the cord) it will proceed (by Prop. XXXVI. Cor.) with a velocity equal 
 to that which it would have acquired in falling perpendicularly from E to A, which will carry it 
 on the opposite side to the height from which it fell. Being brought back again towards A by 
 the force of gravitation, it will acquire a new velocity which will carry it towards C ; and in 
 this manner it will vibrate by the force of gravitation, till the resistance of the air, and the 
 friction of the string, stop its motion. 
 
 PROP. XL. The same pendulum, vibrating in small unequal arcs, per- 
 forms its vibrations nearly in equal times. 
 
 In the circle CGA, the small arcs CA, EA, will differ little from their respective chords in Plate 1. 
 length or declivity. But (by Prop. XXXVI. Cor.) the times in which the chords are passed and'ir.' 
 over are equal;- therefore the times of describing the arcs CA, EA, and also (by Prop. 
 XXXVIII.) of describing their doubles CAD, EAF, will be nearly equal. 
 
33 OF MECHANICS. BOOK II. 
 
 Exp. Two equal pendulums, vibrating in small, but unequal arcs, will, for a long time, keep 
 pace in their vibrations. 
 
 PROP. XL1. If a pendulum vibrate through small arcs of circles of 
 different lengths, the velocity, it acquires at the lowest point, is as the chord 
 of the arc which it describes in its descent. 
 
 Plate 1. Let BA be the pendulum, and CAD, EAF, the arcs through which it vibrates ; and draw 
 
 jnd'lV" *' ie nor ' zon t a l lines EK, CH. The velocity acquired in falling from H to A is (by Prop. 
 XXXVI. Cor.) to that acquired by falling from G to A, as ^/HA to ,/G\ ; that is, (by El. VI. 
 8. Cor.) as CA to GA. For the same reason, the velocity acquired in falling from G to A, is 
 to that acquired in falling from K to A, as GA to EA. Consequently, eat cequali, the velocity 
 acquired in falling from H to A is to that acquired by falling from K to A, as CA to EA. But 
 (by Prop. XXXVI. Cor.) the velocity acquired in falling from H to A is equal to that from C 
 to A ; and the velocity acquired in falling from K to A is equal to that from E to A. Therefore 
 the velocity acquired in descending through the arc CA. is to that through EA, as the chord CA 
 to the chord EA ; and the same may be shewn concerning the remaining half of the vibrations, 
 AF, AD. 
 
 Fig. 17. COR. Hence the lengths of the chords of arcs, through which pendulums move, are measures 
 
 of velocity. 
 
 PROP. XLII. The time of the descent and ascent of a pendulum, sup- 
 posing it to vibrate in the chord of a circle, is equal to the time in which a 
 body, falling freely, would descend through eight times the length of the 
 pendulum. 
 
 For the time of the descent of a body upon the chord is (by Prop. XXXV.) equal to that of 
 the fall through the diameter of the circle, which is twice the length of the pendulum ; but in 
 double that time, that is, in the descent and ascent, or whole vibration, the body would fall 
 (by Prop. XXVII.) through four times the space, that is, through eight times the length of the 
 pendulum. 
 
 PROP. XL1II. The times in which pendulums of different lengths per- 
 form their vibrations, are as the square routs of their lengths. 
 
 ]|Iate 1. Let the two pendulums, AB, CD, be of different lengths. The time in which the first, AB, 
 
 Fl - 18 - vibrates through a chord, is equal to that in which a body (Prop. XXXV.) would fall freely 
 through twice AB, the diameter of the circle of which AB is radius. In like manner, the time 
 in which CD vibrates is equal to that in which a, body would fall through twice CD. But the 
 time* in which d body would fall through these different spaces are (Prop XXVI. Cor. 2.) as 
 the square roots of the spaces ; that is, as the square roots of AB and CD, the lengths of the 
 pendulums ; therefore the vibrations are in the same ratio. 
 
 Fig-. 19. COR. The times in which pendulums of unequal lengths vibrate, are as the square roots of 
 
 the similar arcs through which they move. Let BA, BC, be pendulums of different lengths 
 
CHAP. V. OF PENDULUMS. 3S 
 
 vibrating in the similar arcs FG, DE. Since the times of vibration are as the square roots of 
 the lengths BA, BC, and that similar arcs are as the diameters,, the times of vibration are as the 
 square roots of the arcs, FA, DC, or of their doubles, FG, DE. 
 
 EXP. Two pendulums, the lengths of which are as I to ^, will perform their vibrations in 
 times as 1 to 2; that is, the shorter pendulum will make two vibrations, whilst the longer makes 
 one ; for T : t :: \/Li : ^/ 1. 
 
 PROP. XLIV. The squares of the times, in which a pendulum of a 
 given length performs its vibrations, are inversely as the accelerating forces, 
 or gravities. 
 
 By^Prop. XXVI, where the accelerating force is given, the space described is as the square 
 of the time in wkich it is described. And since, in any given moving body, the velocity is as 
 the accelerating force (Prop. A. p. 13.) where the square of the time, or the time itself, is given 
 (by Prop. II.) the space described will he as the accelerating force. Consequently, where 
 neither the accelerating force, iior the square of the time, is given, the space described will be in 
 the ratio compounded of both. If then the space described be called S, the accelerating force A, 
 
 T* A S 
 
 and the square of the time T 2 , S will be as T*A; whence , or T* is as -T-. But, when 
 
 A \ 
 
 I 
 
 the spaces are equal, S is a given quantity ; whence (since fractions, whose numerators are given, 
 
 S S 
 
 .are inversely as their denominators) is inversely as A. But T z is as ; therefore where S 
 
 A A 
 
 is given, T 2 is inversely as A ; that is, where the spaces described are equal, the squares of the 
 times in which they are described are inversely as the accelerating forces. And if the squares 
 of the times of falling bodies are inversely as their accelerating forces, the squares of the times, 
 in which pendulums vibrate, are in the same ratio, on account of the constant equality between 
 the time of vibration and that of the descent through eight times the length of the pendulum, 
 by Prop. XLII. 
 
 COR. I. Hence, if the same pendulum, at different parts of the earth, performs its vibrations 
 in different times, the forces of gravitation will, in those places, be inversely as the squares of 
 those times. 
 
 COR 2. If the vibrations of pendulums of unequal lengths be performed in the same time, 
 the accelerating forces will be as their lengths. For (by Prop. XLIII. and XLIV.) 
 
 l-:.-.4:l, 
 
 A ft 
 
 therefore when T = t, A : a : : L : I. Hence, as it is known by experiment, that the lengths of 
 pendulums that vibrate seconds are diminished in approaching the equator, the force of gravity 
 must also decrease. 
 
 Ex. At the equator a pendulum vibrating seconds is j'^th of an inch shorter than such a 
 pendulum in the latitude of London, and the length of this pendulum in London is 39.2 inches : 
 therefore gravity under the equator is to gravity here as 391 is to 392. 
 
 5 
 
31 OF MECHANICS. BOOK II. 
 
 LEMMA I. 
 
 -?/ f rom ^ as the centre, with any distance XA, a quadrant of a circle 
 AD B be described, and in the right line AX a body descends with such force, 
 that its velocity in any points M, N, fc. shall be always as MD, NP, &fc. the 
 sines of the arcs AD, AP : the time in which the body will descend from A to 
 X, will be equal to the time in which it would describe the whole arc ADD, 
 with the uniform velocity, expressed by XB, acquired by the falling body when 
 it arrives at X ; also, the time of the fall through any space AM, will be to 
 the time of the fall through any other space AO, as the arc AD to the 
 arc AQ; and the force with which the body is accelerated in any place M, will 
 be as MX, the distance of that place from the centre. 
 
 Let DP be a part of the circumference taken indefinitely small, and therefore not assignably 
 differing from a right line ; join DX ; and draw DL perpendicular to NP. Because the trian- 
 gles MDX, LDP, are similar (having each a right angle, and the angles MDX, LDP, whose 
 common complement is LDX, equal ) MD will be to DX as LD or MN to DP. But, by the 
 hypothesis," M D is as the velocity of the descending body at the point M, that is, as the velocity 
 vith which the indefinitely small line MN is described ; and XD is as the velocity last acquired 
 by the falling body at X, that is, as the uniform velocity with which the arc DP is described. 
 The velocity therefore of the body descending through the indefinitely small line MN, will be to 
 the velocity of the body moving along the arc DP, as MN to DP. Wherefore, since the velocities 
 are proportional to the spaces passed over, the times wherein those spaces MN, DP, are 
 described, will be equal. After the same manner it may he proved, that any other indefinitely 
 small portion of the circumference, PQ, may be described with the velocity XB, in the same time 
 in which the corresponding line NO will be described with the corresponding velocity NP ; 
 and consequently, by composition, the falling body will descend through all the indefinitely 
 small portions of the perpendicular AX, that is, through the whole line, in the time in which all 
 the corresponding parts of the circumference, that is, the whole quadrant AUB, is described with 
 a uniform velocity as XB. 
 
 Moreover, the time in which the falling body descends from A to M, is equal to the time in 
 which the arc AD is passed over ; and the time in which it descends from A to O is equal to 
 the time in which the arc AQ is described ; but the time in which the arc AD is passed over, is 
 to that in which the arc AQ is passed over, (since they are both described with the same velocity) 
 as the arc AD to the arc AQ; therefore the time of descent from A to M, will be to the time of 
 descent from A to O. as the arc AD to the arc AQ; and consequently, by division, the time of 
 descent through AM will be to the time of descent through MO, as the arc AD to the 
 arc DQ. 
 
 Lastly, let the arcs DP, PQ, be equal ; join XP,and from P let fall PS perpendicular to OQ. 
 The time of descent through MN will be equal to that through NO; and, since the triangles 
 LDP, MDX, are similar, and also SPQ, NPX ; LP will be to DP or PQ, as MX to XD or XP ; 
 also PQ is to SQ as XP to XN ; and consequently (El. V. 11.) t.P will be to SQ as XM to XN. 
 But LP is as the increment of the velocity acquired while the body is passing over MN, and 
 
CHAP. V. OF THE CYCLOID. 85 
 
 SQ is as the increment of the velocity acquired in passing over in an equal time the indefinitely 
 small line NO ; and the forces with which the body is accelerated at M and N, are as the 
 increments of the velocities generated in equal times ; the accelerating forces at M and N 
 Will therefore be as the lines LP, SQ ; that is, the force with which the body is impelled at M is 
 to that at N, as the distance XM to the distance XN, or the accelerating forces are as their 
 distances from the centre. 
 
 COR. Hence, conversely, if a body, descending from A to X, is impelled hy a force which is 
 as its distance from the centre X, and that force at the beginning of the motion is expressed by 
 the right line CE (the arc AE being taken indefinitely small) the velocities of the same body 
 in any places M, O, will be expressed by the sines MD, OQ ; and the times by the arcs AD, 
 AQ ; and the increments of the velocities, or, if the arcs increase equally, the accelerating forces, 
 will be expressed by the increments of the sines. 
 
 LEM. II. If a body, moving along the line AX, be impelled by forces Plate 2. 
 proportional to its distance from the point X; from whatever height it falls, ' s ' 
 it will arrive at the point X in the same time ; and this time will be to the 
 time in which it would move over the line AX with the velocity which it 
 acquires by falling from A to X, as half the circumference of a circle to its 
 diameter. 
 
 Let two bodies be let fall from the points A and P at the same time ; and let them be 
 impelled by forces proportional to their distances from the point X; these bodies will come to 
 X at the same time. From X as a centre, with the radii XA and XP, describe the two 
 quadrants AB and PQ ; aiul let the force by which the body A is impelled, or, which is the 
 same thing, its velocity at the beginning of motion, be represented by RS, the sine of the 
 indefinitely small arc AS. It is manifest (from the Cor. of the preceding lemma) that its 
 velocity, after the fall to X, will be properly expressed by XB. But, by hypothesis, the force by 
 which the body at A is accelerated, is to that by which the body at P is accelerated, as AX is to 
 PX, that is, (since the arcs AS and PN are similar) as RS to MN. As therefore RS expresses 
 the first velocity of the body moving from A, MN will express the first velocity of the body 
 moving from P; and consequently (hy the Cor. to the last lemma) XQ will express the velocity 
 of the body moving from P, when it arrives at X. Farther, the time of the fall from A to X (by 
 Lemma I.) is equal to the time in which the arc AB would be described with a velocity as XB ; 
 and the time of the fall from P to X is equal to the time in which the arc PQ would be described 
 with a velocity as XQ. But (because the line XQ is to the line XB as the arc PQ to the 
 arc AB, and that the spaces passed over are proportional to the velocities) the time in which the 
 arc AB is described with the velocity XB is equal to the time in which the arc PQ is described 
 with the velocity XQ. Wherefore the time of the fall from A to X will be equal to the time of 
 the fall from P to X. 
 
 Again, since (by Lem. I.) the time in which a body would fall from A to X is equal to the 
 time in which it would move over the arc AB, with its last acquired velocity at X; and since 
 it is evident, that the time in which a body would move over the arc AB with the velocity at X 
 is to the time in which it would move over AX with the same velocity, as AB is to AX ; the 
 
36 OF MECHANICS. BOOK II. 
 
 time in which a body would fall from A to X is to the time in which it would move over AX 
 with the last acquired velocity as AB to AX. But AB is to AX, as twice AB to twice AX 
 that is, as half the circumference of a circle is to its diameter. Therefore the time in which 
 a body would fall from A to X is to the time in which it would move over AX with its last ac- 
 quired velocity, as half the circumference of a circle is to its diameter. 
 
 pute 2. D EF% vi. if a c i rc ie, a s FCH, be rolled along the line AB, till it has 
 
 l'"t*. 3* 
 
 turned once round ; the point C in its circumference, which at first touched 
 the line at A, will describe the curve line ACXB, which curve is called a 
 Cycloid. The right line AB is its base; the middle point X is its vertex ; a 
 perpendicular, as XD, let fall from thence to the base, is its axis ; and the 
 circle FCH, or any other, as XGD, equal thereto, is called the generating 
 circle. 
 
 LEM. III. If on XD, the axis of the cycloid, as a diameter, the 
 generating circle XGD be described ; and if from a point in the cycloid, as 
 C, the line CIK be drawn parallel to the base, the portion of it CG will be 
 equal to the arc GX. 
 
 Because the generating circles FCH, DGX, are equal (the diameter HF being drawn) KG 
 is equal to CI ; whence, adding GI to both, KI will be equal to CG ; and KI, by construction, 
 is equal to DF ; therefore CG is equal to FD. By the description of the cycloid, the arc CF is 
 equal to the line AF ; and by the construction the arc CF is equal to DG ; therefore AF is equal 
 to DG ; but, by the description of the cycloid AFD is equal to DGX ; consequently, FD is 
 equal to GX; and CG was proved to be equal to FD; therefore CG is equal to GX. 
 
 LEM. IV. A tangent to the cycloid at the point C is parallel to GX, 
 a chord of the circle DGX. 
 
 Draw ck, parallel to the base and indefinitely near to CK meeting the cycloid in c, the axis 
 in fc, and the circle in g. Let CM and GK, parallel to the "axis, meet ck in M and n, and from 
 T, the centre of the circle XGDM, draw the radius TG. Since eg is equal (Lem. III.) to gX, 
 gk being added to both, ck will be equal to \g + gk ; therefore cu, the excess of ck above CK, 
 is equal to Gg + gn, the excess of Xg + gk above XG + GK. And, if we suppose ck to 
 approach towards CK, as Gg and gn vanish, the triangles Ggn and GTK become similar; for 
 the angle gGn is then equal to the angle TGK, since both have the same angle nGT, as their 
 complement. Whence Gg is to gn as TG to TK, and (El. V. 18.) Gg + gn to gn, as 
 TG + TK or DK to TK; but Ore is to gn as GK to TK ; therefore Gg + gn is to Gn as 
 DK is to GK, that is, (El. VI. 8.) as GK to XK. And consequently cu (shown to be equal 
 to Gg+gn") is to Gn, or CM, as GK to XK ; and if the chord Cc be drawn, the triangles 
 CMC, XKG, will be similar ; so that the chord Cc (as the points C and c coincide) becomes 
 parallel to XG ; therefore the tangent of the cycloid at C is parallel to XG. 
 
CHAP. V. OF THE CYCLOID. 37 
 
 LEM. V. If from a point of the cycloid, as L, the line LMK be drawn Plate 2. 
 parallel to the base AB, the arc XL of the cycloid, icill be double of XM ; Flg ' ? ' 
 the chord of the circle corresponding thereto. 
 
 Draw the Hue Sfc parallel and indefinitely near to LK crossing the circle in R, and the 
 chord XM produced, in P; join the points X and R; ou MP let fall the perpendicular RO : 
 and draw MN, XN, tangents to the circle at M and X. Then will the lines XN" and fcS, heing 
 each perpendicular to the diameter DX, be parallel; and the triangles MNX, MPR, having 
 their angles at M vertical, and at P and X alternate, will be similar. But the tangents NX 
 and NM are equal ; (El. Ill, 36.) whence the lines PR and RM are also equal ; the triangle 
 RMP is therefore isosceles; and RO being perpendicular to its base MP, MO (El. I. 6.) 
 is equal to OP ; whence MP is equal to twice MO. The indefinitely small arc LS of the 
 cycloid will not assignably differ from a portion of a tangent drawn through the point L. LS 
 may therefore (Lem. IV.) be said to be parallel to MP, and consequently (from the parallelism of 
 ML and PS) equal to it; it is therefore equal also to twice MO. But LS-is the difference 
 between the cycloidal arcs XL and XS ; and MO is the difference between the chords 
 XM and XR; for since XO and XR are indefinitely near to each other, RO which is 
 perpendicular to one of them, may be considered as perpendicular to both ; the indefinitely 
 small difference therefore between any two arcs of the cycloid is twice that which is between 
 the two corresponding chords of the circle ; and the same is true when the magnitude of the 
 difference is assignable, because such difference is compounded of indefinitely small parts. 
 Now, any arc whatsoever may be considered as a difference between two arcs, and consequently 
 any arc, as XL, is double of the corresponding chord XM. 
 
 COR. Since when the arc XL becomes XB, the corresponding chord XM becomes XD, the 
 diameter of the circle DMX ; it is obvious, that the semicycloid BX, or AX, is equal to twice 
 DX, the diameter of the generating circle DMX. 
 
 LEM VI. If a body descends in a cycloid, the force of gravity, so far 
 as it acts upon the body in causing it to descend along the cycloid, will be 
 proportional to the distance of the body from the lowest point of the cycloid. 
 
 Let the cycloid be AXB, whose base is AB, and its axis DX ; on which last, as a p] ate 2. 
 diameter, describe the generating circle DQX ; draw the chords OX and QX ; through the Fig- 4. 
 points O and Q. and parallel to the base AB, draw the lines LS and MR; draw also the 
 tangents LV and MY. Then because (by Lem. IV.) the tangent LV is parallel to OX, and 
 the tangent MY parallel to QX, it is obvious that gravity exerts the same power upon a body 
 descending in the cycloid at L, (because it then descends in the tangent LV) as it would do 
 upon the same body descending along the chord OX ; and, for the like reason, it exerts the 
 same force upon it when it comes to M, that it would do if it were descending along QX ; but 
 (from Prop. XXXV.) the power or force of gravity upon bodies descending along the chords OX 
 and QX, are as the lengths of those chords ; that is, by Lem. V. (halves being proportional to 
 their wholes-) as the length of the cycloidal arcs LX and MX. The force therefore of gravity 
 upon a body descending in the cycloidal at the point L is to its force upon the, same 
 when at M (as may be said of any other corresponding points) as the space or distance it has to 
 
38 OF MECHANICS. BOOK II. 
 
 move over in the former case, before it reaches the lowest point X, to that which it has to pass 
 over in the latter, before it arrives at the same point. 
 
 PROP. XLV. If a pendulum be made to vibrate in a cycloid, all its 
 vibrations, however unequal in length, will be performed in equal times. 
 
 The force of gravity, (by Lem. VI.) so far as it causes a body to descend in a cycloid, is 
 proportional to the distance of that body from the lowest point. Imagine then that body to be a 
 pendulum vibrating in the cycloid, and from whatever point it sets out, it will (by Lem. II.) 
 come to the lowest point in the same time ; and consequently, since the same may be easily 
 inferred in its ascending from that point, all its vibrations, be they large or small, will be 
 performed in the same time. 
 
 SOHOL. This proposition is demonstrated only on the supposition that the whole mass of the 
 pendulum is concentrated in a point, for it cannot otherwise take place, because as the string 
 varies in its length, the centre of oscillation of a body will vary. On this account, therefore, 
 pendulums vibrating in circular arcs are now always used, for the same arcs will be always 
 described in the same time. 
 
 PROP. XL VI. To make a pendulum vibrate in a given cycloid. 
 
 Plate 2. Let AXB be the given cycloid ; its base AB, its axis DX, and its generating circle DQX, as 
 
 *% 4. before ; produce XD to C, till DC is equal to DX ; through C draw the line EF parallel to AB, 
 and take CE and CF, each equal to AD or DB ; and on the line CE as a base, and with the 
 generating circle AGE equal to DQX, describe the semicycloid CTA, whose vertex will there- 
 fore touch the bise of the given cycloid in A. And on the line CF also as a base, describe an 
 equal semicydoid CB. Let the semicycloids CA, CB, represent thin plates of metal bent to their 
 figure, and on the point C, hang the pendulum CTP by a flexible line equal in length to the 
 line CX. The upper part of its string (as CT, in its present situation in the figure) as it 
 vibrates, will then apply itself to the cycloidal cheeks CA and CB, and a ball at P will oscillate 
 in the given cycloid AXB. 
 
 Draw TG and PH each parallel to the base AB, and draw AG and DH. Then (Lem. V. 
 Cor.) AC is equal to twice AE ; and by construction, twice DC ; that is, (wice AE, is equal to 
 CX ; therefore AC is equal to CX. Also, by construction, CTP is equal to CX ; that is, to 
 ATC ; whence, taking away CT, AT is equal to TP. By Lem. IV. GA is parallel to TP ; 
 and, by construction, AK is parallel to GT; therefore GA is equal to TK, and GT to AK; but 
 (Lem. V.) GA is half TA ; therefore TK is equal to half TA ; since therefore it has been 
 proved that TA is equal to TP, TK is equal to half TP; that is, to KP. Hence it is manifest, 
 that the parallel lines GT, PH, are equally distant from AD, the arc GA equal to the arc DH, 
 the chords GA and DH parallel, and GE equal to HX. And, because GA has been shewn to 
 be parallel to TK, and also to DH, KP and DH are parallel ; whence KD is equal to PH. But 
 (Lem. HI.) GT, that is, AK is equal to the arc AG; and by the description of the semicycloid 
 CTA, AKD is equal to AGE ; therefore KD is equal to EG ; that is, PH is equal to HX. And 
 (by Lem. III.) if PH be equal to HX, P is a point in the cycloid AXB. The ball of the pendu- 
 lum therefore, being at that point, is in the given cycloid. 
 
CHAP. V. OF THE CENTRE OF GRAVITY. 39 
 
 SOHOL. 1. It is easy to conceive, that in a pendulum there must be some one point, on each Plate 1. 
 side of which the momenta of the several parts of the pendulum will be equal, or in which the ^' 
 whole gravity of the pendulum might be collected without altering the time of its vibrations. 
 This point, which is called the centre of oscillation, is different from the centre of gravity ; for 
 if a plane perpendicular to the string of the pendulum AI5, be conceived to pass through the 
 centre of the ball B, bisecting it; the velocity of the lower half, and consequently its momen- 
 tum, will, in vibration, be greater than that of the upper half; consequently the centre of 
 oscillation must be farther from A than the centre of gravity is ; and a plane passing through 
 the centre of oscillation will divide the ball into two unequal parts, so that the greater quantity 
 of matter above it, shall compensate for the greater velocity below it, and the momenta on each 
 side be equal. If the pendulum be an inflexible rod, every where of equal size, it is found, that 
 the distance of the centre of oscillation from the point of suspension is two thirds of the length 
 of the rod. 
 
 If, whilst a pendulum is in motion, it meets with an obstacle at its centre of oscillation 
 sufficient to stop it, the whole motion of the pendulum will cease at once, without any jarring ; 
 for the obstacle resists equal momenta above and below this point ; which is therefore also called 
 the centre, of percussion. 
 
 SOHOL. 2. The vibrations of pendulums are subject to many irregularities, for which no 
 effectual remedy has yet been devised. These are owing partly to the variable density and 
 temperature of the air, partly to the rigidity and friction of the rod by which they are 
 suspended, and principally to the dilatation and contraction of the materials, of which they are 
 formed. The metalline rods of pendulums are expanded by heat, and contracted by cold ; 
 therefore clocks will go slower in summer and faster in winter. The common remedy for this 
 inconvenience is the raising or lowering of the bob of the pendulum (by means of a screw) as the 
 occasion may require. By the last scholium it appears, that a pendulum consisting of a tube of 
 glass or metal, every where uniform, filled with quicksilver, and 58.8 inches long, will vibrate 
 seconds ; for | of 58.8 is equal to 39.2. Such a pendulum will be expanded and contracted at 
 the same time ; for when the tube is extended by heat, the mercury will also be expanded, and 
 by rising in the tube, will raise the centre of oscillation, so that its distance from the point of 
 suspension will be diminished, and the vibrations of the pendulum, which would have been 
 rendered slower by the expansion of the tube, will become quicker by the expansion of the 
 mercury ; and, by adjusting the tube and mercury in such a manner, that these contrary effects 
 may be the same, a clock with such a pendulum would admit of little or no variation for a long 
 time. Phil. Trans. No. 392. p. 40. 
 
 SECT. IV. 
 
 Of the Centre of Gravity. 
 
 PROP. XLV1I. In every body there is a centre of gravity, or a point 
 about which all its parts balance each other. 
 
 Let AB be an inflexible rod, throughout uniform and of the same density; let it be supported Plate 2. 
 at the point C, equally distant from its extreme points A and B, by the prop C. Let A and '&' 
 B be indefinitely small and equal portions of the rod AB. These portions, A and B, tend 
 
40 OF MECHANICS.^ 
 
 f 'S- G - towards the centre of the earth with equal forces of gravitations They would likewise, without 
 obstruction, move with equal velocities ; for if the rod AB be moved on its prop till it come into 
 the position DE, the velocities of the parts A, B, or F, will DC as the spaces over wiiich they 
 pass in the same time ; that is, as the arcs AD, EB or FG j which arcs are as their respective 
 circumferences, or as their diameters or radii ; whence the velocity of the part B is to the 
 velocity of the part A, or F, as BC is to AC, or FC. And the quantities of matter in A and 
 B are by supposition equal. Therefore, if the parts A and B were in motion, they would have 
 equal momenta ; that is, the efforts which A and B make to descend towards the earth are equal. 
 But these efforts counteract each other ; for, whilst the portion A endeavours with a certain 
 force to draw down one arm of the rod, the other portion B endeavours with the same force to 
 draw down the other arm ; that is, since the rod is inflexible, to raise the portion A. Therefore 
 the portion A is acted upon by two equal forces in contrary directions, and consequently must 
 be at rest. For the same reason, the portion B will be at rest. And the same may be shewn 
 concerning any other equal portions, at equal distances from C, in the rod AB. Therefore the 
 rod will be at rest; that is, the parts on each side of the point will balance each other, and C 
 will be the centre nf gravity, 
 
 If the rod were placed oblique to the prop C, indefinitely small and equal parts being taken, 
 as before, at equal distances from C, and resolving each oblique force into a horizontal and 
 perpendicular force (as in Prop. XVI.) it might be shewn, by a similar manner of reasoning, 
 that they would tend towards the earth with equal forces, and consequently, that an equilibrium 
 would be produced. 
 
 Fig. 6. And if, instead of equal portions of the rod, portions of matter were placed at different 
 
 distances, which should be to each other inversely as those distances, as at F and B, the equili- 
 brium would still be preserved ; for the forces with which such portions of matter, so situated, 
 would endeavour to descend, would be equal, when the quantities of matter, multiplied into the 
 velocities with which they are endeavouring to move, that is, into their distances (Prop. XI. 
 Cor.) are equal ; as will be more fully shewn, in treating' of the Mechanical Powers. 
 
 Since, therefore, all the parts of any irregular body may be referred to some one of the above 
 cases, it is manifest, that there is in every body a certain point, the parts on each side of which 
 balance each other. 
 
 PROP. XLVIII. If the centre of gravity, in any body be supported, 
 the whole body is supported ; if this centre be not supported, the body will 
 fall. 
 
 For, when the centre of gravity is supported, the body rests on a prop on which the parts on 
 each side, acting with equal force against each other, will (Prop. XLVII.) be in equilibrio, and 
 neither side will move ; but when this centre is not supported, but the body has a prop under 
 some other point, the parts of the body on one side of that other point will over-balance the 
 parts on the other side, and the body will fall. 
 
 COR. Whenever a body moves by the power of gravitation, or falls, its centre of gravity 
 descends ; for if this centre do not descend, it must be supported ; and if the centre be supported, 
 the whole body is sustained or kept from falling. 
 
CHAP. V. OF THE CENTRE OF GRAVITY. 41 
 
 Ex-. 1. Let a board of a circular form be sustained perpendicularlj on its centre of gravity, 
 it will be at rest in any position. 
 
 2. A beam turning on an axis which passes through its centre of gravity will rest in the 
 same manner. 
 
 3. A beam, whose axis passes through a point which is directly above the centre of gravity, 
 will be at rest only when the beam is parallel to the plane of the horizon, because the centre of 
 gravity will be then fallen as low as possible. 
 
 4. A cylinder, which has its centre of gravity near one of its sides, will roll up an inclined 
 plane, if the side nearest the centre of gravity be placed towards the upper part of the plane; 
 for this centre, endeavouring to descend, will carry the cylinder forward in the ascending 
 direction of the plane. 
 
 5. Let a body, consisting of two equal and similar cones united at their bases, be placed upon 
 the edges of two straight and smooth rods, which at one end meet in angle, and rest upon a 
 horizontal plane, and at the other are raised a little above the plane, the body will roll towards 
 the elevated end of the rules, and appear to ascend, while its centre of gravity descends ; as may 
 be seen by applying a string horizontally above the path of the base of the cones. 
 
 PROP. XLIX. If the line of direction comes within the base on which 
 any body is placed horizontally, the body will be sustained ; otherwise it 
 will fall. 
 
 In the body ABDE let C be the centre of gravity. The line of direction CO (that is, the p) a te Z. 
 line drawn from the centre of gravity towards the centre of the earth) being within the base DE, Fl S- 7 - 
 the body will be supported, because the weight presses upon the base. Also since the body 
 cannot fall towards K without turning round on the point E, the point C must in the motion 
 ascend towards F, contrary to Prop. XLVIII. Cor. But in the position of the body abde, co 
 the line of direction falling out of the base, c in its motion towards k descends, and the body 
 will fall. 
 
 Our own motions and actions are subject to this rule. When a man stands upright, his 
 centre of gravity falls between his feet, and he is supported ; but if he lean forward he throws 
 the line of direction without his base, and he would fall if he did not put forward one of his feet 
 so as to cause it to fall within. Hence a porter, with a load on his back, leans forward that tin- 
 load may not throw the line of direction out of his base behind ; and by an artful adjusting of 
 this point it is that such wonders are performed in horsemanship, and on the tight and slack 
 rope, &c. 
 
 EXP. 1. Any body of a cylindrical or other regular form, so placed upon its base, that it 
 line of direction does not come within the base (which may be seen by a cord and weight 
 suspended from the centre of gravity) will fall; otherwise it will not fall. 
 
 2. Let two bodies be laid upon 'an inclined plane, the one a cube, the other a figure with 
 many sides, and let the line of direction of the former fall within the base, and that of the latter 
 without the base, the former body will slide, the latter roll down the plane. 
 
 DBF. VII. The centre of motion is the point about which a body moves. 
 
 6 
 
42 OF MECHANICS. BOOK II. 
 
 PROP. L. A heavy body suspended on a centre of motion will be at 
 rest, if the centre of gravity is directly under, or above, the centre of motion : 
 otherwise it will move. 
 
 Plate. 2. If a heavy body E, hangs by a string on a centre of motion C, the action of gravitation at E, 
 
 Fig. 8. j g j n jj )e Direction EL, contrary to the direction in which the string acts to prevent the body 
 from falling. In this position, therefore, the opposite forces being equal and in contrarv 
 directions, destroy each other, and the body is at rest. But if the body is at p, one of the forces 
 acts in the. direction pC, and the other in the direction pL ; that is, in directions oblique to 
 each other; whence the body will move in the diagonal of the parallelogram formed by }jC, /?L. 
 And in all cases, since (without the aid of mechanical powers afterwards explained) the force 
 which sustains any body must be equal to its weight, the centre of gravitation can only be at 
 rest when these forces are in the same line of direction, that is, when the centre of gravity is 
 directly under, or directly above, the centre of motion. 
 
 Exp. A circular board, sustained at a point above or below the centre of gravity, will only 
 be at rest when the centre of gravity is at the lowest point, that is, in the line of direction ; or 
 when the centre of gravity is in the same line above the centre of motion. 
 
 SOHOL. If two or more bodies be united, they may be considered as one, and have a com- 
 mon centre of gravity. 
 
 EXP. 1. Let two unequal balls be fixed upon the ends of a wire, they will have a common 
 centre of gravity. 
 
 O tt 
 
 2. A board, which of itself would fall from a table (its centre of gravity lying beyond the edge 
 of the table) may be made, in the same position, to support a vessel of water, hanging upon it near 
 the table ; if a stick, fixed with one end at the bottom of the vessel, and the other in a hole, in the 
 horizontal board, be long enough to push the vessel a little out of the perpendicular ; that is, to 
 bring the centre of gravity of the whole under the table. 
 
 Plate 2 SOHOL. The common centre of gravity of any number of bodies may be thus found. Let 
 
 l-'ig. a. C be the common centre of gravity of two bodies, dividing their distances (see Prop. XLVII.) 
 in such a manner, that AC is to CB, as B to A ; whence A X -VC = B x BC, and consequently 
 if the point C is supported, the bodies A and B balance each other. Suppose a third body, 
 equal to the sum of A and B, placed in their common, centre of gravity C ; from the point C 
 draw a right line to the centre of a third body D, which divide in O, so that OD may be to OC, 
 as A + B is to D ; then is O the common centre of the three bodies, A, B, D. In the same 
 manner may be found the common centre of any number of bodies. 
 
 PROP. Lt. If any number of bodies move uniformly in right lines, 
 whether in the same or different directions, their common centre of gravity is 
 either at rest, or moves uniformly in a right line. 
 
 Plate 2 ^ tvvo D0l '' e9 A and B, move towards each other in the same right line, having their 
 
 Fig. 5. ' common centre of gravity C, and their momenta equal, the velocity of A will be to that of B, as 
 
 the body B to the body A, that is (as was shewn, Prop. XLVII.) as AC to BC. Whence 
 
CHAP. V. OF THE CENTRE OF GRAVITY. 43 
 
 (Prop. VI.) whilst A passes through AC, B will pass through BC, and the bodies will meet in 
 C, which is their centre of gravity during their motion, and at the time of concourse ; therefore 
 the point C remains at rest. 
 
 In the same manner, it may be shewn, that if the bodies recede from each other with uniform 
 motions, the centres of gravity will be at rest. 
 
 Next, suppose that two bodies, A and B, move in different directions AC, BD, describing plate 2. 
 equal spaces AC, CE, and BD, DF, in equal times ; their common centre of gravity L, will move f 'S- 9 - 
 uniformly in a right line. Produce CA, DB, till they meet in G ; make AG to GH, as AC is 
 to BD; draw the right line AH ; and through C and E draw CI, EK, parallel to AH. AC i 
 to HI (El. VI. 2.) as AG to GH ; that is, as AC to BD. Therefore (El. V. 9.) HI is equal to 
 BD, and adding IB to each, HB is equal to ID. In like manner, CE is to IK, as AG to GH : 
 that is, as AC to BD, or CE to DF; therefore (El. V. 9.) IK is equal to DF, and adding KD 
 to each, ID is equal to KF ; but ID was proved to be equal to HB ; therefore, KF is equal to 
 HB. From L, the common centre of gravity of the bodies A and B, draw LM parallel to BD ; 
 Draw GM, and produce it till it cut CI, EK, in the points N and O ; and through these points 
 draw NP, OQ, parallel to BD. AL is to LB (El. VI. 2.) as AM to MH ; and CP to PD, (as 
 CN to MI, that is) as AM to MH ; therefore (El. V. 11.) CP is to PD, as AL to LB ; that is, 
 (because L is the common centre) as B to A. Consequently, P will be the common centre of 
 the bodies when they are found in C and D ; and, in like manner, it may be shewn, that Q 
 will be their common centre, when they are in E and F. But, since ML is to HB, as AM to 
 AH ; that is, as CN to CI ; that is, as NP to ID, and that HB has been proved to be equal to 
 ID, ML is equal to NP; and, in like manner, NP equal to OQ .; Whence (ML, NP, OQ, being 
 parallel to one another) the line LPQ is equal to the line MNO, and the points, P, Q, (any 
 points of the line in which the common centre of gravity is found as the bodies are moving from 
 A to E, and from B to F) will be in a right line. Moreover, since (El. VI. 2.) AC is to CE, as 
 MN to NO, or LP to PQ, and that AC is equal to CE, LP, will be equal to PQ. Therefore the 
 common centre of gravity of the bodies A and B, is always in the same right line, and moves 
 uniformly, or passes over equal spaces in equal times. 
 
 In like manner, the common centre of these two bodies and any third body, or of the three 
 bodies and a fourth, &c. being found, it may be proved that it moves uniformly in a right line. 
 
 COR. 1. Hence it is manifest, that any forces acting upon a system of bodies, must affect 
 the motion of the common centre of gravity of that system, in the same manner as if the same 
 force were similarly applied to a body equal to the sum of all the bodies, placed in the common 
 centre of gravity. And the mutual actions of the pa; ts of a system upon each other, producing 
 (by Prop. III.) equal momenta in contrary directions, cannot change the state of motion or rest 
 of their coaimon centre of gravity. Consequently the law of a system of bodies, as to motion 
 or rest, is the same as that of one body, and is rightly estimated from the motion of its centre of 
 gravity. 
 
 COR. 2. Hence the centre of gravity of a system of bodies, will not he disturbed by their 
 mutual attractions, as the motions thus communicated are always equal and opposite. Hence 
 the centre of gravity of our system of planets is either at rest, or moves uniformly in a straight 
 line. The latter is supposed by Dr. Herschel to be the case. 
 
4t OF MECHANICS. BOOK II. 
 
 CHAP. VI. 
 
 Of Motion as directed by certain instruments, called MECHANICAL POWERS. 
 
 DEF. VIII. That body, which communicates motion to another, is called 
 the Power. 
 
 D'EF. IX. That body, which receives motion from another, is called the 
 
 Weight. 
 
 DEF. X. The Lever is a bar, moveable about a fixed point, called its 
 fulcrum, or prop. It is in theory considered as an inflexible line without 
 weight. It is of three kinds ; the first, when the prop is between the weight 
 and the power ; the second, when the weight is between the prop and the 
 power ; the third, when the power is between the prop and the weight. 
 
 Exr. Let the three kinds of the lever be shewn, as in Plate 2, Fig. 12, 13, 14. 
 
 PROP. LI I. A power and weight acting upon the arms of a lever will 
 balance each other, when the weight is to the power, as the perpendicular 
 distance of the line in which the power acts, from the fulcrum, is to the per- 
 pendicular distance of the line in which the weight acts, from the fulcrum. 
 
 Plate 2. Case 1. When the power acts perpendicularly ; Let AB be the lever, C the prop, P the 
 
 power, W the weight. The force with which any body moves being as its rno:nentum, (Prop. 
 XIII.) and its momentum as the quantity of matter multiplied into the velocity, (by Prop. XI. 
 Cor.) the foice with which the weight VV would move in the first instant of its motion, if no 
 other body counteracted it, would be as its quantity of matter multiplied into its velocity. But 
 because the weight W is suspended 'from the lever AB at the point B, it would move witli the 
 same velocity as this point; which (us was shewn in Prop. XLV1I.) is as the distance of the 
 point B from the prop C, or D. The force therefore, with which the v. eight W would move 
 without any counteracting force, is as its quantity of matter multiplied into the distance of the 
 point of suspension B from the prop C, or U. But the weight will be prevented from 
 descending, if a force equal to that with which it would descend without obstruction, acts upon 
 it in the contrary direction ; that is, if a force be applied to raise the point B of the lever AB, 
 equal to that with which the weight W would draw it downwards. Let the power P be sus- 
 pended from the other extremity of the lever at the point A ; and let the quantity of matter in 
 the power P multiplied into the distance oi A, its point of suspension from C, or 1J, be equal to 
 the quantity of matter in the weight W multiplied into the distance of B from C, or D ; it 
 appears from what has been said concerning the weight, that the force with which the power P, 
 without obstruction, would descend and draw down the point A, is equal to the force with which 
 the weight W would descend and draw down the point B. But, as much force as the power P 
 exerts to draw down the point A, it exerts to raise the point B. Therefore equal and opposite 
 forces are exerted to raise and depress the point B ; and consequently it will continue at rest, 
 and the weight and power will balance each other. 
 
CHAP. VI. OF THE MECHANICAL POWERS. 
 
 Case 2. When the power acts obliquely ; 
 
 Let the weight A hang fiecly from one end of a balance, so as to have its line of direction 
 DA perpendicular to the arm of the balance; and let another weight as 13, be hung at the 
 other end E, in such manner that its line of direction KG, by passing over a pulley at C, 
 may be oblique to the arm of the balnce. If the whole force of gravity in the weight B 
 acting in the direction EC, be denoted by the line EC, it may be resolved into two forces 
 denoted by EF and FC, acting in the directions of these lines ; of which two forces, the 
 latter only, which acts in the direction FC perpendicular to the arm of the balance, resists 
 the force of gravity in the weight A, the other force FE acting in the direction of the 
 line of the lever. Since, therefore, that part of the weight B which acts in opposition 
 to the weight A, is to the whole weight B, as FC to EC ; it is manifest, that in order 
 to make the weight B balance the weight A, it must exceed the weight A, in the same 
 ratio that the line EC exceeds the line FC. If from G the centre of motion be let fall 
 GH perpendicular to EC produced, that line will be the perpendicular distance of the direction 
 EC from G ; and EG, equal to DG, the perpendicular distance of the direction DA ; 
 but the triangles EFC and EHG are similar, consequently (SI. VI. 4.) as EC is to CF, so 
 is EG to HG ; but the weight B is to the weight A, as EC to FC ; B is therefore to A, as EG, 
 or DG, to HG. 
 
 Or more generally, Let C be the centre of motion in the lever KL ; let A and B be any two plate 2. 
 powers applied to it at K and L, acting in the directions KA and LB. From the centre of *' J1> 
 motion C, let CM and CN be perpendicular to those directions in M and N ; suppose CM to 
 be less than CN, and from the centre C, at the distance CN, describe the circle NHD, meeting 
 KA in D. Let the power A be represented by DA, and let it be resolved into the power DG 
 acting in the direction CD, and the power DF perpendicular to CD, by completing the 
 parallelogram AFDG. The power DG, acting in the direction CD from the centre of the circle, 
 or wheel, DHN towards its circumference, has no cft'ect in turning it round the centre, from D 
 towards H, and tends only to csrry it oft' from that centre. It is the part DF only that 
 endeavours to move the wheel from D towards H and N, and is wholly employed in this effort. 
 The power B may be conceived to be applied at N as well as at L, and to be wholly employed 
 in endeavouring to turn the wheel the contiary way, from N towards H and I). If, therefore, 
 the power B be equal to that part of A which is represented by DF, these efforts, being equal 
 and opposite, must destroy each ether's effect ; that is, when the power B is to the power A, 
 as DF to DA, o:-, (because of'tfie similarity of the triangles AFD, DMC) as CM to CD, or as 
 CM to CN, then the powers must be in equilihno; and those powers will sustain eacli other, 
 which are inversely as the distances of their directions from the centre of motion. 
 
 SCIIOL. It is evident, that in the first kind of lever, either the weight may exceed the power 
 or the power may exceed the weight ; but in the second kind, the weight must exceed the power, 
 and in the third, the power must exceed the weight. The second is adapted to produce a slow 
 motion by a swift one; and the third serves to produce a swift motion of the weight, by a slow 
 motion of the power. See Fig. 1C, IS, and 14. 
 
 To the/rsi kind of lever may be reduced several sorts of instruments ; such as the steelyard, 
 whose arms are unequal ; the false balance, whose arms are imperceptibly unequal ; the common 
 balance, whose accuracy depends on its possessing the following properties; (!) The arms 
 must be equal in length and weight. (2.) The centre of motion must be a little above, and 
 
OF MECHANICS. 
 
 BOOK II. 
 
 Plate 3. 
 Fig. 2. 
 
 directly over the centre of gravity. (3.) The points from which the scales are suspended should 
 be in a right line, passing through the centre of gravity of the beam. And (4.) the friction ot 
 the beam on the centre of motion should be as little as possible. Scissars, pincers, snuffers, &c. 
 are formed of two levers, the fulcrum of which is the pin which rivets them. 
 
 To the second kind of lever may be reduced oars and rudders of ships ; cutting knives fixed 
 at one end ; doors moving on hinges, &c. 
 
 To the third kind, we may refer the action of the muscles of animals, laddes fixed at one 
 end, and raised against a wall by a man's arms, &c. sv ' -j 
 
 CtKt^ ^ 
 
 DBF. XI. The wheel and &acxs^is a wheel turning round together 
 
 its -axis ; the power is applied to the circumference of the wheel, and the 
 
 weight to that of the 
 
 by means of cords. 
 
 PROP. LIII. An equilibrium is produced in the wheel and^Htis? when 
 the weight is to the power, as the diameter of the wheel to the diameter 
 of ilie -axis* 
 
 Let AB be the diameter of the wheel, DE that of the axis, W the weight, and P the power, 
 suspended from the points D and 13. When the wheel has performed one revolution, the 
 power P has drawn off as much cord from the wheel as is equal to its circumference, and has 
 therefore moved through a space equal to that circumference. In the same time the weight 
 W is raised through a space equal to the circumference of the axis, upon which the cord, by 
 which the weight is suspended, is once turned round. Therefore the velocity of the power > 
 exceeds the velocity of the weight as much as the circumference, that is, the diameter of the 
 wheel exceeds that of the axis. If then the weight exceeds the power as much as the velocity 
 of the power exceeds that of the weight, that is, as much the diameter, or semi-diameter of the 
 wheel, AB, or CB, exceeds the diameter, or semidiameter of the axis, DE, or CE, the momenta 
 will be equal, and the power and weight will balance each other. 
 
 Or thus ; The axis and wheel is a lever of the first kind ; in which the centre of motion is 
 in C, the centre of the axis ; the weight W, sustained by the rope DW, is applied at the 
 distance DC, the radius of the axis ; and the power P, acting in the direction PB perpendicular 
 to CB, the radius of the wheel, is applied at the distance of that radius ; therefore, Prop. LII. 
 there is an equilibrium, when the power is to the weight, as the radius of the roller to the 
 radius of the wheel. 
 
 COR. 1. Hence it is evident, that by increasing the diameter of the wheel, or diminishing 
 that of the axis, a less power may sustain a given weight. 
 
 COR. 2. The thickness of the rope to which the weight is suspended, ought not to be 
 neglected. 
 
 SCHOL. To the wheel and axle we may refer the capstan, mills, cranes, &c. A drawing 
 and description of a safe and truly excellent crane, invented by Mr. James White, may be seen 
 in the 10th volume of the Transactions of the Society for encouraging Arts and Sciences, in 
 London. 
 
CHAP. VI. OF THE MECHANICAL POWERS. 4? 
 
 DEF. XII. The pulley is a small wheel, moveable about its axis, by 
 means of a cord, which passes over it. 
 
 PROP. LIV. When the axis of the pulley is fixed, the pulley only 
 changes the direction of the power; if moveable pullies are used, an equi- 
 librium is produced, where the power is to the weight as one to the uurnber 
 of ropes applied to them. If each moveable pulley has its own rope, each 
 pulley will double the power. 
 
 If the pulley ED bo fixed upon the beam A, the power and weight, in equilibrio, will be pi a t e 3. 
 equal. But, if one end of the rope be fixed in B, and the other supported by the power P, it F )S- 3. 
 is evident, that in order to raise the weight W one foot, the power must rise two ; for both 
 the ropes BC and CP, will he shortened a foot each ; whence the space run over by the 
 power will be double of that of the weight ; if therefore the power is to the weight as 1 to -2, 
 their momenta will be equal. For the same reason, if there be four ropes passing from the upper 
 to the lower pullies, the velocity of the power will be quadruple to that of the weight, or as 
 4 to 1, &c. In all cases, therefore, when the power is to the weight, as 1 to the number of 
 ropes passing from the upper to the lower pullies, there will be an equilibrium. 
 
 Or thus ; Every moveable pulley hangs by two ropes equally stretched, which must bear Fig. 5. 
 equal parts of the weight ; and therefore when one and the same rope goes round several 
 fixed and moveable pullies, since all its parts on each side of the pullies are equally stretched, 
 the whole weight must be divided equally amongst all the ropes by which the moveable pullies 
 hang 1 . Consequently, if the power which acts on one rope be equal to the weight divided by 
 that number of ropes, the power must sustain the weight. 
 
 If each moveable pulley has its own cord, the first, as appears from what has been said, Fig. 6. 
 doubles the velocity of the power ; and therefore if the power be half of the weight, the 
 momenta will be equal, and the balance will be produced. In like manner, the second pulley 
 caus.es the weight to move with half the velocity with which it would move, if suspended from 
 the first moveable pulley, that is, makes the velocity of the power quadruple of that of the 
 weight ; and so of the rest. 
 
 If in the solid block B, grooves be cut, whose radii are 1,3, 5, 7, &c. and in the block A Plate 12. 
 other jjrooves be cut, whose radii are 2, 4, 6, 8, &c. and a string be fastened to A and passed tl S- * 
 round these grooves, the grooves will answer the purpose of so many distinct pullies, and every 
 point i;i each, moving with the velocity of the string in contact with it, the whole friction 
 will be removed to the two centres of motion in the blocks A and B, which is a great 
 advantage over the common pullies. This pulley was invented by Mr. James White. 
 
 PROP. LV. In the inclined plane the power and weight balance each 
 other, when the power is to the weight, as the sine of the inclination of the 
 plane is to the sine of the angle, which the line of the direction of the 
 power makes with the perpendicular to the plane. 
 
 Let a weight be supported on the inclined plane CA by a power acting in any -given direc- Plate 3- 
 fion PD. Let the whole force, whereby the weight would descend perpendicularly, be rep- Fj S 7 
 
48 OF MECHANICS. BOOK II. 
 
 resented by BP ; and resolving PB into two forces, one of which, BD, is perpendicular to the 
 plane CA, and the other, PD, is in the direction of the power ; the force BD is destroyed by 
 the reaction of the plane, and the force PD will be sustained by an equal power, acting in 
 the direction PD. Therefore, when there is an equilibrium, the power is to the weight, as 
 PD to PB ; that is, as the sine of the angle PBD, or (El. VI. 8.) its equal CAB, to the sine 
 of the angle PJ)B. 
 
 When PD is in the direction of the plane, tins ratio becomes that of CD to CB, or of the 
 . height of the plane CB, to CA its length. 
 
 When the direction of the power PD is parallel to the base of the plane, the ratio of the 
 power to the weight becomes that of ED to EB ; or (El. VI. 8. Cor.) of CB, the height of the 
 plane, to BA the base. 
 
 When the direction of the power coincides with the perpendicular BD, the ratio of the 
 power to the weight becomes that of the sine of a finite angle, to the sine of an angle indefinitely 
 diminished. From which it appears, that no finite power is sufficient to support a weight upon 
 an inclined plane, if that power acts in a direction perpendicular to the plane. 
 
 DBF. XIII. The screw is a cylinder, which has either a prominent part, 
 or a hollow line, passing round it in a spiral form, so inserted in one of the 
 opposite kind, that it may be raised or depressed at pleasure, with the weight 
 upon its upper, or suspended beneath its lower, surface. 
 
 PROP. LVI. In the screw the equilibrium will be produced, when the 
 power is to the weight, as the distance between two contiguous threads, in a 
 direction parallel to the axis of the screw, to the circumference of the circle 
 described by the power in one revolution. 
 
 Plate 3. While the screw is made to perform one revolution, the weight W may be considered, 
 
 F 'ff- 8. as raised up an inclined plane cq, whose height cp is the interval between two contiguous 
 spirals, whose basely is the periphery of the cylinder, and whose length ccj is the spiral liive, 
 hy a power acting parallel to the base of the plane ; for such an inclined plane, involved about 
 a cylinder, will form the spiral line of the screw. A power at p, acting parallel to the base, is 
 in cquilibrio with the weight W to be raised, when the power is to the weight, as the height of 
 the inclined plane, to the base ; or, in this case as pc, the interval between the spirals, to the 
 circumference described by p ; but a power applied at P, which is to that applied at p, as the 
 circumference described by p, to the circumference described by P, has the same effect ; there- 
 fore there is an equilibrium, when the power applied at P is to the weight to be raised, as^c, 
 the interval between two contiguous spirals, to the circumference described by the power P. 
 
 DBF. XIV. The wedge is composed of two inclined planes, whose bases 
 are joined. 
 
 PROP. LXIl. When the resisting forces, and the power which acts 
 on the wedge, are in equilibrio. the weight will be to the power, as the 
 height of the wedge, to a hue, drawn from the middle of the base to one 
 
CHAP. VI. OF THE MECHANICAL POWERS. 49 
 
 side, and parallel to the direction in which the resisting force acts on that 
 side. 
 
 Let the equilateral triangle ABC represent a wedge, whose base, or back, is AC, whose Plate 3. 
 sides are the lines AB and CB, and whose height is the line BP, which bisects the vertical angle F 'S- 9l 
 ABC, and also the base perpendicularly in P. Let E and F represent two bodies, or two 
 resisting forces acting on the sides of the wedge perpendicularly, and whose lines of direction 
 EP and FP meet at the middle point of the base, on which the power P acts perpendicularly, 
 then will EP and FP (Kl. I. 5, and 26) be equal. Let the parallelogram ENFP be completed ; 
 its diagonals PN and EF will bisect each other perpendicularly in H. Now when these forces 
 (which act perpendicularly on the sides and base of the wedge) are in equilibrio, they will be to 
 each other (Prop. XIV.) as the sides and diagonal of this parallelogram ; that is, the sum of the 
 resisting forces will be to the power of P, as the sides EP and FP to the diagonal PN, or as one 
 side EP to half the diagonal F.I; that is (from the similarity of the right-angled triangles BEP, 
 EHP) as BP, the height of the wedge, to EP the Hue which is drawn from the middle of the 
 base to the side AB, and is the direction in which the resisting force acts on that side. 
 
 From the demonstration of this case, in which the resisting forces act perpendicularly on 
 the sides of the wedge, it appears that the resistance is to the power which sustains it, as one 
 side of the wedge AB is to the half of its breadth AP ; because AB is to AP, (El. VI. 8.) as BP 
 is to EP. 
 
 It appears also from hence, that if PN be made to denote the force with which the power P 
 acts on the wedge, the lines PE and PF, which are perpendicular to the sides, will denote the 
 force with which the power P protrudes the resisting bodies in directions perpendicular to the 
 sides of the wedge. 
 
 Let us now suppose, in the second case, that the resisting bodies E and F act upon the 
 wedge in directions parallel to the lines DP and OP, which are equally inclined to its sides, and 
 meet in the point P. Draw the lines EG and FK perpendicular to DP and OP; then making 
 PN denote the force with which the power P acts 011 the wedge, PE and PF will denote the 
 forces with which it protrudes the resisting bodies in directions perpendicular to the sides of tiie 
 wedge, as was observed before ; now each of these forces may be resolved into two, denoted 
 respectively by the lines PG and GE, PK and KF, of which GE and KF will be lost, as they 
 act in directions perpendicular to those of the resisting bodies; and PG and PK. will denote the 
 forces by which the power P opposes the resisting bodies, by protruding them in directions 
 contrary to those in which they act on the wedge ; therefore, when the resisting forces are in 
 equilibrio with the power P, the former must be to the latter, as the sum of the lines PG and 
 PK, is to PN, or as PG is to PH. But (El. VI. 4. 1 ) PG is to PE, as PE to PD ; and PH is to 
 PE, as PE to PB; whence (El. VI. 16.) both the rectangle PG x PD and the rectangle 
 PH X PB, are equal to the square of PE ; these rectangles are therefore equal to one another ; 
 whence their sides (El. VI. 14 ) are reciprocally proportional ; that is, PG is to PH, as PB to 
 PD. Whence it follows from what was shewn above, that, in equilibrio, the resisting forces are 
 to the power, as PB to PD ; that is, as the height of the wedge to the line drawn from the 
 middle of the base to one side of the wedge, and parallel to the direction in which the resisting 
 force acts on that side. 
 
 From what has been demonstrated, we may deduce the proportion of the po'ver to the 
 resistance it is able to sustain in all the cases in which the wedge is applied. First, when in 
 
 7 
 
50 OF MECHANICS. BOOK II. 
 
 cleaving timber the wedge fills the cleft, then the resistance of the timber acts perpendicular! v 
 on the sides of the wedge, therefore in this case, when the power which drives the wedge, is to the 
 cohesive force of the timber, as half the base, to one side of the wedge, the power and resistance 
 will be in equilibrio. 
 
 Secondly ; When the wedge does not exactly fill the cleft, which generally happens because 
 the wood splits to some distance before the wedge ; let ELF represent a cleft into which the 
 wedge ABC is partly driven ; as the resisting force of the timber must act on the wedge in 
 directions perpendicular to the sides of the cleft, draw the line PI) in a direction perpendicular 
 to EL, the side of the cleft, and meeting the side of the wedge in D ; then the power driving the 
 wedge and the resistance of the timber, when they balance, will be to each other as the line 
 PD to PB, the height of the wedge.* 
 
 Thirdly; When a wedge is employed to separate two bodies that lie together on a hori- 
 zontal plane, for instance, two blocks of stone ; as these bodies must recede from each other in 
 horizontal directions, their resistance must act on the wedge in lines parallel to its base CA ; 
 therefore the power which drives the wedge, will balance the resistance when they are to each 
 other as PA, half the breadth of the wedge, to PB its height. 
 
 SCHOL. 1. Since in all the mechanical powers, an equilibrium is produced, when the power 
 is to the weight as the velocity of the weight is to the velocity of the power, in all compound 
 machines there will be an equilibrium, when the sum of the powers are to the weight, as the 
 velocity of the weight is to the sum of the velocities of the powers. 
 
 SCHOL. 2. In the theory of mechanical powers, we suppose all planes and bodies perfectly 
 smooth ; levers to have no weight: cords to be perfectly pliable, and the parts of machines to 
 have no friction. (See Schol. S ) Allowances, however, must be made for the difference 
 between theory and practice. Mr. Ferguson observes, that there are but few compound 
 machines, but what, on account of friction, will require a third part more to work them, when 
 loaded, than what is sufficient to constitute an equilibrium between the weight and the power. 
 
 Fills' Exp. 1. Let A, B, C, be a compound lever, consisting of three levers, in the first of which, 
 
 A, the velocity of the weight is to that of the power, as 1 to 5; in the second, B, as 1 to 4 ; in 
 the third, C, as 1 to 6. The velocity of the weight will be to that of the power, as 1 to 
 5X4x6 = 120; and if the power be to the weight, as 1 to 120, they will balance each other. 
 
 Fig. 16. 2. Let GC and LF be the levers fixed to the supporters RA, SE, and let their shorter arms 
 
 be kept in equilibrio with the longer respectively by the weights fixed at G and L. Let NH be 
 a bar screwed to the fixed parts to keep them steady. If the power C be ten times farther 
 from A the prop, than the weight P, they will be in equilibrio when the power C is to the weight 
 P, as 1 to 10. In like manner, the distance ME being ten times DE, if the power M be J^ of 
 the weight C suspended from D, they will be in equilibrio ; whence M, 1, will balance P, 100. 
 
 3. Exhibit models or draughts of different compound machines, as mills, cranes, the pile- 
 driver, &c. 
 
 * In estimating' the lateral cohesion of woody fibres when separated by a wedge, the pressure on only one 
 tide of the wedge should be reckoned, so that on Mi account the cohesion should be estimated at only half 
 what it is in the text; but on anot/ter account, not mentioned in the text, it should be reckoned much more, 
 for the sides of the cleft are actually levers, in which the pressure of the wedge is the power, the point 
 where the cohesion is just giving way is the place of weight, while the fulcrum is at some distance further 
 from the wedge, more or less, according to the rigidity or flexibility of the timber j so that notwithstanding 
 the cohesive force is erroneously doubled in the text, it is probably much underrated. 
 
CHAV. VII. OF PROJECTILES. 31 
 
 SOHOL. 3. The inequality of the surface on which any body moves occasions an attrition, 
 called friction, which prevents the accurate agreement of many experiments in mechanics, with 
 theory- On this subject the very accurate experiments of Mr. Vince should be consulted, the 
 object of which was, to determine, ( i .) Whether friction be an uniformly retarding force, 
 (a.) The quantity of friction. (5.) Whether friction varies in proportion to the pressure or 
 weight. And (4.) whether the friction be the same, on whichever of its surfaces a body moves. 
 After a great variety of experiments made with the utmost care and attention, Mr. Vince 
 deduces the following conclusions, which may be considered as established facts. 
 
 1. That friction is an uniformly retarding force in hard bodies, not subject to alteration by 
 the velocity ; except when the body is covered with woollen cloth, &c. and in that case the 
 friction increases a little with the velocity. 
 
 2. Friction increases in a less ratio than the weight of the body, being different in different 
 bodies. It is not yet sufficiently known for any one body, what proportion the increase of 
 friction bears to the increase of weight. 
 
 3. The smallest surface has the least friction ; the weight being the same. But the ratio of 
 the friction to the surface is not accurately known. 
 
 See a full account of these experiments, Vol. LXXV, Phil. Trans. 
 
 SCHOL. 4. Wheel carriages are used, to avoid friction as much as possible. A wheel turns 
 round upon its axis, because the several points of its circumference are retarded in succession 
 by attrition, whilst the opposite points move freely. Large wheels meet with less resistance 
 than smaller from external obstacles, and from the friction of the axle, and are more easily 
 drawn, having their axles level with the horses. But in uneven roads, small wheels are used, 
 that in ascents the action of the horse may be nearly parallel with the plane of ascent, and 
 therefore may have the greatest ett'ect ; small wheels are also more conveniently turned. The 
 greater part of the load should be laid on the hinder part of a wheel carnage. 
 
 CHAP. VII. 
 
 Of Motion as produced by the united Forces of PKOJECTION and 
 
 GRAVITATION. 
 
 SECT. I. 
 Of Projectiles. 
 
 PROP. LVIII. Bodies thrown horizontally or obliquely, have a cur- 
 vilinear motion, and the path which they describe is a parabola ; the air's 
 resistance not being considered. 
 
 If a body be thrown in the direction AF, and acted upon by the projectile force alone, it will 
 continue to move on uniformly in the riht line AF, and would describe equal parts of the line 
 AF in equal times, as AC, CD, DE, &c. But if, in any indefinitely small portion of time, in 
 which the body would by the projectile force move from A to_C, it weuld, by the force of gravity, 
 
52 OF MECHANICS. BOOK II. 
 
 have fallen from A to G; by the composition of these forces (Prop. XVI.) it will, at the end of 
 that time, be found in H, the opposite angle of the parallelogram ACGH. [n two such portions 
 of time, whilst it would have moved from A to D by the projectile force, it would (Prop. XXVI.) 
 by gravitation fall through four times AG, that is, AM; and therefore, these forces being com- 
 bined, it will be found at the end of that time in I, the opposite angle of the parallelogram DM. 
 In like manner, at the end of the third portion of time, it would by the projectile force be carried 
 through three equal divisions to E, and by the force of gravitation over nine times AG to N 
 and consequently, by both these forces acting jointly, it will be carried to K, the opposite angle 
 of the parallelogram EN. Therefore the lines CH, JJI, EK ; that is, AG, AM, AN, which are 
 to each other as the numbers 1, 4, 9, are as the squares of the lines AC, AD, AE ; that is, GH, 
 MI, NK, which are as 1,2, 3. And because the action of gravitation is continual, the body in 
 passing from A to H, &c. is perpetually drawn out of the right line in which it would move if 
 the force of gravitation were suspended, and therefore moves in a curve. And II, I and K are 
 any points in this curve in which lines let fall from points equally distant from A in the line 
 AB meet the curve. Therefore the body moves in a parabola, the property of which is (Simp- 
 son's Conic Sections, Bonk I. Prop. XII. Cor.) that the abscissae AG, AM, AN, are to each other 
 as the squares of the ordinates Gil, MI, NK. 
 
 REMARK. Very dense bodies moving with small velocities describe the parabolick track so 
 nearly, that any diviation is scarcely discoverable; but with very considerable velocities the 
 resistance of the air will cause the body projected to describe a path altogether different from 
 a parabola, which will not appear surprising when it is known that the resistance of the air to 
 a cannon ball of two pounds weight, with the velocity of 20UO feet per second is more than 
 equivalent to 60 times the weight of the ball. See Button's Diet. Art. Resistance. 
 
 PROP. LIX. The path which a body thrown perpendicularly upward 
 describes in rising and falling is a parabola. 
 
 A stone lying upon the surface of the earth, partaking of the motion of the earth (here 
 supposed) round its axis, this motion which it has with the earth will not be destroyed by throw- 
 ing it in a direction perpendicular to the surface of the earth. After the projection, therefore, 
 the stone will be moved by two forces, one horizontal, the other perpendicular, and will rise in 
 a direction which may be shewn, as in the last proposition, to be the parabolick curve ; in which 
 it will continue till it reaches the highest point, fiom whence it might be shewn, as in the last 
 proposition, that it will descend through the other side of the parabola. 
 
 PROP. LX. The velocity with which a body ought to be projected to 
 make it describe a given parabola, is such as it would acquire by falling 
 through a space equal to the fourth part of the parameter belonging to that 
 point of the parabola from which it is intended to be projected. 
 
 Plate 3. The velocity of the projectile at the point A (by Prop. LVIII.) is such as would carry it 
 
 fig- 10 > from A to E, in the same time in which it would descend by its gravity from A to N. And the 
 
 velocity acquired in falling from A to N (by Prop. XXVII.) is such as in the same time by an 
 
 uniform motion would carry the body through a space double of AN. Therefore the velocity 
 
 which is acquired by the body in falling to N is to that with which the body is projected at A, 
 
CHAP. VII. OF PROJECTILES. 63 
 
 and uniformly carried forward to E, as twice AN is to AE. But since, from the nature of the 
 
 AE* 
 
 parabola, (Simpson's Conic Sections, Book I. Prop. XIII.) ^ is equal to the parameter of the 
 
 , 
 
 point A, one fourth part of this parameter will be expressed by 4 ' . I . And because the velo- 
 
 cities acquired by falling bodies are (by Prop. XXVI. Cor. 1.) as the square roots of the spaces 
 they fall through, the velocity acquired by a body in falling through AN is to the velocity 
 
 1 A V% 
 
 acquired in falling through * ' ' or one fourth part of the parameter of A, as the square root 
 
 of AN to the square root of ^Arr 5 that is, as ^/AN to ^- -, or AN to AE, or twice AN 
 
 A.N \/ AiN 
 
 to AE. Therefore the velocity acquired by a body in falling from A to N has the same ratio to 
 thfe velocity with which the body is projected or the line AE described, and to the velocity 
 acquired by a body in falling through a fourth part of the parameter belonging to the point A ; 
 consequently (El. V. 11.) these velocities are equal. 
 
 COR. Hence may be determined the direction in which a projectile from a given point, with late 
 a given velocity, must be thrown to strike an object in a given situation. 
 
 Let A be the place from which the body is to be thrown, and K the situation of the object. 
 Raise AB perpendicular to the plane of the horizon, and equal to four times the height from 
 which a body must Jail to acquire the given velocity. Bisect AB in G ; through G draw HG 
 perpendicular to AB ; at the point A raise AC perpendicular to AK, and meeting HG in C ; 
 on C as a centre with the radius CA describe the circle ABD ; and through K draw the right 
 line KEI perpendicular to the plane, of the horizon, and cutting the circle ABD in the points E 
 and I. AE, or Al, will be the direction required. 
 
 For, drawing Bl, BE, since AK is a tangent to the circle, and BA, IK, are parallel to each 
 other, the angie ABE (El. III. 32.) is equal to the angle EAK ; and the alternate angles BAE, 
 AEK, are equal ; therefore the triangles ABE, AEK, are similar j and AB is to AE, as AE to 
 
 AF 2 
 
 EK. Therefore AB x EK = AE S ; and AB = ^ . In like manner, the triangles BAT, 
 
 JoiK. 
 
 AI* 
 
 KAI, being similar, BA is equal to yrp-. Since, then, AB is equal to four times the height from 
 
 llv 
 
 AE 2 AI* 
 
 which a body must fall to acquire the velocity with which it is to be thrown ; -^ (or - its 
 
 E/H. iK. 
 
 equa') is the same. Consequently (by this Prop.) the point K will be in the parabola which the 
 body will descrioe, w!\ich is thrown with the given velocity in the direction AE, or AI, and the 
 body will strike au object placed at K. 
 
 SCHOL. If the velocity with which a projectile is thrown be required, it may be determined 
 from experiments in the following manner. By the help of a pendulum or any other exact 
 chronometer, let the time of the perpendicular flight be taken ; then, since the times of the 
 ascent and descent are equal, the time of the descent must be equal to one half of the time of 
 the iiijjht, consequently, that time will he known ; and, since a heavy body descends from a 
 state of rest at ti^e rite of 16.1 feet in the first second of time, and that the spaces through 
 which bodies descend are as tlie squares of the times ; if we say, as oue second is to 16.1 feet. 
 
M OF MECHANICS. BOOK II. 
 
 so is the square of the number of seconds which express the time of the descent of the projectile, 
 to a fourth proportional, we shall have the number of feet through which the projectile fell, 
 which being doubled, will give us the number of feet which the projectile would describe in the 
 same time with that of the fall, supposing it moved with an uniform velocity, equal to that which 
 it acquired by the end of the fall ; which last found number of feet, being divided by the number 
 of seconds which express the time of the projectile's descent, will give a quotient, expressing the 
 number of feet, through which (lie projectile would move in one second of time with a velocity 
 equal to that which it acquired in its descent, which velocity is equal to the velocity with which 
 the projectile was thrown up ; consequently, this velocity is discovered. 
 
 PROP. LXI. The squares of the velocities of a projectile in different 
 points of its parabola, are as the parameters belonging to those points. 
 
 For (by the last Prop.) the velocities in the several points of the parabola, are equal to the 
 velocities acquired in falling through the fourth parts of the parameters of the points. Therefore 
 the squares of these velocities being (by Prop. XXVI.) as the spaces described, the squares of 
 the velocities in the several points of the parabola are as the fourth parts of the parameters of 
 those points; but the whole parameters are as their fourth parts; therefore the squares of the 
 velocities at the several points of the parabola are as the parameters of those points. 
 
 COR. Hence, setting aside any difference which may arise from the resistance of the air, a 
 projectile will strike a mark as forcibly at the end as at the beginning of its course, if the two 
 points be equally distant from the principal vertex ; for, the parameters belonging to these points 
 being equal, the velocities in these points must also be equal. 
 
 PROP. LXII. When a body is thrown obliquely with a given velocity, 
 if the space through which it must have fallen perpendicularly to acquire that 
 velocity is made the diameter of a circle, the height to which the body will 
 rise is equal to the versed sine of double the angle of elevation. 
 
 Plate 3. Let a body be thrown in the direction BE, with the same velocity which any body would 
 
 Fig. 12. acquire by falling perpendicularly through AB ; if AB is made the diameter of a circle, the 
 greatest height to which the body will rise will be BD. 
 
 Let 1L be a right lii.e drawn in the plane of the horizon, touching the circle in B, and 
 making with the line BE, which is the direction in which the body is thrown, the angle 1BE, or 
 angle of elevation. Because IL touches the circle, and EB drawn in the circle meets it in the 
 point of contact, (El. III. 32.) the angle EBI is equal to the angle EAB. And ECB is double 
 of EAB, (El. HI. 20.) therefore ECB is double of EBI, the angle of elevation. And BD is the 
 versed sine of ECB ; that is, of double the angle of elevation. 
 
 Let BE represent the velocity with which the body is thrown. Then since this velocity is, 
 by supposition, such as might be acquired by falling down AB, if the body was thrown perpen- 
 dicularly upward with the same velocity BE, it would rise to the height BA. Let the oblique 
 motion BE be resolved into two others, one in the direction BD perpendicular to the horizon, 
 and the other in the direction DE parallel to it; then the ascending velocity will be to the 
 horizontal velocity, as BD to DE. and to the whole velocity, as BD to BE. But the part of the 
 
CHAP. VII. OF PROJECTILES. 65 
 
 velocitv BD is the only part winch is employed in raising the body, since the other part DE is 
 parallel to the plane of the horizon. Now, the height of a body ascending perpendicularly with 
 the whole velocity BE, will be to the height when it ascends with the part BD (compare Prop. 
 XXVI. and Prop. XXVIII.) as the square of BE to the square of BD. But because (El. VI. 8.) 
 the triangle EDB is similar to the triangle AEB, BD is to EB ; as KB is to BA; and BD, BE, 
 BA, being continued proportionals, BD is to BA, as the square of BD is to the square of BE. 
 And the perpendicular heights to which the velocities BE and BD will make the body ascend 
 have been shewn to be as the square of BE to the square of BD ; the heights are therefore as 
 BA to BD. Since therefore the first velocity BE would make the body ascend tl.ro.igh BA, the 
 other velocity BD, which is the part of the whole velocity which acts to make the body thrown 
 in the direction BE to ascend, will carry it to the height BD, which is the versed sine of double 
 the angle of elevation. The same might be shewn in any other direction of the body, as BF, 
 or BG. 
 
 DBF. XV. The Random of a projectile is the horizontal distance to 
 which a heavy body is thrown. 
 
 PROP. LXIII. When a body is thrown obliquely with a given velocitv, 
 if the space through which it must have fallen perpendicularly to acquire that 
 velocity is made the diameter of a circle, the random will be equal to four 
 times the sine of-double the angle of elevation. 
 
 If EBI be the angle of elevation, and ECB double that angle, DE will be the sine of double plate 3. 
 the angle of elevation. Let a body be thrown from the point B in the direction BE, with the f 'S- ** 
 velocity which it would acquire in falling through AB ; the random, or horizontal distance at 
 which the body will fall, is equal to four times DE. 
 
 For, since (as in the last Prop.) the velocity BE being resolved in BD, DE, the ascending 
 velocity is BD, and the horizontal DE, if these two velocities were to continue uniform, the 
 spaces described in equal times (Prop. V.) would be as the velocities, and in the same time in 
 which the body by the ascending velocity would rise through BD, by the horizontal velocity it 
 would be carried forward through DE. Of these velocities, the horizontal one DE is uniform, 
 because the force of gravity can neither accelerate nor retard a motion in this direction ; but the 
 ascending velocity is uniformly retarded ; and therefore the body (compare Prop. XXIII. and 
 XXVIH.) will be twice as long in ascending to its greatest height BD, as it would have been if 
 the first ascending velocity had continued uniform ; but on this supposition, the body would 
 have been carried through BD and DE in the same time ; therefore in double the time, that is, 
 in the time of ascent through BD with an uniforoily retarded velocity, it would be carried for- 
 ward through twice DE ; consequently, in the times of descent and ascent together it would 
 move forward through four times DE. Therefore a body thrown from B in the direction BE 
 with such a velocity as might be acquired by falling down AB, the diameter of a circle, will fall 
 at the distance of four times the sine of double the angle of elevation. 
 
 PROP. LXIV. The random of a projectile will be the greatest possible, 
 with a given velocity, when the angle of elevation is an angle of forty-five 
 degrees. 
 
36 OF MECHANICS. BOOK II. 
 
 Tlie velocity being given, the height from whence the body must have fallen to acquire that 
 velocity, or (Prop. XXXV.) the diameter of the circle AB, is a given quantity. And in a 
 given circle the greatest sine is the radius or sine of a right angle; therefore four times the 
 radius is greater than four times any other sine; and consequently, the random which is equal 
 to four times the radius, (which by Prop. LXII will be the case when the double angle of 
 elevation is a right one, or the angle of elevation forty -five degrees) will be the greatest possi- 
 ble randum. 
 
 ESP. This proposition, and the two following, may be illustrated by water spouting from a 
 pipe. 
 
 PROP. LXV. The random of a projectile, whose velocity is given, 
 will be the same at two different elevations, if the one be as much above 
 forty-five degrees as the other is below it. 
 
 Plate 3. If EBI be an angle of SO degrees, and OBI an angle of 60 degrees, because EBI falls 
 
 Fig. 12. short of half a right angle as much as GBI exceeds it, the double of EBI will fall short of a right 
 
 angle as much as the double of GBI will exceed it ; therefore, from the definition of a sine, these 
 
 doubles will have the same sine. Consequently, four times their sines, that is, (by Prop. LXIII.) 
 
 their randoms will be equal. 
 
 PROP. LXVI. The greatest random of a projectile, whose velocity is 
 given, is double the height to which it would rise if it were thrown per- 
 pendicularly with the same velocity. 
 
 Plate 3 ^ a kdy be projected in the direction BF, at an angle of forty-five degrees, and its 
 
 Fig-. 12. velocity be equal to that which a body would acquire in falling down AB (*>y Prop.- LXIV.) 
 the random will be the greatest possible, and will be equal to four limes CF, or twice BA. 
 But the body cast perpendicularly upwards with the same velocity would (by Prop. XXVIII.) 
 rise to the height BA. Therefore the greatest random, with a given velocity, is double the 
 height to which the body, thrown perpendicularly with the same velocity, would rise. 
 
 PROP. LXVII. The randoms of projectiles, whose elevations are given, 
 are as the squares of their velocities. 
 
 Pl:.te 3. If a body be thrown in any direction BE, its random (Prop. LXIII.) will he equal to 
 
 Fig. 12. f nur times DE, or tour times the sine of double the angle of elevation, in a circle whose 
 diameter AH is the height from which the body must fall to acquire the velocity with which it is 
 projected. And because, in the triangle EDO the angle at D being a right angle is always 
 invariable, and that (he angle ECI), which is double of EAD, that is. (",l. Ill 32.) of the given 
 angle of elevation EBI, is given, the triangle ECD in every variation of AB, is always 
 equiangular and similar to itself, and ED is always as EC ; but EC being a radius, is as AB: 
 therefore El), the sine of twice the given angle of elevation, is as A B, the diameter. Con- 
 sequently four times the sine ED. that is, the random, is as AB. But the height AB. from which 
 a body must fall to acquire any velocity, is (by Prop. XXVI.) as the square of that velocity. 
 Therefore the random is as the square of the velocity- 
 
CHAP. VII. OF CENTRAL FORCES. 57 
 
 SECT. II. 
 Of Central Forces. 
 
 PROP. LXVIII. A body which is constantly drawn or impelled toward 
 any point, may be made to describe, round that point as a centre, a curve 
 returning into itself. 
 
 Let T be the centre of the earth, and GDEI its surface. Let a body be projected in any Plate 3. 
 direction GH, which does not pass within the surface of the earth. The projectile force, Fl ' 14- 
 together with the-force of gravity, will make it describe a curve, which, as the projectile force is 
 increased, will recede farther from the perpendicular GE, as GB, GC, GD. It is manifest 
 that the projectile force may be increased, till the body shall pass beyond the surface CDKE, 
 and move in the path GML, GN V, .or some larger curve. 
 
 First; Suppose the projectile force to be such, that the body will _be 'carried in the semi- 
 circle GN, it will continue in the curve of that circle till it returns to G. For, when a body 
 moves in the circumference of a circle (as in fig. 15.) the projectile force, acting in a line which 
 is a tangent to the circle, as GB, acts (El. III. 18 ) in a direction which is perpendicular to the Plate 3. 
 direction BA, in which it is impelled toward the centre. And since if the force which Fl f> 15 - 
 impels the body towards the centre ceased to act in any point, as C the body would move 
 forward in the right line CF, the projectile force in every point of the circumference, acts in 
 a direction perpendicular to the force of gravitation ; consequently, these two forces remaining 
 the same, and acting always in the same direction with respect to each other, the velocity of 
 the body must remain the same ; whence, at the point M, it will have the same power to recede Fl J>- 14< 
 from the centre as at G ; and, retaining this power through every remaining part of its course, 
 it will proceed in the circumference, till it arrive at G, and will continue to revolve in the 
 circle. . .. 
 
 Next; Let the body be projected from G with a force less than that which is required 
 to carry it round in the circumference of the circle GNV ; and let the curve in whichV.it moves 
 be an ellipse, having the earth in its remoter focus. Because the force of projection, as the 
 body proceeds in the first half of its orbit, acts in the direction of a tangent to the curve, whilst 
 the force of gravitation acts in the direction of a right line from the body to the centre of the 
 earth, the directions of these two forces make an acute angle with one another, and consequently, 
 through this part of the course of the body, the force.of gravitation conspiring with the force of 
 projection, the velocity of the body must be increased, and at the same time it must be 
 continually drawn downward toward the earth. At the point in which the forces act in 
 directions perpendicular to each other, the force of gravitation does not conspire with that of 
 projection to bring the body toward the earth ; and afterward, in the latter half of its course, 
 the directions of the forces making an obtuse angle with each other, the force of gravitation is 
 opposed by that of projection in the same degree in which the former was before aided by the 
 latter ; and therefore the body in passing toward G will fly oft' from the earth or rise, as much 
 as it before approached to the earth or descended, and thus will return to the point G with the 
 same velocity with which it set out at first, having lost as much velocity by receding from 
 the earth in the latter part of its course, as it had gained by falling toward the earth in 
 the former part. 
 
 8 
 
58 OF MECHANICS. BOOK II. 
 
 Lastly ; Let the body be projected from G with a force which is greater than sufficient to 
 carry it round in the circle GNV ; and let it perform its revolutions in an elliptic curve, whose 
 greater axis is greater than the diameter of the circle GNV, setting out from G, and having the 
 earth in the nearer focus ; the effect will be the same as in the last case, except that the 
 projectile force will oppose the force of gravitation in the first half of the revolution, and 
 conspire witli it in the latter. 
 
 EXP. Let a ball revolve round the central point of a whirling table. Concerning the 
 construction and use of this machine, see Ferguson's Lectures, Lect. II. 
 
 PROP. LXIX. A body revolving in an orbit, endeavours in every point 
 of its course to fly off from the centre in a right line, which is a tangent 
 to the orbit. 
 
 Plate 3. Let BCD L be a circle in which a body is revolving; when it is arrived at the point B, 
 
 'f>' "* let the force which impels it toward the centre be withdrawn, and the body (by Prop. I ) 
 would fly off from the point 8 in the direction BG; in like manner at C, it would fly off in 
 the right line CF; at D, in DH ; and at L, in LK. The same is manifestly true in an elliptical 
 orbit. Now the same force with which it would fly off, if no other cause prevented it, must 
 make it endeavour to My off in the same manner in every point of the orbit. 
 
 EXP. Whilst a ball is revolving on a whirling table, if the cord which retains it be suddenly 
 cut, the ball will fly off in a right line, which will be a tangent to the orbit in which it 
 moved. 
 
 COR. A body revolving about a centre endeavours to recede from that centre ; for every 
 point of the tangent in which it endeavours to move out of the circle, is farther from the centre, 
 than the point in which the tangent meets the curve. 
 
 DEF. XVI. The force which impels a body toward the centre, when 
 it revolves in an orbit, is called the centripetal force ; that by which it 
 endeavours to recede from the centre, is called the centrifugal force ; and 
 these two forces are called jointly, central forces, 
 
 SCHOL. The projectile and centrifugal forces differ from each other, as the whole from the 
 part. The projectile force is that with which a body would move forward in a tangent to its 
 orbit, if there were no centripetal force to prevent it ; the centrifugal force is that part of the 
 projectile force which carries the body off from the centre while it is describing the tangent. 
 1'late 3. Thus, if the body revolved in the orbit BD, the projectile force is that -which would make it 
 describe the tangent BA, if the centripetal force were to cease acting. But in the mean time, 
 the whole force BA does not carry the body off from the centre C ; when it is arrived at A, 
 it is farther from the centre than it was at B, only by the length AN, and it is that part of the 
 projectile force which, when the whole is resolved into two forces, may be considered as acting 
 in this line AN, which carries the body off from the centre, and is called the centrifugal force. 
 
CHAP. VII. OF CENTRAL FORCES. 59 
 
 PROP. LXX. When bodies revolve in a circular orbit about a centre 
 the centripetal and centrifugal forces are equal. 
 
 If a body revolve in the circle BD, in the time in which it describes the arc BN, it will have Plate 3. 
 been impelled toward the centre through the space AN; for, by the projectile force alone it Fl - 1<5> 
 would have been carried from B to A. The line AN is then the space described by means of 
 the centripetal force, and this force is proportional to AN. But if, when the body was at B, 
 no centripetal force had acted upon it, instead of describing the arc B\, it would have moved 
 along the tangent B A, and the line NA weuld have been the space through which it would 
 have departed from the centre ; therefore the centrifugal force is proportional to NA. Both 
 these forces being then proportional to the same line NA, they are equal to one another. 
 
 LEMMA I. 
 
 Quantities and the ratios of quantities, which, in any finite time, tend 
 continually to equality, and, before the end of that time, approach nearer to 
 each ottier than by any given difference, become ultimately equal. 
 
 If you deny it, let them be ultimately unequal ; and let their ultimate difference be D. 
 Therefore they cannot approach nearer to equality than by that given difference D ; which is 
 contrary to the supposition. If a straight and a curve line, continually diminishing, perpetually 
 approach toward equality, and at the end of any finite time would vanish together, at the 
 instant in which they are vanishing they are equal. 
 
 LEM. II. If in any figure AacE, terminated by the right lines Aa, Plate 3. 
 AE, and the curve ocE, there are inscribed any number of parallelograms ' s ' 
 Aft, Be, Cd, Sfc. contained under equal bases AB, BC, CD, &fc. and the sides, 
 ~Bb, Cc, Dd, &c. parallel to Aa the side of the figure ; and the parallelograms 
 alibi, blicm, cMrfn, 8fc. are completed ; then, if the breadth of these paral- 
 lelograms be diminished, and their number augmented continually, the 
 ultimate ratios, which the inscribed figure AK&LcMrfD, ths circumscribed 
 figure \albmcndolZ, and the curvilinear figure A.abcdE, have to each other, 
 are ratios of equality. 
 
 For the difference of the inscribed and circumscribed figure is the sum of the parallelograms 
 Kl, Lm, M, Do, that is, (because of the equality of all their bases) the rectangle under one of 
 their bases Kb, and the sum of their altitudes Aa , that is, the rectangle AEJn. But this 
 rectangle, because its breadth AB is diminished indefinitely, becomes less than any given rectan- 
 gle. Therefore (by Lem. I.) the inscribed and circumscribed, and much more the intermediate 
 curvilinear figure become ultimately equal. 
 
 LEM. III. Tlie same ultimate ratios are also ratios of equality, when plates. 
 the breadths AB, BC, CD, "c. of the parallelograms are unequal, and are Fi s- 18 - 
 all diminshed indefinitely. 
 
60 OF MECHANICS. BOOK II. 
 
 For, let AF be equal to the greatest breadth; and let the parallelogram VAaf be completed. 
 This will be greater than the difference of the inscribed and circumscribed figures ; but, because 
 its breadth AF is diminished indefinitely, it will become less than any given rectangle. 
 
 COR. Hence the ultimate sum of the evanescent parallelograms coincides in every part 
 "with the curvilinear figure. Much more does the rectilinear figure, which is comprehended 
 under the chords of the evanescent arcs aft, be, cd, &c. ultimately coincide with the curvilinear 
 figure. As also the circumscribed rectilinear figure, which is comprehended under the tangents 
 of the same arcs. And therefore, these ultimate figures (as to their perimeter ocK) are not 
 rectilinear, but curvilinear limits of rectilinear figures. 
 
 PROP. LXXI. The plane of an orbit in which a body revolves passes 
 through the line of projection, and through the centre toward which the 
 centripetal force is directed. 
 
 Jj? ate 9 4 ' -ket ABCF be the orbit in which the body revolves ; S the centre, or point toward which 
 
 the centripetal force is directed ; and AV the line of projection ; the plane of the orbrt will pass 
 through AV and S ; or the orbit lies in the same plane, with the lines AV, B</, 8cc. (lines in 
 which the projectile force acts in different parts of the orbit) and with the centre S. 
 
 For, let ABC D represent a pait of the orbit described by a body impelled toward S. The 
 body beginning to move by the projectile force from A in the direction ABV would, by that 
 force alone, be carried on uniformly in that direction. Suppose the centripetal force to act 
 upon it by separate impulses after equal intervals of time, and that, when the body is carried 
 by the projectile force to B, it receives one impulse from the centripetal force, drawing it out 
 of its course toward S, so that by the action of both forces together at B, it will (by Prop. XIV.) 
 be made to describe BC in the same time in which the projectile force alone would have made 
 it describe Be. The same will take place after equal intervals at C and D. At B, the projec- 
 tile force rs in the direction BV, the centripetal force in the direction BS. Let Be and BG, 
 taken in the direction of these forces, represent their ratio to each other, and the projectile 
 force be to the centripetal as Be to BG. The body (by Prop. XIV.) will describe BC, the 
 diagonal of a parallelogram of which Be and BG are the sides. But (El. XI. 1. and 2.) BC is 
 in the same plane with Be and BG, that is, with AV, the line of projection, and with BS, in 
 which is the centre or point S. The same may be proved concerning the lines CD, and DE. 
 And if the centripetal force acts continually, and not by interrupted impulses, the diagonals 
 AB, BC, &c. will be diminished indefinitely, and the ultimate perimeter ADF (by Lem. III. 
 Cor.) will become a curve line, which, from what has been shewn, must be always in the same 
 plane with the line of projection, and with the centre. 
 
 PROP. LXXII. A body revolving in an orbit describes, by a radius 
 drawn to the point toward which the centripetal force acts, equal areas in 
 equal tinies, and in unequal times areas proportional to the times. 
 
 Plate 4. Let ABCD be part of an orbit described by a body which revolves round the point S, toward 
 
 fig. 2. which it is impelled by a centripetal force. If this force be supposed to act upon the body by 
 
 separate impulses, as at B, C, D, when the body receives the impulse at B, it will be drawn out 
 
 
CHAP. VII. OF CENTRAL FORCES. 61 
 
 of its course toward S, and (by Prop. XIV.) will describe the diagonal EC in the same time in 
 which the projectile force alone would have made it describe Be. After equal intervals, the 
 same will take place at C, and at D. 
 
 Since AB, BC, &c. are the lines described in equal times by the body, the areas described 
 round S by a radius drawn from the body to S, are ASB, BSH, &c. Now AB, Be, expressing 
 spaces passed over in equal times by the uniform motion of the. body acted upon by the projectile 
 force alone, are equal bases of the triangles ASB, BSe, which, being terminated by the same 
 point S, are of the same altitude ; these triangles are therefore (El. I. 58.) equal. And, because 
 the body at B is by the joint action of the projectile and centripetal forces carried forward in 
 the diagonal BC, of the parallelogram Ge. the opposite sides thereof, G, Cc, are parallel ; and 
 Ce is parallel to BS. But BS is the common base of the two triangles BSC, BSe. Therefore 
 these triangles, being upon the same base, and between the same parallels, (El. I. 37.) are equal. 
 Consequently, ASB, which has been proved equal to BSc, is likewise equal to BSC ; that is, the 
 areas described in equal times are equal. And, by composition, any sums of these areas ASC, 
 ASE, are to each other as the times in which they are described 5 that is, universally, the areas 
 are as the times. 
 
 Let the number of these triangles be augmented, and their breadth diminished indefinitely, 
 and (by Lem. III. Cor.) their ultimate perimeter will be a curve lin.e ; and therefore the centri- 
 petal force will act continually ; and the above reasoning still being applicable to those triangles 
 whose breadth is indefinitely diminished, the areas will be as the times. 
 
 COR. 1. The velocity of a body revolving freely about an immoveable centre is inversely as 
 a perpendicular let fall from that centre on a right line that touches the orbit. For since the 
 lines AB, BC, are described in equal times, the velocities will be as other lines, which being 
 (by this Prop.) the bases of equal triangles, must be inversely as the heights of the triangles; 
 therefore the velocities are inversely as these heights, which are measured by perpendiculars 
 let fall from the common vertex, the centre S, to the bases, or the bases produced ; that is, when 
 AB, BC, &c. are indefinitely small, to a tangent to the orbit. 
 
 COR. 2, If the chords AB, BC, of two arcs successively described in equal times by the 
 same body moving freely, are completed into a parallelogram ABCG, and the diagonal BG, in 
 the position which it ultimately acquires when these arcs are diminished indefinitely, be pro- 
 duced toward S, it will pass through S, the centre of the centripetal force; for BG is an 
 indefinitely small part of the radius SB. 
 
 COR. 3. If the chords AB, BC, and DE, EF, of arcs described in equal times be completed 
 into the parallelograms ABCG,DEFZ, the centripetal forces at BandE, will be to each other in 
 the ultimate ratio of the diagonals BG, EZ, when those arcs are indefinitely diminished. For, the 
 motions of the body BC, EF, (by Prop. XVI.) are compounded of the motions Be, BG, and 
 Ef, EZ; but BG and EZ are equal to Cc and F/, which, as appears from this proposition, are 
 generated by the impulses of the centripetal force in B and E, and are therefore proportional to 
 those impulses. 
 
 COR. 4. The forces with which bodies are drawn into curvilinear orbits are to each other as 
 the versed sines, (JB, |ZE, of the indefinitely small arcs AC, DF, described in equal times, 
 which versed sines converge to the centre S. and bisect the chords when those arcs ate diminished 
 indefinitely ; for such versed sines are half the diagonals of a parallelogram, BG, EZ, being 
 bisected by the diagonals AC, FD. 
 
63 OF MECHANICS. BOOK II. 
 
 PROP. LXXIII. When a body describes equal areas in equal times 
 about an immoveable point, or proportional areas in unequal times, it is 
 impelled toward that point by the centripetal force which retains it in its 
 orbit. 
 
 Plate 4. Let AB, BC, &c. be lines described by the revolving body in equal times ; and it may be 
 
 Ft S- 2 - proved as before, that the triangles ASB and BSe being of the same height, and having equal 
 bases AB, Be, are equal. But, by supposition, ASB, BSC, are equal ; therefore BSc and BSC 
 are equal. And these equal triangles, being upon the same base SB, (El. I. 39.) are between 
 the same parallels ; therefore Cc and BS are parallel. Because the body at B is acted upon by 
 two forces, the projectile force in the line Be, and, the centripetal force, and by supposition, 
 these two forces together make it describe BC ; BC is the diagonal of a parallelogram of which 
 Be is one side, and the direction of the centripetal force at B is in the other side. Now, in the 
 parallelogram whose diagonal is BC, and one of its sides Be, Cc must be another ; whence the 
 opposite side, that is, the direction of the centripetal force, must be parallel to Cc ; but Cc and 
 BS have been proved to be parallel, when equal aieas are described in equal times about the 
 point S. Therefore, on this supposition, the body at B is acted upon by the centripetal force in 
 the direction BS. The same may be shewn at every other point C, D, &c. Therefore the 
 centripetal force tends to that point round which the body, by a radius drawn thither, describes 
 equal areas in equal times. 
 
 plate 4. LEM IV. If any arc ACB, of a finite curvature, is subtended by its 
 chord AB, and a straight line AD produced bath ways touch the arc in the 
 point A, the arc, the chord, and the tangent, in their ultimate vanishing state, 
 will be equal. 
 
 The arc being supposed of a finite curvature, or such as may be measured by a circle of a 
 finite diameter; let BAG he the circle of the curvature ; draw the line AC in that circle parallel 
 to the subtense BD, and complete the triangle BAG. Because the angle DAB (El. III. 32.) is 
 equal to the angle ACB, and the alternate angles CAB, ABD, are equal, the triangles, CAB, 
 DAB, are similar. Hence, (El. VI. 4.) AB is to AD, as AC is to BC. The point B approach- 
 ing continually to A, let BA become less than any assignable quantity ; then the finite lines 
 AC, BC, approach nearer to the ratio of equality, than by any given distance ; therefore likewise 
 AB, AD, which are proportional to AC, BC, and much more the intermediate arc, are ultimately 
 equal. 
 
 LEM. V. The nascent or evanescent subtense of the angle of contact, 
 in circles and in all curves which have a finite curvature, is as the square of 
 the conterminous arc. 
 
 Plate 3. Let AD, in the semicircle ADC, be the given arc, AB the tangent, and the angle BAD the 
 
 angle of contact. Draw BD, HG, parallel to AC the diameter; these lines, subtending the 
 angle BAD, are called the subtenses of the angle of contact. The arcs AD and AG, having the 
 common term or limit, the point A, are called conterminous arcs. Draw the lines DC, GC. 
 If these lines be conceived to turn round upon the point C as a centre, so that the two points 
 
CHAP. VII. OF CENTRAL FORCES. 63 
 
 D, G, and with them the two subtenses BD, HG, may approach toward A ; it is manifest, that 
 as these subtenses come nearer to A, they will diminish, and at hist will vanish in A. At the 
 instant of their vanishing, BD will be to HG, as the square of the arc AD is to the square of the 
 arc AG. 
 
 Let ED be drawn parallel to AB, and FG to AH. Then, because AB is a tangeut at the 
 point A, and consequently (El. III. 18.) perpendicular to the diameter AC, ED, which is parallel 
 to AB, is likewise perpendicular to AC ; for the same reason FG is perpendicular to AC. And 
 ADC, AGC (El. III. 31.) are right angles. Therefore (El. VI. 8.) AED is similar to ADC, and 
 AE is to AD, as AD to AC. Therefore (El. VI. 17.) the rectangle of AE, AC, is equal to the 
 square of AD. But AE is equal to BD ; therefore the rectangle BD, AC, is equal to the square 
 of AD. For the same reason, the rectangle of HG, AC, is equal to the square of the chord AG. 
 Consequently, the square of the chord AD is to the square of the chord AG, as the rectangle 
 BD, AC, is to the rectangle HG, AC, that is (El. VI. 1.) as BD to HG. But (by Lem. IV.) 
 the arc AD and the chord AD, are ultimately in the ratio of equality, and also the arc AG and 
 the chord AG. Therefore the square of the arc AD is to the square of the arc AG, at the 
 instant in which they vanish, as BD to HG ; that is, the evanescent subtense of the angle of 
 contact is as the square of the conterminous arc. 
 
 Next ; Let the subtenses be not parallel to the diameter, hut parallel to one another. Let Fig. 16. 
 MN, FG, be the subtenses parallel to the diameter; and AN, OG, two subtenses parallel to 
 each other, but not to the diameter. Because BA is a tangent at the point B, BD (El. III. 16.) 
 is perpendicular to BA ; since, therefore, MN and FG are parallel to BD, they are also per- 
 pendicular to B A, and the angles OFG, AMN, are equal. But because OG and AN are parallel 
 by construction, the angle FOG (El. I. 29.) is equal to the angle MAN. Therefore the triangles 
 FGO, MNA, are similar, and (El. VI. 4.) AN is to OG, as MN is to FG. But it has been 
 proved that MN is ultimately to FG, as the squares of the conterminous arcs; therefore AN is 
 ultimately to OG, as the squares of the conterminous arcs BN, BG. 
 
 Lastly ; Suppose both AN and OG directed to C, the centre of the circle. In this case, pj a te 3. 
 each of these would be a semidiameter, continued from G and N respectively to the tangent BA. Fi g- 16 - 
 In their ultimate state these lines AN, OG, must coincide in the point B, and in the same right 
 line BC ; and therefore will become parallel, and will be, from what has been shewn, ultimately 
 as the squares of the conterminous arcs. 
 
 If GC, DC, be beginning to move from A, they are in their nascent state ; and it is manifest, pig. 17. 
 that the subtenses in this state are the same, and therefore have the same ratio, as in the 
 evanescent state. 
 
 COR. 1. Hence, because the tangents AB, AH, the arcs AD, AG, and their sines ED, FG, 
 became ultimately equal (by Lem. IV.) to the chords AD, AG, their squares also will be 
 ultimately as the subtenses BD, HG. 
 
 COR. 2. The same squares are also ultimately as those versed sines of the arcs, which bisect 
 the chords and converge to a given point. For by the second rase of this proposition these 
 versed sines AE, AF, are as the subtenses BD, HG, or AN, OG. 
 
 COR. 3. Hence these versed sines AE, AF, are as the squares of the times in which a body 
 describes the arcs AD, AG. with given velocities. For the spaces AD, AG, described with 
 given velocities, are as the times, and the squares of the spaces as the squares of the times ; but 
 (by last Cor) the squares of these spaces are as the versed sines AE, AF ; therefore these 
 versed sines are as the squares of tfce times in which the arcs AD, AG, are described. 
 
61 OF MECHANICS. BOOK II. 
 
 LEM. VI. The nascent or evanescent subtense of the angle of contact is 
 equal to the square of the conterminous arc divided by the diameter. 
 
 Plate 3. It has been shewn in the preceding Lemma, that BD is to AD, as AD is to AC. Therefore 
 
 F * lr ' AD* 
 
 BD multiplied into AC is equal to the square of AD. and BD = . But (i>y Lem. IV ) the 
 
 Ai> 
 
 arc AD is ultimately equal to the chord AD ; therefore the nascent or evanescent subtense BD 
 is equal to the square of the arc AD divided by the diameter AC. 
 
 PROP. LXXIV. The centripetal forces of bodies, revolving in different 
 circular orbits about the same centre toward which they tend, are as the 
 squares of the arcs described in the same time, divided by the radii of the 
 circles. 
 
 Plate 3. In the circular orbits END, RLE, let bodies revolve about the centre C, toward which they 
 
 Fig. 16. tend. Let them in the same time describe the indefinitely small arcs BG, RL. Then because 
 the projectile forces would carry them in the same time through the tangents BK, RH, and the 
 spaces through which, at the points G and L, they have been drawn from the tangents toward 
 the centie by the centripetal force, are FG, HL; the centripetal forces mud be as FG and HL. 
 And (by Lem. VI.) the evanescent, or nascent, subtense FG is equal to the square of the arc BG 
 divided by BD, and the evanescent, or nascent, subtense HL is equal to the square of the arc 
 RL divided by RE. Therefore the subtense FG is to the subtense HL as the square of the arc 
 BG divided by BD or its half BC, is to the square of the arc RL, divided by RE, or its half RC. 
 Therefore the centripetal forces, when the arcs are nascent, are in the same ratio ; that is, as 
 the squares of the arcs divided by the radii. 
 
 And this is true, whatever arcs BG and RL be taken, if they be described in the same time ; 
 for the nascent arcs will be as the velocities : and any other arcs BND, RLE, described in any 
 given time, will be also as the velocities ; therefore, the arcs BND, RLE, are as the nascent arcs 
 BG, RL, and their squares are likewise proportional. But the centrifugal forces are as the 
 squares of the nascent arcs, BG, RL, divided by the radii BC, RC ; therefore these forces are as 
 the squares of any other arcs, BND, RLE, divided by the radii of their circles. 
 
 PROP. LXXV. The centripetal forces of equal bodies revolving iu 
 circular orbits, are as the squares of the velocities directly, and the radii of 
 the orbits inversely. 
 
 Because arcs described in the same time are as the velocities, and that the centripetal forces 
 are (by Prop. LXXIV.) as the squares of the arcs described in the same time divided by the 
 radii, these forces are also as the squares of the velocities divided by the radii, that is, as the 
 squares of the velocities directly, and the radii of the otbits inversely. 
 
 COR. Hence the centripetal foices of equal bodies, at equal distances from the centre, are 
 as the squares of the number of revolutions in any given time ; for this number is as the velocity 
 with which the body moves. 
 
CHAP. VII. OF CENTRAL FORCES. 65 
 
 PROP. LXXVI. The centripetal forces of equal bodies revolving in 
 equal circular orbits are inversely as the squares of their periodical times. 
 
 The circular orbits or spaces being equal, the times in which these are described, or the 
 periodical times, are (!y Prop. V.) inversely as the velocities ; and therefore the squares of the 
 periodical times are inversely as the squares of the velocities, or the squares of the velocities are 
 inversely as the squares of the periodical times ; but (by Prop. LXXV.) the centripetal forces 
 are as the squares of the velocities ; therefore these forces are inversely as the squares of the 
 periodical times. 
 
 PROP. LXXVII. The centripetal forces of equal bodies revolving in 
 unequal circular orbits, if the periodical times are equal, are as the radii of 
 the circles. 
 
 Let one body revolve in the circular orbit END, and another, in the same time, in the Plate 3. 
 circular orbit RLE. Because the periodical times are equal, each body in any given part of its '*>' 
 periodical time will describe an equal number of degrees in its respective orbit, that is, will 
 describe similar arcs. The arcs BN, RL, being similar, will be described in equal portions of 
 the periodical time ; therefore (by Prop. LXXIV:) the centripetal forces will be as the squares 
 
 BN* RL 2 
 
 of the similar arcs BN, RL, divided by the radii BC, RC ; that is, as -r 10-7^77 . But because 
 
 isL> KVj 
 
 similar arcs are to each other as the circumferences, or radii, of circles, BN is to RL as BC to 
 
 "R \T 2 T? T 2 "R i~* 2 
 
 RC, and consequently, BN to RL* as BC 2 to RC 2 . Therefore ^ is to - as -: is to 
 
 jiL< KC ISC 
 
 RO* BN 3 RL S 
 
 -T 7 ; that is, as BC to RC. But the centripetal forces (Prop. LXXIV.) are as -p to -77-7 ; 
 K.C/ liO Jvl_/ 
 
 therefore these forces are as BC to RC ; that is, as the radii of the orbits in which the bodies 
 move. 
 
 PROP. LXXVIII. The centripetal forces of equal bodies revolving in 
 circular orbits, are as the radii of the orbits directly, and the squares of the 
 periodical times inversely. 
 
 If the periodical times are equal, and the radii unequal, the forces are (by Prop. LXXVII.) 
 as the radii. If the radii are equal, and the periodical times unequal, the forces (by Prop. 
 LXXVI ) are inversely as the squares of the periodical times. Therefore, if both the radii and 
 periodical times are unequal, the forces will be in the compound ratio of both, or as the radii 
 directly, and the squares of the periodical times inversely. 
 
 PROP. LXXIX. When bodies revolve round the same centre, if the 
 squares of their periodical times are as the cubes of their distances from the 
 centre, the centripetal forces will be inversely as the squares of their dis- 
 tances. 
 
 9 
 
6ti OF MECHANICS. BOOK II. 
 
 Let the distances of the two bodies be expressed by D, d; and the periodical times by P,p ; 
 then, by the supposition, P 2 : p* : : D 3 : d 3 . 
 
 By Prop. LXX.VI1I. the centripetal forces are as the distances directly, and the squares of 
 
 the periodical times inversely ; that is, (taking C, c, for the centripetal forces) C : c : : -^ : 5 
 
 and by supposition P 2 : p* :: D 3 : d 3 ; therefore, substituting D 3 , d } , for P*,^ 1 , 
 
 D d 
 : D> : d~*' 
 
 that is, C : c : : . : ; and, because where the dividend is given, the quotient is inversely as 
 
 he divisor, is to -TJ inversely as D 3 to d*. Therefore C : c : : d* : D z ; that is, the centrip- 
 etal forces are inversely as the squares of the distances. 
 
 SCHOL. 1. Let C, c, express the central forces ; A, a, the arcs described ; V, v, the veloci- 
 ties with which the bodies move ; P, p, the periodical times of their revolutions ; D, d, the radii 
 or distance from the centre ; and N, n, the number of revolutions in a given time ; the pre- 
 ceding Propositions may be thus expressed. 
 
 The bodies being equal, 
 
 
 
 
 A 2 
 
 o 3 
 
 
 >. LXXIV. 
 
 C 
 
 : c 
 
 
 
 
 
 
 
 R 
 
 r 
 
 
 LXXV. 
 
 c 
 
 : c 
 
 y_ s 
 
 "1 
 
 
 
 
 
 D 
 
 d 
 
 
 COR. 
 
 c 
 
 : c 
 
 N* 
 
 : n*. 
 
 
 
 
 
 1 
 
 \ 
 
 
 LXXVI. 
 
 c 
 
 : c 
 
 P 2 
 
 : -=- or 
 P 3 
 
 P 9 
 
 LXXVIL 
 
 c 
 
 : c 
 
 D 
 
 : d. 
 
 
 LXXVIII. 
 
 c 
 
 : c 
 
 77* 
 
 d 
 
 * _.a 
 
 
 P. 
 
 LXXIX. If P* : p :: D 3 : d 3 , C : c : : : - or d : D a . 
 
 SOHOL. 2. Since it was proved (Prop. LXX.) that the centripetal and centrifugal forces are, 
 in circular orbits, equal to one another, the preceding Propositions, being demonstrated respect- 
 ing the centripetal force, are also true of the centrifugal force ; and it may be asserted universal- 
 ly, that the central' forces are in the ratios above expressed. 
 
 These propositions may be confirmed by the following experiments, on the whirling tables. 
 
 Exp. 1. Let two equal balls be placed at equal distances from the centre of motion on the 
 whirling tables ; and let one table revolve twice whilst the other revolves once ; the ball on the 
 table whose number of revolutions is, with respect to that of the other in the same time, as 
 2 to 1 (or the periodical times as 1 to 2) will raise 4 times the weight raised by the other ball ; 
 that is. (according to Prop. LXXV. and Cor.) the radii being equal, C : c : : V* : v* : : N* : w* ; 
 or (by Prop. LXXVJ.) :: p* : P*. 
 
 2. Let two equal balls be placed on tables whose number of revolutions in the same time 
 areas 2to 1 ; let the ball on the table, whose number of the revolutions is 2, be placed at half the 
 distance from the centre, at which the ball on the table, whose number of revolutions is 1, is 
 
CHAP. VII. OF CENTRAL FORCES. 67 
 
 placed ; whence their velocities will be equal. The ball at the distance 1, will raise double the 
 weight raised by the ball at the distance 2; that is, according to Prop. LXXV. the velocities 
 being equal, G : c : : d : D. 
 
 3. Let two equal balls revolve on tables whose periodical times are equal ; and let the distances 
 of the balls from the centre be to each other as 2 to 1 ; the ball which is at the distance 2 
 will raise double the weight raised by the ball which is at the distance 1 : that is, according to 
 Prop. LXXVII. C : c : : D : d. 
 
 4. Let equal balls be placed on tables whose periodical times are as 2 to 1 ; let the ball 
 on the table whose periodical time is 2, be placed twice as far from the centre as the ball whose 
 periodical time is 1 ; the ball whose distance is 2, and periodical time 2, will raise hall 
 the weight raised by the ball whose distance is 1, and periodical time 1 ; that is, according 
 
 to Prop. LXXVIII. C : c : : ^ : jL : . | : I. 
 
 5. Let the equal balls be so placed on different tables, that the distance of one from the 
 
 centre may be to that of the other as 3 to Si ; let that ball which is at the least distance 
 
 revolve twice in the same time in which the other ball revolves once ; the periodical time of 
 
 the ball at the less distance, is to that of the ball at the greater, as 1 is to 2, and the square 
 
 _of the periodical times will be as 1 to 4, and the cubes of the distances are 8, and 31.75 ; but 
 
 1 : 4 : : 8 : 32, therefore the squares of the periodical times being in this case nearly as the 
 cubes of the distances, the weight raised by the ball whose distance is 2, will be to that raised by 
 the ball whose distance is 3-J, as the square of 3J is to the square of 2; that is, nearly as 10 
 to 4, or 5 to 2. 
 
 PROP. LXXX. The centrifugal forces of revolving bodies are as their 
 quantities of matter. 
 
 For the whole centrifugal force of any body is made up of the centrifugal forces of each 
 particle of matter of which it consists ; and therefore the more numerous the particles of matter 
 in any body are, the greater will be its centrifugal force. 
 
 EXP. Let two glass tubes be half filled with water ; into one put some leaden shot, and 
 into the other a few small round pieces of light wood; let the orifice of each tube be closed 
 by a cork ; fasten the tubes to an inclined plane, and let the lower end of it rest upon the 
 centre of a whirling table. On turning the table, the bodies will be carried by their centrifugal 
 forces from the centre ; and the heavier bodies will recede farther from the centre than the 
 lighter. See Ferguson's Lectures. 
 
 COR. Hence, when the revolving bodies are not equal, the centrifugal forces are in the 
 ratios laid down in the preceding propositions multiplied into their quantities of matter. Thus 
 Q, 9, expressing the quantities of matter, and the other expressions remaining as in Prop. 
 LXX1X. Schol. 
 
 C : c :: Q : q 
 QV* gv* 
 ~D~ T 
 C : c : : QN 2 : qn* 
 
 ' 
 
 
 
 
If 
 
 OF MECHANICS. BOOK II. 
 
 C : c : : QD : qd 
 
 QD qd 
 
 C : c : : : 
 
 . Q 9 
 
 Con. Hence the central forces will be equal, whenever the expressions proportional to them 
 are equal ; thus, C = c if QD = qd. 
 
 Any of the above proportions may be confirmed by experiment; for example ; 
 
 EXP. 1. Let the two balls A, B, be as 2 to I ; let the distance of the ball A be to that of the 
 ball B from the centre, as 2 to 1, and the periodical time of the ball A be twice that of the ball 
 B ; their velocities will be equal ; therefore the centrifugal force of A will be to that of B, as 
 
 Q (I 
 
 U is to ^-, that is, as 1 to 1, or A and B will raise equal weights. 
 
 2, 3. Let the same balls revolve about a fixed point, and have their distances reciprocally 
 proportional to their quantities of matter, their centrifugal forces (compare Prop. LXXV. and 
 LXXX.) will be equal, and they will balance each other. This may be shewn by two balls 
 suspended freely and united by a cord, having the point of the cord which is directly above the 
 centre of the table at distances from the balls reciprocally as their weight ; or by two balls 
 united by a wire, and resting in equilibrio on a forked support fixed in the centre of the tables, 
 which will continue in equilibrio when the tables are turned. 
 
 In like manner other cases may be confirmed by experiment. 
 
 LEM. VII. If a body revolves freely in any orbit about an immoveable 
 centre, and in an indefinitely small time describes any nascent arc; and 
 the versed sine of the arc be drawn which may bisect the chord, and being 
 produced may pass through the centre of force ; the centrifugal force, in the 
 middle of this arc, will be as the versed sine directly, and the square of the 
 time inversely. 
 
 Plate 4. Let two bodies revolve round their centre offeree S, s; let QPM, qpm, be the nascent 
 
 * arcs described in any times, T, t ; and let PB, pb, or QR, Aa, be the versed sines bisecting the 
 
 chords, and when produced, passing through S >he centre of force. Supposing the arcs QPM 
 APN, to be described in the same time with different forces, C, c; by Prop. LXXII. Cor 4. 
 QR : Aa : : C : c. Hence, supposing the forces to be equal, QR is equal to Aa described 
 in the same time ; and (!iy Lem. V.) QR or Aa : qr : : Ap z : qp 3 ; that is, since the motion 
 in the arcs is uniform, Aa : qr : ; T 2 : I 2 . Therefore supposing both the times and forces 
 diiTere;\t, and compounding these ratios, QR : qr : : C x T 2 : c x t 3 ; whence 
 
 Plate. 4. Con. 1. If a body P, revolving about the centre S, describes a curve line APQ, and a 
 
 right line ZPR touches that curve in any point P ; and, from any other point Q of the cut ve, 
 Q!l is drawn parallel to the distance SP, meeting the tangent in R; and QT is drawn 
 perpendicular to the distance SP ; the centripetal force will be reciprocally as the quantity 
 
CHAP. VII. OF CENTRAL FORCES. e 
 
 SP 2 x QT 2 
 
 - , if this be taken of that magnitude which it ultimately acquires, supposing the points 
 
 P and Q continually to approach to each other. For QR is equal to the versed sine of double 
 the arc QP, in whose middle is P ; and double the triangle SQP, or SP x QT is proportional to 
 the time, in which that double arc is described (by Prop. LXXII.) and therefore may be used 
 
 for the exponent of the time. Whence C : c : : - - - : - -- ; that is, C is to c recip- 
 
 fir X Q 1 $P" X </t 
 
 SP 2 x QT 2 s/>* X qt* , . F SP* x QT 
 
 rocally as - -^TH - : - - ; or the centrifugal forces are reciprocally as - -- . 
 
 COR. 2. Hence, if any curvilinear figure APQ is given ; and therein a point S is also given, 
 to which a centrifugal force is perpetually directed ; the law of centrifugal force may be found, 
 by which the body P, continually drawn back from a rectilinear course, will be retained in the 
 perimeter of that figure, and will describe the same by a perpetual revolution. That is, we are 
 
 SP 2 x QT' 
 to find the quantity -^-r- - , reciprocally proportional to this force. 
 
 PROP. LXXXI. If equal bodies, revolving in ellipses, describe equal 
 areas in equal times, their centripetal forces are to one another inversely as 
 the squares of their distances from the^bcz of the ellipse toward which they 
 tend. 
 
 Let S be the focus ; let a body P, tending toward S, describe a part of the ellipse PQ ; Plate 4. 
 join SP; draw QR to the tangent YZ. parallel to SP ; join PC, and produce it to G. Complete Flff> ** 
 the parallelogram Q.#Pll, produce Q,r to v, Qu is ordinately applied to GP ; draw DK., a 
 diameter parallel to YZ, and draw 1H from the other focus H to SP parallel to YZ; join HP, 
 and draw QT perpendicular to SP, as also PF to DK. 
 
 EP is equal to the greater semiaxis AC. For, because CS is equal to CH. ES is equal to 
 El, (El. VI. 2.) whence EP is'half the sum of PS, PI ; that is, of PS, PH, for (Simson's Conic 
 Sect. II. 11. Cor.) the angle 1PR is equal to HPZ; whence (El. I. 29.) the angle PIH is equal 
 to PHI, and PI is equal to P1I ; and PS, PH, together, (Simson's Conic Sect. II. 1.) are equal 
 to the whole axis 2AC. EP therefore is equal to AC. 
 
 Putting L for the principal latus rectum of ihe ellipse, L, (by definition) is equal to - r-(fr 
 
 AC 
 
 T. 
 
 AC : CB : : CB : -, whence -f- = L) And L x QR : L x Pv : : QR : Pv ; and QR = Px ; 
 Ai-< 
 
 and Vx : Pv : : PE : PC ; whence L X QR : L x Pv : : PE or AC : PC. And (El. VI. 1.) 
 L x P : Gv x Pt> : : L . Gu ; and (Sims. II. 15.) Gi? x Pv : Qi- 2 : : PC 2 : DC 2 . And (Lem. 
 IV) the points Q and P continually approaching, Qi; 2 is to Q.r 2 ultimately in the ratio of 
 equality. And (since the triangles QTa-, EPF, are similar, for Q.rT = PEF, and QT.r to 
 EFP) Q* 2 or Qt- 2 : QT 2 : : bP 2 or AC 2 : PF 2 . But because (Sims. II. 20. Compare Vince's 
 Con. Sect. II. 10. Cor. 1.) parallelograms about conjugate diameters are equal to the rectangle 
 under the axes, the rectangle PF, DC, is equal to the rectangle ACB, whence PF : AC : : CB : DC, 
 and AC 3 : PF* : : CD 2 : CB 2 , wherefore Qc 2 : QT 2 : : CD 2 : CB. Compounding the following 
 ratios, 
 
70 OF MECHANICS. BOOK II. 
 
 L X QR : L x Pv : : AC : PC, 
 L X Pw : Gu x P : : L : Gt-, 
 GvxPr :Qi> 2 ::PC 2 :CD 2 , 
 Qu 2 :QT 2 ::CD 2 : CB 2 ; 
 
 And, striking out the equal quantities, L x QR : QT* : : AC X L X PC : Gv X CB. 
 Then substitute for AC x L its equal 2CB 2 , and 
 
 L x QR : QT 2 : : 2BC S X PC : Gv x BC* 
 orBC z x2PC:GuxBC 2 
 or 2PC : Gv. 
 
 But the points Q and P continually approaching without end, 2PC and Gv are equal ; 
 wherefore L x QR and QT 3 , proportional to these, are also equal. Multiply these equals into 
 
 gp SP 2 x QT* 
 
 -^r and L x SP 8 will become equal to 7^ . 
 
 yiv IJK 
 
 Therefore ^by Letn. VII. Cor. 1 and 2.) the centripetal force is reciprocally as L x SP a ; 
 that is, since L is a given quantity, as SP 2 , or in a duplicate ratio of the distance SP. 
 
BOOK III. 
 
 OF HYDROSTATICS AND PNEUMATIC'S; 
 
 OR THE LAWS OF 
 
 INCOMPRESSIBLE AND COMPRESSIBLE FLUIDS. 
 
 PART I. 
 
 OF HYDROSTATICS. 
 
 CHAP. I. 
 
 Of the Weight and Pressure of Fluids. 
 
 DBF. I. A FLUID is a body, the parts of which yield to any force 
 impressed upon them, and easily move out of their places. 
 
 PROPOSITION I. 
 
 The weight of fluids is as their quantities of matter. 
 
 Since each particle of any fluid gravitates toward the earth, the greater is the number of 
 particles, that is, the greater the quantity of matter in any mass of fluid, the greater will be 
 the weight of that mass. 
 
 EXP. 1. The different pressures of different columns of fluid in the same vessel at different 
 depths, appear from the different quantities of fluid discharged, at different depths, in the same 
 time, from orifices of the same bore. 
 
 2. If the air be exhausted from a tube in part filled with water, and the tube be closed up, 
 the solidity of the particles of water will be perceived from the sound produced by suddenly lifting 
 up the tube. 
 
 COR. Fluids gravitate in fluids of the same kind. For they cannot lose the property of 
 gravity which belongs to all bodies by such a change of situation. 
 
 EXP. Suspend a stopped phial from one arm of a balance, in a vessel of water, and balance 
 it by weights from the opposite arm of the balance ; upon unstopping the phial under water, a 
 
73 OF HYDROSTATICS. BOOK III. 
 
 quantity of water will rush into it, by which the weight will be increased as much as the weight 
 of water in the phial. 
 
 PROP. II. When the surface of a fluid is level, the whole mass wi 
 be at rest. 
 
 Plate 5. Let ABCD be a vessel containing water, the level surface of which is EF. Conceive the 
 
 whole mass of fluid in the vessel to be divided into tl.i;i strata, or plates, RS, TV, XY, &c. lying 
 horizontally one above another; and into small perpendicular columns GH, IK, LM, &c. 
 contiguous to each other. In the stratum XY, and the columns IK, LM, let m, n, be two 
 adjacent drops. Neither of these drops can move toward the column in which the other is, 
 without driving that other out of its place, because the fluid is supposed incompressible. But, 
 with whatever force the particle m endeavours to displace the particle n, this force is counter- 
 balanced by an equal and contrary effort on the part of n ; because (Prop. I.) they are equally 
 pressed by the equal columns above them ; consequently the particles will be at rest. 
 
 PROP. III. Any part of a fluid at rest presses, and is pressed, equally 
 in all directions. 
 
 For (Def. I.) each particle is disposed to give way on the slightest difference of pressure ; 
 and, by supposition, each particle is pressed by the contiguous particles in such manner as to 
 be kept at rest in its place ; it is therefore pressed with an equal degree of force on all sides ; 
 and, consequently, (Book II. Prop. III.) it presses equally in all directions. 
 
 COR. Hence the lateral pressure of a fluid is equal to the perpendicular pressure. This is 
 one of the most extraordinary properties of fluids, and can be conceived to arise only from the 
 extreme facility with which the component particles move among each other. 
 
 EXP. 1. Into several tubes, bent near their lower ends in various angles, pour a sufficient 
 quantity of mercury to fill the lower parts of their orifices ; then dip them into a deep glass 
 vessel filled with water, keeping the orifice of the longer legs above the surface ; whilst the tubes 
 are descending, the mercury will be gradually pressed upward in the tubes, and the pressure 
 will be equal at any given depth, whatever be the direction- of the pressing column of fluid in the 
 shorter leg of the tube. Oil may be used instead of mercury. 
 
 2. Dip an open end of a tube, having a very narrow bore, into a vessel of quicksilver; then, 
 stopping the upper orifice with the finger, lift up the tube out of the vessel; a short column of 
 quicksilver will hang in the lower end, which, when dipped in water lower than 14 times its 
 own length, will, upon removing the finger, be suspended and pressed upward. 
 
 S. Let a large open tube be covered at one end with a piece of bladder drawn tight ; pour into 
 the tube a quantity of coloured water sufficient to press the bladder in a convex form ; then, dip 
 the covered end of the tube slowly into a deep vessel of water; the bladder, by the upward pres- 
 sure, will become first less convex, then plane, and at last concave. 
 
 4. If the like be done with several tubes, whose covered oi-ifices are cut obliquely at different 
 angles, the lateral pressure will be seen to increase with the depths to which the tubes are 
 immersed. 
 
CHAP. I. THE PRESSURE OF FLUIDS. 7 
 
 5. Let a circular piece of brass, whose upper surface is covered with wet leather, be held 
 close to one orifice of a large open tube, by means of a cord or wire fastened to the middle of 
 the plate, and passing through the tube ; let the plate, thus kept close to the orifice of the tube, 
 be immersed with the tube into a large vessel of water ; when the plate is at a greater depth 
 than 8 times its thickness in the water, the cord or wire may be left at liberty, and the upward 
 pressure of the fluid will keep the plate close to the tube. 
 
 6. Let a small bladder, tied closely about one end of an open tube, having a large bore, be 
 filled with coloured water till the water rises above the neck of the bladder ; upon immersing 
 the bladder into a vessel of water, the bladder will be compressed on all sides, and the coloured 
 water will be raised up in the tube in proportion to the depth to which the bladder is sunk. 
 
 PROP. IV. When a fluid flows through a tube which is wider in some 
 parts than in others, the velocity of the fluid will, in every section of the tube, 
 be inversely as the area of the section. 
 
 Let ADMN, a bended tube larger at 1L than at FG, be filled with water to the height Plate 5. 
 ADFG. Let the water be forced downward in the part ADBP, and consequently be made to fl S- * 
 rise in the other part KHMN. It is manifest, that the water which is forced out of one part of 
 the tube, is driven in the other. Hence equal quantities pass through every section of the tube in 
 the same time ; for if less, or more, water passed through the section FG than through IL in the 
 same time, the quantity of water between FG and IL must be increased or diminished, which 
 cannot be, since no cause is supposed which could increase or diminish it. But if equal 
 quantities pass through unequal parts of the tube in the same time, the water must run propor- 
 tionally faster where the tube is narrower, and slower where it is wider. If, for example, as 
 much water runs through the section FG, as runs in the same time through the section IL, the 
 water must move as much faster at FG than it moves at IL, as the tube is narrower at FG than 
 at IL ; that is, the velocity is inversely as the area of the section. 
 
 COR. The momentum will be the same in every section of the tube ; for the quantity of water 
 at each section is directly as the area of the section, and the velocity is inversely as the area; 
 therefore the velocity is inversely as the quantity of matter ; whence (Book II. Prop. Xf.) the 
 momentum is every where the same. 
 
 SOHOL. Hence we may account for the suspension of the fluid in a tube, the upper part of 
 whose bore is capillary, and the lower of a much larger dimension, as was seen in the experi- 
 ment, Book I. Prop. VII. 
 
 Let there be a tube consisting of two parts DR and RCK, of different diameters ; DR, the pi ate 5. 
 smaller part of the tube, is able (Book I. Prop. VIII.) to raise water higher than the other ; let Kg- 3. 
 then the height to which the larger would raise it be TC, and that to which it would rise in the 
 lesser, if continued down to the surface of the fluid, be XH. If this compound tube be filled 
 with water, and the larger orifice CK be immersed in the same fluid, the surface of the water Fig. 4. 
 will sink no farther than XL, the height to which the lesser part of the tube would have rJled it. 
 But if the tube be inverted, and the smaller orifice XL be immersed, the water will run out till 
 the surface falls to TF; the height to which the larger part of the tube would have raised % 4 
 
 10 
 
7* OF HYDROSTATICS. BOOK III. 
 
 Tig. 8. Let the tube DR be conceived to he continued down to HI ; and let it be supposed that the 
 
 fluids contained in the tube XLHI, and the compound one XLKC, are not suspended by the 
 ring of glass at XL, but that they press upon their respective bases, HI and CK. Let it farther 
 be supposed that these bases are each of them moveable, and that they are raised up or let down 
 with equal velocities ; then will the velocity with which XL, the uppermost stratum of the fluid 
 XLCK, moves, exceed that of the same stratum, considered as the uppermost of the fluid in the 
 tube XLHI, as much as the tube RCK is wider than DR, (by this Prop ) that is, as much as the 
 space MNKC exceeds XLIH. Consequently, the effect of the attracting ring XL, as it acts 
 upon the fluid contained in the vessel XLCK, exceeds its effect, as it acts upon that in XLHI, 
 in the same ratio. Since, therefore, it is able to sustain the weight of the fluid XLHI by its 
 natural power, it Is able, under this mechanical advantage, to sustain the weight of as much as 
 would fill the space MNKC ; but the pressure of the fluid XLCK is equal to that weight, as 
 having the same base and an equal height (as will be shewn by Prop. VI.) Its pressure, there- 
 fore, or the tendency it has to descend in the tube, is equivalent to the power of the attracting 
 ring XL, for which reason it ought to be suspended by it. 
 
 Fig. 4. Again, the height at which the attracting ring in the larger part of the tube is able to sustain 
 
 the fluid is no greater than NF, that to which it would have raised it, had the tube been con- 
 tinued down to MM. For here the power of the attracting ring acts under a like mechanical 
 disadvantage, and is thereby diminished, as much as the capacity of the tube TFNM is greater 
 than that of H1XL ; because, if the bases of these tubes are supposed to be moved with equal 
 velocities, the rise or fall of the surface of the fluid TFXL would be so much less than that of 
 TFMN. And, since the attracting ring TF is able, by its natural power, to suspend the fluid 
 only to the height NF in the tube TFMN; it is in this case able to sustain no greater pressure 
 than what is equal to the weight of the fluid in the space HIXL ; but the pressure of the fluid 
 TFXL, which has equal height, and the same base with it, is equal to that weight ; and therefore 
 is a balance to the attracting power. 
 
 From hence we may clearly see the reason, why a small quantity of water put into a 
 capillary tube, which is of a conical form, and laid in a horizontal situation, will run toward 
 
 Plate 5. the narrower end. For let AB be the tube, ahd CD a column of water contained within it; 
 
 F) fj- s ' when the fluid moves, the velocity of the end D will be to that of the end C reciprocally as the 
 cavity of the tube at U to that at C, (by this Prop.) that is, (El. XII. 2.) reciprocally as the 
 square of the diameter at D, to the square of the diameter at C ; but the attracting ring at D is 
 to that at C, singly as the diameter at D to the diameter at C. Now, since the effect of, the 
 attraction depends as much upon the velocity of that part of the fluid where it acts, as upon its 
 natural force, its effect at D will be greater than at C ; for though the attraction at D be less in 
 itself than at C, yet its loss of force upon that account, is more than compensated by the 
 mechanical advantage it has arising from hence, that the velocity of the fluid in that part is more 
 increased than the force itself is diminished at D. The fluid will therefore move toward B. 
 See on this subject Mr. Vioce's Principles of Hydrostatics, p. 65 9. 
 
 V. la beaded cylindrical tubes, fluids at rest will be at the same 
 height on each side. 
 
 Plate 5. I % the tube ADMN, filled with water to the height AD, the water cannot descend from AD, 
 without rising toward MN. The water in each side of the vessel $nay therefore be considered 
 as two forces actii g upon each other in contrary directions ; and consequently these two masses 
 of fluid will only be at rest when their momenta are equal; that is, (Book II. Prop. XI. Cor.) 
 
CHAP. I. THE PRESSURE OF FLUIDS. 75 
 
 when the quantities of matter are inversely as the velocities, or (Prop. IV.) directly as the area 
 of the section through which it flows. Thus, at the sections BP, KH, the momenta are equal, 
 when the quantities of matter, or cylindrical masses of fluid are as the areas of the sections ; 
 that is, as the bases of the cylinders ADBP, FGHK. But cylinders are as their bases (El. XII. 
 11.) only when- their perpendicular heights are equal. Therefore the momenta of the two 
 cylinders of fluid will be equal, and consequently the mass will be at rest, only when the 
 perpendicular heights of each column are equal. 
 
 EXP. 1. In a bended tube of large but unequal bore, water will rise to the same height on 
 each side. 
 
 2. Let water spout upward through a pipe, having a small orifice inserted into the bottom 
 of a deep vessel ; it will rise nearly to the height of the upper surface of the water in the vessel. 
 The resistance of the air, and of the falling drops, prevents it from rising perfectly to the level. 
 
 COR. If, therefore, a pipe convey a fluid from a reservoir, it can never carry it to a place 
 higher than the surface of the fluid in the reservoir. 
 
 SCHOL. In this demonstration, we do not consider the velocity with which the two columns 
 of fluid are moving, but the velocity with which, if they move at all, they must begin to move. 
 And since, if their perpendicular height is the same, the velocity with which they must begin to 
 move will be inversely as their respective quantities of matter, they cannot begin to move but with 
 equal momenta; and their motions must be in contrary directions, because one column cannot 
 descend without making the other ascend ; therefore those equal momenta would destroy each 
 other. These two columns then, making a continual effort to move with equal momenta in 
 contrary directions, counterbalance each other. 
 
 PROP. VI. The pressure of fluids is proportional to the base, and the 
 perpendicular height of the fluid, whatever be the form of the vessel or 
 quantity of the fluid. 
 
 Case 1. Let the fluid be contained in a perpendicular cylindrical vessel. 
 
 In such a vessel, ABCD, because the whole weight of the fluid, and no other force, presses Plate 5. 
 directly upon the bottom CD, the pressure (by Prop. I.) must be as the quantity ; that is, (El. *' 
 XII. 11, 14.) as the base and perpendicular height of the fluid. 
 
 Case 2. Let the fluid he contained in a perpendicular vessel, the bottom of which is equal 
 to that of the cylinder in the last case, but its top narrower than the bottom. 
 
 Let the vessel DBLP, have the portions of its base LA, CP, each equal to OR. From Prop- Plate 5. 
 I. and III. it appears, that each of these portions are equally pressed by the column DBOR, as ^' ' 
 the base OR. In like manner, every portion of the base LP equal to OR is as much pressed as 
 OR. Therefore the whole base LP is as much pressed as if the vessel was of the cylindrical 
 form FHLP. 
 
 Or thus ; Because (by Prop. V.) if a tube were inserted at NT, of the diameter OR, the 
 water, being at the height DB, would rise to the level FE, there must at NT be an upward 
 pressure toward F sufficient to fill up the columns of fluid FELA ; that is, equal to the weight 
 of as much water as would fill the space PENT. Consequently the re-action, that is, the 
 pressure upon the base LA, must be equal to the weight of as much water as would fill PENT. 
 But the base LA supports this re-action, and likewise the weight of the water NTLA, which 
 
OF HYDROSTATICS. BOOK III. 
 
 are together equal to the weight of DBOR. The hase LA, therefore, sustains a pressure equal 
 to the weight of the column DBOR. And every equal portion of the base may, in the same 
 manner, be shewn to sustain an equal pressure. Therefore, the pressure on the base is the same 
 in ve s sels of the form supposed in this case, as in cylinders of equal bases, and of the same 
 altitude with these vessels. The same may be shewn with respect to a vessel of the form of 
 plate 5, fig. 7. 
 
 Case S. Let the vessel be of the same base and altitude, but have its top wider than 
 the base. 
 
 Plate 5. Let the fluid of the vessel be divided into strata EF, GH, IK, &c. Let us also imagine the 
 
 Fig. & bottom of the vessel C to be moveable, that is, capable of sliding up and down the narrow part 
 of the vessel, from C to GH. Let it further be supposed that this moveable bottom is drawn up 
 or let down with a given velocity, while the vessel itself is fixed and iinmoveable ; it is evident 
 the lowest stratum, which is contiguous to the bottom, will be raised or let down with the same 
 velocity, and will therefore have a momentum proportional to that velocity, and the quantity 
 of matter it contains; but (by Prop. IV. Cor ) the rest of the strata will have the same momen- 
 tum ; consequently, the momentum of all taken together, that is, of the whole fluid, is the same 
 as if the vessel had been no larger in any one part than it is at the bottom, for then the momen- 
 tum of each stratum would also have been as great as that of the lowest. The pressure, there- 
 fore, or action of the fluid, with which it endeavours to force the bottom out of its place, is as 
 the number of strata, that is, the perpendicular height of the fluid, and the magnitude of the 
 lowest stratum, that is, the base. 
 
 Case 4. Let the fluid be in an inclined cylindrical vessel. 
 
 Plate 5. In the inclined cylindrical vessel ABNI, as much as the fluid is prevented from pressing 
 
 fl S- 9 - upon the base NI, by being in part supported by the side of the vessel AN, so far is the 
 pressure upon the base increased by the re-action of the opposite side BI, which is equal to the 
 action of the former, because the fluid, pressing every way alike at the same depth below the 
 surface, exerts an equal force against both the sides. The base NI is therefore pressed with 
 the same force with which it would be pressed, if the fluid contained in the vessel ABNI 
 was included in the vessel EDIO, having an equal base, and the same perpendicular 
 height with the vessel ABNI ; that is, (by the first case) the pressure is as the base NI and 
 altitude CN. 
 
 Since then, the pressure upon the base of vessels, either wider or narrower at the top than 
 the bottom, and likewise the pressure upon the base or vessels inclined to the horizon, is 
 equal to that upon the base of a cylindrical vessel of the same base and height, the sides of 
 which are perpendicular to the horizon ; and since the pressure upon the base of such a cylinder 
 is as its base and height ; the pressure upon the bottom of all vessels filled with fluid is propor- 
 tional to their base and perpendicular height. 
 
 EXP. 1. Let two tubes of different forms be successively applied to the same moveable 
 circular base, suspended by a wire, passing from the centre of the base through the tubes, to 
 the beam of a balance ; when the different tubes are filled to the same height, it will require 
 the same weight at the opposite end of the balance to keep the base from sinking. Hence any 
 quantity of fluid, how small soever, may be made to balance and support any quantity how 
 great soever, which is called the hydrostatical paradox. 
 
 2. Let two tubes, the one cylindrical, the other of the form of a speaking trumpet, have 
 
CHAP. I. THE PRESSURE OF FLUIDS. 77 
 
 their bases of equal diameter, covered with bladder, and inserted in a vessel of water, as in 
 Prop. ill. EXP. 3. the bladder will become plane at the same depth in both ; from whence it 
 appears, that since the upward pressures, at the same depth, are equal, the downward pressures 
 in the two tubes are also equal. 
 
 COH. 1. Hence in different vessels, containing different fluids, the pressures are as the 
 areas of the bases multiplied into the depths, and specific gravities. 
 
 COR. 2. If a cone be filled with a fluid, and standing on its base, the pressure on its base 
 will be equal to three times the weight of the flaid. Let B be equal the base, H equal the 
 
 TJ 
 
 perpendicular height, then the solid content, or weight, will be equal x H, but the pres- 
 
 3 
 
 sure will be B x H. therefore equal to three times its weight. 
 
 COR. 3. A small quantity of fluid may be made to press with a force sufficient to raise a 
 great weight. 
 
 Since (as was shewn in Prop. V k ) as much fluid as will fill the tube DBIV presses upward plate 5. 
 against VM, with a force equal to the weight of as much fluid as would fill the space r '- 6> 
 BHVM ; the base remaining the same, the space BHVM, that is, the weight which may be 
 raised, will (by this Prop.) be as the height VB, which may be increased at pleasure. 
 
 EXP. Let two circular pieces of wood be united by leather in the manner of a pair of 
 bellows ; in the upper board insert a long tube with a large bore; through which pour water into 
 the vessel ; the upward pressure of the water, as it is poured in, will raise a great weight. 
 
 COB. 4. From hence it may be proved, independently of the reasoning in Prop. V. that, 
 in bended vessels, or channels of anj form, fluids rise to the same height, whatever be the dif- 
 ference between the quantities of fluid on each side; for whatever be the form of the channels, 
 the plane which is perpendicular to the lowes^point being considered as the common base, the 
 pressure upon it is equal, when the fluid on each side is of equal altitude ; and the whole mass 
 can only be at rest when the opposite pressures are equal. 
 
 SCHOL. This pressure of the fluid upon the base does not alter the weight of the vessel 
 and fluid considered as one mass, because the action and re-action which cause it, with respect 
 to the weight of the vessel, destroy one another ; the vessel being as much sustained by the 
 action upward, as it is pressed by the re-action downward. 
 
 PROP. VII. The pressure of a fluid upon any indefinitely small part 
 of the side of a vessel which contains it, is equal to the weight of a column 
 of the same fluid, whose base is the part pressed, and whose height is the 
 distance of that part from the surface of the fluid. 
 
 Let ABCD be a vessel filled with fluid ; AB its surface ; and L a point in the side of the plate 5. 
 vessel. The indefinitely small drop which lies next to the point L is pressed downward Fl S- 12< 
 (uy Prop. I ) by a force equal to the weight of a column of water whose base is L, and height 
 LA, the distance of that part from the surface. And (by Prop. III.) this drop is pressed 
 sideways toward L with the same force with which it is pressed downward. Whence the 
 proposition is manifest concerning the point L. And the same may be proved concerning any 
 other points M, N, C, equal to L. The same is evidently true in an inclined vessel. 
 
78 OF HYDROSTATICS. BOOK III. 
 
 PROP. VIII. The pressure of a fluid upon any plane is equal to 
 the weight of a hody which has the same density with the fluid, and is 
 formed by raising perpendiculars upon each indefinitely small part of the 
 plane, equal in height to the distance of that part from the surface of the 
 fluid. 
 
 Plates. It has been proved, in the last proposition, that the pressure up:>n each indefinitely small 
 
 part of the line AC, in the side of the vessel ABCD. is equal to the weight of a column 
 of fluid whose base is the part pressed, and whose height is the distance of that part from 
 the surface AB. Hence, if from the point L a perpendicular LO be raised whose babe is L, 
 and whose length LO is equal to LA, the distance of L from the surface, if this perpendicular 
 consisted of matter of the same density with the fluid in the vessel, the weight of this perpen- 
 dicular column would be equal to the pressure upon the point L. If, in like manner, perpen- 
 diculars, consisting of matter ot the same density with the fluid, were raised upon every point 
 between A, C, they would together fill up the area of the triangle ACD ; and the pressure upon 
 the whole line AC in the side of the vessel ABCD, because it is equal to the sum of the pressures 
 upon all its parts, must be equal to the weight of this triangle ACD. The same may be proved 
 
 Fig. 14. concerning any other lines in the side of the vessel, as HI, EF. Consequently, the pressures 
 upon the whole side will be equal to the weight of as many such triangles as there can be lines 
 drawn upon it in the same manner as AC, HI, EF, are drawn. But all these triangles 
 together would fill up the whole space, or compose a solid, CFGDAE. Therefore the pres- 
 sure upon the side AECF will be equal to the weight of this solid, consisting of matter which 
 has the same density with the fluid in the vessel ; which solid is formed by raising per- 
 pendiculars upon each line of the side, respectively equal to the distance of that point from 
 the surface of the fluid. 
 
 Hate 5. In like manner, if AC is a line drawn in the inclined side of a vessel, in which the water 
 
 F 'g> 13i reaches to the level AB, the pressure upon this line may be estimated as before. SL is the 
 distance of L from the surface. Let therefore a perpendicular LO, equal in length to LS, be 
 raised upon the point L; then, if this perpendicular was a column of matter of the same density 
 with water, the weight of it would be equal to the pressure upon L. For the same reason, if a 
 perpendicular MP is raised upon the point M, and is made equal in length to MT, the distance 
 of M from the surface ; such a perpendicular, consisting of matter of the same density with 
 water, and being of the same size, would have the same weight as the column of water MT. 
 And since (by Prop. I.) the pressure upon M equals the weight of the incumbent water MT, 
 it likewise equals the weight of the perpendicular MP. In like manner, the points N and C are 
 pressed by the weight of the incumbent columns NV and XC, which is equal to the weight of 
 the perpendiculars NQ, CR, supposing those perpendiculars to be equal in height to NV and XC, 
 and to consist of matter whose density is the same with that of the columns NV and XC. 
 Thus the pressure upon the whole line, being made up of the pressures upon all its parts, will 
 be equal to the weight of as many perpendiculars, as can be raised in this manner between 
 A and C. The sum of all those perpendiculars is the triangle ACR, whose weight therefore is 
 equal to the pressure upon the line AC. But if as many such triangles were added together, 
 as there are lines parallel to AC in the whole side of the vessel, all these triangles together 
 would form a solid. And since this solid is the sum of all the pressures upon each point of the 
 
CHAP. II. OF THE MOTION OF FLUIDS. 79 
 
 side, the weight of it, supposing it to consist of matter that has the same density as water, 
 would be equal to the pressure upon the whole side. 
 
 PROP. IX. The pressure upon any one side of a cubical vessel, filled 
 with fluid, is half the pressure upon the bottom. 
 
 The bottom sustains a pressure equal to the whole weight of the fluid in the vessel. And 
 the pressure which the side sustains is equal to the weight of the prism CFGDAE, which pj ate 5. 
 (El. XI. 28.) is half the cube ; therefore the side sustains a pressure equal to half the pressure Fig. 14, 
 upon the bottom. 
 
 Or thus ; Because the pressure upon every part of the vessel at the bottom is equal to the 
 weight of a column whose base is the part pressed upon, and height that of a perpendicular from 
 the bottom to the surface ; if the pressure were the same every where from the top to the 
 bottom, it would be equal to the weight of as many such columns as would correspond to all 
 the parts of the vessel. But the pressure every where diminishes as we approach toward 
 the surface, where it is nothing; the pressure oil the side is therefore only half of that 
 on the bottom of the vessel ; a number of terms in arithmetical progression beginning from 
 nothing being half the sum of an equal number of terms, each of which is equal to the last in 
 the progression. 
 
 COR. 1. The gravity of the fluid in a cubical vessel producing upon each of the four sides 
 a pressure equal to half that upon the bottom, and upon the bottom a pressure equal to 
 itself, produces on the whole a pressure three times as great as itself. 
 
 COR. 2. When the area of the part pressed is given, the pressure is as the perpendicular 
 distance of that part from the surface ; where the depth of the part is given, the pressure is as 
 the area. 
 
 SCHOL. There is a particular point in which the whole pressure against the side acts ; it is 
 called the centre of pressure, and is the same with the centre of oscillation of the side vibrating 
 on the upper line of it as an axis. See Prop. XLVI. Schol. 1. Book 2. 
 
 CHAP. II. 
 
 Of the Motion of Fluids. 
 
 SECT. I. 
 Of Fluids passing through the Bottom or Side of a Vessel. 
 
 PROP. X. The momentum with which any fluid runs out of a given 
 orifice in the bottom or side of a vessel, is proportional to the perpendicular 
 depth of the orifice below the surface of the fluid. 
 
 The pressure of a fluid against any given surface being (by Prop. I. and III.) proportional to 
 
80 OF HYDROSTATICS. BOOK III. 
 
 the perpendicular height of the fluid above that part ; if that given surface be removed, the fluid 
 will be driven through the orifice by this pressure. The force therefore with which the fluid 
 passes through the orifice is as the perpendicular depth of the orifice below the surface of the 
 fluid ; but the momentum is always as the moving force ; therefore the momentum is also as 
 the perpendicular depth of the orifice. 
 
 PROP. XI. The momentum with which any fluid runs out of a given 
 orifice in the bottom or side of a vessel, is as the square of its velocity, or as 
 the square of the quantity of matter. 
 
 The momentum (Sy Book II. Prop. XI.) is in the compound ratio of the quantity of matter 
 and velocity. And it is manifest, that, since the orifice is given, the quantity of fluid discharged 
 will always be as the velocity ; therefore the momentum is as the square of the velocity, or of the 
 quantity of fl-jid. 
 
 PROP. XII. The velocity with which any fluid runs out of an orifice in 
 the bottom or side of a vessel, is as the square root of the perpendicular depth 
 of the orifice from the surface of the fluid. 
 
 Because the momentum is as the square of the velocity, (by Prop. XI.) and as the perpen- 
 dicular depth of the orifice, (by Prop. X.) the square of the velocities (El. V. U.) is as the per- 
 pendicular depth, and, consequently, the velocity as the square root of the perpendicular depth. 
 
 COR. 1. Hence a fluid running out of a vessel which empties itself, and whose horizontal 
 sections are all equal, flows with an uniformly retarded velocity ; for the perpendicular depths 
 are continually diminishing. 
 
 COR. 2. Hence also the surface descends with an uniformly retarded velocity, and the 
 spaces described by it, in equal portions of time, are (Prop. XXVIII. Book II.) as the odd 
 numbers 1, 3, 5, 7, 9, &c. taken backward. 
 
 COR. S. If therefore a cylindrical vessel be divided into portions, continued to the surface 
 of the fluid, which are as the odd numbers, 1, S, 5, 7, &c. a clepsydra or hour glass will be form- 
 ed ; for the surface will descend through these divisions in equal times. 
 
 PROP. XIII. A fluid runs out of an orifice in the bottom or side of a 
 vessel, with the velocity which a heavy body would acquire in falling freely 
 through a space equal to the perpendicular distance of the orifice from the 
 surface of the fluid. 
 
 I'late S. 
 
 Fig. 10. Let ABCD be a vessel filled with any fluid, to the height FG. It is manifest, that at the 
 
 beginning of the fall of each .drop from the upper surface FG, it must be carried downward by 
 its gravity with the same velocity with which any other heavy body would begin to descend. 
 And, if an orifice be made in the vessel at L, any point below the surface, the fluid which passes 
 through that orifice will (by Prop. XII.) move with a velocity which is as the square root of the 
 distance from the surface. But if a body were to fall from the surface to the point L, it would 
 acquire a velocity which would be (by Book II. Prop. XXVI. Cor. 2.) as the square root of 
 this distance. Therefore, since the velocity with which the fluid moves is, at the beginning 
 
CHAP. II. OF THE MOTION OF FLUIDS. 31 
 
 of its motion, equal to that of a falling body, and since at every given distance these velocities 
 have the same ratio, namely, that of the square root of the distance from the surface, that is, 
 (El. V. 9.) are equal, the proposition is manifest. 
 
 COR. Supposing O, V, T, Q, to represent the area of the orifice, velocity, time, and quantity, 
 flowing out in that time respectively; Q will vary as O x V x T, or as O x T x VH, (Prop. 
 XII.) and when T is given, as O x \/H. 
 
 SCHOL. When a fluid spouts from a vessel, it rushes from all sides toward the orifice, which 
 is the cause of the contraction of the stream at the distance from the orifice equal to its diameter, 
 and is called the vena contracta. Now the area of the orifice is to the area of the smallest 
 section of the stream, nearly as \/-2 to 1 ; hence (by Prop. IV.) the velocity at the vena con- 
 tracta is to the velocity at the orifice as i/T to 1. Sir I. Newton found, that the velocity at 
 the vena contracta was that which a body acquires, in falling down the altitude of the fluid above 
 the orifice. We must, therefore, disting(B^^\veen the velocity at the orifice, and at the vena 
 contracta, and in the doctrine of spoil tin^flfHfe-. H is the latter velocity which must be con- 
 sidered, aiMthe point of projection must be assnkt&yfcpm that point. 
 
 PROP. XIV. When two cylindrica^^ssels have their bases, heights 
 and orifices equal, if one of them be always ke^t full, it will discharge double 
 the quantity of fluid discharged in the same time' bjdlBjk other whilst it empties 
 itself. ^P 
 
 For (by Prop. I.) the fluid will continue through the whole time, to run with the same 
 velocity out of the vessel that is kept full. But the fluid will run (Cor. 1. Prop. XII.) with an 
 uniformly retarded velocity out of the vessel^vhich empties itself. And, since both vessels are 
 full at first, the velocity which continues uniform in one vessel, will (by Prop. 1.) be the same 
 with the first velocity in the vessel in which theMuid is uniformly retarded. Therefore the 
 quantity discharged out of the former vessel will Wto the quantity discharged in an e'qual time 
 out of the latter, as the space described by a body moving wilflli uniform velocity, to the space 
 described by a body which sets out with the same velocity, and is uniformly retarded. But (by 
 Book II. Prop. XXVII.) the space described by the former will be double of the space described 
 by the latter. Therefore the quantity discharged out of the former vessel, will be double of the 
 quantity discharged out of the latter. 
 
 PROP. XV. A stream of any fluid which spouts obliquely forms a 
 parabola. 
 
 Each drop in a stream of fluid, spouting obliquely, is a heavy body projected obliquely by the 
 force or pressure which drives it out of the orifice. Therefore (by Book 11. Prop. LVI1I.) every 
 drop of the stream, that is, the whole stream, forms a parabola. 
 
 EXP. Observe the figure formed by a fluid spouting obliquely. 
 
 COR. Hence fluids spouting obliquely are subject to the laws of projectiles laid down, Book 
 II. Ch. VII. Sect. 1. 
 
 11 
 
S3 
 
 OF HYDROSTATICS. 
 
 BOOK III. 
 
 Plate 5. 
 *' ' 
 
 PROP. XVI. When a fluid spouts horizontally from an orifice in the 
 side of a vessel which isJiept full, if a line passing through the orifice perpen- 
 dicular to the horizon, ami intercepted between the surface of the fluid and 
 the horizontal plane that receives it, be made the diameter of a circle, and a 
 line drawn horizontally from the orifice to the circumference, the distance, to 
 which the fluid will spout, will be double of this horizontal line. 
 
 Let AB be the perpendicular ; C, E. or e, the orifice ; ADHB the semicircle drawn on the 
 side; El), (JH, tie, lines drawn horizontally from the orifice to the circumference. The fluid 
 spouts at E (by Prop. XIII.) with the velocity which a heavy body would acquire in falling from 
 A to E ; and this motion, being in a horizontal direction, can neither be accelerated nor 
 retarded by the force of gravitation, and will therefore continue uniform. But beside this, the 
 fluid spouts with the velocity which it acquires in falling after it has passed the orifice. This 
 
 velocity, when the fluid arrives at GB, is the 
 have acquired in filling through an equal 
 descending velocity, and that with w 
 since the horizontal velocity is the 
 
 A to E. and the descending 
 
 velo 
 
 that which any other heavy body would 
 to B. Let this velocity be called the 
 spouts at E the horizontal veio^y. Then, 
 which a body would acquire byWalling from 
 fluid arrives at the plane GB, is the same with 
 from E to B, and since (by Book II. Prop. XXVI.) 
 k of the last acquired velocities 
 squares of tbe.se last acquired 
 lties, are as the lines AE, EB- 
 ean proportional between AE, 
 EB. But the square of the 
 
 > bodies, are as the sq 
 is, (inverting the terms) 
 iorizontal and descending v 
 
 that which a body would acqu 
 the spaces AE, EB, describ 
 of bodies falling through tii 
 velocities, or the squares o 
 
 But in the triangle ADB, right-angled (El. VI. 8.) at D, DE is 
 EB, and the square of AE is to the square of ED, as AE is 
 horizontal velocity is to the square of the last descending velocity as AE to EB. Therefore the 
 square of the horizontal velocity is to the squardM' the last desc&hding velocity, as the square 
 ED to the square EB; whence the horizontal velocity is to the last descending velocity as ED 
 to EB. Now the spaces described in^&esAe time, in uniform motions, are (Book II. Prop. 
 VI.) as the velocities. Conseq^^jipPffieTluid had begun to fall from E with the velocity it 
 has acquired at B, and had falflMmnormly, in the time of descent the spaces described by the 
 horizontal and descending velocities would have been respectively as those v( locities ; that is, 
 as ED to EB. Thus while the fluid was descending till it reached the plane GB, the horizontal 
 velocity would have carried it forward through a space equal to ED, or the horizontal distance 
 would be ED. But the descending velocity being at the first nothing, and continually increasing, 
 the time of dessent (see Book II. Prop. XXVII )is twice what it would have been upon the sup- 
 position that it began to descend from the last acquired velocity. And the horizontal velocity is 
 uniform, and therefore in twice the time, or the true time of descent, the fluid will be carried 
 horizontally to twice the distance ED. Consequently, if BF be made equal to twice DE, whilst 
 the stream is descending from E to GB, it will be carried forward to the point F. The same 
 may be proved concerning any other points, C, e, 
 
 PROP. XVII. If a fluid spouts horizontally out of orifices in the side of 
 a vessel which is kept full, it will spout to the greatest distance from the 
 orifice which is in the middle of the side, and to equal distances from orifices 
 equally distant from the middle. 
 
CHAP. II. 
 
 Let C be the orij 
 from C. 
 
 The distance 
 twice ED. But 
 parallel to the 
 
 Also, since 
 or ed ; and tl 
 III. 14.) are 
 
 Hence, ifj 
 double of C I 
 E and e, u[ 
 
 1 double 
 
 LOTION OF FLUIDS. $ 
 
 bf the side, and E, e, equal .orifices at equal distances Plate 5. 
 
 Fig. 11, 
 
 L spout at C (byTronJtVI.) is twice CH, and at E 
 l-eater than DE, anjpne drawn from the diameter 
 
 is greater than twice ED. 
 
 I whicli the fluid will spout at E and e, are twice ED, 
 Itant from the centre, and parallel to the radius, (El. 
 Ices from E, e, are equal. 
 JGB be drawn perpendicular to the side AB, and GB be 
 \de, the fluid spouting from C will fall upon G, and from 
 
 COR. , Bide of the jet d'ea^^inclined, in any angle to the horizon, and the direction, 
 and velocifB i Re spouting fluid be known, the amplitude, altitude, and time of flight, may be 
 discovered | ' rules investigated in Book II. on Projectiles. 
 
 EXP. ji Blpftr spout from the middle orifice, and from orifices equally distant from the 
 middle, ^J Jli oi' the proposition will be manifest. 
 
 all the propositions respecting the times in which vessels empty themselves, 
 posed to be very small in respect to the bottom of the vessel, otherwise the 
 Sot agree with the theory. 
 
 JEF. II. A river is a stream of water which runs by its own weight 
 down the inclined bottom of an open channel. 
 
 DEF. III. A section of a river is an imaginary plane, cutting the stream, 
 which is perpendicular to the bottom. 
 
 DEF. IV. A river is said to flow umiformly when it runs in such a man- 
 ner, that the depth of the water in any one part continues always the same. 
 
 PROP. XVIII. If a river flows uniformly^he same quantity of water 
 passes in an equal time through every section. 
 
 Let AB be the reservoir, BC the bottom of the river, and ZX, QR, sections of the river. P| ate 5 - 
 Because the river flows uniformly, the same quantity of water which passes through ZX in a 
 given time must pass through QR in the same time ; otherwise the quantity of water in the 
 space ZQXR, must in that time be increased or diminished, and consequently the depth of the 
 water in that space altered ; contrary to the supposition. 
 
 COR. Hence if V, B, D, be the velocity, breadth, and depth respectively, V x B x D will 
 be a given quantity, and V will vary as -. 
 
 PKOP. XIX. The breadth of the channel being given, the water in rivers 
 
 
OF HYDROSTAT 
 
 is accelerated in the same manner with any 
 plane. 
 
 For each drop of the water^K>ves down upon the 
 inclined plane of the sheet of water, next below it, par. 
 
 PROP. XX. The breadth of the chan 
 each drop of water in a river is the same tha 
 from the level of the surface of the water in t 
 drop. 
 
 BOOK III. 
 own an inclined 
 
 m, or upon the 
 
 sriven, , 
 *-" ' . 
 
 ody would a 
 
 velocity of 
 in f 1'iing 
 ce of the 
 
 torn of the 
 rolls down 
 e reservoir 
 in falling 
 II. Prop. 
 GE the 
 equal 
 the 
 
 Let AB be the depth of the reservoir, AP the level ot^te surface, and BC 
 channel. Any drop at E, after it comes out of the reservoir at K (by Prop. 
 the inclined plane KE, parallel to the bottom. And this drop, when it comes ou 
 AB at K (by Prop. XIII.) has the same velocity which a heavy body would 
 from A to K ; and, in rolling down the inclined plane KE, it acquires (by 
 XXXIV.) the same velocity which any heavy body would acquire in falling 
 perpendicular height of the plane. At E the drop will therefore have acquired 
 to that which a body would acquire by falling through AK and GE, that is, througtu 
 perpendicular drawn from the level of the reservoir to the place of the drop. 
 
 Plate 5. COR 1- Hence the breadth of the channel being given, the velocity of each drop of water 
 
 Fig. 15. i n a river is as the square root of its distance from the level of the surface of the reservoir. For, 
 
 if E and R be two drops in different parts of the river, <tnd AP the level, the velocity of the drop 
 
 E is the same that a body would acquire by falling down ME, and that of R the same which a 
 
 body would acquire by falling down HR. Therefore (by Bi..ok II. Prop. XXVI. Cor. 2.) the 
 
 velocity of the drop E is to the velocity of R, as the square root of ME to the square mot of HR. 
 
 Con. 2. Hence the breadth of the channel being given, the water at the bottom of a river 
 
 will run faster than the water at the surface. ^ 
 
 PROP. XXI. The bdth of the channel being given, the depth of (he 
 river continually decreases as it runs. 
 
 The same quantity of watei (by Prop. XVIII.) passes through each of the sections ZX, QR, in 
 the same time. But (by Prop. XX. Cor 2.) the water runs faster at the lower section QR. than 
 at the upper ZX. Therefore the area of the section QR must be as much less than the area of 
 the section ZX, as the velocity at QR is greater than the velocity at ZX. But the breadth of 
 the secti >ns are by supposition equal; therefore their areas are (El. VI. 1.) as their heights. 
 Consequently the heights of the section QR, ZX, will be inversely as the velocities at those 
 sections ; that is, the depth of the water at QR will be as much less than the depth at ZX, as 
 the velocity at QR is greater than the velocity at ZX. 
 
 PROP. XXII. At a given distance from the reservoir, if the river flows 
 uniformly, the velocity of the water will be inversely as the breadth of the 
 . channel. 
 
 
CHAP. II. OF THE MOTION OF FLUIDS. 85 
 
 Because the river flows uniformly, the depth at any given section ZX is always the same ; 
 anil in any given time, the same quantity of water must How through the different sections ZX, 
 QR. as was shewn in Prop. XV'Ili. But a given quantity of water cannot flow in a given time 
 through any section, unless as much as the area is increased, so much the velocity is diminished, 
 and the reverse ; that is, the velocity must be inversely as the area of the section, or the depth 
 being given, as its breadth.* 
 
 PROP. XXIII. The depth of a river being given, the pressure upon 
 any part of the bank will be the same, whatever is the breadth of the river. 
 
 The pressure upon any given part in the bank (by Prop. I. and III.) will be as the distance 
 of that part from the surface ; which remains the same whilst the depth is the same, whatever be 
 the breadth of the river ; therefore tlie pressure will remain the same. 
 
 PROP. XXIV. If the breadth of a river be given, the pressure on any 
 part of the bank will be as the depth of the river. 
 
 For the pressure on any part of the bank is (by Prop. I. and III.) as the depth of that part 
 below the surface, which depth \\ill increase with the depth of the river. 
 
 PROP. XXV. The pressure against any given surface in the bank of a 
 river, if that surface reaches from the bottom to the top of the stream, is equal 
 to the weight of a column of water whose base is the surface, and whose 
 height is half the depth of the stream. 
 
 Let ZQXR be a given surface in the bank, reaching from the bottom EC of the river to'its top pi a te 5. 
 AD. The pressure upon this is (from what was shewn in Prop. IX.) half the pressure on an Fig. 15. 
 equal surface at the bottom XR; which pressure (by Pi-op. I and III.) is equal to the weight of 
 a column of water whose base is the surface ZQ, and whose height is the depth of the stream. 
 Therefore the pressure against the surface ZQXR is equal to the weight of a column whose 
 base is the surface ZQ, and its height half the depth of the stream. 
 
 PROP. XXVI. When a stream which moves with the same velocity in 
 every part strikes perpendicularly upon any obstacle, the force with which it 
 strikes is equal to the weight of a column of the same fluid, whose base is the 
 obstacle, and whose height is the space through which a body must fall to 
 acquire the velocity ol the stream. 
 
 Let a stream of water flow horizontally out of the orifice e. If this -stream were to strike Plate S. . 
 upon an obstacle of the same breadth every way as the orifice or stream, placed perpendicular '"' 
 to the horizon, the stream must strike upon the obstacle with its whole force. But this force is 
 equal to the weight of a column of water whose base is e, and height Ae. And (by Prop. XIII.) 
 Ae is the height from which a body must fall to acquire the velocity with which the stream 
 
 * This and the three preceding propositions can be applied only to straight regular canals of considerable 
 declivity and no great length. 
 
86 OF HYDROSTATICS. BOOK IIL 
 
 spouts from e. Therefore the force with which this stream would strike such an obstacle is 
 equal to the weight of a column of water whose base is e, and height that from which a body 
 must fall to acquire the velocity of the stream. And because no part of the stream, however 
 broad, can strike the obstacle except so much as is contained within a section equal to the 
 surface of the obstacle, no other part of the stream is to be considered in estimating this force. 
 It is also manifest, that if the stream flow horizontally with the same velocity, in any other 
 manner than through an orifice, as in the current of a stream, it will strike an obstacle with the 
 same force. 
 
 PROP. XXVIF. When the obstacle is given, the force with which a 
 stream strikes upon it will be as the square of the velocity with which the 
 stream moves. 
 
 If any stream strikes upon a given obstacle, the force will (by Prop. XXVI.) be equal to the 
 weight of a column of water whose base is the obstacle, and whose height is equal to the space 
 through which a body must fall to acquire the velocity of the stream. Since then the base is 
 given, the weight will be as the height of such a column. But the spaces through which bodies 
 fall to acquire different velocities are (by Book II. Prop. XXVI.) as the squares of those velo- 
 cities. Therefore the height of this column, and its weight, and consequently the force of the 
 stream, which is equal to this weight, will be as the square of the velocity with which the stream 
 moves. 
 
 CHAP. in. 
 
 Of tlie Resistance of Fluids. 
 
 PROP. XXVIII. If a spherical body is moving in a given fluid, the 
 resistance which arises from the re-actiou of the particles of the fluid is, 
 within certain limits of the velocity, as the square of the velocity with which 
 the body moves. 
 
 # 
 
 A spherical body moving in a given fluid, the number of particles which it will meet with in 
 a given time will be as its velocity ; for the space through which it will pass will be as its 
 velocity, and the number of particles it will meet with will be as the space through which it 
 passes. But the re-action of the particles of the fluid, and consequently the resistance, is as the 
 number of particles or quantity of matter by which the resistance is made. Again, if a given 
 quantity of matter is to be moved, the moving force is (by Book II. Prop. IX.) as the velocity 
 communicated ; and the resistance of that given quantity of matter is as the moving force. 
 Therefore the resistance arising from re-action in a given number of particles of fluid is as the 
 respective velocities with which they are moved ; that is, as the velocities with which the bodies 
 which pass through the fluid move. The resistance of the fluid being then as the velocity on a 
 double account, first, because the number of particles moved are as the velocity of the moving body, 
 and secondly, because the resistance of a given number of particles is as the velocity of the 
 moving body ; the resistance will be in the duplicate ratio, or as the square of this velocity. 
 
CHAP. III. THE RESISTANCE OF FLUIDS. 87 
 
 SCHOL. In very swift motions, the resistance of the air increases in a greater ratio ; (see 
 Remark to Prop. LVIII. Book 11.) and in other fluids the same consequence would follow for 
 the same reason, with respect to projected bodies. Besides, the greater the velocity is, the less 
 will be the pressure against the back of the body which will cause a deviation in the law of 
 resistance. 
 
 PROP. XXIX. When a spherical body moves with a given velocity in 
 any fluid, the resistance of the fluid arising from its re-action will be as the 
 square of the diameter of the spherical body. 
 
 A spherical body, in moving through a fluid, displaces a cylindrical column of that fluid, the 
 height of which is the space which the sphere describes, and its base a great circle of the 
 spherical body. Because the velocity is given, the space described in a given time, that is, the 
 length of the column is given ; whence, the quantity of fluid in the column ; that is, the column 
 will be as its base, a great circle of the sphere. And the resistance which the column of fluid 
 makes by re-action to the motion of the sphere will be as its quantity of matter ; it will therefore 
 be as the base of the column, or as the great circle of the sphere, or (El. XII. 2.) as the square 
 of its diameter. 
 
 PROP. XXX. If two unequal homogeneous spheres arc moving in the 
 same fluid with equal velocities, the greater sphere will be less resisted in 
 proportion to its weight, than the lesser sphere. 
 
 The weights of spheres, or their solid contents, are (El. XII. 18.) as the cubes of their 
 diameters; but their resistances (Prop. XXIX.) are as the squares of their diameters; and the 
 cubes of any numbers have a greater ratio to each other than their squares. Therefore the ratio 
 of the weights of spherical bodies is greater than that of their resistances in a given fluid ; that 
 is, the weight of the greater sphere exceeds the weight of the lesser, more than the resistance of 
 a given fluid against the former exceeds the resistance against the latter, provided the spheres 
 are moving with equal \elocit ; es. 
 
 SCHOL. Hence the resistance of the air may be able to support small particles of fluid, but 
 unable to support them when they are collected into larger drops. 
 
 PROP. XXXI. The resistance of a fluid, arising from its re-action, is as 
 the side of the body perpendicularly opposed to it. 
 
 The resistance is as the column, or quantity of fluid removed in a given time, which, as was 
 shewn, (Prop. XXIX.) is as the base of the column ; that is, as the side of the body perpendicu- 
 larly opposed to it. 
 
 PROP. XXXII. When equal spheres move with the same velocity in 
 different fluids, the resistances will be as the densities of the fluids. 
 
 The resistances arising from re-action are as the momenta communicated to the fluid in a 
 given time ; that is, since the spheres move with equal velocities, as the quantities of matter 
 
OF HYDROSTATICS. BOOK HI. 
 
 moved. But, because the spheres are equal, the bases of the columns to which they communicate 
 motion, are equal ; and because the spheres move with efjlial velocity, the length of the columns 
 to which they communicate motion are equal. Hence the columns to which motion is commu- 
 nicated, having their bases and heights equal, are of equal' magnitude ; and consequently, their 
 quantities of matter are as their densities. But it has been shewn, that their momenta and 
 resistances are as their quantities of matter ; therefore their resistances are as their densities. 
 
 SCHOL. Hence drops of water may be sustained in the lower parts of the atmosphere, which 
 cannot be sustained in the higher. 
 
 PROP. XXXIII. The retardation of bodies in a resisting fluid, where 
 the weights of the bodies are given, is as the resistance of the fluid. 
 
 The more a body is resisted by any fluid in which it moves, the greater portion of its 
 momentum is destroyed ; but, because the weight of the body is given, its momentum is as its 
 velocity ; therefore the greater the resistance of the fluid, the greater portion of its velocity is 
 destroyed, that is, the more it is retarded. 
 
 PROP. XXXIV. When the resistance is given, the retardation is inversely 
 as the weights. 
 
 The same resistance will destroy an equal portion of momentum whatever is the weight of 
 the moving body. But when the momentum is the same, the velocity is (by Book II. Prop. 
 XII.) inversely as the quantity of matter. Therefore the velocity destroyed, on the retardation, 
 will be inversely as the quantity of matter in the body in which the momentum is destroyed ; 
 and the weight is as the quantity of matter ; therefore the retardation is inversely as the weight. 
 
 PROP. XXXV. The retardation of spherical bodies, moving with equal 
 velocities in the same fluid, is inversely as their diameters. 
 
 The resistance which spherical bodies meet with in a given fluid is (by Prop. XXIX.) as the 
 squares of their diameters. The retardation, when the weight is given, is (by Prop. XXXIII ) as 
 the resistance ; and when the resistance is given, the retardation (by Prop. XXXV.) is inversely 
 as the weight; that is, (El. XII. 18.) inversely as the cubes of the diameters. Now, when 
 unequal spheres move with the same velocity in the same fluid, the retardations will be unequal, 
 both because the resistances are unequal, and because the weights 'are unequal. The retardations 
 will therefore be directly as the squares of the diameters, and inversely as the cubes of the 
 diameters ; that is, (compounding these ratios) inversely as the diameters. 
 
 PROP. XXXVI. When a body moves in an imperfect fluid which has 
 tenacity, or the parts of which cohere, the resistance of any given portion of 
 the fluid from this cause, is inversely as the velocity of the body ; the 
 resistance, when the velocity is given, is as the quantity of fluid through 
 which the body passes ; and the resistance is always as the time during 
 which the body moves in the fluid. 
 
CHAP. IV. OF SPECIFIC GRAVITIES. 89 
 
 Case 1. Suppose such an imperfect fluid, as soft clay, divided into thin plates ; each plate 
 having a certain portion of tenacity wilt continue to resist the body during the whole time in 
 which it is passing through it; the resistance therefore will be less, the shorter time the bodv 
 takes in passing through it. that is, the greater velocity the body moves with. And this is true 
 concerning every plate which composes the fluid. Therefore the resistance arising from 
 tenacity in a given quantity of fluid, is inversely as the velocity of the body which passes* 
 through it. 
 
 Case2. Again, the velocity of the body being given, the resistance which the body meete 
 with, from what has been said, is also given, and will be as the number of plates or quantity of 
 the fluid. 
 
 Case 3. Lastly, when a body moves for a given time, the resistance (by the second case) 
 is as the number of plates, that is, as the space through which it passes in a given time, that is, 
 (by Book II. Prop. VI.) as the velocity directly. And (by the first case) the resistance is, on 
 account of the tenacity, inversely as the velocity. Therefore as much as the resistance is 
 increased on account of the velocity in one respect, so much it is diminished on account of the 
 velocity in another; and consequently, whatever be the velocity of a body in such a fluid, the 
 resistance which it meets with in a given time will be the same ; whence this resistance will be 
 as the time in which the body moves in the fluid. 
 
 CHAP. IV. 
 
 Of the Specific Gravities of Bodies. 
 
 DBF. V. Tbe density of a body is its quantity of matter when the bulk 
 is given. 
 
 DKF. VI. The specific gravity of a body is its weight, compared with 
 that of another body of the same magnitude. 
 
 Cor. 1. The specific gravity of a body is as its density. For the specific gravity of a body 
 is the weight of a given magnitude, and the weight of a body (by Book II. Prop. XXIV. Cor.) 
 is as its quantity of matter ; therefore the specific gravity of a body is as the quantity of matter 
 contained in a given magnitude, that" is, as its density. 
 
 COR. 2. The specific gravities of bodies are inversely as their magnitudes when their 
 weights are equal. For by the last Cor. the specific gravities of bodies are as their densities, 
 and their densities (from Def. I.) are inversely as their magnitudes when their weights are 
 equal. Therefore the specific gravities are also inversely as their magnitudes when their weight* 
 are equal. 
 
 PROP. A. The weight of a body varies as its magnitude, and specific- 
 gravity conjointly. 
 
 For if the magnitude of any body is varied, its specific gravity remaining the same, the 
 weight must be altered in the same ratio. And if the specific gravity vary while its magnitude 
 
 12 
 
90 OF HYDROSTATICS. BOOK III. 
 
 continues the same, the weight must also vary in the same ratio. Therefore the weight must 
 vary as the magnitude and specific gravity conjointly. ' 
 
 PROP. XXXVII. A fluid specifically lighter than another fluid will 
 float upon its surface. 
 
 For (by Book II. Prop. XXIV.) the lighter fluid will be less powerfully acled upon by the 
 force of gravitation than the heavier ; whence, the heavier will take the lower place. 
 
 EXP. 1. Let a small and open vessel of wine be placed within a large vessel of water ; the 
 wine will ascend. 
 
 2. Let mercury, water, wine, oil, spirits of wine, be put into a phial in the order of their 
 specific gravities ; they will remain separate. 
 
 PROP. XXXVIII. The heights to which fluids, which press freely 
 upon each other, will rise, are inversely as their specific gravities. 
 
 Since (by Prop. VI.) the opposite parts of a homogeneous mass of fluid, in a curved tube 
 or channel, press equally against each- other when they rise to the same height ; in order to 
 preserve the pressure equal when the fluids on each side are different, that which has the least 
 specific gravity, must proportionally rise above the level to preserve the balance ; and the reverse. 
 
 EXP. Into the longer arm of a recurved tube, of equal bore throughout, and open at each 
 end, pour such a quantity of mercury, that it shall rise in each arm about half an inch ; then 
 pour water into the longer arm till the mercury is raised one inch above its former height; the 
 specific gravities of these fluids will be inversely as the heights to which they rise. 
 
 PROP. XXXIX. The force with which a body lighter than any fluid 
 endeavours to ascend in that fluid, is as the excess of the specific gravity of 
 the fluid above the solid. 
 
 Hate 5. 
 
 Fig. 19. Since ABCD, the fluid in a vessel, will be at rest (Prop. Ill ) when every p^rt of an 
 
 imaginary plane SQ, under the surface of the floating body ptei, sustains an eq*ual pressure ; if 
 the solid body be of equal specific gravity with the fluid, th'at is, weighs as much as a quantity 
 of the fluid equal to it in bulk, and whose place it takes up, this imaginary plane being equally 
 pressed by the solid, as if the same space were filled with fluid, the fluid will be at rest, and the 
 solid will neither ascend nor descend. Consequently, if the body be specifically heavier than the 
 fluid, that part of the plane which is directly under the solid being so much more pressed than, 
 the other equal parts of the same plane as the solid body is specifically heavier than the fluid, 
 the body must descend with a force equal to that excess ; and, on the contrary, if the body be 
 specifically lighter than the fluid, that part of the plane which is directly under the solid being so 
 much less pressed than the other equal parts of the same plane, as the body is specifically 
 lighter than the fluid, it must be buoyed up with a force equivalent to the difference of their 
 specific gravities. 
 
 PROP. XL. Any fluid presses equally against the opposite sides of a 
 solid body immersed in it. 
 
CHAP. IV. OF SPECIFIC GRAVITIES. 91 
 
 The opposite sides of the solid are at the same depth; and fluids at the same depth press Plate 1. 
 equally. Thus the opposite sides RM, SN, of any body immersed in a vessel of water ACCD, Tl & 18 - 
 are pressed equally 6y the surrounding fluid. 
 
 COR. No motion of the solid will be produced by these opposite lateral pressures. 
 
 PROP. XLI. A body immersed in a fluid is pressed more upward than 
 it is downward, and the difference of these two pressures is equal to the 
 weight of as much of the fluid as would fill the space which the body fills. 
 
 The body MRNS being immersed in a vessel of water ABCD, its lower part MN must be plate 5. 
 pressed upward just as much as the water itself at the same depth MNT would be if no solid Fi - 18 - 
 were immersed. Now the water at any depth (by Prop. III.) is pressed as much upward 
 as it is pressed downward. And at the depth MNT, the portion of this stratum MN, would, 
 if the solid were away, be pressed downward by a force equal to the weight of the incumbent 
 column of water EMNH. Therefore the force with which MN, that is, the lower part of the 
 solid, is pressed upward, is equal to the weight of as much water as would fill the whole space 
 EHMN. But the solid body RSMN is pressed downward by the weight of the column 
 above it EHRS. Therefore the difference between the two pressures is the difference of the 
 weights of the two columns of water EHMN, and EHRS; that is, the upward pressure upon 
 the solid body RSMN exceeds the downward pressure, by a force equal to the weight of as 
 ttiuch water as would fill the space RSMN, taken up by the solid body. The case will be 
 the same, whatever is the figure of the body immersed. 
 
 PROP. XLII. A body immersed in a fluid, if it is specifically heavier 
 than the fluid, will sink. 
 
 If the body RSMN is specifically heavier than the fluid, it weighs more than a quantity of plate 5. 
 the fluid of the same bulk with it. Hence the column EHMN, consisting of the column of fluid Fi - ls - 
 EHRS and the solid body RSMN, is heavier than the same column would be if it consisted 
 wholly of water. But the upward pressure against MN is (by Prop. III.) equal to the down- 
 ward pressure of the column of water EHMN, and therefore only sufficient to support the 
 weight of that column. It cannot then support the weight of the heavier column, consisting of a 
 fluid and a solid, EHMN ; and that part of this column which is specifically heavier than the 
 fiuid, that is, the solid, will sink, with a force equal to the difference of the weights of the 
 column of fluid EHMN, and the mixed column EHRS, RSMN. 
 
 PROP. XLIII. A body specifically lighter than the fluid, in which it 
 is immersed, will rise to the surface and swim. 
 
 If th* solid RSMN be a body specifically lighter than water, the column EHMN will weigh Piate 5i 
 less as it consists of the column of water EHRS and the solid RSMN, than if it consisted Fig-. 18. 
 entirely of water. Consequently, the upward pressure upon MN, which is equal to the weight 
 of the column of water EHMN, will be equal to more weight than that of the mixed column 
 EHRS, RSMN; and therefore the lighter part of this column, that is, the solid body, will be 
 carried upward with a force equal to the difference of the weights of the column of fluid EHMN, 
 and the mixed column EHRS, RSMN. 
 
93 OF HYDROSTATICS. BOOK III. 
 
 PROP. XLIV. A body which has the same specific gravity with the 
 fluid, in which it is immersed, will remain suspended in any part of the fluid. 
 
 Plate 5. The body RMNS being of the same specific gravity with the fluid, the column EHMN 
 
 presses downward with the same force, whilst this body makes a part of it, as if the column 
 consisted wholly of water, that is, with a force equal to the upward pressure against MX. 
 Therefore the body RSMN, having its lower surface MN, and in like manner all its parts, 
 pressed by equal forces in opposite directions, will remain at rest. 
 
 EXP. Let small glass images made hollow, and of specific gravity somewhat less than 
 water, having a small orifice to receive water, swim in a large glass vessel nearly filled with 
 water and covered over closely with a piece of bladder; by pressing the bladder with the hand, 
 the air on the surface is compressed ; this pressure is communicated to the air in the images, 
 which consequently receive a larger portion of water, and become in specific gravity as heavy 
 as the water, or heavier, and either float in the water, or sink. 
 
 PROP. XLV. A body specifically heavier than the fluid, in which it is 
 immersed, may be supported in it by the upward pressure, if the pressure 
 downward be takeu away; and a body specifically lighter than the fluid, 
 in which it is immersed, will not rise in the fluid, if the upward pressure 
 be taken away. 
 
 Plate S. For, in the first case, the pressure which the solid RSMN sustains from the weight of the 
 
 Fig. 18. fluid bfiiug removed, the solid may press downward with a force equal to, or less than, that of 
 the column of fluid EHMN, that is, than that with which it is pressed upward, according to 
 the degree of depth in the fluid at which the solid is placed. 
 
 In the second case, as the upward pressure agains-t MN is diminished, the downward pres- 
 sure of the mixed column EHMN becomes equal to, or greater than, the upward pressure, and 
 the solid will either float in the fluid, or sink. 
 
 EXP. For the first part of the proposition see Prop. HI. Exp. 2 n;\<\ 5. The second part 
 may be thus confirmed. If a plane and smooth piece of hard wood, or of cork, be closely press- 
 ed down by the hand upon the plane and smooth bottom of any vessel, whilst mercury is pouring 
 into the vessel ; upon removing the pressure of the hand, the downward pressure of the 
 mercury will prevent the wood from rising. 
 
 PROP. XLVI. If a body floats on the surface of a fluid specifically 
 heavier than itself, it will sink into the fluid till it has displaced a portion of 
 fluid equal in weight to the solid. 
 
 riate S. Let ptei be a body, floating on a liquor specifically heavier than itself, it will sink into it till 
 
 Fig- 19- the immersed part, rnei. takes up the place of so much fluid as is equal to it in weight. For, in 
 that case, ei, that part of the surface of the stratum upon which the body rests, is pressed with 
 the same degree of force, as it would be, was the space rnei full of the fluid ; that is, all the 
 parts of that stratum are pressed alike, and therefore the body, after having sunk so far into the 
 fluid, is in equilibrio with it, and wiil remain ai rest. 
 
CHAP. IV. OF SPECIFIC GRAVITIES. 93 
 
 EXP. 1. Place a cube of wood on a small jar, exactly filled with water; a part of its bulk 
 will be immersed, and will displace a quantity of the water ; take a cube out of the water, and 
 put it into a scale, with which an empty vessel in the other scale stands balanced ; then pour 
 water into that vessel till the equilibrium is restored ; that portion of water will fill up the jar 
 in which the cube was placed. 
 
 2. Let a glass jar with a weight sufficient to make it sink in water to about two thirds of its 
 length, be placed first in a large vessel of water, and afterward in one which is very little wider 
 than the jar, and which has in it a small quantity of water, the jar will sink to the same depth in 
 both vessels ; that is, till so much of the vessel is under water as is equal in bulk to a quantity 
 of the fluid whose weight is equal to that of the whole vessel. 
 
 COR. Hence arises a rule for estimating the specific gravities of fluids or solids. For, since 
 (by this Prop.) the weights of the water displaced and of the solid are equal, their specific 
 gravities are inversely as their magnitudes ; that is, the magnitude of the water displaced is to 
 that of the solid, as the specific gravity of the solid is to the specific gravity of the fluid ; or 
 (since the part immersed is equal in magnitude to the fluid displaced) the part immersed is to 
 the whole, as the specific gravity of the solid to the specific gravity of fluid. Consequently, the 
 greater portion of any given solid is immersed in any fluid, the less is the specific gravity of the 
 fluid ; and with respect to solids, inverting the proposition, as the whole is to the part immersed, 
 so is the specific gravity of the fluid to that of the solid ; whence, the greater is the portion of 
 any solid immersed in a given fluid, the greater is its specific gravity. 
 
 PROP. XL VII. A splld weighs less when immersed in a fluid than in 
 open air, by the weight of a quantity of the fluid equal in bulk to the solid. 
 
 If the body immersed were of the same specific gravity with the fluit', (by Prop. XLIV.) it 
 would be supported in the fluid by the upward pressure. The fluid therefore sustains so mucli 
 of the gravity of the body, or takes away so much of its weight, as is equal to the weight of that 
 quantity of fluid which would fill the place taken up by the body. 
 
 Or thus ; A body endeavours to descend by its whole weight ; but (as was shewn, Prop. 
 XLl ) when it is immersed in a fluid, it is supported by a force equal to the weight of an equal 
 bulk of that fluid. And since these two forces act in contrary directions, the weight which the 
 body retains in the tluid will he the difference between them ; that is, it weighs as much less in 
 the fluid than in the air, as the weight of a quantity of the fluid equal in bulk to the solid. 
 
 EXP. Having provided a solid cylinder of lead which exactly fills a hollow cylinder of brass, 
 place in one scale '.he hollow cylinder; under the same scale suspend by a string the solid 
 cylinder, and balance the whole by weights ; then immerse the solid cylinder in water, and the 
 equilibrium will be restored by filling up the hollow cylinder. 
 
 REMARK. In strictness both the solid and fluid should he weighed in vacuo. The error, 
 however, arising from the pressure of the air is very small, and may be neglected, unless where 
 the body to be weighed is very light, and also where great precision is required. 
 
 COR. 1. Hence the specific gravities of different fluids may be compared, by observing how 
 much the same solid (specifically heavier than the fl lici-) loses of its weight in each fluid ; that 
 fluid having the greatest specific gravity in which it loses most of its weight. 
 
94 OF HYDROSTATICS. BOOK III. 
 
 Exp. Let a cubic inch of wood, made sufficiently heavy to sink in water, be immersed 
 successively in different fluids ; it will displace a cubic inch of the fluid in which it is immersed ; 
 and since the cube (by Prop. XL VIII.) weighs less in the fluid, by the weight of a quantity of 
 the fluid equal in bulk to the cube, its loss of weight will be the weight of a cubic inch of the 
 fluid. 
 
 COR. 2. The weights which bodies lose in any fluid are proportional to their bulks. 
 
 EXP. 1. Two balls of equal bulk, one of ivory, the other of lead, will lose equal weight in 
 water. 
 
 2. A piece of copper and a piece of gold being of equal weight in air, the gold outweighs 
 the copper in water. 
 
 COR. 3. If it be known what a cubic inch of any body loses in water, the solid content of 
 any irregular mass of the same kind may be known, by observing how much more or less it 
 loses, than a cubic inch would lose. 
 
 Exp. Weigh a cubic inch and any irregular piece of wood of the same kind, and observe the 
 difference of their weights. 
 
 COR. 4. The weight of a solid body of the same specific gravity with the fluid, or of a 
 portion of the fluid itself, suspended in the fluid, is not perceived, because this weight is sup- 
 ported, and not because the gravity of the body is lost or destroyed. 
 
 PROP. B. If a and 6 be the specific gravities of two fluids which are to 
 be mixed together ; A and B their magnitudes, and c the specific gravity of 
 the compound ; then A : B : : b c : c a, provided the magnitude of the 
 compound be equal to the sum of the magnitudes of the parts when separate. 
 
 Since (by Prop. A ) the weights of bodies are as their magnitudes, and specific gravities, 
 conjointly, the weight of A = A x a, and that of B = B X b ; and the weight of the compound 
 =s A + B X c ; but the weight of the compound must be equal to the sum of the weights of the 
 two parts, A+Bxc=Axa+Bx&> therefore Ac-|-Bc=Aa+IMand Ac Aa=B6 Be ; conquent- 
 ly A : B : : 6 c:c a. 
 
 EXP. Let the specific gravity (to avoid fractions) of gold be 19, of silver 11, and of the 
 compound 14; then the magnitude of the silver in the mixture is to that of the gold, as 
 19 14: 14 11 ::5 : 3. 
 
 Con. Hence the solution of that problem which was investigated by Archimedes, in order to 
 detect the fraud of the artist, who, instead of gold was suspected of having substituted silver, in 
 the crown of Hiero, king of Syracuse. If the proportion of the weights of each body is required, 
 the ratios of their magnitudes, and of the specific gravities, must be taken conjointly ; in this 
 case the weights of A and B are as a x 6 c : b x c a; that is, the weight of the silver is to 
 that of the gold, as 11 X 5 : 19 x 3 : : 55 : 57. 
 
 SOHOL. We easily deduce from this chapter the methods of obtaining the specific gravities 
 
CHAP. IV. 
 
 OF SPECIFIC GRATITES. 
 
 of any bodies, taking rain water as a standard, a cubic foot of which being uniformly found to 
 weigh 1000 avoirdupois ounces. 
 
 The weight which a body loses in a fluid is to its whole weight, as the specific gravity of tha 
 fluid is to that of the body ; where three terms of the proportion being given, the fourth is easily 
 found. Ex. If a guinea weigh in air 129 grains, and on being immersed in water lose 7 of its 
 weight, the proportion will be 7^:129:: 1000 to the specific gravity of a guinea. By this 
 method the specific gravities of all bodies that siuk in water may be found. 
 
 COR. 1. Hence if different bodies be weighed in the same fluid, their specific gravities will 
 be as their whole weights directly, and the weights lost inversely. 
 
 If a body to be examined consist of small fragments, they may be put into a small bucket and 
 weighed ; and then if from the weight of the bucket and body in the fluid, we subtract the 
 weight of the bucket in the fluid, there remains the weight of the body in the fluid. 
 
 COR. 2. If the same body be weighed in different fluids, the specific gravity of the fluids 
 will be as the weights lost. 
 
 EXP. The loss of weight sustained by a glass ball in water and milk is respectively 803 and 
 831 grains ; therefore the specific gravity of water is to that of milk as 803 : 831 ; that is, as 
 1.000 : 1.034. By the same method, the specific gravity of water is to that of spirits of wine, as 
 1.000 to .857. 
 
 TABLE OF SPECIFIC GRAVITIES. 
 
 Platina (pure) 
 
 Fine gold ... 
 
 Standard gold A - 
 
 Mercury ... 
 
 Lead ... 
 
 Fine silver ... 
 
 Standard silver - 
 
 Copper - 
 
 Gun metal - 
 
 Fine brass ... 
 
 Steel - 
 
 Iron 
 
 Pewter .... 
 
 Cast iron 
 
 Load stone 
 
 Diamond ... 
 
 "White lead 
 
 Marble .... 
 
 Green glass 
 
 Flint 
 
 REMARK 1. The above table shews the specific weights of the various substances contained 
 in it, and the absolute weight of a cubic foot of each body is ascertained in avoirdupois ounces, 
 
 23.000 
 19.640 
 18.888 
 
 Ivory 
 Sulphur - ... 
 Chalk - 
 
 1.825 
 - 1.810 
 1.793 
 
 14.019 
 
 Calculus humanus 
 
 - 1.542 
 
 11.325 
 11.091 
 
 Lignum vitae 
 Coal 
 
 1.327 
 - 1.250 
 
 10.535 
 9.000 
 
 Mahogany ... 
 
 Milk 
 
 1.063 
 - 1.034 
 
 8.784 
 
 Brazil wood - 
 
 1.031 
 
 8.350 
 
 Box wood - 
 
 - 1.030 
 
 7.850 
 
 Rain water - 
 
 1.000 
 
 7.645 
 
 Ice .... 
 
 * .908 
 
 7.471 
 
 7.425 
 
 Living men ... 
 Ash 
 
 - .891 
 .800 
 
 4.930 
 3.517 
 
 Maple .... 
 Beech .... 
 
 - .755 
 .700 
 
 S.I 60 
 
 Elm .... 
 
 - .600 
 
 2.705 
 
 Fir ... 
 
 .550 
 
 2.600 
 
 Cork ... 
 
 .240 
 
 2.570 
 
 Common air 
 
 .001 T V 
 
96 OF PNEUMATICS. BOOK III. PART II. 
 
 by multiplying the number opposite to it by 1000 ; thus the weight of a cubical foot of mercury 
 is 14019 ounces avoirdupois, or 8761b. 
 
 REMARK 2. If the weight of a body be known in avoirdupois ounces, its weight in Troy 
 ounces will be found by multiplying it into .91145. And if the weight be given in Troy ounces, 
 it will be found in avoirdupois by multiplying it into 1.0971. 
 
 REMARK 3. Mr. Robertson, late librarian to the Royal Society, was the gentleman who 
 investigated the specific gravity of living men, in order to know what quantity of timber would 
 be sufficient to keep a man afloat in water, supposing that most men were specifically heavier 
 than river water, but the contrary appeared to be the case from trials which he made upon ten 
 different persons, whose mean specific gravity was, as expressed in the table, 0.891, or about -Jth 
 less than common water.* Phil. Trans. Vol. L. 
 
 The scales made use of to determine the specific gravities of bodies, are, called the hydros- 
 tatic balance. 
 
 BOOK III. PART II. 
 
 OF PNEUMATICS. 
 
 CHAP. I. 
 
 Of the Weight and Pressure of the Air. 
 DBF. The Air, or Atmosphere, is that fluid which encompasses the earth. 
 
 PROP. XLVI1I. The air has weight. 
 This appears from experiment. 
 
 Exp. 1. The air being exhausted, by an air-pump, from a glass receiver, the vessel will be 
 held fast by the pressure of the external air. 
 
 2. If a small receiver be placed under a larger, and both be exhausted, the larger will be 
 held fast, while the smaller will be easily moved. 
 
 3. If the hand be placed upon a small open vessel, in such a manner as to close its upper 
 orifice, it will he held down with great force. 
 
 4. The upper orifice of an open receiver being closely covered with a piece of bladder, upon 
 exhausting the receiver, the bladder will burst. 
 
 5. In the same situation a thin plate of glass will be broken. 
 
 6. Pour mercury into a wooden cup, closely placed upon the upper orifice of an open 
 receiver ; when the air beneath is exhausted, the pressure of the external air will force the 
 mercury through the wood, and it will descend in a shower. 
 
 7. On a transferrer let the air be exhausted from a long receiver 5 then let water b 
 admitted through a pipe, by means of a cock ; the water will rise in &jet d'eau. 
 
 * This must have been taken during inspiration, and the buoyancy of different individuals is probably 
 very different. 
 
CHAP. I. OF THE WEIGHT AND PRESSURE OF AIR. 97 
 
 8. Fill a glass tube, about S feet long and closed at one end, with mercury ; then insert the 
 open end in a vessel of mercury ; the mercury will remain suspended in the tube by the pres- 
 sure of trie external air upon the surface of the mercury in the vessel ; when this pressure is 
 removed, by placing the tube and vessel under a receiver, and exhausting the air, the mercury 
 will sink in t!ie tube, and on re-admitting the air, will rise. 
 
 9. If the same immersed tube be suspended from the beam of a balance, the weight neces- 
 sary to counterpoise it, exclusive of the weight of the tube, is equal to that of the mercury 
 sustained in the barometer by the pressure of the atmosphere ; for the weight of the column of 
 air incumbent upon the tube not being counterbalanced by the contrary pressure from below, 
 which is employed in bearing up the mercury within the tube, must press upon the beam. 
 
 10. Let a barometer tube, instead of being hermetically sealed at the top, be closely covered 
 with a piece of bladder ; the mercury will rise to the same height as in a common barometer: 
 and on piercing the bladder with a needle, to admit the air, it will fall. 
 
 SCHOL. Hence the pressure of the atmosphere on or near the surface of the earth is known ; 
 the weight of any column of air being equal to the weight of the column of mercury, of the 
 same diameter, supported in the barometer. And, since the height of this column varies with 
 the weight of the atmosphere, the varieties in the weight of the atmosphere are known by the 
 barometer. 
 
 1 1 . Let the air be exhausted from a glass vessel, and by means of a cock let the vessel be 
 kept exhausted ; weigh the vessel whilst it is exhausted, and when the air is re-admitted ; the 
 difference is the weight of so much air as the vessel contains ; whieh difference will be about 
 324 grains for a thousand cubic inches. 
 
 PROP. XLIX. The air presses equally in all directions. 
 
 EXP. 1 . Let a bladder, filled with air, be placed within a condensing receiver, the condensed 
 '* air will make the bladder flaccid. 
 
 . In a tall phial let an orifice be made about 3 inches above the bottom,- stop this orifice ; 
 f through a cork in the neck of the phial insert a long tube open at each end ; and let its lower 
 
 end be below the orifice in the side of the phial. The mouth of the phial being closed up about 
 the tube, pour water into the tube till it is full. Upon opening the orifice, the water will be 
 discharged till its surface in the tube is level with the orifice ; after which it will cease to flow, 
 because the external lateral pressure of the air balances the perpendicular pressure upon the 
 water in the tube. 
 
 3. If a glass vessel be filled with water and covered with a loose piece of paper, on inverting 
 the glass, the water will be kept from falling by the upward pressure of the air. 
 
 4. If a vessel be perforated in small holes at the bottom, but closed at the top, the upward 
 pressure of the air will keep the water within the vessel; as will appear by successively stop- 
 ping and unstopping a small hole in the top of the vessel. 
 
 5. Two brass hemispherical cups put close together, when the air between them is exhausted, 
 will be pressed together with considerable force. 
 
 6. A syringe being fastened to a plate of lead, and the piston of the syringe being drawn 
 upward with one hand, whilst the lead is held in the other, the air, by its upward pressure, 
 will drive back the syringe upon the piston; whereas if the loaded syringe be hung in a 
 
 13 
 
98 OF PNEUMATICS. BOOK III. 
 
 receiver, and the air be exhausted, the syringe and lead will descend/ but upon re-admitting 
 the air, they will again be driven upward. 
 
 7. If a thin glass vessel, whose aperture is closed, be placed under the receiver of an air- 
 pump, and the air exhausted from the receiver; the vessel will be broken by the pressure of air 
 within. 
 
 PROP. L. The pressure of the atmosphere varies at different altitudes. 
 
 EXP. Put a glass tube, open at both ends, through a cork into a large phial containing a 
 small quantity of coloured water; let the lower end of the tube be in tlie water ; and 1ft the 
 cork and tube be closely cemented to the neck of the bottle. Tiien, blow through the tube, 
 till the quantity of air within the phial is so increased, that the water will rise above the neck 
 of the phial. Let this phial be placed in a vessel of sand, to keep the air within of the same 
 temperature; the water will stand at different heights in the tube, according to the elevation of 
 the place where it is placed ; from whence it appears, that the pressure of the atmosphere varies 
 at different altitudes. 
 
 COR. Hence the proportion of the specific gravity of air to that of water may be determined. 
 If the difference in height of the two places where the above experiment is made be 54 feet, and 
 that difference cause a difference of | of an inch in the height of the water; it follows, that a 
 column of water of | of an inch, or -^ of a foot, is equiponderant to a column of air of 54 feet 
 having the same base ; therefore the gravity of air to that of water, is as 54 to -j-'j-, or 864 to 1. 
 In ascending the mountain of Snowden in Wales, which is 3720 feet perpendicular height, it 
 was found that the barometer sunk 3 T 8 7 inches. See Art. Barometer, Prop. LV1I. 
 
 PROP. LI. The force with which the wind strikes upon the sail of a 
 ship, the velocity of the air, and the dimension of the. sail being given, will 
 be as the square of the sine of the angle of incidence. 
 
 PlateS. Let AD represent the sail of a ship, with its edge toward the eje ; and let a circle be 
 
 Fig. 16. drawn upon the centre K ; whence K will be the middle of the sail, and AD its length. If the 
 
 wind blows perpendicularly against the sail, all the air included within the space FADG will 
 
 strike upon it. But if the sail is inclined in tne position BE, all ttie air, which strikes upon it, 
 
 is included within the space HBEI. 
 
 If it were possible that the sail should be struck with the same quantity of air in the 
 perpendicular position AD as in the oblique position BE, yet the quantity of the oblique stroke 
 would be to the quantity of the direct stroke (by Book II. Prop. XVIII.) as the sine of incidence 
 to the radius; that is, since (supposing LK drawn parallel to BH, the direction of the wind, 
 and BC perpendicular to KL) BKL is the angle of incidence, and BL its sine, as BL to AK. 
 Again, if it were possible that the oblique stroke of the wind upon the sail BE should be equal 
 to the direct stroke upon AD ; yet, the column of air which strikes upon the sail directly, 
 having AD for its base, and the column which strikes obliquely, having BC for its base, the 
 quantity of air which strikes obliquely, is to that which strikes directly, as BC to AD, that is, 
 as BL to AK; but the velocities in either case are supposed to be the same ; therefore the 
 momenta, or forces with which the sails are struck, will be as the quantities of matter, that is, 
 as BL the sine of incidence to AK the radius. 
 
CHAP. II. OF THE ELASTICITY OF AIR. 99 
 
 Thus, the force with which the wind strikes the sail BE obliquely, is to the force with which 
 it strikes an equal sail AD directly, as BL to AK on two accounts ; first, because an oblique ,,,, 
 stroke is to a direct stroke in this ratio ; and secondly, because the quantity of air which strikes 
 the oblique sail is to that which strikes the direct one in the same ratio. Consequently, upon 
 both accounts together, the oblique force is to the direct one as BL x BL to AK x AK, or as 
 the square of BL the sine of incidence to the square of AK the radius. But, the length of the 
 sail, or AD being given, AK the radius is a given quantity. Therefore the force of the wind, in 
 different obliquities of the sail, will be as BL, the square of the sine of incidence. 
 
 CHAP. II. 
 
 Of the Elasticity of the 
 
 PROP. LII. The air is an elastic fluid, or capable of compression and 
 expansion. 
 
 Exp. 1. A blown bladder, pressed with the hand, will return into the form which it had 
 before the pressure. 
 
 2. A flaccid bladder, put under a receiver, when the external air is exhausted, becomes 
 extended by the elasticity of the internal air. 
 
 3. A bladder suspended within the receiver, with a small weight hanging from it which 
 touches the bottom, when the external air is exhausted, by the expansion of the internal air, will 
 raise the weight. 
 
 4. The bladder being put into a box, and a weight laid upon the lid, the weight, on 
 exhausting the air, will be lifted up. 
 
 5. If a tube, closed at one end, be inserted at its open end in a vessel of water, the fluid 
 in the tube will not rise to the level of the water in the vessel, being resisted by the elastic 
 force of the air within the tube. On this principle the diving bell is formed. 
 
 6. If a bladder be inclosed in a glass vessel so closely that the air in the vessel without the 
 bladder cannot escape, but the air within the bladder communicates with the external air 
 through the neck of the vessel; the external air being exhausted, the bladder will be closely 
 pressed by the air in the vessel ; and when the air is re-admitted, the bladder will be 
 distended. 
 
 7. A shrivelled apple, under an exhausted receiver, will have its coat distended by the 
 internal air. 
 
 8. In the u ame situation, the air contained in a fresh egg will expel its contents from an 
 orifice made in its smaller end. 
 
 9. On green vegetables, an-1 other substances, placed in a vessel of water under a receiver, 
 whilst the air is exhausting, bubbles will be raised by the expansion of the internal air. 
 
 10. Beer, a little warmed, will, from the same cause, whilst the internal air is exhausting, 
 have the appearance of boiling. 
 
 11. Let a cylindrical piece of wood (made just specifically heavier than water by fastening 
 to it a small plate of lead) be placed in a vessel of water unuer a receiver ; upon exhausting 
 
490 OF PNEUMATICS. BOOK III. PART II. 
 
 the air the wood will swim ; some particles of air escaping from the wood, and hereby dimin- 
 ish its specific gravity. 
 
 12. Let a glass bulb, having a long neck, he put, with the neck downward, into a vessel of 
 water; put the whole under a receiver, and exhaust the air; on re-admitting the air, that fluid, 
 acting on the surface of the water in the vessel by its elasticity, will cause it to rise in the bulb, 
 or, if the degree of exhaustion be great, nearly fill it. If the air be again exhausted from the 
 receiver, the air, remaining in the bulb, by its elasticity will expel the water from the bulb. 
 
 13. Place a double transferrer upon the air-pump, with two receivers, exhaust one receiver ; 
 then open the pipe between the two receivers ; and the air in the unexhausted receiver will, 
 by its elasticity, be in part driven into the exhausted receiver; and both receivers will have 
 equal portions of air; but this air will be rarer in both than the external air; whence both the 
 receivers will be held fast by the external pressure. 
 
 PROP. LIII. The elastic spring of the air is equivalent to the force 
 which compresses it. 
 
 If the spring with' which the air endeavours to expand itself when it is compressed were 
 less than the compressing force, it would yield still farther to that force ; if it were greater, it 
 would not have yielded so far. Therefore, when any force has compressed the air so that it 
 remains at rest, the spring of the air arising from its elasticity can neither be greater nor less 
 than this force, that is, must be equal to it. 
 
 EXP. Let the air be exhausted from an open tube, whose lower part is inserted in a vessel 
 containing a small quantity of mercury, and let the air within the vessel be prevented from 
 escaping ; this air, by its elasticity, will force the mercury up the tube nearly to the height to 
 which it would he raised by the pressure of the atmosphere. 
 
 PROP. LIV. The space which any given quantity of air fills is inverse- 
 ly, and its density directly, as the force which compresses it. 
 
 Plate 5. EXP. Let there be a bent tube of the form nkg, open at n and closed at g. Let a small 
 
 Fig. 20. portion of mercury be at the bottom fci. Then gi is filled with air compressed by the weight 
 of the atmosphere, equivalent to the weight of a column of -mercury about 29 inches in height. 
 If more mercury be poured into the orifice n, the weight of this mercury is an additional com- 
 pressing force acting upon the air ig. Since (by Prop. V.) the columns of equal heights Ik, hi, 
 balance each other, the air in the space gi is pressed both by the weight of the atmosphere and 
 the column ml. If therefore ml be 29^- inches, the air in gi is pressed with double the weight 
 of the atmosphere, or with two atmospheres; and it will be found, that it will be compressed 
 into the space gh, half the space which the same quantity of air took up when it was pressed 
 only with the weight of the atmosphere ; therefore the space is inversely as the compressing 
 force. And its density (Def. V.) is inversely as its bulk, or the space filled by it. Since 
 therefore, both the compressing force and the density of the air are inversely as the space, the 
 density must be directly as the compressing force. 
 
 PROP. C. The density of air being increased, the elasticity is increased 
 in the same ratio. 
 
CHAP. II. OF THE ELASTICITY OF AIR. 101 
 
 For (!>y Prop. LIII.) the elasticity is equivalent to the compressing force ; and (by Prop. 
 >UV.) the compressing force is as the density ; therefore the elasticity is as the density. 
 
 Exp. 1. Condense the air within a glohular vessel, having a long neck, by blowing through 
 the neck, the increased elasticity of the air within the vessel will force out water. 
 
 2. The glass bulb and vessel, used in Experiment 12, Prop. Lli. being placed within a 
 condensing receiver, and the quantity of air in the receiver increased, water will rise into the 
 bulb. 
 
 3. The quicksilver in the gauge of the condenser will be forced upward in the tube by 
 increasing the density of the air. 
 
 4. Condense the air in different degrees in the condenser, and observe the gauge ; and note 
 the different heights at which a column of mercury is supported by air of different degrees of 
 density. 
 
 PROP. L~V. The air consists of particles, which repel each other with 
 forces which are inversely as the distances between their centres. 
 
 An elastic fluid equally compressed in all directions must have all its particles at equal pi ate 5i 
 distances from each other; for if the distances are unequal, where it is the least, the repelling Fig- 17. 
 force will be the greatest, and the particles will move toward the side where there is less 
 repulsion, till the forces become equal, that is, till the particles are equally distant, or the 
 fluid becomes, every where of the same density. Suppose, then, two equal cubes of air, A and 
 B ; it is manifest, from the nature of the cube, that the number of particles in the whole mass 
 A is equal to the cube of the number of particles in the line de ; and, in like manner, that the 
 number of particles in the mass B is equal to the cube of the number of particles in the line hi. 
 And the density of these two equal cubes of air A and B will be as the number of particles 
 contained in them. Therefore the density of the cube A is to the density of the cube B, as 
 the cube of the number of particles in the line de to the cube of the number of particles in 
 the line hi. But, since these lines de, hi, are of a given length, the number of particles in 
 each will be greater, as the distances between their centres are less, that is, will be inversely 
 as those distances. Whence, the cube of the number of particles in de, hi, will be inversely as 
 the cube of the distance between their centres. And it has been shewn, that the density of the 
 mass A is to the density of the mass B, as the cube of the number of particles in de to the cube 
 ot the number of particles in At. Therefore the density of A is to the density of B inversely, as 
 the cube of the distance between the centres of the particles. 
 
 Also, in compressing any mass, A, every surface, as defg, is pressed closer to the surface 
 next beyond it. And the repulsion of the surface defg against the surface next beyond it will 
 be (all other circumstances being equal) as the number of repelling particles in that surface, 
 that is, as the square of the number of particles in the line de. But the number of particles in 
 the line de is inversely as the distance between their centres. Therefore the square ot the 
 number of particles in de, that is, the number of repelling particles in the surface defg, that is, 
 the repulsion of this surface against the next beyond it, is inversely as the square of the 
 distance between the particles. Again, where the number of particles in each surface is given, 
 if it be supposed that the particles repel each other with a force which is inversely as the 
 distance Between their centres, since the surfaces are at the same distance from each other with 
 the particles which compose them, the repulsion of the surfaces must be in the same ratio. 
 
103 OF PNEUMATICS. BOOK III. PART II. 
 
 Thus, the repulsion in the mass A is to that in the mass B, inversely as the distances of the 
 particles, if only their approach to each other be considered. And it has been shewn that the 
 repulsion is inversely as the square of these distances, if only the number of particles be 
 considered. Therefore on both accounts taken together, the repulsion is inversely as the cube 
 of the distance of the particles. And (by Prop. LIII.) the compression is as the repulsion ; 
 therefore the compression is inversely as the cube of the distance of the particles. 
 
 Now it was shewn above, that the density of A is to the density of B inversely as the cube 
 of the distance of the particles. Therefore, when a fluid consists of particles which repel each 
 other with forces inversely as the distances between (he centres of the particles, the density of 
 the fluid will be as the compressing force. But it was shewn (Prop. LIV.) that the density 
 of the air is as the compressing force. Therefore the air consists of particles which repel each 
 other with forces which are inversely as the distances between their centres. 
 
 SCHOL. From the doctrine of the elasticity of the air, the phenomena of sound may be 
 explained. 
 
 When the parts of an elastic body are put into a tremulous motion, by percussion, or the 
 like, as long as the tremors continue, so long is the air included in the pores of that body, and 
 likewise that which presses upon its surface, affected with the like tremors and agitations. 
 Now, the particles of air being so far compressed together by the weight of the incumbent 
 atmosphere as their repulsive forces permit, it follows, that those which are immediately agitated 
 by the reciprocal motions of the particles of the elastic body, will, in their approach toward 
 those which lie next them, impel these also toward each other, and hereby cause them to be more 
 condensed than they were by the weight of the incumbent atmosphere, and in their return will 
 suffer them to expand themselves again ; hence the like tremors and agitations will be propa- 
 gated to them ; and so on, till having arrived at a certain distance from the body, the vibrations 
 cease, being gradually destroyed by a continual successive propagation of motion to fresh 
 particles of air throughout their progress. 
 
 Thus it is that sound is communicated from a tremulous body to the organ of hearing. 
 Each vibration of the particles of the sounding body is successively propagated to the particles 
 of the air, till it reaches those which are contiguous to the tympanum of the ear, (a fine mem- 
 brane distended across it) and these particles, in performing their vibrations, impinge upon the 
 tympanum, which agitates the air included within it ; which being put into a like tremulous 
 motion, affects the auditory nerve, and thus excites in the mind the sensation or idea of what 
 we call sound. 
 
 Now, since the repulsive force of each particle of air is equally dilTused around it every way, 
 it follows, that when any one approaches a number of others, it not only repels those which 
 lie before it in a right line, but the rest laterally, according to their respective situations ; that 
 is, it makes them recede every way from itself, as from a centre. And this being true of everv 
 particle, the tremors will be propagated from the sounding body in all directions, as from a 
 centre ; and further, if they are confined for some time from spreading themselves by passing 
 through a tube, will, when they have passed through : t, spread themselves from the end in every 
 direction. In like manner, those which pass through a hole in an obstacle, they meet in their 
 way, will afterward spread themselves from thencej as if that was the place where they be- 
 gan ; so that the sound will be heard in any situation whatever, that is not at too great a 
 distance. 
 
 The Utmost distance, at which sound of any kind has been heard, is about 200 miles, which, 
 
CHAP. II. OF THE ELASTICITY OF AIR. 108 
 
 is said to have been observed in the war between England and Holland, in the year 1672. The 
 watch words, Jill's u-ell, given at New Gibraltar, was heard at the Old, a distance of 12 miles. 
 In both these cases, the sound passed over water, which, with respect to conducting sound, is of 
 the greatest consequence. By an experiment made on the river Thames, a person was distinctly 
 heard to read at the distance of 140 feet on the water, on land at that of 76 ; in the latter case 
 no noise intervened, hut in the former there was some occasioned by the flowing of the water 
 against the boats. Watermen observe, that when the water is still, the weather calm, and no noise 
 intervene, a "hisper may be heard across the river. After water, stone may be reckoned the 
 best conductor of sound. Brick has nearly the same properties as stone. 
 
 Since the repulsive force with which the particles of air act upon each other, is reciprocally 
 as their distances (by Prop. LV.) it follows, tliat when any particle is removed out of its place 
 by the tremors of a sounding body, or the vibrations of those which are contiguous to it, it will 
 be driven back again by the repulsive force of those toward which it is impelled, with a velocity 
 proportional to the distance from its proper place, because the velocity will be as the repelling 
 force. The consequence of*this is, that, let the distance be groat or small, it will return to its 
 place in the same time ; for the time a body takes up in moving from place to place will always 
 be the same, whilst the velocity it moves with is proportional to the distance between the places. 
 The time therefore in which each vibration of the air is performed, depends on the degree of 
 repulsion in its particles, and BO long as that is not altered, will be the same at all distances 
 from the tremulous body ; consequently, as the motion of sound is owing to the successive 
 propagation of the tremors of a sounding body through the air, and as that propagation depends 
 on the time each tremor is performed in, it follows, that the velocity of sound varies as the 
 elasticity of the air, but continues the same at all distances from the sounding body. 
 
 The velocity of sound, according to Mr. Derham, is at the rate of 1 142 feet in a second of 
 time. Hence, with a stop watch, may be easily estimated the distance of thunder, for by multi- 
 plying the number of seconds between the flash and clap of thunder by 1142, the distance is 
 given in feet. Or thus, persons in good health have about 75 pulsations at the wrist in a minute, 
 consequently in 75 pulsations, sound flies about 13 miles, that is, one mile in about six pulsa- 
 tions. Example. On seeing the flash of a gun at sea, 54 pulsations at the wrist were counted 
 before the report was heard, consequently the distance of the ship is * 5 4 = 9 miles. 
 
 Moreover, since the undulatory motion of the air, which constitutes sound, is propagated in 
 all directions from the sounding body ; it will frequently happen, that the air, in performing its 
 vibrations, will impinge against various objects, which will reflect it back, and so cause new 
 vibrations the contrary way; now, if the objects are so situated, as to reflect a sufficient number 
 of vibrations hack to the same place, the sound will be there repeated, and is called an echo. 
 And, the greater the distance of the objects is, the longer will be the time before the repetition 
 is heard. And when the sound in its progress meets with objects, at different distances, suffi- 
 cient to produce an echo, tbe same sound will be repeated several times successively, according 
 to the different distances of those objects from the sounding body ; which makes what is called 
 a repented echo. Echoes repeat more by nigl.it than in the dav. 
 
 If the vibrations of the tremulous body are propagated through a long tube, they will be 
 continually reverberated from the sides of the tube into its axis, and by that means prevented 
 from spreading, till they get out of it; whereby they will be exceedingly increased, and the 
 sound rendered much louder than it would otherwise be, as in the Speaking Trumpet. 
 
 The difference of musical tones depends on the different number of vibrations communicated 
 
104 OF PNEUMATICS. BOOK III. PART II. 
 
 to the air, in a given time, bj the tremors of the sounding body ; and the quicker the succession 
 of the vibrations is, the acuter is the tone, and the reverse. 
 
 PROP. LVI. The elasticity of air is increased by heat. 
 
 EXP. To the bottom of a hollow glass ball let an open bended tube be affixed. Let the 
 lower part of the bended tube and part of the ball be filled with mercury ; the external surface 
 will be pressed by the weight of the atmosphere ; and the internal surface will be equally 
 pressed by the spring of the air, inclosed within the vessel. If the ball be immersed in bulling 
 water, the increased elasticity of the included air will raise the mercury ii. the small tube. The 
 same may be shewn by immersing in boiling water a tube, closed at one end, into which a small 
 quantity of mercury has been admitted, inclosing a portion of air within the tube. 
 
 SCHOL. 1. The wind is no other than the motion of the air upon the surface of the globe. 
 The principal cause of the wind is, that the atmosphere is heated over one part of the earth more 
 than over another. For, in this case, the wanner air being rarefied, becomes specifically lighter 
 than the rest ; it is therefore overpoised by it and raised upward, the upper parts of it diffusing 
 themselves every way over the top of the atmosphere ; while the neighbouring inferior air rushes 
 in from all parts at the bottom ; which it continues to do, till the equilibrium is restored. Upon 
 this principle it is, that most of the winds may be accounted for. 
 
 Under the Equator, the wind is always observed to blow from the east point. For, sup- 
 posing the sun to continue vertical over some one place, the air will be more rarefied there; 
 and consequently, the neighbouring air will rush in from every quarter with equal force. But, 
 as the sun is continually shifting to the westward, the part where the air is most rarefied, is 
 carried the same way ; and therefore the tendency of all the lower air, taken together, is greater 
 that way, than any other. Thus the tendency of the air toward the west becomes general, and 
 its parts impelling one another, and continuing to move till the next return of the sun, so much 
 of its motion, as was lost by his absence, is again restored, and therefore the easterly wind 
 becomes perpetual. 
 
 On each side of the Equator, to about the thirtieth degree of latitude, the wind is found to 
 vary from the east point, so as to become north-cast on the northern side, and south-east on the 
 southern. The reason of which is, that as the equatorial parts are hotter than any other, both 
 the noithern and southern air ought to have a tendency that way ; the northern current, there* 
 fore, meeting in this passage with the eastern, produces a north-east wind on that side ; as the 
 southern current, joining with the same, on the other side the Equator, forms a south-east wind 
 there. 
 
 This is to be understood of open seas, and of such parts of them as are distant from the 
 land ; for near the shores, where the neighbouring air is much rarefied, by the reflection of the 
 sun's heat from the land, it frequently happens otherwise ; particularly on the Guinea coast, the 
 wind always sets in upon the land, blowing westerly instead of easterly. This is because the 
 deserts of J)frica lying near the Equator, and being a very sandy soil, reflect a greater degree 
 of heat into the air above them ; which being thus rendered lighter than that which is over the 
 sea, the wind continually rushes in upon the land to restore the equilibrium. 
 
 That part of the ocean, which is culled the Rains, is attended with perpetual calms, the wind 
 scarcely biowiii<; sensibly either one way or otiier. For this tract being placed between the 
 westerly wind blowing from the ocean toward the coast of Guinea, and the easterly wind 
 
 
CHAP. II. OF THE ELASTICITY OF AIR. 10* 
 
 blowing from the same coast to the westward thereof, the air stands in equilibrio between both, 
 and its gravity is so much diminished thereby, that it is not able to support the vapour it con- 
 tains, but lets it fall in continual rain, from whence this part of the ocean has its name. 
 
 There is a species of winds, observable in some places within the Tropics, called by the sailors 
 Monsoons, or Trade Winds, which, during six months of the year, Wow one way ; and the 
 remaining six the contrary. The occasion of them in general is this; when the sun approaches 
 the northern Tropic, there are several countries, as Arabia, Persia, India, &c. which become 
 hotter, and reflect more heat than the seas beyond the Equator, which the sun has left ; the 
 winds therefore, instead of blowing from thence to the parts under the Equator, blow the con- 
 trary way ; and when the sun leaves those countries, and draws near the other Tropic, the 
 winds turn about, and blow on the opposite point of the compass. 
 
 From the solution of the general trade winds, we may see the reason, why, in the Atlantic 
 ocean, a little on this side the thirtieth degree of north latitude, there is generally a west, or 
 south-west wind. For, as the inferior air, within the limits of those winds, is constantly rushing, 
 toward the Equator, from the north-east point, or nearly so, the superior air moves the contrary 
 way; and therefore, after it has reached these limits, and meets with air, that has little or no 
 tendency to any one point more than to another, it will determine it to move in the same 
 direction with itself. 
 
 In our own climate, we frequently experience, in calm weather, gentle breezes blowing from 
 the sea to the land, in the heat of the day; which phenomenon is very agreeable to the principle 
 laid down above ; for the inferior air over the land being rarefied by the beams of the sun, 
 reflected from its surface, more than that which impends over the water, the latter is constantly- 
 moving on to the shore, in order to restore the equilibrium, when not disturbed by stronger 
 winds from another quarter. 
 
 From what has been observed, nothing is more easy than to see, why the northern and 
 southern parts of the world, beyond the limits of the trade winds, are subject to such variety of 
 winds. For the air, upon account of the lesser influence of the sun in those parts, being unde- 
 termined to move toward any fixed point, is continually shifting from place to p!ace, in order to 
 restore the equilibrium, whenever it is destroyed, by the heat of the sun, the rising of vapours 
 or exhalations, tiie melting of snow upon the mountains, or other circumstances. 
 
 EXP. Fill a large dish with cold water ; into the middle of this put a water plate filled with 
 warm water. The first will represent the ocean ; and the other an island rarefying the air 
 above it. Blow out a wax candle, and if the place be still, on applying it successively to every 
 side of the dish, the smoke will be seen to move toward the plate. Again, if the am'.ient water 
 be warmed, and the plate filled with cold water, let the smoking wick of the candle be held over 
 the plate, and the contrary will happen. 
 
 SOHOL. 2. Heat expands all bodies, solid as well as fluid. 
 
 EXP. 1, 2. Water may be rarefied into steam, and will become exceedingly elastic, acting 
 with great power, as in the eolipile, and in steam engines. See Art. X. Prop. LVH. 
 
 3. Metals expand by heat, and the degrees of their expansion are measured by the PYRO- 
 METER, winch is an instrument invented to render the smallest expansions sensible. 
 
 Various machines have been contrived for this purpose, by Ferguson, Dessaguliors, Ue Luc, rlate 12> 
 &c. but the general principle may be thus illustrated. Let abc be a lever, whose fulcrum is b, Fig-. 13.' 
 14 
 
106 OF PNEUMATICS. BOOK III. PART II. 
 
 acting upon another lever cde, whose fulcrum is d ; this again acts upon a third lever rfg. whose 
 fulcrum is/, and let x be a metallic rod, one end of which rests against an immoveable obstacle 
 P, and the other end against the lever abc, at a. If a lamp be put under this rod, the heat will 
 increase its length, and put the levers in motion. Now by the principle of the lever, 
 
 Vel. of a : Vel. of c : : ab : be 
 
 Vel. of c : Vel. of e : : cd : de 
 
 Vel. of e : Vel. of g : : ef :fg. 
 Therefore Vel. of a : Vel. of g : : ab x cd x ef : be x de X fg. 
 
 Hence, if ab, cd, ef, be small in proportion to be, de,fg, a trifling increase in the length x 
 will produce a very considerable motion in the point g, which may be measured upon the 
 graduated arc yz. 
 
 Ex. If ab, cd, ef, be each equal to 1, and be, de, fg, each equal to 15, then if the rod 
 increase but the 3375th part of an inch, the point g will describe 1 inch ; consequently by 
 dividing each inch in the graduated arc into 20 parts, an expansion in the rod of less than a 
 60 thousandth part of an inch easily becomes visible. 
 
 Mr. Ferguson found the expansion of metals to be in the following proportion; iron and 
 steel 3; copper 4j- ; brass 5; tin 6; and lead 7. An iron rod 3 feet lonjr, is about ^ of an 
 inch longer in summer than m winter. See Ferguson's first Lecture and Supplement ; Uessagu- 
 lier's Exp. Phil. Chamber's Cyclopaedia, by Dr. Rees. 
 
 4 Mercury expanding or contracting by an increase or decrease of heat in the air, is made 
 the measure of heat in thermometers. See Art. VIII. Prop. LV11. 
 
 SCHOL. 3. It is found by experiment, that air is necessary to the existence of sound, of 
 animal life, of fire, and of explosion. 
 
 Exp. 1. Let a bell ring under an exhausted receiver, and in a condenser. 
 
 2. Let a lighted caudle be extinguished under a receiver. 
 
 3. Let gunpowder fall upon red hot iron placed within an exhausted receiver. 
 
 SCHOL. 4. The elasticity of the air affords a method of determining the depth of the sea 
 where a line cannot be used. 
 
 A wine glass immersed in water with its mouth downward will not become filled, because the 
 spring of air will prevent the water from entering beyond a certain point. The diving bell is 
 constructed on this principle. 
 
 PROP. LVII. To explain the nature and use of sundry Hydraulic and 
 Pneumatic Instruments. 
 
 I. The SYPHON. 
 
 Plate 5. Let DEC be a bended tube, having one leg longer than the other. This instrument, used 
 
 fig 23. f or Drawing ofl' liquors, is called the syphon. If the shorter leg of the tube be inserted in a 
 vessel of fluid, and if by sucking with the mouth a vacuum be produced in the tube, or if the 
 tube be filled with the fluid before it is used, the fluid will run off' from the vessel. The cause 
 of which may be thus explained ; the orifice C, of the longer leg, is exposed to the pressure of 
 the atmosphere ; also, since the fluid within the shorter leg is supported by the surrounding 
 
CHAP. II. OF THE ELASTICITY OF AIR. 107 
 
 fluid in the vessel, the pressure upon the orifice D is that of the atmosphere. The two equal 
 orifices are then acted upon by equal pressures ; the difference of the lengths of the columns of 
 atmosphere being too small to cause anj perceptible difference in their pressure. But these 
 equal pressures are counteracted by the pressures of two unequal columns of fluid ED, EC. 
 If, therefore, the pressures of the columns of atmosphere be more than sufficient to balance those 
 of the columns of fluid, that which acts with the lesser force, that is, the lesser column DE, is 
 more pressed against the column CE, than the column CE is pressed against DE at the vertex 
 E. Consequently, the column EC must yield to the greater pressure, and flow off through the 
 orifice C. 
 
 EXP. 1. Draw off water by a syphon. 
 
 2. Whilst mercury is passing off from a vessel by a syphon, let the air be exhausted from the 
 vessel, and the fluid will cease to run. 
 
 3. Intermitting fountains are natural syphons. 
 
 II. The SYRINGE. 
 
 Let a hollow cylindrical tube have a small orifice at one end ; at the other end insert a 
 solid cylinder so exactly fitted to the tube, that no air can pass along its sides, and fix a handle 
 to the solid cylinder. If that end of this instrument which has the smaller orifice, be inserted 
 in water, and the solid cylinder, or piston, be drawn back, a vacuum will be produced within 
 the syringe ; and the pressure of the atmosphere on the surface of the water, meeting with no 
 opposite pressure, will force the water into the tube, from whence it may be forcibly expelled, by 
 pushing down the piston. 
 
 III. The COMMON PUMP. 
 
 In this useful instrument, a handle, acting upon a pin as a lever of the first kind, draws up pi a t e 5. 
 a piston AD, fitted to the shaft or barrel of the pump, as described in the syringe. This piston F> 21. 
 has a hole, over which is a valve of leather, loaded with lead, opening upward. Toward the 
 lower part of the shaft is inserted a plug C, which also has in it a hole, and a valve which opens 
 upward. When the piston, or sucker, is drawn up from the plug, a vacuum is produced in the 
 shaft between D and C, into which the air contained in the lower part of the pipe expands itself. 
 By repeated strokes the air escapes through the upper valve, and the vacuum becomes so 
 perfect, that the external air, pressing without counteraction, upon the surface of the water, 
 in the well or reservoir in which the shaft is supposed to be inserted, forces the water 
 through the valves at C and D, into the space AD ; from whence it is prevented from return- 
 ing downward, by the valves, which are closely pressed down by the incumbent fluid. If 
 therefore the handle he repeatedly lifted up, the column of water will increase upon every 
 stroke, till it ris.es to the level of the spout, and is discharged. But if the height be more than 
 34 feet, the water cannot be raised ; for such a column is equal to the weight of a column of the 
 atmosphere of the same diameter. 
 
 IV. The FORCING PUMP. 
 
 In this pump, the piston is one entire cylinder, as in the syringe. The water is raised into pi ate 5. 
 the pipe between A and D, as in the common pump ; from hence it is forced, by tke downward Fi S- 32, 
 
108 OF PNEUMATICS. BOOK III. PART II. 
 
 pressure of the piston, or forcer, through a tube inserted in the side of the main shaft. In this 
 side-tube a valve is inserted at E to prevent the water from returning, and when a sufficient 
 quantity is raised, it is discharged by the spout. 
 
 The common engine for extinguishing fires consists of two such forcing pumps, which con- 
 vey the water into a reservoir made air-tight, into which a pipe is inserted. As this reservoir 
 fills witii water, the air within it is proportionally condensed, and therefore forces the water up 
 a cylinder from which it is conveyed, at pleasure, by leathern pipes. 
 
 V. The CONDENSER. 
 
 This instrument, which is used to force air into any vessel, is a syringe, having a solid piston, 
 and a valve in the lower part of its barrel which opens downward. By thrusting down the 
 piston, the air is forced through the valve, which is afterward held close by the elasticity of the 
 condensed air. When the piston is lifted up, a vacuum is produced, till it is raised above a 
 small hole in the barrel, when the air rushes in, and is again discharged through the valve. 
 
 ARTIFICIAL FOUNTAINS are formed by the help of a condenser, which throws any quantity 
 of air into a vessel in part filled with water; which, by its elasticity, forces the water up into 
 pipes from which it is conveyed at pleasure. 
 
 The AIR-GUN is an instrument, in the form of a gun, by which a quantity of condensed air 
 is suddenly set free, and drives a ball through the barrel with great force. 
 
 VI. The AIR-PUMP. 
 
 Plate 12. This instrument, the use of which is to exhaust the air from any vessel, has two strong 
 r>s ' 12- barrels A, A, which communicate with a cavity in D ; within each of which, near the bottom, 
 is fixed a valve opening upward, and two pistons, one in each barrel, having a valve which like- 
 wise opens upward. These pistons are moved hy means of a cog wheel in the piece TT, to the 
 axis of which the handle I) is fixed, and whose teeth catch in the racks of the pistons CC, and 
 move them upward or downward. PQR is a circular brass plate, having at its centre the orifice 
 K of a concealed pipe that communirates with the cavity. In the piece D at V is a screw that 
 closes the orifice of another pipe, for the purpose of admitting the external air when required. 
 Upon W is placet! the short barometer gauge for the purpose of shewing the degrees of exhaustion. 
 When the handle is turned, one of the pistons is raised, ancf a vacuum produced in its barrel. 
 By means of the pipe, which passes from the orifice K in the plate upon which the receiver LM, 
 or vessel to be exhausted, stands, to the part of the barrel beneath the lower valve, the air con- 
 tained in the receiver, communicating with the bairel, raises the lower valve by its elastic 
 spring, and expands into the vacuum. Thus a part of the air in the receiver is extracted. 
 By turning the handle the contrary way, the same effect is produced in the other barrel ; whiist 
 the first piston, being depressed, the air which had passed from the receiver is compressed, and 
 escapes through the valve in the piston. This operation is continued till the air is nearly 
 exhausted from the receiver; for it can never be perfectly exhausted, since at each stroke only 
 sue!) a part of the air which remained is taken away, as is to tiie quantity before the stroke, as 
 the capacity of the barrel, to that of the receiver, pipe and bairel taken together? which 
 may be easily proved in the following manner. 
 
 Let II = the content of the receiver and pipe, B = the content of the barrel. 
 
 If L = the quantity of air in R before the stroke, and I = the quantity exhausted by it ; and 
 
CHAP. II. OF TUB ELASTICITY OF AIR. 109 
 
 since, the piston being raised, the air is uniformly diffused through R and B, and that in iJ 
 extracted by the stroke, consequently L : J : : R + B : B, or*:L::B:R + B; that is, the 
 quantity of air extracted is to the quantity before the stroke, as the capacity of the barrel is to 
 that of the receiver, pipe and barrel taken together. 
 
 COR. 1. Let L, M, N, &c. be the quantities of air, before any successive strokes ; I, m, ft, 
 &c. the quantities exhausted by each stroke ; and L : I : : R + B : B : : M : m : : N : n, &c. by 
 division L : L I (M) : : M : M in (N) : : N : N n (O) &c. or L : M : : M : N : : N : O, &c. 
 therefore L, M, N, &c. are in decreasing geometric progression, whose common ratio is that 
 
 2T ^\f 
 
 of R + B : R. If R = 2B, then R + B : R : : 3 : 2, and M = , N = ^, &c. anil the quan- 
 
 O O 
 
 ties of air are equal to L, f L, L, -/ r L, &c. 
 
 COR. 2. Since L : M : : I : m ; M : N : : m : n, &c. I, m, n, &c. are in a decreasing geo- 
 metric progression, whose common ratio is that of R + B : R. If, as in the last Cor. 
 
 2/ 2?n 
 
 R -{- B : R : : 3 : 2, then m = , = , &c. and the quantities of air exhausted by the succes- 
 
 O i> 
 
 sive strokes are I, \ I, I, -^ I, &c. 
 
 COR. 3. If R be to B in any finite ratio as 3 : 2, the receiver can never be perfectly ex- 
 hausted by any finite number of turns ; for let the number of turns be n, and Q the last remain- 
 der, then Q = L x f | n , supposing L to be the quantity of air in the receiver at first ; and this 
 quantity L x f I" is finite, since n is finite. 
 
 VII. The BAROMETER. 
 
 (1) If a glass globe be exhausted of air, and balanced at one end of a beam, upon admitting 
 the air tbe globe preponderates. This experiment not only, in common with others beforemen- 
 tioned, shews that the air has weight, but also what that weight is. The density of air was 
 found, by Mr. Haukshee, to be 885 times less than that of water, when the barometer stood at 
 29J inches. Hence as a cubic inch of water weighs 253.18 grains Troy, a cubic inch of air weighs 
 0..^86 grains. And if mercury be 14 times heavier than water, the specific gravity of air is to 
 that of mercury as 1 to 885 X 14 = 12390. 
 
 (!) If a glass tube AB, of about 32 or 33 inches long, hermetically sealed at one end, be Plate 12. 
 filled with mercury, and then inverted into a bason 1) of the same fluid, the mercury in the tube Fl &' 4 ' 
 will stand at an altitude above the surface of that in the bason between 28 and 31 inches. A 
 tube thus fillet!, and graduated from 28 to 31 inches, is called a barometer. The height of the 
 mercury in the tube above the surface of the mercury in the bason is called the standard alti- 
 tude, which, in this country, fluctuates between 28 and 31 inches; and the difference, between 
 the greatest and least altitudes, is called the scale of variation. 
 
 Now the mercury in the barometer tube will subside, till the column be equivalent to the 
 weight of the external air upon the surface of the mercury in the bason, and is therefore a true 
 criterion to measure that weight, and chiefly directed to that purpose, in order to foretell the 
 changes in the weather. 
 
 If each inch of the scale of variation AD, (fig. 5, made larger for the sake of perspicuity) be Plate 12* 
 divided into ten equal parts, marked 1. 2, 3, increasing upward, and a vernier LM, whose &* 
 length is yjths of an inch, be likewise divided into ten equal parts, increasing downward, and 
 so placed as to slide along the graduated scale of the barometer, the altitude of the mercury in 
 
OF PNEUMATICS. BOOK HI. PART II. 
 
 the tube, above tbe surface of that in tbe bason may be found, in inches and hundredth parts of 
 an inch, by this process. If the surface E of the mercury in the tube do not coincide with 
 a'division in the scale of variation, place the index of the vernier M even with this surface, and 
 observing where a division of the vernier coincides with one of the scale, the figure in the vernier 
 will show what hundredth parts of an inch are to be added to the tenths immediately below the 
 index. If the surface of the mercury be between 6 and 7 tenths above 30 inches, and the index 
 of the vernier being placed even with it, and the figure 8 upon the vernier coincide with a 
 division upon the scale, the altitude of the barometer will be 30 inches T ^- and y^ of an inch. 
 For each division of the vernier being greater than that of the scale by T ^ of an inch, (for the 
 tenth part of a tenth of an inch is the hundred part of an inch) and there being eight divisions, 
 the whole must be T f^ of an inch above the number 6 in the scale, and the height of the mercury 
 is therefore 30.68 inches. 
 
 COR. 1. Hence, if the atmosphere were homogeneous, its altitude would be easily found. 
 For by the former part of this article, when the mercury stood at 29| inches, the density of the 
 air was to that of mercury as 1 to 12390 ; consequently the altitude of a homogeneous atmos- 
 phere would be equal to 12390 x 29' = 5.77 miles. The real height of the atmosphere may be 
 determined from the beginning and end of twilight. See Book VII. Prop. XXXIX. 
 
 COR. 2. The barometer has been applied to the measuring of the heights of towers, moun- 
 tains, &c. Since 12390 inches of air, near the surface of the earth, is equal to one inch of 
 mercury; 1239 inches, or about 103 feet of air, must be equal to -^ of an inch of mercury. 
 Therefore if a barometer be carried up any great eminence, the mercury will descend -^ of an 
 inch for every 103 feet that the barometer ascends. This corollary supposes that the atmos- 
 phere near the surface of the earth is every where of the same density, which is so far from 
 being true, that the conclusions drawn from the supposition deviate from fact even in small 
 altitudes, as appears from the following observations made by Dr. Nettleton. 
 
 Perpen. Altitudes. Lowest Station. Highest Station. Diff. 
 
 Town of Halifax 102 29.78 29.66 0.12 
 
 Coal mine 236 29.50 29.23 0.27 
 
 Halifax-hill 507 30.00 29.45 0.55 
 
 See Abr. Phil. Trans. Vol. vi. 
 
 M. De Luc, Sir George Shuckburgh, and General Roy,' have considered this subject very 
 attentively, and have laid down certain rules, which, with proper corrections, on account of the 
 difference of the temperature of the air, will hold good for all altitudes within our reach. See 
 De Luc on the Modifications of the Atmosphere. Phil. Trans. Vol. LXXVII. 
 
 COR. 3. When the mercury in the barometer stands at the altitude of 30 inches, the pressure 
 of the air upon every square inch is rather more than 15lb. avoirdupois. Now, supposing the 
 surface of a middle sized man to be 14J square feet, the pressure upon him, when the air is 
 lighest, will be 13.2 tons, and when heaviest, it will be 14.3 tons, the difference of which is 
 2464lb. The difference of pressure must affect us in regard to our health and animal spirits, 
 especially when the change takes place suddenly. 
 
 For a description of the different kinds of barometers, see Parkinson's Hydrostatics, p. 97. 
 
CHAP. II. OF THE ELASTICITY OF AIR. 
 
 VIII. The THERMOMETER. 
 
 The thermometer is an instrument calculated for measuring the temperature of the air, and 
 other bodies contiguous to it, as to heat and cold, being usually a cylindrical glass tube, contain- 
 ing air, water, oil, spirits of wine, mercury, &c. which fluids are found to occupy different 
 portions of the tube in different temperatures, and these portions being measured, exhibit the 
 different expansions of the included fluid. 
 
 AB represents a glass tube, whose end A is blown into a bulb ; this bulb and part of the tube ! ate 12 - 
 being filled with quicksilver, the least change of the bulk of quicksilver, and consequently of the 
 temperature of contiguous bodies, is shewn by the rise or fall of the surface in the tube, which 
 is indicated by the scale ab affixed to the frame of the instrument. 
 
 The thermometer chiefly used in Great Britain, is that constructed by Fahrenheit ; in which 
 there are 180 divisions between the freezing and boiling water points, the freezing point being 
 reckoned 32 above zero, or the commencement of the scale ; consequently the boiling water 
 point is 21k. .* 
 
 A good thermometer must possess the following properties ; the capacity of the tube should 
 be very small and regular, and its upper end must be hermetically sealed. The empty space 
 must be as free as possible from air. The scale must be well adjusted, and accurately divided 
 according to the capacity of the tube. Thermometers with small bulbs, and tubes in proportion, 
 are the most to be depended upon, for a large volume of mercury is not sufficiently sensible to 
 the change of temperature. 
 
 Since the thermometers of Fahrenheit and Reaumeur are those mostly in use, it will be often 
 found convenient to be able readily to convert the degrees on Fahrenheit's scale into those of 
 Reaumeur, and vice versa ; and as one degree on Reaumeur's scale is equal to 2.25, or to |> of 
 Fahrenheit ; and as the former scale places the freezing point at zero, and the latter places it at 
 32; the* following canons will reduce the degrees on the one to the corresponding ones on the 
 
 other. 
 
 u f\^j 
 
 1. To convert the degrees of Fahrenheit into those of Reaumeur ; X 4 = R ; thus 
 
 y 
 
 the 167 of Fahrenheit answers to the 60 of Reaumeur. 
 
 R x 9 
 
 2. To convert the degrees of Reaumeur to those of Fahrenheit ; [- 32 = F. 
 
 Thus the 40 of Reaumeur answers to the 122 of Fahrenheit. See No. 4, Appendix to 
 Lavoisier's Chemistry. 
 
 It is evident, that the thermometers hitherto described, are limited in their extent. The 
 mercurial thermometer extends no farther than the heat of boiling mercury, which answers to 
 6 )0 of Fahrenheit's scale ; but the heat of solid bodies in the state of ignition exceeds that of 
 bjiling mercury. To remedy this defect, Mr. \Vedgewood has contrived a thermometer for 
 measuring tiie higher degrees of heat, by means of a distinguished propeity of argillaceous bodies, 
 viz. the diminution of their bulk by fire. This diminution commences in a dull red heat, and 
 proceeds regularly as the heat increases, till the clay becomes vitrified. This thermometer, 
 
 * The scale on Reaumeur's thermometer, which is principally used on the continent, begins at the 
 freezing point, and proceeds both ways, from or zero. From freezing to boiling water are 80 degrees. 
 For the construction, uses, &c. of this and several oilier thermometers, see Parkinson's Hydrostatics, p. 
 154169. 
 
OF PNEUMATICS. 
 
 BOOK III. PART II. 
 
 therefore, marks witli precision, the different degrees of ignition from the red heat visible only 
 in the dark, to the heat of an air furnace. Its construction is extremely simple. It consists of 
 two rulers fixed upon a flat plane, a little farther asunder at one end than at the other, leaving 
 an '-open longitudinal space between them. Small pieces of alum and clay, mixed together, are 
 made of such a size as just to enter at the wide end ; they are then heated in the fire along with 
 the body, whose heat we wish to determine. The fire, according to the degree of heat it con- 
 tains, contracts the earthy body, so that applied to the wide end of the gauge, it will slide on 
 toward the narrow end, less or more, according to the degree of heat to which it has been 
 exposed. Each degree of Mr. Wedgewood's thermometer answers to 130 degrees of Fahren- 
 heit ; and the scale begins from 1077 of Fahrenheit. Hence the following 
 
 TABLE. 
 
 Fahrenheit's Wedgewood'* 
 
 scale. scale. 
 
 Extremity of Wedgewood's scale - 32277 - 240 
 
 Cast iron melts - - - 21877 - 160 
 
 Least welding heat of iron - 12777 - 90 
 
 Fine gold melts - 5237 - - 32 
 
 Fine silver melts - 4717 - 28 
 
 Brass melts ... - - 3807 - 21 
 
 Red heat fully visible in day light - 1077 - 
 
 Red heat fully visible in the dark - 947 - - 1 
 
 MERCURY BOILS, also expressed oils - 600 
 
 Lead melts ------- 540 C Note. If these three metals be miced 
 
 ,/- ft 1 together by fusion, in the proportion of 5, 
 
 Bismuth melts - J 1 a, and 3, the mixture will melt in a heat 
 
 Tin melts ....... 408 Lbelow boiling 1 water. 
 
 Nitrous acid boils - 
 
 Cow's milk boils - ... 213 
 
 WATER BOILS - - 212 
 
 Heat of the human body - - 92 to 99 
 
 Oil of olives begins to congeal ... 
 
 WATER FREEZES and snow melts - 
 
 Milk freezes - ... 
 
 Urine and vinegar freeze .... 
 
 Strong wine freezes - - 
 
 A mixture of snow and salt freezes - - - to 4 
 
 MERCURY FREEZES - - - - - 39 or 40 
 
 Cold produced at Hudson's Bay, by a mixture of 
 
 vitriolic acid and snow 69 
 
 IX. The HYGROMETER. 
 
 The hygrometer is an instrument for measuring the degrees of moisture in the air; of 
 which there are various kinds ; for whatever contracts, and expands by the moisture, and dry. 
 ness of the atmosphere, is capable of being formed into a hygrometer. Such are most kinds of 
 
CHAP. II. OF THE ELASTICITY OF AIR. 113 
 
 wood ; catgut, twisted cord ; the beard of wild oat, &c. The following are very simple in their 
 construction, and will serve to explain the principle of the instrument. 
 
 1. Stretch a catgut or a common cord, ABD, along a wall, passing it over a pulley B ; fixing 5 ' late 12. 
 it at one end A, and to the other hanging a weight E, carrying a small index F. Against the same '*>' 
 wall, fit a metal plate HI, divided into any number of equal parts, and the hygrometer is complete. 
 
 For it is known, that moisture sensibly shortens catgut, cord, &c. and that as the moisture 
 evaporates, they return to their former length. Hence the weight E, with the index, will ascend 
 when the air is moist, and descend when it becomes drier ; and the divisions on the scale will 
 shew the degrees of moisture or dryness. This hygrometer may be made more sensible and 
 accurate by straining the catgut over several pullies placed in a parallel or any other position. 
 
 3. The sponge hygrometer is constructed as follows ; BC is the beam of a balance ; to the Plate 12. 
 end B is hung a piece of sponge, so cut as to contain as large a superficies as possible, which %' 7 ' 
 must be exactly balanced on the other side by another thread of silk D, on which is strung some 
 very small leaden shot at equal distances, so adjusted as to cause an index E to point to G, the 
 middle of the graduated arc FGH, (made large for distinction's sake) when the air is in a middle 
 state between the greatest moisture and the greatest dryness. Under this silk, strung with shot, 
 is placed a shelf I, for that part of the shot to rest upon which is not suspended. When the 
 moisture imbibed by the sponge increases its weight, it will raise the index, and vice versa when 
 the air is dry. 
 
 To prepare the sponge, it may be proper first to wash it in water very clean, and, when dry 
 again, dip it in water or vinegar in which there has been dissolved sal amoniac, or salt of tartar; 
 after which let it dry again. Salt of tartar, or any other salt, or pot-ashes, may be put into the 
 scale of a balance, and used instead of the sponge. 
 
 X. STEAM ENGINE. 
 
 The steam engine is a machine which derives its moving power from the elasticity and 
 condensation of the steam of boiling water. The high importance of this machine to the 
 mechanical arts of life, especially where immense powers are required, has given birth to many 
 considerable improvements both in its consti uction and mode of operation. 
 
 The following is a description of one of the earliest steam engines, which, as it exhibits the 
 general principles of the machine, will be deemed sufficient in a work only introductory to 
 science. A history of the steam engine, from its first construction by Capt. Savary, down to the 
 present time, in which are included all the great improvements made by the ingenious Mr. 
 Watt, of Birmingham, will be found in the Encyclopredia Brittannica, Vol. XVH. Part u. 
 
 H represents the boiler on its furnace ; E the cylindrical vessel of iron, in which the piston p late ,} 2 ' 
 OO moves up and down. The cavity between the piston and bottom of the cylinder is made air- 
 tight. F is a cock to admit the steam into the cylinder. IK is a lever, attached to the piston 
 at 1, and at K to the piston of the pump which works on that side. PQ is a solid piston moving 
 in the pipe RM, and loaded with a heavy weight at P. ABC is the main pipe that receives the 
 water forced from RM through a valve at C, opening outward. N is an air vessel communi- 
 cating with the main pipe. At D is a valve opening upward, and at M is the water to be 
 raised. 
 
 The engine is represented at the end of a forcing stroke, which is likewise its position when 
 at rest. Suppose the main pipe ABC to be filled with water, and the water in the boiler H to 
 15 
 
OF PNEUMATICS. BOOK III. PART II. 
 
 boil strongly. The cock F being opened, the steam rushes into the cylinder, and being much 
 lighter than air, rises to the top, and expels the air though a valve in the bottom of the cylinder. 
 F is then shut, and the cock G communicating with the main pipe is opened, which, by spouting 
 cold water against the bottom of the piston, condenses the stream. A vacuum being thus 
 obtained, the pressure of the atmosphere forces the piston down to the bottom of the cylinder ; 
 the level- IK is moved, and the piston PQ with its weight is raised, and the water ascends in the 
 pipe MR upon the principle of the common pump. The cock G being now shut, and F opened, 
 the steam enters the cylinder, and counteracts the pressure of the atmosphere on the piston 
 OO; consequently, the weight P prevails, and drives down the piston RQ, forcing the water 
 through the valve C into the main pipe and its air vessel. The ue of the air vessel is to prevent 
 the main pipe from bursting by the sudden entrance of the water ; for the air at N being elastic, 
 gives way to the stroke, and its re-action during the time of elevating the piston PQ continues 
 the motion of the water, so that its velocity is no more than half what it would have been if it 
 had been impelled by starts, and rested during the raising of the piston. By opening the cock 
 G, and shutting F, (which is done by a single operation) the steam is again condensed, the 
 pressure of the atmosphere again prevails, and thus the work may be continued at pleasure. 
 
 The power of some of the steam engines, constructed by Messrs. Boulton and Watt, is thus 
 described as taken from actual experiment. An engine, having a cylinder of 31 inches in 
 diameter, and making 17 double strokes per minute, performs the work of 40 horses, working 
 night and day, (for which three relays, or 120 horses, must be kept) and burns 11,000 pounds of 
 (Staffordshire coal per day. A cylinder of 19 inches, making 25 strokes of 4 feet each per 
 minute, performs the work of 12 horses, working constantly, and burns 3700 pounds of coal 
 per day. These engines will raise more than 20,000 cubic feet of water, 24 feet high, for every 
 hundred weight of good pit coal consumed by them. 
 
 XI. The HYDROMETER. 
 
 Hate 12. The Hydrometer, an instrument usually applied to find the specific gravities of liquids, is 
 Fig. 3. thus constructed ; AB is a hollow cylindrical tube of glass, ivory, copper, &c. joined to a hollow 
 ball D, at the bottom of which is a smaller ball E, containing some quicksilver, or shot, by 
 which the instrument is so poised, that it swims vertically in a liquid. The stem AB is gradu- 
 ated in such a manner, that the figures exhibit the magnitudes of the parts below, and conse- 
 quently, the specific gravities of the different fluids in which it descends to those figures. Thus 
 if the parts immersed in water, and spirits of wine, be as 10 to 11.1, then the specific gravity of 
 the water will be to that of the spirits of wine as 11.1 to 10. 
 
 To make this instrument of more service, there has been added a little plate, or dish, at the 
 top of the tube, upon which inay be placed weights, as convenience may require. For example ; 
 if the whole instrument float, immersed in spirits to a certain point, it will require an additional 
 weight to sink it to the same depth in water. Suppose the instrument to weigh 10 dwts. and to 
 be adjusted to rectified spirits of wine, it will then require an additional weight of 1.6 dwt. to 
 sink it to the same point in water. Consequently, the specific gravity of water is to that of 
 rectified spirits of v/ine as 11.6 to 10, or as 10 to 8.6. 
 
BOOK IV. 
 
 OF MAGNETISM. 
 
 DBF. I. A HE earth contains a mineral substance which attracts iron, steel, 
 and all ferruginous substances ; this is called a natural magnet. 
 
 DBF. II. The same substance has the power to communicate its properties 
 to all ferruginous bodies ; those bodies, after having acquired the magnetical 
 properties, are called artificial magnets. 
 
 Those magnets are alsu made without the assistance of the natural magnet, 
 as will hereafter be shewn. * 
 
 SCHOL. The property of attraction in the magnet was that by which it was first discovered. 
 Every substance that contains iron, is more or less attracted by the magnet ; and so universally 
 is this truly important metal disseminated ; that there are very few substances which are not in 
 some degree capable of being attracted by the magnet. In this way iron is found to enter into 
 the composition of animals, vegetables, minerals, and even into that of the atmosphere. On this 
 subject, see Cavallo on Magnetism, Chap. vi. Part i. 
 
 DBF. III. Those points in a magnet, which seem to possess the greatest 
 power, are called tlie poles of the magnet. 
 
 DBF. IV. The magnetical meridian is a vertical circle in the heavens, 
 which intersects the horizon in the points to which the magnetical needle, 
 when at rest, is directed. 
 
 DBF. V. The axis of a magnet is a right line, which passes from one 
 pole to the other. 
 
 DBF. VI. The equator of a magnet is a line perpendicular to the axis, 
 and exactly between the two poles. 
 
 SCHOL. The distinguishing and characteristic properties of a magnet, are, (1.) Its attractive 
 and repulsive powers. (2.) The force by which it places itself, when freely suspended, in a 
 certain direction toward the poles of the earth. (3.) Its dip or inclination toward a point 
 below the horizon. (4.) The property which it possesses of communicating the foregoing powers 
 to iron and steel. 
 
lift OF MAGNETISM. BOOK IV. 
 
 I)EF. VII. The direction of the dipping needle in any place is called 
 the magnetical line. 
 
 PROP. I. That mineral substance which is called the loadstone, or 
 magnet, has the property of attracting iron ; but no other body whatever, 
 unless it has a mixture of iron. 
 
 EXP. 1. The action of the magnet on iron may be shewn on needles, steel filings, &c. 
 
 2. Let a needle be suspended from a loadstone, and a string passing through its eye be 
 fastened to the beam of a balance placed under it 5 the degree of force with winch it is attracted, 
 may be measured. 
 
 SCHOL. Some philosophers have supposed that iron is not the only substance attracted by 
 the magnet. Mr. Kirwan says, that nickel, when purified from iron, becomes more, instead of 
 less magnetic, and acquires the properties of a magnet. Mr. Cavallo instituted a number of 
 experiments, with a view of ascertaining whether any other bodies than ferruginous ones, were 
 attracted by the magnet. After all, he does not decide positively on the question. 
 
 PROP. II. The action, and re-action of the magnetic power, are mutual 
 and equal. 
 
 A piece of iron, or steel, or other ferruginous substance, being brought within a certain 
 distance of one of the poles of a magnet, is attracted by it, so as to adhere to the magnet, and 
 not to suffer itself to be separated without an evident effort. This attraction is also mutual, for 
 the iron attracts the magnet, as much as the magnet attracts the iron ; since if they be placed 
 on pieces of wood or cork, so as to float upon the surface of vvaier, it will be found that the iron 
 advances toward the magnet, as well as the magnet advances toward the iron ; or, if the iron 
 be kept steady, the magnet will move toward it. 
 
 SCHOL. The strength of magnetic attraction varies according to different circumstances 5 
 such as, the strength of the magnet ; the weight and shape of the body presented to it ; the 
 magnetic, or unmagnetic state of that body ; the distance between it and the magnet, &c. 
 
 The attraction is strongest near the surface of the magnet, and diminishes as it recedes 
 from it; the law of this diminution has not yet been ascertained. 
 
 The four following experiments, accurately made by Professor Muschenbroek, will exhibit 
 some of the irregularities respecting magnetic attraction. In these experiments, the magnet 
 was suspended to one scale of an accurate balance, and under it there was successively placed 
 on a table at different distances, another magnet, or piece of iron, and at each distance, the 
 degree of attraction between the iron and the magnet was ascertained by weights put into the 
 other scale.' The results were as follow ; 
 
BOOK IV. 
 
 OF MAGNETISM. 
 
 117 
 
 Exp. 1. In this ex- 
 
 EXP. 2. A spherical 
 
 Exp. 3. Instead of 
 
 EXP. 4. A globe of 
 
 periment a cylindrical 
 
 magnet, of the same di- 
 
 the cylindrical magnet, 
 
 iron of the same diam- 
 
 magnet, weighing 16 
 
 ameter as the last, but 
 
 the cylinder of iron was 
 
 eter as the magnet, 
 
 drams, was suspended 
 
 of greater strength, was 
 
 placed on the table, and 
 
 was now placed on the 
 
 to the scale ; and on the 
 
 affixed to the scale, and 
 
 under the globular mag- 
 
 table. 
 
 table a piece of iron of 
 
 the cylindrical magnet 
 
 net. 
 
 
 the same shape and 
 
 used in the preceding 
 
 
 
 weight. 
 
 experiment, was placed 
 
 
 
 
 on the table. 
 
 
 
 Dist. in Attract. 
 
 Dist. in Attract. 
 
 Dist. in Attract 
 
 Dist. in Attract. 
 
 inches. in grains. 
 
 inches. in grains. 
 
 inches. in grains 
 
 inches. in grains. 
 
 6 - - 3 
 
 6 - - 21 
 
 6 - - 7 
 
 8 . - 1 
 
 5 - - - 3i 
 
 5 - - -27 
 
 5 - - -94 
 
 7 - - - 2 
 
 4 - - 44 
 
 4 - 34 
 
 4 - - 15 
 
 6 - ' - si 
 
 3 - - - 6 
 
 3 - - -44 
 
 3 - - 25 
 
 5 - - - 6 
 
 2 - - 9 
 
 2 - - 64 
 
 2 - 45 
 
 4 - - 9 
 
 1 - - - 18 
 
 1 - 100 
 
 1 - 92 
 
 3 - 10 
 
 - 57 
 
 - - 260 
 
 - - 340 
 
 2 - - 30 
 
 
 
 
 1 - - 64 
 
 
 
 290 
 
 COR. It appears from the second and third experiments, that, when in contact, a magnet 
 attracts another magnet with less force than it does a piece of iron. This has been confirmed 
 by many other experiments. But the attraction between the two magnets begins from a greater 
 distance than between the magnet and iron ; hence it must follow a different law of decrement.* 
 
 PROP. III. The attraction and repulsion of magnetism is not sensibly 
 affected by the interposition of bodies of any sort except those which are 
 ferruginous. 
 
 EXP. 1. Suppose a magnet placed at an inch distance from a piece of iron requires an ounce 
 of force to remove it, or, which is the same thing, suppose that the attraction toward each other 
 is equal to one ounce ; it will he found that the same degree of attraction remains constantly 
 unaltered, though a plate of other metal, glass, paper, &c. be interposed between the magnet and 
 the iron, or though they be inclosed in separate boxes of glass or other matter. 
 
 2. Move steel filings placed on a brass plate, in water, &c. by holding a magnet under the 
 vessel. 
 
 3. Sprinkle steel dust on a sheet of paper, under which is placed a magnet, or two magnets, 
 having their poles opposite to each other, and at the distance of about an inch. 
 
 4. A needle under an exhausted receiver will be attracted at the same distance, as in the 
 open air. 
 
 SCHOL. 1. Heat weakens the power of a magnet; and a white heat destroys it entirely. 
 Hence it appears, from this cause alone, besides others which may concur, the power of a magnet 
 must be continually varying. 
 
 SCHOL. 2. The attractive power of a magnet may be increased considerably by gradually 
 adding more weight to it; for it is found that a magnet will keep suspended on one day a little 
 
 * Mr Coulomb lias ascertained that the force of both magnetic and electric influence, like gravitation, is 
 inversely as the square of the distance. 
 
118 OF MAGNETISM. BOOK IV. 
 
 more weight than it did the preceding ; which, additional weight being added to it on the follow- 
 ing day, it will be found that the magnet can keep suspended a weight still greater, and so on, 
 as far as to a certain limit. 
 
 On the contrary, by putting a very small weight of iron to it, the magnet may gradually lose 
 much of its strength. 
 
 SOHOL. 5. Among natural magnets, the smallest generally possess a greater attractive power, 
 in proportion to their size, than those which are larger. There have been natural magnets not 
 exceeding 20 or 30 grains, which would lift a piece of iron that weighed 40 or 50 times more 
 than themselves. A small magnet, worn by Sir I. Newton in a ring, weighing but about 3 
 grains, is said to have taken up 746 grains, or nearly 250 times its own weight. And Mr. 
 Cavallo has seen one of 6 or 7 grains weight, which was capable of lifting a weight of SOO grains. 
 But magnets of two pounds and upwards, seldom lift up ten times their own weight of iron. 
 
 PROP. IV. The magnetic power may be communicated from the load- 
 stone ; and from one piece of iron to another, which then becomes an artificial 
 magnet ; and this communication of power is without apparent loss of power 
 in the loadstone. 
 
 EXP. 1. Take a bar of soft iron, about three feet long and one inch thick, (some kitchen 
 pokers will answer for this experiment) and place it upright, or rather in the magnetical line. 
 Then present a magnetic needle to the various parts of the bar from top to bottom, and the 
 lower half of the bar will be possessed of the north polarity, capable of repelling the north, and 
 of attracting the south pole of the needle, and the upper half is possessed of the south polarity. 
 The attraction is strongest at the very extremities of the bar, it diminishes as it recedes from 
 them, and vanishes about its middle point. 
 
 If the bar is turned upside down, the south pole will become north, and the north will become 
 the south pole. In the southern parts of the globe, the lower part is a south pole ; or more 
 generally, the extremities of the bar will acquire the polarities corresponding to the nearest poles 
 of the earth. 
 
 If an iron bar be left a long time in the direction of the magnetic line, or even in a perpen- 
 dicular posture, it will sometimes acquire a great magnetic power. Tongs, pokers, &c. by being 
 often heated, and set to cool again in an erect posture, frequently acquire a considerable mag- 
 netic virtue. Magnetism is often communicated to iron and steel by repeated blows of the 
 hammer ; by some experiments of Mr. Cavallo, it appears that this effect is often produced on 
 brass, hence it is necessary carefully to examine the brass before it is used in the construction 
 of theodolites, &.c. 
 
 hlat 13. 2. Place two magnets A and B in a right line, so that the nor*h end of the one is opposed to 
 the south end of the other, and at such a distance that the bar to be touched may rest upon them. 
 Take now two other bars. D and E, and apply the north end of D,* and the south end of E to 
 the middle of the untouched bar C, elevating their other ends so as to make an acute angle with 
 the said bar. 
 
 Separate the bars D and E, drawing them different ways along the surface of C, but preserv- 
 ing the same elevation ; then removing the bars D and E to the distance of a foot or more from 
 
 The north ends of magnetic bars are generally marked with n cross or straight line, as are also the 
 north ends of the horse-shoe, or any other shaped magnets. 
 
BOOK IV. OF MAGNETISM. 119 
 
 the bar C, and bringing the north and south ends into contact, and then apply them again to the 
 middle of C. This process being repeated several times to each surface of the bar C, it will be 
 found to have acquired a strong and permanent magnetism. 
 
 3. Take twelve bars, sis of soft steel, and six of hard, the former to be each three inches 
 long, J of an inch broad, ^ of an inch thick; with two pieces of iron, each half the length 
 of one of the bars, but of the same breadth and thickness. The 6 hard bars to be each 5 
 inches long, \ an inch broad, and -/ of an inch thick, with two pieces of iron of half the length 
 but of the same breadth and thickness of one of the hard bars ; and let all the bars be marked 
 with a line quite round them at one end ; then take an iron poker and tongs, or two bars of 
 iron, the larger they are, and the longer they have been used, the better; and fixing the poker 
 upright, or rather in the magnetical line, between the knees, hold to it, near the top, one of the 
 soft bars, having its marked end downward, by a piece of sewing silk, which must be pulled 
 tight by the left hand that the bar may not slide ; then grasping the tongs with the right hand, a 
 little below the middle, and holding them nearly in a vertical position, let the bar be stroked by 
 the lower end from the bottom to the top about 10 times on each side, which will give it a mag- 
 netic power sufficient to lift a small key at the marked end ; which end, if the bar were sus- 
 pended on a point, would turn toward the north, and is therefore called the north pole ; and the 
 unmarked end, for the same reason, is called the south pole. Four of the soft bars being Plate 13. 
 impregnated after this manner, lay the other two parallel to each other, at a quarter of an inch Vl S- 2t 
 distance, between the two pieces of iron belonging to them, a north and a south pole against 
 each piece of iron ; then take two of the four bars already made magnetical, and place them 
 together so as to make a double bar in thickness, the north pole of one even with the south pole r | ate 13. 
 of the other ; and the remaining two being put to these, one on each side, so as to have two '*' J 
 north, and two south poles together, separate the north from the south poles at one end by the 
 interposition of some hard substance I, and place them perpendicularly with that end downward 
 on the middle of one of the parallel bars AC, the two north poles toward its south end, and the 
 two south poles toward its north end. Slide them three or four times backward and forward 
 the whole length of the bar; then removing them from the middle of this bar, place them on the 
 middle of the other bar BD as before directed, and go over that in the same manner ; then turn 
 both the bars the other side upward, and repeat the former operation ; this being done, take the 
 two from between the pieces of iron ; and placing the two outermost of the touching bars in 
 their stead, let the other two be the outermost of the four to touch these with ; and this process 
 being repeated till each pair of bars have been touched three or four times over, which will give 
 them a considerable magnetic power. 
 
 When the small bars have been thus rendered magnetic, in order to communicate the mag- 
 netism to the large bars, lay two of them on the table, between their iron conductors, as before ; 
 then form a compound magnet with the six small bars, placing three of them with the north 
 poles downward, and the three others with the south poles downward. Place these two parcels 
 at an angle, as was done with four of them, the north extremity of the one parcel being put 
 contiguous to the south extremity of the other, and, with this compound magnet, stroke four of 
 the large bars, one after another about twenty times on each side, by which means they will 
 acquire some magnetic power. 
 
 When the four large bars have been so far rendered magnetic, the small bars are laid 
 aside, and the large ones are strengthened by themselves, in the manner as was done with the 
 small bars. 
 
190 OF MAGNETISM. BOOK IV. 
 
 To expedite the operation, the bars ought to be fixed in a groove, or between brass pins, 
 otherwise the attraction anil friction between the bars will be continually deranging them, when 
 placed between the conductors. 
 
 This whole process may be gone through in about | an hour, and each of t'le large bars, if 
 well hardened, will lift about 28 ounces Troy, and they are fitted for all the purposes of 
 magnetism, in navigation and experimental philosophy. The half dozen being put into a case 
 in such a manner as that no two poles of the same name may be together, and their irons with 
 them as one bar, they will retain the virtue they have received ; but if their power should, hy 
 making experiments, be ever so much impaired, it may be restored without any foreign assistance 
 in a few minutes. 
 
 This method of communicating magnetism was sent to the Royal Society by Mr. Canton, in 
 the year IT 51. 
 
 SCHOL. 1. The magnetic virtue may be readily communicated by the horse-shoe magnet, 
 much in the same way as in the preceding experiment. 
 
 SCHOI.. 2. A small compass needle may be touched by being put between the opposite 
 poles of two magnetic bars. Whilst it is receiving the magnetism, it will be violently agitated, 
 moving backward and forward as if it were animated ; and when it has received as much 
 magnetism as it can acquire in this way, it becomes quiescent. 
 
 Another method of communicating magnetism to a compass needle, is by means of the 
 combined horse-shoe magnet, from the centre of which draw that half of the needle which is to 
 have the contrary pole; from a considerable distance draw the needle over it again. This 
 repeated twenty times or more, and the same for the other half, will sufficiently communicate 
 the power. 
 
 PROP. V. Two magnets having a free motion, will attract when differ- 
 ent poles are directed toward each other, and repel when the adjacent poles 
 are of the same name. 
 
 EXP. A needle turning on its centre will be attracted or repelled by another, as different, 
 or the same poles are brought near to each other. 
 
 SCHOL. 1. If the magnetic powers are very unequal, or the two bodies are forcibly brought 
 together, they will attract with the same poles. 
 
 EXP. I. Suspend a magnet by a thread, and let a small needle be brought near it, making 
 poles of the same name contiguous. 
 
 2. Bring two very unequal needles into contact at the same poles, suspended in the same 
 manner, they will cohere. 
 
 SCHOL. 2. The following experiments will shew the attraction of the magnet on ferruginous 
 bodies which are not magnetic. 
 
 Properly speaking, however, the magnet has no action upon unmagnetic bodies, for any 
 ferruginous body becomes magnetic, on being presented to the magnet, and then is attracted 
 by it. 
 
 EXP. 1. Place a magnetic needle upon a pin stuck on a table, and when it stands steady, 
 
BOOK II. OF MAGNETISM. 1S1 
 
 place an iron bar 8 inches long, and an inch thick upon the table, so that one cud of it may 
 be on one side of the north pole of the needle, and near enough to draw it a little out of its 
 natural direction. In this situation approach gradually the north pole of a magnet to the other 
 extremity of the bar, and you will see that the needle's north end will recede from the bar in 
 proportion as the magnet is brought nearer to the bar. 
 
 The reason of this phenomenon is, that, by the approach of the north pole of the magnet, in 
 the first case, the extremity of the iron bar next to it acquires a south polarity, and consequently, 
 the opposite extremity acquires a north polarity, by which the needle is repelled ; but in the 
 second case, when the north pole of the magnet is brought near the bar, the end of the bar 
 next to it acquires a south polarity, and the opposite end, acquiring the north polarity, causes 
 the north end of the needle to recede. 
 
 2. Tie two pieces of soft iron wire, AB, AB, each to a separate thread AC, which join at Plate 13. 
 top, and suspend them on a pin so that the wires may hang at some distance from the wall. '^' 
 Then bring the marked end D of a magnetic bar just under them, and it will be seen that the 
 
 wires repel each other more or less in proportion to the distance of the magnet. The same may 
 be shewn by means of the south pole of the magnet. 
 
 If the wires be of soft iron, they will, on removing the magnet, soon collapse ; but if steel 
 wires, or two sewing needles be used, they will retain their magnetic virtue, and continue to 
 repel each other. 
 
 3. Take four pieces of steel wire, or four common sewing needles, tie threads to them, and 
 join them two and two, as in the last experiment; then bring the same pole of the magnet under 
 both pairs, by which means they will acquire a permanent magnetism, and the wires of each pair 
 will repel each other. After putting the magnet aside, bring one pair of the wires near the 
 other pair, so that their lower extremities may be level, and the four wires will repel each other, 
 and form a kind of square. 
 
 4. Strew some iron filings upon a sheet of paper laid on a table, and place a small artificial 
 magnet among them, then give a few gentle knocks to the table with the hand, so as to shake the 
 filings, and they will dispose themselves round the bar in the manner represented by the figure ; P j ate jg 
 many particles clinging to one another, and forming themselves into lines, which, at the very Fig 6. 
 poles, are in the same direction with the axis of the magnet; a little sideways of the poles 
 
 they begin to bend, and then they form complete arches, reaching from a point in the north 
 half of the magnet to a point in the other half, which is possessed of the south polarity. 
 
 5. Tie a thread to one end of a bit of soft iron wire AB, about four inches long, and suspend Plate 15. 
 it freely ; let a bar of soft iron CD be so supported, as to have one of its extremities C Fig. *" 
 about | of an inch distant from the lower extremity B of the wire. Bring now either pole of 
 
 a strong magnet EF under it, and the end B of the wire will recede from C, because they 
 are both possessed of the same polarity ; but if the magnet be applied to the upper part of 
 the wire, in the situation Gil, then the end B of the wire will be attracted by the extremity C 
 of the iron bar, because, supposing G to be the north pole of the magnet, C acquires a south 
 polarity, and attracts the end B; because B being farthest from the north pole G, acquires also 
 the north polarity. 
 
 SCHOL. 3. Hence methods are easily devised to ascertain whether a body possesses any 
 magnetism, and in case it does, to find out the poles. 
 
 16 
 
OF MAGNETISM. BOOK IV. 
 
 EXP. 1. To ascertain whether a body has any attraction toward the magnet. 
 
 If the body contain an evident quantity of iron, it will be perceived as soon as His brought 
 in contact with the magnet, as a certain force will he required to separate them. 
 
 If the body be not sensibly attracted by the magnet in this way ; let it be placed, by 
 means of a piece of cork or wood, upon some water, or mercury, in a common soup plate, in 
 in which situation let a magnet be brought sideways to it, and the attraction will be manifest by 
 the body coming toward the magnet. 
 
 2. To ascertain whether a given body has any magnetism. 
 
 The only difference in this experiment from the last is, that instead of a magnet, must be 
 used a piece of soft clean iron, about one inch long, and of half an ounce weight. 
 
 3. Jl magnetic body being given to find out itf> poles. 
 
 Present the various parts of the surface of the magnetic body successively to one of the poles 
 of a magnetical needle, and the parts possessed of a contrary polarity will be discovered by the 
 needle's standing perpendicularly towards them. Then present the various parts of the surface 
 of the same body to the other pole of the needle. 
 
 DBF. VIII. There is a point between the two poles where the magnet 
 has no attraction nor repulsion ; this point is called the magnetic centre, 
 though it is not always exactly between the poles. 
 
 PROP. VI. If a magnet be cut through the middle, or any way broken 
 in two, each piece will become a complete magnet, and the parts which were 
 contiguous will become opposite poles. 
 
 Plate 13. EXP. 1. Take a magnetic bar AB, six or eight inches long, and i of an inch thick, having 
 
 ' ' only two poles A and B. The magnetic centre of this bar will be in, or very near, its middle C. 
 
 Now if. by a smart stroke of a hammer, part of the magnet be broken off as FB, it will be found 
 
 that the part of the fragment contiguous to the fracture has acquired the contrary polarity, and 
 
 a magnetical centre E will be generated. 
 
 At first the magnetic centre of this fragment is nearer to the fracture F, but in time it 
 advances toward the middle of the fragment. The original centre C of AF, after the fracture, 
 will likewise advance nearer to the middle of it. 
 
 2. A steel bar, of the same size as that mentioned in the last experiment, being made quite 
 hard, may be broken into two parts, and so pressed together as to appear whole. In this situa- 
 tion it may be rendered magnetic by the application of very powerful magnets to its extremities; 
 and the whole bar will be found to have two poles at its extremities, and one magnetic centre in its 
 middle ; but if the parts be separated, each will be found to have two poles and a magnetic ceutre. 
 
 COR. Hence it is seen that ; the magnetic centre may be removed ; it maybe removed 
 also, by striking a magnetic bar, by heating it, by hard rubbing, &c. 
 
 PROP. VII. Magnetism requires some time to penetrate through iron. 
 
 EXP. Place a bulky piece of iron, weighing 40 or 50 pounds, so near a magnetic needle as to 
 draw it a little out of its direction, apply one of the poles of a strong magnet to the other 
 extremity of the iron, and several seconds will be required before the needle can be affected by 
 
BOOK IV. OF MAGNETISM. 123 
 
 it. The interval is greater or less according to the size of the iron and the strength of the 
 magnet. 
 
 DBF. IX. A magnet is said to be armed, when its poles are surrounded 
 with plates of iron or steel. 
 
 PROP. VIII. A magnet will take up much more iron when armed than 
 it cau alone. 
 
 As both magnetic poles together attract a much greater weight than a single one, and as the 
 two poles of a magnet are generally in opposite parts of its surface, in which situation the same 
 piece of iron cannot be adapted to them both at the same time; therefore it has been common, 
 to place two broad pieces of soft iron to the poles of a magnet, and projecting on one side, 
 because in that case, the pieces of iron beiug rendered magnetic, another piece of iron could be 
 conveniently adapted to their projections, so that both poles may act at the same time. Those 
 pieces of iron called the armature are generally held fast upon the magnet by means of a silver 
 or brass box. Thus AB represents the magnet CD, CD represents the armature or pieces of Plate 13. 
 iron, the projections of which are DD, and to which the piece of iron F is made to adhere. The fl S' 4- 
 dotted lines represent the brass box having a ring E at top by which the armed magnet may be 
 suspended. In this manner the two poles of the magnet, which are at A and B, are made to 
 act at DD. 
 
 For this purpose, and to avoid armature, artificial magnets have been constructed in the 
 shape of a horse-shoe, having their poles in the truncated extremities. 
 
 PROP. IX. A magnetical needle, accurately balanced on a pivot or 
 centre, will settle in a certain direction, either duly, or nearly north and 
 south, called the magnetical meridian. 
 
 This is known by long eiperience. 
 
 The directive power of the magnet is the most wonderful and useful part of the subject. By 
 it mariners are enabled to conduct their vessels tbrough vast oceans in any given direction ; by 
 it miners are guided in their works below the surface of the earth ; and travellers conducted 
 through deserts, otherwise impassable. 
 
 The usual method is to have an artificial magnet suspended, so as to move freely, which will 
 always place itself in or near the plane of the meridian north and south ; then by looking upon 
 the direction of this magnet the course is to be directed so as to make any required angle with 
 it. Thus, suppose that a vessel setting off from any place in order to go to another which is due 
 west of the former; in that case, the vessel must be so directed that its course may he always 
 at right angles with the situation of the magnetic needle, the north end of which must be to the 
 right hand. A little reflection will show how the vessel may be steered in any other direction. 
 An artificial steel magnet, fitted for this purpose in a proper box, is called the mariners' compass, 
 or sea compass, or, simply, the compass ; which instrument is too well known to need any par- 
 ticular description. The mariner's compass, with the addition of sights, divided circles, &c. for 
 observing azimuths, and amplitudes of the heavenly bodies, is called the azimuth compose. 
 
OF MAGNETISM. 
 
 BOOK IV. 
 
 DBF. X. The deviation of the horizontal needle from the meridian, or 
 the angle which it makes with the meridian, when freely suspended in a 
 horizontal plane, is called the declination or variation of the needle. 
 
 PROP. X. There is generally a small variation in the direction of the 
 magnetic needle, which differs in degree at different places and times. 
 
 This is known by observing the different points of the compass at which the sun rises or sets, 
 and comparing them with the true points of the sun's rising or setting, according to astronomical 
 tables. Thus, if the magnetic amplitude is 80 eastward of -the north, aud the true amplitude is 
 82 toward the same side, then the variation of the needle is west. The variation may be 
 estimated from the azimuths in the same way. 
 
 SOHOL. I. A needle is continually changing the line of its direction, traversing slowly to 
 certain limits toward the east and west. The first good observations, on the variations were 
 made by Burrowes about the year 1580, when the variation, at London, was 11 15' east, and 
 since that time the needle has been moving to the westward at that place ; also by the observa- 
 tions of different persons it has been found to point, at different times, as in the following table. 
 
 Years. 
 
 Observers. 
 
 Variation E. or W. Years. 
 
 Observers. 
 
 Variation E. or W. 
 
 
 
 1 
 
 
 
 o / 
 
 
 1580 
 
 Burrowes. 
 
 11 15 East. 
 
 1723 
 
 Graham. 
 
 14 17 
 
 West. 
 
 1622 
 
 Gunter. 
 
 5 56 
 
 1747 
 
 _____ 
 
 17 40 
 
 
 1634 
 
 Gellibrand. 
 
 4 3 
 
 1774 
 
 Royal Soc. 
 
 21 16 
 
 
 1640 
 
 Bond. 
 
 3 7 
 
 1775 
 
 Royal Soc. 
 
 21 43 
 
 
 1657 
 
 Bond. 
 
 
 
 1776 
 
 Royal Soc. 
 
 21 47 
 
 
 1665 
 
 Bond. 
 
 1 23 West. 
 
 1777 
 
 Royal Soc. 
 
 22 12 
 
 
 1666 
 
 Bond. 
 
 1 36 
 
 1778 
 
 Royal Soc. 
 
 22 20 
 
 
 1672 
 
 
 
 2 SO 
 
 1779 
 
 Royal Soc. 
 
 22 28 
 
 
 1683 
 
 
 
 4 30 
 
 1780 
 
 Royal Soc. 
 
 22 41 
 
 
 1692 
 
 
 
 6 00 
 
 
 
 
 
 By this table it appears, that from the first observation in 1580 till 1657, the change in the 
 variation at London was 11 15' in 77 years, which, at a mean rate, is nearly 9' a year. And 
 from 1657 to 1780, it changed 22 41', which is at the rate of 1 1' a year nearly. 
 
 r 
 
 1550 
 
 1 
 
 8 East. 
 
 At Paris the j 
 
 1640 
 
 3 
 
 Variation of , 
 
 1660 
 
 
 
 the Needle j in 
 
 1681 
 
 2 2 West. 
 
 was 
 
 1759 
 1760 
 
 18 10 
 18 20 
 
 At St. Helena f 
 the Variation! . 
 of the Needle^ 
 
 was 
 
 1600 
 1623 
 1677 
 1692 
 
 8 East. 
 
 6 
 
 40 
 
 1 West. 
 
 Near the equator, in long. 40 east, the highest variation from the year 1700 to 1756, was 
 17 15' west ; and the least 16 30' W. In lat. 15 N. and long. 60 VV. the variation was 
 constantly 5 E. In lat. 10 S. and long. 60 E. the variation decreased from 17 W. to 7 15' 
 
BOOK IV. 
 
 OF MAGNETISM. 
 
 135 
 
 W. In lat. 10 S. and long. 5 W. it increased from 2 15' to 12 45' W. In lat. 15 N. and 
 long. 20 W. it increased from 1 W. to 9 W. In the Indian seas the irregularities were 
 greater, for in 1700, the west variations seem to have decreased regularly from long. 50 E. to 
 long. 100 E; but in 1756 the variation decreased so fast, ihai there was east variation in long. 
 80, 85, and 90 E. and jet, in long. 95 and lOQo E. there was west variation. 
 
 In the year 1775, in lat. 58 17' S. and long. 348 16' E. it was 16' W. In lat. 2 24' 
 N. and long. 32 12' W. it was 14' 45" W. In lat. 50 6' 30." N. and long. 4 0' W. it was 
 19 28' W. 
 
 SCHOL. 2. The variation of the needle is affected by heat and cold. The following is the 
 result of observations made by Mr. Canton at different hours of the day, and also the mean 
 variation for each month in the year. 
 
 the Year. 
 
 Morning 
 
 Afternoon 
 
 lination 
 
 observed at different Hours of the 
 
 The mean Vj 
 
 
 same day. 
 
 
 
 JUNE 27, 1759. 
 
 
 
 
 Degrees of the 
 
 
 Hours. 
 
 Miii. 
 
 Declin. West. 
 
 Thermometer. 
 
 
 r 
 
 18 
 
 12 2' 
 
 62 
 
 January 
 
 6 
 
 4 
 
 18 58 
 
 62 
 
 February 
 
 8 
 
 30 
 
 18 55 
 
 65 
 
 March 
 
 9 
 
 2 
 
 18 54 
 
 67 
 
 April 
 
 10 
 
 20 
 
 18 57 
 
 69 
 
 May 
 
 Jl 
 
 40 
 
 19 4 
 
 68-- 
 
 June 
 
 
 
 
 
 July 
 
 r o 
 
 50 
 
 19 9 
 
 70 
 
 August - 
 
 1 1 
 
 38 
 
 19 8 
 
 70 
 
 September 
 
 j s 
 
 10 
 
 19 8 
 
 68 
 
 October 
 
 \ r 
 
 20 
 
 18 59 
 
 61 
 
 November 
 
 9 
 
 12 
 
 19 6 
 
 59 
 
 December 
 
 -11 
 
 40 
 
 18 51 57i 
 
 
 7 8 
 . 8 58 
 
 11 17 
 
 12 26 
 
 13 
 13 21 
 13 14 
 12 19 
 11 43 
 10 36 
 
 8 9 
 6 58 
 
 PROP. XI. A needle which, before it receives the magnetic power, rests 
 on its centre parallel to the horizon, on becoming magnetical will incline 
 toward the earth ; this is called the inclination or dip of the needle. 
 
 EXP. Let a small dipping needle be carried from one end of a magnetic bar to the other : 
 when it stands over the south pole, the north end of the needle will be directed perpendicularly 
 to it ; as the needle is moved, the dip will grow less, and when it comes to the magnetic centre 
 it will be parallel to the bar; afterward the south end will dip, and the needle will stand perpen- 
 dicular to the bar when it is directly over the north pole. 
 
 SOHOL. 1. This property of the magnetic needle was first discovered accidentally by Kobert 
 Norman, a compass maker at RadcliS'e, about the year 1576. He Delates, that it being his 
 custom to finish and hang up the needles of his compasses, before he touched them, he found 
 that immediately after the touch, the north point would always dip or incline downward pointing 
 in a direction under the horizon ; so that, to balance the needle again he was forced to put a 
 piece of wax on the south end as a counterpoise. The constancy of the effect lid him to meas- 
 ure the angle which the needle would make with the horizon, and he found it at London to be 
 71 50'. 
 
OF MAGNETISM. BOOK!\. 
 
 It is not yet absolutely ascertained whether the dip varies at the same place ; it is now, ant! 
 has been since the year 1772, about 72, according to several observations made by Mr. Nairne 
 and the Royal Society. The trifling difference between the first observations of Mr. Norman, 
 and these last of Mr. Nairne, &c. leads us to suppose that the dip is unalterable at the same 
 place. 
 
 It is certain, however, that the dip is different in different latitudes, and that it increases in 
 going northward. It appears from a table of observations made with a marine dipping needle of 
 Mr. Nairne, in a voyage toward the nortli pole in 1773, that 
 
 In latitude 60 18', the dip was 75 0'. 
 In latitude 70 45, the dip was 77 52. 
 In latitude 80 12, the dip was 81 52. 
 In latitude 80 27, the dip was 82 2|, 
 
 See Phil. Trans. Vol. IA\. 
 
 SCHOL. 2. The phenomena of the compass, and the dipping needle, and of the magnetism 
 acquired by an iron bar in a vertical position, leave no room to doubt but that the cause exists in 
 the earth. Dr. Halley supposed that the earth has within it a large magnetic globe, not fixed 
 within to the external parts, having four magnetic poles, two fixed and two moveable, which 
 will account for all the phenomena of the compass and dipping needle. This would make the 
 variation subject to a constant law, whereas we find casual changes which cannot be accounted 
 for upon this hypothesis. This the doctor supposes may arise from an unequal and irregular 
 distribution of the magnetieal matter. The irregular distribution also of ferruginous matter in 
 the shell may likewise cause some irregularities. 
 
 Mr. Cavallo's opinion is, that the magnetism of the earth arises from the magnetic substances 
 therein contained, and that the magnetic poles may be considered as the centres of the 
 polarities of all the particular aggregates of the magnetic substances ; and as these substances 
 are subject to change, the poles will change. Perhaps it may not be easy to conceive how 
 these substances can have changed so materially, as to have caused so great a variation in the 
 poles, the position of the compass having changed from the east toward the west about 
 33 in 200 years. Also the gradual, though not exactly regular change of variation, shows 
 that it cannot depend upon the accidental changes which may take place in the matter of the 
 earth, 
 
 Mr. Churchman of America, says, there are two magnetic poles in the earth, one to the 
 north and the other to the south, at different distances from the poles ot the earth, and revolv- 
 ing in different times ; and from the combined influence of these two poles, he deduces rules for 
 the position of the needle in all places of the earth, and at all times, past, present, or to come. 
 
 The north magnetic pole, he says, makes a complete revolution in 426 years 77 days 
 9 hours, and the south pole in about 54J9 years. In the beginning of the year 1777, the north 
 magnetic pole was in 7G 4' north latitude; and in longitude from Greenwich 140 east; and 
 the south was in 72 south latitude, 140 east from Greenwich. 
 
BOOK Y. 
 
 OF ELECTRICITY. 
 
 DEF. I. 1 HE earth, aud all bodies with which we are acquainted, 
 are supposed to contain a certain quantity of an exceedingly elastic fluid, 
 which is called the electric fluid. 
 
 SCHOL. This certain quantity belonging to all bodies, may be called their natural share; 
 and so long as each body contains neither more nor less than this quantity, it seems to He 
 dormant, and to produce no effect. 
 
 DEF. II. When any body becomes possessed of more or less than its 
 natural quantity, it is said to be electrified, and is capable of exhibiting 
 appearances which are ascribed to the power of electricity. 
 
 SCHOL. This equilibrium could never be disturbed, or, if it was disturbed, would be 
 immediately restored, and therefore be insensible'; but that some bodies do not admit the pas- 
 sage of the electric fluid through their pores, or along their surfaces, though others do. 
 
 DEF. III. When a body has acquired an additional quantity of electric 
 matter ; or lost a part of what naturally belonged to it, and it is at the same 
 time surrounded by bodies through which it cannot pass, it must remain in 
 that state, and is said to be insulated. 
 
 PROPOSITION I. 
 
 The ELECTRIC FLUID, being excited, becomes perceptible to the senses. 
 
 EXP. 1. Let a long glass tube be rubbed with the hand, or with a leathern cushion ; the 
 electric fluid, being thus excited, will attract light substances, and give a lucid spark to the 
 finger, or any metallic substance, brought near it. 
 
 The glass tube is called the electric, and all those bodies which are capable, by any means, 
 to produce such effects, are called electrics. The hand, or any other body that rubs an electric, 
 is called the rubber. 
 
 2. As the exciting of a tube is very laborious for the operator, and the electricity procured 
 by that means is small in quantity ; globes and cylinders are used for this purpose. These, 
 by a proper apparatus, are made to revolve on their axes, and a rubber of leather is applied to 
 
128 OF ELECTRICITY. BOOK V. 
 
 the equatorial parts of the revolving glass, which become electrical by the friction. The 
 electricity of the globe, or cylinder, is received by a metallic conductor insulated on a glass 
 supporter. 
 
 ri!- te q 13 ' -^ c y'' n( l er or globe thus fitted up is called an electrical machine. C represents a glass 
 cylinder about 1 foot in diameter and 20 inches long, which is turned by means of a wheel ; 
 the rubber or cushion is supported behind the cylinder by two upright springs that appear 
 beneath, and are fastened to two cross bars of glass. B is a metallic conductor, supported on 
 two pillars of glass ; from the end nearest to the cylinder issue several points, and at the other 
 end the ball E projects by means of a wire. Sparks given by the conductor of a machine of this 
 construction and magnitude are from 12 to 14 inches long. A chain D must connect the rubber 
 with the earth. 
 
 SCHOL. 1. In all experiments in electricity the greatest care should be taken to keep every 
 part of the apparatus clean, and as free as possible from dust ant! moisture. When the weather 
 is clear, and the air dry, especially in clear frosty weather, the electrical machine will always 
 work well. But in very hot, or damp weather, the machine is not so powerful. 
 
 Before the machine is used, the cylinder should be first wiped very clean with a soft linen 
 cloth ; and afterward with a clean hot flannel, or old silk handkerchief. 
 
 Sometimes it will be necessary to apply to the rubber a very small quantity of amalgam 
 made with one part of zinc, and four or five of mercury. 
 
 SOHOL. 2. Respecting the theory of electricity, there are two different hypotheses, one that 
 there is only one fluid, and the other that there are two. Dr. Franklin's hypothesis is the 
 former, and it depends on the following principles, (i) That all terrestrial bodies are full 
 of the electric fluid. (2) That the electric fluid violently repels itself, and attracts all other 
 matter. (3) By exciting an electric the equilibrium of the electric fluid contained in it is 
 destroyed, and one part contains more than its natural quantity, and the other less. (4) Con- 
 ducting bodies, connected with that part which contains more electric fluid than its natural 
 quantity, receive it, and are charged with more than their natural quantity ; this is called 
 positive electricity ; if they be connected with that part which has less than its natural quantity, 
 they part with some of their own, and contain less than their natural quantity ; this is 
 called negative electricity. (5) When one body positively and another negatively electrified 
 are connected by any conducting substance, the fluid in the. body which is positively electrified 
 rushes to that which is negatively electrified, and the equilibrium is restored. These are the 
 principles of positive and negative electricity The other hypothesis is, that there are two 
 distinct fluids, which was suggested by M. Du Faye, upon his discovery of the different proper- 
 ties of excited glass, and excited resins, sealing-wax, &c, The following are the principles of 
 this theory. (1) That the two powers arise from two different fluids which exist together in 
 all bodies. (2) That these- fluids are separated in non-electrics, by the excitation of electrics, 
 and from thence they become evident to the senses, they destroying each other's effects when 
 united. (3) When separated they rush together again with great violence, in consequence ot 
 their strong mutual attraction as soon as they are connected by any conducting substance. 
 These are the principles of vitreous and resinous electricity. 
 
 PROP. II. The electric fluid passes easily along the surfaces of some 
 bodies ; whilst other bodies do not convey it ; the former are called Con- 
 ductors, the latter Non conduc(ors } or Electrics. 
 
BOOK V. OF ELECTRICITY. 1S9 
 
 EXP. The metallic cylinder being fixed upon glass supporters, and placed near the electric 
 machine, will, by means of the pointed wires, receive the electric fluid from the glass cylinder, 
 and the fluid will be diffused over the whole surface of the metallic cylinder, from whence it 
 cannot pass through the glass supporters which are electric, but may be conveyed away by any 
 metallic or other conducting substances, brought near, or into contact with it. This metallic 
 cylinder is called the Prime Conductor, or the Conductor. 
 
 PROP. III. Some conductors are more perfect than others; and the 
 electric fluid passes through that which is most perfect. 
 
 EXP. The fluid will pass through a wire held in the hand. 
 
 SCHOL. 1. The following bodies are conductors and electrics, disposed in the order of their 
 degrees of perfection : CONDUCTORS ; gold, silver, copper, brass, iron, tin, quicksilver, lead, the 
 semi-metals, ores, charcoals, water, ice, snow, salts, soft stones, smoke, steam : NON-CONDUC- 
 TORS, or ELECTRICS ; glass, and all vitrifications, even those of metals ; precious stones, resins, 
 gums, amber, sulphur, baked wood, bituminous substances, wax, silk, cotton, feathers, wool, hair, 
 paper, air, oil, hard stones. Many electrics become conductors, when heated, and all when 
 moistened. 
 
 SCHOL. 2. Glass vessels, made for electrical purposes, are often rendered very good electrics, 
 by use and time, though they might be very bad ones when new. And some glass vessels, 
 which had been long used for excitation, have sometimes lost their power almost entirely. 
 Dr. Priestley mentions several instances of very long tubes which, when first made, answered 
 the purposes of electricity admirably, but after a few months they have become almost 
 useless. 
 
 SCHOL. 5. An exhausted glass vessel on being rubbed shews no signs of electricity upon 
 its external surface. But the electric power of a glass cylinder is the strongest when the 
 air within is a little rarefied. If the air be condensed, or the cylinder be filled with some 
 conducting substance, it is capable of being excited. Nevertheless a solid stick of glass, sealing- 
 wax, sulphur, &c. may be excited. 
 
 SCHOL. 4. The same substance, by different preparations, is sometimes a conductor, and at 
 others an electric. A piece of wood just cut from a tree is a good conductor ; let it be baked, 
 and it become? an electric; burn it to charcoal, and it is a good conductor ag.iiu ; lastlv, let 
 this coal be reduced to ashes, and these will be impervious to electricity. Such changes are also 
 observable in many other bodies ; and very likely in all substances there is a gradation from the 
 best conductors to the best non-conductors of electricity. 
 
 PROP. IV. Non-conductors retain the fluid on a small part of their 
 surface where the friction has acted ; conductors diffuse it over all their sur- 
 face, and therefore cannot confine it, unless they be surrounded entirely by 
 non-conductors, or be insulated. 
 
 EXP. Observe the partial distribution of the fluid on an excited electric, and its universal 
 diffusion over a conductor. If a finger, or any other conductor, be presented to an excited glass, 
 cylinder, tube, &c. it will receive a spark, and in that spark, a small part only of the electricity 
 
 17 
 
OF ELECTRICITY. BOOK V. 
 
 oi the electric; because the excited electric being a non-conductor, cannot convey the electri- 
 city of all its surface to that point to which the conductor has been presented. But if any 
 conducting substance be brought to a charged metallic conductor, it will receive in one spark 
 nearly the whole of the electricity accumulated upon it. The small part which remains is very 
 trifling in comparison of the first spark, and is called the residuum. 
 
 DBF. IV. A body is said to be positively electrified, when it has thrown 
 upon it a greater quantity of the electric fluid than its natural share. 
 
 DBF. V. A body is said to be negatively electrified, when it has a less 
 quantity of the electric matter than is natural to it. 
 
 PROP. V. The electric fluid may be excited by rubbing, by pouring a 
 melted electric into another substance, by heating and cooling, and by evapo- 
 ration. 
 
 EXP. 1. In working the electrical machine, the fluid is excited by friction. Rubbing is the 
 general mean by which all electric substances that are at all excitable may be excited. Whether 
 they be rubbed with electrics of a different sort, or conductors, they always shew signs of elec- 
 tricity, and in general stronger when rubbed with conductors, and weaker when rubbed with 
 electrics. 
 
 2. When sulphur is melted into an earthern vessel, if the vessel be supported by a conduct- 
 ing substance, the sulphur, when cold and separated from the vessel, is strongly electrical, and 
 will attract light bodies. 
 
 3. If sulphur be melted into glass vessels, when cold, the glass, whether supported by elec- 
 trics or not, will be positively electrified, and the sulphur negatively. 
 
 4. Melted sealing-wax, when poured into sulphur, becomes positively electrified, and the 
 sulphur negatively. 
 
 5. Melted sealing-wax poured into glass cups acquires a negative electricity ; upon being 
 separated the glass is positive. 
 
 6. Sulphur, melted into cups, shows no signs of electricity till it is separated from the cup, 
 when the cup is negative and the sulphur is positive. 
 
 7. If a stick of sealing-wax be broken into two pieces, the extremities that were contiguous 
 will he found electrified, one positively, and the other negatively. 
 
 8. The tourmalin, a stone which is generally of a deep red, or purple colour, about the size 
 of a walnut, and found in the East Indies, while kept in the same degree of heat, shows no signs 
 of electricity, but will become electrical by increasing or diminishing its heat, and stronger in 
 the latter than in the former case. (I.) Its electricity does not appear all over its surface, but 
 only on two opposite sides, which may be called its poles, and they are always in one right line 
 with the centre of the stone, and in the direction of the strata ; in which direction the stone is 
 absolutely opaque, though on the other side it is semitransparent. (2.) Whilst the tourmalin 
 is heating, one of its sides (call it A) is electrified plus ; the other (call it B) minus. But when 
 it is cooling A is minus, and B is plus. (3.) If this stone be excited by friction, then both its 
 sides at once may be made positive. (4.) If a tourmalin be cut into several parts, each piece 
 will have its positive and negative poles, corresponding to the positive and negative sides of the 
 tone from which it was cut. 
 
BOOK V. OF ELECTRICITY. 181 
 
 SCHOL. These properties are now found to belong to several hard and precious stones, as 
 well as to the tourmalin. 
 
 9. Electricity may be produced by the evaporation of water in this manner ; Upon an insu. 
 lating stand, as a wine glass, place an earthern vessel, as a crucible, a basin, &c. and put into it 
 three or four lighted coals. Let a wire be put with one end among the coals, and with the other 
 let it touch a very sensible electrometer. Then pour in a spoonful of water at once upon the 
 coals, which will occasion a quick evaporation 5 and at the same time the electrometer will 
 diverge. For a description of the electrometer, see Prop. XII. Schol. 
 
 PROP. VI. The electric fluid may be lodged in electrics, or in insulated 
 conductors, in a greater quantity than naturally belongs to them, or they may 
 be positively electrified. 
 
 EXP. In working the machine, the cylinder acquires more than its natural quantity of fluid 
 by excitation, the conductor, by communication ; for, while there is a free conveyance of fluid 
 from the earth to the rubber, by means of a conducting supporter, the conductor will be highly 
 electrified. 
 
 SOHOL. The electric matter with which the prime conductor is loaded, is not produced by 
 the friction of the cylinder against the rubber. Jt is only collected bv that operation from the 
 rubber, and all the bodies that are contiguous to it. If, therefore, the rubber be well insulated, 
 the friction of the cylinder will produce but little electricity ; for in that case the rubber can 
 only part with its own share, which is very inconsiderable. In this situation, if the finger be 
 presented to the rubber, sparks will be seen to dart from it to the rubber, to supply the place of 
 that electric matter which had passed from it to the cylinder; if the conductor be also insulated, 
 these sparks will cease as soon as it is fully loaded. 
 
 PROP. VII. The electric fluid being accumulated on any body will pass 
 to any conductor brought near to the body ; if it pass from, or be received 
 by, pointed wires, it wili be conveyed in a continued stream ; if it pass from, 
 or be received by, a surface which has no sharp points, it will be discharged 
 with an instantaneous explosion or spark. 
 
 EXP. 1. Receive the fluid from the conductor upon a pointed wire, and upon a brass ball. 
 2. The fluid will be diffused through tKe surrounding atmosphere, by wires placed upon the 
 conductor. 
 
 COR. Hence arises the necessity of keeping the whole surface of the conductor free from 
 points. 
 
 SCHOL. When a conductor is electrified by communication, its whole electric power is dis- 
 charged at once, on the near approach of a conductor communicating with the earth ; whereas 
 an excited electric, in the same circumstances, loses its electric power only in the parts near to 
 the conductor. 
 
132 OF ELECTRICITY. BOOK V. 
 
 PROP. VIII. If conductors be insulated, they will retain a greater or 
 less quantity of the electric fluid (the power of the machine being given) pro- 
 portional to the extent of surface in the conductor. 
 
 EXP. Observe the difference in the magnitude and distance of sparks taken from a small 
 conductor, and of those taken from a large one. 
 
 There is a limit, beyond which this Proposition will not hold true, but which experiment has 
 not yet ascertained. For it i* certain, that if the conductor be very long, it will discharge itself 
 tfver the cylinder hack to the rubber long before it is fully charged. The late Mr. G. C. Morgan, 
 whose memory will be ever dear to the editor of this work, asserts, that by the most powerful 
 excitation of a cylinder 14 inches in diameter, the spark afforded by a conductor 8 inches in 
 diameter, and 12 feet long, did not equal half the length of that procured from the same cylinder 
 with a conductor of equal diameter but shortened to 6 feet And he thinks that a conductor of 
 half that length even, and about 16 inches in diameter, would have yielded a longer spark than 
 
 either of the preceding. See Morgan's Lect. on Elect. Vol. I. p. 54, &c. 
 
 * 
 
 PROP. IX. A body may be deprived of part of its natural portion of 
 electric fluid, or be negatively electrified. 
 
 EXP. If the rubber which communicates the fluid to the glass cylinder, and from thence to 
 the conductor be insulated, because by working the machine a quantity of its fluid is conveyed 
 away, and it cannot receive a fresh supply through its supporter, it will be in an exhausted 
 or negative state. 
 
 SOHOL. If negative electricity be required, then the chain which connects the rubber with 
 surrounding objects, and consequently with the earth, the great reservoir of the electric fluid, 
 must be removed from the insulated rubber, and hung to the prime conductor ; for in this case 
 the electricity of the conductor will be communicated to the ground, and the rubber will appear 
 strongly negative. Another conductor may be connected \\ith the insulated rubber, and then as 
 strong negative electricity may be obtained from this as positive can be in the case before men- 
 tioned. 
 
 The patent machine of Mr. Nairne is admirably adapted for the purposes both of positive and 
 negative electricity. 
 
 PROP. X. When bodies are negatively electrified they receive the fluid 
 from other bodies brought near them. 
 
 EXP. 1. Let two insulated conductors, one of which is connected with the glass cylinder, the 
 other with the rubber, be electrified ; whilst they are in this state let them be brought near each 
 other; a spaik will pass from that which (by Prop. VI.) is positively, to that which (by Prop. 
 IX.) is negatively electrified. 
 
 2. Let two persons standing on glass feet be electrified, first, both positively, or both nega- 
 tively, they will not, on contact, communicate the fluid to each other; but let them be electrified, 
 the one positively and the other negatively, by making a communication from one to the con- 
 ductor, and from the other to the rubber, on contact, the former will give, and the latter receive 
 a spark. 
 
BOOK V. OF ELECTRICITY. 133 
 
 PROP. XL From a pointed body positively electrified the fluid will be 
 seen to stream out, toward any electrified body brought near it, in a conical 
 pencil of rays ; whereas in passing from the uuekctrified body to a pointed 
 body negatively electrified, it will form a globular flame, or star, about its 
 point. 
 
 EXP. I. Observe, in a dark room, the different appearances of the electric fluid at the 
 extremity of a pointed wire, when the point is presented to an insulated conductor positively, 
 and when it is presented to one negatively, electrified j or when sucli a wire is fixed upon a 
 conductor positively or negatively electrified. 
 
 2. Within a luminous conductor electrified positively, (viewed in a dark room) the fluid will 
 he seen passing in the form of a pencil from one wire, and received in the form of a star upon 
 the other ; and the reverse if it be electrified negatively. 
 
 PROP. XII. If two bodies be electrified, both positively, or both nega- 
 tively, they repel each other; but if one be electrified positively, and the 
 other be negatively or not at all electrified, they attract each other. 
 
 EXP. 1. Light feathers, or hair, connected with the conductor, appear repellent, but are 
 attracted by bringing any non-electrified body near them. 
 
 2. The hair of a person electrified becomes repellent. 
 
 3. In the graduated electrometer the ball is repelled according to the degree in which the 
 conductor is electrified. 
 
 4. Downy feathers, paper figures, threads of flax, thistle down, gold leaf, brass dust, or other 
 light bodies brought near to the conductor, are alternately attracted and repelled. This will not 
 take place if the bodies be laid on a plate of glass. 
 
 5. Two bells being suspended by wires from a brass rod connected with the conductor, and 
 a third by a silk cord, and two small balls of brass suspended by a silken thread between the 
 bells, the fluid will be communicated from the conductor to the outer bells, and by the balls to 
 the middle bell, and from theuce conveyed by a chain to the earth ; the balls in receiving and 
 communicating the fluid are attracted and repelled successively, and produce ringing. 
 
 6. Let water flow from a capillary tube, from which, before it is electrified, it passes in 
 drops ; upon being electrified, the particles of fluid will be separated, and their motion accel- 
 erated. 
 
 These appearances will be presented, whether the conductor be positively or negatively 
 electrified. 
 
 7. Mr. Symmer, in the year 1T59, presented to the Royal Society some papers upon the 
 electricity of silk stockings. He had been accustomed to wear two pair of silk stockings, a 
 \vhite pair under black. When these were pulled off together, no signs of electricity appeared, 
 but on pulling off the black from the white, he heard a snapping noise, and in the dark 
 perceived sparks of fire. On this subject he has related a number of very curious experi- 
 ments on the attraction and repulsion of the stockings, and upon their different states of 
 electricity. 
 
1** OF ELECTRICITY. BOOK V. 
 
 COR. Since it is found that rubbed glass electrifies any insulated conductor positively, it 
 may he determined whether any body is electrified positively or negatively, by bringing it near 
 to a j.-ith-ball, or down-feather, positively electrified, and observing whether the ball or feather 
 be attracted or repelled by the body. 
 
 F,XP. Bring a pith-ball or down-feather, suspended by a silken thread and positively electri- 
 fied by any rubbed glass surface, near to another pith-ball or feather suspended by a flaxen 
 thread from a conductor connected with the cylinder ; then bring the same near to a conductor 
 connected with the rubber. 
 
 Plate 13. SCHOL. The. electrometer is an instrument invented to measure the degree of electrification 
 Ft S io - of any body. Small degrees of electricity are shown by the divergence of two very small pith- 
 balls, a, b, suspended upon parallel threads, straws, &c. These balls presented to a body in its 
 natural state will not be affected ; but if the body be electrified, they will be attracted by it and 
 diverge. 
 
 Plate 13. Another very useful and common electrometer consists of an upright stick, AB, to which is 
 '' 11- affixed a graduated semicircle ; D is a pith-ball stuck upon the end of a fine straw, which by 
 means of an axis at C, is moveable in a plane parallel to that of the semicircle. This electro- 
 meter is fixed upright on a prime conductor ; and when it is not electrified, the radius will hang 
 down, and according to the intensity of the electric state given to the conductor, the repulsion 
 must cause the ball to ascend. The ascent will be marked by the graduations, 
 a Mr. Cavallo has invented a very sensible electrometer, well adapted for the observation of 
 
 the presence and quality of natural and artificial electricity. ABC is the brass case containing 
 the instrument. When the part AB is unscrewed, and the electrometer taken out, it appears as 
 represented in ABDC. A glass tube, CDNM, is cemented into the piece AB. The upper part 
 of the tube is shaped tapering to a small extremity, which is entirely covered with sealing-wax. 
 Into this tapering part a small tube of glass is cemented, the lower extremity, being also covered 
 with sealing wax, projects a small way within the tube CDNM. Into this smaller tube, a wire 
 is cemented, which, with its under extremity, touches a flat piece of ivory H, fastened to the 
 tube by means of a cork. The upper extremity of the wire projects about a quarter of an inch 
 above the tube, and screws into the brass cap EF, which cap is open at the bottom, and serves 
 to defend the waxed part of the instrument from the rain. From H are hung two fine silver 
 wires, having very small corks at the lower ends, which, by their repulsion, show the electricity. 
 JM, and K.N, are two slips of tin-foil stuck to the inside of the lass, and communicating with 
 the brass bottom AB. They serve to convey away that electricity, which, when the corks touch 
 the glass, is communicated to it, and might disturb their free motion. 
 
 When this instrument is used to observe artificial electricity, it is set on a table, and elec- 
 trified by touching the brass cap EF with an electrified body ; in this state, if any electrified 
 substance is brought near the cap, the corks of the electrometer, by their converging, or diverg- 
 ing more, will show the species of electricity. 
 
 When it is to be used to try the electricity of fogs, &c. it must he unscrewed from its case, 
 and held a little above the head by the bottom AB, so that the observer may conveniently see 
 the corks, which will immediately diverge if there is any sufficient quantity of electricity in the 
 air, the nature of whicb may be ascertained by bringing an excited piece of sealing-wax toward 
 the brass cap EF. 
 
BOOK V. OF ELECTRICITY. 135 
 
 PROP. XIII. From the sharp points of electrified bodies there proceeds 
 a current of air. 
 
 EXP. 1. A wire, with sharp points bended in opposite directions, and suspended on the point 
 of a perpendicular wire inserted in the conductor, will be carried round by the current proceed- 
 ing from the points. 
 
 2. Let several pieces of gilt paper be stuck like vanes into the side of a coik, through the 
 centre of which a needle passes ; suspend the whole by a magnet, and present one of the vanes 
 to the point of a wire inserted in the conductor ; they will be put into motion. 
 
 PROP. XIV. Some bodies, upon being rubbed, are electrified positively, 
 and others negatively ; and the same bodies are capable of being electrified 
 positively, or negatively, as they are rubbed with different substances. 
 
 EXP. Smooth glass becomes positively electrified by being rubbed with any substance hith- 
 erto tried, except the back of a living cat ; rough glass becomes positively electrified b) being 
 rubbed with dry oiled silk, sulphur, and metals ; negatively, with woollen cloth, sealing-wax, 
 paper, the human hand. White silk becomes positively electrified by being rubbed with black 
 silk, metals, black cloth ; negatively, with paper, hairs, the hand. Black silk will be positively 
 electrified with red sealing-wax ; negatively, with hare's skin, metals, the hand. Sealing-was 
 will be positively electrified with the hand, leather, woollen cloth, paper, hare's skin. Baked 
 wood will be positively electrified with silk ; negatively, with flannel. If these and other sub- 
 stances, being electrified, be brought near to a pith-ball or down-feather, as described Prop. XII. 
 Cor. Exp. it will appear whether they are electrified positively or negatively. 
 
 PROP. XV. Bodies insulated, if placed within the influence of an 
 electrified body, will be electrified, at the part adjacent to that body, in the 
 manner contrary to that of the electrified body. 
 
 Exp. 1. Bring a conductor (without pointed wires) near to the glass cylinder, whilst the 
 machine is working; if the conductor be not insulated, it will be negatively electrified till it is 
 brought sn near as to receive sparks from the cylinder; if the conductor be insulated, it will, 
 in the same situation, be electrified negatively, in the parts nearest the cylinder, and positively 
 in the parts more remote ; as may be seen by bringing a;i excited glass tube (which is positively 
 electrified) near to a ball suspended from the conductor. Compare Prop. XII. Cor. 
 
 2. Let two pith-balls be so suspended by flaxen threads as to 'be in contact when unelectri- 
 fied ; on being brought near to a body electrified positively, they will repel each other being 
 electrified negatively : if the balls be suspended in the same manner by silken threads, they will, 
 in the same situation, be positively electrified. 
 
 3. Let PC be an electrified prime conductor, and AB a metallic body placed within its Plate I. 1 }, 
 atmosphere, but beyond the striking distance. Now from the principles aLcady explained, Flf >' '' 
 it is evident that the electrical atmosphere of the prime conductor must be ]>'i^itire or negative. 
 
 (I.) If it be positive, then the adjacent part A of the metallic body AB, wiil be found to be 
 electrified negatively ; the remote part B, will be electrified positively ; and there wiil be a 
 certain point D, in its natural state, or not electrified at all. (i.) If the prime conductor be 
 
130 OF ELECTRICITY. BOOK V. 
 
 charged with negative electricity, then A will be positive, B negative, and still some point, as D, 
 will be found unelectrified, which is called the neutral point. 
 
 Karl Stanhope has demonstrated, by a considerable number of experiments, that the neutral 
 point D is the fourth point of a harmonical division of the line CAB. Consequently, the 
 points C, A, and B, being given, the neutral point D may be always found. For by the pro- 
 portion assumed by his lordship, as the whole line BC is to the part CA, so is the remote part 
 BO to the middle term DA ; therefore, by composition, 
 
 BC + CA (BA + SAC) : C A : : BD + DA (BA) : AD. 
 Thus, if BA be 40 inches, and CA 36, then AD is equal to 12f inches. 
 
 COR. 1. From the nature of this proposition, it is evident that the neutral point D can never 
 be farther from A than half the distance between A and B, supposing the electrified conductor 
 PC to be removed to an infinite distance. 
 
 COR. 2. It is likewise evident, that the evanescent position of the neutral point D must 
 be A, when the end A of the metallic body AB comes into contact with the charged body PC. 
 
 SCHOL. From the above considerations, Lord Stanhope has, with great ingenuity, proved 
 by an elaborate mathematical demonstration, illustrated and confirmed by a great variety of 
 experiments, that the density of an electrical atmosphere superinduced upon any body must be 
 inversely as the square of the distance from the charged body. 
 
 4. Let a circular plate composed of rosin and sulphur, or of sealing-wax, be negatively 
 electrified by rubbing it with flannel ; whilst it is in this state, let a metallic plate of the 
 same form and size, having a glass handle fastened to its centre, be placed, by means of the 
 handle, on the electrified plate ; then receive a spark from the metallic plate with the finger ; 
 after which the metallic plate, being removed by the glass handle, will be found to be positively 
 electrified. This instrument is called an electrophorus. 
 
 5. Let one side of a plate of glass be electrified positively, the other side will attract light 
 bodies, being negatively electrified. 
 
 G. Let a plate of glass be placed between two metallic plates about two inches in diameter 
 smaller than the plate of glass, and let the plates be supported by a conductor ; upon positively 
 electrifying the upper metallic plate, by means of a wire connected with the prime conductor, 
 the fluid not being able to pass along the glass, will be accumulated upon the part contiguous to 
 the upper metallic plate; whilst the lower metallic plate, being within the electric influence of 
 the upper, will be negatively electrified. 
 
 PROP. XVI. When any electric substance is electrified, it will continue 
 in that state till some conductor conveys away the accumulated or restores the 
 deficient fluid ; which will be done more or less rapidly, according to the 
 degree of conducting power in the conductor, and the number of points in 
 which it touches the electric. 
 
 Kxi>. 1. When the metallic plate in the electrophorus is electrified (as decribed Prop. 
 XV. Kxp. 4.) by setting it upon the electric plate, touching it with the finger, and separating 
 it successively, many sparks may be obtained, without again exciting the electric plate ; for 
 this plate being negatively electrified, the metallic plate on being touched with the hand, 
 becomes positively electrified (by Prop. XV.) and the electric plate remains long in its negative 
 
BOOK V. OF ELECTRICITY. 187 
 
 state, because not being a conductor, its deficiency will be slowly supplied from the air where its 
 surface is not covered. 
 
 2. If a glass vessel, a common drinking-glass, for instance, held in the hand, receive the 
 electric fluid on the inside from a wire, or chain, fixed on the conductor, pith-balls, placed under 
 the vessel upon a conducting supporter, will continue long in motion. 
 
 3. Let a plate of glass be electrified in the manner described in Prop. XV. Exp. G. Because 
 one side of the plate is positively electrified, and the other negatively, if a communication is 
 made from one metallic plate to the other by means of some conductor, part of the accumulated 
 fluid will suddenly pass to the side which is deficient; upon a second application of the plates of 
 metal to the glass, there will be a second explosion. 
 
 SCHOL. AB is an electric jar, coated with tin-foil on the inside and outside, within three Pf* 16 !>> 
 inches of the top, having a wire with a round brass knob K, at its extremity. This wire passes '*' 
 through the cork D, tfefci. rtops the mouth of the jar, and, at its lower end, is bended or branched 
 so as to touch the ii;jid coating in several places. Coated jars may be made of any form 
 and size, and are called Leyden Phials, or Leyden Jars. 
 
 A number of jars combined, make what is termed an electrical battery ; they all stand in a 
 box, the bottom of which is covered with tin ? thus all their outsides are connected ; and by 
 means of wires and brass rods, their insides are also connected. 
 
 The discharging rod consists of a glass handle A, and two curved wires BB, which move by p] ate \^ 
 a joint C, fixed to the brass cap of the glass handle A. The wires BB are pointed, and the points Fig. IS. 
 enter the knobs DD, to which they are screwed, and may be unscrewed from them at pleasure. 
 By this construction, the balls or points may be used as occasion requires. The wires being 
 moveable at the joint C, may be adapted to smaller or larger jars at pleasure. 
 
 PROP. XVII. If a glass plane, or cylindrical vessel, coated on both 
 sides with tin-foil, or any other conducting substance, be charged,, that is, 
 positively electrified on one side, and consequently negatively electrified on 
 the other ; a communication being made from one side to the other by 
 some conductor, the plane, or vessel, will be suddenly discharged, with an 
 explosion. 
 
 There is a strong attraction (compare Prop. XII. and XV.) between the fluids on opposite 
 sides of the glass, or the fluid which is accumulated on one side makes a powerful effort toward 
 the other side where the fluid is deficient ; but the substance of the glass itself being impervious 
 to the electric fluid, the accumulated fluid cannot pass to the deficient side till a commu- 
 nication is made between them by some conducting substance. When such a communication 
 is made, because the metallic coating touches the whole surface of the electrified glass, the 
 whole quantity of redundant fluid easily passes from the side which was positively electrified 
 to the other. 
 
 EXP. 1. Let a plate of glass, coated with tin-foil, (except about 1^ inch from the edge) be 
 charged, as described in Prop. XV. Exp. 6. Upon waking a communication from one side to 
 the other by the discharging rod, there will be a sudden discharge. 
 
 2. Let the same be done with the Leyden Phial. 
 
 18 
 
138 OF ELECTRICITY. BOOK V. 
 
 3. Charge a jar coated on the insiile with water, shot, or brass dust, and held on the 
 outside by the hand, then discharge it in a dark room. 
 
 4. If two equal circular brass plates, one of which is suspended by a long metallic rod from 
 the conductor parallel to the horizon, and the other, supported by a conductor, is placed parallel 
 and opposite to the first, be electrified ; the plate of air between them will be charged by the 
 brass plates. 
 
 5. Let one coated jar be suspended by a wire under another; let the upper jar be charged 
 by taking sparks from the conductor ; the lower uninsulated jar will be charged with the fluid 
 which passes from the side negatively electrified of the upper jar. 
 
 6. Discharge, in a dark room, a jar imperfectly coated. 
 
 Cou. 1. A coated jar cannot be charged unless its outer surface be connected with some 
 conductor. For without such a conductor, the fluid cannot pass from or to the outer surface> 
 which is necessary in order to charge the jar. 
 
 COR. 2. When a coated glass vessel is charged, the charge of electric fluid is in the glass, 
 and not in the coating. 
 
 EXP. Lay a plate of glass between two metallic plates, as described Prop. XV. Exp. 6. 
 Having charged the plate of glass, remove the upper plate of metal by a glass handle, with some 
 non-conducting substance, as silk ; remove the electrified glass plate, and place it between two 
 other plates of metal unelectrified and insulated ; the plate of glass thus coated afresh will still 
 be charged. 
 
 SCHOL. The discharge of a plate of glass, Leyden Phial, &c. is made by restoring the 
 equilibrium which was destroyed by the charging ; and it is effected by forming a communication 
 between the overloaded and the exhausted side ; and if the communication be made by metal, or 
 other good conductors, the equilibrium will be restored with violence, the redundant electricity 
 on one side will rush with great rapidity through the metallic communication to the exhausted 
 side, and a large explosion will be made, that is, the Hash of electric light will be very visible, 
 and the report will be loud. 
 
 PROP. XVIII. If tbe conductor be electrified positively, that side of 
 the jar with which it has a communication will he electrified positively, the 
 other negatively. 
 
 Exp. 1. Charge one jar on the inside positively, and another negatively, and observe, in a 
 dark room, the different appearances of the fluid, upon the point of a wire brought near to the 
 ball which is connected with the inner side of each jar : when the point is presented to the jar 
 positively electrified on the inner side, it will exhibit the appearance of a star; when presented 
 to the other, that of a pencil. 
 
 2. Observe the different appearances, in a dark room, when with the same charged jar the 
 point is presented toward the side positively, and toward the side negatively, electrified. 
 
 3. Between two jars, charged one negatively and the other positively, suspend by a silken 
 string a cork ball, from which short threuiU hang freely; the ball will pass with a rapid motion 
 from one to the other, and, beiir^ first attracted toward the jar positively electrified, then toward 
 the other, it will receive the fluid from the former, and communicate it to the latter, till both are 
 discharged. If both be charged in the same manner, the cork will remain at rest. 
 
BOOK V. OF ELECTRICITY. 139 
 
 4. If, after a jar is charged, the uncoated part of the jar be moistened by the breath, or bj 
 steam, the jar placed upon a conductor will be gradually discharged, and the fluid will be seen, 
 in a dark room, to flash strongly from one side to the other ; if the jar be insulated, the flashes 
 will be greatest on the side positively electrified. 
 
 5. Let a discharging rod be applied without its balls to a charged jari in such manner as to 
 discharge the jar gradually ; the point which approaches toward the side positively electrified, 
 will, in a dark room, exhibit a star; the other point, a pencil. 
 
 6. Within the receiver of an air-pump place two well polished brass balls, the lower sup- 
 ported on a brass stem by the plate of the pump, the other fixed on a stem which is moveable 
 in the neck of the receiver ; let the balls be brought within the distance of four or five inches 
 from one another ; then let the upper ball be connected with the conductor, and electrified posi- 
 tively ; a lucid atmosphere will, in a dark room, appear on the lower surface of the upper ball ; 
 whereas if the upper ball be negatively electrified, the lucid atmosphere will be seen on the 
 lower ball. 
 
 PROP. XIX. The electric fluid can be conveyed through an insulated 
 conductor of any length, and its passage from one aide of a charged jar to the 
 other, is apparently instantaneous, through whatever length of a metallic, or 
 other good conductor, it is conveyed. 
 
 Exp. 1. Let a long wire, passing round a room, suspended by silk cords, be a part of the 
 circuit of communication from one side of a charged jar to the other; the discharge will be appa- 
 rently at the same instant in which the communication from one side to the other is completed. 
 
 2. Let any number of persons make a part of the circuit of communication; the fluid will 
 pass instantaneously through the whole circuit. 
 
 SCHOL. The shock of the Leyden jar has been transmitted through wires of several miles in 
 length, without taking any sensible space of time. Dr. Priestley relates several curious experi- 
 ments made with a view of ascertaining this point soon after the invention of the Leyden Phial. 
 See Priestley's Hist, of Elect. 
 
 PROP. XX. The sudden discharge of a charged jar gives a painful 
 sensation to any animal, placed in the circuit of communication, called the 
 electric shock. 
 
 The discovery of the effects of electricity, as exhibited by the Leyden jar, immediately drew 
 the attention of all the philosophers in Europe. The account which some of them gave of the 
 experiments to their friends, border very much on the ludicrous. M. Muschenbrock, who tried 
 the experiment with a glass bowl, told M. Reaumur, in a letter written soon after the experi- 
 ment, that he felt himself struck in his arms, shoulder, and breast, so that he lost his breath ; 
 and it was two days before he recovered from the effects of the blow and the terror. He added, 
 that he would not take a second shock for the whole kingdom of France. 
 
 M. Allamand, who made the experiment with a common beer glass, said, that he lost his 
 breath for some moments, and then felt such an intense pain all along his right arm, that he was 
 apprehensive of bad consequences ; but it soon went off without any inconvenience. 
 
140 OF ELECTRICITY. BOOK V. 
 
 Notwithstanding the parade made by these philosophers, the shock was probably, not by any 
 means, stronger than what many children of 6 or 7 years old would bear without the smallest 
 hesitation. Their descriptions must have arisen from terror, or love of the marvellous. 
 
 Con. The force of the electric shock may be increased, by increasing the surface of the 
 coated glass. 
 
 EXP. 1. A battery being charged, a fine metallic wire brought into the circuit will be melted. 
 
 2. If a plain piece of metal be placed upon one of the rods of the discharger, and upon the 
 other a needle with the point opposite to the surface of the metal, upon discharging the battery, 
 the surface of the piece of metal will be marked with coloured circles, occasioned by thin 
 laminae of metal raised in the explosion. 
 
 3. If a piece of gold-leaf be put between two pieces of glass, and the whole fast bound 
 together, the metal will be melted, and a metallic stain will be seen in both glasses. 
 
 4. If a shock be sent through a needle, it will give it magnetic polarity. 
 
 5. An animal or plant may be killed by being placed in the circuit of a battery. 
 
 SCHOL. Persons, not thoroughly conversant in electricity, should be very cautious in using 
 large batteries ; they should be sure that they are perfect masters of a small force, before they 
 meddle with a greater. Such a force of electricity as may he accumulated in batteries is not to 
 be trifled with, since the consequences, if not fatal, may be great and lasting. A large shock, 
 taken through the arms and breast, which an operator is most in danger of receiving, might 
 possibly injure the lungs, or some other vital part ; and if the shock were taken through the 
 head, which may easily happen when a person is stooping over the apparatus in order to adjust 
 it, it might aftect his intellects for the remainder of life. 
 
 PROP. XXI. If the circuit be interrupted, the fluid will become visible, 
 and where it passes, it will leave an impression upon any intermediate body. 
 
 EXP. 1. Let the fluid pass through a chain, or through any metallic bodies placed at small 
 distances from each other ; the fluid, in a dark room, will be visible between the links of the 
 chain, or between the metallic bodies. 
 
 2. If the circuit be interrupted by several folds of paper, a perforation will be made through, 
 them, and each of the leaves will be protruded by the stroke from the middle toward the out- 
 ward leaves. 
 
 3. Let a card be placed under wires which form the circuit, where the circuit is interrupted 
 for the space of an inch ; the card will be discoloured. If one of the wires be placed under the 
 card, and the other above it, the direction of the fluid may be seen. 
 
 4. Spirits of wine, or gunpowder, being made part of the circuit, may be fired. 
 
 5. Inflammable air may be fired by an electric gun. 
 
 PROP. XXII. The atmosphere is electrified, sometimes positively, and 
 sometimes negatively. 
 
 EXP. Let a kite be sent up into the air with cord (consisting of copper thread twisted with 
 twine ;) let the lower end of the cord be insulated by a silk line ; a metallic conductor sus- 
 pended from the lower end of the cord will be positively or negatively electrified. The air at 
 
BOOK V. OF ELECTRICITY. 
 
 some distance from houses, trees, masts of ships, &c. is generally electrified positively ; particu- 
 larly in frosty, clear, or fogzy weather. For the particular construction of the electrical kite. 
 and other instruments used with it, see Cavallo's Elect. Vol. n. Chap. i. 
 
 SCHOL. The following general laws have been deduced by Mr. Cavallo, from a great number 
 of experiments made during two years in almost every degree of the atmosphere from 15 to 80 
 of Fahrenheit's thermometer. 
 
 1. The air appears to be electrified at all times ; its electricity is constantly positive, and 
 much stronger in- frosty than in warm weather ; but it is by no means less in the night than in 
 the day time. 
 
 2. The presence of clouds generally lessens the electricity of the kite. 
 
 3. When it rains, the electricity of the kite is generally negative, and very seldom positive. 
 
 4. The aurora boreahs seems not to aft'ect the electricity of the kite. 
 
 5. The electrical spark, taken from the string of the kite, or from any insulated conductor 
 connected with it, especially if it does not rain, is very seldom longer than A of an inch, but it 
 is exceedingly pungent. When the index of the electrometer is not higher than 20 the person 
 that takes the spark will feel the effect of it in his legs; it appearing more like the discharge of 
 an electric jar, than the spark taken from a prime conductor. 
 
 6. The electricity of the kite is in general stronger or weaker, according as the string is 
 longer or shorter ; but it does not keep any exact proportion to it. The electricity, for instance, 
 brought down by a string of an hundred yards, may raise the index of the electrometer to 20 
 when with double that length of string, the index of the electrometer will not go higher than 25. 
 
 7. When the weather is damp, and the electricity is pretty strong, the index of the electro- 
 meter, after taking a spark from the string, or presenting the knob of a coated phial to it, rises 
 surprisingly quick to its usual place, but in dry and warm weather it rises exceedingly slow. 
 
 PROP. XXIII. The electric fluid and lightning are the same substance. 
 
 Their properties and effects are the same. Flashes of lightning are generally seen to form 
 irregular lines in the air ; the electric spark, when strong, has the same appearance. Lightning 
 strikes the highest and most pointed objects ; takes in its course the best conductors ; sets fire 
 to bodies ; sometimes dissolves metals ; rends to pieces some bodies ; destroys animal life ; in 
 all which it agrees (as has been shewn) with the phenomena of electric fluid ; both causes have 
 the same power of making iron magnetic. Lightning has been known to strike men with blind- 
 ness. Dr. Franklin produced a similar effect on a pigeon by the electrical fluid. Lastly, the 
 lightning being brought from the clouds to an electrical apparatus, by a kite or wire, will exhibit 
 all the appearances of the electric fluid. 
 
 EXP. Take a Leyden phial, 5 inches in diameter, and 13 inches in height ; on the inside let 
 the coating rise till its upper edge be 2i inches from the rim of the vessel; on the outside let 
 the coating rise no higher than 1 inch from the bottom. When the phial is thus coated, let it 
 be charged, and a spark will pass from the tin-foil on the outside to that on the inside ; but its 
 form v, ill resemble that of a tree, whose trunk will increase in magnitude and brilliancy, and 
 consequently in po'wer, as it approaches the edge, owing to ramifications which it collects from 
 all parts of the glass. Within two inches of the edge, it becomes one body, or stream, and 
 along that interval its greatest force acts. 
 
14JI OF ELECTRICITY. BOOK V. 
 
 \\ lien two clouds, or the two correspondent parts of a cloud, have their equilibrium restored 
 by a discharge, the appearances are exactly similar to those of the preceding experiment. Each, 
 extremity of the flash is formed by a multitude of little streams, which gather into one body, 
 whose power is undivided in that interval only which separates the positive from the negative. 
 
 PROP. XXIV. Buildings may be secured from the effects of lightning, 
 by fixing a pointed iron rod higher than any part of the building and contin- 
 uing it, without interruption, to the ground, or the nearest water. 
 
 The electric fluid will, by means of the pointed rod, be gradually conveyed from the cloud 
 to the earth by a continued stream, and thus prevent the effects of a sudden and violent 
 explosion. 
 
 EXP. Let a board, shaped like the gable end of a house, be fixed perpendicularly upon a 
 horizontal board ; in the perpendicular board let a hole be made, about an inch square and \ 
 inch deep ; in this hole let a piece of wood nearly of the same dimensions be so inserted as to 
 fall easily out of its place, and let a wire be fastened diagonally to this square piece of wood ; 
 let another wire, terminated by a brass ball, be fastened to tlte perpendicular board, with its ball 
 above the board, and its lower end in contact with the diagonal wire in the square piece of 
 wood ; let the communication be continued by a wire to the bottom of the perpendicular board. 
 If the wires in this state be made part of a circuit of communication, on discharging the jar the 
 square piece of wood will not be displaced ; but if the communication be interrupted by chang- 
 ing the direction of the diagonal wire, the square piece of wood will, upon the discharge, be 
 driven out of its place. 
 
 If insfead of the upper brass ball, a pointed wire be placed above the perpendicular board, 
 the discharge may be drawn off without an explosion. 
 
 SOHOL. The following directions are given by Earl Stanhope, to persons erecting conductors 
 for lightning ; 
 
 (1.) The rods must be made of such substances as are, in their nature, the best conductors 
 of electricity. 
 
 (2.) The rods must be uninterrupted, and perfectly continuous. 
 
 (3.) They must be of sufficient* thickness. 
 
 (4.) They must be perfectly connected with the common stock, that is, the earth, or nearest 
 water. 
 
 (5.) The upper extremity of the rods must be finely tapered, and as acutely pointed as 
 possible. 
 
 (6.) The rods must be very prominent, and several feet above the chimneys. 
 
 (7.) Each rod must be carried in the shortest convenient direction from its upper end to the 
 common stock. 
 
 (8.) There should be no prominent bodies of metal on the top of the building proposed to be 
 secured, but such as are connected with the conductor by some proper metallic communication. 
 
 (9.) There should be a sufficient number of substantially erected high and pointed rods. 
 See "Principles of Electricity," by Charles Viscount Mahon, now Earl Stanhope. To the 
 
 * Perhaps J inch. 
 < Y So many that no part of the building may be more than 30 or 40 feet from one. 
 
BOOK V. OF ELECTRICITY. 143 
 
 same work, the reader must be referred for an account of a discovery made by his lordship in 
 the science of electricity, which he denominated the " returning stroke," by which, he asserts, 
 that persons may be killed, and other vast mischief ensue by lightnings at the distance of 
 several miles from the flash. It is proper also to observe, that several respectable electricians, 
 though willing to admit the fact as discovered by Earl Stanhope, yet do not seem to think that 
 the danger attending the returning stroke can ever be great or formidable. See Cavallo's 
 Elecf. Vol. n. and in. Morgan's Lectures on Elect. Vol. 11. Dr. Huttou's Diet. Art. Re- 
 turning Stroke. 
 
 PROP. XXV. The electric fluid passes easily through a vacuum. 
 
 The air being a non-conductor, in proportion as it is removed, the effort of the electric fluid 
 on the surface of the body positively electrified to pass to the next conductor, meets with less 
 resistance, and therefore is diffused over a greater space. 
 
 ESP. t. Let a jar be charged in vacua. 
 
 2. Let a luminous conductor be placed in the circuit, and observe the fluid passing through 
 it. 
 
 3. Let a vacuum be made a part of the circuit in discharging a phial. 
 
 4. Make a vacuum in a double barometer, and let the fluid pass from one leg to the otl\,er by 
 connecting one of the vessels of mercury with the conductor. 
 
 5. The electric fluid may be made to pass through a large tube three feet in length, and four 
 or five inches in diameter, if, being well exhausted, one end of it be connected with a large 
 conductor. The preceding experiments are to be performed in a dark room. 
 
 SCHOL. 1. From the resemblance between these electrical appearances, and the atmospherical 
 phenomena of the Aurora Borealis, meteors, &c. it is inferred, that these phenomena are pro- 
 duced by the electric fluid. 
 
 SCHOL. 2. The success of the foregoing experiments depends, it is highly probable, upon the 
 air in the jar, tube, &c. being rarefied in a high degree ; for Mr. W. Morgan, a gentleman 
 deeply skilled in calculations and political arithmetic, has shewn that a perfect vacuum is 
 absolutely impermeable to the electric fluid. See Phil. Trans, vol. LXXV. 
 
 PROP. XXVI. Some fishes have the property of giving shocks analo- 
 gous to those of artificial electricity ; namely, the Torpedo, the Gymnotus 
 electricus, and the Silurus electricus. 
 
 If the torpedo, whilst standing in water, or out of water, but not insulated, be touched with 
 one hand, it generally communicates a trembling motion or slight shock to the hand. If the 
 torpedo be touched with both hands, at the same time, one hand being applied to its under, and 
 the other to its upper surface, a shock will be received exactly like that occasioned by the 
 Leyilen Phial. \Vhen the hands touch the fish on the opposite surfaces, and just over the 
 electric organs, then the shock is the strongest ; but no shock is felt, if both hands are placed 
 upon the electric organs of the same surface ; which shews that the upper and lower surfaces 
 of the electric organs are in opposite states of electricity, answering to the plus and minus sides 
 of a Ley den Phial. 
 
144 OF ELECTRICITY. BOOK. V. 
 
 The shock given by the torpedo, when in air, is about four times as strong as when in water ; 
 and when the animal is touched on both surfaces by the same hand, the thumb being applied to 
 one surface, and the middle finger to the opposite, the shock is felt much stronger than when the 
 circuit is formed by both hands. 
 
 This power of the torpedo is conducted by the same substances which conduct electricity, 
 and is interrupted by those substances which are non-conductors of electricity. A circuit may 
 be made of several persons joining hands, and the shock will he felt by them all at the same 
 time ; but the shock will not pass through the least interruption of continuity, not even the 
 distance of the two hundredth part of an inch. 
 
 No electric attraction or repulsion could be ever observed to be produced by the torpedo, 
 nor, indeed, by any of the electric fishes. The shocks of the torpedo seem to depend on the 
 will of the animal. 
 
 The gymnotus electricus, or electrical eel, possesses all the electric properties of the torpedo, 
 but in a superior degree. When small fish are put into the water wherein the gymnotus is kept, 
 they are generally stunned or killed by the shock, and then they are swallowed, if the animal be 
 hungry. 
 
 The strongest shock of the gymnotus will pass a very short interruption of continuity in the 
 circuit. When the interruption is formed by the incision made by a pen-knife on a slip of tin- 
 foil that is pasted on glass, and that slip is put into the circuit, the shock, in passing through 
 that interruption, will shew a small but vivid spark, plainly to be seen in a dark room. 
 
 The gymnotus seems also to be possessed of a sort of new sense, by which he knows whether 
 the bodies presented to him are conductors or not. This fact was ascertained by a great number 
 of experiments made by Mr. Walsh. 
 
 The silurus electricus is known to have the power of giving the shock, but we have a very 
 imperfect account of its properties. 
 
 A fourth electrical fish was found on the coast of Johanna, one of the Comora islands, in 
 lat. 12 13' south, by William Paterson ; and an account of it was published in the 76th vol. of 
 the Phil. Trans. 
 
 SCHOL. When electricity is strongly communicated to insulated animal bodies, the pulse 
 is quickened, and perspiration increased ; and if they receive, or impart electricity on a 
 sudden, a painful sensation is felt at the place of communication. But what is more extraordi- 
 nary is, that the influence of the brain and nerves upon the muscles seems to be of an electric 
 nature. 
 
 We are indebted for this discovery to M. Galvini, a learned Italian, who has denominated 
 that part of science, ANIMAL ELECTRICITY. We shall, without pretending to enter at large on 
 the subject, give the result of the principal observations hitherto made, together with three or 
 four illustrative experiments. 
 
 1. The nerve of the limb of an animal being laid bare, and surrounded with a piece of 
 tin-foil, if a communication be formed between the nerve thus armed, and any of the 
 neighbouring muscles, by means of a piece of zinc, strong contractions will be produced in 
 the limb. 
 
 2. If a portion of the nerve which has been laid bare be armed as above, contractions will be 
 produced as powerfully, by forming the communication between the armed and bare part of the 
 uerve, as between the armed part and muscle. 
 
BOOK V. OF ELECTRICITY. 
 
 3. A similar effect is produced by arming a nerve, and simply touching the armed part of 
 it with the metallic conductor. 
 
 4. Contractions will take place if a muscle be armed, and a communication be formed by 
 means of the conductor between it and a neighbouring nerve. The same effect will be produced 
 if the communication be formed between the armed muscle and another muscle which is contigu- 
 ous to it. 
 
 5. Contractions may be produced in the limb of an animal by bringing the pieces of metal 
 into contact with each other at some distance from the limi>, provided the latter make part of a 
 line of communication between the two metallic conductors. 
 
 6. Contractions can be produced in the amputated leg of a frog, by putting it into water, 
 and bringing the two metals into contact with each other at a small distance from the limb. 
 
 7. The influence which has passed through, and excited contractions in, one limb, may be 
 made to pass through, and excite contractions in, another limb. 
 
 8. The heart is the only involuntary muscle, in which contractions can be excited by these 
 experiments. 
 
 9. Contractions are produced more strongly the farther the coating is placed from the origin 
 of the nerve. 
 
 10. Animals which were almost dead have been found to be considerably revived by exciting 
 this influence. 
 
 11. When these experiments are repeated upon an animal that has been killed by opium, 
 or by the electric shock, very slight contractions are produced ; and no contractions whatever 
 will take place in an animal that has been killed by corrosive sublimate, or that has been starved 
 to death. 
 
 12. Zinc appears to be the best exciter when applied to gold, silver, molybdena, steel, or 
 copper. The latter metals, however, excite but feeble contractions when applied to each other. 
 Next to zinc, in contact with these metals, tin and lead, and silver and lead, appear to be the 
 most powerful exciters. 
 
 EXP 1. Place the limb of an animal, a frog for instance, upon a table ; hold with one hand 
 the principal nerve previously laid bare, and in the other hold a piece of zinc; let a small plate 
 of lead or silver be then laid upon the table at some distance from the limb, and a communica- 
 tion be formed, by means of water, between the limb and the part of the table where the metal 
 is lying. If now, the silver be touched with the zinc, contrtctions will be produced in the limb 
 the moment that the metals come into contact with each other. The same effect will be pro- 
 duced, if the two pieces of metal be previously placed in contact, and the operator touch one of 
 them with his finger. 
 
 2. Let two amputated limbs of a frog be taken ; let one of them be laid upon a table, and its 
 foot be folded in a piece of silver ; let a person lift up the nerve of this limb with a silver probe, 
 and another person hold in his hand a piece of zinc, with which he is to touch the silver includ- 
 ing the foot ; let the person holding the zinc in one hand, catch with the other the nerve of the 
 second limb, and he who touches the nerve of the first limb is to hold in the other hand the foot 
 of the second; let the zinc now be applied to the silver, including the foot of the first, and 
 contractions will be immediately excited in both limbs. 
 
 3. Take a living flounder, lay it flat in a pewter-plat**, or upon a sheet of tin-foil, and put a 
 piece of silver, as a shilling, or a half crown, upon the fish. Then by means of a piece of metal, 
 
 19 
 
146 OF ELECTRICITY. BOOK V. 
 
 complete the communication between the pewter-plate, or tin-foil, and the silver piece, on doing 
 which the animal will give evident tokens of being; affected. 
 
 O O 
 
 4. Let a person lay a piece of zinc upon his tongue, and a half crown, or other silver under 
 it ; on forming a communication between those two metals, by bringing their two edges into 
 contact, he will perceive a peculiar sensation, a kind of cool sub-acid taste, not exactly like, and 
 yet not much different from that produced by artificial electricity. See Cavallo's Elect. Vol. in. 
 
 SCHOL. 2. Electricity has been administered for various diseases. M. Cavallo has taken 
 great pains in ascertaining the cases in which electricity has been successfully applied. We are 
 informed by that gentleman, that rheumatic disorders, even of long standing, are relieved, and 
 generally quite cured. Deafness, the tooth-ach, swellings in general, inflammations of every sort, 
 palsies, ulcers, cutaneous eruptions, the St. Vitus' dance, scrofulous tumours, cancels, abscesses, 
 nervous hcad-achs, the dropsy, gout, agues, and obstructions, have all been considerably relieved, 
 and in many instances perfectly cured, by the application of electricity. A full account of the 
 method of administering electricity in the cases above mentione'l, with an accurate description 
 of the instruments used, may be seen in the 2d Vol. of Cavallo's Complete Treatise of 
 Electricity. 
 
 PROP. XXVII. There is a considerable analogy and difference between 
 magnetism and electricity. 
 
 The power of electricity is of two sorts, positive and negative ; bodies possessed of the same 
 sort of electricity, repel each other, and those possessed of different sorts attract each other. In 
 magnetism, every magnet has two poles ; poles of the same name repel each other, and the con- 
 trary poles attract each other. 
 
 In electricity, when a body in its natural state is brought near to one electrified, it acquires 
 a contrary electricity, and becomes attracted by it. In magnetism, when a ferruginous sub- 
 stance is brought near to one pole of a magnet, it acquires a contrary polarity, and becomes 
 attracted by it. 
 
 One sort of electricity cannot be produced by itself. In like manner, no body can have one 
 magnetic pole without the other. 
 
 The electric virtue may be retained by electrics, but it easily pervades non -electrics. The 
 magnetic virtue is retained by ferruginous bodies, but it easily pervades other bodies. 
 
 On the contrary, the magnetic power differs from the electric, in that it does not affect the 
 senses with light, smell, taste, or noise, as the electric does. 
 
 Magnets attract only iron, whereas the electric power attracts bodies of every sort. 
 
 The electric virtue resides on the surface of electrified bodies, but the magnetic is internal. 
 
 A magnet loses nothing of its power, by magnetising other bodies, but an electrified body 
 loses part of its electricity by electrifying other bodies. See Cavallo's Magnetism, Part n> 
 Chap. n. 
 
BOOK VI. 
 
 OF OPTICS ; OR, THE LAWS OF LIGHT AND VISION. 
 
 CHAP. I. 
 
 Of Light. 
 
 DEF. I. ALIGHT is that which, proceeding from any body to the eye, 
 produces the perception of seeing. 
 
 DEF. II. A Hay of Light is any exceedingly small portion of light as it 
 comes from a luminous body. 
 
 DEF. III. A body, which is transparent, or affords a passage for the rays 
 of light, is called a Medium. 
 
 DEF. IV. Rays of light which, coming from a point, continually separate 
 as they proceed, are called Diverging .Rays. 
 
 DEF. V. Rays which tend to a common point are called Converging 
 Rays. The divergency, or convergency, of rays, is measured by the angle 
 contained between the lines which the rays describe. 
 
 Di.F. VI. Rays of light are parallel, when the lines which they describe 
 are parallel. 
 
 DEF. VII. A Beam of light is a body of parallel rays ; a Pencil of rays, 
 is a body of diverging or converging rays. 
 
 DEF. VIII. The point, from which diverging rays proceed, is called the 
 radiant point ; that to which converging rays are directed, is called the focus. 
 
 If the rays proceed from B; BD, BA, BC, BE, are diverging rays, and B is the radiant; if Plate (5. 
 the rays tend toward B, DB, AB, &c. are converging rays, and B is tiie focus. Fl ' * 
 
 If the rays AC, BC, converge to the focus C, passing on from thence in a right line, they Fig. 2. 
 become diverging, and C becomes a radiant. 
 
 DEF. IX. A ray of light, bent from a straight course in the eame medium, 
 is said to be inflected. 
 
148 OF OPTICS. BOOK VI. 
 
 PROPOSITION I. 
 Kays of light consist of particles of matter. 
 
 For, like all matter with which we are acquainted, they are capable of being inflected out of 
 their course by attraction. 
 
 EXP. 1. If a beam of light be admitted into a dark room through a small hole, and the edge 
 of a knife be brought near the beam, the rays, which would otherwise have been in a straight 
 line, will be inflected toward the knife. The edge of any other thin plate of metal &c. produces 
 the same eft'ect. 
 
 2. The shadow of a small body, as a hair, a thread, &c. placed in a beam of the sun's light, 
 will be much broader than it ought to be if the rays of light passed by these bodies in right lines. 
 
 S. A beam of light passing through an exceedingly narrow slit, not above -jig- part of an inch 
 broad, will be split into two, and leave a dark space in the middle. 
 
 PROP. II. Every visible body emits particles of ligbt from its surface 
 in all directions, which, passing without obstruction, move in right lines. 
 
 Wherever a spectator is placed with respect to a luminous body, every point of that part of 
 the surface which is turned toward him is visible to him ; the particles of light are, therefore, 
 emitted in all directions, and those rays only are intercepted in their passage by an interposed 
 object, which would be intercepted upon the supposition that the rays move in right lines. 
 
 EXP. 1. Let a portion of a beam of light be intercepted by any body, the shadow of that 
 body will be bounded by right lines passing from the luminous body, and meeting the lines 
 which terminate the opaque body. 
 
 2. A ray of light, passing through a small orifice into a dark room, proceeds in a straight line. 
 
 3. Rays will not pass through a bended tube. 
 
 SCHOL. Rays of light are properly represented by right lines. 
 
 PROP. III. The rays of light move with great velocity. 
 
 The velocity of light is much greater than that of sound ; for the flash of a gun, fired at a 
 considerable distance, is seen some time before the report is heard. The clap of thunder is not 
 heard till some time after .the lightning has been seen. 
 
 This proposition is proved by observations made on the satellites of the planet Jupiter, and 
 on the abberration of the rays of light from the fixed stars, as will be shewn in treating upon 
 Astronomy ; from whence it will be seen, that the velocity is at the rate of 200,000 miles in one 
 second of time. 
 
 PROP. IV. The particles of light are exceedingly small. 
 
 Otherwise their velocity would render their momentum too great to be endured by the eye 
 without pain. 
 
CHAP. I. OF LIGHT. 149 
 
 EXP. 1. If a candle be lighted, and there be no obstacle to obstruct the progress of its rays, 
 it will fill all the space within two miles every way before it has lost the least sensible part of its 
 substance. 
 
 2. Rays of light will pass without confusion through a small puncture in a piece of paper, 
 from several candles in a line parallel to the paper, and form distinct images on a sheet of paste- 
 board placed behind the paper. 
 
 PROP. V. The quantities of light, received from a luminous body upon 
 a given surface, are inversely as the squares of the distances of the surface 
 from the luminous body. 
 
 Let ABD, EFG, be two concentric spherical surfaces ; of which let ELFI, AH*BK, be two plate 6. 
 similar portions. Let the rays CE and CF, with the rest proceeding from the centre C, fall f> S' & 
 upon the portion ELFI, and cover it ; it is evident from inspection, that the same rays at the 
 distance CH will cover the portion AHBK only; now these rays, being the same in number at 
 each place, will he as much thinner in the former, than they are in the latter, as ELFI is larger 
 than AHBK; but these spaces being similar portions of the surfaces of spheres, have the same 
 ratio to each other, that the surfaces themselves have ; that is, they are to each other as the 
 squares of their radii CL, CH ; the density of the rays is therefore inversely as the squares of 
 these radii, or of their distances from the luminous point C. 
 
 EXP. The light, passing from a candle through a square orifice, will diverge as it proceeds, Plate 12. 
 and will illuminate surfaces which will be to each other as the squares of their distances from fl S- 8l 
 the candle. Thus at the distance AF the candle will illuminate the square BF, at the distance 
 AO it will illuminate the surface CO equal to four times BF, and at the distance AS it will 
 illuminate the surface DS equal to nine times BF, but AF, AO, and AS, are as 1, 2, and 3, 
 consequently the illuminated surfaces are as the squares of the distances. 
 
 PROP. VI. If the distance between rays diverging from different radi- 
 ant points be the same, the distances of the radiant points are inversely as 
 the divergency of the rays. 
 
 Let D and E he two different radiants ; and let the rays diverging from D describe the lines Plate S. 
 DA, DB, arid the ravs diverging from E describe the lines EA, EB; so that, at the points A fl S-*- 
 and B, the distance between the former rays shall be the same with the distance between the 
 latter, and let EC, DC, be the perpendicular distances of the radiants, E, D. At the point E 
 make the angle ZEC equal to ADC, which is half AI3B ; whence ZEC and ADC (El. V. 7) 
 have the same ratio to AEC. But if these angles are small, they are very nearly in the propor- 
 tion of their tangents ZC, AC. And because the angle ADC is equal to the angle ZEC (El. I. 28.) 
 AD is parallel to ZE ; and because these lines are parallel, (Kl. I. 29) the angles CAD, CZK, 
 are equal ; whence the two triangles ZEC, ADC, are equiangular, and (El. VI. 4 ) EC is to DC, 
 as ZC to AC, or (from what was shewn above) as ADC to AEC; that is, the distance of the 
 radiant K is to the distance of the radiant D, as half the angle of divergency of the rays whicb. 
 proceed from D is to half the divergency of the rays which proceed from E, or as the whole 
 angle of divergency ADB to the whole angle of divergency AEB; that is, the distances of the 
 radiants are inversely as the divergency of the rays. 
 
130 OF OPTICS. BOOK VI. 
 
 PROP. VII. If the distance between converging rays tending to differ- 
 ent foci be the same, the distances of the foci are inversely as the convergence* 
 of the rays. 
 
 Plate ; 6. L et AD, BD, be lines described by rays converging to the focus D, and AE, BE, lines 
 
 described by other rays converging to E, and let the distance AB, at the points A and B, be the 
 same between the former and the latter rays. The angles ADB, AEB, are in this case the 
 angles of convergency ; and EC, DC, are distances of the foci to which they respectively tend. 
 Now it was proved in the last Prop, that EC is to DC as ADB is to AEB. Therefore the 
 distances of the foci are inversely as the convergency of the rays. 
 
 PROP. VIII. If rays proceed from a radiant at an infinite distance, their 
 divergency is considered as nothing, and the rays are considered as parallel. 
 
 Since (by Prop. VI.) the divergency of rays is inversely as the distance of the radiant, when. 
 the distance of the radiant is infinitely great, the angle of divergency is infinitely small, and the 
 rays may be considered as parallel. 
 
 Con. Hence all the rays which come from the centre, or any other given point, of the sun's 
 surface, are considered as parallel. 
 
 PROP. IX. If rays tend to a focus at an infinite distance, their conver- 
 gency is considered as nothing, and the rays are considered as parallel. 
 
 Since (by Prop. VII ) the convergency is inversely as the distance of the focus, when that 
 distance is infinitely great, the angle of convergency is infinitely small. 
 
 CHAP. II. 
 
 Of Refraction. 
 
 SECT. I. 
 OF THE LAWS OF REFRACTION. 
 
 DBF. X. A ray of light bent from a straight course by passing out of one 
 medium into another, is said to be refracted. 
 
 DEF. XI. The JLngle of Incidence is that, which is contained between 
 the line described by the incident ray, and a Hue perpendicular to the surface 
 on which the ray strikes, raised from the point of incidence. 
 
 DBF. XII. The Jingle of Refraction is that, which is contained between 
 
CHAP. II. OF REFRACTION. 151 
 
 the line described by the refracted ray, and a line perpendicular to the 
 refracting surface at the point in which the ray passes through that surface. 
 
 DBF. XIII. The Jingle of Deviation is that, which is contained between 
 the line of direction of an incident ray, and the direction of the same raj- 
 after it is refracted. 
 
 AC is a ray of light ; HK. the surface of the refracting medium ; CF the refracted ray ; OP Plate ti. 
 the perpendicular ; ACO the angle of incidence ; PCF the angle of refraction, and FCL the Fl - 6> 
 angle of deviation. 
 
 SOHOL. The radiant point and focus may be either real or imaginary. If the rays rn, ro, p] a te 6. 
 diverging from the radiant r, suffer refraction and move on in the directions of the lines 'S- 5 - 
 A, oB, which produced in the contrary direction would meet in R, this radiant point is 
 imaginary. 
 
 If the rays Ip, Ly, tending toward the point F, be refracted at p and q, and acquire a direc- 
 tion toward/, tiie focus F is imaginary. 
 
 PROP. X. The attracting force of any medium, acting upon a ray of 
 light, is every where perpendicular to the refracting surface. 
 
 If the medium be uniform in all Its parts, its immediate power upon the ray of light will be Plate G 
 equally strong in every point of a plane drawn parallel to the refracting surface ; though its 
 strength may be different in the next parallel plane, and so onward as far as that power is 
 extended on each side of the surface of the medium. The extent of this power will therefore 
 be terminated by two planes, parallel to each other and to the refracting surface. Let R be a 
 particle of light, acted upon by the refractive power of the medium whose refracting surface is 
 DC. It is evident that the refractive power at O will move the particle R in the direction RO ; 
 and taking any two points D, C, at equal distances on each side of O, the powers at D and C 
 being equal, and acting at equal distances, RD, RC, equally inclined to RO, cannot move R in 
 any direction but that of RO. The same may be shown of the powers at every point of the line 
 DC, and in every line parallel to DC, that is, of the whole power of the medium. 
 
 PROP. XI. A ray of light, in passing out ofe a rarer into a denser medi- 
 um, is refracted toward a perpendicular to the surface of the denser, raised 
 frewn the point in which the ray meets the medium ; in passing out of a denser 
 into a rarer medium, it is refracted from the same perpendicular. 
 
 Let a ray of light, AC, pass obliquely out of a rarer medium X, into a denser medium Z; piate 6, 
 let HK be the plane surface of the denser medium ; from the point C, in which the ray AC Fl S- 6 - 
 passes into the denser medium, raise the perpendicular OCP; the ray will be refracted out of 
 the direction ACL. toward the perpendicular OCP. 
 
 Because the ray is more attracted by the denser medium than by the rarer, it will be accel- 
 erated on entering the medium Z ; for whilst the ray is so near the surface of the medium Z as to be 
 within its attraction, and more attracted toward the denser than toward the rarer, this attraction 
 conspire* with the motion of the ray, and, consequently, increases its velocity. And, since the ar 
 
I5a OF OPTICS. BOOK VI. 
 
 tion of the attracting force of the medium Z, must (by Prop. X.) be in the direction of a line OOP 
 perpendicular to its surface, if the oblique motion of the ray in the direction AC be resolved into 
 two others, AD parallel to the surface HK, and AB, or DC, perpendicular to it, the parallel 
 motion AD cannot be accelerated or retarded by the attraction which acts in the direction OC ; 
 the change of velocity, therefore, which the ray receives from the attracting force, must be made 
 in the perpendipular part of its motion DC. Take CG greater than DC representing the 
 perpendicular motion of the ray after passing into the denser medium ; and take CE equal to 
 AD representing the parallel part of the motion of the ray, which, because it is parallel to AB, 
 remains the same when the ray enters the denser medium. The ray, therefore, at its entering 
 the medium Z, may be considered as acted upon by two forces CE, CG, and consequently 
 (Book II. Prop. XIV.) will describe CF the diagonal of a parallelogram, the sides of which are 
 CE, CG. Now, of these sides, CE remaining the same, whilst CG becomes greater than CD, the 
 angle GCF (from the nature of the parallelogram) will be less than the angle NCL, equal (El. I. 
 15.} to ACD. Therefore the ray, after it has passed into the denser medium, makes a less angle 
 with the perpendicular OCP than AC, the ray before it passes into the denser medium ; that is, 
 the ray, in passing out of the rarer into the denser medium, is refracted toward the perpendic- 
 ular. On the contrary, whilst the ray of light FC is passing out of the denser medium Z into 
 the rarer medium X, it is more attracted by the denser than by the rarer medium, and is there- 
 fore more drawn toward the former than toward the latter ; whence the attraction opposes the 
 motion of the ray, and will retard it as much, as in passing out of the rarer into the denser 
 medium it was accelerated ; and, consequently, the effect will be the reverse of that which was 
 shewn in the former case. 
 
 SCHOL. 1. Although there is no doubt that refraction is performed gradually, and in time, 
 during which the light really describes a curve line extending quite through the refracting space, 
 and connecting the refracted with the incident ray, which are tangets to this curve at its respec- 
 tive extremities, yet both the time and space are so small, that experiment has never been able to 
 render even tiie space perceptible, so that the incident and refracted rays are commonly consid. 
 ered as funning a perfect angle precisely at the surface separating the two mediums. 
 
 SCHOL. 2. The principles of optics are demonstrated upon the supposition that light is a 
 homogeneal substance ; and though light will appear to be compounded of_several kinds of rays, 
 yet the principles of refraction and reflection &c. are mathematically true when applied to rays 
 of any one sort. 
 
 Plate 6. E XP j. Let a perpendicular cylindrical vessel be so placed that the sun, shining upon its 
 
 side NA, may cast the shadow of the side to a point L in the bottom of the vessel. This shadow 
 is terminated by SNL, a ray which passes, in a right line, by the edge of the vessel. If the 
 vessel be filled with water, the shadow will recede, as the water is poured into the vessel, from 
 the point L, which terminated it when the vessel was empty, toward the side NA, on which the 
 sun shines, and will be terminated by the ray ONC ; that is, the ray SNL, which first termi- 
 nated the shadow, by passing out of the air into the water, is refracted toward AN, a line 
 drawn perpendicular to the surface of the water, at the point in which the ray enters the water; 
 or the angle of refraction is less than the angle of incidence. 
 
 2. Let a small bright object be laid upon the bottom of a cylindrical vessel NBAL at C. 
 Let the spectator's eye be so placed at S, as just to lose sight of the object at C ; that is, so that 
 a ray passing in a right line from the remote edge of the object toward the eye at S will be 
 
CHAP. II. OF REFRACTION. 158 
 
 intercepted by the edge of the vessel, or that the first ray which is not intercepted passes in the 
 direction ONC above the eye. Whilst the eye continues in the same situation, if the vessel be 
 filled with water, the object will become visible ; that is, the ray which passed from the remote 
 edge of the object, in a right line CNO, by the vessel, in entering the air is refracted into the 
 direction NS, toward the eye, or from the perpendicular BNA. 
 
 PROP. XII. All refraction is reciprocal. 
 
 The ray AC, in passing out of the medium X into Z, is refracted into CF, because it is Plate. 6. 
 accelerated at its entrance into Z by the greater attraction of the denser medium : and the *'' 6- 
 ray FC in passing out of Z into X is refracted into CA, because it is retarded by the same 
 attraction. Since, then, the acceleration and retardation are produced by the same degree 
 of attraction in opposite directions, they will be equal to one another, and the refractions 
 produced by them will be equal, but in opposite directions ; that is, if the refracted ray becomes 
 the incident ray, the incident ray will become the refracted one ; or, the refractions are 
 reciprocal. 
 
 LKM. Tlie space passed through during any portion of the time a piate i. 
 body is Jailing is altrays proportional to the difference of the squares of the '''S- 10 - 
 velocities at the beginning and end of that portion of time. 
 
 For (B. II. Prop. XXVI.) AFG : Afg : : FG 2 :fg* ; therefore AFG Afg arfYGg : FG 
 f s > : : AFG : FG*, and ADE : Ade : : DE Z : rfe 3 ; therefore ADE Ade or rfDEe : DE* 
 de* :: ADE : DE^but AFG : FG :: ADE: DE S ; therefore fFGg : FG fg* : : dDEe : DE* 
 -de 2 . 
 
 PROP. XTII In any two given mediums, the sine of any one angle of 
 incidence has the same ratio to the sine of the corresponding angle of 
 refraction, as the sine of any other angle of incidence has to the sine of its 
 corresponding angle of refractiom. 
 
 Let AC represent the velocity of a ray of light obliquely incident on the plane surface FG of plate 16, 
 a denser medium W at the point C. Resolve this motion, which is constant and invariable in the ^'S' * 
 same medium, into AB perpendicular to the refracting surface and BC parallel to it. Of these (as 
 was shown in Prop. XI.) only the perpendicular motion AB is accelerated, the parallel motion 
 BC continuing unaffected by the attraction. Next (since most probably the intensity of the re- 
 fractive power is greatest precisely at the refracting surface, and gradually diminishes on each 
 side to the limits of its action, thus producing a variable acceleration.) suppose the refracting 
 space to be divided into strata by planes parallel to the refracting surface at the point thiough 
 which the ray passes, and so near to each other that the refractive power, and, of course, the 
 acceleration may be considered uniform between them. Now (by Lemma) the space passed 
 through by a motion uniformly accelerated is proportional to the difference of the squares of 
 the initial and final velocities. But the space passed through by the perpendicular part of the 
 motion of any ray while passing through the same stratum, is always the same, (that is, the 
 thickness of the stratum) whatever the obliquity of the ray may be; therefore the difference 
 between the square of the perpendicular velocity of any ray at its entering any one stratum and 
 
 20 
 
454 OF OPTICS. BOOK VI. 
 
 the square of its perpendicular velocity at its leaving the same stratum is the same as the dif- 
 ference between the squares of the perpendicular velocities of any other ray at its entering and 
 leaving the same stratum, however different the obliquity, and, of course, the actual perpendicu- 
 lar velocities of the rays may be. And as the number of these differences, with reference to any 
 single ray, is equal to the number of strata assumed, the sums of an equal number of equal 
 differences must be equal. Therefore the whole differences produced in passing through all the 
 strata or whole refracting space, or the difference between the squares of the perpendicular velo- 
 cities in one medium until the rays enter the refracting space, and in the other after emerging 
 from it, is always the same, however the actual perpendicular velocities may vary with the obliqui- 
 ty of incidence, and however the refracting force may vary during refraction. Take CD = BC to 
 represent the parallel motion of the ray after refraction, as that continues uualtered, and draw DB 
 in the denser medium perpendicular to the surface to represent the perpendicular velocity of the re- 
 fracted ray. The square of DE must exceed the square of AB by a certain quantity which con- 
 tinues constant in all positions of the ray AC. Now since CD* = BC*. and DE 2 exceeds 
 AB 1 by a constant quantity, CD a + DE 2 or CE* must exceed BC* + AB or AC 2 by the 
 same constant quantity. Therefore as AC and AC 2 are constant quantities, so CE S and CE 
 are also constant ; and as AC expresses the direct velocity of the incident ray which never 
 varies with the angle of incidence, so CE expresses the direct velocity of the refracted ray, 
 which is the same at all angles. On the centre C, and with CB or Cf>, representing the paral- 
 lel velocity, as a radius describe a circle ; it is plain that AC is the secant of the angle ACB or 
 cosecant of the angle of incidence, and CE the secant of the angle DCK or cosecant of the 
 angle of refraction, and as these are constant quantities, however the radius or parallel mo- 
 tion may vary, the cosecants of incidence and refraction in the same two mediums have a con- 
 stant ratio to each other; but sines are inversely as the cosecants of the same angles, therefore 
 the sines also have always the same ratio to each other. When light passes from a denser me- 
 dium into a rarer, the same reasoning may easily be adapted to the retardation the differences 
 of the squares both of the perpendicular and direct velocities of the rays before and after re- 
 fraction being the same at all angles of inclination, and the direct velocity of the refracted ray- 
 being less than that of the incident ray by a certain constant quantity, &c. the terms of the 
 ratio being reversed. 
 
 If a, b, d, e, be read instead of A, B, D, E, respectively, the figure exhibits an example of a 
 much greater angle of incidence. 
 
 COR, 1. Hence, when the angle of incidence is increased, the corresponding angle of refrac- 
 tion will also be increased ; because the ratio of their sines cannot continue the same, unless 
 they be both increased ; and if two angles of incidence be equal, the angles of refraction will 
 also be equal. 
 
 COH. 2. Hence the angle of deviation varies with the angle of incidence. 
 
 Plate 6.; SCHOL. 1. If a ray of light, AC, pass obliquely out of air into water, AD, the sine of the angle 
 
 Fig. 6. O f incidence ACD is to NS,the sine of the angle of refraction NCF, nearly as 4 to 3 ; therefore, 
 supposing the sines proportional to the angles, the sine of FCL, the angle of deviation, is as the 
 diflerence between AD and NS, that is, as 4 3, or 1 ; whence the sine of incidence is to the 
 sine of the angle of deviation as 3 to 1. In like manner it may be shewn, that, when the ray 
 passes obliquely out of water into air, the sine of the angle of incidence will be to that of deviation, 
 as NS to AD NS, that is, as 4 to 1. In passing out of air into glass, the sine of the angle of 
 
CHAP. II. OF REFRACTION. 193 
 
 incidence is to that of 'refraction, as 3 to 2, and to that of deviation, as 3 to 3 2, or 1 ; and 
 in passing out of glass into air, the sine of the angle of incidence is to that of refraction, as 2 
 to 3, and to that of deviation as 2 to 1. 
 
 COR. 3. Hence a raj of light cannot pass out of water into air at a greater angle of incidence 
 than 48 36', the sine of which is to radius as 3 to 4. Out of glass into air the angle must not 
 exceed 40 1 1', because the sine of 40 11' is to radius as 2 to 3 nearly ; consequently, when 
 the sine has a greater proportion to the radius than that mentioned, the ray will not be refracted. 
 
 SCHOL. 2. It must be observed, that when the angle is within the limit, for light to be 
 refracted, some of the rays will be reflected. For the surfaces of all bodies are for the most 
 part uneven, which occasions the dissipation of much light by the most transparent bodies ; 
 some being reflected, and some refracted, by the inequalities on the surfaces. Hence a person 
 can see through water, and his image reflected by it at the same time. Hence also, in the 
 dusk, the furniture in a room may be seen by the reflection of a window, while objects that are 
 without are seen through it. 
 
 EXP. Upon a smooth hoard draw the right line BCD, and on its extremities erect the per- Plate 16; 
 pendiculars BA and DE in opposite directions ; on the middle C, of line BD, as a centre, and Fl ^' *' 
 with any extent greater than BC, intersect the line BA suppose in A, and with an extent great- 
 er than CA in the ratio of 4 to 3, intersect the line DE, perhaps in E. Then if pins be 
 stuck perpendicularly at A, C, and E, and the board be dipped in the water as far as the line 
 BD, the pin at E will appear in the same line with the pins at A and C. This shews, that 
 the ray which comes from the pin E is so refracted at C, as to come to the eye along the line 
 CA; whence the sine of incidence is to the sine of refraction as 4 to 3. If other pins 
 were fixed along CE, they would all appear in AC produced ; which shews that the ray is bent 
 at the surface only. The same may be shewn, at different inclinations of the incident ray, by 
 means of two moveable rods turning upon the centre C, which always keep the ratio of the 
 sines, as 4 to 3. Also the sun's rays, coinciding with AC, may be shewn to be refracted in the 
 same manner. 
 
 PROP. XIV. Rays of light, which pass perpendicularly out of one 
 medium into another, suffer no refraction. 
 
 When AC, the incident ray, coincides with OC, the perpendicular, the action of the medium Plate 6. 
 Z or X to accelerate or retard the motion of the ray, being perpendicular to its surface, cannot *"' 6 - 
 turn the ray out of its perpendicular path. 
 
 PROP. XV. When parallel rays pass obliquely out of one medium 
 into another through a plane surface, they will continue parallel after 
 refraction. 
 
 Let AB, CD, be parallel rays, falling on the plane surface RED of a medium of different P i ate 6. 
 density; because they make equal angles of incidence with their respective perpendiculars F'g 9- 
 OP, ST, they will suffer an equal degree of refraction ; that is, the angles of refraction EBP, 
 FDT, will be equal ; whence the retracted rays BE, DF, will be parallel. 
 
t6 OF OPTICS. BOOK VI. 
 
 PROP. XVI. Through a plane surface, if diverging rays pass out of a 
 rarer into a denser medium, they are made to diverge less ; and if they pass 
 out of a denser intu a rarer medium, they are made to diverge, more : if 
 converging rays pass put of a rarer into a denser medium, they will be made 
 to converge less ; if out of a denser into a rarer, to converge inure. 
 
 Plate 4. J- 16 * tlie diverging rays AB, AE, AF, pass out of a rarer into a denser medium, through the 
 
 Fig. 10. plane surface Gil, and let the ray AB be perpendicular to that surface ; the rest being refracted 
 toward their respective perpendiculars IK, LM, and that the most which falls the farthest from 
 B, they will proceed in the directions E.V and FO, diverging in a less degree from the ray AP, 
 than they did before refraction ; whereas, lad they proceeded out of a denser into a rarer 
 medium, they would have been refracted from their perpendiculars EK, FM, and those the most 
 which were the most oblique, and theiefore would have diverged more than before. Again, 
 Fig. 11. j e t thg con verging rays AB, CD, EF, pass out of a rarer into a denser medium, through the 
 plane surface GH, and let the i ay AB be perpendicular to that surface ; the other rays being 
 refracted toward their respective perpendiculars IK, LM, and EF being refracted more than 
 CD, they will proceed in the directions DN, FN, converging in a less degree toward the raj 
 AN, than they did before ; whereas, had the first medium been the denser, they would have 
 been refracted the other way, and therefore have converged more. 
 
 DEF. XIV. A Lens is a round piece of polished glass, which has both 
 its sides spherical, or one spherical and the other plan e. 
 
 A lens may either be convex on both sides, plano-convex, concave, plano-concave, or 
 convex on one side, and concave on the other ; which last is called a meniscus. 
 
 In plate 6, fig. 12. Sections of these, formed by a plane passing perpendicularly through 
 their centres, are represented. 
 
 DEF. XV. The Axis of a Lens is a right line passing through its centre, 
 perpendicular to both its surfaces, and the extremities of the axes are its 
 poles. 
 
 Each kind of lens is generated by the revolution of a section of the lens about this line. 
 Thus, in the first lens, if acb, adb, revolve about erf, the convex lens will be formed. 
 
 DEF. XVI. In every beam of light, the middle ray is called the Axis. 
 
 DEF. XVII. Rays are said to fall directly upon a lens, if their axis coin- 
 cides with the axis of the lens ; otherwise, they are said to fall obliquely. 
 
 DEF. XVIII. The point, in which parallel rays are collected by passing 
 through a lens, is called the Focus of parallel rays of that lens. 
 
 PROP. XVII. Through a convex surface of the denser medium, parallel 
 
CHAP. II. OF REFRACTION. 157 
 
 rays, passing out of a rarer into a denser medium, will become converging ; 
 diverging rays will be made to diverge less, to become parallel, or to con- 
 verge, according to the degree of divergency before refraction, or of the con- 
 vexity of the surface ; rays converging toward the centre of convexity 
 will suffer no refraction ; rays converging to a point beyond the centre of 
 convexity will be made more converging ; and rays converging toward a 
 point nearer the surface than the centre of convexity, will be made less converg- 
 ing by refraction ; and when the rays proceed out of a denser into a rarer me- 
 dium, through a concave surface of the denser, the contrary occurs in each case. 
 
 Let AB, ID, be parallel rays entering a denser medium through the convex surface CDE, p] a te 6. 
 whose centre of convexity is L ; and let one of these, ID, be perpendicular to the surface. This F '- I3 - 
 will pass on through the centre without suffering any refraction, but the other, being oblique to 
 the surface, will be refracted toward the perpendicular LB, and will therefore be made to pro- 
 ceed in some line, as PK, converging toward the other ray, and meeting it in K, the focus. 
 Had one ray diverged from the other, suppose in the line MP, it would, by being refracted 
 toward its perpendicular LB, have been made either to diverge less, be parallel, or to converge. 
 Let the line ID be produced to K ; and if the ray had converged, so as to have desciibed the 
 line NBL, it would then have been coincident with its perpendicular, and have suffered no 
 refraction. If it had proceeded with less convergency toward any point beyond L in the line 
 IK, it would have been made to converge more by being refracted toward the perpendicular LB, 
 which converges more than it ; and had it proceeded with more convergency than BL, that is, 
 toward' any point between D and L, being refracted toward the perpendicular, it would have 
 been made to converge less. 
 
 And the contrary happens, when rays proceed out of a denser into a rarer medium, through 
 a concave surface of the denser. For being now refracted from their respective perpendiculars, 
 as they were before toward them, if they are parallel before refraction, they diverge afterward ; 
 if they diverge, their divergency is increased ; if they converge in the direction of their perpen- 
 diculars, they suffer no refraction ; if they converge less than their respective perpendiculars, 
 they are made to converge still less, to be parallel, or to diverge ; if they converge more, (heir 
 convergency is increased. All which may clearly be seen by the figure, imagining the rays AB, 
 ID, &c. bent the contrary way in their refractions to what they were in the former cases. 
 
 Exp. Let parallel, diverging, and converging rays pass through a convex lens ; the several 
 cases of this proposition will be confirmed. 
 
 If CDE11 be a convex lens, whose axis is IK, let L be the centre of the first convexity CDE, 
 and M that of the other CHE ; and let the ray AB be parallel to the axis ; through B draw the 
 line LN, which will be perpendicular to the surface CDE at that point. The ray AB in 
 entering the denser substance of the lens will be refracted toward the perpendicular, and there- 
 fore proceed after it has entered the surface at B in some direction inclined toward the axis, as 
 BP. Through M the centre of convexity of this surface^ and the point P draw the line MR, 
 which passing through the centre will be perpendicular to the surface at P, and the ray now 
 entering a rarer medium vvill be refracted ffom the perpendicular into some direction as PK. 
 In like manner, and for the same reasons, the parallel ray ST on the other side the axis, and 
 
158 OF OPTICS. BOOK VI. 
 
 also all the intermediate ones, as XZ, &c. will meet it in the same point, unless the rajs AB 
 and ST enter the surface of the lens at too great a distance from the axis IK, the reason of 
 which will be afterward explained. 
 
 The point K where the parallel rays AB, ST, &c. are supposed to be collected bj passing 
 through the lens CE, is called the focus of parallel rays of that lens. 
 
 If the rays come diverging from a point equally distant from the surface as the focus of 
 parallel rays, they will be rendered parallel ; if from a point farther from the surface than L, 
 they will be brought to a point beyond L ; if from a point nearer than L, they will diverge less 5 
 as may be inferred from Prop. XII. 
 
 If the rays come diverging toward L, they will suffer no refraction ; if toward a point 
 beyond I/, they will become more converging ; if toward a point nearer the surface than L, they 
 will become less converging ; as is sufficiently explained in the proof of this proposition. 
 
 Plate 6. SCHOL. If the rays AB, CD, EF, be parallel to each other, but oblique to GH, the axis of the 
 
 Tig. 14. j ens IK, or if the diverging rays CB, CF, proceed as from some point C, which is not situated 
 in the axis of the lens, they will be collected into some point as L, not directly opposite to the 
 radiant C, but nearly so ; for the ray CD, which passes through the middle of the lens, and falls 
 upon the surface of it with some obliquity, will itself suffer a refraction at D and N ; but it will 
 be refracted the contrary way in one place from that in the other; and these refractions will be 
 equal in degree, if the surfaces are parallel, as we may easily perceive if we imagine ND to be 
 a ray passing out of the lens both at N and D, for it is evident the line ND has an equal incli- 
 nation to each surface at both its extremities. Upon which account the difference between the 
 situation of the point L, and one directly opposite to C, is so small, that it is generally neg- 
 lected ; and the focus is supposed to be in that line, in which a ray that would pass through 
 the middle point of the lens, were it to suffer no refraction, would proceed. 
 
 PROP. XVIII. When rays pass out of a rarer into a denser medium, 
 through a concave surface of the denser, if the rays are parallel before refrac- 
 tion, they are made to diverge ; if they are divergent, they are made to 
 diverge more, to suffer no refraction, or to diverge less, according as they 
 proceed from some point beyond the centre, from the centre, or from some 
 point between the centre and the surface ; if they are convergent, they are 
 either made less converging, parallel, or diverging, according to their degree 
 of convergency before refraction ; and the reverse, in passing out of a denser 
 into a rarer medium through a convex surface of the denser. 
 
 Plate 6. Let MF, 01, be two parallel rays entering a concave and denser medium, the centre of 
 
 Fig. 15. w hose convexity is H, and the perpendicular to the refracting surface at the point F is LH ; the 
 ray Ol, if we suppose it perpendicular to the surface, will proceed on directly without refraction, 
 but the obliqof ray MF, being refracted toward the perpendicular II L, will recede from the 
 other ray OL If the ray MF had proceeded from a point in 01 farther from the surface than 
 II, it would have been bent nearer to the perpendicular, and therefore have diverged more ; if it 
 had diverged from the centre H, it would have fallen in with the perpendicular I1L, and not 
 have been refracted at all ; and had it proceeded from a point nearer the surface than the centre 
 
CHAP. II. OF REFRACTION. 159 
 
 H, it would, by being refracted toward the perpendicular HL, have proceeded in some line 
 nearer it than it otherwise would have done, and so would diverge less than before refraction. 
 Lastly, if it had converged, it would have been rendered less converging, parallel, or diver<*in<, 
 according to the degree of convergency, which it had before it entered into the refracting 
 surface. 
 
 If the same rays proceed out of a denser into a rarer medium through a convex surface of the 
 denser, the contrary happens in each supposition ; the parallel rays are made to converge ; thos e 
 which diverge less than their respective perpendiculars, that is, those which proceed from a point 
 beyond the centre, are made less diverging, parallel, or converging, according to the degree in 
 which they diverge before refraction ; those which diverge more than their respective perpen- 
 diculars, that is, those which proceed from a point between the centre and the refracting surface, 
 are made to diverge still more. And those which converge, are made to converge more. All 
 which may easily be seen by considering the situation of the rays with respect to the perpendic- 
 ular HL. 
 
 EXP. Let parallel, diverging, and converging rays pass through a concave lens; the several plate 6. 
 cases of this proposition will be confirmed; thus, let ABCD represent a concave lens, ED its Fi fT- ls ' 
 axis, FH the radius of the first concavity, IK that of the second ; produce HF to L, and let MF 
 be a ray of light entering the lens at the point F. This ray being refracted toward the perpen- 
 dicular FL, will pass on to some point, as K in the other surface, more distant from the axis 
 than F, and being there refracted from the perpendicular IK, will be diverted farther still from, 
 the axis, and proceed in <he direction KN. as from some point O, on the first side of the lens. 
 In like manner other rays, as PQ parallel to the former, will proceed after refraction at both 
 surfaces as from the same point O ; which upon that account will be the imaginary radiant of par- 
 allel rays of this lens. 
 
 If the rays diverge before they enter the lens, their imaginary radiant is then nearer the lens 
 than that of the parallel rays. If they converge before they enter the lens proceeding toward 
 some distant point in the axis, as E, they are then rendered less converging : if they converge 
 to a point at the same distance from the lens with the focus of parallel rays, they then go out 
 parallel ; if to a point at a less distance, they remain converging, but in a less degree than before 
 they entered the lens. 
 
 SCHOL. If the lens is plane on one side, and convex or concave on the other, the refraction 
 is similar, but in a less degree. In a meniscus, if the convexity on one side is equal to the con- 
 cavity on the other, the two sides will produce equal and contrary effects, and the inclination 
 of the rays to each other will be the same after refraction as before. If the convexity is greater 
 than the concavity, the meniscus will have the effect of a lens which has i's convexity equal to 
 the excess of the convexity of the meniscus above its concavity ; and the reverse, if its concavity 
 exceeds its convexity. 
 
 PROP. XIX. When diverging rays are made to converge by passing 
 through a convex lens, as the radiant approaches toward the lens on one 
 side, the focus departs from it on the other ; and the reverse. 
 
 For, the nearer the radiant point is to the lens, the more the rays which fall upon the lens 
 diverge before refraction ; whence (the power of the refracting medium being given) they will 
 
160 
 
 OF OPTICS. 
 
 BOOK VI. 
 
 Plate 6. 
 Fig. 13. 
 
 Plate 6. 
 Fig. 16. 
 
 converge the less after refraction, and have their focul point at the greater distance fiom the 
 surface : on the contrary, the more remote the radiant point is from the lens, the less the 
 incident rays will diverge, and consequently the more will the refracted rays converge, and 
 the nearer will the focus be to the surface, till, at an infinite distance of the radiant, the rays are 
 collected in the focus of parallel rays. 
 
 PROP. XX. When the radiant point is at that distance from the surface, 
 at which parallel rays coming through it from the. other side would be 
 collected, rays flowing from that point become parallel on the other side. 
 
 It manifestly follows from Prop. XII. that if the parallel rays AB, ID, ST,in passing through 
 CDE, are brought to a focus in K, rays from K as a radiant point will, after refraction, proceed 
 in the parallel lines BA, DI, TS. 
 
 PROP. XXI. When rays pass out of one medium into another of 
 different density through a plane surface, if they diverge, the distance of the 
 imaginary radiant will be to that of the real radiant ; if they converge, 
 the distance of the real focus will be to that of the imaginary focus of the 
 incident rays, as the sine of the angle of incidence is to that of the angle of 
 refraction. 
 
 This proposition admits of four cases. 
 
 Case 1. Of diverging rays passing out of a rarer into a denser medium. 
 
 Let X represent a rarer, and Z a denser medium, separated from 'each other by the plane 
 surface AB ; suppose CD and CE to be two diverging rays proceeding from the point C, the 
 one perpendicular to the surface, the other oblique ; through E draw the perpendicular PK. 
 The ray CD being perpendicular to the surface, will proceed on in the right line CQ, but the 
 other falling upon it obliquely at E, and there entering a denser medium, will suffer a refraction 
 toward the perpendicular EK. Let then EG be the refracted ray, and produce it back till it 
 intersects DC produced also in F ; this will the imaginary radiant. On the centre E with the radius 
 EF, describe the circle AFBQ. and produce EC to H ; draw HI the sine of the angle of inci- 
 dence, and GK that of refraction ; equal to this is FP or CM, which let be drawn. Now if we 
 suppose the points 1) and E contiguous, or nearly so, then will the line HE be almost coincident 
 with FD, and therefore FD will be to CD, as HE to CE or (El. VI. 4.) as HI to CM ; that is, 
 the distance of the imaginary radiant of the ray EG is to the distance of the real radiant, as the 
 sine of the angle of incidence is to that of the angle of refraction.* 
 
 Whereas IE is to ME, or ND to CD, as HI to CM, that is, as the sine of Ihe angle of incidence to that 
 of the angle of refraction, which (Prop. XIII.) is always the same, the line IN is in all inclinations of the 
 ray CE, at the same distance from CM. Consequently, had CE been coincident with CD, the point H had 
 fallen upon N j and because the circle passes through both H and F, F would also have fallen upon N j 
 upon which account the imaginary radiant of the ray EG would have been there. But the ray CE being 
 oblique to the surface DB, the point H is at pome distance from N ; and therefore the point F is necessarily 
 so too, and the more so, the greater that distance is. Hence it is manifest, that no two rays flowing from 
 the radiant point C, and falling with different obliquities on the surface BD, will, after refraction there, 
 proceed as from the same point ; therefore, strictly speaking, tliera is no one point in the line D productd. 
 
CHAP. II. OF REFRACTION. 161 
 
 Case 2. Of diverging rays proceeding out of a denser into a rarer medium. 
 
 Let X be the denser, Z the rarer medium. FD and FE twj diverging rays proceeding from 
 the point F; and supposing the perpendicular PK drawn as before, VI 1 will he the sine of the 
 angle of incidence of the oblique ray FE, which in this case being refracted from the perpendic- 
 ular, will pass on in some line as ER, which being produced back to the tircumferroce of the 
 circle will cut the ray FD somewhere, suppose in C ; this therefore will be the imaginary focus 
 of the refracted ray ER ; draw RO, the sine of the angle of refraction, to which HI will be et|ual ; 
 but here also FP or its equal CM, is to Hf, as EC to EH, or (if the points D and E be consid- 
 ered as contiguous) as DC to DF; that is, the sine of the angle of incidence is to the sine of the 
 angle of refraction, as the distance of the imaginary radiant to that of the real radiant. 
 
 Cor. The image R of a small object G, placed under water, is one fourth nearer the 
 surface than the object. And hence the bottom of a pond, river, &c. is one third deeper than 
 it appears to a spectator. If a river appears to be only 4i feet deep, it will he 6 feet ; a person 
 not apprized of this, might venture into water at the hazard of his life. And hence also we 
 have the reason of the common phenomenon of a shilling, or other object, placed in an 
 empty vessel, appearing to be elevated higher and higher as the vessel is filled with water. 
 Suppose the vessel empty, CQ its side opaque, G the object ; if the eye be at H, the 
 object will be hidden by the side CQ; but by filling the vessel with water up to DB, it 
 will become visible, and seen at R, the ray EG being refracted into HE. And if the eye be so 
 placed as to see the object at G when the vessel was empty, while, it is filling the object will 
 appear to rise gradually. Hence appears the reason why a straight stick HER, (Fig. 16) 
 immersed obliquely into water, appeals bent at the surface l)B ; the image of the part within 
 lying above the object. 
 
 Case 5. Of converging rays passing out of a denser medium into a rarer. 
 
 Next ; let Z be the denser, X the rarer medium, and GE the incident ray ; this will be Plate 6. 
 refracted from the perpendicular into a line as EH ; then all things remaining as before, GK, or fl S- 1G - 
 its equal FP, or CM will be the sine of the angle of incidence, and HI that of refraction ; but 
 these lines, as before, are to each other, as DC to DF ; that is, the focal distance is to the dis- 
 tance of the imaginary focus, as the sine of the angle of incidence to that of the angle of refraction. 
 
 Case 4. Of converging rays passing out of a rarer into a denser medium. 
 
 Let Z be the rarer, X the denser medium, and RE the incident ray ; this will be refracted 
 toward the perpendicular into a line, as EF; C will be the imaginary focus, and F the real one; 
 HI, which (El. I. C.) is equal to RO, will be the sine of the angle of incidence, and FP that of 
 
 that can more properly be called the focus of rays flowing from C, than another ; for those which enter the 
 refracting surface near D, will, after refraction, proceed, as has been observed, from the parts about N j those 
 which enter near E, will flow as from the parts about F ; those which enter about T, as from some points in 
 the line DF produced, &c. And it is farther to be observed, that when (he angle DCE becomes large, the line 
 NF increases apace ; whence those rays which fall near T proceed, after refraction, as from a more diffused 
 space, than those which fall at the same distance from each other near the point D. Upon which account it 
 is usual, with optical writers, to suppose the distance between the" points where the rays enter the plane 
 surface of a refracting medium, to be inconsiderable with regard to the distance of the radiant point, if they 
 diverge ; or to that of their imaginary focus, if they converge ; and unless there be some particular reason 
 to the contrary, they consider them as entering the refracting medium in a direction as nearly perpendicular 
 to iU surfaces as may be. 
 
 21 
 
103 OF OPTICS. BOOK VI. 
 
 the angle of refraction ; but these are to each other (El. VI. 2.) as DP to DC ; and therefore the 
 focal distance is to that of the imaginary focus, as the sine of the angle of incidence is to that of 
 the angle of refraction. 
 
 PROP. XXII. When parallel rays fall upon a spherical surface of dif- 
 ferent density, the focal distance will be to the distance of the centre of con- 
 vexity, as the sine of the angle of incidence is to the difference between that 
 sine and the sine of the angle of refraction. 
 
 Case 1. Of parallel rays passing out of a rarer into a denser medium through a convex sur- 
 face of the denser. 
 
 Plate 6. Let A B represent a convex surface; Cits centre of convexity; HA and DB two parallel 
 
 rays, passing out of a rarer medium into a denser, the oi;c perpendicular to the refracting sur- 
 face, the other oblique ; draw CB ; this being a radius, will be perpendicular to the surface at 
 the point B; and the oblique ray DB being in this case refracted toward the perpendicular, 
 will proceed in some line, as BF, meeting the other ray in F, which will therefore be the focal 
 point ; produce CB to N, then will DBN, or (El. I, 29.) its equal BCA be the angle of incidence, 
 and FBC that of refraction. Now, since any angle has the same sine with its supplement to 
 two right ones, the angle of FCB iieing the supplement of ACB, which is equal to the angle of 
 incidence, may here be taken for that angle ; and therefore, as the sides of a triangle have the 
 same relation to each other, as the sines of their opposite angles have, KB being opposfte to this 
 angle, and FC being opposite to the angle ot refraction, they may here he considered as the sines 
 of the angles of incidence and of retraction ; and for the same reason CB may be considered as 
 the sine of the angle CFB, which angle being, together with the angle FBC, equal to the external 
 one ACB (El. I. 32.) it is itself equal to the difference between those two last angles ; and 
 therefore the line FB is to CB, as the sine of the angle of incidence is to the sine of an angle 
 which is equal to the difference between the angle of incidence and of refraction. Now because 
 in very small angles as these are, for we suppose in this case also the distance AB to vanish, 
 (the reason of which will be shown in the note) their sines will have nearly the same ratio to 
 each other that they themselves have, the distance FB will be to CB as the sine of the angle of 
 incidence is to the difference between that sine and the sine of the angle of refraction ; but 
 because BA vanishes, FB and FA are equal, and therefore FA is to CA in that ratio.* 
 
 * It appears from the above proposition, that the focal distance of the oblique ray D!l is such, that the 
 line KF shall be to the line CB or CA as the sine of the angle of incidence to the sine of an angle equal to the 
 difference between the angle of incidence and refraction ; therefore so long as the angles BCA &c. are small, 
 so long the line FB is nearly of the same length, because small angles have nearly the same ratio to each 
 oilier that their sines have. But when the point B is removed far from A, so that the ray DB enters the 
 surface, suppose about O, the angles BCA &c. becoming large, the sine of the angle of incidence begins to 
 bc:ir a considerably less ratio to the sine of an angle which is equal to the difference between the angle of 
 incidence and refraction than before, and therefore the line BF begins to bear a much less ratio to BC ; 
 wherefore its length decreases apace ; upon which account those rays which enter the surface about O, not 
 .only meet nearer the centre of convexity than those which enter at A, but are collected into a more diffused 
 space. Hence it is, that the point where those only which enter near A, are collected, is reckoned the 
 true focus j and the distance AB in all demonstrations relating to the foci of parallel rays entering a spher- 
 ical surface, whether convex or concave, is supposed to vanish. 
 
CHAP. II. OF REFRACTION. 163 
 
 Case 2. Of parallel rays passing out of a denser into a rarer medium through a concave sur- 
 face of the denser. 
 
 Let AB be the concave surface of the denser medium, C the centre of convexity, and HA ? ate ,S' 
 and DB two parallel rays. Through B, the point where the oblique ray DB enters the rarer 
 medium, draw the perpendicular CN ; and let the ray DB, being in this case refracted from the 
 perpendicular, proceed in the direction BM ; produce BM back to H ; this will ho the imagi- 
 nary radiant, and DBN, or its equal ACB, will be the angle of incidence, and CBM, or its equal 
 (El. I. 15.) IIBN that of refraction ; then because NBD and DBH together are equal to NBH, 
 the angle of refraction, therefore BCA, which is equal to the first, and AHB, which is equal to 
 the second, are together equal to the angle of refraction; and therefore, since one of them, 
 BCA, is equal to the angle of incidence, the other, AHB, is the difference between that angle 
 and the angle of refraction. Now the sine of the angle BCA, the angle of incidence, is to the 
 sine of the angle AHB, as BH to BC ; but the distance AB vanishing, HB is to CB, as HA to 
 CA; that is, the sine of the angle of incidence is to the sine of an angle which is the difference 
 between the angle of incidence and refraction, or because the angles are small, to the difference 
 between the sine of the angle of incidence and that of refraction, as the distance of the focus 
 from the surface is to that of the centre from the same. 
 
 Case 3. Of parallel rays passing out of a rarer into a denser medium through a concave 
 surface of the denser. 
 
 Let AB be the concave surface of the denser medium, and let LB and FA be the incident 
 rays. Now whereas, when DB was the incident ray, and passed out of a rarer into a denser 
 medium, as in Case the first, it was refracted into a line BF, this ray LB having the same incli- 
 nation to the perpendicular, will also suffer the same degree of refraction, and will therefore 
 pass on afterward in the line FB produced, toward P. So that whereas, in that case the point 
 F was the real focus of the incident ray DB, the same point will, in this case, be the imaginary 
 radiant of the incident ray LB ; but it was there demonstrated, that the distance FA is to CA, 
 as the sine of the angle of incidence is to the difference between that and the sine of the angle 
 of refraction, therefore the radiant distance of the refracted ray BP is to the distance of the 
 centre of convexity in that ratio. 
 
 Case 4. Of parallel rays passing out of a denser into a rarer medium through a convex sur- 
 face of the denser. 
 
 Let AB be the convex surface of the denser medium, and let LB and FA be the incident 
 rays, as before. Now whereas, when DB was the incident ray passing out of a denser into a 
 rarer medium, it was refracted into BM, as in Case the second, having a point as II in the line 
 MB produced for its imaginary radiant; therefore LB, for the like reason as was given in the 
 last case, will in this he refracted into BH, having the same point II for its real focus. So that 
 here also the focal distance will be to that of the centre of convexity, as the sine of the angle of 
 incidence is to the difference between that and the sine of the angle of refraction. 
 
 COR. t. Hence, the sines of the angles of incidence and of refraction of parallel rays being 
 given, and also the distance of the centre of convexity from the surface, the focus of any lens 
 may he easily found. 
 
 COR. 2. The distance of the centre of a glass sphere, from the principal focus of rays refract- 
 ed by it, is nearly equal to a radius and a half of the sphere. For in this case it will be 
 FA : CA : : 1 : 21 -2R, (where I and R signify the sines of incidence and refraction) therefore, 
 
164 OF OPTICS. BOOK VI. 
 
 FA = jp^j = r- And if the sphere be of water instead of glass, FA = = 2CA = the 
 
 diameter of the sphere. 
 
 PROP. XXIII. "When diverging or converging rays enter into a medium 
 of different density through a spherical surface, the ritio compounded of that 
 which the focal distance bears to the distance of the radiant point (or of the 
 imaginary focus of the incident rays, if they converge) and of that, which the 
 distance between the same radiant point (or imaginary focus) and the centre, 
 bears to the distance between the centre and the focus, is equal to the ratio, 
 which the sine of the angle of incidence bears to the sine of the angle of 
 refraction ; that is, FD x CA : DA x CF : : FB : BG. 
 
 Case 1. Of diverging rays passing out of a rarer into a denser medium through a convex 
 surface of the denser, with such a degree of divergency, that they shall converge after refraction. 
 Plate 6. Let BD represent a spherical surface, and C its centre of convexity ; and let there be two 
 
 Fig 1 . 19. diverging rays, Ali and AD, proceeding from the radiant point A, the one perpendicular to the 
 surface, trie other oblique. Through the centre C produce the perpendicular ray AD to F, and 
 draw the radius CB and produce it to K, and let BF be the refracted ray ; then will F be the 
 focal point ; produce AB to H, and through the point F draw the line FG parallel to CB. AB 
 being the incident ray, and CK. perpendicular to the surface at the point B, the angle ABK., or 
 which is equal to it, because of the parallel lines CB and FG, FGH is the angle of incidence. 
 Now, since the supplement of any angle to two right ones has the same sine with the angle 
 itself, the sine of the angle FGB, which is the supplement of FGH to two right ones, may be 
 considered as the sine of the angle of incidence ; for which sine the line FB, as the sides of a 
 triangle have the same ratio to each other, that the sines of their opposite angles have, may be 
 taken. Again, the angle FBC is the angle of refraction, or its equal, because alternate to it, 
 BFG, to which BG being an opposite side, may be taken as the sine. But FB is to BG in a 
 ratio compounded of FB to BA, and of B A to BG ; for the ratio that any two quantities bear to 
 each other is compounded of the ratio which the first bears to any other, and of the ratio which 
 that other bears to the second. Now FB is to BA, supposing BD to vanish, as FD to DA ; and 
 BA is to BG, because of the parallel lines CB and FG, as AC to CF. That is, the ratio com- 
 pounded of FD, the focal distance, to DA, the distance of the radiant point, and of AC, the 
 distance between the radiant point and the centre, to CF, the distance between the centre and 
 the focus, is equal to that which the sine of the angle of incidence bears to the sine of the angle 
 of refraction.* 
 
 Since the focal distance of the oblique ray AR is such that the compound ratio of FO to BA and of AC 
 1o CF shall be the same, whatever be the distance between B and D ; it is evident, that AC being always of 
 the same length, die more the line AB lengthens, the more FB must lengthen too, or else FC must shorten ; 
 but if BF lengthens, CF will do so too, and in a greater ratio with respect to its own length than BF will, 
 therefore the lengthening of UF will conduce nothing toward preserving the equality of the ratio; but aa 
 AB lengthens, BF and CF must both shorten, which is the only possible wav in which the ratio can be 
 continued the same. And it is also apparent, that the farther B moves from D toward O, the faster AB 
 lengthens, and therefore the farther the rays enter from D, the nearer to the refracting surface is the place 
 
CHAP. II. OF REFRACTION. 165 
 
 Case 2. Of converging rays passing out of a rarer into a denser medium through a concave 
 surface of the denser with such a degree of convergency, that they shall diverge after refraction. 
 
 Let the incident rays be HB and FD passing out of a rarer into a denser medium through "j^^ 
 the concave surface BD, and tending toward the point A, fiom whence the diverging rays 
 flowed iu the other case ; then the oblique ray HB having its angle of incidence HBC equal to 
 ABK, the angle of incidence in the former case, will be refracted into the line BL such, that its 
 angle of refraction KBL will be equal to FBC, the angle of refraction in the former case ; that 
 is, it will proceed after refraction in the line FB produced, having the distance of the imaginary 
 radiant the same as the focal distance FD with the diverging rays AB, Al), in the other case. 
 But, by what has been already demonstrated, the ratio compounded of FD, the focal distance, to 
 DA, in this case, the distance of the imaginary focus of tiie incident rays and of AC, the dis- 
 tance between the same imaginary focus and tiie centre, to CF, the distance between the centre 
 and the imaginary radiant, is equal to that which tiie sine of the angle of incidence bears to the 
 sine of the an^le of refraction. 
 
 Case 3. Of diverging rays passing out of a rarer into a denser medium through a convex 
 surface of the denser, with such a degree of divergency as to continue diverging. 
 
 Let AB, Al), be the diverging rays and let their divergency be so great, that the refracted Plate 6. 
 ray liL shall also diverge from the other ; produce LB back to F, which will be the imaginary ' s ' 
 radiant ; draw the rauius CB, and produce it to K, produce BA likewise toward G, and draw 
 FG parallel to BC. Then will ABK be the angle of incidence, whose sine BF may be taken 
 for, as being opposite to the angle BGF, which is the supplement of the other to two right ones. 
 And LBC is the angle of refraction, or its equal K8F, or which is equal to this, BFG, as being 
 alternate ; therefore BG. the opposite side to this may be taken for the sine of the angle of 
 refraction. But BF is to BG, for the like reason as was given in Case the first, in a ratio com- 
 pounded of BF to BA, and of BA to BG. Now BF is to BA, (DB vanishing) as DF to DA, 
 and because of the parallel lines FG and BC, the triangles CBA and AGF are similar, there- 
 fore BA is to AG, as CA to AF ; consequently, BA. is to BA together with AG, that is, to BG, 
 as CA is to CA together with AF, that is, BF. Therefore the ratio compounded of DF, the focal 
 distance, to DA, the distance of the real radiant point, and of CA, the distance between the real 
 radiant point and the centre, to CF, the distance between tiie centre and the imaginary radiant, is 
 equal to that which the sine of the angle of incidence bears to the sine of the angle of refraction. 
 
 SCHOL. By making HB and CD the incident rays, the proposition may be proved of con- 
 verging rays passing out of a rater into a denser medium, through a concave surface of a denser, 
 with such convergency, that they shall continue to converge. Also, by the same method of 
 reasoning as in the preceding cases, the proposition may be proved, of diverging rays passing 
 out of a denser into a rarer medium through a concave surface of the denser, and of converging 
 rays passing out a denser into a rarer medium through a convex surface of the denser. And all 
 
 where they meet, but the space thej are collected in is the more diffused ; and therefore is this case, as 
 well as those taken notice of in the t*o preceding notes, different rays, though flowing from the same point, 
 will constitute different foci ; and none are so effectual as those which enter at or very near the point D. 
 And since the same is observable of converging as well as of diverging rays, none, except those which enter 
 very near that point, are usually taken into eonsideration j upon which accourr it is, that the distance UB 
 in determining the focal distances of diverging or converging fays entering a convex or concave surface, is 
 supposed to vanish. 
 
OF OPTICS. BOOK VI, 
 
 those cases which are the converse of the preceding, admit of a similar proof; for, when rays 
 pass out of one medium into another, the sine of the angle of incidence has the same ratio to the 
 sine of the angle of refraction, as the sine of the angle of refraction lias to the sine of the angle 
 of incidence, when they pass through the same lines of direction the contrary way. 
 
 Case 4. Of rays passing out of a denser into a rarer medium, from a point between the 
 centre of convexity and the surface. 
 
 1'late 6. Let AB, AD, be two rays passing out of a denser into a rarer medium, from the point A, 
 
 Pig- 20. wi,;,;), j s taken between C, the centre of convexity, and the refracting surface BD : through B 
 draw CK, and let BL be the refracted ray ; produce BL back to F, and draw FG parallel to 
 BC. Then will ABC be the angle of incidence, of which BF, being opposite to its alternate 
 and equal angle BGF, is the sine. LBK will be the angle of refraction, or its equal FBC, of 
 which BG, being opposite to its supplement to two right angles BFG, is the sine. But, BF is 
 to BG in the compound ratio of BF to BA, and of BA. to BG; and (BD vanishing) BF is to 
 BA as DF to DA, and because the lines CB and FG are parallel, BA is to BG as CA to CF. 
 
 Case 5. Of rays passing out of a rarer into a denser medium from a point between the centi>e 
 of concavity and the surface. 
 
 Plate 6. Let AB, AD, be two diverging rays passing out of a rarer into a denser medium through the 
 
 *' 21. refracting surface BD, whose centre of concavity is C, a point beyond that from whence the 
 rays flow. Through B draw CK, and let BL be the refracted ray, produce it back to F, and 
 draw FG parallel to BC, meeting BA in G. ABC will be the angle of incidence, of which BF, 
 being opposite to its supplement to two right angles BGF, is the sine. The angle of refraction 
 is LBK, or its equal FBC, of which BG, being opposite to its alternate and equal angle BFG, is 
 the sine. But BF is to BG in the compound ratio of BF to BA and of BA to BG ; and (BD 
 vanishing) BF is to BA as DF to DA ; and because of the parallel lines CB and GF, the trian- 
 gles AFG and ABC are similar. BA therefore is to AG, as CA to AF; consequently, BA is 
 to BA AG, that is, to BG, as C \ is to CA AF, that is, to CF. 
 
 In like manner the Proposition may be proved of rays passing out of a denser into a rarer 
 medium toward a point between the centre of convexity and the surface, and in all other sup- 
 posable cases. 
 
 COR. Hence the distance of the radiant point, or of the imaginary focus, being given, and 
 also the radius of convexity, and the sines of the angles of incidence and refraction in the two 
 mediums, the focus of any lens may be thus found. 
 
 Let it be required to determine the focal distance of diverging rays passing out of air into glass 
 through a convex surface, arid let the distance of the radiant point be 20, and the radius of con- 
 vexity be 5 ; let the fo^al distance be expressed by x ; then, because by the preceding Proposi- 
 tion the ratio compounded of that which the focal distance bears to the distance of the radiant 
 point, (that is, in this supposition, of je to 20 ;) and of the ratio which the distance of the same 
 radiant point fiom the centre bears to the distance between the centre and the focus (in this 
 case, of 25 to x 5) is equal to the ratio which the sine of the angle of incidence bears to the 
 sine of the angle of refraction, (that is, of 17 to 11) we shall have in the instance before us the 
 
 x ' ^0 ) 
 following proportion, " . > ::]':!!, and compounding them in one, which is done by 
 
 ^0 tXt - O J 
 
 multiplying the two first parts together, we have 25.*? : 20.r 100 : : 17 : 1 1, x *= l %%- 
 
CHAP. II. OF REFRACTION. 167 
 
 SECT. II. 
 
 Of Images produced by Refraction* 
 
 PROP. XXIV. Rays of light flowing from the several points of any 
 object, farther from a convex lens than its principal focus, by passing through 
 the lens, will be made to converge to points corresponding to those from 
 which they proceeded, and will form an image. 
 
 Let ABC be a luminous or illuminated object. From every point, as A, B, C, rajs diverge Plate 6. 
 in all directions. Let some of these rays fall upon a convex lens GHK placed in a hole GK, '*' 
 in the window shutter of a dark room ML, at a greater distance from the object than the prin- 
 cipal focal distance of the lens. BH being the axis, will (by Prop. XIV.) pass through the lens 
 without refraction in the direction BHE. But the collateral rays BG, BK, made equally con- 
 vergent by the lens, will cross the axis at E ; that is, all the rays \vhich come from the point B 
 in tlie object, will be united behind the lens in the focus E. In like manner, among the rays 
 AG, AH, AK, which diverge from the point A, whilst AH the axis (as was shown Prop. XVII. 
 Schol.) may be considered as if it went straight through the lens, the other rays will be made to 
 converge, and will be united in a focus atF; and also, the rays from C will be united in D. 
 The same may he shown concerning every other point in the object. Consequently, there wilt 
 be as many correspondent fuel in the image as there are radiant points in the object; and these 
 foci will be disposed in the same manner with respect to one another as the radiants, and will 
 therefore form an image. The object must be farther from the lens than its principal focus, else 
 the rays from the several radiants would not converge, but either become parallel or diverging, 
 (by Prop. XVII.) whence no image would be formed. 
 
 EXP. 1. Let the rays of the sun pass through a convex lens into a dark room, and fall upon a 
 sheet of white paper placed at the distance of the principal focus from the lens. 
 
 2. The rays of a candle, in a room from which all external light is excluded, passing through 
 a convex lens, will form an image on white paper. 
 
 PROP. XXV. The image produced by rays of light passing through a 
 convex lens is inverted. 
 
 The focus in which the rays, that come from any point A, or B, are united, is, in the axis AIIF, Plate 6. 
 or BHE, of the beam, whether it fall directly or o'uliquely upon the lens. But the axis (by Prop. *' s ' 22> 
 XVII.) is the middle ray of a cone of rays whose base is the surface of the lens, and vertex the 
 radiant point. Every axis, therefore, as AHF, BHE, must pass through H, the middle point of 
 the lens, and consequently must cross one another in that point ; from which it is manifest, that 
 the rays from the lowest point C of the object will become the highest point of the image D ; 
 and that the image will be, with respect to the original object, inverted. 
 
 PROP. XXVI. The image will not be distinct, unless the plane surface, 
 on which it is received, be placed at the distance of the principal focus of the 
 lens. 
 
OF OPTICS. BOOK VI. 
 
 For otherwise the rays which come from a single point in the object, will not have its cor- 
 responding point in the image, but will be spread over a larger surface. 
 
 PROP. XXVIT. As an object approaches a convex lens, its image departs 
 from it, and as the object departs, the image approaches. 
 
 Plate 6. As t ], e object ABC approaches the lens, the several radiants approach it; and consequently 
 
 (by Prop. XIX.) the several foci which form the image FED recede; and the reverse. But the 
 image can never be nearer the lens than its focus of parallel rajs, since this is the place of the 
 image, when the object is infinitely distant. 
 
 PROP. XXVIII. When the object is placed parallel to the image, the 
 diameter of the object is to the diameter of the distinct image, as the dis- 
 tance of the object from the lens, is to the distance of the image from the lens. 
 
 Plate 6. The radiant A (as appears from Prop. XXV.) is represented by its fi-cus in the point F 
 
 where the line AH, produced behind the lens, cuts a plane passing through the focus of parallel 
 rays parallel to the lens. In like manner the radiant C is represented by its corresponding 
 focus in the point D, in which the same plane is cut by the line CH produced. If therefore 
 the distance of the extremities of the object, or its length, be AC, the length of its image will be 
 DF. Since therefore AC is parallel to.DF, the alternate angles ACD, CDF, (El. I. 29) are 
 equal; and also the alternate angles CAP, API) ; whence (K\. VI. 4 ) AC is to FD, as CH to 
 UH, as BH is to EH, that is, the height of the object is to that of the image, as the distance of 
 the object from the lens, to the distance of the image from the lens. Any diameter or line 
 drawn across the object may be proved, in like manner, to have the same ratio to any -corres- 
 ponding diameter or line drawn across the image. 
 
 PROP. XXIX. "When the image appears confused, it is larger than 
 when it is distinct. 
 
 For the rays, in this case, are not received upon the white surface exactly at the distance 
 1 from the lens at which they are brought to a focal point, but either at a distance greater or less ; 
 and in either case the. rays which come from any radiant points at the extremities A and 
 C, will not be collected into points on the plane at F and D, but be spread over a small 
 circular space round these points ; whence the confused image will be larger than the dis- 
 tinct image. 
 
 PROP. XXX. The object and distinct image are similar surfaces. 
 
 Though the side of any object which is toward the lens be not a plane surface, yet the light 
 is reflected from it in the same manner as if the figure of the object were drawn upon the plane 
 surface ol a piece of convas, and diilerently shaded. Therefore the side of the object next to 
 the lens may be considered as a plane figure. And since (by Prop. XXV'Ill.) the height of the 
 object is to that of the picture, as the distance of (be object from the lens, to the distance of the 
 image from the lens, and alsg the breadth of the object in any part, to the breadth of the image 
 in the corresponding part in the ratio of these distances; it follows (Kl. V. 11.) that the height 
 of the object is to the height ot the image, as the breadth of the object in any part is to the 
 
CHAP. II. OF REFRACTION. 169 
 
 breadth of the image in its corresponding part; that is, the object and image are similar 
 surfaces. 
 
 PROP. XXXI. The diameter of an image formed by rays passing from 
 ' a given object through a convex lens, increases as the object approaches the 
 lens, and decreases as the object recedes from the lens. 
 
 The diameter of the image (by Prop. XXVIII ) increases as its distance increases, and 
 decreases as its distance decreases. And (by Prop. XXVII.) the distance of the image increases 
 as the distance of the object decreases, and the reverse. Whence the diameter of the image in- 
 creases as the distance of the object decreases, and decreases as the distance of the object 
 
 increases. 
 
 v 
 
 PROP. XXXII. When the distance of the object is given, the diameter 
 f the image is as the diameter of the object. 
 
 If the object AC remains at the distance BH from the lens, the image (by Prop. XXVII.) Plate 6. 
 will remain at the distance EH ; whence the ratio of BH to EH, and consequently (by Prop. Fl S- 22. 
 XXVIII.) the ratio of the diameter AC to its correspondent diameter DF, is given, or is invari- 
 able. Consequently, if AC increases or decreases, DF must proportionally increase or decrease ; 
 that is, the diameter of the image is directly as the diameter of the object. 
 
 PROP. XXXIII. When the diameter and distance of the object are 
 given, the diameter of the image will be as its distance from the lens. 
 
 If the diameter and distance of the object are given, it is manifest that the diameter of the 
 image cannot be varied without changing the lens. But if, instead of the lens GHK, one less 
 convex, or more convex, be used, the rays will be brought to a focus, and the image (by Prop. 
 XXIV.) will be formed at a greater or less distance from the lens. And since (by Prop. 
 XXVIII.) the distance of the object is always to the distance of the image, as the diameter of the 
 object to the diameter of the image ; the first and third terms remaining invariable, the second 
 and fourth must be increased or diminished proportionally ; that is, the diameter of the image 
 will be directly as its distance from the lens. 
 
 PROP. XXXIV. When the diameter and distance of the object are 
 given, the area of the image is as the square of its distance from the lens. 
 
 Because the surface of the image (by Prop. XXX ) is similar to the surface of the object, 
 whilst the surface of the object remains the same, the image, in every variation of its mag- 
 nitude, must be similar to itself. But the areas of similar surfaces (El. V], 20. Cor. 1.) are 
 as the squares of their homologous sides, that is, as the squares of their heights or breadths. 
 Therefore the area of the image is always as the square of its diameter. And the diameter of 
 the image, when the diameter and distance of the object are given, is (by Prop. XXXIII.) as its 
 distance from the lens ; therefore the square of its diameter, or its area, is as the square of its 
 distance from the lens. 
 
 82 
 
470 
 
 OF OPTICS. 
 
 BOOK VI. 
 
 Plate 6. 
 Fig. 22. 
 
 Plate 6. 
 Fig. 23. 
 
 Plate 6. 
 t'\ S - 22. 
 
 PROP. XXXV. Though the distance of the object from the lens be 
 varied, the image may be preserved distinct without varying the distance of the 
 plane surface which receives it. 
 
 This will be the case, if as much as the image is brought forward by the removal of the ob- 
 ject, it is thrown backward by diminishing the convexity of the lens, and the reverse ; or the 
 image may be preserved distinct without changing the lens, by increasing or diminishing the 
 distance of the lens from the plane surface which receives the image, in the same ratio as the 
 distance of the object from the lens is increased or diminished ; which may be done either by 
 moving the lens or the plane surface. 
 
 PROP. XXXVI. The distances of the object and image, and the diam- 
 eter of the object being given, the diameter of the image will not be altered by 
 changing the area of the lens. 
 
 The height of the image DF is the distance between the two extreme foci F and D ; the 
 former of which is always in the axis AHF of the cone which has A for its vertex, and the latter 
 in the axis CHD of the cone whose vertex is C, which axes cross each other in H. Since there- 
 fore DF, the height of the image, is the distance between these two lines AHF, CHD, where 
 they meet the plane surface, the height of the image will be the same, whether the whole area, 
 GHK is open, or only a small part of it at H. 
 
 PROP. XXXVII. When the object is near the lens, but not so near as 
 the principal focus, in order to make the image distinct, the area of the lens 
 must be small. 
 
 If the object was as near to the lens as the principal focus, or nearer, no image (by 
 Prop. XVII.) could be formed. But let the object A be at a distance from the lens NP, very- 
 little greater than that of the principal focus; then the extreme rays AN, AP, of the cone NAP, 
 diverging more than the rays. AD, AE, if the plane surface, which is to receive the rays, is 
 placed where these rays are collected to a focus, the extreme rays AN, AP, diverging more, will 
 not be collected, and the image on the plane surface will be confused. To prevent this, the 
 extreme rays must be excluded by diminishing the area of the lens, .or of the hole where it 
 is placed. If the radiant A were at a greater distance, this would be unnecessary. Supposing 
 the lens at SR, the extreme rays AN, AP, would pass above or below the lens, and only the 
 middle rays, which are brought to a focus on the plane surface, would pass through the lens. 
 
 PROP. XXXVIII. The brightness of an image, when its distance from 
 the lens is given, is as the area of the lens. 
 
 When the whole area GHK is open, the entire cone of rays AGK passes through the lens 
 from the point A, and is brought to a focus at F; but when the area is diminished to a small 
 surface at H, the greatest part of the cone is excluded, and no rays, but the axis AH and those 
 which are near it, can pass through the lens ; whence it is manifest, that the focal point F must 
 be more illuminated by the rays from A when the area of the lens is GHK, than when the area 
 is diminished. The same may be said of every other cone of rays, and of every other point in 
 
CHAP. II. OF REFRACTION. 171 
 
 the image. Therefore the whole image, although (by Prop. XXXVIJ.) made more distinct by 
 diminishing the area, will be made fainter or less bright. 
 
 PROP. XXXIX. Tbe brightness of the image, when the area of the 
 lens and the distance of the object are given, is inversely as the square of 
 its distance from the lens. 
 
 The area of the lens and distance of the object being given, the number of rays which 
 pass through the lens and form the image, is given. Now the same number of rays spread over 
 a larger surface will not illuminate it so strongly as they would a smaller surface ; that is, the 
 brightness will be inversely as the illuminated area ; and the at ea of the image is (by Prop. 
 XXXIV.) as the square of its distance from the lens ; whence its brightness is inversely as the 
 square of its distance. 
 
 PROP. XL. The heat at the focus of a burning glass, when the area 
 of the glass is given, is inversely as the square of the focal distance. 
 
 The distance of the burning spot, that is, the image of the sun, from the lens, is the focal 
 distance, because the sun's rays are parallel. And because the heat and the brightness at the 
 focus are as the number of rays collected, the heat is as the brightness. But the brightness 
 (by Prop. XXXIX.) is inversely as the square of the distance of the image from the lens ; there- 
 fore the heat is in the same ratio, that is, in this case, inversely as the square of the focal dis- 
 tance of the glass. 
 
 * 
 
 PROP. XLI. The heat at the focus of a burning glass, when the focal 
 distance is given, is as the area of the glass. 
 
 The brightness is (by Prop XXXVIII.) as the area of the lens, and the heat is as the bright- 
 ness ; therefore the heat is also as the area of the lens. 
 
 PROP. XL II. The heat at the focus of a burning glass is to the com- 
 mon heat of the sun, inversely as the area of the focus to the area of the 
 glass. 
 
 The brightness, or the heat, must be inversely as. the space or area over which the rays 
 which cause them are spread, that is, inversely as the area of the focus to the area of the 
 glass. 
 
 SOHOL. This proposition supposes all the rays which fall upon the lens to pass through it to 
 the focus. 
 
 EXP. Most of the preceding Propositions, from Prop. XIX. to XXXIX. may be confirmed, . ^ 
 in a room from which all external light is excluded, by placing a convex lens x, fixed in a frame Fig 9. 
 which moves perpendicularly upon an oblong bar of wood, or table BI), at distances, such as the 
 Propositions require, from a lighted candle Q placed perpendicularly on the same bar of wood, 
 and receiving the images upon white paper q. Upon this bar of wood, on one side of a line over 
 which the convex lens is placed, let a line perpendicular to the last mentioned .line be divided 
 
OF OPTICS. BOOK VI. 
 
 into parts 1 , 2, 5, 4, &c. each equal to the distance of the focus of parallel rays ; and on the 
 other side of the lens let a line be divided in the same manner, and let the first division 
 which is farther from the lens than the focus be subdivided into parts respectively equal to 
 
 1, ! % i, &c. of the distance of the focus of parallel rays ; if a candle be placed over the division 
 
 2, it will form a distinct image on a paper held over the division ; if the candle be over 3, 
 the image will be at |, &c. whence it appears, that the distances of the correspondent foci vary 
 reciprocally. Prop. XL, XLI, XLII, may be confirmed by holding a large double convex 
 lens, or burning glass, in the sun's rays, and receiving the image on white paper, or other 
 substances. 
 
 CHAP. III. 
 
 Of Reflection. 
 
 SECT. I. 
 
 OF THE LAWS OF REFLECTION. 
 DBF. XIX. A Hay of Light, turned back into the same medium in which 
 
 it moved before its return, is said to be reflected. 
 
 DBF. XX. The Angle of Reflection is that, which is contained between 
 the line described by a reflected ray and a liue perpendicular to the reflecting 
 surface at the point of reflection. 
 
 Mate 6. Let AC be the incident ray falling upon the reflecting surface DE, CB will be the reflected 
 
 Fig. 25. j-ajj QC the perpendicular, AGO the angle of incidence, and OCB the angle of reflection. 
 
 PROP. XLIII. The reflection of light from transparent bodies is either 
 partial or total ; the partial reflection happens either at the first or second 
 surface, the total, at the second surface only. 
 
 When a beam of light falls upon a thick piece of polished glass, all the rays will not pass 
 through it; but some of them will be reflected from the first surface of the glass, where the beam 
 enters. At the second surface, some of the rays will also be reflected. These partial reflec- 
 tions happen, whatever is the obliquity of the rays. The total reflection happens when the angle 
 of incidence, or the obliquity, is greater than 41 degrees. All the light which then comes to 
 the second surface will be reflected. See Cor. S. Prop. XIII. 
 
 SCHOL. The rays of light are not reflected by striking upon the solid parts of bodies. 
 At least as many rays are reflected from the second surface when the light passes out of glass 
 into air, as from the first when it passes out of air into glass ; but if the reflection were caused 
 by the striking of the rays upon solid parts of bodies, since glass is denser than air, that is, 
 has more solid parts in a given space, a greater quantity of rays would be reflected from the first 
 surface than from the second. Besides, it seems improbable that, at the second surface, with 
 one degree of obliquity, the rays should meet with nothing but pores or interstices in the air, 
 
CHAP. III. OF REFLECTION. 173 
 
 and pass freely into it, and that with a greater degree of obliquity, it should meet with nothing 
 but solid parts, and be totally reflected. Again, since water is denser than air, if the reflection 
 were owing to the striking of the rays upon the solid body, it might be expected that ihe light 
 would be more perfectly reflected in passing out of glass into water than into air ; whereas, it 
 is found, that if water be placed behind the glass instead of air, rays will not be reflected at the 
 second surface, though their obliquity be greater than 41 degrees ; hence also it is manifest, 
 that the reflection is not owing to the striking of the rays upon the second surface of the glass ; 
 for then it would be the same, whatever were the medium beyond it. 
 
 PROP. XLIV. Reflection is caused by the powers of attraction and 
 repulsion in the reflecting bodies. 
 
 Supposing that bodies attract those rays which are very near them, and repel those a little Plate 6. 
 farther from them ; and calling the space contiguous to the surface of the glass, where the rays F 'S- 24* 
 are attracted, the attracting surface ; and the space next to this, the repelling surface ; the pro- 
 position may be thus proved. 
 
 Let GG, MM, be the repelling surface of a piece of glass, and Ro a ray falling upon it. This 
 ray when it enters the surface will be retarded by the repulsion, and consequently, refracted 
 from the perpendicular at . And this repulsion increasing till the ray gets into the surface of 
 attraction, the ray will be constantly retarded, that is, turned out of its straight course at b, c, d, 
 &c. till it becomes parallel to MM at/, the limit of the repelling surface. And in this situation 
 of the ray, the repelling force, which had retarded, will now constantly accelerate it, and con. 
 sequently it will be continually refracted toward the perpendicular, at g, h, i, &c. till it emerge 
 from the surface at I ; when, the repelling force ceasing to act, the ray will proceed in a right 
 line. Thus the ray, by reflection, is made to describe the curve aft. 
 
 PROP. XLV. The angle of reflection is equal to the angle of incidence, 
 and their complements are also equal. 
 
 In all cases of reflection, the rays (by Prop. XLIV.) describe such a curve at afl. And 
 since they describe one half of this curve by being retarded, and one half by being similarly 
 accelerated, one half will be similar to the other ; whence one half will make the same angle 
 with a perpendicular at/ as the other half makes with it. And the bending of the rays is ma 1*3 
 within so very small a compass, that is, the curve afl is so small, that it may be neglected, as in. 
 fig. 25, where the angle ACO is equal to the angle BCO, and consequently, the angle ACD equal 
 to BCE. 
 
 Exp. 1. Having described a semicircle on a smooth board, and from the circumference let 
 fall a perpendicular bisecting the diameter, on each side of the perpendicular cut off equal parts 
 of the circumference; draw lines from the points in which those equal parts are cut off to the 
 centre ; place three pins perpendicular to the board, one at each point of section in the circum- 
 ference, and one at the centre ; and place the board perpendicular to a plane mirror. Then 
 look along one of the pins in the circumference to that in the centre, and the other pin in the 
 circumference will appear in the same line produced with the first, which shows that the ray 
 which comes from the second pin, is reflected from the mirror at the centre to the eye, in the 
 tame angle in which it fell on the mirror. 
 
 I w 
 
174 OF OPTICS. BOOK Vt. 
 
 2. Let a ray of light passing through a small hole into a dark room, be reflected from a plane 
 mirror ; at equal distances from the point of reflection, the incident and the reflected ray will 
 be at the same height from the surface. 
 
 PROP. XL VI. All reflection is reciprocal. 
 
 Plate 6. If the ray AC, after it has been reflected in the line CB, is turned back again in the direc- 
 
 Vl & 2S - lion CB, it will be reflected (by Prop. XLV.) into AC. Therefore if ACO is the angle of 
 
 incidence, OCB will be the angle of reflection ; and if OCB be the angle of incidence, ACO will 
 
 be the angle of reflection. 
 
 o 
 
 SCHOL. Sir Isaac Newton explains the cause of reflection by supposing, that light, in its pas- 
 sage from the luminous body is disposed to be alternately reflected by, and transmitted through 
 any refracting surface it may meet with ; that these dispositions which he calls Fits of easj 
 reflection and easy transmission, return successively at equal intervals ; and that they are com- 
 municated to it, at its first emission out of the luminous body it proceeds from, probably by 
 some very subtle and elastic substance diffused through the universe, in the following manner. 
 As bodies falling into water, or passing through the air, cause undulations in each, so the rays 
 of light may excite vibrations in this elastic substance. The quickness of these vibrations 
 depending on the elasticity of the medium, (as the quickness of the vibrations in the air, which 
 propagate sound, depend solely on the elasticity of the air, and not upon the quickness ot those 
 in the sounding body) the motion of the particles of it may be quicker than that of the rays ; 
 and therefore when a ray, at the instant it impinges upon any surface, is in that part of a vibra- 
 tion of this elastic substance which conspires with its motion, it may be easily transmitted, and 
 when it is in that part of a vibration which is contrary to its motion, it may be reflected. He 
 farther supposes, that when light falls upon the first surface of a body, none is reflected there, 
 but all that happens to it there is, that every ray that is not in a fit of easy transmission, is there 
 put into one, so that when they come at the other side, (for this elastic substance easily pervad- 
 ing the pores of bodies, is. capable of the same vibrations within the body as without it) the rays 
 of one kind shall be in a fit of easy transmission, and those of another in a fit of easy reflection, 
 according to the thickness of the body, the intervals of the fits being different in rays of a different 
 kind. 
 
 PROP. XLVII. Rays of light reflected from a plane surface, have the 
 same degree of inclination to each other that their respective incident ones 
 have. 
 
 Plate 6. Tne angles of reflection of the rays AC, AI, AK, being equal to that of their respective 
 
 Fig. 26. incident ones, it is evident that each reflected ray will have the same degree of inclination to 
 the surface DE, arid from whence it is reflected, that its incident one has ; but it is here sup- 
 posed that all those portions of surface, from whence the rays are reflected, are situated in the 
 same plane ; consequently, the reflected rays FC, LI, MK, will have the same degree of incli- 
 nation to each other that their incident ones have, from whatever part of the surface they are 
 reflected. 
 
 COR. Parallel rays falling on a reflecting plane surface are reflected parallel. 
 
CHAP. III. OF REFLECTION. 175 
 
 PROP. B. If a plane mirror revolve upon an axis, the angular velocity 
 of the reflected ray is double that of the mirror. 
 
 Let DE be a mirror, AC an incident ray, and CB a reflected raj. If the mirror turn upon ! ate ? ; 
 an axis at C so as to come into the situation FG, then the incident ray AC will be reflected into 
 the line CH. Now the angle BCH, which expresses the angular velocity of the reflected ray 
 BC, is doable of the angle DCF, which is the angular velocity of the mirror. For the angle 
 ACD = BCE = BCH + HCE. (Prop. XLV.) For the same reason ACF = HCG = HCE + 
 (ECG) FCD. Therefore the angle DCF (ACD ACF) = BCH DCF, consequently the 
 angle BCH = 2DCF ; that is, the angular velocity of the reflected ray is double that of the 
 mirror. 
 
 SCHOL. Upon this Proposition depend the construction and use of Hadley's quadrant. 
 
 PROP. XLVIII. Parallel rays reflected from a concave surface are made 
 converging. 
 
 Let AF, CD, EE, represent three parallel rays falling upon the concave surface FB. whose Plate 6. 
 centre is C. To the points F and B draw the lines CF, CB ; these being drawn from the Y>s ' 27 ' 
 centre will be perpendicular to the surface at those points. The incident ray CD also passing 
 through the centre will be perpendicular to the surface, and therefore will return after reflection 
 in the same line ; but the oblique rays AF and EB will be reflected into the lines FM, BM, 
 situated on the contrary side of their respective perpendiculars CF and CB. They will there- 
 fore proceed converging after reflection toward some point, as M, in the line CD. 
 
 PROP. XL1X. Converging rays falling upon a concave surface are made 
 to converge more. 
 
 Let GB, HF, be the incident rays. Now, because these rays have larger angles of incidence Plate 6. 
 than the parallel ones, AF, EB, in the foregoing case, their angles of reflection will also be F 'S- ^ 7 - 
 larger than theirs; they will therefore converge after reflection, suppose in the lines FN, and. 
 BN, having their point of concourse N farther from C than the point M, to which the parallel 
 rays AF and EB converged in the foregoing case. 
 
 PROP. L. Diverging rays, falling upon a concave surface, if they 
 diverge from the focus of parallel rays, become parallel ; if from a point 
 nearer the surface than that focus, will diverge less than hefore reflection ; 
 if from a point between that focus and the centre, will converge after reflec- 
 tion to some point, on the contrary side of the centre, and farther from the 
 centre than the point from which it diverged ; if from a point beyond the 
 centre, the reflected rays will converge to a point on the contrary side, hut 
 nearer to it than the point from which they diverged ; if from the centre, they 
 will be reflected thither again. 
 
 Let the incident diverging rays be MF, MB, proceeding from M, the focus of parallel rays ; Plate 6. 
 
 l''isr. 27. 
 
176 OF OPTIC 9. BOOK VI. 
 
 then as the parallel rays AF and EB were reflected into the lines FM and BM, these rajs will 
 now on the contrary be reflected into them. (By Prop. XL VI.) 
 
 Let them diverge from N, a point nearer to the surface than the focus of parallel rays, they 
 will then be reflected into the diverging lines HF and BG, which the incident rays GB and HF 
 described, which were shown to be reflected into them in the foregoing proposition ; but the 
 degree wherein they diverge, will be less than that wherein they diverged before reflection. 
 
 Let them proceed diverging from X, a point between the focus of parallel rays and the 
 centre, they then make less angles of incidence than the rays MF and MB, which became par- 
 allel by reflection; they will consequently have less angles of reflection, and proceed therefore 
 converging toward some point, as Y ; which point will always fall on the contrary side of the 
 centre, because a reflected ray always falls on the contrary side of the perpendicular with respect 
 to that on which its incident one falls ; and therefore will be farther distant from the centre 
 than X. V 
 
 If the incident ones diverge from Y, they will after reflection converge to X, those which 
 were the incident rays in the former case being the reflected ones in this. 
 
 Lastly, if the incident rays proceed from the centre, they fall in with their respective per- 
 pendiculars, and for that reason are reflected thither again. 
 
 EXP. Place a concave mirror at proper distances from an open orifice or a convex, or a 
 concave lens, through which a beam of solar rays passes, according to the three preceding pro- 
 positions. 
 
 PROP. LI. Parallel rays reflected from a convex surface are made 
 diverging. 
 
 Plate 6. ket -^' ^' ^' ^ e * nree P ara l' e ' rav8 falling upon the convex surface BF, whose centre of 
 
 Fig. 28. convexity is C, and let one of them, GD, be perpendicular to the surface. Through B, D, and 
 F, the points of reflection, draw the lines CV, CG, and CT, which because they pass through the 
 centre will be perpendicular to the surface at those points. The incident ray GD, being per- 
 pendicular to the surface, will return after reflection in the same line, but the oblique ones AB 
 and EF in the lines BK and FL situated on the contrary side their respective perpendiculars 
 BV and FT. They will therefore diverge after reflection, as from some point M in the line GD 
 produced. 
 
 PROP. LIT. Diverging rays reflected from a convex surface are made 
 more diverging. 
 
 ^ et ^^' G ^' k* ^ e ' nc '^ en * ra 7 s> These having larger angles of incidence than the parallel 
 ones AB and EF in the preceding case, their angles of reflection will also be larger than theirs ; 
 they will therefore diverge after reflection, suppose in the lines BP and Ft), as from some point 
 N farther from C than the point M ; and the degree wherein they will diverge, will exceed that 
 wherein they diverged before reflection. 
 
 PROP. LIII. Converging rays reflected from a convex surface, if they 
 tend toward the focus of parallel rays, will become parallel ; if to a point 
 nearer the surface than that focus, will converge less than before reflection j 
 
CHAP. III. OF REFLECTION. 177 
 
 if to a point between that focus and the centre, will diverge as from a point 
 on the contrary side of the centre farther from it than the point toward which 
 they converged ; if to a point beyond the centre, they will diverge as from 
 a point on the contrary side of the centre nearer to it than the point toward 
 which they first converged ; and if toward the centre, they will proceed, on 
 reflection, as from the centre. 
 
 Let the converging rays be KB, LF, tending toward M, the focus of parallel rays ; then, as Plate 6. 
 the parallel rays AB, EF, were reflected into the lines BK and FL, those rays will now on the J ^' 
 contrary be reflected into them. 
 
 Let them converge in the lines PB, OF, tending toward N, a point nearer the surface than 
 the focus of parallel rays, they will then be reflected into the converging lines BG and FG, in 
 which the rays GB, GE, proceeded, which were shown to be reflected into them in the last pro- 
 position immediately foregoing ; but the degree wherein they will converge, will be less than 
 that wherein they converged before reflection. 
 
 Let them converge in the lines RB and SF proceeding toward X, a point between the focus 
 of parallel rays and the centre ; their angles of incidence will then be less than those of the rays 
 KB and LF, which became parallel after reflection; their angles of. reflection will therefore be 
 less, on which account they must necessarily diverge, suppose in the lines BH and FI, from 
 some point, as Y ; which point will fall on the contrary side the centre with respect to X, and 
 will be farther from it than X. 
 
 If the incident rays tend toward Y, the reflected ones will diverge as from X, those which 
 we.re the incident ones in one case, being the reflected ones in the other. 
 
 And lastly, if the incident rays converge toward the centre, they fall in with their respective 
 perpendiculars; on which account they proceed after reflection, as from thence. 
 
 f' 
 
 EXP. Illustrate the three preceding propositions by receiving upon a convex mirror, a solar 
 
 ray passing through an open orifice, or a concave or convex lens. 
 
 PROP. LIV. When rays fall upon a plane surface, if they diverge, the 
 focus of the reflected rays will be at the same distance behind the surface, that 
 the radiant point is before it ; if they converge, it will be at the same dis- 
 tance before the surface, that the imaginary focus of the incident rays is 
 behind it. 
 
 Case 1. Of diverging rays. Let AB, AC, be two diverging rays incident in the plane Plate 6. 
 surface DE, the one perpendicularly, the other obliquely; the perpendicular one AB will be & ^ 
 reflected to A proceeding as from some point in the line AB produced ; the ohlique'one AC 
 will be reflected into some line CF such tiiat the point G, where the line F<^roduced intersects 
 the line AB produced also, shall be at an equal distance from the surface DE with the radiant 
 A. For the perpendicular ('II being drawn, ACH and HCF will be the angles of incidence and 
 reflection, which being equal, their complements ACB and FCE are so too ; but the angle BCG 
 is equal (El. I. 15.) to FCE ; therefore in the triangles ABC and GBC the angles at C are 
 equal, the side BC is common, and the angles at B are also equal v to each other as being right 
 
 23 
 
178 OF OPTICS. BOOK VI. 
 
 ones; therefore (El. I. 26.) the lines AB arid BG, opposite to the equal angles at C, are also 
 equal, and consequently the point G, the focus of the incident rays AB, AC, is at the same 
 distance behind the surface, that the point A is before it. 
 
 Case 2. Of converging rays. Supposing FC and AB to be two converging incident rays, C A 
 and BA will be the reflected ones, (the angles of incidence in the former case being now the 
 angles of reflection, and the reverse) having the point A for their focus ; but this, from what was 
 demonstrated above, is at an equal distance from the reflecting surface with the point G, which 
 in this case is the imaginary focus of the incident rays FC and AB. What is here demonstrated 
 of the ray AC, holds equally of any other, as A I, AK, &c. 
 
 SCHOL. The case of parallel rays incident on a plane surface, is included in this proposition ; 
 for in that case we are to suppose the radiant to be at an infinite distance from the surface, and 
 then by the proposition, the focus of the reflected rays will be so too ; that is, the rays will be 
 parallel after reflection, as they were before. 
 
 PROP. LV. When parallel rays are incident npon a spherical surface, 
 the focus (or imaginary radiant) of the reflected rays will be the middle point 
 between the centre of convexity and the surface. 
 
 Plate 7. Case 1. Of parallel rays falling upon a convex surface. Let AB, DH, represent two parallel 
 
 Fl ff- * rays incident on the convex surface BH, the one perpendicularly, the other obliquely ; and let 
 C be the centre of convexity ; suppose HE to be the reflected ray of the oblique incident one 
 DH proceeding as from F, a point in the line AB produced. Through the point II draw the 
 line CI, which will be perpendicular to the surface at that point, and the angles DHI and IHE, 
 being the angles of incidence and reflection, will be equal. But HCF is equal (El. I. 29.) to 
 DHI, and CHF (El. I. 15.) to IHE; wherefore the triangle CFH is isosceles ; and consequently, 
 the sides CF and FH are equal ; but supposing BH to vanish, FH is equal to FB, and therefore 
 upon this supposition FC and FB are equal ; that is, the imaginary radiant of the reflected rays 
 is the middle point between the centre of convexity and the surface. 
 
 Plate 7. Case 2. Of parallel rays fulling upon a concave surface. Let AB, DH, he two parallel rays 
 
 f 'S- 2> incident, the one perpendicularly, the other obliquely, on the concave surface. BH, whose centre 
 of concavity is C. Let BF and HF be the reflected rays meeting each other in F ; this will be the 
 middle point between B and C. For drawing through C the perpendicular CH, the angles DUG 
 and FHC, being the angles of incidence and reflection, will be-equal, to the former of which the 
 angle HCF is equal, as alternate ; and therefore the triangle CFH is isosceles. Wherefore CF 
 and FH are equal ; but if we suppose BH to vanish, FB and FH are also equal, and therefore 
 CF is equal to FB ; that is, the focal distance of the reflected rays is the middle point between 
 the centre and the surface.* 
 
 It is here observable, that the farther the line DH is taken from AB, the nearer the point P falls to 
 the surface. For the farther the point II recedes from B, the larger the trhngle CFH will become ; and 
 consequently, since it is always an isosceles one, and the base CH, being the radius, is every where of the 
 same length, the equal legs CF and FH will lengthen ; but CF cannot grow longer unless the point F 
 approach toward the surface. And the farther H is removed from B, the faster F approaches to it. This 
 is the reason, that whenever parallel rays are considered, as reflected from a spherical surface, the distance 
 of the oblique ray from the perpendicular ray is taken so small with respect to the focal distance of that 
 surface, that without any physical error, it may be supposed to vanish. Hence it follows, that if a 
 
CHAP. IIT. OF REFLECTION. 179 
 
 SOHOL. The converse of these two cases may be demonstrated in a similar manner, by 
 making the incident rays the reflected ones. 
 
 PROP. LVI. When rays fall upon any spherical surface, if they diverge, 
 the distance of the focus of the reflected rays from the surface is to the dis- 
 tance of the radiant point from the same (or, if they converge, to that of the 
 imaginary focus of the incident rays) as the distance of the focus of the reflect- 
 ed rays from the centre is to the distance of the radiant point (or imaginary 
 focus of the incident rays) from the same. 
 
 Case I. Of diverging rays falling upon a convex surface. Let RB, RI), represent two Plate 7. 
 diverging rays flowing from the point R as from a radiant, and falling the one perpendicularly, fl S- * 
 the other obliquely, on the convex surface BD, whose centre is C. Let DE be the reflected 
 ray of the incident one Rl) ; produce ED to F, and through R draw the line RH parallel to FE, 
 till it meets CD produced in II. Then will the angle RHD be equal (El. I. 29.) to EDFI the 
 angle of reflection, and therefore equal also to RDH, which is the angle of incidence ; wherefore 
 the triangle DRH is isosceles, and consequently DR is equal to RH. Now the lines FD and 
 RH being parallel. (El. VI. 2.) FD is to RH, or its equal RD, as CF to CR ; but BD vanishing, 
 FD and RD differ not from FB and RB ; wherefore FB is to RB also, as CF to CR ; that is, 
 the distance of the imaginary radiant from the surface is to the distance of the real radiant point 
 from the same, as the distance of the imaginary radiant from the centre is to the distance of the 
 real radiant from thence. 
 
 Case 2. Of converging rays falling upon a concave surface. Let KD and CB be the oonverg- Plate 7. 
 ing incident rays, having their imaginary focus in the point R. which was the radiant in the Fl ^' 4 ' 
 foregoing case. Then as RD was in that case reflected into DE, KD will in this be reflected 
 into DF; for, since the angles of incidence in both cases are (El. I. 15.) equal, the angles of 
 reflection will be so too ; so that F will be the focus of the reflected rays ; but it was there 
 demonstrated, that FB is to RB as CF to CR, that is, the distance of the focus from the surface 
 is to the distance (in this case) of the imaginary focus of the incident rays, as the distance of 
 the focus from the centre is to the distance of the imaginary focus of the incident rays from the 
 same. 
 
 Case 3. Of converging rays falling upon a convex surface, and tending to a point between Plate ". 
 the imaginary radiant of parallel rays and the centre. Let BD represent a convex surface, ''*' 
 whose centre is C, and focus of parallel rays is P ; and let AB, KD, be two converging rays 
 incident upon it, and having their imaginary focus at R, a point between P and C. Now because 
 
 number of parallel rays, as AB, CD, EG, &c. fall upon a convex surface, and if BA, DK, the reflected rays 
 of the incident ones AB, CD, proceed as from the point F, those of the incident ones CD, EG, namely, UK, 
 GL, will proceed as from NT, those of the incident ones EG, HI, as from O, &c. because the further the 
 incident ones CD, EG, &c. are from AB, the nearer to the surface are the points F,/,/, in the line BF, from 
 which they proceed after reflection ; so that properly the imaginary radiants of the reflected rays BA, UK, 
 GL, &c. are not in the line AB produced, but in a curve line passing 1 through the points F, N, O, Sec. 
 
 The same is applicable to the case of parallel rays reflected from a concave surface, as expressed by the 
 dotted lines on the other half of the figure, where PQ, RS, TV, are the incident rays ; QF, S/, V/, the 
 reflected ones intersecting each other in the points X, Y, and F ; so that the foci of those rays are not in the 
 line FB, but JB a curve passing through those points. 
 
i80 OF OPTICS. BOOK VI. 
 
 KD tends to a point between the focus of parallel rays and the centre, the reflected ray DE will 
 diverge from some point on the other side the centre, suppose F ; as was shown Prop. LIH. 
 Through D draw the perpendicular CD, and produce it to H, then will KDH and IIDE be the 
 angles of incidence and reflection, which being equal, their vertical ones RDC and CDF will be 
 so too, and therefore the vertex of the triangle RDF is besected by the line DC ; whence (El. VI. 
 3.) FU and DR, or BD vanishing, FB and BR are to each other as FC to CR ; that is, the dis- 
 tance of the imaginary radiant of the reflected rays is to that of the imaginary focus of the 
 incident ones, as the distance of the former from the centre is to the distance of the latter from 
 the same. 
 
 Plate 7. Case 4. Of diverging rays falling upon a concave surface, and proceeding from a point 
 
 tig- 5. between the focus of parallel rays and the centre. Let RB, RD, be the diverging rays incident 
 upon the concave surface BD, having their radiant in the point R, the imaginary focus of the 
 incident rays in the foregoing case. Then as KD was in that case reflected into DE, RD will 
 now be reflected into DF. But it was there demonstrated that FB and RB are to each other as 
 CF to CR ; that is, the distance of the focus is to that of the radiant, as the distance of the 
 former from the centre is to the distance of the latter from the same.* 
 
 SCHOL. I. If the reflected ray be made the incident one, those cases which are respectively 
 the converse of the foregoing may be demonstrated in the same manner. 
 
 SOHOL. 2. Let it be required to find the distance of the imaginary radiant of diverging rays 
 incident upon a convex surface, whose radius of convexity is 5 parts, and the distance of the real 
 radiant from the surface is 20. 
 
 Call the focal distance sought #, then will the distance of the focus from the centre be 5 x t 
 and that of the radiant from the same 25 ; therefore (by Prop. LV1.) we have the following pro- 
 portion, x : 20 : : 5 x : 25, and x = \/. 
 
 If in any case it should happen, that the value of x should be a negative quantity, the focal 
 point must then be taken on the contrary side of the surface. 
 
 SECT. II. 
 
 Of Images produced by Reflection. 
 
 PROP. LVII. When any point of an object is seen by reflected light, it 
 appears in the direction of that line which the ray describes after its last 
 reflection. 
 
 Plate 7* j n t] ie case o f diverging rays falling upon a convex surface, the farther the point D is taken from B, 
 
 the nearer the point F, the imaginary radiant of the reflected rays, approaches to B, while the radiant R 
 remains the same. For it is evident from the curvature of a circle, that the point D may be taken so far 
 from B, that the reflected ray DE shall proceed as from F, G, H, or even from B, or from any point between 
 B and K, and the farther it is taken from B, the faster the point, from which it proceeds, approaches toward 
 H. The like is applicable to any of the other cases of diverging or converging rays incident upon 9 -spheri- 
 cal surface. This is the reason that, when rays are considered as reflected from a spherical surface, the 
 distance of the oblique rays from the perpendicular one is taken so small, that it may be supposed to vanish. 
 From hence it follows, that if a number of diverging rays are incident upon the convex surface BD at the 
 several points B, D, D, &c. they shall not proceed after reflection as from any point in the line RB produced, 
 but as from a curve line passing through the several points F,f,f, &c. The same is applic^le in all the 
 other cases. 
 
CHAP. III. OF REFLECTION. 181 
 
 Since reflection gives the same direction to the rays as if they had originally come from the 
 place from which the reflected rays diverge, an object seen by reflection must appear in that 
 place. The visible image must therefore consist of imaginary radiants diverging from thence. 
 
 PROP. LVIII. In all mirrors, plane or spherical, the place of the ima- 
 ginary radiant, when it is determined, is the intersection of the perpendicular 
 from the radiant to the mirror, and any reflected ray. 
 
 Let D be a radiant in any object DE, and DP a ray from this radiant reflected in the line p |ate 7, 
 FC ; draw DI the perpendicular from the radiant to the mirror, and produce CF, DI, till they * 
 meet in L ; this point will be the imaginary radiant. Because the ray 1)1 falls perpendicularly 
 upon the mirror, it will be reflected back in the same line ID, and therefore will appear to come 
 from some point in DI produced. And since (from Prop. LVII.) all rays which diverge from 
 the same real radiant before reflection, must diverge from the same imaginary radiant after 
 reflection, any other ray from D, as FC, must appear to diverge from the same imaginary radi- 
 ant with the ray DI, that is, from some point in DI ; but the ray FC (by Prop. LVII.) appears 
 after reflection to proceed in the line FC ; it must therefore appear to come from some point in 
 FC, and also from some point in DI, that is, from the point L, in which DI intersects FC. The 
 imaginary radiant of the rays DI, DF, after reflection is therefore L, the intersection of the 
 perpendicular and the reflected ray. 
 
 DBF. XXI. The passage of reflection, is the incident ray added to the 
 reflected ray ; as DF + FC. 
 
 PROP. LTX. In plane mirrors, the distance of the last image from the 
 mirror is equal to the distance of the object from it, and the distance of any 
 point in the last image from the eye is equal to the passage of reflection. 
 
 The distances of the imaginary radiants L, M, behind the mirror, are (by Prop. LIV.) respec- Plate 7. 
 tively equal to the distances of the corresponding real radiants D, E, before the mirror ; therefore F '^' *' 
 the distance of the last image L, M, made up of imaginary radiants between L ant! M, corres- 
 ponding to real radiants in the object DE, is equal to the distance of that object. The distance 
 of L, the highest point of the image, from C, any given place of the eye, is CFL, equal to DFC 
 the passage of reflection, because (by Prop. LIV.) LF is equal to DF. The same may be 
 shown of M, or any other point in the image. 
 
 PROP. LX. In plane mirrors, the image is equal and similar to the 
 object. 
 
 If D be the highest point of the object, the highest point of the image is (by Prop. LVII.) i;i plate 7- 
 the perpendicular DIL ; and if E be the lowest point of the object, the lowest point of the.image Fig. 7. 
 is in the perpendicular EZM. But DIL and EZM are parallel (El. XI. G.) because they are 
 both perpendicular to the plane surface AB. Consequently, the distance between these lines, 
 that is, the heights of the object and image, DE, LM, are equal. In like manner it may be 
 shown, that any diaaieter of the object is equal to its corresponding diameter in the image ; 
 -vhence the object and image are in all respects equal, and consequently similar surfaces. 
 
183 OF OPTICS. BOOK Vl. 
 
 CHAP. IV. 
 
 Of Vision. 
 
 SECT. I. 
 OF THE LAWS OF VISION. 
 
 PROP. LXI. When the rays which come from the several points in 
 any object enter the eye, they will paint an inverted image upon the 
 retina. 
 
 Plate/. Let ABA be a section of an eye. AB, BA, is the tunica sclerotica, a white coat which en- 
 
 Fl - 10> compasses the globe of an eye, except the fore-part between A and A. The fore-part AA is 
 covered by a transparent coat, a little more protuberant than any other part of the eye, called 
 the tunica cornea. In the cavity of the eye is placed a convex lens Ce, consisting of a hard 
 transparent substance, called the crystalline humour. This humour is kept in its place and 
 fixed to the coats by certain ligaments all around it at ee called ligamin'a ciliaria. Under the 
 tunica cornea, and at some little distance from it, is the iris, o, o, which has in it a small 
 orifice, called the pupil of the eye. This iris is tinged with a variety of colours, from which the 
 eye is said to be blue, hazel, black, &c. It consists of muscular fibres, which can contract or 
 dilate the pupil. The remaining part between the cornea and the crystalline humour is filled 
 with a thin transparent fluid, like water, called the aqueous humour. SeN is a white coat, 
 which consists of the fibres of the optic nerve woven together like a net ; this coat is called the 
 retina. Between the sclerotica and the retina is another coat, which is called the choroides. 
 The cavity of the eye, between the crystalline humonr and the retina, is filled with a transpa- 
 rent substance, neither so fluid as the aqueous humour nor so hard as the crystalline, called the 
 vitreous humour. The optic nerve MBB is inserted at N. 
 
 Mr. Harris has given a table of the dimensions of the parts of the human eye, of which the 
 following are the principal particulars. 
 
 In* 
 
 Diameter from the cornea to the choroides ..... -95 
 
 Radius of the cornea - - - - '335 
 
 Distance of the cornea from the first surface of the crystalline - -106 
 
 Radius of the first surface of the crystalline .... -331 
 
 Radius of the back surface of the crystalline ... 25 
 
 Thickness of the crystalline .... . -373 
 
 Fig. 11. Through the pupil the rays which diverge from the several points of any object ABC pass 
 
 into the cavity. The rays emitted from any point of an object beyond the nearest limit of dis- 
 tinct vision, on entering the cornea are rendered converging. The aqueous humour, having 
 about the same degree of convexity as the anterior surface of the cornea, probably does not 
 much change their convergency, but after passing through the pupil they are rendered more and 
 more converging at both surfaces of the crystalline humour, and are thus finally thrown upon a 
 single point of the retina. Consequently at DEF, or somewhete upon the retina, as upon a 
 piece of white paper in a dark room, an inverted image of the object (by Prop. XXV.) will be 
 painted. 
 
CHAP. IV. OF VISION. 183 
 
 SCHOL. The refractive powers of the aqueous and vitreous humours have been found by ex- 
 periinentto be about the same as common water; and that of the crystalline is a little greater 
 that is, the sine of incidence is to that of refraction, out of air into the aqueous humour, as 4 to 
 3, out of the aqueous into the crystalline as 13 to 12, and out of the crystalline into the vitreous 
 as 12 to 13. 
 
 EXP. Take off the sclerotica from the back part of the eye of an ox, or other animal, and 
 place the eye in the hole of the window shutter of a dark room, with its fore-part toward the 
 external objects; a person in the room will, through the transparent coat, see the inverted 
 image painted upon the retina. 
 
 DBF. XXII. The optic axis, is the axis of the crystalline humour con- Plater, 
 tinned to the object at which we look. The axis PO of the crystalline hu- >s ' 
 uiour GPH, continued to the point B, is the optic axis directed toward that 
 point. 
 
 DBF. XXIII. That point of the retina, upon which the optic axis con- 
 tinued back would fall, is called the middle of the retina. If OP be continu- 
 ed back to K, the point E is the middle of the retina. 
 
 PROP. LXII. The images upon the retina are the cause of vision. 
 
 It is found from experience, that when the image upon the retina is bright, the object is 
 clearly seen ; and when the image is faint, the object appears faint; also, that when the image 
 is distinct, the object is seen distinctly; and when the image is confused, the object appears 
 confused. Hence it may be concluded, that these images are the cause of vision. 
 
 COR. It is manifest that a different conformation of the eye, or some parts of the eye, is 
 necessary for distinct vision at different distances. Some think the change is in the length of 
 the eye ; others that it is a change in the figure or position of the crystalline humour, and others 
 that it is a change in the cornea. Any of these changes would produce the effect. 
 
 PROP. LXIII. The point in any object toward which the optic axis is 
 directed, is seen more distinctly than the rest. 
 
 It is known from experience, that when the eye is turned directly toward any part of an ob- 
 ject, that is, when the optic axis is directed toward that point, though the whole object, if it be 
 not very large, will be seen, that part on which we look directly will appear most distinct, and 
 the other portions of the object being drawn on parts of the retina, somewhat nearer to the crys- 
 talline humour than the middle point of the retina will appear a little confused. 
 
 PROP. LXIV. Objects appear erect, although their images on the re- 
 tina are inverted. 
 
 This is known by experience, and is not inconsistent with the explanation above given of 
 vision. For it is not the image in the retina, but the object itself, which we see, and we judge 
 
OF OPTICS. BOOK VI. 
 
 of its relative position, by moving the point of distinct vision along the object, and determine 
 that part to be the highest which requires the eye to be the most lifted up in order to see it 
 distinctly. 
 
 ' PROP. LXV. Au object may appear single, although it is seen by both 
 eyes at once. 
 
 Plate. 7. If both eyes are turned directly to the object C, that is, if the optic axes AC, BC, meet iu 
 J ff- |] 1C object, it will appear single. But if, whilst one eye is turned toward C, the corner of the 
 other is pressed with the finger so as to alter the position of its optic axis, the object will then 
 appear double. For when one eye is turned toward an object, and the other turned a differ- 
 ent way, the same object will be seen by each eye in a different direction ; that is, one eye will 
 see it in one place, and the other in another, from whence it must appear double. But, if both 
 eyes are directed the same way, that is, to the place of the object, though two objects may be 
 said to be seen, yet as both of them are alike, and seen in the same place, they will appear but 
 as one. If whilst both eyes are directed toward C, another object D be placed at some consid- 
 erable distance directly beyond it, the object D will appear double ; for since the eyes see the 
 object D without being turned directly toward it, the place of D is indeterminate; to the right 
 eye it appears on the right hand of C, and to the left eye on the left of C ; that is, being seen in 
 two places, it must appear double. If the sight be directed to the farther object D, the nearer 
 object C will appear double. For, the object C is seen by the right eye in the direction of a 
 line which passes on the left side of D, and by the left eye in the direction of a line which pass- 
 es on the right of D. In both cases, one of the objects appears double, when the eyes are 
 not directly fixed upon it, that is, when the optic axes do not meet in it; and the other object 
 appears single, when the eyes are both directly fixed upon it, that is, when the optic axes meet in 
 the object. 
 
 EXP. 1. View a nearer and a more distant object at the same time, according to the prop- 
 osition. 
 
 2. Let a pasteboard, having a hole in it, be fixed between the eyes of a spectator and two 
 candles, so placed, that when the right eye is shut, the left eye may see only one candle, and 
 when the left eye is shut, the right eve can see only the other, although both candles are visible; 
 if both eyes be fized steadfastly upon the hole, they will appear as one candle placed at the 
 hole. 
 
 PROP. LXVI. There is one part of the retina where no perception of 
 the object is conveyed to the mind by the image formed upon it. 
 
 This is found by experiment. Place t>vo small circles of white paper upon a dark coloured 
 wall at the height of the eye, and at the distance of near two feet from each other. If the spec- 
 tator, at a proper distance, shuts his right eye and looks with the left directly at the paper on 
 his right hand, he will not see the left hand paper, although the objects round it are visible. 
 Hence it is to be inferred, that the rays from the left hand paper fall upon a part of the 
 retina which is insensible. 
 
 SOHOL. It is supposed that this part of the retina is that where the optic nerve is inserted ; 
 and because the coat called the choroides touches the retina m all other parts but is discontin- 
 
CHAP. IV. OF VISION. 185 
 
 ued here, it has been conjectured that the seat of vision is not the optic nerve, but the choroides ; 
 but this point remains undetermined. 
 
 COR. Hence an object cannot become invisible to both eyes at once ; because the image 
 cannot fall upon the optic nerve of each eye at the same time. An object seen with both eyes, 
 is said to appear about T V or -^ brighter than with one eye alone. Harris's Optics, p. 116. 
 
 PROP. LXVII. If the crystalline humour has either too much or too 
 little convexity, the sight will be defective. 
 
 In persons who are short sighted, the humours of the eye are too convex, and bring the rays 
 to a focus before they reach the retina, unless the object be brought near to it; in which case 
 (by Prop. XXVII.) the image is cast farther back. In others, the humours of the eye have so 
 little convexity, that the focal point lies behind the retina ; whence, unless the object is remov- 
 ed to a great distance from the eye, the vision will be indistinct. 
 
 DEF. XXIV. The optic angle in viewing any object is the angle at the Plater, 
 eye subtended by the diameter of the object. The angle AOC, subtended ' s ' 
 by AC, the diameter of the object, is the optic angle. 
 
 PROP. LXVIII. The apparent diameter of any object is proportional 
 to the diameter of the image of that object in the retina. 
 
 To an eye placed at O, the apparent magnitude of the object ABC is that visible extension Plate 7". 
 which lies between A and C. If two lines AO, CO. are drawn from these points crossing one Vl S- 11> 
 another at the eye, they will be the axes of the pencils which come from A and C, and will con- 
 tain between them the diameter AC ; and the points A, C, will be represented on the retina 
 (by Prop. LXI.) at F and D ; consequently, DF will be the diameter of the image ; and this 
 diameter is contained between the two lines AO, CO, produced to the retina. Now it is mani- 
 fest, that the visible extension contained between AO and CO, that is, the apparent diameter of 
 the object, is as the angle AOC ; and that the diameter of the image contained between DO and 
 FO is as the angle DOF. But the angles DOF, AOC, (El. I. 15.) are equal. Therefore the 
 apparent diameter of the object, and the diameter of the image, are each of them proportional to 
 the same angle, and consequently proportional to one another. 
 
 SCHOL. 1. When we speak of an optic angle, it is not meant that we see the point in which 
 the optic axes meet; but, since by experience we learn to judge of such distances as are not 
 very great, by the sensations accompanying the different inclinations of the eye, which are anal- 
 ogous to the optic angle, we express these different inclinations of the eye, by that angle. In 
 like manner, although the eye does not see a pencil of rays, whilst the breadth of the pupil 
 bears a sensible proportion to the distance of the focus from which the rays diverge to the 
 aye, we have sensations from which experience enables us to judge of the place of that focus. 
 So, the magnitude of an image upon the retina being always proportional to that of the visual 
 angle of the object, though that angle is not actually measured by the eye, the difference of 
 .sensations accompanying different magnitudes of the image enable us to distinguish different 
 
 24 
 
186 OF OPTICS. HOOK VI. 
 
 visual angles. Thus it appears, that the use of lines and angles in optics, has its foundation 
 in nature. 
 
 SCHOL. 2. The angle subtended by the least visible object, called by the writers on optics 
 the minimum visibile, cannot be accurately ascertained, as it depends upon the colour of the ob- 
 ject, and the ground upon which it is seen; it depends also upon the eye. Mr. Harris 
 thinks the least angle for any object to be about 40" ; and at a medium not less than two 
 minutes. 
 
 To the generality of eyes the nearest distance of distinct vision is about 7 or 8 inches. 
 Taking 8 inches for that distance, and 2 minutes for the least visible angle, a globular object of 
 less than T ^ part of an inch cannot be seen. Harris's Optics, p. 120 124. 
 
 PROP. LXIX. When the diameter of an object is given, its apparent 
 diameter is inversely as its distance from the eye. 
 
 The apparent diameter of an object (Prop. LXVIII.) is as the diameter of its image upon 
 the retina; and (Prop. XXXI.) the diameter of the image, when the object is given, is inversely 
 as the distance of the object. Therefore the apparent diameter of the object is also inversely as 
 the distance of the object. The same may be proved of any apparent length whatsover. 
 
 COR. T. Hence the apparent diameter of an object may be magnified in any proportion ; 
 for the less the distance of the eye from the object, the greater will be its apparent diameter. 
 But without the help of glasses, an object brought nearer the eye than about five inches, though 
 it appears larger, will at the same time be seen confusedly. 
 
 Plate T- COR. 2. Hence parallel lines, as the sides of a long room, or two rows of trees, as ABC, 
 
 Fjg. 15. DEP ( seen obliquely, appear to converge more and more, as they are farther extended from the 
 eye ; for the apparent magnitude of their perpendicular intervals, as AD, BE, CF, &c. are per- 
 petually diminished. 
 
 COR. 3. An horizontal plane AI seems to ascend. For the visual rays cut a perpendicu- 
 lar DR to the horizon, in points that are higher and higher, or nearer to the horizontal line OG, 
 according as they proceed from points in AI that are more remote from A. 
 
 COR. 4. It is for a like reason that a ceiling DH appears to descend, and that faster than 
 the floor ascends, as the distance of the eye of the spectator to the ceiling is greater than the 
 distance of the eye from the floor. 
 
 PROP. LXX. The apparent diameter of an object, whose distance is 
 given, is directly as its real diameter. 
 
 The apparent diameter of an object (by Prop. LXVIII.) is as the diameter of its image ; and 
 the diameter of the image (by Prop. XXXIII.) when the distance of the object is given, is as 
 the diameter of the object. Therefore the apparent diameter of an object, whose distance is giv- 
 en, is as its real diameter. 
 
 PROP. LXXI. The apparent diameters of different objects at different 
 distances from the eye will be equal, if their real diameters are as their 
 distances. 
 
CHAP. IV. OF VISION. 187 
 
 For (by Prop. XXXIV.) the diameters of their respective images upon the retina, will be 
 equal ; and their apparent diameters (by Prop. LXVIII.) are as the diameters of their images. 
 
 PROP. LXXII. The apparent length of an object, seen obliquely, is as 
 the apparent length of a subtense of the optic angle perpendicular to the 
 optic axis. 
 
 If DF is an object seen obliquely, and DG an object seen directly, that is, if DF is oblique, Plate r. 
 and DG perpendicular to the optic axis OR, then supposing them to subtend the same angle F '&" 3 ' 
 DOF, their images upon the retina (Def. XXIV.) will be equal, whence (by Prop. LXVIII.) 
 their apparent diameters will be equal. Consequently, the greater the subtense GD is, the 
 greater will be the apparent length of the object DF ; and the reverse. 
 
 COR. Hence an object appears shortened by being seen obliquely. 
 
 PROP. LXXI1I. "When equal objects in the same right line are seen 
 obliquely, their apparent lengths are inversely as the squares of tlieir distan- 
 ces from the eye. 
 
 Let the eye be at O ; and in the line BC at different distances from the eye take equal Plate 7. 
 spaces DF, df. The apparent length of DF (by Prop. LXXII.) is proportional to the apparent tl %' ** 
 length of GD, the subtense of the optic angle DOF, perpendicular to the optic axis OR. In like 
 manner, the apparent length of df\s proportional to that of gd, the subtense of dOf. Now GD, 
 gd, are subtenses also of the angles GFD, gfd ; and as the side Of is to the side OF, so is the 
 sine of the angle OF/, that is, of its supplement OFB, to the sine of the angle O/F, or O/B. 
 Hence, since small angles are to one another nearly as their sines, if these are small angles, Of 
 will be to OF, as the angle OFB to the angle O/B ; that is, O/ will be to OF, as the subtense GD 
 to the subtense gd, or GD is to gd inversely as OF to Of ; that is, the subtenses of the optic 
 angles, and consequently, from what has been shown, the apparent diameters DF, df, are 
 inversely as their distances from the eye. This proportion arises from the different degrees of 
 obliquity at which the eye sees the equal spaces DF, df. But if their obliquities with respect 
 to the eye were the same, the apparent length of DF (by Prop. LXIX,) would be to that of df 
 inversely as their distances. Since then the apparent lengths of DF, df, are inversely as 
 their distances on account of their different obliquities, and also inversely as their distances on 
 account of their different distances ; on both accounts taken together, they are in the ratio 
 compounded of the inverse lutio of their distances, and the same, that is, inversely as the 
 squares of their distances. 
 
 PROP. LXXIV. The apparent diameter of an object is not changed by 
 contracting or dilating the pupil. 
 
 For when the distance is given, the diameter of the image (by Prop. XXXVI. Schol. 2.) re- 
 mains the same, whatever be the area of the pupil, and consequently (by Prop. XXXIII.) the 
 apparent diameter of the object. 
 
 PROP. LXXV. An object appears larger when it is seen confusedly, 
 than when it is seen distinctly. 
 
188 OF OPTICS. BOOK VI. 
 
 For the confused image (by Prop. XXIX.) is larger than the distinct image, and consequent- 
 ly (by Prop. LXVIII.) the apparent magnitude of the object is greater when it is seen confusedly 
 than when it is seen distinctly. 
 
 COK. Hence objects appear magnified when seen through a mist; the drops of which refract 
 the rays so differently, that they cannot be collected into one focus. 
 
 PROP. LXXVI. A spectator in motion sees an object at rest, moving 
 the contrary way. 
 
 Plate 7. If while an object at P is at rest, the eye be carried parallel to PQ in the direction from Q 
 
 Fig. 14. to war j p } its image on the retina will move from p to q ; the same effect will be produced, if 
 
 the object moves from P toward Q ; and if the velocity of the object and the eye, in each case, 
 
 be the same, the apparent velocity will be the same also. 
 
 COR. 1. An object moving along a line PK will appear at rest to a spectator moving along the 
 line QG, parallel to PK ; if the motion of the object be quicker or slower than that of the spec- 
 tator, it will have an apparent motion direct or retrograde ; if the two motions are in contrary 
 directions, the apparent motion of the object will vary with the real motion of the spectator. 
 
 Cor. 2. If the earth be supposed to move round its axis from west to east, while the heavenly 
 bodies are at rest, they appear to us to move the contrary way, there being nothing in this 
 case to indicate to us our own motions. And therefore no argument drawn from the apparent 
 diurnal motions of the stars and planets can be of any support to either the Ptolemaic or Pytha- 
 gorean systems. 
 
 SCHOL. A person riding, or walking slowly, though he perceives the change of situation of 
 adjacent bodies, yet being sensible of his own motion, and having time to reflect in the intervals 
 of these apparent changes, those bodies appear to keep their places. But if he runs or rides 
 very swiftly, he cannot help fancying, that the bodies, which he is looking at, are moving toward 
 him. The deception is still stronger when he sits at his ease in a swift sailing vessel. 
 
 PROP. LXXVII. The same degree of velocity appears greatest, when 
 the motion is in a line perpendicular to the optic axis ; and when the motion 
 is in other directions, the apparent velocity will be as the cosine of the angle 
 of inclination to the said perpendicular. 
 
 Plate 7. If two bodies set out at the same time from P, the one moving alone the line PQ, perpen- 
 
 dicular to the optic axis Q</, and the other along the line PS, oblique to it, and if their velocities 
 be such, as to pass over the lines PQ, PS, in the same time, it is manifest, that their apparent 
 velocities will be the same ; for the images of each will pass over the same space pq, on the 
 retina, in the same time. The real velocities being, in this case, as PQ to PS, it is manifest, 
 that when the velocities are equal, the apparent velocity of the body which moves in PQ is to 
 that of the body moving in PS, as PS- to PQ ; that is, as radius to tha cosine of the angle QPS ; 
 but PS is always greater than PQ ; whence the proposition is manifest. 
 
 PROP. LXXVIIT. If objects at different distances from the eye move 
 in parallel lines, nearly at right angles to the optic axis, and if their velocities 
 
CHAP. IV. OF VISION. 189 
 
 are proportional to these distances, their apparent velocities will be equal ; 
 and if their real velocities are equal, their apparent velocities will be recipro- 
 cally as their distances from the eye. 
 
 Let an object move from Q to P, in the same time (Tiaf another moves from G to H, their PUte 7. 
 real velocities are as QP to GH, that is, (El. VI. 2.) as QO to GO, the distances from the eye ; f ' s ' 14 ' 
 and their apparent velocities will be equal ; for the space qp upon the retina will be passed over 
 in the same by the image of each. If the velocities of the objects G, Q, be equal, the object G 
 will arrive at K, and its image describe the space qk upon the retina, in the same time that the- 
 image of the object Q describes the space qp ; whence the apparent velocities of these two 
 objects are as qk and qp, or as GK (or QP) and Gil ; that is, (El. VI. 2.) the apparent velocif v 
 of the object G is to that of Q, as QO to OG. 
 
 SOHOL. It is here supposed, that the spectator makes no allowance for the different distances. 
 
 PROP. LXXIX. The apparent velocities of bodies moving in parallel 
 lines at different distances from the eye, are directly as the real velocities, 
 and reciprocally as the distances. 
 
 Let two bodies move from Q, G, in the parallel lines QP, GH ; let the velocity of the object ^! ate 7 - 
 Q be called V, and that of G, v ; and let their apparent velocities be called A, a. If the two 
 objects be conceived to move in the same line GK, whatever be their velocities, V : v : : A :a ; 
 and supposing the velocity of the object Q, to be the same in QP as before in GK., A in QP : A in 
 GK : : GO : QO, by Prop. LXXVIII. and A in GK : a : : V : v; whence, compounding these ratios, 
 
 A: : :GO x V:QO x t>, that is, : : ^y^p 
 
 SCHOL. 1. We judge of the distance of any object by the visible length of the plane which 
 lies between the eye and the object. When this method fails us, we compare the known magni- 
 tude of the object, with its present apparent magnitude; or we compare the degrees of distinct- 
 ness with which we see the several parts of an object ; or we observe whether the change of the 
 apparent place of an object when viewed from different stations, or its parallax, be great or 
 small, this change being always in proportion to the distance of the object. On this principle, 
 we may judge of the distance of a near object by observing the change which is made in its 
 apparent situation, upon viewing it successively, with each eye singly. Or, since it is the dif- 
 ference of the apparent place of an object, as viewed by each eye separately, which makes an 
 object to be seen double unless we turn both eyes directly toward it, and since in doing this, 
 where the distance is very small, we turn the eyes very much toward each other, and less at a 
 greater distance; the different sensations accompanying the different degrees in which the eyes 
 are turned toward each other, afford, hy habit, a rule for judging of the distance of objects. 
 
 SOHOL. 2. In objects placed at such distances as we are used to, and can readily allow for, 
 we know by experience how much an increase of distance will diminish their apparent magni- 
 tude, and therefore instantly conceive their real magnitude, and neglecting the apparent, suppose 
 them of the size they would appear if they were less remote; but this can only be done, where 
 we are well acquainted with the real magnitude of the object ; in all other cases, we judge of 
 magnitudes by the angle which the object subtends at the known, or supposed distance ; that is, 
 we infer the leal magnitude from the apparent magnitude in comparison with the distance of the 
 object. 
 
190 OF OPTICS. BOOK VI. 
 
 SECT. II. 
 Of Vision as affected by Refraction. 
 
 PROP. LXXX. When any small object is seen through a refracting 
 medium, it appears in the direction of that line which the rays describe after 
 their last refraction. 
 
 Plate 7. The ray DE from any small object D, in passing through a glass prism, the end of which is 
 
 Fig. 16. ACB, will be refracted toward a perpendicular, and will describe the line EF ; and when it 
 goes out of the prism, it will be refracted from a perpendicular, into the line FG, which is its 
 direction after its last refraction ; and the object D will be seen in this direction at L instead 
 of D. For the image in the retina will be in the place in which it would have been, if the naked 
 eye had been looking at an object really placed at L, the last direction of the rays. 
 
 Fig. 17. COR. Hence an object seen through a glass cut into different surfaces inclined to one another, 
 
 will appear at one view in many different places. The object A seen from the point F through 
 the glass CEDB, will appear at H, A, G; the last direction of the rays AC, AD, AB, after 
 refraction. 
 
 Exp. View any object through a multiplying glass. 
 
 SOHOL. If a hair be placed across a small hole made in a thin board, and an eye situated in 
 the dark, look through the hole at a number of caudles, the hair will appear multiplied : for a 
 shadow of the hair is cast upon the eye by each of the candles. 
 
 PROP. LXXXI. An object seen through a denser medium terminated 
 by plain and parallel surfaces, appears nearer, brighter, and larger, than with 
 the naked eye. 
 
 Plate 7. Let AB represent the object, CDEF the medium, and GH the pupil of an eye. Let RK, 
 
 Fig. 18. JIL ? t^ ^ wo ra y g proceeding from the point R, and entering the denser medium at K and L ; 
 these rays will here by refraction be made to diverge less (by Prop. XVII.) and to proceed 
 afterward, suppose in the lines Ka, L6; at a and b, where th^y pass out of the denser medium, 
 they will be as much refracted the contrary way, proceeding in the lines oc, bd, parallel to their 
 first directions ; (by Prop. XI.) produce these lines back, till they meet in e, this will be the 
 apparent place of the point R, and it is evident from the figure, that it must he nearer the eye 
 than that point ; and because the same is true of all other pencils flowing from the object AB, 
 the whole will be seen in the situation fg, nearer to the eye^than the line AB. As the rays 
 RK, R!J, would not have entered the eye, but have passed by it in the directions Kr, Lt, had 
 they not been refracted in passing through the medium, the object appears brighter. The rays 
 Ah, Bi, will be refracted at h and i into the less converging lines hk, il, and at the other surface 
 into ArM, IM, parallel to Ah and Bi produced, (by Prop. XI.) so that the extremities of the 
 object will appear in the lines MA.-, Ml, produced, namely, in / and g, and under as large an 
 angle /Mg, as the angle AgB, under which an eye at q would have seen it, had there been no 
 medium interposed to refract the rays ; and therefore it appears larger to the eye at GH, beins 
 
CHAP. IV. OF VISION. 191 
 
 seen through the interposed medium, than otherwise it would have done. But it is here to be 
 observed, that the nearer the point e appears to the eye on account of the refraction of the ravs 
 RK, RL, the shorter is the image fg, because it is terminated by the lines M/ and Mg, upon 
 which account the object is made to appear less; and therefore the apparent magnitude of an 
 object is not much augmented by being seen through a medium of this form. 
 
 Farther, it is apparent from the figure, that the effect of a medium of this form depends 
 wholly upon its thickness ; for the distance between the lines Rr and PC, and consequently the 
 distance between the points e and R depends on the length ot the line Krj; again, the distance 
 between the lines AM and/M, depends on the length of the line hk, but both Krt and At depend 
 on the distance between the surfaces CB and DF, and therefore the effect of this medium 
 depends upon its thickness. 
 
 4 
 
 PROP. LXXXII. In viewing objects through a convex or concave lens, 
 the object itself is not seen, but its last image, consisting of all the imaginary 
 radiants from which Uie rays appear to diverge after the last refraction. 
 
 If AC be an object nearer the convex lens GIL than its principal focus,, the ravs which Plater, 
 diverge from any point B in this object, will, in passing through the lens (by Prop. XVII.) be F '^' 8t 
 made to diverge less, and the imaginary radiant (by Prop. VI.) will be more distant than the 
 real one. Hence the rays BG, BL, after refraction will appear to diverge from the radiant E, 
 farther from the lens than the real radiant B. The same happening to the rays from every 
 other point of the object, there will be in DEF as many imaginary radiants as there are real radi- 
 ants in the object ABC, which, taken together, will compose the last image. And since all the 
 rays fall upon the eye as if they had diverged from this last image, the eye will be affected by 
 the object ABC in the same manner in which it would be affected without the lens, by an object 
 in all respects like L>EF, that is, the eye perceives the last image. 
 
 PROP. LXXXIII. An object seen through a convex or concave lens 
 will appear erect, if the object and its image are on the same side of the lens, 
 but inverted if they are on contrary sides. 
 
 It appears from the last Prop, that all the rays wlijch diverge from B before refraction, will P ] at -, 
 appear to diverge from E after refraction ; and the like may be said of any other points, A and Fig. 8, 9. 
 D or C and F. Now AI is (he axis of the beam which comes from A, and therefore, with the 
 rest of the rays of the beam after refraction, will appear to diverge from the point D in the same 
 right line with A. The same may be shown of FCI. Now these right lines only cross one 
 another at the lens. Consequently, the highest point both of the object and image is in DI, the 
 upper side of the angle DIP, and the lowest points of both in FI, the lower side of the same 
 angle ; that is, the image, which is visible, is erect, or in the same position with the object. 
 
 If the object ABC is more remote than the principal focus of the convex lens E, there will 
 be (by Prop. XXIV.) a distinct image formed on the other side of the lens. If the rays thus 
 collected are not received upon a plane surface, they will proceed straight forward ; those which 
 'had converged diverging from the focus ; whence an inverted image will be presented to an eye 
 placed beyond the focus. In the same manner this, and the preceding proposition, way be 
 proved concerning an object seen through a concave lens. 
 
192 OF OPTICS. BOOK VI. 
 
 EXP. 1. Observe the image of a candle whose rays have passed through a convex lens, and 
 are received at the focus on a white surface, whilst the object is on the same, or on the contrary 
 side of the lens. 
 
 2. An inverted image will be produced without a lens, by solar rays passing through a very 
 si/.all hole into a darkened room ; and if the edge of a knife be applied to one side of the hole, and 
 moved slowly over it, the parts of the image situated opposite to the covered side will be first 
 concealed ; from whence it is manifest, that the rays cross one another in passing through the 
 hole. 
 
 PROP. LXXX1V. The apparent magnitude of an object seen through 
 a lens placed close to the eye, or to the object, is equal to its apparent magni- 
 tude when seen without the lens. 
 
 Plate 7. If the eye be placed close to the lens at I, the diameter of the object of refracted vision DF, 
 
 is to the diameter of the object of plane vision AC (by Prop. XXVIII.) as El to IB, that is, as 
 their respective distances from the vertex of the lens. Therefore (as appears from Prop. LX1X.) 
 their apparent diameters, when seen from I, are equal. 
 
 If the lens is placed close to the object, the real radiants touching the lens, the imaginary- 
 radiants, that is, the last image, will also touch the lens; whence their diameters, or apparent 
 magnitudes, will be equal. 
 
 PROP. LXXXV. If an object seen through a convex lens is nearer to 
 the lens than its principal focus, it will appear brighter than to the naked eye, 
 distinct, and in an erect position. 
 
 In this case, the rays which come from any point (by Prop. XVII.) will be brought nearer 
 by refraction, and consequently a greater number will enter the eye, than if there had been no 
 refraction ; whence it is manifest, that the object will appear brighter. Because the rays of 
 each pencil diverge after refraction, but less than before, they come to the eye, as they would if 
 the object were at a moderate distance and no lens were used, and therefore will be seen dis- 
 tinctly. And because the refracted rays (by Prop. XVII.) diverge less than the incident rays, 
 that is, (by Prop. VI.) the imaginary radiants are more remote than the real ones, the last image, 
 as DEF, which is formed by these imaginary radiants, is farther from the lens than the object, and 
 on the same side of the lens ; and consequently, since the extreme axes DAI, FC1, only cross 
 one another at the lens, the image will be in the same position with the object and appear erect. 
 
 PROP. LXXXVI. If an object seen through a convex lens is farther 
 from the lens than its principal focus, the object will appear brighter than to 
 the naked eye, confused, and in an erect position. 
 
 If the eye be placed between the lens and the distinct image, whilst the eye is nearer the 
 lens than the place of the image, the rays being made convergent by the lens, will be closer 
 together, and therefore a greater number of them will enter the eye, and <he object will appear 
 brighter, than if it had been seen without a lens ; because the rays come to the eye in a converg- 
 ing state, which from one and the same point they do not in plane vision,, they will give % con- 
 
CHAP. IV. OF VISION. 198 
 
 fused image. And because no image is formed till the rajs come to the retina, the object will 
 
 appear erect. 
 
 . PROP. LXXXVII. If an object seen through a convex lens is in the 
 principal focus, it will appear brighter than to the naked eye, and erect. 
 
 The rays of each particular beam, becoming in this case (by Prop. XVII.) parallel after refrac- 
 tion, are brought nearer together than if there had been no lens ; consequently, more rays will 
 enter the eye from every point, and the object will appear brighter ; and no image being formed 
 before the rays come to the retina, the object will appear erect. 
 
 PROP. D. A minute object, when seen through a lens of very small 
 principal focal length, appears magnified and distinct, if the object be placed 
 in the principal focus. 
 
 The angle under which the object appears, will be to that which it subtends, when seen by 
 the naked eye ; as the distance at which it is viewed by the naked eye distinctly, is to the prin- 
 cipal focal length of the lens. 
 
 If the principal focal length of the lens be ^ of an inch, and the distance at which the eye 
 can see distinctly be eight inches, it follows that the lens will magnify 240 times in diameter. 
 
 PROP. LXXXVIII. When an object, seen through a convex lens, is 
 nearer than the principal focus, it is magnified, unless the lens touches the eye 
 or the object; and as the eye departs from the leas, its apparent magnitude 
 will decrease. 
 
 If the object ABC continues in the same place, or does not change its distance BI from the Plate 7, 
 lens, the last image DF will (by Prop. XX VI I.) remain at the same distance El ; therefore the Fl &' 8- 
 real diameter of DF (by Prop. XXXIII.) will be invariable, wherever the eye is placed. If the 
 eye be at the vertex of the lens I, the apparent diameters of the last image DEF, seen through 
 the lens, and that of the object ABC, seen with the naked eye, are manifestly the same. Both 
 these apparent diameters decrease as the eye recedes from the lens toward O ; but the apparent 
 diameter of the last image decreases^ the inverse ratio of OE to IE, and that of the object in 
 the inverse ratio of OB to IB. But OE has a less ratio to IE than OB to IB ; for IE and 1 B 
 are unequal quantities, of which IE, the distance of the image, is always (by Prop. XVII. and 
 VI ) the greater, which are equally increased, but not proportionally ; therefore the apparent 
 diameter of the image decreases slower than that of the object, as the eye recedes from the lens. 
 Consequently, when the eye is at any distance from the lens, the last image, or the object of re- 
 fracted vision, will appear greater than the object seen by the naked eye. As the eye departs 
 from the lens, the apparent magnitude of the object, from what has been said, must continually 
 decrease, till at an infinite distance it vanishes. 
 
 ,. PROP. LXXXIX. If an object seen through a convex lens is more 
 remote than the principal focus, and the eye on the other side of the lens 
 is nearer than the place of the image, the object appears magnified, aud its 
 
 25 
 
191 OF OPTICS. BOOK VI. 
 
 apparent magnitude will be inversely as the distance of the eye from the 
 image. 
 
 Plate/. Suppose the eye at the side of the lens GL; it might successively see both the object and 
 
 the image without looking through the lens ; and in this situation (by Prop. XXVIII.) the real 
 diameter of the object is to that of the image, as their respective distances from the lens or the 
 eye; consequently (by Prop. XXXIV.) their apparent diameters will be equal. Next, suppose 
 the eye close to the image at F, E, or D, the apparent diameter of the image would manifestly 
 be infinite. Also in this situation of the eye, the apparent diameter of the object would be infi- 
 nite ; for, if the eye be at F, the rays from the point C are the only rays collected into the 
 eye s which appear diffused over the whole surface, and would do so if the lens were ever so 
 large; and the same would be true of the points B or A, if the eye was at E or D ; that is, the 
 apparent diameter of the object seen through the lens is infinite. Since then the object and the 
 image appear equal when the eye is close to the lens, and that both appear infinite when the 
 .eye is close to the image ; they must have increased equally as the eye was moving from the 
 lens to the image, and their apparent diameters must always have been equal. Hence, the 
 object in every station of the eye when it dees not touch the lens is magnified. And because 
 the apparent diameter of the object seen through the lens is every where equal to that of the 
 image, and that of the image (by Prop. XXXI.) inversely as the distance of the eye from the 
 picture, the apparent diameter of the object seen through the lens, is inversely as the distance 
 of the eye from the image. 
 
 PROP. XC. If an object seen through a convex lens is placed in the 
 principal focus, its apparent magnitude will not be altered by withdrawing 
 the eye from the lens. 
 
 Since in this case the rays from the object come parallel to the eye, both the imaginary ra- 
 diants and the image (by Prop. XVH.) are infinitely distant. Therefore the apparent magni- 
 tude of the object cannot be diminished by receding from the imaginary radiants, nor increased 
 by approaching to the image, but will always remain the same. 
 
 PROP. XCI. If a convex lens is moved whilst the eye and object re- 
 main fixed, the apparent magnitude of the object .will increase, till the lens 
 is at the middle point between them, after which it will decrease till the lens 
 reaches the object ; provided the eye is never farther from the lens than the 
 place of the image. 
 
 AVhen the lens is at either extreme (by Prop. LXXX1V.) the object is not magnified ; but 
 between the extremes (by Prop. LXXXVI1I, and LXXXIX.) it is magnified ; therefore when it 
 is equally distant from the two extremes, it is most magnified, and must increase in its apparent 
 magnitude as the lens moves from the eye toward the middle station, and decrease, as it 
 moves from that middle station toward the object * 
 
 PROP. XCII. The apparent magnitude of an object, seen through a 
 concave lens, decreases as the eye, or the object, departs from the lens. 
 
 * The reasoning here appears not perfectly conclusive. 
 
r 
 
 CHAP. IV. OF VISION. 195 
 
 If the eye touches the vertex of the lens I, the apparent diameters of the object and the last Plater. 
 image are equal. As the eye recedes from the lens, its distance both from the object ABC and l 'S-- 
 last image DEF increases, and consequeutly, the apparent magnitude of both decreses. But 
 the distance IE from the last image increases faster than the distance IB from the object, as 
 was shown in Prop. LXXXV1II. Therefore (by Prop. ^^X.) the apparent diameter of the 
 last image, or the object of refracted vision, is diminished as the eye recedes from the lens, 
 more than that of the object seen by the naked eye. Again, as the object departs from the lensr, 
 the image departs with it ; whence its visible diameter decreases.* 
 
 PROP. XCIII. When the eye and object are fixed, if a concave lens be 
 moved from the eye, the apparent magnitude of the object will decrease till it 
 reaches the middle point between them, and increase as it moves on toward 
 the object. 
 
 When the lens is at each extreme, the apparent magnitude of the object seen through the 
 lens (by Prop. LXXXIV.) is the same as when seen with the naked eye. In all other stations 
 of the lens, the object appears diminished; therefore it must appear most of all diminished 
 when the lens is in the middle station, and it must decrease whilst it is approaching to that sta- 
 tion, and increase whilst it is departing from thence toward the object. 
 
 EXP. View a candle through a convex or concave lens, in the manner described Prop. 
 XL1I. varying the position of the object, or lens, according to the preceding Propositions, from 
 Prop. LXXXIV. 
 
 PROP. XCIV. Convex lenses assist the sight of those persons whose 
 eyes are not sufficiently convex, and concave lenses, that of those whose eyea 
 are too convex. 
 
 For convex lenses enable the former to bring the rays from objects to a focus nearer to the 
 crystalline than can be done by their eyes ; and concave lenses enable the latter to bring the 
 rays to a focus at a greater distance ; and thus to produce a distinct image upon the retina. 
 
 SCHOL. Were there no other use of the science of dioptrics, says Mr. Molyneux, than that of 
 spectacles, the advantage that mankind receive thereby is inferior to no other benefit, not abso- 
 lutely requisite to the support of life. For as the sight is the most noble and extensive of all 
 our senses; as we make the most frequent use of our eyes in all the actions and concerns of 
 life, surely that instrument which relieves the eyes when decayed, and supplies their defects, 
 must be estimated as the greatest of all advantages. Forlorn indeed must have been the situa- 
 tion of many young, and almost all old people, before this admirable invention. The same 
 author concludes, that spectacles were first used about the end of the 13th century; and he as- 
 cribes the invention to Friar Bacon. 
 
 * The reasoning here appears not perfectly conclusive. 
 
496 OF OPTICS. BOOK VI. 
 
 SECT. in. 
 
 Of Vision as affected by Reflection. 
 
 PROP. XCV. If a plane mirror and the object seen in it are both perpen- 
 dicular to the horizon, the object appears erect. 
 
 Plate 7. The object DE, and the mirror AB, being both perpendicular to the horizon, the lines DI, 
 
 F 'g- 7. EZ, in which (by Prop. LVII.) the highest and the lowest points of the object appear, being 
 both perpendicular to the surface AB, are parallel to each other, and do not meet. Therefore 
 the line 1)1, which is highest at the object, is also highest at the image, and EZ will be lowest 
 at both ; therefore the image is not inverted with respect to the object ; and each point of the 
 ' image LM (by Prop. LIV.) is equally distant from the surface of the mirror with its corres- 
 
 ponding point in the object DE ; therefore LM, DE, are parallel, (El. I. 50.) and since the ob- 
 ject is erect, the image will be so too. 
 
 PROP. XCVI. When the ohject is parallel to a plane mirror, the length 
 or breadth of that part of the mirror, upon which the image appears, is to 
 the length or breadth of the object, as any reflected ray is to the passage 
 of reflection. 
 
 Plate/. If the object DE is parallel to the mirror AB, and the image LM is seen by the eye at C, 
 
 fl S- 7- then FN, the length of that part of the mirror which is taken up by the image, subtends the angle 
 LCM, under which the image appears. For since all the visible length of the image is manifestly 
 included within the angle LCM, there cannot be more of the mirror taken up by that visible 
 length, than is included within the same angle. Now the length of tiie image LM is equal (by 
 Prop. LX.) to the length of the object DE. And (El. VI. 2.) FN is to LM, as FC to CL, or 
 (bv Prop. LIX ) CFD ; that is, the length of that part of the mirror which is taken up by the 
 image is to the length of the image, or (by Prop. LX ) the length of the object, as any reflected 
 ray is to the passage of reflection of that ray. In the same manner it may be shown, that the 
 breadth of that part of the mirror taken up by the image, is to the breadth of the object in the 
 same ratio. 
 
 COR. Hence, in plane mirrors the object, and the part of the surface on which it appears, 
 are similar. 
 
 PROP. XCV1I. If, at a certain distance from the mirror, the whole ob- 
 ject cannot be seen by reflection, the whole will become visible either by 
 bringing the eye nearer to the mirror, or removing the object farther from 
 it. 
 
 For (by Prop. XCVI.) the less the ratio of the reflected ray is to the passage of reflection, so 
 
 much the less will be the ratio of the length of that part of the mirror on which the whole object 
 
 Plate 7. will appear, to the length of the object. If therefore the reflected ray CF decreases by bringing 
 
 fl S'7- the eye nearer to the mirror, since it is diminished faster than the greater quantity DFC, the 
 
 passage of reflection, when equal parts are taken from each, the ratio of the reflected ray CF to 
 
CHAP. IV. OF VISION. 197 
 
 the passage of reflection DFC (El. V. 8.) diminishing, the ratio of the length of that part of the 
 mirror, OH which the whole object will be seen, to the object, is diminished ; that is, (the oliject 
 being given) the length of the part of the mirror on which the whole object is seen, will be di- 
 minished ; consequently, the whole object, which at a certain distance of the eye from the mir- 
 ror was not visible, on a nearer approach of the eye may-become visible. 
 
 If the object DE be removed farther from the glass, the ratio of the reflected ray FC to 
 the passage of reflection Dt'C will also be diminished, because DFC will be increased whilst 
 FC remains the same; and consequently the length of the part of the mirror, on which the 
 whole image is seen, is diminished, and a less surface of glass is required in order to see the 
 whole image. 
 
 PROP. XCVIII. If a spectator sees himself entirely in a plane mirror 
 placed parallel to him, the mirror must be half as long as himself. 
 
 When a spectator is looking at himself, the incident ray is his distance from the mirror, 
 and the reflected ray is equal to it, and is the distance of the mirror from him. The passage of 
 reflection is therefore equal to twice his distance from the mirror ; and consequently the re- 
 flected ray is to the passage of reflection as 1 to 2 ; whence (by Prop. XCVI.) the length of the 
 glass, in which he can see himself entirely, must be to his own, length as 1 to 2, or the length of 
 the glass must be half his own length. 
 
 If the mirror be at all shorter than this, the spectator will not be able to see himself, whether 
 he is nearer to the glass, or farther from it. If he approaches toward the glass, the object, being 
 himself, approaches as fast as the eye, so that, though (by Prop. XCVII.} he might see more of 
 himself by the approach of the eye, he will see just as much less of himself on account of the 
 approach of the object. In the same manner it may be shown, that if he recedes from the mir- 
 ror, he will not be able to see himself entirely. 
 
 PROP. XCIX. Objects perpendicular to the horizon, seen in a plane 
 mirror parallel to the horizon, appear inverted. 
 
 By Prop. LIV. each imaginary radiant is at the same distance behind the mirror, that the 
 real radiant is before it ; hence, if the mirror be below the object and the eye, the object will 
 have its lowest part nearest the surface of the mirror, and its highest part farthest from it, and 
 therefore will in this situation appear inverted ; if the mirror be above the object and the eye, 
 the object will have its highest part nearest the surface, and its lowest part farthest from it, and 
 therefore will, in this situation also, appear inverted. 
 
 PROP. C. If a plane mirror be inclined to the horizon at an angle of 
 forty- ftve degrees, an object parallel to the horizon will appear erect in 
 the mirror, and an object perpendicular to the horizon will appear parallel 
 to it. 
 
 Let the object CD parallel to the horizon be seen in AB, a mirror so placed as to incline to Plilte ^ 
 the horizon in an angle of forty-five degrees. At whatever distance any radiant C is from the Fig 19- 
 mirror, at tiie same distance (by Prop. LIV.) is the corresponding radiant C in the image, or 
 CE will be equal to cE. in like manner, DB will be equal to dE. Thus every radiant in the 
 
193 OF OPTICS. BOOK VI. 
 
 image is at the same distance behind the mirror, as the object is before it 5 whence the image cd 
 makes half a right angle cBA with the mirror on one side, whilst the object makes with it half 
 a right angle CBA on the other; whence cBC is a right angle ; that is, the image is perpendicu- 
 lar to the object or horizon, and appears erect in the mirror. 
 
 By making cd the object, and CD the image, it may be shown in like manner that when the 
 object is erect, it will appear in the mirror parallel to the horizon. 
 
 PROP. CI. If an object be placed between two plane mirrors inclined to 
 one another at any angle, several images may be seen. 
 
 Hate 7. Let the object F be placed between the two plane mirrors CB, CA, making with one anotli- 
 
 *''20. er the angle BCA. From the object F, draw FD perpendicular to the mirror CA, meeting CA 
 in K, and make KD equal to FK. The image of F will (by Prop. LIV.) appear at D. In like 
 manner, if FG be drawn perpendicular to CB, the object will be seen in G, as far behind the 
 mirror as F is before it. Thus two images of the same object, but of different sides or surfaces 
 of it, will be seen. 
 
 Again, since some of the reflected rays, which diverge from the image D in all directions fall 
 upon the opposite mirror CB, the image D may be considered as an object placed before the 
 mirror CB ; and consequently, when the rays which diverge from D are reflected from CB, if 
 DHE be drawn perpendicular to CB, and if EH is taken equal to HI), these reflected rays wilt 
 (by Prop. LVIII.) represent the image of D at E, as far behind the mirror as D is before it. In 
 like manner, some of the rays from this second image E will fall upon the mirror CA, and the 
 image of E, or a third image of the object, will appear. And thus, as long as the image repre- 
 sented in one mirror is before the other, so long a new image of the last image will be produced. 
 And all these images, beginning from CA, and being successive representations of D, will be 
 images of the side of the object F toward C A. Besides these, there will be another set of im- 
 ages beginning from CB, which will be formed in the same manner, and represent the side of the 
 object toward CB, the first of which will be G, and the second L. 
 
 PROP. CII. The images which appear in two plane mirrors inclined to 
 one another, are in the circumference of a circle, the radius of which is the 
 distance of the object from the vertex of the angle contained between the 
 mirrors. 
 
 Plate. 7. Since FK ' s e( l ual t K D, KC common, and the angles at K right angles, CF, the distance 
 Fig. 20. of the object from the vertex of the angle made by the inclination of the mirrors to one another, 
 is (El. I. 4.) equal to CD. In like manner it may be proved that CE is equal to CD. There- 
 fore CF and CE are equal to one another. Thus all the straight lines drawn from C to G, L, 
 or any other image in the mirrors, may be proved to be equal to CF. Consequently, if C be 
 made the centre of a circle, and CF its radius, the circumference will pass through the points 
 1), E, G, L, and every other image which appears in the mirrors, that is, all the images, are 
 in the circumference of a circle, whose radius is the distance of the object from the vertex 
 of the angle maue by the inclination of the mirrors to one another. 
 
 PROP. CHI. If two plane mirrors are parallel to one another, and an 
 
HAP. IV. OF VISION. 199 
 
 object is placed between them, innumerable images of that object may be seen 
 in each, standing in a right line. 
 
 If the two mirrors CA, CB, were separated at C, so as to be less inclined, or nearer parallel J* te JJ* 
 to one another, the angle of inclination being diminishedrit-is manifest from Prop. CII. that the 
 number of images will be increased. At the same time the circumference of the circle in which 
 the images are placed will be enlarged, because the vertex C is farther removed from F, or FC 
 is increased. Consequently, if the mirrors CA, CB, are so far separated, that the vertex is 
 infinitely distant, the images become innumerable, and they are placed in a straight line. 
 
 PROP. CIV. In spherical mirrors, concave or convex, when the place 
 of the image is determinate, the object and the image are in the same situa- 
 tion, if they are both on the same side of the centre, and in contrary situations 
 if they are on opposite sides. 
 
 Let AFB be an object placed nearer the concave mirror SGV than its principal focus, and pi ate 7. 
 let C be the centre of concavity. The rays from A, F, B, being reflected, will (by Prop. HI.) Fig. 23. 
 diverge, and the distances of the corresponding imaginary radiants I, E, M, may be determined 
 by Prop. LVI. The real and the imaginary radiants are, in this case, on the same side of C the 
 centre of concavity. Now, the imaginary radiant which corresponds to the real one A, is (by 
 Prop. LVIII.) in CAI the perpendicular drawn from A to the surface ; and the same with 
 respect to B, and all the other radiants. And these perpendiculars, and all the rest, being 
 drawn from the centre, do not cross each other but at the centre; consequently they are in the 
 same position with respect to each other at the object and the image; and that which is the 
 highest at the object will be the highest at the image, and the reverse. Since therefore (by Prop. 
 LVIII.) every point of the object appears in its perpendicular at the image, the highest point in 
 the object will appear the highest in the image, and the reverse; that is, the object and image 
 will be in the same situation. In like manner it may be shown, that if the object AFB is placed Plate 7. 
 before a convex mirror, and its image IM is on the same side of the centre, they will be both in F 'S' 22 - 
 the same situation. If the object AFB is farther from the concave mirror SGV than its princi- Fig. 21. 
 pal focus, and if M, E, I, be the places of the several foci, to which the rays from A, F, B, (by 
 Prop. L.) will converge, a distinct image of the object will appear upon a paper placed at M, E, 
 I; and if the paper is taken away, and ttie eye is more remote from the mirror than MEI, the 
 rays will diverge from these foci, and become the last image. But, because the extreme perpen- 
 diculars ICH, MCK, in which (by Prop. LVIII ) the points A and B will appear, cross each 
 other at the centre C, between the object and image, A, the highest point of the object, will 
 appear at I, the lowest point of the image, and the reverse; that is, the image with respect to 
 the object, will be inverted, or they will be in contrary situations. The same may be shown in 
 like manner with respect to the convex mirror. 
 
 PROP. CV. In spherical mirrors, concave or convex, the diameter of 
 the object is to the diameter of the image, as the distance of the object from 
 the centre to the distance of the image from the centre, and also as the dis- 
 tance of the object from the surface to the distance of the image from the 
 surface. 
 
200 OF OPTICS. BOOK VI. 
 
 Plate 7. If the eye is any where in the line FG, or that line produced, FG is the optic axis ; whence 
 
 ' the visible length of the object AB, and also of the image IM, is proportional (as appears from 
 Prop. LXVIII.) to a subtense of the optic angle perpendicular to FG. The visible extensions, 
 or lengths of the object and of the image being then perpendicular to the same line FG, are par- 
 allel to one another. Hence in all the cases, the angles ACB, ICM, are equal, and also the 
 angles CAB, CIM. Therefore (El. VI. 4.) AB, the visible length of the object, is to MI, that of 
 the image, as AC. the distance of the object from the centre, is to 1C, the distance of the image. 
 Again, since the object AFB consists of real radiants, and the image MEI of imaginary radi- 
 ants when the rays diverge, and of fuci when they converge, after reflection ; and since when 
 they diverge, FG, the distance of the object from the surface, is (by Prop. LVI.) to EG, the 
 distance of the image from the surface, as FC, the distance of the object from the centre, is to 
 EC, the distance of the image from the centre ; but (by El. VI. 2.) AB is to MI, as FC to EC ; 
 therefore AB is likewise to MI, as FG to EG; that is, the diameter of the object is to that of the 
 image, as the distance of the object from the surface to that of the image from the surface. 
 
 PROP. CVI. If the eye is close to a concave or convex mirror, the 
 apparent diameter of the object is equal to the apparent diameter of the image. 
 
 If the eye is at G, the real diameters of the object AFB and image MEI are (by Prop. C V.) 
 as their respective distances from the eye. Therefore (by Prop. LXX1.) their apparent diam- 
 eters will subtend the equal angles AGB, IGrM, and will be equal. 
 
 PROP. CVII. If the eye is placed in the centre of a concave mirror, it 
 can see nothing in the mirror but Us own image. 
 
 For the eye. in this situation, is in the place of its own image, and therefore rays will be 
 reflected to it from every point of the surface. 
 
 PROP. CVIII. If an object is nearer to a concave mirror than its prin- 
 cipal focus, the image appears behind the mirror, farther from the mirror and 
 larger than the object, erect, and distinct. 
 
 Plate 7. The object AFB being nearer to the concave mirror than the principal focus, the rays vvhicli 
 
 'S- 23> diverge from each point in the object before the mir ror'will diverge after reflection (by Prop. L.) 
 less than before from the imaginary radiants M, E, I ; whence the image formed by them will 
 (Prop. VI.) be farther from the mirror than the object; and consequently (by Prop. CV.) it will 
 be larger than the object. Because the object is nearer the mirror than its principal focus, it is 
 likewise nearer than the centre ; whence (by Prop. CIV.) the image will be erect. Lastly, the 
 image will be seen distinctly, because the rays from it diverge as from objects at a moderate 
 distance. 
 
 PROP. CIX. If an object touches a concave mirror, the image will touch 
 it likewise, and they will be equal. 
 
 For the real radiants being close to the mirror, the imaginary radiants are so too; whence 
 (by Prop. CV.) their diameters will be equal. 
 
CHAP. IV. OF VISION. , SOl 
 
 PROP. CX. If an object is placed in the focus of a concave mirror, the 
 image is at an infinite distance behind the mirror, larger than the object, erect, 
 and distinct. 
 
 When the object AFB is in the principal focus of the-concave mirror SGV, the image is at 
 an infinite distance. It will (by Prop. CV.) be larger than the object, on account of its remote- 
 ness. It will be seen erect, (by Prop. CIV.) because it is on the same side of the centre with 
 the object. And since the rays of each beam are parallel, it will be seen as distinctly, as the 
 naked eye sees very remote objects. 
 
 PROP. CXI. If the object is farther from a concave mirror than its focus, 
 and the eye is nearer than the place of the image, the object will appear con- 
 fused, behind the mirror, erect, and magnified. 
 
 The rays which diverge from A, F, B, in an object more remote from the concave mirror Plate T. 
 8GV than its focus, will (by Prop. L.) be collected, and on a surface of white paper form an f 'S- 31> 
 inverted image. If the eye is auy where between B and G, the rays from every radiant are 
 converging when they come from the mirror to the eye, whence it will appear confused, because 
 the eye is not accustomed to see rays in this state. The rays AH, AG, diverging from A, will, 
 after reflection, converge toward I ; but if the eye is nearer to the mirror than I, the reflected 
 ray GI will not cross its perpendicular HI in any place before the eye, since they sre in a state 
 of convergency ; consequently, the apparent place of this, or any other point of the image, will 
 be indeterminate. It will be seen erect, because MEI, the inverted image of the object, will be 
 drawn inverted on the retina. Lastly, because the rays of each beam converge after reflection, 
 as HI, GI, they will appear to come, not from points, but from circular spots larger than the 
 points of the object, the image will appear confused, and be larger than the object. 
 
 PROP. CXIL If an object is farther from a concave -mirror than its 
 principal focus, and the eye is farther from the mirror than the place of the 
 image, the image appears before the mirror, inverted, and distinct. 
 
 Rays coming from M, E, I, an object at a greater distance than the principal focus from the Plate 1. 
 mirror SGV, will (by Prop. L.) converge to AFB, and would paint an inverted image upon F ' s ' 2I ' 
 a surface of white paper placed there. From thence they will diverge, (the paper being taken 
 away ;) whence to the eye placed any where beyond AFB, the image will appear inverted. 
 And it will be seen distinctly, because the rays come to the eye diverging, as from an object at 
 a moderate distance. 
 
 SCHOL. 1. The inverted images of objects may be represented in a dark room by a concave 
 mirror, which receives rays passing from external objects through a hole in a window-shutter, 
 and collects them into a focus on a surface of white paper. 
 
 SOHOL. 2. A concave mirror, collecting the parallel rays of the sun into a focus, will act as 
 a burning glass. 
 
 PROP. CXIII. When an object is placed before a convex mirror, its 
 
 26 
 
202 OF OPTICS. BOOK VI. 
 
 image appears behind the mirror, nearer the mirror, and less than the object, 
 distinct, and erect. 
 
 Plate 7. If AFB is an object placed before tbe convex mirror SGV, the rays which before reflection 
 
 tig. 22. diverge from A, F, B, will (by Prop. LVIII.) after reflection diverge from as many radiants 
 I, K, M, behind the mirror, forming the image. And because the reflected rays (by Prop. LII.) 
 diverge more than the incident ones, the image IM (by Prop. VI,) will be nearer the mirror 
 than the real radiants or object AFB ; whence (by Prop. C V.) the image will be less than the 
 object. And because the reflected rays come to the eye in a state of divergency, the image will 
 be seen as distinctly as any \isible object, seen by such diverging rays, at the same distance- 
 Lastly, if the object AFB were at an infinite distance from the mirror, the rays proceeding from 
 any point in it would fall parallel upon the mirror, and therefore would upon reflection form the- 
 image in the principal focus, that is, in the middle point between G and C, or on the same side 
 of C with the object ; whence (by Prop. CIV.) it would be erect. At any finite distance of the 
 object, the image, being still nearer the surface, must therefore be erect. 
 
 PROP. CXIV. When either the eye or the object departs from a con- 
 vex mirror, the apparent diameter of the image decreases. 
 
 Plate 7 If the object AFB continues in its place, the image IM will (by Prop. LVI.) be always at 
 
 Fig. 22. the same distance from the mirror; and (by Prop. CV.) the real diameter will be invariable; 
 consequently (by Prop. LXIX.) the apparent diameter of the image will be inversely as the 
 distance of the eye. 
 
 If the object AFB departs from the mirror, the ratio of FG, the distance of the object, to GE, 
 that of the image, (El. V. 8.) will increase; whence (by Prop. CV.) the ratio of the diameter of 
 the object to that of the image will likewise increase; that is, the image will become less with 
 respect to the object : but. the eye remaining in the same place, the apparent diameter of the 
 image will (by Prop. LXX.) be as its real diameter; consequently, the apparent diameter will 
 decrease. 
 
 EXP. The Propositions in this section may be confirmed, by placing an object before a Plane, 
 icave, or Convex Mirror, according to the terms of the respective Propositions. 
 
 Concave, or 
 
 CHAP. V. 
 
 Of Colours. 
 
 SECT. I. 
 OF THE DIFFERENT REFR \NGIBILITY OF LIGHT. 
 
 DBF. XXV. Rays oflight are differently refrangible) when at the same or 
 equal angles of incidence, some are more turned out of the way than others. 
 
CHAP. V. OF COLOURS. 203 
 
 DBF. XXVI. Rays are differently reflexible, when some are more easily 
 reflected than others. 
 
 DEF. XXVII. Light is called homogeneous, when all the rays are equally 
 refrangible ; and heterogeneous, when some rays are more refrangible than 
 others. 
 
 DEF. XXVIII. The Colours of homogeneous rays are called primary 
 or simple colours ; those of heterogeneous, secondary or mixed. 
 
 PROP. CXV. The rays of the sun are not all equally refrangible ; and 
 those rays which have a different degiee of refrangibility, have likewise a 
 different colour. 
 
 If a beam of light SF from the sun passes into a dark room through F, a round hole in a win- Plate 7. 
 dow-shutter KG, and is received upon a white surface, a white round image will be seen. If a F 'f>- ^ 4/ 
 glass prism ABC is so placed as to receive the beam of light, the rays of this beam, from their 
 refraction in passing through the prism, will be turned upward, and the refracted image PT will 
 be oblong, havinjr its breadth equal to the diameter of the circular picture O. If all the rays 
 were equally refracted upward, it is manifest that such a refraction would not change the form 
 of the picture. Since therefore the refracted image is oblong, it must be formed by rays differ- 
 ently refrangible, which fall with equal angles of obliquity upon BC, the first side of the prism, 
 but are some of them, in refraction, turned more out of the way than others ; those rays which 
 go to P, the upper part of the image, being most refrangible, and those which go to T, the lower 
 part, being least refrangible. 
 
 This oblong image is of different colours in different parts, the whole image being made up of 
 rays of seven different colours, in the following order, beginning with those which are most refran- 
 gible ; violet, indigo, blue, green, yellow, orange, red. This refracted picture consists of several 
 round pictures so near each other, that each higher circle mixes in part with that below it, whence 
 the colours near the upper and lowet,edge of each circle are blended. The sides of these circles 
 being very near to each other appear like right lines. 
 
 Ex?. 1. Observe the prfcmatic image formed by the refraction of the rays in passing through, 
 a single pi ism. 
 
 2. Toj^aratethe several colours as much as possible, make the hole F in the window- 
 shutter very small, and collect the rays which pass through it, into a focus L, by a convex lens 
 MN. LetTRe rays \vjych have passed through the lens be now received upon a prism placed 
 near the lens; the rays will lie refracted upward into an oblong image. 
 
 3. To prove that the prismatic image is produced by the different refrangibility of the rays, 
 and by no other cause, let a second prism DH be placed beyond the first abc, at right angles to it. 
 The rays passing through this second prism are refracted sideways ; those which were most 
 refracted upward by the first prism, are most refracted sideways by the second ; but, the rays 
 not being spread in breadth, the image remains of the same form. 
 
 PROP. CXVI. Those rays of light, which are most refrangible, are also 
 most reflexible. 
 
204 OF OPTICS. BOOK VI. 
 
 Plate 7. If the beam of light passing through F falls upon a prism ABC, whose sides AC, AB, are equal, 
 
 Fig. 26. and the angle at A a right angle ; when the obliquity of these rays, as they are to pass out of the 
 prism at its base BC, is less than 40 degrees, (by Cor. 3. Prop. XIII.) the greatest part of the 
 beam will pass through, but some rays will be reflected at the surface BC. Those rays which 
 pass through the base (by Prop. CXV.) form an oblong coloured image at HG, between the most 
 refrangible ray MH, and the least refrangible ray MG. If the rays which are reflected from M 
 are made to pass through another prism XYV, they will also form a faint oblong coloured image 
 pt. Now, if the prism ACB is turned slowly round upon its axis in the direction ACB, the 
 obliquity of the rays FM to the base BC, will continually increase till all the rays will be reflect- 
 ed at M. Consequently, the image pt will become much brighter than before. And this total 
 reflection will not be produced at once, but the most refrangible rays MH will be first entirely 
 reflected ; for the violet colour in HG will first disappear, and the same colour at p will first 
 become brighter. In like manner, as the prism ABC is turned round, each different sort of ray 
 will be reflected sooner, as it has a greater degree of refrangibility. Hence it appears, that the 
 rays of the sun have different degrees of reflexibility, and that those which are most refrangible 
 are also most reflexible. 
 
 PROP. CXVII. Homogeneous ligbt is refracted regularly without any 
 dilatation of the rays. 
 
 Exp. When the rays of any colour in the oblong image, as green, are separated from the 
 rest, in the manner described in Prop. CXV. if some of these rays are transmitted through a 
 small hole in a thin board, and refracted by a prism placed in the other side, the image formed 
 by these rays after refraction will not be oblong, but circular. 
 
 PROP. CXVIII. The confused appearance of objects seen through 
 refracting bodies, is owing to the different refrangibility of light. 
 
 EXP. Small objects placed iu a sun-beam and viewed through a prism, will be seen confus- 
 edly; but if they are placed in a beam of homogeneous light, separated by a prism, they will 
 appear as distinct through a prism, as when viewed by the t*ked eye. 
 
 SCHOL. Although the 13th Proposition (in which it was shown^hat when a ray of .the sun 
 i passing^Hit of one medium, into another, the ratio of the sine of incidence to the refracted sine 
 will not be changed by changing the obliquity of the incident ray) proceeds upon the supposition, 
 that all rays are equally refrangible, and therefore is not exactly true ; the demonstration is 
 sfrictly applicable to any one sort of rays, as the red ones, which are equally refrangfUe. 
 
 SOHOL. 2. Since, all other circumstances being equal, the same c^l^ namely, the passing 
 of the rays out of one given medium into another, will turn the violet raysinore out of the way 
 than the red rays ; the attracting force which acts upon both being the same, it is probable that 
 any single ray of the least refrangible sort contains a greater quantity of matter than any single 
 ray of the most refrangible sort. 
 
 PROP. CXIX. The colours of homogeneous light can neither be chang- 
 ed by refraction nor reflection. 
 
CHAP. V. OF COLOURS. 205 
 
 EXP. 1. Let a beam of homogenous light pass through a round hole in a pasteboard, and theu 
 be refracted by a prism on the other side, the colour of the rajs will remain the same. 
 
 2. Red lead, viewed in homogeneous red light, will be red, but if placed in green, or any 
 other homogeneous light, it will take the colour of the rays which fall upon it. 
 
 PROP. CXX. The whiteness of the sun's light arises from a due mix- 
 ture of all the primary colours. 
 
 EXP. If the oblong picture PT fall upon the convex lens MN, the rays, which were separated plate 7. 
 PT, will, by passing through the lens, be collected into a focus at G, and form a round image of Fl - 2 "' 
 the sun upon a piece of paper DE. This image formed of all the primary sorts of rays, is white. 
 That the whiteness of the image is owing to the due mixture of all the sorts of rays, appears 
 from hence, that, if any of the colours be intercepted at the Sens, the hnage loses its whiteness. 
 The paper being removed from DE to de, the rays, having crossed at G, will form the prismatic 
 image tp, inverted, but distinct; from whence it appears, that the colours are not changed by 
 being mixed at the focus. 
 
 PROP CXXI. The colours of all bodies are either the simple colours 
 of homogeneous light, or such compound colours as arise from a mixture of 
 homogeneous light. 
 
 Each sort of light having a peculiar colour of its own, which no refraction or reflection can 
 alter, since bodies appear coloured only by reflected light, their colours can be no other than the 
 colour of some single homogeneous light, or of a mixture of different sorts of light. 
 
 PROP. CXXII. Water, air, glass, or any other transparent substance, 
 when drawn into thin plates, becomes coloured. 
 
 EXP. 1. If a soap-bubble be blown up, and set under a glass that the motion of the air may 
 not affect it, as the water glides down the sides and the top grows thinner, several colours will 
 successively appear at the top, and spread themselves from thence in rings down the side of the 
 bubble, till they vanish in the same order in which they appeared. At last a black spot appears 
 at the top, and spreads till the bubble bursts. 
 
 2. If a piece of plane polished glass is placed upon the object glass of a long telescope, and 
 the interval between them is filled up with water, as the glasses are pressed together the same 
 colours arise at the point of contact, and spread themselves in circular rings round that point in 
 the same order as the soap-bubble. 
 
 3. A convex and concave lens, of nearly the same curvature, being pressed closely together, 
 exhibit rings of colours about the point where they touch. Between the colours there are dark 
 rings, and when the glasses are very much compressed, the central spot is dark. Sir I. Newton 
 found the thickness of the air between the glasses, where the colours appeared, to be as 1, S, 
 5, 7, &c. and the thickness where the dark rings appeared, to be as 0, 2, 4, 6, 8, &c. The 
 coloured rings must have appeared from the reflection of the light; and the dark rings from the 
 transmission of it Consequently, the rays were transmitted when the thickness of the air was 
 0, 2, 4, 6, 8, &c. and reflected at the thicknesses 1, 3, 5, 7, &c. Sir I. Newton, therefore, sup- 
 posed, that every ray of light, in its passage through any'refracting surface, is put into a certain 
 
306 OF OPTICS. BOOK VI. 
 
 state, which, in the progress of the ray, returns at equal intervals, and disposes the ray, at every 
 return, to be easily transmitted through the next refracting surface, and between the returns, to 
 be easily reflected by it. These he calls fits of easy transmission and reflection. See Schol. 
 Prop. XLVI. 
 
 4. Two pieces of plate glass wiped clean, and rubbed together, will soon adhere with a con- 
 siderable force, and exhibit various ranges of colours. One of the most remarkable circumstance! 
 attending this experiment, is the facility with which the colours may be removed, or even made 
 to disappear, by heats too low to separate the glasses. A touch of the finger immediately causes 
 the irregular rings of colours tn contract toward their centre, in the part touched. 
 
 COR. 1. From these experiments it appears plain, that the colours of bodies depend in some 
 degree, upon the thickness and density of the particles that compose them. 
 
 COR. 2. Hence, if the density, or size of the particles in the surface of a body be changed, 
 the colour is likewise changed. 
 
 SCHOL. 1. When the thickness of the particles of a body is such, that one sort of colour is 
 reflected, other colours will be transmitted, and therefore the body will appear of the first colour. 
 And, in general, a less thickness is found to be necessary to reflect the most refrangible rays, as 
 violet and indigo, than those which are least refrangible, as red and orange. % 
 
 SCHOL. 2. Sir I. Newton, from a great variety of experiments on light and colours, concludes 
 that every substance in nature is transparent, provided it be made sufficiently thin. Gold, when 
 reduced into thin leaves, transmits a bluish-green light. If we suppos^caMv body, therefore, as ^ ' 
 gold, for instance, to be divided into a vast number of plates, so thin as to be altnostperfeortry \ 
 transparent; it is evident, that all, or the greatest part of the rays will pass through the upper 
 plates, and when they lose their force, will be reflected from the under ones. They will then 
 have the same number of plates to pass through, which they had penetrated before ; and thus, 
 according to the number of those plates, through which they are obliged to pass, the object 
 appears one colour or another, just as the rings of colour appeared in Kxp. 3, according to the 
 distance of the glasses, or the thickness of the plates of air between them. 
 
 The philosopher who has of late years most distinguished himself on the subject of light and 
 colours, is Mr. Delaval, who, by a great variety of well conducted experiments, has shown that 
 colours are exhibited, not by reflected, but by transmitted light. This he pioved by covering 
 coloured glass, and other transparent coloured media, on the further surface, with some substance 
 perfectly opaque, when he found that they reflected no colour, but appeared perfectly black. 
 
 He concludes, therefore, that, as the fibres of mineral and animal substances are found, when 
 cleared of heterogeneous matters, to be perfectly white, the rays of light are reflected from these 
 white particles, through the coloured media with which they are covered ; that these media 
 serve to intercept and impede certain rays in their passage through them, while a free passage 
 being left to others, they exhibit, according to these circumstances, different colours. 
 
 Mr. Delaval instituted other experiments with coloured fluids put into phials of flint glass, 
 in the form of a parallelopiped. The bottom, and three sides of each phial, were covered with 
 a black varnish, the neck and the front being left uncovered. On exposing them to the incident 
 light, he found, that, from the parts of the phials, which were covered, no light was reflected, but 
 it was perfectly black, while the light transmitted through the uncoated parts of the phials was 
 of different colours. The same fluids spread thinly on a white ground, exhibited their proper 
 
CHAP. V. OF THE RAINBOW. 207 
 
 colours; the light, indeed, being in this case reflected from a white ground, and transmitted 
 through a coloured medium 
 
 From these, and many other experiments, Mr. Delaval concludes, (I) That the colouring 
 particles do not reflect any light. (2) That a medium, such as Sir I. Newtiin described, is dif- 
 fused over the anterior ami further surfaces of the plates, whereby objects are reflected equally 
 and regularly, as in a mirror. 
 
 Our author next considers the colouring particles themselves, unmixed with other media. 
 For this purpose, he reduced several transparent coloured liquors to a solid consistence by 
 evaporation with a gentle heat, which does not injure the colouring matter; and in this state 
 also the colouring particles reflect no light, but are perfectly black. 
 
 To determine the principle on which opaque bodies appear coloured, it must be recollect- 
 ed, first, that all the coloured liquors appeared such only by transmitted light ; and, second- 
 ly, that these liquors spread thinly upon white ground, exhibited their respective colours ; 
 he therefore concludes, that all coloured bodies, which are not transparent, consist of a substra- 
 tum of s6me white substance, which is thinly covered with the colouring particles. 
 
 On extracting by means of spirits of wine, the colouring matter from the leaves, wood, and 
 other parts of vegetables, he found that the basis was a substance perfectly white, lie also ex- 
 tracted the colouring matter from different animal substances, as flesh, feathers, &c. when the 
 same conclusion was obtained. Flesh consists of fibrous vessels, containing blood, and is per- 
 fectly white when divested of blood by ablution, and the red colour proceeds from the light which 
 is reflected from the white fibrous substance, through the red transparent covering formed by 
 the blood. The result was the same from an examination of the mineral kingdom. 
 
 Some portions of light are reflected from every surface of a body, or from every different 
 medium into which it enters. Thus transparent bodies reduced to powder, and water in the 
 shape of froth, appear white, which is no other than a copious reflection of light from all the 
 surfaces of the minute parts, and from the air interposed between them. 
 
 For a full investigation of this curious and interesting subject, the reader must be referred to 
 the Memoirs of the Manchester Society, vol. II. 
 
 SECT. II. 
 
 Of the Rainbow. 
 
 PROP. CXX1II. When the rays of the sun fall upon a drop of rain and 
 enter into it, some of them, after one reflection and two refractions, may come 
 to the eye of a spectator, whose back is toward the sun, and his face toward 
 the drop. 
 
 If the sun shines upon XY, a drop of rain, in any lines SF, SD, SA, &c. the greatest part of p[ ate & 
 the rays will enter the drop, and passing on to the second surface, will be transmitted through Fig. I. 
 the drop. But at PG in the second surface some few rays will be reflected, and proceed in 
 some such lines as NR, NQ; and coming out of the drop in the lines RV, QT, they may fall 
 upon the eye of a spectator, placed in those lines with his face toward the drop. These rays 
 are refracted when they enter the drop, reflected from the second surface, and again refracted 
 when they come out of the drop. 
 
208 OF OPTICS. BOOK VI. 
 
 DBF. XXIX. When rays of light reflected from a drop of rain come 
 to the eye, those rays, which excite a perception of light, are called ef- 
 fectual. 
 
 PROP. CXXIV. When rays of light come out of a drop of rain, they 
 will not be effectual, unless they are parallel and contiguous. 
 
 Most of the rays, which enter the drop between X and A, passing out of the hinder surface 
 between P and G, only a few rays are reflected, and come out of the drop through the nearer 
 surface between A and Y. None of these, only the rays which are parallel to one another will 
 be effectual, because if they diverge, they will be so far asunder when they come to the eye, that 
 only a very few of them can enter the pupil, and no perception of colours will be excited. Also, 
 unless several parallel rays be very near each other, the rays will be too few to create anj 
 perception. 
 
 PROP. CXXV. When rays of light come out of a drop of rain after one 
 reflection, those will be effectual which are reflected from the same point, and 
 enter the drop near one another. 
 
 Plate 8- Any rays Afi, CD, when they have passed out of the air into a drop of water, will be re- 
 
 * fracted toward the perpendiculars BL, DL, by Prop. XI. And as the ray AB falls farther from 
 the axis than the ray CD, AB will be more refracted than CD ; so that these rays, though paral- 
 lel to one another at their incidence, may describe the lines BE, DE, after refraction, and be 
 both of them reflected from one and the same point E. Now all rays which are thus reflected 
 from one and the same point, when they have described the lines EF, EG, and after reflection 
 emerge at F and G, will be so refracted, when they pass out of the drop into the air, as to de- 
 scribe the lines FH, Gl, parallel to one another. If these rays were to return from E in the 
 lines EB, ED, and were to emerge at B and D, they would be refracted into the lines of their 
 incidence BA, DC, (bv Prop. XII.) But if these rays, instead of being returned in the lines 
 EB, ED, are reflected from the same point E in the lines EG, EF, the lines of reflection EG and 
 EF will be inclined both to one another and to the surface of the drop, just as much as the 
 lines EB and ED are. First, EB and EG make just the same angle with the surface of the 
 drop; for the angle BEX, which EB makes with the surface of -the drop, is the complement of 
 incidence, and the angle GEY, which EG makes with the surface, is the complement of reflec- 
 tion ; and these two are equal to one another, by Prop. XLV. In the same manner we might 
 prove that ED and EF make equal angles with the surface of the drop. Secondly, the angle 
 BED is equal to the angle PEG, or the reflected rays EG, EF, and the incident rays BE, DE, 
 are equally inclined to each other. For the angle of incidence BEL is equal to the angle of re- 
 flection GEL, and the angle of incidence DEL is equal to the angle of reflection FHL, by 
 Prop. XII. Consequently, the difference between the angles of incidence is equal to the dif- 
 ference between the angles of reflection, or BEL DEL is equal to GEL FEL, or BED to 
 GEF. Since therefore either the lines EG, EF, or the lines EB, ED, are equally inclined both 
 to one another and to the surface of the drop, the rays will be refracted in the same manner, 
 whether they were to return in the lines EB, ED, or are reflected in the lines EG, EF. But if 
 they were to return in the lines EB, ED, the refraction, when they emerge at B and D, would 
 
CHAP. V. OF THE RAINBOW. SOU 
 
 make them parallel. Therefore if they are reflected from one am! the same point K in the lines 
 EG, EF, the refraction, when they emerge at G and F, will likewise make them parallel. 
 
 Farther, in order to render the rays which emerge at F and G effectual, they must not 
 only emerge in a direction parallel to each other, but must enter the drop nearly at the 
 tame place. 
 
 Let XY be a drop of rain, AG the axis or diameter of the drop, and SA a ray of light, Plate 8. 
 that comes from the sun and enters the drop at the point A. This ray SA, because it is per- ^S- ! 
 pentlicular to both the surfaces, will pass straight through the drop in the line AGH without 
 being refracted, (by Prop. XIV.) But any collateral rays that fall about SB, as they pass through 
 the drop, will be made to converge to their axis, and passing out at N will meet the axis at II ? 
 (by Prop. XI.) Rays which hill farther from the axis than SB, such as those which fall about SC, 
 will likewise be made to converge ; but then their focus will be nearer to the drop than II, 
 (by Prop. XI.) Suppose therefore I to be the focus to which the rays that fall about SC will con- 
 verge ; any ray SC, when it has described the line CO within the drop, and is tending to the 
 focus I, will pass out of the drop at the point 0. The rays, that fall upon the drop about SD 
 more remote still from the axis, will converge to a focus still nearer than I, suppose at K, (by 
 Prop. XXI. note.) These rays therefore go out of the drop at P. The rays, that fall still more 
 remote from the axis, as SE, will converge to a focus nearer than K, as suppose at L ; and the 
 ray SE, when it has described the line EO within the drop, and is tending to L, will pass out at 
 the point O. The rays, that fall still more remote from the axis, will converge to a focus still 
 nearer. Thus the ray SF will, after refraction, converge to a focus at M, which is nearer than 
 L, and having described the line FN within the drop, it will pass out at the point N. Now here 
 we may observe, that as any rays SB or SC fall farther above the axis SA, the points N, or O, 
 where they pass out behind the drop, will be farther above G, or, that as the incident ray 
 rises from the axis SA. the arc GNO increases, till we come to some ray SD, which pass- 
 es out of the drop at P, and this is the highest point where any ray, that falls upon the side 
 AX, can pass ont ; for any rays SE, or SF, that full higher than SD, will not pass out in any 
 point above P, but at the points O. or N, which are below it. Consequently, though the arc 
 GNOP increases, whilst the distance of the incident ray from the axis SA increased, till we came 
 to the ray SD ; yet afterward the higher the ray falls above the axis 8 A, this arc PONG will 
 decrease. > 
 
 As there are many rays which pass out of the drop between G and P, so (by Prop. XLIII.) 
 some few rays will be reflected from thence ; and consequently, the several points between G 
 and P, which are the points where some of the rays pass out of the drop, are likewise the points 
 of reflection for the rest, which do not pass out. Therefore in respect of those rays which are 
 reflected, we may call GP the arc of reflection, and may say that this arc of reflection in- 
 creases, as the distance of the incident ray from the axis SA increases, till we come to the ray 
 SD ; the arc of reflection is GN for the ray SB, it is GO for the ray SC, and GP for the ray SD. 
 But after this, as the distance of the incident ray from the axis SA increases, the arc of reflection 
 decreases ; for OG, less than PG, is the arc of reflection for the ray SE, and NG is the arc of 
 reflection for the ray SF. 
 
 From hence it is obvious, that some one ray, which falls above SB, may be reflected from the 
 same point with some other ray, which falls below SD. Thus, for instance, the ray SB will be 
 reflected from the point N, and the ray SF will be reflected from the same point ; and conse- 
 quently, when the reflected rays NR, NQ, are refracted as they pass out of the drop at R and 
 
 27 
 
310 OF OPTICS. BOOK VI. 
 
 Q, they will be parallel, by what has been shown in the former part of this Prop. But since tha 
 intermediate rays which enter the drop between SF and SB, are not reflected from the same 
 point N, these two rays alone will be parallel to one another when they come out of the drop 
 and the intermediate rays will not be parallel to them. And consequently, these rays RV, QT, 
 though they are parallel, after they emerge at R and Q, will not be contiguous, and for that rea- 
 son will not be effectual, (by Prop. CXXIV.) The ray SU is reflected from P, which has been 
 shown to be the limit of the arc of reflection ; such rays, as fall just above SO and just be- 
 low SD, will be reflected from nearly the same point P, as appears from what has been al- 
 ready shown. These rays therefore will be parallel, because they are reflected from the same 
 point P; and they will likewise be contiguous, because all of them enter the drop at one and 
 the same place, very near to D. Consequently such rays, as enter the drop at I) and are re- 
 flected from P, the limit of the arc of reflecfon, will be effectual, (by Prop. CXXIV.) since 
 when they emerge at the part of the drop between A and Y, they will be both parallel and 
 contiguous. 
 
 PROP. CXXVI. When rays which are effectual emerge from a drop of 
 rain after one reflection and two refractions, those which are most refrangible 
 will, at their emersion, make a less angle with the incident rays than those 
 do which are least refrangible ; by wnich means, the rays of different colours 
 will be separated from one another. 
 
 Platens. Let FIIj G i be effectual violet rays emerging from the drop at F, H ; and FN, GP, effectu. 
 
 al red rays emerging from the same drop at the same poiats. The violet ravs ( >y Prop. CXXIV. ) 
 are parallel among themselves, because they are effectual ; for the same reason the red rajs are 
 parallel among themselves; but on account of the difference of refrangibility of the violet and 
 red rays, the violet ray GI is not parallel to the red ray GP, but they diverge from the point 
 G ; and so of the rest. Both the violet ray GI and the red ray GP are refracted from the per- 
 pendicular LO, hut (by Prop. CXV.) GI more than GP; whence the angle 1GO is greater than 
 the an^le PGO. If the incident ray AB he continued in the direction ABK, and if JG ami PG 
 be continued backward till they meet AB in K and W, the angle 1KA is that wMch the violet 
 or most refrangible ray makes at its emersion with the incident ray. and PW.\ that which the 
 red or least refrangible ray makes with the same. And the ajigle 1KA (El. I. 16) is less than 
 the exd-rior angle PWA. The same may be proved concerning the rays Ffl, FN, or any 
 other rays which emerge respectively parallel to GI and GP. But (by Prop. CXXIV.) all the 
 effectual violet rays are parallel to GI, and all the effectual red rays are parallel to GP. There- 
 fore tl'.e effectual violet rays at their emersion make a less angle with the incident rays than the 
 effectual red rays. And universally the more refrangible rays, at their emersion, make a less an- 
 clc witli the incident ravs, than those which are less refrangible. And since the effectual rays GI, 
 GP, of different colours make different angles with the incident ray AK at their emersion, 
 they will be separated from one another; so that if the eye was placed in the beam FGHI, it 
 would receive only rajs of one colour from the drop XY, and in FGiNP only rays of another 
 colour. 
 
 SCHOL. The angle which the effectual red rays make with the incident rays is found to be 
 42 20', that of the violet rays 40 17'. 
 
CHAP. V. OF THE RAINBOW. 
 
 EXP. Let a glass globe filled with water be exposed to the rays of the sun ; iet the eye 
 of the spectator be so situated, that the least refracted ray from the drop, coming to the eye, 
 shall make an angle of about 42 with the line passing through the eye aud the sun, the red rays 
 oi.ly will be seen ; if the place of the eye be changed so as to enlarge this angle, the red will 
 disappear; but if the angle be lessened, the colours of tue more refrangible rays will appear. 
 
 PROP. CXXYII. If a line is supposed to be drawn from the centre of 
 the sun through the eye of the spectator, the angle which any effectual ray 
 afti'r two refractions and one reflection makes with the incident ray, will be 
 equal to the angle which it makes with that line. 
 
 Let 1 be the place of the eye of the spectator ; QT a line drawn from the centre of the sun pi a t e & 
 through the eye; and AB a ray coming from the centre of the sun. These two lines AB, QT, Fig. 2. 
 on account of the gieat distance of the sun, may be looked upon as parallel. Therefore (El. I. 
 29.) the alternate angles AKI,KIT, or GIT, are equal. 
 
 PROP. CXXV11I. When the sun shines upon the drops of rain as they 
 are falling, the rays which come from those drops to the eye of the spectator, 
 after one reflection and two refractions, produce the innermost or primary 
 rainbow. 
 
 Let TFY be the innermost or primary rainbow, the outer part of which TFY is red, the in- plate g 
 ner part VDX violet, and the intermediate parts, reckoning from the red to the violet, orange. Fig. 3. 
 yellow, green, blue, indigo. Suppose the spectator's eye at A ; and let A I be an imaginary line 
 from the centre of the sun to the eye of the spectator. If a beam of light S coming from the 
 sun falls upon any drop F, and the effectual rays which emerge at F make an angle FAI of 42 
 2' with the line A I, these rays (by Prop. C XX VI I.) make the same angle with the incident 
 rays, and consequently are red. Hence the drop F will appear red ; for all the other ray s 
 which emerge from F, and would be effectual if they fell upon the eye, being refracted more than 
 the red rays, will pass above the eye. If another beam of light 8 falls upon the drop D, and 
 the effectual rays emerging at H make an angle of 40 17' with the incident rays, the drop D 
 will be of a violet colour; for all the other rays which emerge from H, and would be effectual 
 if they came to the eye, being refracted less than the violet rays, will pass below the eye. The 
 intermediate drops between F and D will for the same reasons be of the intermediate colours- 
 And that which has been proved concerning the drops in the line FD, may be shown of any 
 other set of drops in which the angles made by the emerging and incident rays are equal. Thus", 
 wherever a drop of rain is placed, if the angle which the effectual rays make with AI is equal to 
 the angle FAI, or is 42 ', any such drop will appear red. If FAI was turned round upon 
 the line AI, so that one end of this line should always be at the eye, and the other at 1 opposite 
 to the sun, in this revolution the drop F would describe a circle, of which I would be the cen. 
 tre, and TFY an arc. And since in this revolution the angle FAI continues the same, if the 
 sun was to shine upon this drop as it revolves, the effectual rays (by Prop. CXXVII.) would 
 make the same angle with the incident rays in whatever part of the arc TFY the drop may 
 happen to be ; aud consequently in whatever part of the arc the drop F is, it will appear ted. 
 Mow aa innumerable drops are falling at once in right lines from the cloud, whilst one drop is 
 
OF OPTICS. BOOK. VI. 
 
 at F, there will be others at T, Y, and every other part of the arc, which will appear red in the 
 same manner that F would have done in the supposed circular revolution. Therefore, when the 
 sun shines upon the rain, there will be a red arc TFY produced opposite to the sun. In like 
 manner a violet arc VDX will be produced, and other intermediate arcs of the several interme- 
 diate colours, which will together make up the primary rainbow. 
 
 SCHOL. Cascades and fountains, whose waters, in their fall, are divided into drops, will 
 exhibit rainbows to a spectator, properly situated, during the time of the sun's shining. Thi 
 appearance is also seen by moon light, though seldom sufficiently vivid to render the colours 
 distinguishable. Dr. Gregory, in his excellent Economy of Nature, says, that he once saw a 
 lunar bow ; it was in autumn, the night was uncommonly light but showery, and the colours much 
 more vivid than he could have conceived. There were not so many colours distinguishable as in 
 the solar bow. Coloured bows have been seen on grass formed by the refraction of the sun's 
 rays in the morning dew. 
 
 Artificial rainbows may be produced by candle light on the drops of water ejected by a small 
 fountain, or jet d'eau, or from the stream emitted from an reolipile. But tha most natural and 
 pleasing is by means of the air fountain, the jet of which i perforated with a great num- 
 ber of very fine holes, from which the water spouts so as to form a kind of fluted column. The 
 rainbow is formed by the sun's rays, for the spectator has only to place the spouting streams 
 directly in the sun's beams, with his own back to the sun, and being in a direct line with the 
 sun and centre of the jet, by stooping his head to a certain degree, he will discover the beautiful 
 appearance of the natural prismatic colours, and a small rainbow, on the same principle as 
 those which are seen in the time of rain and sun-shine. 
 
 PROP. CXXIX. The primary rainbow is never a greater arc than a 
 semicircle. 
 
 Plate 8. Since the line AI is drawn from the sun through the eye of the spectator, and through I the 
 
 centre of the rainbow, this centre is always opposite to the gun. And since the angle FAI is 
 an angle of 4 2', F, the highest part of the bow, is 42 2' from I, its centre. If therefore the sun 
 is more than 42 2' above the horizon, I, which is opposite to it, must be more than 42 2' below 
 the horizon, and no primary rainbow will be seen. As much as the altitude of the sun is less 
 than 42 /, so much will the highest point F of the rainbow be above the horizon ; and when the 
 sun is in the horizon, 1, the centre of the bow, will also be in the" horizon on the opposite side, 
 and half the circle will be visible ; but when the sun is set, no bow can be seen. 
 
 PROP. CXXX. When the rays of the sun fall upon a drop of rain, 
 some of them after two reflections and two refractions may come to the eye 
 of a spectator, who has his back toward the sun and his face toward the 
 drop. 
 
 PlateS. If parallel rays from the sun, ZV, YW, fall upon the lower part of the drop of rain BOW, 
 
 t>S ' S ' they will be refracted toward the perpendicular VL, WL, in entering the drop, and proceed in 
 
 the direction VH, WL At HI some part of these rays will (by Prop. XLIII.) be reflected 
 
 into the directions HF, IG. And some of these rays will be again reflected at F, G, into 
 
 tjie directions FD, GB ; which rays, when they emerge out of the drop at 8 and U, will be 
 
CHAP. V. OF THE RAINBOW. 
 
 refracted from the perpendiculars, and mar come to the eye of a spectator whose back is toward 
 the sun and his face toward the drop. 
 
 PROP. CXXXI. Those rays which are parallel to one another after they 
 have been once refracted and once, reflected in a drop of rain, will be effectual 
 when they emerge after two refractions, and two reflections. 
 
 The contiguous raysZV, YW, being refracted toward the perpendiculars VL, WL, when they p] ate g_ 
 enter the drop, will (hy Prop. XVIII.) become convergent ; and because these rays fall upon Fig. 5. 
 the drop very obliquely, their focus will not be far from the surface VW. If this focus is at K, 
 the ravs, after they have passed the focus, will diverge from thence in the directions KII, Kl ; 
 and if KI is the focal distance of the concave reflecting surface HI, the reflected rays HF, IG, 
 (by Prop. L.) will be parallel. These rays are reflected again from the concave surface FG, and 
 will meet in a focus at E, so that GE will be the focal distance of this reflecting surface ; and 
 because HI, FG, are parts of the same sphere, the focal distances GE, KF, are equal. When 
 the rays have passed the focus E they will diverge in the lines EB, ED. Now, if the rays VK, 
 WK, when they have met at K, were to be turned back in the directions KV, KW, on emerg- 
 ing at V and W, they would (by Prop. XX.) be refracted into the lines of incidence, and become 
 parallel. But since GE is equal to IK, the rays ED, EB, which diverge from K, fall in the 
 same manner upon the drop at D and B, as the rays KV, KW, would fall upon it at V and W? 
 and ED, EB, have the same inclination to the refracting surface UB, as KV, KNV, would have 
 to VW ; whence the rays ED, EB, emerging at D and B, will be refracted in the same manner, 
 and will have the same situation with respect to one another, as KV, KW, would have, that is, 
 will be parallel to one another; having been contiguous before their entrance into the drop, they 
 will therefore (by Prop. CXXIV.) be effectual. 
 
 PROP. CXXXII. When effectual rays emerge from a drop of rain after 
 two reflections and two refractions, those which are most refrangible will at 
 their emersion make a greater angle with the incident rays than the least 
 refrangible will make with them ; by which means the rays of different col- 
 ours will be separated. 
 
 Let BM, BA, a violet and a red ray, emerge from B; the angle which the violet ray BM p] a t e 8. 
 makes with the incident ray YW is YrM ; and that which the red ray BA makes with the same *'' s - 
 is YSA. And since BSY, the external angle of the triangle BrS, is (El. I. 16.) greater than the 
 internal angle BrS or BrY ; YrM, the complement of BrS, is greater than YSA, the complement 
 of BSY. Consequently, since the emerging rays make different angles with the same incident 
 ray, the refraction which they suffer at emersion will separate them from one another. 
 
 SOHOL, The angle which the violet rays make with the incident ones is found to be 54 7', 
 And that of the red rays 50 57'. 
 
 PROP. CXXXIII. If a line is supposed to be drawn from the centre of 
 the sun through the eye of the spectator, the angle which, after two refractions 
 and two reflections, any effectual ray makes with the incident ray, will be 
 equal to the angle which it makes with that line. 
 
S14 OF OPTICS. BOOK VL 
 
 Plate 8. If YW be an incident ray, and BA an effectual ray, and AO a line drawn from the centre 
 
 fig S. -of the sun through A, the eye of the spectator, YW and AO may be considered as parallel: 
 whence the alternate angles YSA, SAO, (El. I. 29.) will be equal. 
 
 PROP. CXXXIV. When the son shines upon the drops of rain as they 
 are falling, the rays which come from those drops to the eye of the spectator, 
 after two reflections and two refractions, produce the outermost or secondary 
 rainbow. 
 
 Plate 8. When the sun shines upon a drop of rain E in the outer edge of the secondary rainbow 
 
 Fig. 4. CBD, the effectual violet ray EA (by Prop. CXXXII. Schol.) makes an angle EA1 of J4 7" 
 with AI, a line drawn from the sun through the eye of the spectator, and therefore (by Prop. 
 CXXXII1.) makes the same angle with the incident ray SB. Therefore if the spectator's eye is 
 at A, all the rays except the violet, will (by Prop. X.) make a less angle with A I than EA, and 
 fall above the spectator's eye. In like manner it may be shown, that from the drop F only red 
 rays will come to the spectator's eye, the rest falling below it; and that the rays emerging 
 from the intermediate drops between E and F, and coming to A, will emerge at intermediate 
 angles, and present to the eye the intermediate colours. If EA1 be conceived to turn round 
 upon the line AI, in such a revolution of the drop E, the angle EAI would remain the same, and 
 consequently the emerging rays would make the same angle with the incident rays. But in 
 such a revolution the drop E would describe a circle, of which I would be the centre, and CBD 
 an arc. Consequently, since the emerging rays make the same angle with the incident ones 
 when the drop is at any other part of the arc, as at E, the colour of the drop will be violet to an 
 eye placed at A, in whatever part of the arc the drop is placed. Now, since there are innumer- 
 able drops of rain falling at once, whilst one drop is at E, there will be others in all parts of the 
 arc, which will all appear violet-coloured, for the same reason that E would have appeared of 
 this colour in any other part of the arc. In like manner, as the drop F appears red at F, and at 
 any part,of the arc FD, so will any other falling drop when it comes to any part of that arc. 
 The intermediate arcs are formed in the same manner with the violet arc CBD, and the red arc 
 FD ; and thus the whole secondary rainbow is produced. 
 
 PROP. CXXXV. The colours of the secondary raiuhow are fainter than 
 those of the primary, and are ranged in the contrary order. 
 
 At every reflec'.ion many rays pass out of the drop without being reflected ; consequently 
 the secondary raiabow, which is produced after two reflections, is formed by fewer rays than the 
 first, which ia produced after one reflection. 
 
 Plate 8. Again, in the primary b.>\v, the violet rays, when they emerge effectually, make a less angle 
 
 T> *- 3 ' w ith the incident rays (by Prop. CXXV1.) and therefore (by Prop. CXXVII ) with the line 
 AI, than the red rays. But the rays are here only once reflected, and the angle which the 
 effectual rays make with A I is the distance of the coloured drop from 1, the centre of the bow. 
 Therefore the violet arc in the primary bow will I e nearer to the centre of the bow than the red 
 arc ; that is, the innermost colour will be violet, and the outermost red. But in the secondary 
 rainbow, the rays are twice reflected : and (by Prop. CXXXII.) the violet rays, which emerge 
 so as to be effectual after two reflections, make a greater angle with t'ne incident rays, that is, 
 with the line AI, than the red ones ; which angle is a distance of the violet arc from I, the centre 
 
CHAP. VI. OF THE RAINBOW. 215 
 
 of the bow. Therefore the violet arc in the secondary bow will be farther from the centre of 
 the bow than the red arc ; that is, the outermost colour is violet, and the innermost red. 
 
 PROP. CXXXVI. The secondary rainbow is never a greater arc than 
 a semicircle. 
 
 This is proved in the same manner as Prop. CXXIX. with this difference, that, since the 
 rays of the highest colour in the secondary bow make an angle of 54 7' with At, this bow will 
 begin to appear when the altitude of the sun is less than 54 T ; and when the sun is in the hori- 
 zon on one side, this bow will have its centre in the horizon on the other side at the distance of 
 54 T from it* highest point. 
 
 CHAP. VI. 
 
 Of Optical Instruments. 
 SECT. I. 
 
 Of Telescopes. 
 
 DF.P. XXX. An Astronomical Telescope consists of two convex 1 lenses, 
 whose distance from each other is ftqual to the sum of their principal foci ; 
 that lens which is toward the object, is called the object-glass ; that which is 
 next the eye, is called the eye-glass. 
 
 If NL is one convex len, whose focal distance is MF, and BD another, whose focal distance Plate S. 
 Is CF ; and if these are so placed that the distance between them is equal to MF added to CF, Fl ' 7 ' 
 that is, MC, they form an astronomical telescope. 
 
 PROP. CXXXVII. Very remote objects, seen through an astronomical 
 telescope, appear distinct and inverted. 
 
 Let PM, PL, PN, be rays coming (by Prop. VIII.) parallel from the middle point in a very p[ ate g. 
 distant object ; let AN, AM, AL, come from the highest point, and QN, QM, QL, come from t'ig- 7. 
 the lowest point. These parallel rays will (by Def. XVIII.) be collected into the focus, and 
 there form an image of the object, which (by Prop. LXXXII.) forms the object of refracted 
 vision. But by the construction of the telescope, GFK is the focus ot the eye-glass. Conse- 
 quently the rays which diverge from any point O in this image will, (by Prop. XX.) after they 
 have passed through the eye-glass, become parallel. Therefore if the eye is at any point on the 
 other side of the eye-glass, the object of refracted vision may be seen as distinctly as any very 
 remote object can be seen by the naked eye ; and because the image is the object of vision (by 
 Prop. XXV.) it will be seen inverted. 
 
 PROP. CXXXVIII. The apparent diameter of an object, seen through 
 an astronomical telescope, is to the apparent diameter of the same object seen 
 
S16 OF OPTICS. BOOK VI. 
 
 by the iiaked eye at the station of the object-glass, as the distance of the 
 image from the object-glass is to its distance from the eye-glass. 
 
 Mate 8. jf the image, formed bj the object-glass NL, were received upon a paper at EFG, the 
 
 apparent diameter of the object seen by the naked eye at M, (he station of the object-glass, 
 would be (by Prop. LXXX1V.) equal to the apparent diameter of the image seen from the same 
 station. Now the real diameter of the image is given, because its distance MF from the lens 
 is given. Consequently, the apparent diameter of the image (by Prop. LX!X.) will be inversely 
 as the distance of the eye from it. If the eye is placed at C, the station of the eye-glass, and 
 consequently its distance from the image is FC, the image will appear to the eye in that station 
 bigger than at the station M (by Prop. LXXXIX,) in the inverse ratio of the distances FC, 
 MF ; that is, the apparent magnitude of the image at C will be to that at M, as MF to FC. 
 But the apparent magnitude of the image seen from M is equal to that of the object seen l>y the 
 naked eye. Therefore the image seen from C appears bigger than the object, in the ratio of 
 MF to FC. This would still be the case (by Prop. LXXXIV.) if the eye-glass were placed 
 between the eye and the image, touching the eye. And since the image is in the focus of the 
 eye-glass, the apparent magnitude (by Prop. XC.) is the same, whether the eye is close to the 
 lens, or at any distance from it. Therefore wherever the eye is, the apparent diameter of the 
 object, seen with the telescope, is to the apparent diameter of the same object seen by the naked 
 eye at the station of the object glass, as A1F to FC, or as the distance of the distinct image from, 
 the object-glass, to its distance from the eye-glass ; that is, as the focal distance of the object- 
 glass is to the focal distance of the eye-glass ; consequently if the former be divided by the 
 latter, the quotient will express the magnifying power ; thus, if MF : FC : : 10 : 1, the telescope 
 will magnify 10 times in diameter. 
 
 PROP. CXXXIX. A telescope will not magnify an object, unless the 
 focal distance of the object-glass is greater than the focal distance of the eye- 
 glass. 
 
 Plate 8. The rays which come from distant objects being neatly parallel, the image GFE (by Def. 
 
 Fl S' 7- XVIII.) will be in the focus of the object-glass, which, by the construction of the telescope, is 
 also the focus of the eye-glass. But the apparent diameter of an object seen through a teles- 
 cope, is to its apparent diameter when seen by the naked eye (by Prop. CXXXVII1.) as the 
 distance of the image from the object-glass, to its distance from the eye-glass ; that is, by what 
 has been just proved, as the focal distance of the object-glass, to the focal distance of the eye- 
 glass. Consequently, if MF, the focal distance of the object-glass, is greater than FC, the focal 
 distance of the eye-glass, the object will be magnified ; if MF be equal to FC, the object will 
 appear as to the naked eye ; if MF be less than FC, the object will appear diminished. 
 
 COK. 1. Hence the object-glass of a telescope should be less convex than the eye-glass. 
 
 Cou. 2. An object will be equally magnified by two telescopes of very different lengths, if 
 the ratio of the focal distances of the object-glass and eye-glass be the same in each. 
 
 COR. 3. If a telescope is inverted, objects seen through it will be diminished ; for the objeef- 
 glass which has the greater focal distance then becomes the eye-glass. 
 
CHAP. VI. OF OPTICAL INSTRUMENTS. 217 
 
 PROP. CXL. The visible area, or space which may be seen at one view 
 through a telescope, is as the area of the eye-glass. 
 
 If GFE is any image, its distance from the object-glass being equal to the focal distance of Plate 8. 
 the lens, the area of the image (by Prop. XXXVI.) is given ; but the quantity of this image F '' 7 ' 
 which can be seen at one view must be greater or less, according to the magnitude of the hole 
 through which it is seen ; that is, must be as the area of the eye-glass. 
 
 PROP. CXLI. The brightness of an object seen through a telescope 
 depends upon the area of the object-glass, but not the visible area. 
 
 The brightness of the image, that is, of the object of refracted vision, is (by Prop. XXXVIII.) 
 as the area of the lens which forms it, that is, of the object-glass. But (by Prop. XXXVI. 
 Schol. 2.) the magnitude of the image is the same, whether the area of the object-glass is great 
 or small ; and consequently, if we look at it through an eye-glass of a given area, the quantity 
 to be seen at once will not be altered by any change in the area of the object-glass. 
 
 PROP. CXLII. The distance of the eye from the eye-glass, should be 
 equal to the principal focal distance of the eye-glass. 
 
 Since the image GFE is in the focus of the lens DCB, wherever the eye is placed on the Plate 8. 
 other side of the glass, the image will appear equally magnified. But when the eye is just as * 
 far from the eye-glass as its focal distance, the visible area will be the greatest ; for, in that case, 
 (by Def. XVIII.) none but rays parallel, before the refraction, to MC the axis of the telescope, 
 and therefore to the sides of the cylindrical tube in which the lenses are placed, can reach the 
 eye, and consequently, no rays can come from the inner surface of this tube to the eye to make 
 it visible ; whereas in any other station of the eye, oblique rays from that surface would make 
 the sides of the tube visible; whence the area of vision, which remains the same, being (by Prop. 
 CXL.) always as the area of the eye-glass, will be in part occupied by the sides of the tube, and 
 the object will be seen only through the remaining part. 
 
 DEF. XXXI. A telescope, consisting of four convex lenses, is a double 
 Astronomical Telescope. 
 
 Let the two lenses NML, and B, placed at the distance MB, equal to the sum of their focal Plate 8. 
 distances, form one telescope, and the two lenses C, D, placed at the distance CD, equal to the Fi ' 8 ' 
 sum of their focal distances, form another. If these two telescopes are fixed at the distance CB 
 from each other, so as to be both used together, they form a double telescope; the lens LMN 
 next to the object, is called the object-glass, and the lens B next the object-glass is called the 
 first eye-glass, C the second, and D next to the eye the third. 
 
 PROP. CXLIII. An object seen through a double telescope appears 
 distinct and erect. 
 
 The parallel rays which fall upon the object-glass NML (by Prop. CXXXVII.) form a dis- Plate 8. 
 tinct inverted image at GFE, the focus of the object-glass. This image being also in the focus Flg- 8> 
 of the first eye-glass B, the rays of each beam from the several points of this image will become 
 
 23 
 
218 OF OPTICS. BOOK VI. 
 
 parallel by passing through B; whence, falling parallel on the second eye-glass C, they will 
 form a distinct inverted image at KIH, the focus of this second eye-glass ; and because KIH is 
 also the focus of the third eye-glass D, the rays from this image, after passing through this third 
 eye-glass, will come to the eye parallel to each other. Consequently, the object will be seen 
 distinctly ; and because the second image is inverted with respect to the first, which is inverted 
 with respect to the object, the second image, or object of refracted vision, is in the same situation 
 as the object itself. 
 
 PROP. CXLIV. A double telescope magnifies an object in the ratio of 
 the focal distance of the object-glass, to the focal distance of the first eye- 
 glass. 
 
 Plate 8. '-Th e fi rs * telescope MB magnifies the object in the ratio of MF to FB: and the second teles- 
 
 Fig. 8. cope CD is commonly made up of two lenses of equal convexities, which will not alter the 
 apparent magnitude of the objects. Therefore, when both are used together, the object is only 
 magnified by the first in the ratio of MF, the focal distance of the object-glass, to FB, the focal 
 distance of the first eye-glass ; consequently, the magnifying power is found by dividing MF 
 by FB. 
 
 SOHOL. 1. The different reTrangibility of the rays of light makes refracting telescopes imper- 
 fect ; for those rays which are most refracted by passing through the lens, will be brought to a 
 focui and form an image nearer to the object-glass than those which are less refracted ; and 
 consequently the several sorts of rays are not properly collected in one focus to produce a per- 
 fectly white image, but each has its own focus, producing a confused and coloured image. 
 
 Of two refracting telescopes which magnify equally, the shorter will give a more imperfect 
 image than the longer. For the image appearing equal in both, but being farther from the object- 
 glass in the longer than the shorter, must be in reality larger or more magnified ; whence the 
 defect arising from the different retrangibility of the rays will be more visible in the longer than 
 in the shorter telescope. Hence, reflecting telescopes are more perfect than refracting ones ; 
 for when all the rays aic reflected, their angles of incidence and reflection being equal, they will 
 all meet in a focus at the same distance. 
 
 SOHOL. 2. To remedy the defect of refracting telescopes, arising from the different refrangi- 
 Lility of rays of light, a compound object-glass was invented by Mr. Dollond, consisting partly 
 of white flint glass, and partly of crown glass, which have different refracting powers. These 
 refract contrary ways ; and the excess of refraction in the crown glass is made such, as to des- 
 troy the colour caused by the flint glass. A telescope thus formed is called achromatic. Let 
 Fig. 10.' ABED represent a double concave lens of white flint glass, and AGDF a double convex of 
 crown glass ; then the part of the lenses which are on the same side of the common axis, viz. 
 ACB and AFG may be conceived to act like two prisms which refract contrary ways; and if 
 the excess of refraction in the crown glass be such as precisely to destroy the divergency of 
 colour caused by the flint glass, the incident ray 811 will be refracted to X without any produc- 
 tion of colour; the same is true of the ray s/i, and of all the other incident rays; and conse. 
 quently the whole focal image, formed by this compound object-glass, will be achromatic, or free 
 from colour which might arise from refraction. It will therefore bear a larger aperture, and 
 greater magnifying power, and of course enlarge objects much more than a common refracting 
 telescope of the same length. The great impediment to the construction of large achromatic 
 
CHAP. VI. OF OPTICAL INSTRUMENTS. 219 
 
 telescopes, is the want of a flint glass of an uniform density. Dr. Blair has, within these few 
 years, discovered that certain fluids, particularly those which contain the muriatic acid, may be 
 formed into lenses. With these he has produced achromatic telescopes, which seem as perfect 
 as the thing will admit of. See Transactions of the Edinburgh Royal Society. Also, Encyc 
 Brit. Art. Telescope, Vol. XVIII. Part i. 
 
 SCHOL. 3. The construction of the eye has excited a suspicion, that that might be an achromat- 
 ic instrument ; but as the successive refractions are all in the same direction, that notion cannot be 
 maintained, unless one of the humours is found to refract the red rays more than the violet, or 
 that all the humours refract all the rays equally. 
 
 PROP. CXLV. To explain the construction and use of several kinds of 
 telescopes. 
 
 I. Of Galileo's Telescope. 
 
 Galileo's telescope consists of a convex object-glass and a concave eye-glass, so placed that 
 the distance between them is the difference of their focal distances. 
 
 In this telescope ZYX, a convex lens, is placed at the distance from BA, a concave lens Hate 8. 
 of YC, the difference between YF,the focal distance of ZX, and CF, the focal distance of BA. Fi - 9 - 
 
 From a distant object let rays fall upon the convex lens YZ, from which they will proceed 
 toward the focus of this lens at FG. But the concave lens AB, the focus of which is at FG, 
 renders the converging rays parallel when they reach the eye ; whence an image will be formed 
 upon the retina. And the pencils of rays being made more diverging by passing through the 
 concave lens, the visible image is seen under a larger angle than the object, and appears magni- 
 fied. Also, because the pencils which form the image only cross one another once, the image 
 appears erect. 
 
 II. Of Sir Isaac Newton's Telescope. 
 
 In the tube ABCD, toward the end BC, let the concave mirror GH be placed perpendicular viateS. 
 to DC, the lower side of the tube. If an object, which is at such a distance that rays coming fl S- 10. 
 from the same point may be considered as parallel to one another, be placed before the open end 
 of the tube AD, these parallel rays will be reflected from the concave mirror GH, and becom- 
 ing convergent, would (by Prop. CXII.) form an inverted picture of the object upon a paper 
 held at the focus of the mirror. But if the converging rays, before they reach the focus, fall 
 upon a plane mirror K, placed at an angle of 45 degrees with DC, the side of the tube, or with 
 the axis of the telescope, they will be reflected from thence, and meet before it at L, forming 
 an image perpendicular to the object, or parallel to the axis of the telescope. If this image be 
 placed in the focus of the convex lens L, fixed in the side of the telescope, the eye will see it 
 distinctly through the lens. 
 
 The image seen from the station of the eye-glass L, either with or without the glass, will (as 
 in the refracting telescope, see. Prop. CXXXVIII.) appear as much larger than when seen from 
 the concave mirror, that is, as much larger than to the naked eye, as the distance of the image 
 from the eye-glass is less than its distance from the mirror, or as its distance from the mir- 
 ror is greater than its distance from the lens. 
 
 III. Of Gregory's Telescope. 
 
220 OF OPTICS. BOOK VI. 
 
 Plate 8. In the tube TTYY, let a concave mirror EN be placed. Any parallel rays OO 5 PP, form 
 
 ' llp an object A, falling upon this mirror, will, after reflection, (by Prop. CXI.) form an inverted 
 image at C, its focus. Let C be more remote from a second smaller concave mirror PO (placed 
 parallel and opposite to the first mirror EA in such manner that their axes shall be in the same 
 straight line) than its focus. The rays which diverge from the several points of the image at C, 
 and fall upon the mirror PO, will (by Prop. L.) converge after reflection ; and consequently, if 
 they pass through a hole NM in the first mirror EA, they will form a second image which will 
 be inverted in respect of the first, and in the same position with the object. If, whilst these 
 rays are converging, they pass through a plano-convex lens Spo (placed in a smaller tube joined 
 to the larger, they will be brought to a focus, sooner than they would otherwise have been, form- 
 ing the second image F. This erect image is seen by the eye at O, through a meniscal eye-gtass 
 LL, whose convexity is greater than its concavity. For the magnifying power of this telescope, 
 see Muschenbroek. Intro, ad Phil. Nat. or Priestley's Optics, page 736. 
 
 SCHOL. 1. In the telescopes made by Dr. Herschel, the object is reflected by a mirror, as 
 in the Gregorian telescope, and the rays are intercepted by a lens at a proper distance, so that 
 the observer has his back to the object, and looks through the lens at the mirror. The magnify- 
 ing power will be the same as in the Newtonian telescope ; but there being no second reflector, 
 the brightness of the object, viewed in the Herschel telescope, is greater than that in the New- 
 tonian telescope. 
 
 The tube of Dr. Herschel's grand telescope is 39 feet 4 inches in length, 4 feet 10 inches in 
 diameter, every part of which is made of iron. The concave surface of the great mirror is 48 
 inches of polished surface in diameter, its thickness is 3 inches, and its weight is upward of 
 2000lb. This noble instrument was, in all its parts, constructed under the sole direction of Dr. 
 Herschel; it was begun in the year 1785, and completed August 28, 1789, on which day was 
 discovered the sixth satellite of Saturn. It magnifies 6000 times. 
 
 SCHOL. 2. Dr. Priestley observes, that the easiest method of finding the magnifying power 
 of any telescope, by experiment, is to measure the diameter of the aperture of the object-glass, 
 and that of the little image of it which is formed at the place of the eye. For the proportion 
 between these gives the ratio of the magnifying power, provided no part of the original pencil be 
 intercepted by the bad construction of the telescope. For, in all cases, the magnifying power 
 of telescopes or microscopes, is measured by the proportion of the original pencil, to that of the 
 pencil which enters the eye. Another method, is to observe at- what distance you can read any 
 book with the naked eye ; and then removing the book to the farthest distance at which you can 
 distincly read it by the help of the telescope. The book chosen for this purpose should be such, 
 that the connexion of the subject should not assist the observer ; as tables of logarithms, &c. 
 Much depends on the steadiness with which the instrument is fixed. 
 
 SECT. II. 
 Of Microscopes. 
 
 Plate s. DBF. XXXII. A single Microscope is one convex lens placed between a 
 small object and the eye. 
 
CHAP. VI. OF OPTICAL INSTRUMENTS. 
 
 DBF. XXXIII. A double Microscope consists of two convex lenses, of 
 which the object-glass is more convex than the eye-glass ; and the distance 
 between them is equal to the distance of the image from the object-glass, 
 added to the focal distance of the eye-glass. 
 
 Let AB, a convex lens, be the object-glass, and EF, another convex lens, be the eye-glass, plate 8. 
 Let the small object KL be farther from the object-glass than its focus ; an image MDN of the Fl - 12- 
 object will (by Prop. XXIV.) be formed behind the glass ; let the distance of this image from the 
 object-glass be ID, and let its distance from the eye-glass be equal to the focal distance of the 
 eye-glass ; the distance of the two glasses from each other will be ID + DX, or IX, that 
 is, the distance of the image from the object-glass, added to the focal distance of the eye- 
 glass. 
 
 Some compound microscopes are made with three glasses, so that the rays after passing 
 through AB the object-glass, and EF the eye-glass, are again made converging by a second eye- 
 glass, and therefore brought sooner to a focus, than by the first, and the field of vision will be 
 much greater than if only one lens was used. 
 
 COR. Hence it appears, that the difference between the microscope and telescope is, that 
 in the telescope the rays of each pencil fall upon the object-glass nearly parallel, and are united 
 in its focus ; but in the microscope they fall upon it very much diverging from one another, 
 and therefore form the image in a place beyond the focus, and consequently larger than the 
 object. 
 
 PROP. CXLVI. An object seen through a double microscope appears 
 distinct and inverted. 
 
 The pencils of rays issuing from the objects KL, being transmitted through the object-lens P | ate g< 
 AB, their foci will be in JVIN; where there will be an inverted image of the object, which is Fig. 12., 
 viewed through another lens, or eye-glass EF, the focus of which is at MN; hence a distinct and 
 direct image is formed upon the retina, and it is seen inverted. 
 
 PROP. CXLVII. The apparent diameter of an object seen through a 
 double microscope is to that of the same object seen by the naked eye at the 
 limit of distinct, vision, in the compound ratio of the distance of the image from 
 the object-glass, to its distance from the eye-glass, and of the limit of distinct 
 vision to the distance of the object-glass from the object. 
 
 The first part of this Proposition is demonstrated as Prop. CXXXVIII. And it is manifest, 
 (from Prop. LXIX.) that if CI, the distance of the object-glass from the object, is less than the 
 limit of distinct vision, the apparent diameter of the object will be as much greater than that of 
 the object at the distance at which the naked eye can see it distinctly, as 1C is less than that 
 distance. Therefore the object is magnified, because the distance of the image from the object- 
 glass is greater than its distance from the eye-glass, and also because the distance from the ob- 
 ject is less than the limit of distinct vision. The magnifying power of the microscope is then in 
 the ratio compounded of these two ratios. Suppose ID = 6IC 5 and the eye-glass EF to be one 
 
S22 OF OPTICS. BOOK VI. 
 
 inch focus, and the limit of distinct vision to be seven inches then the diameter of the object 
 KL will be magnified 6 x 7 42; consequently 1764 times in surface. 
 
 PROP. CXLVIII. When the same eye-glass is used, the magnifying 
 power of the microscope will be increased by increasing the convexity of the 
 object-glass. 
 
 For in order to keep the image in the focus of the eye-glass, when the convexity of the object- 
 glass is increased, the object-glass must be brought nearer to the object ; the consequence of 
 which will be that the ratio of the limit of distinct vision to the distance of the object-glass from 
 the object will (El. V. 8.) be increased ; whence (by Prop. CXLV1I.) the ratio of the apparent 
 diameter of the object of refracted vision to that of the object seen by the naked eye will also be 
 increased. 
 
 SOHOL. The aperture of the object-glass in a microscope must be small, else the outermost 
 rays, diverging too much, will hinder the distinctness of vision; but, on account of the smallness 
 of the aperture, the object will appear faint, and it will appear necessary, in order to remedy 
 this, to illuminate the object as much as possible. 
 
 PROP. CXLIX. To describe the construction and use of the Solar 
 Microscope. 
 
 In a dark room, let a round hole be made in a window shutter about three inches in diameter, 
 through which the sun may cast a cylinder of rays into the room. In this hole let a tube 
 be fixed, containing a convex lens of about two inches in diameter, and three inches focal dis- 
 tance ; the object, placed between two concave glasses, at the distance of about two inches and a 
 half from the first convex lens; and a second convex lens, whose focal distance is a quarter of 
 an inch, placed at this distance from the object. Let a plane mirror, connected with the tube, 
 and moveable by means of a wheel, receive the sun's rays on the outside of the shutter, and 
 convey them into the tube. The rays, passing through the first lens, will strongly illuminate the 
 object, from which they will pass through the second lens, and form an inverted image of the 
 object, magnified in the ratio of the distance of the object from the lens to that of the image from 
 the lens, that is. in this case, supposing the distance of the distinct picture to be twelve feet, or 
 144 inches, in the ratio of i : 144. Consequently, the diameter of the object will be magnified 
 576 times. 
 
 SECT. III. 
 
 f 
 
 Of the Magic Lantern. 
 PROP. CL. To describe the construction of the Magic Lantern. 
 
 Plate 8. In the side of a lantern let a tube be inserted, consisting of two parts, one moveable upon the 
 
 *' 13 ' other. In the moveable part let a convex lens GG be fixed; in the immoveable part let an 
 
 object EE, painted with transparent colours upon a piece of thin glass, be placed ; and in the 
 
 fixed part of the tube, a convex lens DD. This lens will cast a strong light from the candle 
 
 upon the object EE. And when the rays which diverge from the several points of the object 
 
CHAP. VI. OF OPTICAL INSTRUMENTS. 233 
 
 are, by the lens GG, made to converge, they will (by Prop. XXV.) form an inverted image of 
 the object at K.L, upon auy white surface ; provided that the object is farther from the lens than 
 its focus, and that the whole apparatus is placed in a dark room. The image KL will be larger 
 than the object EE, in proportion as the distance of the image from the lens is greater than the 
 object. A concave reflector AB may be placed within tlie-lautern, behind the candle, to increase 
 the illumination of the picture EE. If the object be placed in an inverted position, its image 
 will appear erect. 
 
 SECT. IV. 
 Of the Camera Obscura. 
 
 PROP. CLI. To describe the construction and use of the Camera Ob- 
 scura. 
 
 Let CD be a convex lens, and HK a plane mirror inclined at an angle of 45 degrees. An 
 inverted image of the object AB would be formed at EG, where the foci of the rays from the 
 object are found after refraction ; but the rays being intercepted by the plane mirror HK, are Plate 8. 
 reflected (by Prop. C.) to MM, the focal distance before it, making an angle with the mirror of Fl - ** 
 45 degrees ; whence the image will be in a position perpendicular to the object, at the top of the 
 box, where, if the rays be received on a sheet of oiled paper, or a plate of glass unpolished on one 
 side, it will be distinctly visible. 
 
BOOK TIL 
 
 OF ASTRONOMY. 
 
 fe 
 
 PART I. 
 
 OF THE MOTIONS OF THE HEAVENLY BODIES. 
 
 CHAP. I. 
 
 Of the Solar System in General. 
 
 DBF. I. J. HE Solar System consists of the Sun, which is a luminous 
 body ; seven primary planets, Mercury, Venus, the Earth, Mars, Jupiter, 
 Saturn, and the Herschel ; eighteen secondary planets, the Earth's Moon, 
 Jupiter's'four Satellites, Saturn's seven, and six belonging to the Herschel ; 
 and an uncertain number of comets ; all which are opaque. 
 
 SCHOL. Upon entering the subject of Astonomy, it will be proper briefly to describe the 
 different systems which have been invented, in order to solve the natural appearances of the 
 heavenly motions. 
 
 Ptolemy supposed the earth to be perfectly at rest, and the sun, nnon, planets, comets, and 
 fixed stars to revolve about it every day ; but that besides this diurnal motion, the sun, moon, 
 planets, and comets, had a motion in respect to the fixed stars, and were situated in respect to 
 the earth, in the following order ; the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn. 
 The revolutions of these bodies he supposed to be made in circles about the earth placed a little 
 out of the centre. This system will not solve the phases of Veaua and Mercury, and therefore 
 cannot be true. 
 
 The system received by the Egyptians was this ; the earth was supposed immoveable in the 
 centre, about which revolved, in order, the Moon, Sun, Mars, Jupiter, and Saturn ; anil about the 
 Sun revolved Mercury and Venus. This disposition will account for the phases of Mercury and 
 Venus, but not for the apparent motions of Mars, Jupiter, and Saturn. 
 
 Another system was that of Tycho Brahe, a Danish nobleman, who was anxious to reconcile 
 the appearances of nature with some passages of the Scriptures, taken in their literal interpre- 
 tation. In his system, the earth is placed immoveable in the centre of the orbits of the sun and 
 moon, without any rotation about its axis ; but he made the sun the centre of the orbits of the 
 other planets, which, therefore, revolved with the sun about the earth. Objections to this svstetn 
 are, the want of that simplicity by which all the apparent motions may be solved ; and the 
 necessity of supposing that all the heavenly bodies revolve about the earth every day; also to 
 
CHAP. II. OF THE EARTH. 225 
 
 suppose that a body should revolve in a circle about its centre without any central body is phy- 
 ' sically impossible. 
 
 Some of Tycho's followers, seeing the absurdity of a diurnal revolution of the heavenly 
 bodies about the earth, gave a rotatory motion to the earth, and this was called the Semi-Tycho- 
 nick system. 
 
 The system, which is now universally received, is called the Copernican. It was formerly 
 taught by Pythagoras, 500 years before Christ; and afterward rejected, till revived by Coperni- 
 cus in the sixteenth century. Here the sun is placed in the centre of the system, about which 
 the planets revolve from west to east, in the following order; Mercury, Venus, the Earth, Mars, 
 Jupiter, Saturn, and the Herschel planet; beyond which, at immense distances, are placed the 
 fixed stars. The moon revolves round the earth ; and the earth turns about an axis. The other 
 secondary planets move round their respective primaries from west to east at different distances, 
 and in different periodical times. 
 
 According to this doctrine, the Sun S is the centre of the system ; Mercury a, Venus b, the p. e 1 9 ' 
 Earth t, Mars e, Jupiter /, and Saturn h. revolve in elliptical orbits round the sun ; the moon d 
 revolves about the earth, and the satellites of Jupiter, Saturn, and the Uerschel, revolve about 
 their primaries ; and the planes of their orbits are inclined to one another.* 
 
 This doctrine, being admitted as true, will account for the apparent motions, and other phe- 
 nomena, of the heavenly bodies, as will be seen in the following Chapters. 
 
 CHAP. II. 
 
 Of the Earth. 
 
 SECT. I. 
 
 Of the Globular Form of the Earth, and its diurnal Motion about its Axis, 
 and of the Appearances which arise from these. 
 
 PROPOSITION I. 
 
 The earth is of a globular form. 
 
 For, ^ The shadow of the earth projected on the moon in an eclipse is always circular; 
 which appearance could only be produced by a spherical body. 2. The convexity of the surface 
 
 * Dr. Herscliel, on the 13th of March, 1781, while pursuing a plan which he had formed of observing, with 
 telescopes, of his own construction, every part of the heavens, discovered, in the neighbourhood of H. Gemi- 
 norum, a planet far beyond the orbit of Saturn, which had never before been visible to mortal eyes. He has 
 since discovered six secondaries belonging to this new planet, which planet is called either the HEHSCBII, 
 from the name of its indefatigable and truly great discoverer ; or the Georgium Sidus, or Georgian Planet, in 
 honour of the present king, who has distinguished himself as the patron of Dr. Herschel. The planet is 
 denoted by this character $ ; an H, as the initial of the name, the horizontal bar being crossed by a pec- 
 pendicular line, forming a kind of cross, the emblem of Christianity, denoting, perhaps, that its discovery 
 was made in the Christian era, as all the other planets were known long before that period. 
 
 29 
 
326 OF ASTRONOMY. BOOK VII. Pxnr 1. 
 
 of the sea is visible ; the mast of an approaching ship being seen before its hull. 3. The north 
 pole becomes more elevated by travelling northward, in proportion to the space passed over. 
 4. Navigators have sailed round the earth, and by steering their course continually westward, 
 arrived, at length, at the place from whence they departed. 
 
 DBF. II. The Axis of the earth is an imaginary line passing through the 
 centre, about which its diurnal revolution is performed. 
 
 DBF. III. The Poles of the earth are the extremities of this axis. 
 
 DBF. IV. The Equator is the circumference of an imaginary great circle 
 passing through the centre of the earth, perpendicular to the axis, and at equal 
 distances from the poles. 
 
 DBF. V. If the axis of the earth be produced both ways, as far as the 
 concave surface of the heavens, in which all the heavenly bodies appear to be 
 placed, it is then called the Axis of the Heavens ; its extremities are called 
 the Poles of the Heavens ; and the circumference produced by extending tha 
 plane of the equator to the same concave surface, is called the Equator in the 
 Heavens. 
 
 DBF. VI. Circles drawn through the poles of the earth or heavens, per- 
 pendicular to the plane of the equator, are called Secondaries of the equator. 
 
 DBF. VII. The sensible Horizon is an imaginary circle, which, touching 
 the surface of the earth, separates the visible part of the heavens from the 
 invisible. The rational Horizon is a circle parallel to the former, the plane 
 of which passes through the centre of the earth. 
 
 SCHOL. Since (by Book VI. Prop. LXIX.) the apparent diameter of an object is inversely 
 as its distance, if the distance be increased in such manner that it may be looked upon as infi- 
 nite, the apparent magnitude becomes a point. Hence AF, the semidiameter of the earth, 
 viewed at the different distances o, O, R, diminishes, till at the distance of O, a fixed star, it 
 
 Plate 9. becomes a point, and the star appears in the same place in the heavens, whether viewed from 
 
 f 'S- 4< the visible horizon SET, or rational horizon H8R. 
 
 DBF. VIII. The Poles of the Horizon are two points, the one of which, 
 over the head of the spectator, is called the Zenith ; the other, which is under 
 his feet, is called the Nadir. 
 
 DBF. IX. Circles drawn through the zenith and nadir of any place, cut- 
 ting the horizon at right angles, ave called Verticle Circles. 
 
CHAP. IT. OF THE EARTH. 3S7 
 
 !)EF. X. A verticle circle passing through the poles of the heavens, is a 
 Meridian, and is said to be the meridian of any place through which it passes. 
 
 DBF. XI. The meridian of any place passing through the poles, and 
 falling perpendicularly upon the horizon, cuts 4t in two opposite cardinal 
 points, called North and South. 
 
 DBF. XII. A Meridian Line is the common intersection of the plane of 
 the meridian and the plane of the horizon. 
 
 COR. Hence any line which lies due north and south in a horizontal plane, may be con- 
 sidered as part of the meridian line. 
 
 SCHOL. 1. To draw a meridian line, perpendicular to a horizontal plane, erect a wire, or 
 stile, seven or eight inches long, and as it is not easy to determine precisely the extremity of 
 the shadow, it will be best to make the stile flat at top, and to drill a small hole through it, 
 noting the lucid point projected by it, mark, at several different times before noon, these lucid 
 points, and through them draw concentric circles about the middle point of the wire's station ; 
 observe in the afternoon when the lucid points again touch these circles ; and find the middle 
 point of each arc between the points already taken ; a line drawn through these middle points, 
 ani! the common centre, will be the meridian line ; for, since at equal distances from noon, the 
 sun is at th6 same height, or in verticals equally distant from the meridian, the circle drawn 
 through the zenith at equal distances from these verticals is the meridian. This should be done 
 about tne summer solstice, between the hours of 9 and 11 in the morning, and 1 and 3 in the 
 afternoon. 
 
 SOHOL. 2. To observe the transit of any heavenly body over the plane of the meridian ; 
 place, in this plane a telescope, having two cross hairs before its object-glass, one vertical, the 
 other horizontal, and observe when the vertical hair passes through the centre of the heavenly 
 body; or hanging; two plumb-lines exactly over the meridian line, place your eye close to one 
 of the threads in such manner, as that it shall cover the other thread, and observe when the 
 body is behind the threads. 
 
 DEF. Xllf. The Altitude or Depression of any heavenly body above or 
 below the horizon, is the arc of a vertical circle intercepted between the body 
 and the horizon, or the angle at the centre measured by that arc. 
 
 SOHOL. The altitude of any heavenly body is found by the help of a quadrant thus ; bring 
 the quadrant into such a situation that the star may be seen through the sights ; then the angle, 
 contained between the string of the plummet and the side of the quadrant, on which the sights 
 are not placed, is the altitude of the star. 
 
 DBF. XIV. The Prime Vertical is that which crosses the meridian at 
 right angles in the zenith and nadir, cutting the horizon in the cardinal points 
 East and West. 
 
'228 OF ASTRONOMY. BOOK VII. PART I. 
 
 DEF. XV. The Azimuth of a heavenly body, is the arc of the horizon 
 intercepted between the meridian and a vertical circle passing through that 
 body ; it is eastern or western, as the body is east or west of the meridian. 
 
 Plate 9. SCHOL. The azimuth of any star may be thus found. Let AC be a given meridian line. 
 
 Above any point A in this line, let a cord with a plummet be hung; let another cord with a 
 plummet be hung at E, so that the star and the two cords shall lie in one and the same right 
 line. Let the perpendiculars AD, BE, represent the cords, and draw AB. From the point B 
 to any point C, in the meridian line AC, taken at pleasure, draw tht right line BC ; then with 
 a scale of equal parts measure the three lines AB, AC, BC. In the triangle, therefore, ABC, 
 there will be given all the sides, from whence will be found the angle BAC, equal to the azimuth 
 required. 
 
 For if the meridian line be supposed to be continued to F, and the line BA to G, the angle 
 F \G will be the azimuth of the star ; but the angle FAG will be equal to the angle at the vertex 
 BAC ; therefore the angle BAC will be equal to the azimuth. 
 
 DEF. XVI. The Amplitude of a heavenly body at its rising is the arc 
 of the horizon intercepted between the point where the body rises, and the 
 east ; its amplitude at setting is the arc of the horizon intercepted between 
 the point where the body sets, and the west ; it is northern, or southern, as 
 the body rises, or sets, to the north or south of east or west. 
 
 DEF. XVII. If a heavenly body rises, or sets, when the sun rises, it is 
 said to rise or set cosmically ; if it rises, or sets, when the sun sets, it is said 
 to rise or set achronically ; it is said to set or rise heliacally, when it ap- 
 proaches so near the sun as to become invisible, or recedes so far from him as 
 to become visible. 
 
 DEF. XVIII. The Latitude of a place upon the surface of the earth is its 
 distance from the earth's equator ; it is measured by the arc of the geographi- 
 cal meridian of the place intercepted between the place and the equator ; 
 latitude is either northern or southern. 
 
 DEF. XIX. Parallels of Latitude are circles on the surface of the earth, 
 drawn parallel to the equator. 
 
 PROP. II. A degree in the equator is to a degree in any parallel of lati- 
 tude, as radius to the cosine of latitude. 
 
 % 
 
 Plate 9. ^ e * EPQ be a geographical meridian. EQ the equator, and FB a parallel of latitude. The 
 
 Fig. 8. circumference EQ is to the circumference FB, and any part of EQ, to any similar part of FB, 
 
 as CQ, or CB, the radius of EQ, to AB the radius of FB ; and AB is the cosine of the arc BQ, 
 
 which is the latitude of the parallel FB. Therefore a degree in EQ is to a degree in FB, as 
 
 radius to the cosine of latitude. 
 
CHAP. II. OF THE EARTH. 
 
 DBF. XX. The Longitude of a place is the distance between the meri- 
 dian of that place, and the meridian of some other place, taken at pleasure, 
 and called the first meridian ; it is measured by the arc in the equator inter- 
 cepted between these two meridians. Longitude is either eastern or western, 
 and is measured 180 degrees each way. 
 
 PROP. III. The altitude of one pole, and the depression of the other, 
 at any place on the earth's surface, is equal to the latitude of that place. 
 
 Let R be a place upon the earth's surface ; Z, N, its zenith and nadir ; P, S, the poles of the Plate 9. 
 heavens, and F, s, the poles of the earth ; EE, the celestial equator, ee, the terrestrial, and HO Fi - 2> 
 the horizon. The latitude of the place is eR, or the equal arc EZ ; and PO is the elevation of 
 one pole, and HS the depression of the other. Because ZO is the distance of the zenith from 
 the horizon, it is an arc of 90 degrees } and because EP is the distance of the pole from the 
 equator, it is also an arc of 90 degrees ; ZO and EP are therefore equal. Take from each of 
 these the common arc ZP, and the remainders EZ and PO are equal. But HS and PO are 
 equal, because they subtend the equal angles HTS, PTO ; therefore the elevation of one pole 
 PO, and the depression of the other HS, are equal to the latitude of the place EZ. 
 
 COR. Hence the circumference of the earth may be measured, by measuring the length on 
 the surface of the earth passed over in a line which lies north and south while the pole gains one 
 degree of elevation, and multiplying this length by 360. A degree of latitude contains 69i Eng- 
 lish miles, whence 24930 miles is the measure of the circumference of the earth, and the radius 
 3985 ; but, as will be shown hereafter, the earth is a spheroid, whose polar diameter is to the 
 equatorial, as 229 to 230. 
 
 PROP. IV. The elevation of the equator at any place is equal to the 
 complement of its latitude. 
 
 Because ZO is equal to EP (each being an arc of 90 degrees) EZ is equal to PO, that is, Plate 9. 
 (by Prop. III.) to the latitude of the place. But EH, the elevation of the equator, is the comple- Fig- 2 ' 
 ment of EZ ; it is therefore equal to the complement of the latitude of the place. 
 
 PROP. V. The earth revolving daily round its axis from west to east, 
 the heavenly bodies will appear to a spectator on the earth to revolve in the 
 same time from east to west. 
 
 Let RCBF be the earth, T its centre, HTO the rational horizon to a spectator at R, whose 
 zenith is Z : let a star appear in the horizon at H. The earth revolving from west to east, that 
 is, in the order of the letters, R, C, B, F, in a fourth part of one revolution, the spectator will 
 be carried from R to C ; consequently, his horizon will become ZN, and the star which appeared 
 in his horizon at H, when he was at R, will now appear nearly in the zenith. When another 
 fourth part of the revolution is completed, the spectator will be at B, and N being now his 
 zenith, and HO his horizon, the star will he set with respect to him, and will not rise till he is 
 again in the station R, that is, till the earth has completed one revolution. Thus whilst the 
 
330 OF ASTRONOMY. BOOK VII. PART I. 
 
 earth has turned once round upon its axis from west to east, all the heavenly bodies in the con- 
 cave sphere of the heavens will appear to have turned round from east to west. 
 
 PROP. VI. The alternate succession of day and night is the effect of the 
 revolution of the earth round its axis. 
 
 For, all the heavenly bodies appearing (Prop. V.) to move from east to west, while the earth 
 revolves from west to east, the sun will appear, in each revolution, to rise above the horizon in 
 the east, and after describing a portion of a circle, to set in the west, and will continue below 
 the horizon, till by the revolution of the earth it again appears in the east ; and thus day and 
 night will be alternately produced. 
 
 SOHOL. The time of noon is found, by observing the instant when the centre of the sun is 
 cut by the perpendicular hair in a meridian telescope, as described (Def. XII. Schol. 2.) or by a 
 sun dial. 
 
 SECT. II. 
 
 Of the annual Motion of the Earth round the Sun. 
 
 PROP. VII. The earth revolving round the sun in 365 days, 6 hours, 
 9 minutes, IS seconds, the sun appears to revolve round the earth in the same 
 time. 
 
 Plate 9. Let S represent the sun, BAC the orbit of the earth, and FGHE the starry concave. Whilst 
 
 Fig. 8. the earth is moving from A through B to C, it is manifest that, to a spectator on the earth, the 
 
 sun must appear to move from E through F to G, in the great circle of the heavens formed by 
 
 the plane of the earth's orbit. In like manner, while the earth is passing from C to A, the sun 
 
 will appear to pass from G to E. 
 
 SCHOL 1. It is manifest that the circle in which the sun appears to move is the same, in 
 which the earth would appear to move to a spectator in the sun. Hence the apparent place of 
 the sun being found, the true place of the earth in its orbit is known. 
 
 SCHOL. 2. The orbit in which the earth revolves round the su.n is not a circle but an ellipse, 
 having the sun in its foci. For the computations of the sun's place, upon this supposition, allow- 
 ing for the disturbing forces of the planets, are found to agree with observations. See Prop. 
 XXXIII. 
 
 DEF. XXI. The circle which the sun appears to describe annually in the 
 concave sphere of the heavens, is called the Ecliptic. 
 
 DEF. XXII. A portion of the heavens, about 16 degrees in breadth, 
 through the middle of which passes the ecliptic, is called the Zodiac. 
 
 SCHOL. Within this zone lie the orbits of all the planets. 
 
 DEF. XXIII. The stars in the Zodiac are divided into 13 Signs ; Aries, 
 
 
CHAP. I. OF THE EARTH. 231 
 
 Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, 
 Aquarius, Pisces. Figures, representing these signs, are drawn upon the plate 9. 
 celestial globe, in that portion of its spherical surface, which corresponds to Flg ' 8> 
 the portion of the concave sphere of the heavens^ in which the stars belonging 
 to each sign are respectively placed. 
 
 PROP. VIII. The axis of the earth in every part of the earth's revolu- 
 tion about the sun, makes, with the plane of its orbit, that is, of the ecliptic, 
 an angle of 66 degrees. 
 
 Let BA represent the plane of the ecliptic or earth's orbit, seen edgeways ; S the sun ; and Plate 9. 
 Pp produced the axis of the equator. If the earth be at S, its axis is not perpendicular to the ' ' 
 plane of the ecliptic, but makes an angle with it, PSA, about 66 SO'. In any other part of its 
 orbit, as M, or X, the axis of the earth is still inclined to the plane of the ecliptic in the same 
 angle. 
 
 Coa. 1. The axis, in any one part of the orbit, is in a position parallel to that in which it was plate 9. 
 at any other part of the orbit. Supposing the line FG to represent"the situation of the axis of Fl &- & 
 the earth when at DFG, and to be parallel to the line HI ; then when the earth is at dfg, or any 
 other part of its orbit, its axis fg, will still be parallel to the same line HI, therefore fg is par- 
 allel to FG. 
 
 COR. 2. The planes of the equator and ecliptic make with each other an angle of 23J de- 
 grees nearly. 
 
 SOHOL. The obliquity of the ecliptic is not permanent, but is continually diminishing, by 
 the ecliptic approaching nearer to a parallelism with the equator, at the rate of about a second 
 in a year, or from 50" to 55" in 100 years. The inclination at this time is 23 28' 5", nearly. 
 The diminution of the obliquity of the ecliptic to the equator is owing to the action of the planets 
 upon the earth, especially the planets Venus and Jupiter. The whole variation, it is said, can 
 never exceed one degree, when it will again increase. 
 
 The obliquity of the ecliptic may be thus found. Observe with a good instrument, very 
 accurately divided, the meridian altitude of the sun's centre, on the days of the summer and 
 winter solstice, then the difference of the two will be the distance between the tropics ; the 
 half of which will be the obliquity sought. 
 
 By the same method, the declination of the sun for every day in the year may be found, and 
 a table constructed. See Prop. XVIII. 
 
 DBF. XXIV. The ecliptic being divided into twelve equal parts, each 
 of these parts is called a Sign', and the names of tiie signs in the ecliptic 
 are the same with those in the zodiac, but do not exactly correspond with 
 them. 
 
 a 
 
 DF.F. XXV. The two points in which the ecliptic cuts the equator are 
 called the Equinoctial Points ; the vernal equinox is at the first degree of 
 Aries in the ecliptic ; the autumnal, at the first of Libra. 
 
233 OF ASTRONOMY. BOOK VII. PART 1. 
 
 SCHOL. The moment of time in which the sun enters the equator may be found by obser- 
 vation, the latitude of ; lie place of the observer being known. For in the equinoctial day, or 
 near it, with an instrument exactly divided into degrees, minutes, and parts of minutes, take the 
 meridian altitude of the sun ; if it be equal to the altitude of the equator, or to the complement 
 of the latitude, the sun is then in the equator; but if it is not equal, mark the difference, which 
 will be the declination of the sun. The next day, again observe the meridian altitude of the 
 sun, and gather from thence his declination. If these two declinations be of different kinds, as 
 the one south and the other north, the equinox happens some time between the two observations; 
 if they be both of the same sort, the sun has either not entered the equinoctial, or has passed it. 
 And from these two observations of the sun's declination, the moment of the equinox is thus 
 investigated. 
 
 Plate9. Let CAB be a portion of the eliptic, EAQ an arc of the equator, and let their intersection 
 
 be in A. Let CE be the declination of the sun at the time of the first observation, OD his de- 
 clination in the second observation ; the arc CO will be the motion of the sun in the ecliptic for 
 one day. In the spherical triangle AEC, right-angled at E, we have the angle A which the 
 equator and the eliptic make, as also CE, the declination of the sun, known by observation, by 
 which may be found the arc CA. And in the same manner in the triangle AOD the side AO is 
 found ; and thence the arc CO, which is the sum or difference of the arcs CA, AO. Therefore 
 as CO is to CA, so is 24 hours to the time between the first observation, and the moment of the 
 ingress of the sun to the equinox. 
 
 DBF. XXVI. The points of the ecliptic which are at the greatest dis- 
 tance from the equator, are called the Solstices ; and the ciicles which pass 
 through these points parallel to the equator, are called the Tropics ; the sum- 
 mer solstice is at the first of Cancer, the winter solstice at the first of Capri- 
 corn ; the northern tropic is called the tropic of Cancer, the southern, of 
 Capricorn. 
 
 COR. The sun is once in the year at each of the tropics, and twice at the equator. 
 
 DBF. XXVII. Circles which pass through the poles at right angles to 
 the equator, or any other great circles, are called Secondaries to that circle : 
 the secondary which passes through the equinoctial points, is called the 
 Equinoctial Colure. 
 
 DBF. XXVIII. That pole which is nearest the tropic of Cancer, is called 
 the North Pole ; that which is nearest the tropic of Capricorn, is called the 
 South Pole. 
 
 DKF. XXIX. An imaginary line passing through the centre of the eclip- 
 tic, and perpendicular to the plane of it, \s the Jlxis of the Ecliptic ; its ex- 
 tremities are the Poles of the Ecliptic, and all circles, passing through these 
 poles, and perpendicular to the ecliptic, are its secondaries. 
 
 COR. The axis of the ecliptic makes an angle of 23i degrees nearly with that of the equator. 
 Compare Prop. VIII. Cor. 2. and Schol. 
 
CHAP. II. OF THE EARTH. S33 
 
 DBF. XXX. The Polar Circles are described by the revolution of the 
 poles of the ecliptic about the poles of the equator ; that which is next to the 
 north pole, is called the Arctic circle ; the opposite, the Antarctic circle. 
 
 DBF. XXXI. The Declination of any heavenly body is its distance 
 from the equator; this is either northern or southern. The degrees of decli- 
 nation of any body are reckoned upon a secondary of the equator passing 
 through that body. 
 
 DBF. XXXII. The Right Ascension of any heavenly body is its distance 
 from the first of Aries reckoned upon the equator ; this is measured, by ob- 
 serving the arc which is intercepted between Aries and a secondary to the 
 equator passing through the sun or star. 
 
 DBF. XXXIII. The Latitude of any heavenly body is its distance from 
 the ecliptic, and the degrees of latitude are reckoned on a secondary of the 
 ecliptic, passing through the body. 
 
 DBF. XXXIV. The Longitude of any heavenly body is its distance 
 from the, first of Aries ; and is measured on the ecliptic by the arc intercepted 
 between the first of Aries and the secondary of the ecliptic, which passes 
 through the body ; the longitude increases, as the body recedes from Aries, 
 through the whole revolution, till it reaches 360, or comes again to Aries. 
 
 DBF. XXXV. Two bodies are said to be in Conjunction with each other, 
 when they have the same longitude, or are in the same secondary of the eclip- 
 tic on the same side of the heavens, though their latitude be different ; they 
 are said to be in Opposition, when their longitudes differ half a circle, or they 
 are on opposite sides of the heavens. 
 
 PROP. IX. The axis of the heavens is perpendicular to the planes of 
 all the circles which the heavenly bodies describe in their apparent diurnal 
 motions. 
 
 For the heavenly bodies, from the revolutions of the earth round its axis, appear to move from 
 east to west in circles perpendicular to the axis. 
 
 COR. 1. The planes of all these circles are parallel to the equator. 
 COR. 2. The axis passes through the centres of the circles. 
 
 DBF. XXXVI. The celestial sphere is called right, oblique, or parallel, 
 as the celestial equator is at right angles, oblique, or parallel to the horizon. 
 
 PROP. X. In all places on the equator, the poles lie in the horizon, and 
 all the circles of daily motion make right angles with the horizon. 
 
 so 
 
234 OF ASTRONOMY. BOOK V II. PART I. 
 
 For these places (by Def. XVIII.) having no latitude, the poles (by Prop. IIT.) are neither 
 elevated above nor depressed below the horizon ; and since the equator is 90 degrees from the 
 poles, it is at right angles to the horizon, and also all circles parallel to it. 
 
 PROP. XI. Those who live at the equator are in a right sphere ; and. 
 consequently, their days and nights are always equal. 
 
 Plate 9. The great circle of the celestial equator and its parallels (by last Prop.) make right angles 
 
 Fig. 10. w jt n f| ie horizons of all places in the earth's equator; therefore (by Def. XXXVI.) the inhabi- 
 
 tants of those places live in a right sphere. Hence,/ because the celestial axis PTS is in the 
 plane of their horizon, ami that this axis is at right angles to the plane of the equator, and 
 (by Prop. X.) passes through its centre and through that of all circles parallel to the equator, 
 the plane of the horizon also passes through the centres of these circles : and consequently di- 
 vides the equator and its parallels into two equal parts. One half of these circles will therefore 
 always be above the horizon, and the other half below it. But each of the heavenly bodies in 
 their daily motion describes some one of those circles, and the diurnal motion of the earth is 
 uniform ; therefore any heavely body will, in this situation, be just as long above the horizon as 
 below it. And because this will be the case with respect to the sun, as well as any other body, 
 in whatever part of the heavens he is seen, the days and nights at the equator will always be of 
 equal length. 
 
 PROP. XIT. At the poles of the earth, one celestial pole is in the zenith, 
 and the other in the nadii ; the equator coincides with the horizon, and all 
 the circles of daily motion are parallel to the horizon. 
 
 For the latitude of the poles is 90 degrees from the equator, and the circles of daily motion 
 are parallel to the equator. 
 
 PROP. XIII. Those who live at either pole are in a parallel sphere ; 
 they see the heavenly bodies carried round them iu circles parallel to the hor- 
 izon, and their day and their night continues each half year. 
 
 Plate9. An inhabitant at P has the equator EQ in the horizon, and all its parallel circles also parallel 
 
 Fl 10 - to the horizon. Therefore each of the heavenly bodies, in its apparent daily motion, being in 
 some one of these circles, must describe a path parallel to the horizon ; so that those which are 
 above the horizon will never set by this motion, and those which are below it will never rise. The 
 sun, therefore, in this situation, will not rise or set by the diurnal n ntion of the earth. But 
 from the annual motion of the earth, the sun daily changes its apparent place in the heavens 
 till it has described the circle of the ecliptic CL ; one half of which is above the horizon, and 
 the other half below it, because these circles have a common centre T, the centre of the earth. 
 Therefore, for one half of the year the sun will be in some part of CT, that half of the ecliptic 
 which is above the horizon, and will daily revolve in circles above the horizon ; and for the 
 other half, it will be in some part of TL, and will perform its daily revolutions in circles, 
 below the horizon. 
 
 PROP. XIV. la any place between the poles and the equator, one ce- 
 
CHAP. II. OF THE EARTH. 235 
 
 lestial pole will be elevated, and the other depressed, at an angle less than 
 a right angle ; and the celestial equator will make an angle less than a 
 right angle with the horizon. 
 
 For, since the place is not in the equator, it has some latitude ; and since it is not at either 
 of the poles, its latitude is less than 90 degrees ; whence (by Prop. III.) the poles are elevated, 
 or depressed, in an angle less than a right angle ; and consequently the equator, which is per- 
 pendicular to the axis, makes an angle less than 90 degrees with the horizon. 
 
 PROP. XV. Those who live on any part of the surface of the earth be- 
 tween the equator and either pole, are in an oblique sphere, and have all the 
 circles of daily motion oblique to their horizon. 
 
 Let HO be the horizon of a place which lies between the earth's equator and either of its plate 9. 
 poles ; the celestial equator EQ, and all its parallel circles, will be oblique to the horizon ; and Vl S- 10 - 
 therefore each of the heavenly bodies, being in some one of these circles, will appear to move in 
 a path oblique to the horizon. 
 
 PROP. XVI. When the sun, in his annual apparent course, is in the 
 points in which the ecliptic cuts the equator, the day and night will be of the 
 same length at all places on the surface of the earth ; but when the sun is in 
 any other part of the ecliptic, the days will be longer as the sun's declination 
 toward the elevated pole increases, and shorter as its declination toward the 
 depressed pole increases. 
 
 The plane of the horizon HO, of any place, passing through T, the centre of the sphere and n late 9 
 also through the centre of the equator, divides the equator EQ into two equal parts, one half Fig. 10. 
 above, and the other half below the horizon. When therefore the sun has no declination, or is 
 in the equator, it will appear in its daily revolution to describe the equator EQ, and, therefore, 
 during one half of the revolution, it will be above the horizon, and, during the other half, below it. 
 
 But suppose the sun to have its declination toward P, the elevated pole, equal to Em ; its 
 diurnal apparent revolution will be in the circle nun, the centre of which is in a part of the axis 
 above the horizon ; whence the plane of the horizon does not pass through the centre, and con- 
 sequently the circle mm is divided into two unequal parts, the greater above the horizon, and 
 the less below it. Therefore the sun, describing the circle mm, with an uniform velocity, in its 
 apparent diurnal revolution, will be longer in describing the part above the horizon, than the 
 part below it. And this difference manifestly increases, as the circle of the sun's apparent diur- 
 nal motion recedes from the equator, that is, as the sun's declination toward P increases. In 
 like manner, it may be shown, that the days will be shorter, as the sun's declination toward the 
 depressed pole increases. 
 
 Or thus. Let AB represent the plane of the ecliptic seen edgeways; S the sun in the focus pi a te9. 
 of the orbit ; MO, KL, XY, the earth in different parts of its orbit. If FI, the axis of the eclip- !'> 7. 
 tic BA, were also the axis of the earth, that is, if the planes of the equator and ecliptic were 
 coincident, it is manifest that the sun, the apparent annual motion of which is in the plane of 
 the ecliptic, would at all times of the year appear to move in the circle of the equator, and to 
 
236 OF ASTRONOMY. BOOK VII. PART 1. 
 
 be equally distant from the poles, and consequently could produce, by its apparent motion, no 
 varieties in the length of days and nights. But the earth's axis being inclined to the plane of 
 its orbit, as Pp, when the earth is at MO, the pole P will be toward the sun, and the pole p turn - 
 ed frotn it, and the reverse when the earth is arrived at XY. When the earth is in the middle 
 station between B and A, in either part of its orbit, both the poles will be in the circle illumi- 
 nated as at KL. v 
 
 In the position MO, since the sun must always illuminate one half of the globe, the light will 
 pass beyond the pole P as far as F, and will extend toward the pole^ no farther than I. Con- 
 sequently, in the diurnal revolution of the earth round its axis, while the earth remains in 
 this position, all the parts of the globe between F and G will be illuminated, and all the parts 
 between I and H will be dark. Farther, in this position greater portions of those parallels, which 
 lie between the equator and the circle FG, will at any instant be in the illuminated, than in the 
 dark, hemisphere ; and, on the contrary, greater portions of those which lie between the circle 
 HI and the equator, will at any instant be in the dark, than in the enlightened, hemisphere. 
 Consequently, any given place on the side of the equator toward P, will, in one diurnal revolu- 
 tion, be longer in the light than in the dark, and the reverse on the side toward p- The differ- 
 ence between the length of day -light and night will decrease on either side of the equator, as we 
 approach toward it ; and at the equator, the illuminated and dark portions of the circle being 
 always equal, the days and nights will be of equal length. The contrary to all this will take 
 place in the situation XY. Continual variations will take place, while the earth passes from 
 MO to KL, and from KL to XY. But in the situation KL, the illumination extending exactly 
 to both poles, all the parallel circles are half illuminated, and half dark; consequently, any 
 place upon the globe will, in a diurnal revolution, have equal portions of light and darkness ; 
 that is, day and night will be every where of equal length. This must happen twice in every 
 annual revolution. 
 
 COR. 1. Allbodie?, which are on the same side of the equator with the spectator, continue 
 longer above the hoiizon than below it, and vice versa, 
 
 COR. 2. As the orbits of the moon and planets are inclined to the equator, a variation of 
 the times of their continuance above and below the horizon will take place. 
 
 SCHOL. 1. "When the sun is very near either of the tropics, the days do not appear of different 
 lengths, for the circles of apparent diurnal motion are so near to each other, that they cannot be 
 sensibly distinguished. 
 
 SCHOL. 2. The different degrees of heat at different seasons of the year are owing partly to 
 the different lengths of the days, and partly to the different degrees of obliquity with which the 
 rays fall upon the atmosphere at different altitudes of the sun. 
 
 PROP. XVII. When the sun, or any other heavenly body, is in the 
 equator, it rises in the east, and sets in the west. 
 
 For it then rises and sets in the points in which the equator cuts the horizon ; that is, be- 
 cause the equator is at right angles to the meridian, which passes through the north and south 
 points, in the points of east and west. 
 
 COR. In north latitude, those bodies, which have north declination, rise between the east 
 and north ; those, which have soutn declination, rise between the east and south. 
 
CHAP. II. OF THE EARTH. S37 
 
 PROP. XVIII. When the declination of the sun is toward the elevated 
 pole, its meridian altitude is equal to its declination added to the elevation of 
 the celestial equator ; when its declination is toward the depressed pole, its 
 meridian altitude is equal to its declination, subtracted from the elevation of 
 the equator. 
 
 Let HO be the horizon, T the earth, P and S the celestial poles, Z the zenith, N the nadir, plate 9. 
 EQ the equator. If the sun be at C, having its declination toward the elevated pole P, when it F 'S- 10 - 
 arrives at the meridian PS, its meridian altitude CH is equal to the sum of CK, its declination, 
 and EH, the elevation of the equator. If the sun be at I, having its declination toward the 
 depressed pole S; when it arrives at the meridian, its altitude IH is equal to the difference of 
 EH, the elevation of the equator, and El, the sun's declination, as appears from the figure. 
 
 PROP. XIX. When the declination of a heavenly body toward the 
 elevated pole is equal to the latitude of any place, the body will pass through 
 the zenith of that place ; and when its declination toward the depressed pole 
 is equal to the latitude, it will pass through the nadir. 
 
 Any star or planet which passes through Z, the zenith, in its apparent diurnal revolution, Plate 9. 
 must describe the circle Zz ; whence its distance from the equator or declination will be EZ. %' 
 But EZ is the distance of the zenith from the equator, which, because the elevation of the equa- 
 tor is equal to the complement of latitude, (Prop. IV.) is equal to the latitude. In like manner 
 the reverse may be proved. 
 
 PROP. XX. A heavenly body, seen from any place, will never set from 
 the diurnal motion of the earth, if the complement of its declination toward 
 the elevated pole be equal to, or less than, the latitude of the place ; and it 
 will never rise, if the complement of its declination toward the depressed pole 
 be equal to, or less than, the latitude. 
 
 Let PD, which is the complement of declination of a body at D, and also the distance of the pi a t e 9 
 body at D from the pole, be equal to PO, the elevation of the pole, or (by Prop. III.) the lati- Fig. 10. 
 tude ; it is manifest, that the body at its lowest depression will be no farther from the pole than 
 the horizon is ; that is, will never be below it. In like manner the reverse may be shown. A 
 parallel of declination, as DO, at a distance from the elevated pole equal to the latitude of the 
 place, is called the circle of perpetual apparition ; and a parallel, as H/J, at the same distance 
 from the other pole, the circle of perpetual occultation. 
 
 SCHOL. The latitude of a place may be found, by observing the greatest and least altitude of 
 a fixed star that never sets. 
 
 Let A be a star near the north pole, which in its daily motion describes the circle AB with- pj a t e 9. 
 out setting. A quadrant being placed in the plane of the meridian, or along the meridian line, fl S- 10> 
 observe its altitude when it is at A, and afterward when at B ; the difference of these altitudes 
 . is AB. And since the star, in its revolution about the pole, is always at equal distances from 
 it, if AB he bisected iu P, this point will be the pole, and consequently PO will be the elevation 
 
238 OF ASTRONOMY. BOOK VII. PART I. 
 
 of the pole. But since the lengths of the arcs AO, BO, have been found by observation, their 
 difference AB, and the half of this difference, AP, or BP, is known ; and PO is equal to 
 BP + BO, or to AO PA. Whence the elevation of the pole, that is, the latitude, is equal to 
 the sum of the least altitude added to half the difference of the greatest and least altitude, or it 
 is equal to the remainder arising from subtracting half the difference of the greatest and least 
 altitudes from the greatest altitude. 
 
 Or, the latitude may be found from the sun's meridian altitude and declination. If the sun's 
 meridian altitude, found by a quadrant, be CFI, this altitude is equal to the sun's declination CE. 
 added to the elevation of the equator EH. Therefore, if CE, the declination toward the elevated 
 pole, be taken from the meridian altitude, the remainder EH. will be the elevation of the equator. 
 But since the elevation of the equator is the complement of latitude, the latitude is the comple- 
 ment of the elevation of the equator. This elevation, therefore, being found, the latitude of the 
 place is known. 
 
 Or, the latitude of a place is equal to the sun's meridian zenith distance, added to his decli- 
 nation, when he passes the meridian between the zenith and the equator. 
 
 Exp. I. To find the latitude from an observation of the sun's altitude, Aug. 7, 1776, at the 
 Observatory at Cambridge. 
 
 Apparent meridian altitude of the sun's lower limb 55 46' 8" 
 
 Sun's apparent semi-diameter, from the Ephemeris 15 50 
 
 Apparent altitude of sun's centre - - 54 1 58 
 
 Deduct for refraction - - - . - - - -00 41 
 
 Altitude of the sun's centre ...... 54 1 17 
 
 Zenith distance of the sun's centre is found by subtracting? 
 the last altitude from 90 J 
 Add the sun's declination 16 13 57 
 
 Latitude of the place ... . 52 12 40 
 
 Exp. 2. Dec. 1, 1793. The observed meridian altitude of Sirius was 59 50'; required the 
 latitude. 
 
 Observed latitude ... 5& 50' S. 
 
 Therefore, Zenith distance - - - - 30 10 N. 
 
 Declination of Sirius - - 16 27 S. 
 
 Consequently, the latitude required - 13 13 N. 
 
 DBF. XXXVII. The two tropics and two polar circles upon the sufface 
 of the earth, divide it into five parts, called Zones ; the torrid zone lies be- 
 tween the two tcopics ; the temperate zones between the tropics and polar 
 circles ; and the frigid zones between the polar circles and the poles. 
 
 PROP. XXI. At any place in the torrid zone the sun is vertical twice 
 every year. 
 
 The sun in passing from the equator to the tropic of Cancer, 23 degrees from the equator, 
 has every northern declination from to 25^ ; and every place between the equator and the 
 
CHAP. II. OF THE EARTH. 239 
 
 tropic of Cancer has some northern latitude between and 23* ; therefore, in some part of its 
 course from the equator to the tropic of Cancer, the sun must have a declination equal to the 
 latitude of every place between the equator and the tropic ; whence it must be oiice in the 
 zenith of every such place in its course toward the ttopi- of Cancer. For the same reason it 
 must be once in the zenith of every such place in its course from the tropic to the equator. The 
 like may be shown on the southern side of the equator. 
 
 PROP. XXII. The sun is vertical once every year at the places which 
 lie in the tropics. 
 
 For the sun's declination is then 2S|- degrees, equal to the latitude of the tropics. 
 
 PROP. XXIII. At the polar circles, the longest day and the longest, 
 night is 4t hours. 
 
 "When the sun is in the tropic of Cancer, the complement of its declination toward the elevated 
 pole is 66 degrees, equal to the latitude of the arctic polar circle ; on this day, therefore, (by 
 Prop. XX.) the sun will not set. When the sun is in the tropic of Capricorn, the complement 
 of its declination toward the depressed pole will be 66| degrees, equal to the latitude of the 
 artic pole ; whence the sun will not rise during that day. The same may be shown with respect 
 to the antarctic circle. 
 
 PROP. XXIV. The longest day, and the longest night, are each of 
 them more than 24 hours within the frigid zone. 
 
 For, while the sun's complement of declination toward the elevated pole is less than the lati- 
 tude of the place, the sun will nut set ; while the complement of declination toward the depressed 
 pole is less than the latitude of the place, it will not rise ; but this must be the case with respect 
 to every place withia the frigid zones, in some part of tiie sun's course toward the tropics. 
 
 PROP. XXV. The sun is never vertical to any place in either of thfc 
 temperate zones. 
 
 For the latitude of all places in the temperate zone is greater than any declination of the sun. 
 
 PROP. XXVI. The longest day, and the longest night, in any part of 
 the temperate zones, are less than 24 hours ; and the days and nights will be 
 longer, the nearer the place is to the polar circles. 
 
 For the complement of the sun's declination can never be less than, or equal to, the latitude 
 of any place in the temperate zo ;es; whence the sun will rise and set every day within these 
 zones. But he farther any place is removed from the equator, the nearer the latitude approaches 
 to an equality with the complement of the sun's greatest declination, when the dav is 24 hours ; 
 that is, at the polar circles. 
 
 PROP. XXVII. At different places, the hour of the day differs in pro- 
 portion to the difference of longitude ; 15 degrees of longitude making the 
 
240 OF ASTRONOMY. BOOK VII. PAKT I. 
 
 difference of one hour in time, 15' one minute of time, 15'' one second of time ; 
 and it is seen at any given place sooner than at places which lie to the west 
 of it, arid later than at places which lie to the east of it. 
 
 The sun in its daily apparent motion, which is from east to west, must arrive at the meridian 
 of any given place, as London, sooner than it will arrive at the meridian of any place which lies 
 to the west of London, and later than at the meridian of any place to the east of London, that 
 is, since it is noon at any place when the sun is in its meridian, it will be noon at London sooner 
 than at places west, and later than at places east of it. 
 
 For example, if any place lies 15 degrees east of London, that is, has 15 degrees of eastern 
 longitude from London taken as the first meridian, the sun will be one hour sooner at its meridian 
 than at the meridian of London j for, since the sun every day appears to make a complete revo- 
 lution from any meridian to the same, in 24 hours, it will in every hour describe a 24th part of 
 the circle, that is, 15. And since a minute of a circle is a 60th part of a degree, and a second 
 of a circle a 60th part of a minute, and 15' the 60th part of 15, and 15" the 60th part of 15', 
 the sun will move at the rate of 15' in every 60th part of an hour, and 15" in every 60th part of 
 a minute, that is, in every minute or second of time. Consequently, it will be noon one minute 
 or one second sooner at a place which is 15' or 15" east of London, than at London. 
 
 PROP. XXVIII. The difference of longitude at two places may be 
 found by observing, at the same time from both places, some instantaneous 
 appearance in the heavens. 
 
 If the eclipse of Jupiter's innermost satellite, on the instant of its emersion into the shadow 
 of Jupiter, be observed by two persons at different places, it will be seen by both at the same 
 instant. But if this instant be half an hour, for example, sooner at one place than at the other, 
 because the places differ half an hour in their reckoning of time, their difference of longitude (by- 
 Prop. XXVII.) is 7 30'. 
 
 SCHOL. From tables of eclipses correctly calculated for any place, the longitude of any place 
 may be found by one observer. But such observations can only be made with certainty by land, 
 on account of the motion of a ship at sea. In order to determine accurately the longitude at 
 sea, it is necessary to have a clock which shall not be sensibly affected by difference of climate, 
 difference of gravity at different places, or the motion of the ship. Such a clock, set for the 
 meridian of London, would constantly show the hour of the day at London, \fliich it is easy to 
 compare with the hour of the day where the ship is, found by observations on the sun or stars. 
 
 PROP. XXIX. Those who live in opposite semicircles of the same 
 meridian, but in the same circle of latitude, have opposite hours of the day, 
 but the same seasons. 
 
 Being both on the same side of the equator and at the same distance from it, when the sun's 
 declination makes it summer or winter in one of the places, it will be the same at the other ; 
 but because they arc distant from each 180 degrees of longitude, when it is noon at one place it 
 will be midnight at the other ; these are called Fericeci, 
 
CHAP. H. OF THE EARTH. 
 
 PROP. XXX. Those who live in opposite circles of latitude, but in the 
 same semicircle of the meridian, have opposite seasons of the year, but the 
 same hour of the day. 
 
 When the sun has declination toward the north pole, itwiH be summer to those who live in 
 the northern circle of latitude, and winter to those who live in the southern circle of latitude. 
 But, having the same longitude, their hours of the day will be the same; these are called 
 Jlntceci. 
 
 PROP. XXXT. Those who live in opposite circles of latitude and oppo- 
 site semicircles of the meridian, have both opposite seasons of the year, and 
 opposite hours of the day. 
 
 Because they are in opposite latitudes, they will have opposite seasons ; and because they are 
 in opposite semicircles of the meridian, they will have noon when it is midnight at the other 5 
 these are called Antipodes. 
 
 DBF. XXXVIII. Twelve secondaries to the celestial equator being con- 
 ceived to be drawn at equal distances from each other, that is, dividing the 
 equator into 24 equal parts, and the meridian of any place being made one of 
 these secondaries, they are called Hour-Circles of that place. Compare 
 Prop. XXVII. 
 
 PROP. XXXII. If the celestial sphere had an opaque axis, the shadow 
 of the axis would always be opposite to the sun ; and when the sun was on 
 one side of any hour-circle, the shadow of the axis would fall upon the oppo- 
 site side of the same hour-circle. 
 
 For all the hour-circles, being secondaries to the equator, pass through the poles, and the ce- 
 lestial axis is in the plane of every hour-circle. And the shadow of any opaque body, being op- 
 posite to the sun, is in the plane with the sun. Therefore in whatever hour-circle the sun is, 
 the shadow of the supposed opaque axis would be in the plane of that circle and opposite to the 
 sun ; that is, while the sun is in one semicircle of any hour-circle, the shadow of the axis would 
 fall upon the opposite semicircle. 
 
 COR. Hence as the sun performs its apparent course from east to west, the shadow of the 
 supposed axis would move from west to east. 
 
 SCHOL. The gnomon of a sun-dial represents the supposed axis, and hence its shadow is 
 a measure of time. 
 
 To construct a Horizontal Dial. 
 
 In every sun-dial the gnom.on, when fixed, is parallel to the earth's axis. Now when the sun 
 is in the meridian of any place, the 12 o'clock hour-circle is perpendicular to the plane of the 
 Tiorizon, and the arc from the pole to this plane is equal to the latitude of the place ; aiid the one 
 
 31 
 
OF ASTRONOMY. BOOK V EL PART I. 
 
 o'clock hour-circle makes an angle at the pole with it of 15, and forms the hypothenuse of a 
 right-angled triangle to the above perpendicular, and the base is the arc measuring the angle 
 between 12 and 1 o'clock; therefore we have, by Spherical Trigonometry, Rad : Sin. L:: 
 tan. 15 : tan. of the hour-angle between 12 and 1 o'clock. If instead of 15, we substitute SO, 45, 
 &c. we get the angles between 12 and 2, 3, &c. o'clock ; the same may be done for the half 
 hours or other divisions. 
 
 Note. The rational and sensible horizons are, in this case, supposed coincident, which, on 
 account of the sun's great distance, will not occasion any sensible error. 
 
 To construct a Vertical South -Dial. 
 
 In doing this we must conceive a plane passing through the centre of the earth perpendicu- 
 lar both to the horizon and meridian ; and on the south side, lines must be drawn from the cen- 
 tre to the points where the hour-circles cut that plane. In finding these points, we say, as 
 Had : co. s. Lat :: tan. 15 : tan. of the hour-angle between 12 and 1 o'clock.. For the arc of the 
 meridian, from the pole to the plane, is equal to the complement of latitude. The other hour- 
 angles, &c. must be obtained in the same way as in the last. 
 
 On the subject of Dialing, see Furguson's Lectures. 
 
 PROP. XXXIII. The orbit in which the earth revolves about the sun, 
 is elliptical. 
 
 It is known from observation, that the apparent motion of the sun, that is, the real motion of 
 the earth, in the ecliptic, is not uniform. But by the universal law of bodies revolving about a 
 centre, if its orbits were circular, its velocity must be uniform ; since (Book II. Prop. LXXII.) 
 Plate 9. it must describe equal areas in equal times. Whereas, if its orbit be an ellipse, and the sun be 
 Fl S' 8. placed in one of the/oci, the same law will require (see Book II. Prop. LXV1I1.) that its veloc- 
 ity should not be uniform, but that in passing through its greatest distance C, to its least dis- 
 tance A, it should be accelerated, and in passing from the least distance A to the greatest 
 C, it should be retarded. Since then the motion of the earth is in fact thus retarded and ac- 
 celerated in different parts of its orbit, it is manifest, that its orbit is elliptical. 
 
 DBF. XXXIX. The greatest distance of the earth or any other planet from 
 the sun, is called its Aphelion ; its nearest distance, its Perihelion ; the lon- 
 ger axis of the ellipse is called the Linea Jlpsidum, the aphelion is also call- 
 ed the Summa Apsis, and perihelion the Ima Apsis. 
 
 DBF. XL. The Eccentricity of the earth, or any planet, is the distance 
 between the sun and the centre of the elliptical orbit. 
 
 PROP. XXXIV. The sun is eight days longer in performing its appa- 
 rent course through the six northern signs, than through the six southern 
 signs. 
 
 Plate 9. Let ABCD be the orbit of the earth, S the sun, and EFGH the ecliptic. While the earth 
 
 *'' 8> moves in its orbit from B through C to D, the sun appears to move in the ecliptic from F through 
 
 G to H, passing through the six northern signs; and while the earth passes from D through A 
 
CHAP. II. OF THE EARTH. 343 
 
 to B, the sun appears to move from H to F, through the six southern signs. Now the line HP 
 bisects the circle EFGH. but divides the ellipse ABCD unequally. And, while the sun appears 
 to pass through the northern signs, the earth passes through more than half of its orbit; and 
 while the sun appears to pass through the southern signs, the earth passes through less than half 
 of its orbit. Therefore, it the velocity of the motion of the earth were uniform, the sun must 
 appear to be longer in passing through the six northern than the six southern signs. But whilst 
 the earth is passing through the greater part of its orbit BCD, it is farther from the sun, and 
 consequently moves slower than in the lesser part DAB. On both these accounts, the sun's ap- 
 parent motion is slower in the northern signs than the southern ; the difference is found by ob- 
 servation to be about eight days. 
 
 
 
 PROP. XXXV. The apparent diameter of the sum is greater in winter 
 than summer. 
 
 It is found by observation, that the diameter of the sun in winter is 32 minutes, 47 seconds ; 
 in summer, 31' 40''. And his mean apparent diameter is 32.' 12", according to Sir I. Newton, 
 in his theory of the Moon. 
 
 COR. Hence it appears, that the earth, at the winter solstice, or Capricorn, is in its peri- 
 helion. 
 
 SCHOL. 1. The difference between summer and winter in the degrees of heat, is owing 
 chiefly to the different heights to which the sun rises above the horizon, and the different lengths 
 of the days. When the sun rises highest, in summer, its rays fall less obliquely, and conse- 
 quently, more of them fall on the earth's surface than in winter ; and when the days are long, 
 and the nights short, the earth and air are more heated in the day than they are cooled in the 
 night, and the reverse. 
 
 SOHOL. 2. The doctrine of the Sphere having been explained in the preceding propositions, 
 some of the more useful Problems to be performed on the Terrestrial and Celestial Globes are 
 here subjoined. 
 
 PROBLEM I. To find the latitude of any place. Bring the place to the graduated side of the 
 fixed brass meridian ; the degree, under which it is found, is its latitude. All places under the 
 same degree are in the same latitude. Thus the latitude of London is Slj north, that of the 
 Cape of Good Hope 34 south. 
 
 PROB. II. To find the longitude of any place. Bring the place to the fixed meridian ; the 
 distance of this meridian from the first meridian, measured on the equator, is the longitude of 
 the place. The longitude of Boston in New England is 70\ west, or 4 hours, 42 minutes in 
 time. That of Rome 12-i east, or 50 minutes in time. 
 
 PROB. III. To rectify either globe to the latitude of any place, the zenith, and the sun's 
 place. If the place be in the northern hemisphere, raise the north pole above the horizon ; but 
 if the place be in the southern hemisphere, raise the south pole. Then move the meridian up 
 and down in the notches, till the degree of the place's latitude, counted upon the meridian, be- 
 low the pole, cuts the horizon : and then the globe is adjusted to the latitude of the place. 
 
 Having elevated the globe according to the latitude of the place, count the same number of de- 
 grees upon the meridian, from the equator toward the elevated pole, and that point will be the zenith 
 
344 OF ASTRONOMY. BOOK VII. PART I. 
 
 or vertex of the place. To this point of the meridian screw the quadrant of altitude, so that its 
 graduated edge may be joined to the said point; then is the globe rectified for the zenith. 
 
 Bring the sun's place in the ecliptic to the meridian, and set the hour-index to 12 at noon : 
 and then the globe will be rectified for the sun's place. 
 
 PROB. IV. To determine the difference of time in different places. Find the longitude of 
 each place, and reduce the difference into time, allowing an hour for every 15 degrees, and pro- 
 portionally for lesser parts; the difference of time will be found. If the place lies westward of 
 another, it has its noon later than that other ; if eastward, sooner. The longitude of Rome is 
 12^ east, that of Constantinople 29, the difference is 17|, consequently the difference of time 
 between Rome and Constantinople is 1 hour 10 minutes. 
 
 PROB. V. The latitude and longitude of any place being known, to find the place on the 
 globe. Bring the degree of the equator which expresses the given longitude to the fixed merid- 
 ian, then find the given latitude on the meridian ; under this point is the place sought. 
 
 PROB. VI. To find the distance between any two places, and their bearing, or relative situa- 
 tion with respect to the points of the compass. Rectify the globe to the latitude of one of the 
 places, and bring that place to the fixed meridian. Then fix the quadrant of altitude to the up- 
 permost point of the meridian, and putting its lower end between the horizon and the globe, 
 slide it along, till it passes through the other place. The number of degrees on the quadrant 
 between the two places, will give their distance, allowing 69| English miles for each degree ; 
 and the number of degrees upon the horizon between the meridian and the quadrant, will 
 give the bearing of the second place with respect to the first. Thus the bearing of the Lizard 
 Point from the island of Bermudas is nearly E. N. E. 
 
 PROB. VII. To find the right ascension and declination of the sun, or any star. On the 
 celestial globe find the day of the month under the ecliptic, against which is the sun's place, or 
 find his place by an ephemeris ; bring that point under the meridian, and the degree which is over 
 the point is the sun's declination, and the degree of the equator then under the meridian will be 
 the sun's right ascension. A star's declination and right ascension are found, by bringing the 
 star on the globe to the meridian, and proceeding as with respect to the sun. The sun's decli- 
 nation, April 19, is 11 14' north, and his right ascension 27 30'. The right ascension of Sirius 
 is 99, its declination 16 20' south. 
 
 PROB. VIII. To find what stars pass over, or near, the zenith of any place. Having found 
 the latitude of the place on the terrestrial globe, all those stars on the celestial globe, which 
 pass under the same degree of the meridian with the given latitude, become vertical at that 
 place. 
 
 PHOB. IX. To find what stars never rise, nor never net, in any yiven place. The globe 
 being rectified for the given place, those stars which do not pass under the wooden horizon never 
 set ; those which do not come above it, never rise. 
 
 PROB. X. To represent the appearance of the heavens at any time. Rectify the globe to the 
 latitude ; bring the sun's place in the ecliptic to the meridian, and set the horary index to the 
 upper 12th hour; then turn the globe till the index points to the given hour. The north pole 
 of the globe must be set to the north in the heavens, then will all the stars upon the globe cor- 
 respond to their places in the heavens, so that an eye at the centre of the globe would refer 
 every star upon its surface to the place of the stars in the heavens. By comparing, therefore, 
 the stars in the heavens with their places on the globe, a person will easily get acquainted with 
 all the stars. 
 
CHAP. II. OF THE EARTH. 343 
 
 Example. The situation of the stars at London on the 9th of February, at nine o'clock in the 
 evening, is as follows ; Sirius, or the dog-star, is on the meridian, its altitude 22; Procyon, the 
 little dog-star, 16 toward the east, its altitude 43 ; about 24 above this last, and a little more 
 toward tlie east, are Castor and Pollux ; 3. 65 E. and 35 in height is Regulus, or Cor Le- 
 onis; exactly in the east, and 22 high, is Deneb in the X,ion's-tail ; 30 from the east toward 
 the north, Arcturus is about 3 above the horizon ; directly over Arcturus and 31 above the 
 horizon, is Cor Caroli ; in the north-east are the stars in the extremity of the Great Bear's-tail. 
 
 Reckoning westward, we see the constellation Orion ; the middle star of the three in his belt 
 is S. 20 W. its altitude 35 ; nine degrees below the belt, and a little more to the west is Rigel, 
 the bright star in his heel ; above his h^lt in a straight line drawn from Rigel, between the middle 
 ami most northward in his belt, and 9 above it, is the bright star in his shoulder ; S. 49 W. and 
 45-^ above the horizon is Aldebaran, the southern eye of the Bull ; a little to tlie west of Aldeba- 
 ran, are the Hyades ; the same altitude, and about S. 70 VV. are the Pleiades ; in the W. by 
 S. is Capella in Auriga, its altitude 73; in the north-west, and 42 high, is the constellation 
 Cassiopeia ; and almost in the north, near the horizon is the constellation Cvgnus. 
 
 PROB. XI. The latitude of a place being given, to find the time of the sun's rising and set- 
 ting, on any given day, at that place. Having rectified the globe according to the latitude, bring 
 the sun's place in the ecliptic to the graduated edge of the meridian, and set the horary index to 
 the upper 12. Then turn the globe to bring the sun's place to the eastern part of the horizon, 
 the index will point to the hour at which the sun rises; on the western side, to the time of its 
 setting. On the 5th of June the sun rises at Sh. 40min. and sets at 8h. 20min. The length of 
 the day is 16h. 40min. that of the night 7h. 20min. 
 
 PROB. XII. To find all the places on the globe to which the sun will be vertical on a given 
 day. Bring the sun's place to the fixed meridian, and observe the point of the sun's declina- 
 tion ; all the places which, in turning the globe round, pass under that point, have the sun verti- 
 cal on the given day. 
 
 When the sun's declination is equal to the latitude of any place, then the sun will be vertical 
 to the inhabitants of that place. 
 
 PKOB. XIII. To find the sun's amplitude. The globe being rectified for the latitude of the 
 place, bring the sun's place to the eastern side of the horizon ; the arc of the horizon intercepted 
 between that point and the eastern point, is the sun's amplitude at rising. Thus on the 24th of 
 May, the sun's amplitude at rising is 30 east, and he sets with 36 western amplitude. The 
 amplitude increases with the latitude of the place. 
 
 PROB. XIV. To find the sun's meridian altitude. The globe being rectified for the latitude, 
 zenith, and sun's place, the number of degrees contained between the sun's place and the zenith 
 is the distance of the sun from the vertex at noon ; the complement of which to 90 degrees, is, 
 the sun's altitude. The meridian altitude of the sun on the 17th of May, at London, is 57 55'. 
 The altitude being given, bring the quadrant of altitude to meet the sun's place and the inter- 
 section of the quadrant and horizon will show the azimuth. Thus, on the 21st of August, at 
 Lond'in, wlien the sun's altitude is 36 in the forenoon, the azimuth is 60 from the south. 
 
 PROB. XV. To find the fita.ce of any heavenly body upon the globe, its longitude and latitude 
 being given. Place the first degree of the quadrant of altitude upon that degree of the ecliptic 
 which expresses the given longitude, and the 90th degree on the pole of the ecliptic ; the point 
 of the globe which is under that degree of the quadrant which expresses the given latitude, is the 
 place of the body ; for the quadrant represents a secondary of the ecliptic, an arc of which 
 
46 OF ASTRONOMY. BOOK VII. PART I. 
 
 between the body and the ecliptic is its latitude, and the arc of the ecliptic between the secon- 
 dary and the first degree of Aries its longitude. 
 
 PROB. XVI. To find the place of any heavenly body upon the globe, its right ascension and 
 declination being given. Bring that point of the equator which expresses the given right ascen- 
 sion to the meridian : the place sought is under that degree in the meridian, north or south, 
 which expresses the given declination. 
 
 PROB. XVII. To find all those places where it is noon, at any given hour of the day, at any 
 given place. Bring the given place to the brass meridian, and set the index to the uppermost 
 12 ; then turn the globe till the index points to the given hour, and it will be noon to all the 
 places under the meridian. When it is 4h. 50m. in the afternoon at Paris, it is noon at New 
 Britain, St. Domingo, Terra Firma, Peru, Chili, and Terra del Fuego. As the diurnal motion 
 of the earth is from west to east, it is plain that all places which are to the east of any meridian 
 must necessarily pass by the sun before a meridian which is to the west can arrive at it. 
 
 PROB. X VIII. The hour being given at any place, to tell what hour it is in any other part of 
 the world. Bring the place where the time is given, under the meridian ; set the hour index to 
 the given time, and turn the globe till the other place comes under the meridian, and the horary 
 index will point to the hour required. Thus, when it is nine in the morning at London, it is 
 half past four in the afternoon at Canton, in China ; when it is three in the afternoon at Lon- 
 don, it is 18 minutes past ten in the forenoon at Boston, in America. 
 
 PROB. XIX. To find the antceci, the periceci, and the antipodes of any place. Bring the 
 given place to the meridian, then, in the opposite hemisphere, and in the same degree of latitude 
 with the given place will be the antceci. The given place remaining under the meridian, set the 
 index to 12, and turn the globe till the other 12 is under the index ; then the periceci will be 
 under the same degree of latitude with the given place, and the antipodes of the given place will 
 now be under the same point of the brazen meridian where the antceci stood before. 
 
 PROB. XX. To find the two days on which the sun is in the zenith of any given place between, 
 the tropics. That parallel of declination which passes through the given place, will cut the 
 ecliptic line upon the globe in two points, which denote the sun's place ; against which, on the 
 horizon, will be found the days required. The sun is vertical at Barbadoes, April 24, and 
 August 18. 
 
 PROB. XXI. The time and place being given, to find all those places where the sun is rising 
 setting, culminating ; and also where it is day light, twilight, or dark-night. Find the place 
 where the sun is vertical at the given hour, rectify for the latitude of that place, and bring it to 
 the meridian. Then all the places that are in the west semicircle of the horizon, have the sun 
 rising; those in the east simicircle, having it setting ; those under the meridian above the hori- 
 /.on, have it culminating ; and all places above the horizon, have the sun so many degrees above 
 the horizon as the places themselves are. Those places that are below the horizon, but within 
 18 of it, have twilight ; those lower than 18 have dark-night ; and to those under the meridian 
 it is midnight. 
 
 PROB. XXII. To find the place of the moon, or any planet, for any given time. Take the 
 nautical almanack, or White's Ephemeris, and against the given day of the month will be found 
 the degree and minute of the sign which the moon or planet possesses at noon. The degree thus 
 found, and marked in the ecliptic on the globe, you may proceed to find the declination, right 
 ascension, latitude, longitude, altitude, amplitude, azimuth, rising, setting, &c. 
 
 PHOB. XXIII. To find the planets which are above t lie horizon at sun-set, upon any given 
 
CHAP. II. OF THE EARTH. 247 
 
 day and latitude. Find the sun's place for the given day, bring it to the meridian, set the index 
 hour to !2, and elevate the pole for the given latitude, then bring the sun's place to the western 
 semicircle of the horizon, and observe what signs are in that part of the ecliptic above the hori- 
 zon, then look to the ephemeris for the day, and it will be seen what planets are in those signs, 
 for such will be visible on the evening of that day. 
 
 PROE. XXIV. To find whether Venus be a morning or an evening star. Rectify the globe 
 for the latitude and sun's place ; find the situation of Venus by an ephemeris, and stick there a 
 small black patch ; bring the sun's place to the edge of the eastern horizon. If Venus be in 
 antecedently that is, for instance, in Taurus when the sun is in Gemini, she will be a morning 
 star. 
 
 But, if Venus be in conseqmntia, that is, for example, in Gemini when the sun is in Taurus, 
 she will set after the sun, and be an evening star. 
 
 PROB. XXV. To find all the places to which a lunar eclipse will be visible at any instant. 
 Find the place to which the sun is vertical at the given time, and bring that place to the zenith, 
 and the eclipse will be visible to all the hemisphere below the horizon, because the moon is oppo- 
 site to the sun. 
 
 SCHOL. 1. Since lunar eclipses continue, in general, for a considerable time together, they 
 may be seen in more places than in one hemisphere of the earth ; for, by the earth's rotation 
 about its axis, during the time of the eclipse, the moon will rise to several places after its com- 
 mencement. 
 
 SCHOL. 2 We cannot by a globe only, determine the places to which a solar eclipse is visible, 
 because that eclipse does not happen to the whole hemisphere next the sun, nor does it happen 
 at the same time to those places where it is visible. Calculations are therefore necessary. 
 
 SECT. III. 
 Of Twilight. 
 
 PROP. XXXVI. The atmosphere by refracting the rays of light in- 
 creases the apparent altitude of all the heavenly bodies, except when they are 
 in the zenith. 
 
 Let AB be a portion of the earth's surface, ab the upper part of the atmosphere over it, HO Plate 16. 
 the horizon of the place A, S the sun or a star seen from A. The rays of light proceeding from ft S- 2> 
 S on entering the atmosphere are refracted toward the perpendicular, (by Prop. XI. B. 6.) and as 
 the air increases in density in approaching the earth, the light is constantly passing from a rarer 
 to a denser medium, and of course constantly refracted more and more toward a perpendicular, 
 describing a curve, the character of which has not been determined. Now (by Prop. LXXX. B. 6.) 
 the body S will appear at s in a line which is a tangent to this curve in the point A, and as the 
 refraction is toward the perpendicular the point s or apparent place of the body S will be nearer 
 the zenith or have a greater altitude than (he body itself. As in all other cases of refraction the 
 larger the angle of incidence (or the less the altitude) the greater is the refraction or angle 
 of deviation. In the horizon this anjjle is about S3', so that the whole of the sun's disc is appar- 
 ently above the horizon when it is really below it. By this means we enjoy the sight of the sun, 
 in latitude 42 about 6m. longer every day in clear weather than we should do without this 
 
348 
 
 OF ASTRONOMY. 
 
 BOOK VII. PART I. 
 
 refraction. This elevation diminishes very fast in the first degrees of altitude, and when the body 
 is in the zenith, the rays coincide with the perpendicular, and (by Prop. XIV. B. 6.) suffer no 
 refraction. 
 
 Tables of refractions have been calculated by various astronomers, as Sir I. Newton, Mr. 
 Thomas Simpson, Dr. Bradley, Mr. Mayer, &c. The following specimen is taken from Dr. 
 Bradley's table, which is esteemed the most correct, and chiefly used by astronomers. For the 
 method of calculating these tables, see Mr. Simpson's Dissert, p. 46, 4to. Gregory's Astron. 
 Vol. I. Pr. 66 j and Vince's Astron. Vol. I. 4to. ch. 7. 
 
 MEAN ASTRONOMICAL REFRACTIONS IN ALTITUDE. 
 
 AI>P. 
 
 Alt. 
 
 Refraction. 
 
 App. 
 
 Alt. 
 
 Refraction 
 
 App 
 Alt. 
 
 Refraction. 
 
 App 
 Alt. 
 
 Refraction. 
 
 App 
 Alt. 
 
 Refraction. 
 
 App 
 
 Alt. 
 
 Retraction. 
 
 o 
 
 / ft 
 
 o 
 
 / // 
 
 
 
 / '/ 
 
 o 
 
 / // 
 
 o 
 
 II 
 
 O 
 
 II 
 
 
 
 35 
 
 11 
 
 4 47 
 
 23 
 
 2 14 
 
 35 
 
 21 
 
 48 
 
 51 
 
 78 
 
 12 
 
 1 
 
 24 29 
 
 12 
 
 4 23 
 
 24 
 
 2 7 
 
 36 
 
 18 
 
 50 
 
 48 
 
 80 
 
 10 
 
 2 
 
 18 35 
 
 13 
 
 4 3 
 
 25 
 
 2 2 
 
 37 
 
 16 
 
 52 
 
 44 
 
 82 
 
 8 
 
 3 
 
 14 36 
 
 14 
 
 3'45 
 
 26 
 
 1 56 
 
 38 
 
 13 
 
 55 
 
 40 
 
 85 
 
 5 
 
 4 
 
 11 51 
 
 15 
 
 3 SO 
 
 27 
 
 51 
 
 39 
 
 10 
 
 58 
 
 35 
 
 88 
 
 2 
 
 5 
 
 9 54 
 
 16 
 
 3 17 
 
 28 
 
 47 
 
 40 
 
 8 
 
 60 
 
 33 
 
 89 
 
 1 
 
 6 
 
 8 28 
 
 17 
 
 3 4 
 
 29 
 
 42 
 
 41 
 
 1 5 
 
 62 
 
 30 
 
 90 
 
 
 
 7 
 
 7 20 
 
 18 
 
 2 54 
 
 30 
 
 S8 
 
 42 
 
 1 S 
 
 65 
 
 26 
 
 
 
 8 
 
 6 29 
 
 19 
 
 2 45 
 
 31 
 
 35 
 
 43 
 
 1 ] 
 
 68 
 
 23 
 
 
 
 9 
 
 5 48 
 
 20 
 
 2 35 
 
 32 
 
 31 
 
 44 
 
 59 
 
 70 
 
 21 
 
 
 
 10 
 
 5 15 
 
 21 
 
 2 27 
 
 33 
 
 28 
 
 45 
 
 57 
 
 72 
 
 18 
 
 
 
 
 
 22 
 
 2 20 
 
 34 
 
 24 
 
 
 
 75 
 
 15 
 
 
 
 PROP. XXXVII. The atmosphere above the horizon is enlightened 
 by the rays of the sun, when the sun itself is below the horizon. 
 
 Plate 9. Let ADL be the surface of the earth ; CBM, the surface of the atmosphere ; A, any place 
 
 Fig. 11. upon the earth : PABN, the sensible horizon. When the sun is at G, any point below the hori- 
 zon, it cannot be directly seen by a spectator at A. But, because rays from the sun at G can 
 pass to the part of the atmosphere above the sensible horizon of the place A, this part of the 
 atmosphere will be illuminated before the sun rises, or after it sets, and will become visible by 
 reflection to the spectator at A ; that is, twilight will be produced. 
 
 Con. The atmosphere is the cause of the apparent elevation of the heavenly bodies above their 
 true places. 
 
 PROP. XXXVIII. When the evening twilight ends, or the morning 
 twilight begins, a ray of the sun reflected from the highest part of the atmos- 
 phere describes, after reflection, a line, which is the plane of the sensible 
 horizon. 
 
 Plate 9. As the sun is depressed, the extreme ray of light from the sun gradually recedes from C 
 
 Fig. 11. toward B, till at last it touches the horizon at B, from whence it is reflected in the direction BA, 
 the plane of the horizon. 
 
CHAP. II. OF THE EARTH. 
 
 PROP. XXXIX. If the time of the beginning of the morning or end of 
 the evening twilight is known, the height of the atmosphere may be deter- 
 mined. 
 
 It is found, that at the beginning of the morning, or tire end of the evening twilight, the sun Plate 16. 
 is 18 degrees below the horizon. And the sun's centre being 18 below the horizon, when the *>' 
 first ray of light appears, his upper limb must be depressed 18 16' = 17 44', and if the sun. 
 were a point, it would be only 17 44' below the horizon at the beginning of twilight, or, subtract- 
 ing the refraction which takes place in passing from S to D, 17 1 1'. Hence AED = 17 11', 
 and its half AEB = 8 35-%', and if the rays 1)B, BA, were not refracted, the upper part of the 
 atmosphere would be at B, but the ray SD tangent to the surface at D is bent into a curve D6, 
 and at b is reflected and moves in a similar curve from b to A. Draw be tangent to the curve 
 bA in b to cut AB in c, and Aa parallel to cb to cut EB in a. Hence Bcb = BAa = the hori- 
 zontal refraction 33', since the tangent at b lies in the direction of the ray on its reaching the 
 atmosphere, and it is evident that the real height of the atmosphere falls between HB and Ha, to 
 calculate which we have in the triangle AEB all the angles, viz. EAB = 90, AEB = 8 33^' 
 ABE = 81 24^', and AE = 1 (radius of the earth) to find EB = 1.01155, and HB = .01135. 
 And in the triangle AE, we have EAo = 89 27', EaA = 8157i', AEa= S35|', to find 
 Ea = 1.00989, and Ha = .00989. Hence Hfr is greater than 0.00989 and less than 0.01135, the 
 radius of the earth being 1. If most of the refraction take place very near the earth at D and 
 A, the point b will fall near a. On the contrary, if it take place very near the upper part of the 
 atmosphere, the point 6 will fall near B. If we suppose, as usual, that the curve Ab is the 
 arc of a circle, we shall have, after drawing the chord Ab, bAc = f 6cB = 16^', (as teB is 
 the external angle of the isosceles triangle A6c). Hence EA6 == 89 4S| ', AEfr = 8 35 J', EftA = 
 81 41', and,asEA = 1, we find the sideEfc of the triangle En6 to be 1.0106. Hence Hft=0.0106, 
 or about 42 statute miles. 
 
 SOHOL. Dr. Keill, in his Lectures on Astronomy, observes, that it is entirely owing to the 
 atmosphere that the heavens appear bright in the day time. For without it, only that part of the 
 heavens would be luminous in which the sun is placed ; and if we could live without air, and 
 should turn our backs to the sun, the whole heavens would appear as dark as in the night. In 
 this case also we should have no twilight, but a sudden transition from the brightest sun-shine to 
 dark night immediately upon the setting of the sun, which would be extremely inconvenient, if 
 not fatal to the eyes of mortals. See Keill's Astron. Lect. xx. 
 
 PROP. XL. The twilight is longest in a parallel sphere, and shortest in 
 a right sphere ; and in an oblique sphere, the nearer the sphere approaches 
 to a parallel, the longer is the twilight. 
 
 In a parallel sphere, the twilight will continue till the sun's declination toward the depressed 
 pole is 1 8, but in this sphere his declination is never more than 23-J decrees ; whence the twi- 
 light will only cease, whilst the sun's declination is increasing from 18 to 23|, and decreas- 
 ing again till in its decrease it becomes 18. The twilight is here caused by the annual 
 motion of the earth. In a right sphere, the sun appears to be carried, by the daily motion of the 
 eatth, in circles perpendicular to the horizon ; whence it is carried directly downward by the 
 whole daily motion, and will arrive at 18 below the horizon, the soonest possible; whereas, in 
 
 32 
 
230 
 
 OF ASTRONOMY. 
 
 BOOK V1T. PAUT I. 
 
 Plate 9. 
 Fig. 2. 
 
 Plate 9. 
 Fig. 13. 
 
 Mate 9. 
 Fig. 14. 
 
 an oblique sphere its path is oblique to the plane of the horizon, anJ therefore will be longer 
 before it is descended 18 degrees below the horizon; and tiie difference of the time of twilight 
 will increase with the degree of obliquity. As the sun sets more obliquely at some parts of the 
 year than others, the twilight varies in its duration. 
 
 SECT. IV. 
 
 Of the Equation of Time. 
 
 PROP. XLI. The time in which the sun completes one apparent diur- 
 nal revolution, is greater than that in which the earth revolves round its axis. 
 
 If the earth turns round its centre T in the direction RCB, and at the beginning of one 
 revolution the sun was seen at Z, in the meridian ; from its apparent annual motion it will, after 
 the diurnal revolution of the earth is completed, be seen advanced in its orbit toward E. The 
 earth, therefore, must perform more than one revolution, and the spectator at R, after returning 
 to the station from which he set out, must advance forward to e. before the sun will be again in 
 the meridian. 
 
 PROP. XLII. The obliquity of the ecliptic to the equator would cause 
 the daily increments of the sun's right ascension to be unequal, although the 
 sun's motion in the ecliptic were uniform. 
 
 Let E be the first degree of Aries, EQ an arc of 90 of the equator, EC the same of the 
 ecliptic, and CQ an arc of the solstitial colure, between Cancer and the equator. At E the sun 
 has neither longitude nor right ascension ; these may therefore be considered as equal, when the 
 sun sets out from Aries. At C, the longitude is equal to the right ascension ; for both EC and 
 EQ are by supposition 90 degrees of great circles of the same sphere. But if the sun be any 
 where between the iirst of Aries and the first'of Cancer, as at S, (he longitude will be greater 
 than the right ascension ER. For SR 'being an arc to the secondary of the equator passing 
 through the sun, ES is the longitude, and ER the right ascension ; but ES is greater than ER, 
 because the angle at R is a right angle, but the angle akS an acute angle. Now, if the sun be 
 supposed to move uniformly in the ecliptic, or to describe equal arcs in equal times, the daily 
 increments of longitude will be equal to one another ; and consequently, since at the two ex- 
 tremes E and C the longitude and right ascension are equal, and the longitude is supposed to 
 increase uniformly, if the right ascension also increased uniformly, they would at all times he 
 equal. But at S, R, or any other points in the same secondary, between the first of Aries and 
 Cancer, the longitude is greater than the right ascension ; the daily increments of right ascen- 
 sion are therefore unequal. 
 
 The longitude and right ascension are equal when the sun is at C and at E, the former being 
 90 the latter 180 from Aries both on the ecliptic and equator. But between C and E, the Ion- 
 gitude is less than the right ascension ; because ES, opposite to the right angle R, is greater than ER 
 opposite to the acute angle S, and consequently the point S is nearer Aries than the point R. 
 But, the sun being supposed to move uniformly, or to increase its longitude equally every day, 
 if the right ascension also increased equally every day, since the longitude and right ascension 
 
CHAP. II. OF THE EARTH. 251 
 
 are equal at C and E, they would be always equal. But at S, or any where between Cancer 
 and Libra, the longitude is less than the right ascension ; consequently, the daily increments of 
 right ascension are not equal. In like manner it may be shown, that in the third quarter the 
 sun's longitude is greater than its right ascension, and in the fourth, less. 
 
 PROP. XL11I. If the plane of the ecliptic coincide with that of the 
 equator, the daily increments of the sun's right ascension would nevertheless 
 be unequal. 
 
 | 
 
 Because (by Prop. XXXIV.) the apparent annual motion of the sun is not uniform, it would 
 in some days describe a longer arc than in others ; that is, since its right ascension and longitude 
 would in this case be the same, the daily increments of its right ascension would be unequal. 
 
 DBF. XLI. A Natural Day is the time the sun takes in passing from the 
 meridian of any place, till it comes round to the same meridian again. 
 
 PROP. XLIV. Any place upon the earth's surface describes more than 
 a circle round the earth's axis in a natural day ; and the arc which it des- 
 cribes more than a circle in any day, is the sun's increment of right ascension 
 for that day. 
 
 While the earth revolves round its axis, any place upon the earth's surface describes a circle ; 
 but (by Prop. XLIII.) while the sun completes its apparent diurnal revolution, anv place on the 
 earth's surface will move through one circle and an arc of a second ; therefore any such place 
 describes more than a circle (Def. XLI.) in a natural day. 
 
 And since both a meridian, and a secondary of the equator, passing through the poles, are rlate y - 
 perpendicular to the equator, (Def. VI. and X.) if the sun at S be in SR, the meridian of any llg> lo> 
 given place, it is also in a secondary of the equator passing through that place. In like manner, 
 if the sun be at T, and TV be after a natural day the situation of the meridian of the given place, 
 the sun will be in TV, which will be both the meridian of the place, and a secondary of the equa- 
 tor. "Whence, RV being part of the equator, since ER was the sun's right ascension when it 
 was at S, and EV is its light ascension when it is arrived at T. RV must be the increment of 
 the sun's right ascension for the natural day in which it is advanced from S to T. And, because 
 SR, TV, are both perpendicular to the equator, and any place, in one diurnal revolution of the- 
 earth, describes a circle parallel to the equator, RV taken in this circle will always be the same 
 arc with RV in tlie equator, and therefore will be equal to the sun's daily increment of risjht 
 ascension. 
 
 PROP. XLV. The natural days are not equal to one another. 
 
 For any natural day is the time in which the earth performs one revolution round its axis, 
 and such a portion of a second, as is equul to the sun's increment of right ascension for that day ; 
 but the sun's daily increments of right ascension are unequal ; (by Prop. XLII. and XLIII.) 
 therefore the additional portion of the second revolution will sometimes be greater and some- 
 times less, and consequently the tim.es in which the natural days are completed will be unequal. 
 
853 OF ASTRONOMY. BOOK VII. FART I. 
 
 DBF. XLIJ. The Equation is the difference between mean time and appa- 
 rent time. 
 
 PROP. XLVI. If the sun were to move uniformly round the equator 
 in the same time in which it appears to describe the ecliptic, its apparent 
 daily motion would be a measure of mean time. 
 
 For the natural days in that case being liable to no variation, either from the obliquity of the 
 sun's orbit, or the irregularity of its motion, must be equal. 
 
 PROP. XLVII. The portion of time which passes between the arrival 
 of the sun in the ecliptic to the meridian of any place, and its supposed arri- 
 val at the same meridian, if it were to move uniformly in the equator, is 
 the equation. 
 
 For (by the last Prop.) it would be noon by mean time at any place, when the sun, if it 
 moved uniformly in the equator, was arrived at the meridian of that place ; and it is noon at the 
 same place by apparent time, when the sun in the ecliptic arrives at the same meridian ; there- 
 fore the difference between these two arrivals is the difference between mean time and apparent 
 time, or the equation. 
 
 PROP. XL VIII. In the time which passes between the arrival of the sou 
 in the ecliptic to the meridian of any place, and its supposed arrival at the 
 same meridian, if it were to move uniformly in the equator, an arc of the 
 equator passes under the meridian, which is equal to the difference between 
 riate 9. the right ascension of the sun, as it moves in the ecliptic, and the right ascen- 
 sion which the sun would have, if it moved uniformly in the equator. 
 
 Let the sun be at S ; and let EC be the ecliptic, EQ the equator, and E the first of Aries ; 
 then if a secondary of the equator passes through the sun, SR, being at right angles to EQ, is 
 an arc of that secondary, and (by Def. XXXII.) ER is the sun's right ascension, and the point 
 R is the point in which the right ascension ends ; which being in. the secondary of which SR is a 
 part, that is, the secondary passing through the sun, arrives at the meridian at the same time 
 with the sun. If the sun were to move uniformly in the equator, and were arrived at P, EP 
 would be its right ascension, and consequently P would be the point in which its right ascen- 
 sion would end, which point P must arrive at the meridian at the same time with the sun, be- 
 cause the sun is supposed to be in that point. Therefore RP, the distance of the two points 
 R and P> is an arc of the equator, (passing under the meridian in the time specified in the Prop, 
 osition) which is equal to the difference between the real and supposed right ascensions of the 
 sun, when he arrives at the meridian by his real motion in the ecliptic, and when he arrives at 
 the same meridian by an uniform motion in the equator. 
 
 PROP. XLIX. The right ascension of the sun, if it were to move uni- 
 formly in the equator, would at any time be equal to the longitude which it 
 would have at that time if it were to move uniformly in the ecliptic, or to its 
 mean longitude. 
 
CHAP. II. OF THE EARTH. 353 
 
 For, on this supposition, the sun, describing the equator with an uniform velocity in the 
 same time in which it actually describes the ecliptic, its velocity would be the same with the 
 mean velocity in the ecliptic. Consequently, the distance of the sun from the first of Aries in 
 the equator would at any time be the same with its distance from the same point in the ecliptic, 
 if it were to move uniformly therein with its mean velocity ; that is, its right ascension in the 
 equator would always be equal to its mean longitude in the ecliptic. 
 
 PROP. L. An arc of the equator, equal to the difference between the 
 sun's right ascension and its middle longitude, at any given time and place, 
 converted into time, is the equation. 
 
 It has been shown (Prop. XLVIII.) that in the portion of time which passes between the 
 arrival of the sun in the ecliptic to the meridian of any place, and its supposed arrival at 
 the same meridian, if it were to move uniformly in the equator, an arc of the equator passes under 
 the meridian, which is equal to the difference of the right ascension of the sun as it moves in the 
 ecliptic, anil the right ascension which it would have if it moved uniformly in the equator. And 
 it has been proved, (Prop. XLV1I.) that this portion of time is the equation, and (Prop. XLIX.) 
 that the right ascension which the sun, at any given time and place, would have if it moved uni- 
 formly in the equator, is equal to its mean longitude in the ecliptic. Therefore, in the equa- 
 tion, an arc of the equator passes under the meridian, equal to the difference of the right ascen- 
 sion of the sun in the ecliptic, and its mean longitude. Consequently, if this arc be converted 
 into time, that is, if for 15 degrees be taken an hour, for 15' one minute of time, for 15" one 
 second of time, the equation of time will be found. 
 
 PROP. LI. If the sun's mean longitude be greater than its right ascen- 
 sion, mean time follows apparent time ; if its mean longitude be less than 
 its right ascension, mean time precedes apparent. 
 
 If the right ascension of the sun, as before supposed in the equator EP, that is, (by Prop. n a te 9. 
 XLIX.) its mean longitude be greater than the sun f s real right ascension ER, the supposed place F 'ff- !. 
 of the sun in the equator P, will be to the east of the point R, where the sun's real right ascen- 
 sion ends. Therefore when this point R, at apparent noon, is come to the meridian, the point 
 !' will not be arrived at the meridian; and mean noon will be later than apparent noon. There- 
 fore, when the sun's middle longitude is greater than its right ascension, mean time follows ap- 
 parent. In like manner the reverse may be proved. 
 
 COR. Hence in the former case the equation is to he subtracted from the apparent time 
 found by the dial, and in the latter, to be added to it, in order to obtain the mean time. 
 
 SCHOL. It appears from the foregoing Propositions, that the difference between mean and 
 apparent time, depends upon two causes ; (I) the obliquity of the ecliptic with respect to the 
 equator; and (2) the unequal motion of the earth in an elliptical orbit. Tha obliquity of the 
 ecliptic to the equator would make the sun and clocks agree on four days of the year, viz. when 
 ihe sun enters Aries, Cancer, Libra, and Capricorn. But the other cause which arises from his 
 unequal motion in his orbit would m^ke the sun and clocks agree only twice a year, that is, when 
 he is in perigee and apogee ; consequently, when these two points fall in the beginnings of 
 
OF ASTRONOMY. 
 
 BOOK VII. PART I. 
 
 Cancer and Capricorn, or of Aries and Libra, they will concur in making the sun and clocks 
 agree in those points. But the apogee, at present, is in the ninth degree of Cancer, and the 
 perigee in the ninth degree of Capricorn ; and, therefore, the sun and clocks cannot be equal at 
 the beginnings of these signs, nor indeed, at any time of the year, except when the equation re- 
 sulting from one of the causes is just equal to that arising from the other, one being positive and 
 the other negative, which happens about the 15th of April, the 15th of June, the Slst of August, 
 and the 24th of December ; at all other times the sun is too fast or too slow for equal time by a 
 certain number of minutes and seconds, which at the greatest is 16' 14", and happens about the 
 first of November; every other day throughout the year having a certain quantity of this diller- 
 cnce belonging to it, which, however, is not exactly the same every year, but only every fourth 
 year, for which reason it is necessary, where great accuracy is required, to have four tables of 
 this equation, viz. one for each of the four years in the period of leap years. The following 
 concise table, adapted to the second year after leap year, will always be found within about a 
 minute of the truth, and therefore sufficiently accurate for common clocks and watches. 
 
 TABLE FOR THE EQUATION OF TIME. 
 
 I f 
 
 Equation 
 in 
 
 Minutes. 
 
 g f 
 
 ! * 
 
 Equation 
 in 
 Minutes. 
 
 5 6 1 
 
 1 ? 
 
 1 s 
 UI 
 
 ft 
 
 .- 
 
 1 S 
 
 I * 
 
 Equation 
 in 
 Minutes. 
 
 Jan. 1 
 3 
 5 
 7 
 9 
 12 
 15 
 18 
 2t 
 25 
 31 
 Feb. 10 
 21 
 27 
 Mar. 4 
 8 
 12 
 15 
 19 
 22 
 25 
 28 
 
 4+ 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 
 Apr. I 
 
 4 
 7 
 11 
 15 
 
 * 
 
 19 
 24 
 SO 
 'lay 13 
 29 
 June 5 
 10 
 15 
 * 
 
 20 
 25 
 29 
 July 5 
 11 
 28 
 
 44 
 3 
 
 2 
 
 1 
 
 
 Aug. 9 
 15 
 20 
 
 24 
 28 
 
 31 
 
 * 
 
 ^ept. 3 
 6 
 9 
 12 
 15 
 18 
 21 
 24 
 27 
 30 
 Oct. 3 
 6 
 10 
 14 
 19 
 
 3 + 
 
 4 
 3 
 2 
 1 
 
 
 Oct. 27 
 Nov. 15 
 
 20 
 24 
 27 
 30 
 
 Dec. 2 
 
 g 
 
 7 
 c 
 
 11 
 13 
 
 16 
 18 
 20 
 
 22 
 
 24 
 
 
 
 26 
 28 
 30 
 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 
 1 
 2 
 3 
 4 
 3 
 2 
 1 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 . 
 10 
 11 
 12 
 13 
 14 
 15 
 
 1 + 
 2 
 3 
 4 
 5 
 6 
 
 1 + 
 2 
 3 
 
 Those columns that are marked + , show that the clock or watch is faster than the sun ; and those mttrked , lhat it is slower. Set 
 Ferguson's Astronomy, ch. 13. Phil. Trans, vol. 54. 
 
CHAP. III. OF THE INFERIOR PLANETS. 355 
 
 CHAP. III. 
 
 Of the Inferior Planets, Mercury and Venus. 
 
 DEF. XLIII. The Elongation of any planet is its apparent distance from 
 the sun. 
 
 DEF. XLIV. The Nodes of the orbit of a planet are the two points, in 
 which the orbit cuts the plane of the ecliptic ; and a right line, drawn from 
 one node to the other, is the Line of the Nodes. 
 
 DEF. XLV. The Limits of the orbit of a planet are two points in the 
 middle between the two nodes. 
 
 DEF. XLVL An inferior planet is in its inferior conjunction, when it is 
 nearer the earth than the sun is, and in its superior conjunction, when it is 
 farther than the sun is from the earth, and both in the same secondary. 
 
 Let A be the place of the earth in its orbit ABO, E the place of Venus in its orbit EHG, Plate 9. 
 S the sun, and FD an arc of a circle in the heavens. Venus will be in its inferior con- ^'ff- 1 "- 
 junction when it is at E, and at its superior when it is at G, and both are in the same 
 secondajy. 
 
 PROP. LII. An inferior planet is at its greatest elongation, when a 
 line drawn from the earth through the planet is a tangent to the orbit of the 
 planet. 
 
 When the planet is at E, being in conjunction with the sun, it has no elongation. As it pi a te9. 
 moves from E toward X its elongation increases, till at X, when AF is a tangent to the orbit of Ft S- 12 - 
 Venus, its apparent place is F, and its elongation FQ, which is the greatest elongation it can 
 have ; for in passing from X to G its elongation decreases, till at G it becomes nothing. This 
 will be true in elliptical as well as circular orbits. 
 
 COR. If the orbits of these planets were circular, the distance of each, from the sun, would 
 be to the earth's distance, as the sine of its greatest elongation to the radius. 
 
 The orbits are not circles, but ellipses, having the sun in one focus ; for on this supposition 
 their computed places are always found to agree with their observed places. 
 
 PROP. L1I1. The inferior planets are never in opposition to the sun. 
 
 For in opposition the earth is between the sun and the planets, which can never happen when 
 the orbit of the planet is nearer to the sun than that of the earth. 
 
 DEF. XLVII. A planet is in Quadrature, when it is 90 degrees in the 
 celestial sphere distant from the sun. 
 
 PROP. LIV. The inferior planets are never in quadrature. 
 
 The greatest angle of elongation is that contained by AQ, drawn from the earth through the Plate 9. 
 
 Tig. 12. 
 
256 OF ASTRONOMT. BOOK VII. PART *; 
 
 sun, and AF a tangent to the orbit of the planet. Now if QAF were a right angle, AF would be. 
 (El. III. 18.) a tangent to the earth's orbit; but AF is a tangent to an orbit less than that of 
 the earth ; it therefore makes an angle with AQ less than a right angle j that is, QF, the greatest 
 elongation, is less than 90 degrees. 
 
 COR. Hence the inferior planets never appear far from the sun. 
 
 PROP. LV. While Venus is moving from the superior conjunction to 
 the inferior, it sets after the sun ; while it is moving from the inferior con- 
 junction to the superior, it rises before the sun. 
 
 Plate 9. 
 
 Fig. 12. Whilst Venus is moving from G, its superior conjunction, through P, to E, its inferior con- 
 
 junction, being in the eastern part of its orbit, the sun will be westward of Venus ; therefore 
 Venus, if far enough from the sun, will be seen in the west after the sun is gone down, whence 
 it is then called the evening star. But on the western side of its orbit, the sun being eastward 
 of it, Venus will set before the sun, and consequently rise before it, whence it is then called 
 the morning star. 
 
 PROP. LVI. The greatest elongation of an inferior planet on one side 
 of the sun is not always equal to that on the other. 
 
 For since the planet moves in an elliptical orbit, at the time of its greatest elongation on one 
 side it may be in its aphelion ; and at its greatest elongation on the other side, it may be in some 
 part nearer the sun ; hence its real distance from the sun at its elongations being unequal, its 
 apparent distances will be so likewise. 
 
 COR. The greatest elongation of Venus being found greater than that of Mercury, the latter 
 must be nearer the sun than the former. 
 
 PROP. LVII. The apparent velocity of the inferior planets is greatest at 
 the conjunctions. 
 
 Since the plane of the orbit of Venus is oblique to that of the earth, those parts of this orbit, 
 which are viewed by a spectator on the earth directly, would appear longer than other equal parts 
 Fig. 12.' viewed obliquely ; whence its motions, if uniform, must appear unequal. If the orbit EPGH 
 of Venus be seen obliquely by an eye placed at A, the parts about E and G, or near the con- 
 junctions, will be seen directly, for AE is perpendicular to a tangent at E ; but the parts about 
 X and P will be seen obliquely; whence the Proposition is manifest. 
 
 SCHOL. The time when an inferior planet will come again into a given situation with res- 
 pect to the sun and the earth, may be thus found. Whilst Venus performs one revolution, the 
 earth, whose periodical time is longer than that of Venus, will not have completed its revolu- 
 tion. Before Venus and the earth can be again in the inferior conjunction, Venus must, there- 
 fore, besides its entire revolution, describe an arc equal to that which the earth has passed over ; 
 consequently, the number of degrees passed over by each, or their angular motions, in the same 
 time, will be reciprocally as their periodical times ; that is, as the periodical time of the earth is 
 to the periodical time of Venus, so is the angular motion of Venus (which is equal to four right 
 angles added to the angular motion of the earth between two inferior conjunctions) to the angu- 
 
. III. OF THE INFERIOR PLANETS. 337 
 
 lar motion of the earth in the same time; whence (El. V. 17.) as the difference between the 
 periodical times of the earth and Venus, is to the periodical time of Venus, so are four right 
 angles, or 360, to the number of degrees over which the earth passes in her orbit from one 
 inferior conjunction to another. This is only true upon the supposition that the planets moved 
 in circular orbits, in which case the following general rule woukl apply to the finding of the time 
 from conjunction to conjunction, or from opposition to opposition of any two planets. " Multi- 
 ply their periodic times together, and diyide the product by their difference, and you have the 
 time sought." For let P = the periodic time of the earth, p = that of the planet (suppose an 
 
 SCO 
 inferior), t = time required ; then P: 1 day :: 360 : p-, the angle described by the earth in 
 
 360 360 SGO 
 1 day ; for the same reason is the angle described by the planet in 1 day ; hence ^j- 
 
 is the daily angular velocity of the planet from the earth. Now if they set out from conjunction, 
 
 360 360 
 they will return into conjunction again, after the planet has gained 360 ; hence 
 
 : 360 : : 1 day : t ? . For a superior planet t ^ . 
 
 DEF. XL VIII. The apparent motion of a gjanet, if seen from the earth, 
 is called its Geocentric Motion ; if seen from the sun, its Heliocentric Motion. 
 
 PROP. LVIII. When the inferior planets are passing from their sta- 
 tionary point through their superior conjunction, to their stationary point ou 
 the other side, their geocentric motion is direct, or they appear to move from 
 west to east. 
 
 When Venus is at X, it appears to a spectator at A to be in the point F of the concave plate 9. 
 sphere of the heavens; when it has moved forward in its orbit to H, G, N, P, it will appear sue- Fi - 12 - 
 cessively in the several points F, Q, R, D. But the motion from F to D is from west to east ; 
 for whilst the sun and earth are on the same side of the planet, it must appear to move in the 
 same direction, whether it be viewed from the earth or the sun, because the spectator at either 
 station views the concave side of the planet's orbit; but from the sun it is always seen to move 
 from west to east ; therefore its apparent geocentric motion in this situation is direct, or in 
 conseyuentia. 
 
 PROP. LIX. While the inferior planets are moving from the stationary 
 point on one side, to the stationary point on the other, through their inferior 
 conjunction, their geocentric motion is retrograde, or from east to west. 
 
 While the planet is in this situation, the convex side of its orbit is toward a spectator on the 
 earth, but its concave side toward a spectator at the sun; hence the former will see the planet 
 move in a diiection contrary to that in which it will appear to the latter to move. Thus, when 
 the planet is at P, it will appear in the heavens at D, and as it passes through K to X, it appears 
 to move from D through R, Q, P, to F ; but the motion from D to F is from east to west ; there- 
 fore the apparent motion of the planet in this part of its orbit is retrograde, or in antecedent. 
 
 S3 
 
358 OF ASTRONOMY. BOOK VII. 
 
 PROP. LX. When the inferior planets are at their greatest elongation, 
 they have the same apparent motion as the sun. 
 
 For if at any time a planet be moving apparently slower or faster than the sun, its elongation 
 must evidently be increased or diminished, and of course cannot then be the greatest. 
 
 At either of the greatest elongations P, of the planet Venus, AD is a tangent to the orbit; 
 which so nearly coincides with a small arc of the curve, that a spectator's eye placed at the 
 earth could not distinguish the tangent from this part of the curve ; consequently, a planet, 
 while it is in this part bf the currc. will appear to lie in the tangent AD ; that is_ to be station- 
 ary in the point D. 
 
 SCHOL. As seen from the sun the motion of these, and indeed of all the planets, appears to 
 be constantly direct ; the appearance of the stations and retrogradations is entirely owing to the 
 situation and motion of the earth. And from these causes the apparent motion of Mercury and 
 Venus appears to us to be in looped curves. In order to represent the apparent paths of these 
 planets, Ferguson removed from an Orrery the balls representing the Sun, Mercury, and Venus, 
 and put in their places black-lead pencils, with the points upward, and fixed a circular piece of 
 paste-board, so that it remained constantly in the same position during the motion of the machine, 
 with its centre resting on the terrestrial ball. Then, the paste-board being gently pressed on 
 the pencils, and the pencils representing Mercury and Venus, together with the terrestrial ball, 
 being made to revolve about the pencil in the sun's place, by turning the winch of the machine, 
 Plate 14. the three pencils described on the paste-board a diagram, of which Plate 14. Fig. 1. is a copy, 
 F 'S' * reduced to a less size. The solar pencil marked the dotted circle of months, which represents 
 the sun's apparent motion in the ecliptic, and is the same every year. The pencil in Mercury's 
 place traced the curve, having 23 loops and crossing itself the same number of times, which repre- 
 sents that planet's apparent motion for 7 years. The pencil in the place of Venus described the 
 curve, having 5 loops, which represents the apparent path of Venus for 8 years ; after which time 
 this planet moves again very nearly in the same apparent path. The ecliptic encloses these curves, 
 in the figure, and dotted lines ate drawn from the earth to the ecliptic, to show the apparent or 
 geocentric motion of Mercury for one year, which is easily traced from A, where it commences, 
 through B, C, &c. At B Mercury is stationary, and the dotted line, drawn through B, shows its 
 place in the ecliptic ; from B to C it is retrograde ; at C stationary ; from C it proceeds directly. 
 And universally, on each side of the loops the planets appear tn be stationary; in the part of the 
 loops nearest to the earth, retrograde; and in the rest of the path, direct. 
 
 PROP. LXI. The heliocentric latitude of an inferior planet is the 
 greatest when the planet is in one of its limits. 
 
 For the planet is then (Def. XLV.) at its greatest distance from the ecliptic, and therefore 
 will have the greatest latitude, as seen from the sun. 
 
 PROP. LXTI. The geocentric latitude of an inferior planet is directly 
 as its heliocentric latitude, and inversely as its distance from the earth. 
 
 The apparent length of a line drawn from the planet to the plane of the ecliptic ; that is, its 
 geocentric latitude, is (by Book VI. Prop. LXIX. LXX.) directly as its real length, and inversely 
 
. IV. OF THE SUPERIOR PLANETS. 359 
 
 as the distance of the spectator's eye. But the real length of a line drawn from the planet to 
 the plane of the ecliptic, is its heliocentric latitude ; and the spectator's eye is at the earth ; 
 whence the proposition is manifest. 
 
 COR. When Venus is in its inferior conjunction, its heliocentric latitude is less than its 
 geocentric ; for it is then farther from the sun than from The earth. The contrary takes place 
 with resptci to Mercury. 
 
 PROP. LXIII. The sun enlightens only one half of a planet, and only 
 one half of a planet is visible, at once. 
 
 This is sufficiently manifest from the spherical form of the planets, and the rectilinear motion 
 of light. 
 
 SCHOL. Venus and Mercury, in passing from the inferior to the superior conjunction, arc 
 observed to have all the phases of the moon from new to full. 
 
 DBF. XLIX. The hemisphere of a planet which is toward the earth is 
 called its Disk, because it appears like a plane circle. 
 
 PROP. LXIV. The inferior planets are invisible in their inferior con- 
 junction ; their whole disk is illuminated, when they are in their superior 
 conjunction ; and they are more or less illuminated, as they are nearer or 
 fartber from their superior conjunction. 
 
 When Venus, or Mercury, is in its superior conjunction, the whole of its enlightened hemis- 
 phere is toward the earth, and its entire disk is visible ; as it passes toward its inferior conjunc- 
 tion, its enlightened hemisphere turns, by degrees, from the earth, till, at the inferior conjunction, 
 it is wholly turned from the earth, and the planet becomes invisible. 
 
 PROP. LXV. If an inferior planet is in one of its nodes at the time of 
 its inferior conjunction, it will appear as a spot in the disk of the sun. 
 
 When the planet is in the nodes, it will be in the plane of the ecliptic ; and if at the same 
 time it be in its inferior conjunction, it will neither appear above nor below the sun, as it docs 
 when in conjunction in other parts of its orbit, but on the sun's disk. 
 
 CHAP. IV. 
 
 Of the superior planets, Mars, Jupiter, Saturn, and the Herschel. 
 
 DBF. L. The superior planets are Mars, Jupiter, Saturn, and the Her- 
 schel. 
 
 PROP. LXVI. The superior planets are sometimes in conjunction with 
 the sun, sometimes in opposition, and sometimes in quadrature. 
 
260 OF ASTRONOMY. BOOK VII. PA 
 
 Tliis is known from observation. 
 
 Plate 10. j je t g fog the sun ; QPO a part of the orbit of Jupiter ; P the planet ; adg, or nkg, the earth's 
 orbit. When the earth is at d, the sun at S, and the planet at P, the planet is in conjunction ; 
 when the earth is at fr, the sun at S, and the planet at P, the planet is in opposition ; when the 
 eartli is at n or g, and the planet at P, the planet will be in quadrature. 
 
 COR. Therefore the superior planets move in orbits farther distant from the sun, than the 
 orbit of the earth. 
 
 PROP. LXVII. The apparent diameter of a superior planet is greatest 
 when the planet is in opposition. 
 
 For, when the planet is in conjunction, its distance from the earth is greater than when it is 
 in opposition, by the diameter of the earth's orbit. 
 
 PROP. LXVIII. If a superior planet were at rest, its apparent geocen- 
 tric motion in any given time would be proportional to the angle subtended 
 by the arc described by the earth in that time, and seen from the planet. 
 
 Plate 10. When the earth is at a, (he planet P appears in the right line aPA, and among the stars in the 
 
 *"' lp heavens at A ; when the earth is at b, the planet appears at B. Therefore while the earth moves 
 
 from a to 6, the planet appears to move from A to B. But this arc AB is proportional to the 
 
 angle APB, that is, to the opposite angle aP6, which is the angle which the arc ab would subtend 
 
 to an eye placed at the planet P. 
 
 PROP. LXIX. The geocentric velocity of a superior planet is greatest 
 at its conjunction and opposition. 
 
 Arcs of a given length near the points d and k, when the planet is seen from the earth in 
 conjunction or opposition, would appear grealer than arcs in any other part of the earth's orbit, 
 viewed from the planet P, because the former are seen directly, the latter obliquely ; conse- 
 quently, these arcs would subtend greater angles; whence the apparent velocity of the planet, 
 as viewed from the earth, is greater at the conjunction or opposition, than at any other time. 
 
 PROP. LXX. When a line drawn from a siiperior planet to the earth 
 is a tangent to the earth's orbit, the planet has the same apparent motion, as 
 it has viewed from the sun. 
 
 At the time described, the earth viewed from the planet is at its greatest elongation from the 
 sun, (by Prop. LII.) and therefore has the same apparent motion with it (by Prop. LX.) But the 
 apparent motion of the earth or sun viewed from the planet is equal to that of the planet viewed 
 from either the earth or sun, which consequently must be the same. 
 
 PROP. LXXI. When a superior planet is passing from one of its sta- 
 tionary situations to the other through the conjunction, its geocentric motion 
 is direct ; when through the opposition, retrograde. 
 
. IV. OF THE SUPERIOR PLANETS. 361 
 
 While the earth is moving from a through d to g, the suu and planet being both on the same Plate 10. 
 side of the earth, the motion of the earth will produce an apparent motion in the sun and planet >s ' 
 the same way, and both will appear to move from A toward G. But while the earth moves 
 from g to n through k, the sun and planet being on contrary sitles of the earth, the motion of the 
 eartli in its orbit will produce an apparent motion of the aiin, and planet in contrary directions ; 
 that is, whilst the sun appears to move from west to east, the planet will appear to move frqin 
 east to west in the direction G, H, I, &c. 
 
 SOHOL. The progress, regress, and stations of the superior planets must take place, notwith- 
 standing the motion of the earth, since this only changes the place of the conjunctions and 
 oppositions, on which these depend, as was shown, Prop. LX. Schol. 
 
 PROP. LXXII. Mars sometimes appears round, sometimes gibbous ; 
 Jupiter, Saturn, aud the Herschel, always appear round. 
 
 When Mars is either in opposition or conjunction, his whole illuminated hemisphere is 
 toward the earth, but when it is in quadrature, some part of his illuminated disk is turned from 
 the earth. The same must happen in the revolutions of Jupiter, Saturn, and the Herschel, 
 about the sun ; but their great distance renders the difference between the perfect and imperfect 
 illumination of their disk imperceptible. 
 
 SCHOL. The following particulars respecting Mars are given by Dr. Herschel after long and 
 accurate observations. 
 
 The axis of Mars is inclined to the ecliptic 59 42'. 
 
 The node of the axis is in 17 47' of Pisces. 
 
 The obliquity of the ecliptic on the globe of Mars is 28 42'. 
 
 The point Aries on the martial ecliptic answers to our 19 28' of Sagitarius. 
 
 The figure of Mars is that of an oblate spheroid, whose equatorial diameter is to the polar 
 one as 1355 to 1272, or as 16 to lo nearly. 
 
 The equatorial diameter of Mars, reduced to the mean distance of the earth from the sun is 
 9" 8'". 
 
 And that planet has a considerable, but moderate atmosphere, so that its inhabitants, proba- 
 bly, enjoy a situation, in many respects, similar to ours. Phil. Trans, vol. Ixxiv. Part 2. 
 
 As the most important particulars respecting the stationary positions of the planets, being 
 erroneous, have been expunged from the preceding propositions in this edition, it ii thought 
 proper to introduce the following 
 
 PROB. To determine the position of tiro planets when they appear mutually stationary, sup- 
 posing them to move in circular orbits. 
 
 Let be the sun, I the inferior, and S the superior planet. Now as no two planets can ever pi a t e ig. 
 be moving in the same right line, it is very obvious that whenever two appear mutually station- Vig. 4. 
 ary, or, which is the same thing, cither appears stationary from the otlier,) while that appear- 
 ance continues, a right line passing through their centres must be so moved as to continue 
 parallel to itself. Let IS be that right line, and vV a right line passing through the same two 
 planets after a certain portion of time, taken so small that the parts of the orbits described in 
 the interval may not difler sensibly from lijit lines; if the planets appear mutually stationary 
 the line vV must be parallel to IS, and the lines Ii and Ss drawn perpendicular to them must be 
 
- 
 then ab = -cd = VI a- 3 , and ac 
 
 63 OF ASTRONOMY. BOOK VII. PART 1. 
 
 equal. Now taking ISV from the right angles ISs and QSV respectively, VS = QSI or the 
 elongation of I from the sun, and in the same manner it may be shown that vis = the supplement 
 of the elongation of S from the sun. But If and SV express the cotemporaneous motion of I and 
 S respectively, and they are also the secants of ilv and sSV respectively, li or Ss being radius. 
 Therefore as the velocity of the inferior is to the velocity of the superior, so is the secant of the 
 elongation of the superior to the secant of the elongation of the inferior, and since secants are 
 inversely as the cosines of the same angles, as the velocity of the inferior is to the velocity of 
 the superior, so is the cosine of the elongation of the inferior to the cosine of the elongation of 
 the superior. Now from the triangle Q\S it is plain that, at all times, the sine of the elonga- 
 tion of I is to that of S, as the distance of I from the sun is that of S. Therefore as the relative 
 distances are given, and the velocities easily computed from the distances and periods, which 
 are likewise given, the problem is reduced to this, viz. The ratio of the sines and the ratio of 
 the cosines of two angles being given, to find those two angles. Put d = Ql the distance and 
 v = Iv the velocity of the inferior planet, and D = Q$ the distance and V = VS the velocity 
 of the superior planet. Let acb and ace (fig. 5.) be the two angles, ab, de, their sines, and ac, 
 dc, their cosines respectively, radius being 1 j then d : D : : ab : de, and v : V : : ac r dc. futde = x, 
 
 a*jc* 
 , and v : V:: ll -- : ^1 *; 
 
 I f* _ V * ~ 
 whence by reduction, a? or de = D f - -- ^-yj- = sine of elongation of the superior planet, 
 
 I t' 2 _ V 2 
 and ab = d IT - -- = sine of elongation of the inferior planet, radius being 1. That 
 
 is, in words, 
 
 Divide the difference of the squares of the velocities by the difference of the squares of the 
 reciprocal products of the velocities into the distances, and multiply the square root of the quo- 
 tient by the distance of either planet from the sun, and the product will he the natural sine, 
 radius being 1, of the elongation of that planet from the sun, when it appears stationary, seen 
 from the other. 
 
 This relation between the distances, velocities, and angles does not strictly subsist for any 
 assignable time, but as it is gradually approached, and gradually destroyed, it apparently con- 
 tinues for a considerable time, especially with reference to the superior planets. 
 
 CHAP. V. 
 
 Of the Moon. 
 
 SECT. I. 
 
 Of the Variations in the Appearance of the Moon. 
 
 DBF. LT. When the moon is at its greatest distance from the earth in its 
 orbit, which is elliptical, or at its higher apsis, it is said to be in its Apogee; 
 when at its least distance, or lower apsis, in its Perigee. 
 
CHAP. V. OF THE MOON. S63 
 
 DEF. LIT. When the moon is in conjunction with the sun, it is New 
 Moon; when in opposition, it is Fall Moon ; its conjunction and opposition 
 are called by the common name of its Syzygies. 
 
 DKF. LIU. A Periodical Month is the time in which the moon describes 
 its orbit ; a Synodical Month is the time which passes between one new moon 
 and the next. 
 
 PROP. LXXII1. A synodical month is longer than a periodical month. 
 
 Because the moon moves in the same direction with the sun, while the moon performs one 
 revolution in its orbit, the sun, by its apparent annual motion, is advanced in the ecliptic ; con- 
 sequently, the moon must pass beyond the point in which it has completed its revolution before 
 it comes again into conjunction with the sun. 
 
 Let S be the sun, BA a part of the earth's orbit, md, MD, the diameter of the moon's orbit Plate 10. 
 when the earth is at B, or A. While the earth is at A, if the moon be at D, it will be in con- '^' 
 junction, and if the earth continued in the same place, after one revolution in its orbit, it would 
 be again in conjunction ; hut if, during the revolution of the moon, the earth is removed to B, 
 the moon at the end of the revolution will be at d, a point which is not between the earth and 
 sun ; it must therefore move on from d to e before it will be in conjunction. 
 
 PROP. LXX1V. The moon, at its conjunction, is invisible ; at its oppo- 
 sition, its whole disk is enlightened ; at its quadrature, it is half enlightened ; 
 between the conjunction and quadrature, it is horned ; and between the quad- 
 rature and opposition, it is gibbous. 
 
 Let QTL be a part of the earth's orbit, S the sun, T the earth, ACEG the moon's orbit. Plate 10. 
 When the moon is at E, or in conjunction, because it is between the earth and sun, its illumi- *>' 
 nated hemisphere will be wholly turned from the earth, consequently, its disk will be dark. At 
 A, being in opposition, its illuminated hemisphere will be wholly toward the earth, and its whole 
 disk will be visible. At C, or G, the apparent distance of the moon from the sun \vill be 90 
 degrees; for a right line from C or G will make a right angle with the line TS, in which the 
 sun appears ; whence the moon at each of them is half way between the opposition and conjunc- 
 tion, that is, in the middle state between the perfect illumination and the entire darkness of its 
 disk ; consequently, its disk is half enlightened. In passing from C to E, and from E to G, its 
 disk will be less than half illuminated, or it will appear horned; in passing from G to A, and 
 from A to C, its disk will be more than half illuminated, or it will appear gibbous. 
 
 Con. The earth must be a satellite to the moon, and subject to the same changes, but more 
 than 13 times larger than the moon appears to us. At new moon to us, the earth appears full 
 to her. 
 
 DEF. L1V. A circle supposed to be drawn upon the surface of the moon, 
 separating the illuminated from the dark hemisphere, is called the Circle of 
 Illumination ; a circle which separates its visible from its invisible hemis- 
 phere, is called the Circle of the Disk. 
 
264 OF ASTRONOMY. BOOK VII. PART I. 
 
 PROP. LXXV. If the centres of the sun, the earth, and the moon, are 
 joined by straight lines forming a triangle, the external angle of this triangle 
 at the moon is equal to the angle contained between the circle of illumiuutiou 
 and the circle of the disk. 
 
 Plate 10. Let S be sun, T the earth, mrnq the moon, and O its centre. Let STO be the supposed 
 Fig-. 3. triangle. Draw the line rq perpendicular to SO, and nm perpendicular to TOP. The angle 
 SOP will be equal to the angle mOcj, contained between rq, which represents the circle of illumi- 
 nation, and nm, which represents the circle of the disk. Because the angles SOq, TOm, are by 
 construction right angles, they are equal ; taking S0?n from each, the remaining angles POS, 
 mOq, are equal. Consequently the remaining angles POS, mOq, are equal. 
 
 Plate 10. SOHOL. This Proposition serves to explain all the different phases of the moon. For exam- 
 ple ; when the moon is moved from O to A, the line SO coincides with SA, and TO with TA ; 
 therefore TO, OS, lie in the same line, and the external angle POS is nothing ; whence the two 
 circles of illumination and of the disk coincide; and because the di^k is then turned from the 
 sun, it is wholly dark. When the moon is in quadrature, the line SO will be a tangent to the 
 moon's orbit; whence SOP will be a right angle, and the two circles will be at ri^ht angles to 
 each other, and the disk will appear half illuminated. If the angle POS be less or greater than 
 a right angle, the circle of illumination will make an angle with that of the disk, less or greater 
 than a right angle ; whence the illuminated part will appear horned or gibbous. Lastly, when, 
 the moon is in opposition, the lines SO, ST, become one and the same line ; whence the circles 
 coincide, and the whole disk is illuminated. 
 
 PROP. LXXVI. The horns of the moon, after the conjunction, are 
 turned toward the east ; before it, toward the west. 
 
 The sun, after the conjunction, setting before, that is, to the west of the moon, illuminates the 
 west side, of the moon's disk ; whence its horns, which are toward the dark part of the disk, are 
 toward the east. The reverse takes place, before the conjunction, when the moon is seen in the 
 east, before the sun rises. 
 
 PROP. LXXVII. When the moon is horned, its obscure part is visible 
 by the reflection of the rays of the sun from the earth. 
 
 Plate 10. When the moon is at D or F, near the conjunction, the enlightened hemisphere of the earth 
 Fig. 2. w jjj jj e ^ward the moon, and reflect the rays of the sun upon it. 
 
 PROP. LXXVIII. The moon always has nearly the same side toward 
 the earth. 
 
 This is proved by observation. 
 
 COR. 1. Hence, if the moon revolves about its axis, its periodical time must be equal to that 
 of its revolution in its orbit round the earth. This is also found to be the case with the fifth 
 satellite of Saturn as it regards its primary. 
 
 COR. 2. Though the year is of the same absolute length, both to the earth and moon, yet the 
 
X 
 
 CHAP. V. OF THE MOON. 265 
 
 number of days in each is very different ; the former having 365i natural days, but the latter 
 only about 12^, every day and night in the moon being as long as 29 on the earth. 
 
 SCHOL. This proposition would be true without any limitation, if the angular velocity of the 
 moon about the earth were equal to the angular velocity about her axis, for then the same face 
 must be always exactly turned toward the earth. But the angular velocity of the moon about 
 the earth is not uniform, while that about its axis is uniform, and therefore the same face is not 
 always turned to the earth. Since, however, the opposite face of the moon is never seen, the 
 time of the moon's rotation about her axis must be equal to the mean time in her orbit. See 
 Prop. LXXX. and LXXXI. 
 
 PROP. LXXIX. The moon appears to have two librations, one upon 
 a line perpendicular to its axis, called its libration in latitude ; the other upon 
 its axis, called its libration in longitude. 
 
 This appears from observation, some small portions of the surface of the moon being visible 
 in some parts, and invisible in other parts, of its orbit ; that is, in consequence of her libration 
 in latitude we sometimes see one pole, and sometimes the other. And by the libration in longi- 
 tude, more of the western limb is sometimes seen, and at others, more of the eastern. The 
 inclination of her axis to her orbit is 6 49'. 
 
 COR. The axis of the moon being almost perpendicular to the ecliptic, there will be but a 
 small difference in its seasons. 
 
 SCHOL. The moon's axis being so nearly perpendicular to the ecliptic, that the small obli- 
 quity of her orbit cannot cause the sun sensibly to decline from the equator; hence we learn, 
 that the inhabitants of the moon must devise some method, different from ours, for ascertaining 
 the length of their year. We know the length of our years by the return of the equinoxes ; 
 but these are not discoverable in the moon, where the days and nights are equal. 
 
 The Lunarians may, perhaps, measure their year by observing when either of the poles of our 
 earth begins to be enlightened, and the other to disappear, which happens at our equinoxes : 
 they being conveniently situated for observing large tracts of land about the poles of the earth, 
 ivhich, at present, are unknown to the most experienced of our navigators. 
 
 PROP. LXXX. The librations of the moon may be explained on the 
 supposition that the moon has a revolution round its axis. 
 
 Let III represent the plane of the moon's orbit, E the earth, and CMD the moon: and let P' ate ' 
 AB be the axis round which the moon revolves, and A be called its north pole, and B its soutli Flf '' 
 pole. CMD will in this situation be the visible hemisphere, and CD the plane of the disk. By 
 the libration in latitude the line AB appears to vibrate on the line IH, so that sometimes the 
 point A is visible, and sometimes the point B. This variation attends the moon's revolution in 
 its orbit ; in one half of the orbit the pole A is always visible, and in the other half the pole B. 
 It must therefore arise from the inclination of the axis of the moon to the plane of its orbit. 
 When the moon is at I, if the axis AB is inclined to IH, the plane of the orbit, making with it 
 the angle AIH to a spectator at E, the visible hemisphere will be CND, and the pole A will be 
 
 34 
 
266 OF ASTRONOMY. BOOK VII. PART I. 
 
 within the disk ; but when the moon is at H, the visible hemisphere will be CMD ; and the axis 
 AB being always parallel to itself, the pole B will be within the disk. 
 
 Plate 10. Again, let the moon be at A, the earth at T. If a circle, whose plane is perpendicular to Tc, 
 F 'S 5. pass through the line dcs, this circle will be that of the disk, and dbqs will be the visible hemis- 
 phere. But the moon has an apparent motion, by which the disk is changed, so that at one time 
 rb is the circle of the disk, and rqb the visible hemisphere, at another/} and/ciy. In the former 
 case, sr becomes visible, and db invisible ; in the latter case, df becomes visible, and qs invisible. 
 This libration in longitude arises from the elliptical form of the moon's orbit. If the moon has 
 a rotation round its axis, it has been shown that its revolution will be completed in the time of 
 one revolution in its orbit; and because the motion round the axis is uniform, one quarter of a 
 revolution will be completed in one fourth part of the periodial time. But, the moon's orbit be- 
 ing supposed elliptical, and the earth placed at T one of the foci, the moon will move slower at 
 A its apogee, than at P its perigee, and its velocity will continually increase in moving from A 
 Plate 10. to P, and decrease in moving from P to A. If therefore when the moon is at A, dbqs is the vis- 
 F 'S' S - ible hemisphere, after it has moved from A to B, through that quarter of its orbit, in which it 
 moves with its less velocity, and consequently takes up more than a quarter of its periodical 
 time, ds will not he perpendicular to Tc, but the point s will have turned from west to east 
 more than a quarter round the axis C ; hence the point s will not be visible when the moon 
 is at B, and fp instead of ds will be the circle of the disk. In passing from B to P, its excess 
 of velocity will make up for its defect in passing from A to B ; and at B it will have completed 
 half its orbit in half its periodical time; but in half its periodical time, it will have revolved half 
 round its axis, therefore at P, ds will again be perpendicular to Tc, and dxs will again be the 
 visible hemisphere. The reverse will take place in passing from P to A through D, when br will 
 be the circle of the disk, and s will be within it. 
 
 PROP. LXXXI. The moon revolves about its axis in the same time in 
 which it revolves about the earth. 
 
 Plate 10. Without such a revolution, the phenomena of its librations could not happen. If the point 
 *'' 4 - A were visible in one part of the moon's orbit, it would be always visible, without a rotation 
 about an axis, oblique to the plane of the orbit, to produce an apparent motion of the point A, or 
 the libration of latitude. If ds were perpendicular to cT, when the moon is at A, it would be 
 Fig. 5. so in every other part of the orbit ; and therefore dxs would always be the illuminated hemis- 
 phere, if there were not a revolution about the axis to produce, in the manner above explained, 
 the libration of longitude. These librations therefore prove the existence of this revolution ; and 
 it has been shown, that if there be such a revolution, its periodical time is the same with that of 
 the moon in its orbit. 
 
 Fig. 2. COR. Hence it is evident, that one half of the moon is never in darkness ; the earth (Prop 
 
 LXXIV. Cor.) constantly affording it a strong Hglit, during the absence of the sun; but the 
 other half has a fortnight's light and darkness by turns. 
 
 PROP. LXXXII. The orbit of the moon is an ellipse. 
 
 It is only on this supposition that the libration of longitude can be explained ; from this phe- 
 nomenon, therefore, the elliptical form of the orbit may be inferred. 
 
CHAP. V. OF THE MOON. 267 
 
 SCHOL. The moon revolves about the earth in an ellipse, and with the earth revolves about 
 the sun. In Fig. 2, Plate 14, the broad curve line ABCDE represents as much of the earth's Plate 14. 
 annual orbit, as it describes in 32 days, which are marked with numeral figures The small &' 2> 
 circles at o, 6, c, d, e, show the moon's orbit in proportion to that of the earth, and the narrow 
 curve abcdef represents the moon's path in the heavens, or with respect to the sun, for 32 days, 
 reckoning from a new moon at a. 
 
 The earth being supposed to move from A, and the new moon at the same time from a ; 
 7 days afterward the earth will be at B, and the moon in quadrature at b ; when the earth is at 
 , the moon is full at c ; when the earth is at D, the moon is in quadrature at d ; and when 
 the earth is at E, the moon is again new, or in conjunction at e, 
 
 From this figure it appears, that the moon's path is always concave toward the sun. For, if 
 we suppose the concavity of the earth's orbit to be measured by the length of a perpendicular 
 Cg, falling from the place of the earth at C, when the moon is full, on the right line bgd, con- 
 necting the places of the earth at the end ot the first and third quarters of the moon, the length 
 of the perpendicular will be about 640000 miles ; when the rnoon is new, it is about 240000 
 miles nearer to the sun than the earth is ; therefore the length of a perpendicular from its place at 
 that time on the same right line will be about 400000 miles, and shows the concavity of that 
 part of the path. 
 
 PROP. LXXXTII. The moon's surface is irregular. 
 
 If the surface were every where regular, the limit between the enlightened and dark parts of 
 the disk being the circle of light, that is, a perfect great circle of a sphere, would be exactly de- 
 fined when the moon is horned, half enlightened, or gibbous, contrary to observation. 
 
 SCHOL. The face of the moon, as seen through a telescope, appears diversified with hills 
 and vallies. This is proved by viewing her at any other time than when she is full ; for then 
 there is no regular line bounding light and darkness ; but the edge or border of the moon ap- 
 pears jagged ; and even in the dark part near the borders of the lucid surface there are seen 
 some small spaces enlightened bv the sun's beams. 
 
 Besides, it is moreover evident, that the spots in the moon taken for mountains and vallies, 
 are really such from their shadows. For in all situations of the moon, the elevated parts are 
 constantly found to cast a triangular shadow in a direction from the sun ; and the cavities are 
 always dark on the side next to the sun, and illuminated on the opposite side. Hence astrono- 
 mers are enabled to find the height of the lunar mountains. Dr. Keil, in his Astronomical Lec- 
 tures, has calculated the height of St. Catherine's hill to be nine miles. Since, however, the 
 loftiest mountains upon the earth are but about three miles in height, it has been long considered 
 as very improbable that those of a planet SB much inferior in size to the earth should exceed in 
 such vast proportion the highest of our mountains. 
 
 By the observations of Dr. Herscliel, made in November, 1779, and the four following 
 months, we learn, that the altitude of the lunar mountains has been very much exaggerated. 
 His observations were made with great caution, by means of a Newtonian reflector, b feet 8 
 indies long, and with a magnifying power of 222 times, determined by experiment; and the 
 method which he made use of to ascertain ttie altitude of those mountains, which, during that 
 time, he had an opportunity of examining, seems liable to no objection. 'I he rock situated near 
 
368 OF ASTRONOMY. BOOK VII. PART I. 
 
 Lacus Niger, was found to be about one mile in height, but none of the other mountains, which 
 he measured, proved to be more than half that altitude; and Dr. Hersthel concludes, that with 
 a very few exceptions, the generality of the lunar mountains do not exceed half a mile in their 
 perpendicular elevation. See Keil's Astron. Lect. x. Phil. Trans. Vol LXX. 
 
 To Dr. Herschel also we are indebted for an account of several burning volcanoes, which he 
 saw at different times in the moon. In the 77th vol. of the Phil. Trans, he says, April 19, 10 
 hours, 36 minutes sidereal time, " I perceive three volcanoes in different places of the dark part 
 of the new moon. Two of them are nearly extinct ; or, otherwise, in a state of going to break 
 out, which perhaps may be decided next lunation. The third shows an actual eruption of fire, 
 or luminous matter." On the next night, Dr. Herschel saw the volcano burn with greater vio- 
 lence than on the preceding evening He considered the eruption to resemble a small piece of 
 burning charcoal when it is covered by a thin coat of white ashes, which frequently adhere to it, 
 when it has been some time ignited, and it had a degree of brightness about as strong as that 
 with which such a cole would be seen to glow in faint day light. It is not yet determined 
 whether there is an atmosphere belonging to the moon. The question is fully discussed in the 
 Encyclo. Brit* Vol. II, p. 457, &c. See also Phil. Trans, for the year 1792. 
 
 SECT. II. 
 
 
 
 Of Eclipses. 
 
 DBF. LV. An Eclipse of tJie Moon happens when the earth, passing 
 between the sun and moon, casts its shadow on the moon. 
 
 PROP. LXXXIV. The moon can only be eclipsed at the full, or in 
 opposition. 
 
 For it is only when the moon is in opposition that it can come within the shadow of the earth, 
 which must always be on that side of the earth which is from the sun. 
 
 PROP. LXXXV. The moon can only be eclipsed when, at the full, 
 it is in or near one of its nodes. 
 
 The earth being in the plane of the ecliptic, the centre of its shadow is always in that plane - 
 if therefore the moon be in its nodes, that is. in the plane of the ecliptic, the shadow of the 
 earth will fall upon it ; also, since this shadow is of considerable breadth, it is partly above and 
 partly below the plane of the ecliptic ; if therefore the moon in opposition be so near one of its 
 nodes, that its latitude is less than half the breadth of the shadow, it will be eclipsed. But, be- 
 cause the plane of the moon's orbit makes an angle of more than 5 degrees with the plane of the 
 ecliptic, it will frequently have too much latitude at its opposition to come within the shadow 
 of the earth. 
 
 PROP. LXXXVI. The sun is larger than the earth, and tlie shadow of 
 the earth is a cone, the base of which is on the surface of the earth. 
 
 If the earth were larger than, or equal to, the sun, it is manifest, that its shadow would ei- 
 ther perpetually ^enlarge, or be always of the same dimension ; but in this case, the superior 
 
CHAP. V. OF THE MOON. 26<J 
 
 planets would sometimes come within it, and be eclipsed, which never happens. Therefore the 
 sun is larger than the earth, and produces a shadow from the earth of a conical form, which does 
 not extend to the orbit of Mars. 
 
 PROP. LXXXVII. The moon is eclipsed by a section of the earth's 
 shadow. 
 
 Let CDE be the earth, CME the cone of its shadow, and FH a part of the moon's orbit plate 10. 
 passing thro.igh the shadow; it is manifest that as the moon describes this part of its orbit, it Fl >- ? 
 passes through the circular section FGHL. 
 
 DBF. LVI. The moon's Horizontal Parallax is the angle which a semi- 
 diameter of the earth would subtend; if it were viewed directly from the 
 moon. 
 
 LEMMA. Half the angle of the cone of the earth's shadow may be taken 
 as equal to the angle of the sun's apparent semidiumeter. 
 
 Let AFBG be the sun, HED the earth, HMD or BMA the angle of the cone of the earth's pute 10. 
 shadow, and CMD half this angle, SA, a semidiameter of the sun, drawn from its centre to the Fl &- 6> 
 point of contact of the tangent AM, is perpendicular (El. Ill 16.) to AM, and is therefore seen 
 directly from D ; and it subtends the angle SDA ; it must therefore appear large in proportion 
 to the magnitude of this angle. But in the triangle SDM, the external angle SDA is equal to 
 the two angles CSD, CMD ; of which, CSD, the angle in which the semidiameter of the earth 
 is viewed from the sun, is so small, that without any sensible error it may be reckoned as noth- 
 ing, and SDA be said to be equal to SMD. 
 
 PROP. LXXXVIII. The semidiameter of the section of the earth's 
 shadow, which eclipses the moon, is equal to the difference between the hori- 
 zontal parallax of the inoou, and the sun's apparent semidiameter. 
 
 CT, being a semidiameter of the earth drawn from the point of contact of the tangent CM, pi ate JQ, 
 is perpendicular to CM. CT will therefore be seen directly from the point F in the moon's Fig- 7. 
 orbit, subtending the angle CFT, which is (Def. LVI.) the moon's horizontal parallax. FMG is 
 the semi-angle ot the cone of the earth's shadow, equal to the angle of the sun's apparent semi- 
 diameter, because MC produced would be a tangent to the circle of the sun's disk. FG is the 
 semidiameter of the section FGHL of the earth's shadow through which the moon passes in an 
 eclipse ; and FTG the angle which this semidiameter will subtend when it falls upon the moon, 
 and is viewed from the earth. Now the angle CFT is equal to the two angles FMG, F I'G, 
 (El. I. 32.) consequently, FTG is equal to CFT FMG; but by the preceding Lemma, FMT 
 may be taken for the angle of the sun's apparent semidiameter; and CFT is the moon's horizon, 
 tal parallax ; whence the Proposition is manifest. 
 
 COR. Hence the section of the earth's shadow, by which the moon is eclipsed, is broader 
 than the moon; for the semidiameter of the shadow, by this Prop, is 61' i(i' = 45', and the 
 diameter of the moon is about 51'. 
 
870 
 
 OF ASTRONOMY. 
 
 BOOK VII. PART I. 
 
 p! ate io' ^ EF ' ^^k An ec lipse of the moon is partial, when only a part of its 
 disk is within the shadow of the earth ; it is total, when all its disk is within 
 the shadow ; and it is central, when the centre of the earth's shadow falls 
 upon the centre of the moon's disk. 
 
 PROP. LXXXIX. The moon, at the full, will not be eclipsed, if its 
 latitude is equal to, or greater than, the sum of its own semidiameter, added 
 to the semidiameter of the earth's shadow. 
 
 Plate 10. 
 Fig. 10. 
 
 . 9. 
 
 Plate 10. 
 Fig. 9. 
 
 Fig. 9. 
 
 Because a circle or ellipse appears as a right line when the eye is in the same plane with it, 
 let OO represent the plane of the ecliptic, RR the plane of the moon's orbit, and N the node. 
 At the full moon, if the earth's shadow be at A, the moon F must be in the same part of the 
 heavens, because it is in opposition. But because only one half of the shadow of the earth, or 
 about 45', is on the same side of the ecliptic with the moon, and only one half of the moon's 
 breadth, or about 16', is on the side of its orbit toward the earth's shadow; if the centre of the 
 moon is more than 61' from the centre of the shadow, the moon will pass clear of the shadow 
 and will not be eclipsed. 
 
 Let GE be an arc of the moon's orbit, AB an arc of the ecliptic, and Cc an arc in a secon- 
 dary of the ecliptic equal to the moon's latitude. If this arc be equal to, or greater than, Ct and 
 cl, the sum ot the seinidiameters of the earth's shadow, and of the moon, it is manifest that the 
 shadow cannot pass over any part of the moon's disk. 
 
 PROP. XC. The moon, at the full, will be partially eclipsed, if its lati- 
 tude is less than the sum, but greater than the difference, of its own seraidiam- 
 and that of the earth's shadow. > 
 
 If Cc, the latitude of the moon, be supposed less than Ct, c, the sum of the semidiameters of 
 the shadow and the moon, /, the lower edge of the moon, will be below t, the upper edge of the 
 shadow; whence the side of the moon toward the ecliptic will be eclipsed. But, because Cc, 
 the moon's latitude, added to co, its semidiameter, is greater than Ct, the semidiameter of the 
 earth's shadow, the upper edge of the moon o cannot come within the shadow ; whence the 
 eclips*e will be partial. And because in this case the moon's latitude, together witli its semi- 
 diameter, is greater than the semidiameter of the shadow, the moo'n's latitude is greater than the 
 difference of the semidiameters of the shadow and the moon ; or, because Cc + co is greater 
 than Ct, Cc is greater than Ct co. 
 
 PROP. XCI. The moon will, at the full, be totally eclipsed, if its lati- 
 tude is less than the difference between its own semidiameter, and that of the 
 earth's shadow. 
 
 If Cc + co be less than Ct, that is, if Cc be less than Ct cl, the upper edge of the moon 
 may come within the shadow of the earth. 
 
 PROP. XCII. The moon is centrally eclipsed, only when, in opposition, 
 it is in one of its nodes. 
 
CHAP. V. OF THE MOON. 271 
 
 The node N, being the common intersection of the moon's orbit, and the plane of the ecliptic, 
 is in both. Therefore when the moon is in the node, its centre is in the plane of the ecliptic, in 
 which is the centre of the earth's shadow, and consequently at the full these centres coincide. 
 
 SOHOL. 1. Both the moon and the shadow moving from west to east, the moon would always 
 be in eclipse while it was at, or near, its nodes, if it moved with the same velocity as the earth ; 
 but because it moves much faster than the shadow of the earth, it soon passes from its opposition 
 out of the shadow. 
 
 SCHOL. 2. The semiangle ot the cone of the earth's shadow being known, the length of the 
 shadow may be found. 
 
 The semiangle CMT being equal to the sun's apparent semidiameter, or about 16', and CT plate 10. 
 a semidiameter of the earth being perpendicular to the tangent CM, if TM be radius, CT * 
 is the sine of the angle CMT Therefore, as the sine of an angle of 1C' is to radius, so is CT to 
 TM, the height of the cone ; that is, as 1 to 217. Whence the length of the shadow is about 
 217 semidiameters of the earth. 
 
 PROP. XCIII. The moon in a total eclipse, is not wholly invisible. 
 
 This is known by observation ; and the phenomenon is produced by the reflection of rays of 
 light falling upon the earth's atmosphere, toward the shadow, and consequently toward the moon 
 in the shadow. 
 
 DEF. LVIIT. An Eclipse of the Sun happens when the moon, passing 
 between the sun and the earth, intercepts the sun's light. 
 
 PROP. XCIV. The sun can only be eclipsed at the new moon. 
 
 For it is only when the moon is in conjunction, that it can pass directly between the sun and 
 the earth. 
 
 PROP. XCV, The sun can only be eclipsed, when the moon, at its con- 
 junction, is in or near one of its nodes. 
 
 For unless the moon is in or near one of its nodes, it cannot appear in or near the same plane 
 with the sun ; without which it cannot appear to us to pass over the disk of the sun. At every 
 other part of its orbit, it will have so much noithern or southern latitude, as to appear either 
 above or below the sun. If the moon be in one of its nodes, having no latitude, it will cover the 
 whole disk of the sun, and produce a. total eclipse, except when its apparent diameter is less than 
 that of the sun ; if it be near one of its nodes, having a small degree of latitude, it will only 
 pass over a part of the sun's disk, or the eclipse will be partial. 
 
 PROP. XCVI. In a total eclipse of the sun, the shadow of the moon 
 falls upon that part of the earth where the eclipse is seen. 
 
 Let SL be the sun, TR the moon, VM a part of the surface of the earth. A spectator placed Plate 10. 
 any where between V and M, will not see any part of the sun, because the moon will intercept ' "' 
 all the rays of light which come to him directly from the sun ; and it is manifest, that, in this 
 situation, the moon, being an opaque body, will cast its shadow upon VM, the part of the earth 
 where the eclipse is total. 
 
78 OF ASTRONOMY. BOOK VII. PART I. 
 
 PROP. XCVII. The shadow of the moon is a cone terminated in a 
 point. 
 
 Because (by Prop. LXXXVIH. Cor.) the diameter of the moon is less than the diameter of 
 the earth's shadow, where it eclipses the moon, and this diameter (by Prop. LXXXVI ) is less 
 than the diameter of the earth, the diameter of the moon is much less than that of the sun ; con- 
 sequently, its shadow will be a cone, terminated in a point. The tangents LAR, SAT, terminate 
 in A ; and only the rays that would pass within these tangents are intercepted by the moon ; 
 therefore RTA is the form of the moon's shadow. 
 
 LEMMA.. Half the angle of the cone of the moon's shadow is equal to the 
 angle of the apparent semidiameter of the sun. 
 
 Plate 10. Let FBGA be the sun ; HED the moon ; the cone HMD the moon's shadow ; CMD the semi- 
 angle of this cone; SA the semidiameter of the sun, and SDA the angle which this semidiameter 
 would subtend, if viewed from D. It may be proved, as in Prop, LXXXVIH. Lerrma, that 
 CMD is equal to SDA ; for the distance of the moon from the sun is so nearly equal to that of 
 the earth from the sun, that the apparent semidiameter of the sun, as seen from the earth or 
 moon, may be considered as equal. 
 
 PROP. XCVIII A semidiameter of the moon's shadow, where it falls 
 upon the earth, is equal to the difference between the apparent semidiameters 
 of the moon and sun. 
 
 Plate 10. Let CDE represent the moon, CME the cone of its shadow; FG the semidiameter of the 
 Fl ' "? moon's shadow where it falls upon the earth in a solar eclipse ; CT the semidiameter of the 
 moon, CFT its angle viewed from F, and FTG the angle of the apparent semidiameter of 
 the moon's shadow viewed from T. 
 
 In the triangle TFM, the external angle TFC (El. I. 32.) is equal to the two angles FTG, 
 FMG. Therefore FTG is equal to the difference between TFC and FMG ; and FMG is (by 
 the preceding Lemma) equal to the angle of the sun's apparent semidiameter, and TFC is the 
 angle of the moon's apparent semidiameter ; whence the Proposition is manifest. 
 
 PROP. XCIX. In a partial eclipse of the sun, & penumbra, or imperfect 
 shadow of the moon, falls upon that part of the earth where the partial eclipse 
 is seen. 
 
 I'late 10. A spectator at N, or P, might see the whole sun ; for a ray passing from the most remote 
 F'- 11- s i(] e of the sun, S or L, would not be intercepted by the moon. But at any point in NM, VP, 
 the spaces between the moon's shadow and the points N, P, the spectator would only see a part 
 of the sun ; thus at G, or D, he would see that half of the sun which lies without the tangents 
 DRC, GTC; consequently, in all places between the points N, M, and P, V, there will be less 
 light from the sun, than if it were not at all eclipsed. This deficiency of light is called the 
 moon's penumbra. 
 
 PROP. C. The moon's penumbra is an increasing cone, and its dark- 
 ness increases toward the shadow of the moon. 
 
CHAP. V. OF THE MOON. 278 
 
 While a spectator is in the space between N and M, or P and V, lie is in the penumbra ; but Plate 10. 
 at the points N, P, passes out of it. Therefore the tangents NS, PL, are the limits of the F ' s< : 
 penumbra. If tangents be supposed drawn round the spherical surfaces of the sun and moon, 
 they will form two cones, having their common vertex at F, and increasing, the one toward the 
 points L, S, the othertoward the points N, P. And as the spectator moves from N toward M, 
 or from P toward V, a greater and greater portion of the sun continually becomes invisible to 
 to him ; whence the penumbra increases in darkness toward M and V. 
 
 PROP. CI. The seraiangle of the moon's penumbra is equal to the angle 
 of the sun's apparent semidiameter. 
 
 Let SL be the sun, and TR the moon. By Prop. C. the tangents SFN", LFP, terminate the i>j a te 10. 
 cone of the penumbra. CE, drawn from the centre of the sun to that of the moon, bisects TFR, Fl S' 11 - 
 the angle of this cone ; whence EFT is its semiangle. LC being the semidiameter of the sun, 
 LTC is the angle under which this semidiameter would appear from the moon T. I say, TFE is 
 equal to LTC. For, in the triangle TCF, the external angle EFT is equal to the two internal and 
 opposite angles FTC, FCT ; that is, to the two angles LTC, TCE. Therefore EFT is equal to 
 LTC, TCE. But TCE, being the angle which the moon's semidiameter would subtend, if 
 viewed from the sun, is so small that it may be neglected ; whence EFT may be considered as 
 equal to LTC. 
 
 PROP. CII. The semidiameter of the moon's penumbra, in that part 
 through which the earth passes in an eclipse of the sun, is equal to the sum of 
 the apparent setnidiameters of the sun and moon. 
 
 Let CD be the moon, CDAB its penumbra, and CMS the angle of the cone of the penumbra; Plate 10. 
 and let AEBF be the section of the penumbra through which the earth passes in an eclipse of the . K' * 2> 
 sun, AB its diameter, and AT its semidiameter. MLT, drawn from the vertex through the 
 centre of the moon, will besect the angle CMD. Therefore (by Prop. CI.) CML, the semiangle 
 of the penumbra, is equal to the sun's apparent semidiameter. And CL is the moon's apparent 
 semidiameter, as seen from the earth A, subtending the angle CAL ; and AT, the semidiameter 
 of the penumbra, seen from L, subtends the angle ALT. Now the angle ALT is equal to the 
 two angles CML, CAL ; whence the truth of the Proposition is manifest. 
 
 DBF. LIX. The disk of the earth is that hemisphere of the earth, which 
 is seen, as a circle, from the moon. 
 
 PROP. GUI. At the new moon the whole disk of the earth is enlight- 
 ened. 
 
 For, the moon being then between the sun and the earth, the earth viewed from the moon will 
 appear in opposition, and consequently its enlightened hemisphere will be toward the moon. 
 
 PROP. CIV. The semidiameter of the earth's disk is equal to the 
 moon's horizontal parallax. 
 
 35 
 
OF ASTRONOMY. BOOK VII. PART I. 
 
 The moon's horizontal parallax is the apparent semidiameter of the earth, as viewed from 
 the moon, that is, it is equal to the semidiameter of the disk, since the disk is a hemisphere of 
 the earth viewed from the moon. 
 
 PROP. C V. If the latitude of the moon, when new, is equal to, or greater 
 than, the sum of the semidiameter of the penumbra and the moon's horizontal 
 parallax, there will be no eclipse of the sun ; if less, there will be an eclipse, 
 either partial, or total. 
 
 Let ACB be a P art of the ecli P tic let GNE be the P lane of the moon's orbit ; let N be the 
 node, DL the earth's disk, and ol a section of the moon's penumbra. Because the centre of the 
 moon's penumbra is always in a right line passing through the centres of the sun and the moon, 
 the distance of c, the centre of the moon's penumbra from the plane of the ecliptic, must always 
 be the same with the distance of the centre of the moon from the same plane ; that is, with the 
 latitude of the moon. And because the centre of the earth, or the earth's disk C, is in the eclip- 
 tic, Cc, the distance between the centres of the penumbra and of the earth's disk, is always equal 
 to the latitude of the moon. Now if Cc, or the latitude of the moon, be equal to, or greater 
 than, cl, the semidiameter of the penumbra, together with tC, the semidiameter of the earth's disk, 
 or the moon's horizontal parallax, then no part of the penumbra will fall upon the disk, that is, 
 there will be no eclipse. If Cc, the latitude of the moon, be less than tC + Ic, the edge of the 
 penumbra will be nearer the ecliptic than the edge of the disk, and there will be a partial eclipse. 
 And if Cc be less than Ct, the shadow of the moon will pass over some part of the disk of the 
 earth, and where this happens, the eclipse will be total. 
 
 PROP. CVI. If the moon, when ew, is in one of its nodes, the eclipse 
 of the sun will be central. 
 
 For then the centres of the earth, sun, and moon, being all in the plane of the ecliptic, the 
 centre of the moon will pass between the sun's centre, and that of the earth. 
 
 SCHOL. 1. The penumbra of the moon in a central eclipse will not cover the whole disk of 
 the earth. 
 
 The semidiameter of the moon's penumbra, being equal to the sum of the apparent semidiam- 
 eters of the sun and moon, that is, about 16' 23" + 15' 3?" or 3-</ at the medium ; its diameter 
 is about 64'; whereas the diameter of the earth's disk is about 120'; whence the penumbra 
 cannot cover the whole disk. 
 
 SCHOL. 2. The height of the shadow of the moon is about 60 semidiameters of the earth. 
 The semiangles of the earth's shadow, and of the moon's shadow, being each equal to the sun's 
 apparent semidiameter, the angles are equal to one another, and these cones are similar. There- 
 fore as the semidiameter of the base of the earth's shadow (that is, of the earth) is to the semi- 
 diameter of the base of the moon's shadow, (that is, of the moon) so is the height of the earth's 
 shadow to the height of the moon's shadow. Now the semidiameter of the earth is to that of the 
 moon nearly as 100 to 28, and the height of the earth's shadow is about 217 semidiameters of 
 the earth ; whence the height of the moon's shadow is equal to about 60| semidiameters of the 
 earth ; for 100 : 28 : : 217 : 60| nearly. 
 
CHAP. V. OF THE MOON. 
 
 DEF. LX. An eclipse of the sun is said to be annular, when, at the time 
 of the eclipse, a ring of the sun appears round the edges of the moon. 
 
 PROP. CVII. A central eclipse of the sun will be an annular one, if 
 the distance of the moon from the earth at the time of the eclipse be greater 
 than its mean distance. 
 
 SL being the sun, TR the moon, TAR the moon's shadow, and EA the height of this shadow, Plate 10. 
 which is about 60i semidiameters of the earth ; if at the time of a central eclipse PN is a part Fl * 
 of the surface of the earth, those who live in the parts PV, MN, being in the penumbra, will (by 
 Prop. XCIX.) sec a partial eclipse ; and those who live between V and M, being in the shadow, 
 will (by Prop. XCVI.) see a total eclipse. But if the distance of the moon from the earth be 
 equal to EA, or G0 semidiameters of the earth, (which is the moon's mean distance,) A will be 
 the only point from which the eclipse will appear total. And if the moon's distance be greater 
 than EA, as EO, the shadow not reaching the earth, there will be no total eclipse. Consequently, 
 though a spectator at O would see a central eclipse, (because the centres C, E, are in the same 
 line with the point of vision S) yet the eclipse would not be total, because the spectator is not in 
 the shadow of the moon. Hence it must appear annular ; for let ORX be a tangent to the moon 
 drawn from the eye at 0; and it will fall upon the sun at X, and the part XL of the sun will be 
 visible ; in like manner parts of the sun equal to XL will be visible all round the moon, forming 
 a ring. 
 
 COR. Hence it appears, that in an annular eclipse it is the penumbra of the moon which falls 
 upon the earth. 
 
 DEF. LXI. The Lunar Ecliptic Limit is the least distance that the 
 moon can be at from one of its nodes, without being eclipsed at the time of 
 opposition ; the Solar Ecliptic Limit is the least distance the moon can be at 
 from one of its nodes, without eclipsing the sun at the time of conjunction. 
 
 PROP. CVIII. The solar ecliptic limit is greater than the lunar. 
 
 By Prop. CV. it is found, that 92' is the least latitude the moon can havej when new, with- pj ate 10 
 out eclipsing the sun.* If, therefore, Nc be the distance of the moon from the node, when its Fig. 9. 
 latitude, or cC, is 92', in the triangle CcN, the angle cCN being a right angle, because cC is 
 perpendicular to the plane of the ecliptic ; the angle cNC being about 5 30', the inclination of 
 the moon's orbit to the plane of the ecliptic ; and the side cC being 92', Nc will be found by 
 trigonometry to be about 16. In the same manner, supposing Cc, the latitude of the moon, to be 
 61', according to Prop. LXXXIX. the length of the side Nc, or the lunar ecliptic limit, will be 
 found to be about 12.* Whence the truth of the Proposition is manifest. 
 
 COR. There are more eclipses of the sun, in a course of years, than of the moon ; for, the 
 sun will be eclipsed, if, when the moon is new, it is within 16 of one of the nodes, but the moon 
 
 The moon's latitude may sometimes be several minutes less than that mentioned in the text, without 
 producing an eclipse. 
 
376 OF ASTRONOMY. BOOK VII. PAUT I. 
 
 only when at the full it is within 12 of one of the nodes ; the sun may be eclipsed while the 
 moon is in 64 degrees of its orbit, but the moon only while it is in 48 degrees of its orbit. 
 
 SOHOL. Every eclipse of the moon will be visible, if the moon be above the horizon at the 
 time of the eclipse ; because that part of the moon on which the shadow of the earth falls, must 
 appear obscured wherever the disk of the moon is visible. But the sun may be eclipsed, and yet 
 the eclipse be invisible in places to which the sun is above the horizon, because there can be no 
 eclipse of the sun, except in those parts of the earth which are within the shadow or penumbra 
 of the moon, and neither of these is large enough to cover the whole disk. Hence, in any 
 given place, more eclipses of the moon than of the sun will be seen in a course of years ; for 
 though there are more eclipses of the sun than of the moon, many of the former are not visible 
 at any one place while the sun is above the horizon ; but all the latter are visible at the same 
 place while the moon is above the horizon.* 
 
 PROP. CIX. When the moon is near the first of Aries, and is moving 
 toward the tropic of Cancer, the time of its rising will vary but little for sev- 
 eral days together. 
 
 If the moon were to move in the equator, its motion in its orbit, by which it describes a 
 revolution, in respect of the sun, in 29 days 12 hours, would carry it every day eastward from 
 the sua about 12 11' ; whence its time of rising would vary daily about 50 minutes. But, 
 because the moon's orbit is oblique to the equator, nearly coinciding with the ecliptic, different 
 parts of it make different angles with the horizon, as they rise or set, those parts which rise with 
 the smallest angles setting with the greatest, and the reverse. Now the less this angle is, the 
 greater portion of the orbit rises, in the same time. Consequently, when the moon is in those 
 parts which rise or set with the smallest angles, it rises or sets with the least difference of time 5 
 and the reverse. But in northern latitudes, the smallest angle of the ecliptic and horizon is made 
 when Aries rises and Libra sets, and the greatest when Libra rises and Aries sets ; and there- 
 fore, when the moon rises in Aries, it rises with the least difference of lime. Now the moon is 
 in opposition in or near Aries, when the sun is in- or near Libra, that is, in the autumnal 
 months ; when, the moon rising in Aries, whilst the sun is setting in Libra, the time of its rising 
 is observed to vary only two hours in 6 days in the latitude of London. This is called the 
 harvest moon. 
 
 SCHOL. This circumstance takes place every month ; but as it does not happen at the time 
 of full moon, there is no notice taken of it. When the moon's right ascension is equal to six 
 signs, that is, when she is in or about the beginning of Libra, there is the greatest difference of 
 the times of rising, viz. about an hour and 15 minutes. Those signs which rise with the least 
 angle set with the greatest, and the contrary ; therefore, when there is the least difference in the 
 times of rising, there is the greatest in setting, and vice versa. 
 
 The following table shows the daily mean difference of the moon's rising and setting, on the 
 parallel of London, for 28 days ; in which time the moon finishes her period round the ecliptic, 
 and gets 9 degrees into the same sign from the beginning of which she set out. 
 
 * For the calculation and projection of lunar and solar eclipses, see the Problems immediately preceding 
 the Tables at the end of this book. 
 
CHAP. VI. 
 
 OF THE SATELLITES. 
 
 1 
 
 w> n? 
 <S' <g 
 
 w fft 
 
 Rising 
 Diff. 
 
 Setting 
 Diff. 
 
 e 
 
 P 
 *- 
 
 en S? 
 
 ok' "g 
 
 Rising 
 Diff. 
 
 Setting 
 Diff. 
 
 
 
 2 
 
 H. M. 
 
 H. M. 
 
 ? 
 
 w n 
 
 H. M. 
 
 n. ST. 
 
 1 
 
 2B 13 
 
 1 5 
 
 50 
 
 15 
 
 Vf 17 
 
 46 
 
 5 
 
 2 
 
 26 
 
 1 10 
 
 43 
 
 16 
 
 .-.v 1 
 vtv * 
 
 40 
 
 8 
 
 3 
 
 a 10 
 
 1 14 
 
 37 
 
 17 
 
 14 
 
 35 
 
 12 
 
 4 
 
 23 
 
 1 17 
 
 32 
 
 18 
 
 27 
 
 30 
 
 15 
 
 5 
 
 itjj 6 
 
 1 16 
 
 28 
 
 19 
 
 X 10 
 
 25 
 
 16 
 
 6 
 
 19 
 
 1 15 
 
 24 
 
 20 
 
 23 
 
 20 
 
 17 
 
 7 
 
 =2. 2 
 
 15 
 
 20 
 
 21 
 
 CM ~ 
 
 17 
 
 16 
 
 8 
 
 15 
 
 1 15 
 
 18 
 
 22 
 
 20 
 
 17 
 
 15 
 
 9 
 
 28 
 
 1 15 
 
 17 
 
 23 
 
 JJ 3 
 
 20 
 
 15 
 
 10 
 
 "I 12 
 
 1 15 
 
 22 
 
 24 
 
 16 
 
 24 
 
 15 
 
 11 
 
 26 
 
 1 14 
 
 30 
 
 25 
 
 29 
 
 SO 
 
 1 14 
 
 12 
 
 / 8 
 
 1 13 
 
 39 
 
 26 
 
 n is 
 
 40 
 
 1 13 
 
 IS 
 
 21 
 
 1 10 
 
 47 
 
 27 
 
 26 
 
 5C 
 
 1 7 
 
 14 
 
 Vf 4 
 
 1 4 
 
 56 
 
 28 
 
 25 9 
 
 1 00 
 
 1 58 
 
 EXP. Let small patches be placed on the ecliptic of a globe, as far from one another as the. 
 moon moves from any point of the ecliptic in 24 hours, that is, about 13i degrees; then, while 
 the globe is turned round, observe the rising and setting of the patches in the horizon ; the hour 
 index will show the difference of time at which the moon rises or sets in different parts of its 
 orbit. 
 
 CHAP. VI. 
 
 Of the satellites of Jupiter, Saturn, and the Herschel planet. 
 
 PROP. CX. Any satellite is at its greatest elongation from its primary, 
 when a line drawn from the earth through the satellite is a tangent to the 
 orbit of the satellite. 
 
 Let FIE be a part of the orbit of the primary planet, AXBT the earth's orbit, S the sun, Plate 10. 
 RGNL the orbit a satellite. If the earth is at X, and the satellite at L or N, so that a line XL Flg% 14- 
 or XN, drawn from the earth, is a tangent to the orbit KGNL, it may be shown, as before con- 
 cerning the planets, that L or N is the greatest elongation of the satellite. 
 
 PROP. CXI. Any satellite appears iu inferior conjunction with its pri- 
 mary, when the satellite is between the earth and the primary, and in superior 
 conjunction, when the primary is between the satellite and the earth. 
 
 If the earth is at X, and the planet at I, the outermost satellite will be in conjunction with 
 its primai-y when they both appear in the same line MIV, and in its inferior conjunction at M, 
 and its superior at V. 
 
 PROP. CXII. The apparent motion of any satellite, as it passes from its 
 
7 OF ASTRONOMY. BOOK VII. PART I. 
 
 greatest elongation on one side of its primary through the superior conjunc- 
 tion, to its greatest elongation on the other side, is direct. 
 
 As the satellite passes from L, its greatest elongation on one side, through V, its superior 
 conjunction to M, its geocentric motion is from west to east, or in consequential as was shown 
 concerning the planets. 
 
 PROP. CX1II. The apparent motion of any satellite, as it passes from 
 its greatest elongation on one side of its primary through the inferior conjunc- 
 tion, to its greatest elongation on the other side, is retrograde. 
 
 As the satellite passes from N through M to L, its geocentric motion will be from east 
 to west, or in antecedeniia, as was proved concerning the planets. 
 
 COR. The satellites are sometimes to the west and sometimes to the east of their primaries. 
 
 PROP. CXIV. The greatest elongations of a satellite on each side are 
 equal. 
 
 For by observation it is found, that the angles LXI, NXI, are equal with respect to all the 
 satellites. 
 
 Con. Hence it appears that the orbits of the satellites are circular, or nearly so, having 
 their primaries at the centre of their orbits. 
 
 SOHOL. The satellites move round the primary in orbits, that are nearly circular ; but round 
 Plate 15. the sun in curves of a different kind. Let ABCDE, &c. to T, (Plate 15, Fig. 1.) be as much of 
 ' e> ' Jupiter's orbit as is described by that planet in 18 days. Then the curves a, b, c, d, (according 
 to Ferguson) represent the paths of the four satellites. 
 
 If we suppose Jupiter to move from A, the first satellite from a, the second from b, the third 
 from c, and the fourth from d<; or each of the satellites from the point of conjunction with the 
 sun, as seen from the primary; then, at the end of one day, Jupiter will be at B, and the satel- 
 lites at 1, in their respective courses. At the end of the second day Jupiter will be at C, and the 
 satellites at 2 in the curves they respectively describe, and so on ; the capital letters showing Jupi- 
 ter's place in its path at the end of each day, the figure under each of them the number of the 
 day, and the like figures on the paths of the satellites, their places at the same time. The first 
 satellite appears to be stationary at -f near C, as seen from the sun ; retrograde from + to 2, 
 at 2 stationary again ; thence direct till beyond 3 ; twice stationary and once retrograde be- 
 tween 3 and 4. This satellite intersects its own path every 42J hours, making loops as in the 
 diagram at 2, 3, 5, &c. soon after every conjunction. The second crosses its own path every 
 3 days 13 hours, as at 4, 7, 11, &c. making only 5 loops, and as many conjunctions while the 
 first makes 10. The ihird satellite at the end of every 7 days 4 hours, makes an angle in con- 
 junction with the sun,as at 7, 14. The fourth satellite is always progressive, making neither loops 
 nor angles ; its first conjunction is at e, at the end of 16 days 18 hours. The first, second, and 
 third are, according to the figure, nearly in the same relative situation, every seventh day. 
 
 The path of Saturn's first satellite about the sun is looped, but not those of the second, third, 
 fourth, and fifth. 
 
CHAP. VI. OF THE SATELLITES. 
 
 PROP. CXV. The satellites of Jupiter, Saturn, and the Herschel, are 
 eclipsed by their respective primaries. 
 
 The planet I, being an opaque body, casts a shadow IV, opposite to the sun. Therefore, Plate 10. 
 \vhen one of its satellites in describing the arc VO comes to V,jt will be eclipsed by falling into Fl - 14> 
 this shadow. If the earth is at A in its orbit, a spectator from the earth will lose sight of the 
 satellite, when it is thus eclipsed at V ; and then, as it emerges from the shadow, it becomes 
 again visible, till, at O, it passes behind its primary. If the earth be at X, the satellite will be 
 eclipsed, and in occulation at the same time. 
 
 PROP. CXVJ. When one of the satellites passes between the sun and 
 its primary, it eclipses the sun. 
 
 A satellite at M will be between the sun and its primary, and occasion an eclipse of the sun 
 on that part of the primary where the shadow of the satellite passes, which shadow will appear 
 as a dark spot on the disk of the planet to an inhabitant of the earth. 
 
 SCHOL. Dr. Herschel has discovered, that the fifth satellite of Saturn is, in its rotation, 
 subject to the same law that our moon obeys, that is, it turns round its axis in the same time in 
 which it revolves about the planet. Hence the doctor thought it natural to conclude, that all 
 the secondary planets might be governed by the same laws to which those are subject. This 
 theory he thinks considerably confirmed by certain observations, which he made on the satellites 
 of Jupiter, and which he communicated to the Royal Society, June 1, 1797. 
 
 The following table will give the periodical times and distances of Jupiter's satellites, and 
 the angles under which their orbits are seen from the earth, as its mean distance from Jupiter. 
 
 Satellites. Dayi. h. min. Dist. in miles. Angles of orbit. 
 
 1 1 18 28.6 266.000 3' 55" 
 
 2 S 13 17.9 423.000 6 14 
 
 3 73 59.6 676.000 9 58 
 
 4 16 18 5.1 1.189.000 17 30 
 
 These satellites of Jupiter are of great use in astronomy. (1.) In determining the distance 
 of Jupiter from the earth. (2.) They afford a method of demonstrating that the motion of light 
 is progressive, and not instantaneous, as was once supposed. See Prop. CXVII. And 
 (3.) The most considerable advantage is derived from the eclipses of the satellites of Jupiter, in 
 ascertaining the longitude of different places on the earth. 
 
 EXP. Suppose two observers of an eclipse, the one at London, the other at the Cape of Good 
 Hope ; the eclipse will appear to both at the same moment of time ; but being situated under dif- 
 ferent meridians, they count different hours, according to which their uifference of longitude is 
 found. Thus, if at the Cape of Good Hope, an emersion of a satellite is observed at lOh. 46' 
 4i>" apparent time, and the same is seen at Greenwich at 9h. 33' 12", the difference of which 
 times is Ih. 13' 35", the longitude of the cape east of Greenwich in time, or 18 23' 5". 
 
 Note. The third satellite is the largest of all ; the first and fourth are nearly of the same 
 size ; the second is the smallest. 
 
280 OF ASTRONOMY. BOOK VII. PART I. 
 
 OF THE SATELLITES OF SATTTRlf. 
 
 Satellite! 
 1 
 
 2 
 
 Periodical times. 
 
 1 day. 2lh. &' 
 2 17 41 
 
 27" 
 22 
 
 Distance in miles. 
 170.000 
 
 217.000 
 
 3 
 
 4 
 
 12 
 
 25 
 
 12 
 
 303.000 
 
 4 
 
 15 
 
 22 
 
 41 
 
 13 
 
 704.000 
 
 5 
 
 79 
 
 7 
 
 48 
 
 _ 
 
 2.050.000 
 
 6 
 
 1 
 
 8 
 
 53 
 
 8 
 
 135.000 
 
 r 
 
 
 
 22 
 
 37 
 
 22 
 
 lor.ooo 
 
 Plate 14. Fig. 3. Plate 14, is a view of the proportional magnitudes of the orbits of the moon, Jupiter's 
 F 'Sf' 3 - four satellites, and the five first of vSaturn. Mm represents the moon's orbit, the earth being 
 supposed to be at E ; J Jupiter, 1, 2, 3, 4, the orbits of the four satellites; Sat. Saturn, 1, 2, S, 
 4, 5, the orbits of the five first satellites. 
 
 The 6th and 7th satellites were discovered by Dr. Herschel, in the years 1787 and 1788. 
 To prevent mistakes, he calls them the 6th and 7th, though nearer to the planet than the other 
 five. Dr. Herschel observes, that Saturn has probably a considerable atmosphere. It turns on 
 an axis perpendicular to the ring, in lOh. 16' 0.44", and is flattened at the poles, so that the 
 equatorial diameter is to the polar as 11 to 10. Phil. Trans. Vol. 80, Parti, and II. and Vol. 84. 
 
 OF THE SATELLITES OF HERSCHEL. 
 
 Dr. Herschel has at different times discovered six satellites belonging to the new planet, two 
 of which he described in the 77th and 78th Vols. of the Phil. Trans, where he says, " I confess 
 that this scene appeared to me with additional beauty, as the little secondary planets seemed to 
 give a dignity to the primary one, which raises it into a more conspicuous situation among the 
 great bodies in our system." 
 
 The following is the arrangement of the six satellites. 
 
 Satellites. When discovered. Periodical times. 
 
 1 Jan. 1, 1790 5 d. 21 h. 25' 0" 
 
 2 Jan. 11, 1787 8 17 I 19 
 
 3 Mar. 26, 1794 10 23 40 
 
 4 Jan. 11, 1787 13 11 5 l 
 
 5 Feb. 9, 1790 38 1 49 
 
 6 Feb. 28, 1794 107 16' 40 
 
 It will be hardly necessary," says Dr. Hersehel, " to add, that the accuracy of these periods 
 depends entirely upon the truth of the assumed distances ; some considerable difference, there- 
 fore, may be expected, when observations shall furnish us with proper data for more accurate 
 determinations." See Phil. Trans. 1798. 
 
 PROP. CXVII. A ray of light is about 8 minutes in coming from the 
 sun to the earth. 
 
 Plate 10. Let A be the sun, BECD the earth's orbit, F the planet Jupiter, and HNG the orbit of its 
 Fig. 13. inner satellite. Let FGH represent the shadow of Jupiter. While the satellite is between H 
 
 and G, it is eclipsed ; when it comes to II, it emerges, and becomes visible to a spectator at B. 
 
 From comparing the times of the apparent entrance and emersion of the satellite, with tables 
 
CHAP. VII. OF COMETS. 
 
 calculated for the mean distances of the earth from the satellite, the visible emersion at the least 
 distance is found to happen about 8 minutes sooner, and at the greatest distance about 8 min- 
 utes later, than by the tables ; consequently, the ray of light is about 16 minutes in passing- 
 through the earth's orbit, or 8 minutes in coming from the sun to the earth. 
 
 COR. The diameter of the earth's orbit being 194,000,000 miles, the velocity of light will be 
 
 194.000 .(100 
 
 ; = 202,088 miles in a second. 
 
 lt> x t>0 
 
 PROP. CXVIII. Jupiter is surrounded by cloudy substances, subject to 
 frequent changes in their situation and appearance, called his Belts. Saturn 
 is encompassed with a Ring, whose greatest apparent diameter is to that of 
 the planet, as 9 to 4. 
 
 These are known from observation. The Belts of Jupiter are sometimes of a regular form ; pi a te 10. 
 sometimes interrupted and broken ; and sometimes not at all to be seen. The plane of Saturn's Fl S- 15 - 
 ring is inclined to the plane of the ecliptic at an angle of 31 degrees ; which appears like two 
 arms to the planet, and which is only visible when the sun and the earth are both on the same 
 side of its plane. On account of its inclination, it always appears oblique to the eye, and there- 
 fore elliptical j whence the part behind Saturn is invisible, and the part before cannot be distin- 
 guished from the planet. The ring, being opaque, can only be visible when the sun's rays are 
 reflected from its broad surface to the earth, that is, when the sun and the earth are both on the 
 same side of the plane of the ring. 
 
 The latter discoveries of Dr. Herschel have shown, that what was supposed to be a single 
 broad flat ring of Saturn, is divided into two parts, lying exactly in the same plane, and revolv- 
 ing about an axis perpendicular to that plane, in lOh. 32' 15". The dimensions of these con- 
 centric rings, and the space between them, he states to be as in the following table. 
 
 Mile*. 
 
 Inner diameter of the smaller ring ... 146,345 
 
 Outside diameter of ditto .... 184,393 
 
 Inner diameter of larger ring .... 190,248 
 
 Outside diameter of ditto --.... 204,883 
 
 Breadth of the inner ring ...... 20,000 
 
 Breadth of the outer ring ..... 7,200 
 
 B read th of the vacant space ..... 2,830 
 
 CHAP. VII. 
 
 Of Comets. 
 PROP. CXIX. Comets are opaque and solid bodies. 
 
 A comet, at a given distance from the earth, shines much brighter when it is on the same side 
 of the earth *ith the sun, than when it is on the contrary side ; from whence it appears that it 
 owes its brightness to the sun. Tie resistance of the comet of 1680 to the action of the great 
 
 56 
 
383 OF ASTRONOMY. BOOK VII. PART I, 
 
 heat, to which it was probably exposed in its near approach to the sun, furnishes evidence in fa- 
 vour of its being a fixed and solid body. 
 
 PROP. CXX. The comets describe very eccentric ellipses about the 
 sun, placed in one of their foci. 
 
 They arc observed to approach toward, and afterward recede from, the sun, and to describe 
 paths in the heavens, which agree with elliptic orbits ; it is therefore most probable, that agreea- 
 bly to the general analogy of nature, they move in such orbits, and have the sun in one of the 
 foci of the ellipse. The calculations framed upon this supposition, by which the returns of 
 comets have been foretold, having, as far as observations have been made, been found to agree 
 with the phenomena, strongly confirm the truth of the Proposition. 
 
 SCHOL. I. Comets are often accompanied with a luminous train, called the tail, which i* 
 conjectured to be smoke rising from the body in a line opposite to the sun. The body of the 
 comet is supposed to be surrounded by an atmosphere ; the sun is also supposed to he surrounded 
 by an ether, or a subtle fluid, extending to a great distance from the sun, which may be consid- 
 ered as the solar atmosphere. From the heat which the comet lias acquired by approaching to- 
 ward the sun, and by the reflection of the sun's rays from the solid body and atmosphere of 
 the comet, the parts of the solar atmosphere where the comet passes are more heated, and con- 
 sequently more rarefied or specifically lighter than elsewhere. The pa> ts thus rarefied will be 
 put into motion ; and since there will be a constant succession of fresh portions of the sun's 
 atmosphere within that of the comet, there will be a perpetual stream of this rarefied matter. 
 This stream will impel the particles of the comet's atmosphere, and make them move along with 
 it, thus producing the. smoke which, reflecting the sun's rays, forms the visible tail. And this 
 stream of rarefied solar atmosphere will move those parts of this atmosphere which have the 
 least specific gravity, that is, directly from the sun. 
 
 SCHOL. 2. Of all the comets, the periods of only three are known with any degree of cer- 
 tainty. The first of these comets appeared in the years 1531, 1607, and 1682; and is expected 
 to appear every 75th year. The second of them appeared in 1532 and 1661, and was expected 
 to return in 1789, and every 129th year afterward. The third, having last appeared in 1680, 
 and its period being no less than 575 years, cannot return until the year 2225. This comet, at 
 its greatest distance, is about 11 thousand two hundred millions of miles from the sun ; and at 
 its least distance from the sun's centre, which is 49,000 miles, is within less than a third part of 
 the sun's semidiameter from his surface. In that part of its orbit which is nearest the sun, it 
 moves at the rate of 880,000 miles in an hour. 
 
 SCHOL. 3. Dr. Hulley, who saw the comet which appeared in 1682, observes, "that there 
 are many things which make me believe, that the comet which Apian saw in the year 1531, was 
 the same with that which Kepler and Longomontanus more accurately described in the year 
 1607 ; and which I myself have seen return, and observed in the year 1682. All the elements 
 agree, and nothing seems to contradict this opinion besides the inequality of the periodic revolu- 
 tions ; which inequality is not so great, but that it may be owing to physical causes. For the 
 motion of Saturn is so disturbed by the rest of the planets, especially Jupiter, that the pe- 
 riodic time of that planet is uncertain for some whole days together. How much more, there- 
 fore, will a comet be subject to such like errors which rises almost four times higher than Saturn, 
 and whose velocity, though increased but very little, would be sufficient to change its orbit 
 
CHAP. VII. OF COMETS. 388 
 
 from an ellipse to a parabola. And I am the more confirmed in my opinion of its being the 
 same ; for, in the year 1456, in the summer time, a comet was seen passing retrograde between 
 the earth and sun, much after the same manner; which, though no body made observations upon 
 it, yet, from its period, and manner of transit, I cannot think different from those I have just 
 now mentioned. And since looking over the history of comets, I find, at an equal interval of 
 time, a comet to have been seen about Easter in the year 1305, which is another double period of 
 151 years before the former. Hence I think I may venture to foretel that it will return again 
 in the year 1758." 
 
 Dr. Halley computed the effect of Jupiter upon this comet in 1682, and found that it would 
 increase its periodic time above a year ; in consequence of which, he predicted its return at the 
 end of the year 1758, or the beginning of 1759. M. Clairaut computed the effects of both Sa- 
 turn and Jupiter, and found that the former would retard its return in the last period 100 days, 
 and the latter 511 days ; and he determined the time when the comet would come to its perihe- 
 lion to be on April 15, 1759; observing, that he might err a month from neglecting small quan- 
 tities in the computation. The comet did pass the perihelion on March 13, within 33 days of 
 the time computed. Now, if Dr. Halley meant the time of its passing the perihelion, and we 
 add 100 days for the action of Saturn, which he did not consider, it will bring it very near to 
 the time in which it passed the perihelion, and prove his computation of the effect of Jupiter to 
 have been very accurate. But if he meant the time when the comet would first appear, his 
 prediction was accurate, for it was seen on December 14, 1758. Dr. Halley, therefore, had the 
 glory, first to foretel the return of a comet, and the event answered, in a remarkable manner, 
 his prediction. He farther observed, that the action of Jupiter, in the descent of the comet 
 toward its perihelion 1682, would tend to increase the inclination of its orbit ; and accordingly 
 the inclination in 1682 was found to be 22' greater than in 1607. 
 
 Dr. Halley suspected the comet in 1680, to have been the same which appeared in 1106,531, 
 and 44 years before Christ. He also conjectured, that the comet observed by Apian, in 1532, 
 was the same as that observed by Hevelius in 1661 ; if so, its period was 129 years, and it 
 ought to have returned in 1789, but it did not appear. M. Mechain having collected all the 
 observations in 1532, and calculated the orbit again, found that it differed materially from that 
 calculated by Dr. Halley, which renders it extremely doubtful whether this was the comet which 
 appeared in 1661 ; and this doubt is increased by its not appearing in 1789. 
 
 SOHOL. 4. From the beginning of our era to this time, it is probable, according to the best 
 accounts, that there have appeared about 500 comets. Before that time above 100 others are 
 recorded to have been seen, but it is probable that not above half of them were comets. And 
 
 when we consider that many others may not have been perceived, from being too near the sun ; 
 
 from appearing in moon ligtit ; from being in the other hemisphere ; from being too small to 
 be perceived ; or which may not have been recorded, we might imagine the s.aole number to be 
 considerably greater ; it is, however, highly probable, that of the comets which are recorded to 
 have been seen, the same may have appeared several times, and therefore the number may be 
 less than is stated. Miss Caroline Herschel, the sister of Dr. Herschel, has discovered several 
 comets within the last 15 years, accounts of which are in the different volumes of the Philosophi- 
 cal Transactions. 
 
 On the subject of Comets, see Mr. Vince's very excellent Complete System of Jlstronom ;/," 
 Vol. I. Quarto. 
 
884 OF ASTRONOMY. BOOK VII. PART I. 
 
 CHAP. VIII. 
 
 Of the Sun. 
 
 PROP. CXXI. The spots, which appear upon the sun's disk, adhere to 
 its surface. 
 
 If one of these spots appears upon the eastern limb or edge of the sun's disk, it moves from 
 thence tow-art- the western edge, and arrives at the western edge in about 1S days. Here the 
 spot disappears ; and in about 13} days more, it is seen again upon the eastern edge ; and so 
 continues to go round, completing its apparent revolution in 27 days ; during one half of which 
 time we see it on the disk of the sun, and during the other half it disappears ; which could not 
 happen, if the spots did not adhere to the surface of the sun. Let A be the centre of the sun's 
 e ' disk, D its eastern, and C its western edge ; HEG the orbit of an opaque body moving round it, 
 and B the eye of the spectator at the earth. If two lines BD and BC are supposed to be drawn 
 from the spectator's eye B, so as to touch the sun at D and C, then DBC, the angle contained 
 between these lines, is the angle under which the sun's diameter appears to a spectator on the 
 earth. EG is the only part of the supposed body's orbit that is within this angle DBC, and 
 consequently, if the body was in any other part of its orbit, except EG, it would not appear upon, 
 the sun's disk. But EG is less than half its orbit ; and the body would not take up half the time 
 of a revolution to describe EG. Therefore such a body would not be seen upon the sun's disk, 
 as the spots are for half the time of a revolution. But if the orbit HEG is not greater than 
 LDFC, or is close to the sun ; that is, if the spot adheres to the sun's surface, then half its orbit 
 DFC, will be within the angle DBC, and consequently, the spot will appear upon the sun's disk 
 during one half of its revolution ; but during the other half of its revolution, while it describes 
 CLD, it will disappear, because then it will be behind the sun, and so will be concealed from 
 the earth, which agrees with the phenomena. 
 
 PROP. CXXII. The sun is a spherical body, which revolves upon its 
 axis from west to east. 
 
 The spots which appear in the sun's disk, adhere to its surface, (by Prop. CXXI.) and those 
 spots revolve; therefore the sun revolves round its axis. 
 
 Whatever side of the sun is turned toward the earth in this rotation, it always appears to be 
 a flat bright circle ; but all the sides of it could not appear in this manner unless it was a sphere ; 
 therefore the sun is a spherical body. 
 
 SCHOL. A real revolution of a spot, and consequently of the sun round its axis, is completed 
 in 25 days, two days less than its apparent revolution, in consequence of the earth's motion in its 
 orbit in the same direction in which the spot moves. 
 
 PROP. CXXIII. The axis of the sun is inclined to the plane of the 
 ecliptic. 
 
 Each spot upon the sun must describe a circle round the sun, either coinciding with its equa- 
 tor, or parallel to it. If therefore the sun's axis were perpendicular to the plane of the ecliptic. 
 
CHAP. IX. OF PARALLAXES, &c. 283 
 
 the plane of the sun's equator would be in the plane of the ecliptic ; and a spectator on the earth, 
 whose eye is in the ecliptic, would see the spots describing right lines, either in the sun's equa- 
 tor, or parallel to it; but the spots are sometimes seen to describe lines oblique to the plane ot 
 the ecliptic ; therefore the axis of the sun is inclined to the plane of the ecliptic. This inclina- 
 tion is observed to be an angle of about 8| degrees. When the sun is in that part of the ecliptic 
 where its equator crosses the plane of the ecliptic, the spots appear to describe right lines parallel 
 to the sun's equator. 
 
 SOHOL. The following particulars respecting the sun are related by Sir I. Newton. 
 
 1. That the density of the sun's heat, which is proportional to his light, is 7 times as great in 
 Mercury as with us, and that water there would be all carried off in the shape of steam ; for, he 
 found, by experiments with the thermometer, that a heat 7 times greater than that of the SUA'S 
 beams in summer will serve to make water boil. 
 
 2. That the quantity of matter in the sun is to that in Jupiter nearly as 1 100 to 1, and that 
 the distance of that planet from the sun is in the same ratio to the sun's semidiameter ; conse- 
 quently, that the centre of gravity ol the sun and Jupiter is nearly in the superficies of the sun. 
 
 3. That the quantity of matter in the sun is to that in Saturn as 2360 to 1, and that the 
 distance of Saturn from the sun is in a ratio but little less, than that of the sun's semidiameter. 
 And hence the common centre of gravity of Saturn and the sun is a little within the sun. 
 
 4. By the same method of calculation it will be found, that the common centre of gravity of 
 all the planets cannot be more than the length of the solar diameter distant from the centre of 
 the sun. 
 
 5. The sun's diameter is equal to 100 diameters of the earth, and therefore its magnitude 
 must exceed that of the earth one million of times. 
 
 6. If 360 degrees (the whole ecliptic) be divided by the quantity of the solar year, it will 
 give 59' 8", which therefore is the medium quantity of the sun's apparent daily motion ; hence 
 his horary motion is equal to 2' 27". By this method the tables of the sun's mean motion are 
 constructed, as found in astronomical books. 
 
 CHAP. IX. 
 
 Of the Parallaxes, Distances, and Magnitudes of the Heavenly Bodies. 
 
 DBF. LXII. The Parallax of the heavenly bodies is the change of their 
 apparent situation with respect to each other, as the spectator views them from 
 different stations. 
 
 DBF. LX1II. The Diurnal Parallax is the distance between the appar- 
 ent place of a heavenly b:>dy, as viewed from the surface of the earth, and its 
 apparent place, as viewed from the centre of the earth. 
 
 Let DAB be the earth, C its centre, A the station of a spectator on the surface of the earth ; Plate 10. 
 and F, G, H, different places of the moon, or any other heavenly body; TO, NM, LI, are its F 'S- 17 - 
 different parallaxes, and THO, or AHC ; MGN, or AGC, &c. angles of parallax. 
 
 SCHOL. If a spectator in his first station at A, sees a planet at G, its apparent place in the 
 
OF ASTRONOMY. BOOK VII. PART I. 
 
 heavens will be N; if now, by the diurnal rotation of the earth, he comes into the station P, the 
 planet will appear at M, which is the place in which it would have appeared if viewed from C 
 the centre ; thus, in all cases, the parallas which arises from the diurnal motion is the same 
 which would arise from a change of station from the surface to the centre; for in either case, the 
 change of the spectator's line of view ia the same. Hence appears the propriety of the above 
 definition of the diurnal parallax. 
 
 PROP. CXXIV. The parallax of any planet is always proportional to 
 the angle which a semidiameter of the earth, drawn from the station of the 
 spectator upon the surface to the centre, would subtend, if viewed from the 
 planet. 
 
 Plate 10. If the planet be at H, and the spectator at A, AHT will be his line of view ; on changing the 
 Fig. 17. 8 t a tion of the spectator from A to C, the line of view will become CHO ; whence TO will be the 
 
 parallax. But TO subtends and is proportional to THO, or (El. I. 15.) AHC, the angle which 
 
 the earth's semidiameter would subtend, if viewed from the planet H. 
 
 PROP. CXXV. The parallax of a planet depresses its apparent place, 
 by the parallactic arc. 
 
 Plate 10. If the planet be viewed from C, its apparent place is O ; if from A, its apparent place is T, 
 * farther from Z the vertex than O, by the parallactic arc TO. 
 
 COR. When the altitude of a body is observed, it must be corrected by parallax and refrac- 
 tion, adding the former, and subtracting the latter, in order to get the true altitude, or the 
 altitude above the rational horizon at the centre of the earth. See table of refractions, Schol. 1. 
 Prop. XXXIX. 
 
 PROP. CXXVI. The diurnal parallax of any planet, at a given dis- 
 tance from the earth, is greatest when the planet is in the horizon, and 
 decreases as the altitude of the planet increases. 
 
 The parallax (by Prop. CXXIV.) is proportional to the angle which AC would subtend, if 
 seen from the planet H ; but this given line, viewed from the given distance of the planet, would 
 continually diminish in its apparent magnitude (by Book VI. Prop. LXXIII.) as the degree of 
 obliquity at which it is viewed increases, that is, as the planet advances from H toward E ; 
 therefore the parallax is greatest in the horizon, and decreases as the planet approaches the 
 vertex. The parallactic angle AGC is less than AHC, and AFC less than AGC. 
 
 PROP. CXXVII. To find the parallax of the moon, or any planet. 
 
 Plate 10. Let HMO be an are of the horizon ; APVM an arc in the meridian ; P the elevated pole; 
 
 Fig. 18. v the vertex ; E the apparent place of the planet, as seen from the surface of the earth, and S its 
 place, as seen from the centre ; then ES is the diurnal parallax in the vertical circle VE. Before 
 the planet comes to the meridian, observe its altitude, at E, above the horizon, whence the com- 
 plement of its altitude, VE, will be known ; at the same time observe its distance from the 
 meridian, or its azimuth, EVM. After the planet has passed the meridian, observe when it ha 
 
CHAP. IX. OF PARALLAXES, &c. 287 
 
 the same altitude as at the first observation ; that is, when eV is equal to EV. Now, if E is the 
 apparent place of the planet when at the time of the first observation it is viewed from the earth's 
 surface, and S would be its place, at that time, if viewed from the centre ; and if e is its appar- 
 ent place when viewed at the second observation, from the surface, and s would be its place, at 
 that time, if viewed from the centre ; the parallax ES is equal to the parallax es, since the alti- 
 tude was the same at both observations, and consequently SV is equal to sV. So that if PS is 
 the secondary of the equator which passed through the planet at the first observation, and Ps the 
 secondary which passed through it at the second observation, the planet between the times of the 
 first and second observation, must have described the arc Ss in a circle of daily motion. From 
 the time which has passed between the two observations, the arc Ss (by Prop. XXVII.) may be 
 found, and consequently the angle SPs. Now, because the angle EVM is known, PVS, its sup- 
 plement to two right angles, is known ; and, because at the two observations the planet was at 
 equal altitudes, that is, at equal distances from the meridian, the meridian bisects the angle 
 SPs, which is known ; whence its half VPS is found.* Also, if the latitude of the place be 
 known, PV, the distance of .the elevated pole from the vertex, or the complement of its distance 
 from the horizon, that is, (by Prop. III.) the complement of latitude, is known. Therefore in 
 the spherical triangle PVS, two angles and one side are known ; whence the length of SV may 
 be determined. Take SV and EV, which is already known, and SE, the planet's parallax, will 
 be found. The moon's mean parallax has been found to be 57' 11". 
 
 Or thus ; observe when the planet, whose parallax is to be found, and any fixed star in con- 
 junction with it, cross the meridian at the same instant ; observe the same planet and star after 
 three hours, and remark how much sooner the planet reaches a line placed perpendicularly in the 
 telescope than the star. As 24 hours is to this difference of time, so will 360 degrees be to the 
 arc which subtends the angle of the parallax ; whence the arc and angle will be known. 
 
 PROP. CXXVIII. Any parallax of a planet being given, to find any- 
 other parallax. 
 
 The parallactic angle AFC being given, it is required to find the angle AUG. Having meas- Pl ate 1< 
 ured the angle ZAL, let the angle ZAH, the apparent distance of the planet from the zenith, be 
 also measured. Then, in the triangle CAP, the sine of the angle C.\F is to the sine of the angle 
 CFA, as the side CF is to the side AC. Again, in the triangle C AH, the sine of the a:igle C AH 
 is to the sine of the angle CII\, as CH is to AC. But CH is equal to CF ; therefore the sine 
 of the angle CAP is to the sine of the angle CFA, as the sine of the angle CAH is to the sine of 
 the angle CHA; hut the three first terms are known, therefore the fourth, namely, the angle 
 CI1A, may be found. 
 
 PROP. CXXIX. At a given altitude of different planets, their dinrnal 
 parallaxes are inversely as their distances from the centre of the earth. 
 
 Let one planet be at /, where its altitude is fA.p, and another at /(, having an equal alti- plate 10. 
 tude Iu\p. If the planet/ is viewed from A at the surface of the earth, the line of view is A/r, Fi - 17 - 
 
 * The moon's declination sometimes varies 12' or IS' an hour, which would render the consecutive dis- 
 tances from the meridian at equal altitudes materially unequal, especially in high latitudes, and thus rendev 
 thts method of finding the moon's parallax totally useless. 
 
388 
 
 OF ASTRONOMY. 
 
 BOOK VII. PATVT I. 
 
 Plate 11. 
 Fig. 1. 
 
 and r is its apparent place in the heavens; viewed from C, its apparent place would be t: 
 whence, its parallax (by Prop. CXXV.) is rt. In the same manner it may be shown, that rs, 
 which is less than rt, is the parallax of the planet h. But (by Prop. CXXIV.) the parallax of 
 each planet is proportional to the angle which AC would subtend, if viewed from the planet: 
 and, since AC is given, and also the degree of obliquity at which it is viewed, the apparent length 
 of AC, or the angle which AC would subtend, at either planet, would be (by Book VI. Prop. 
 LXIX.) inversely as the planet's distance from C. Therefore the parallaxes of these planets 
 are inversely as their distances from the centre of the earth. 
 
 PROP. CXXX. The diurnal parallax of a planet in a vertical circle 
 generally produces a parallax of declination, and also, if the planet is not in 
 the meridian, of right ascension. 
 
 Let HQ be the horizon ; EC an arc of the equator, which cuts the horizon at C ; P the pole 
 of the equator; Z the zenith; ZV a vertical circle; F the apparent place of a planet in the ver- 
 tical circle ZV, as viewed from the surface of the earth, and I its apparent place, as viewed 
 from the centre ; then (by Def. LXI1I.) FI is the diurnal parallax in a vertical circle. When, 
 the apparent place is F, PFA is a secondary of the equator passing through it, and when it is I, 
 PIB is the secondary which passes through it. Therefore AF is the declination of the planet 
 when it appears at F, and BI its declination when it appears at I ; the difference of which, DI, 
 is the change of the apparent declination arising from the different station of the spectator, at 
 the surface or centre of the earth. When the apparent place is F, the distance of A from the 
 first of Aries is the right ascension ; when it is I, the distance of B from the first of Aries is the 
 right ascension ; for PFA and PIB are secondaries of the equator passing through the planet. 
 The difference of right ascension, therefore, produced by the parallax FI, is AB. If the planet 
 ia in the meridian PZH, and if L be its apparent place, as viewed from the surface, and N, as 
 viewed from the centre of the earth, LN will be its diurnal parallax ; LE its declination, as 
 viewed from the surface ; NE its declination, as viewed from the centre ; and NL its parallax of 
 declination. But, because PZH is a secondary of the equator, in whatever part of this vertical 
 circle the planet appears, its right ascension will be the distance of the point E from the first of 
 Aries ; that is, the diurnal parallax, in this case, makes no parallax of right ascension. 
 
 PROP. CXXXI. The diurnal parallax of a .planet in a vertical circle 
 generally produces a parallax of latitude, and also, if the vertical circle be 
 not a secondary of the ecliptic, of longitude. 
 
 Plate 11. Let HQ be the horizon ; P the pole of the ecliptic ; EC an arc of the ecliptic, which cuts the 
 *'' * horizon at C ; and ZV a vertical circle ; and this Proposition may be proved in the same man- 
 ner as the last. 
 
 PROP. CXXXIT. The semidiameter of the earth is to the distance of 
 any planet from the centre of the earth, as the sine of the planet's parallax is 
 to the sine of its apparent distance from the vertex. 
 
 Hate 10. 
 Fig. 17. 
 
 If a planet is at F, and the spectator at A, where the line of view is AFf , the planet will 
 appear at L, and ZAL will be the angle of its apparent distance from the vertex Z. Let the 
 
CHAP. IX. OF PARALLAXES, &c. 389 
 
 parallax IL, or the angle AFC proportional (by Prop. CXXIV.) to IL, be found. In the plane 
 triangle ACF, (the sides being to one another as the sines of the opposite angles) AC, the semi- 
 diameter of the earth, is to FC, the distance of the planet from the centre of the earth, as the 
 sine of the angle AFC, the angle of the parallax, is to the sine of the angle FAC, or of its sup- 
 plement to two right angles ZAL, the angle of the planet's apparent distance from the vertex. 
 
 COR. When the horizontal parallax is taken, the semidiameter of the earth AC is to HC, 
 the distance of the planet, as the sine of the horizontal parallax AHC is to the sine of HAC or 
 radius. 
 
 PROP. CXXXIII. To measure the distance of the moon from the earth. 
 
 Let H be the moon in the sensible horizon observed by a spectator at A, and C the centre of piatelO. 
 the earth. In the triangle AHC, let the angle AHC, the moon's horizontal parallax, be found, F 'g- _W- 
 (by Prop. CXXVU.) The angle HAC is a right angle, and AC, the semidiameterof the earth, is 
 known to be 3985 miles. Hence AC, the sine of AHC. 57' 1.", is to 3985, as AH, taken as 
 radius, to the number ef miles in AH, the moon's distance from the earth; the moon's mean dis- 
 tance is thus found to be 2-0,000 English miles. 
 
 SCHOL. According to Mr. de la Lande, the horizontal semidiameter of the moon is to its 
 horizontal parallax for the mean radius of the earth, as 15' is to 54' 57".4, or very nearly as 
 3 to 11; hence the semidiameter of the moon is T 3 r of the radius of the earth. And as the mag- 
 nitudes of spherical bodies are as the cubes of their radii, the magnitude of the moon is to that 
 of the earth, as 3 3 to H 3 , that N, as 1 : 49. 
 
 PROP. CXXXIV. To determine the relative distances of the inferior 
 planets from the sun. 
 
 Let S be the sun, EHG tho orbit of Venus, and LCM the orbit of Mercury. Let AXF be a plate 9. 
 tangent to the orbit of Venus, and let the elongation of Venus, that is, the angle XAS, be found Fig' 12. 
 by observation. Then as radius to the sine of the angle XAS, so is AS to XS or ES. In like 
 manner, if the elongation of Mercury, or the angle CAS, he observed ; as radius to the sine of 
 CAS, so is AS to CS or LS. If AS, the sun's distance from the earth, be supposed to be divid- 
 ed into 1000 equal parts, the distance of Mercury will in this manner be found to be 387, and 
 that of Venus 723. 
 
 PROP. CXXXV. To determine the relative distances of the superior 
 planets from the sun. 
 
 Let S be the sun, nkg the orbit of the earth, OPQ the orbit of Mars, and NKG a part of a p ]ate J Q 
 great circle in the heavens, in which the planet appears to have a retrograde motion ; let P fig. 1. 
 be the place of Murs. Whilst the earth is passing in its orbit from k to n. Mars will ap. 
 pear to move from K to N. The angle of retrogradation KFN is then known by observa- 
 tion. To this the vertical angle nl'S is equal. In the triangle nSP, the angle at K is a right 
 angle; the angle nPS is the angle of retrogradation which is known, whence the other angle 
 wSP is known, and the ratio of the sides of the triangle to each other is known ; whence 
 the ratio of Sn to SP is found. If the mean distance of the earth from the sun be called 
 
 37 
 
390 OF ASTRONOMY. BOOK VII. PART I. 
 
 1000, that of Mara will be found to be 1523, that of Jupiter 5201, and that of Saturn 
 9538. 
 
 PROP. CXXXV1. To find the parallax of the sun by the transit of 
 Venus. 
 
 Plate 11. In order to explain the general principles of this operation, it must be understood, that 
 the periodical times of Venus and the earth, and thejproportions and positions of their orbits, 
 are accurately known from previous observations. From these elements are computed the 
 horary motion of both planets, the latitude of Venus, the direction and4ength of its path across 
 the sun's disk, and the duration of the transit as viewed from the earth's centre. Then let 8 be 
 the sun, B'EV a part of the orbit of Venus, its apparent motion being retrograde, or from left to 
 right, (see Prop. LIX.I and O W a part of the earth's orbit, and let the transit be observed from some 
 place on the earth's surface, where the sun, for the greater advantage, is on the meridian, about 
 the middle of the transit, and, the earth being at O, let b represent the situation of that place at 
 the beginning of the transit, when Venus at B' is seen just entering on the sun's disk at B. If 
 that place should continue stationary with respect to the earth's centre, Venus must reach e f in 
 its orbit at the end of the transit when it is apparently passing off from the sun's disk at E, but 
 during this time the place is carried from b to the eastward by the earth's diurnal motion to the 
 situation e where it is at the end of the transit, so that the planet passes off from the sun's 
 disk when it has only reached E' in its orbit, the duration of the transit as computed for the 
 earth's centre, being shortened by the motion of the place of observation in the ratio of BV to 
 B'E'. Now as one hour is to this difference between the computed and observed durations, so 
 is the heliocentric horary motion of Venus from the earth (or the difference between the helio- 
 centric horary motions of Venus and the earth) to the arc E'' or the angle eE6 subtended at the 
 sun by the line eb. This line may be computed from the latitude of the place of observation and 
 the observed duration of the transit, and by comparing it with the semidiameter of the earth, 
 allowing for the obliquity of the angles at b and e, we may obtain the angle subtended at the 
 sun by the semidiameter of the earth, or the sun's horizontal parallax. Thus an angle so small 
 as to be scarcely measurable, if it were directly accessible, is determined with great accuracy by 
 observing the time in which an arc, subtending it, is described by a motion so slow, as to afford 
 an interval of time amply sufficient for determination. In fact, it is found that if the line eb 
 should be equal to the earth's semidiameter, and perpendicular to. eE, Venus, with only the ex- 
 cess of its heliocentric horary motion above that of the earth, would be more than five minutes 
 in passing over the arc E'e' subtending the angle 6Ee, which, would then be exactly equal to the 
 sun's horizontal parallax. 
 
 But lest there should be some error in the computed length of the transit, it ought to be 
 observed at some place near the meridian opposite to the former, and at such a distance from the 
 enlightened pole that the beginning may be observed before sunset and the end after sunrise, so 
 that during the transit the observer may be carried in a direction, with respect to the sun and 
 planet, opposite to that in which the former was carried. Let W now denote the earth dur- 
 ing the transit, and b the place of the observer at the beginning when the planet at B' is 
 apparently just entering on the sun's disk at B; if the observer, as before, should continue at 
 by Venus would perform the transit, describing the computed chord of the sun's disk, while 
 moving from B' to e' in its orbit, but during this time the observer is carried from b to e by 
 
CHAP. IX. OF PARALLAXES, &c. 391 
 
 the uiurnal motion of the earth, so that Venus must- proceed to E' in its orbit before the ob- 
 server at e can see it pass off from the sun's disk at E, the length of the transit being in- 
 creased bj the motion of the observer in the ratio of BY to B'E'. From this difference between 
 the computed and observed durations the parallax is ascertained in the same manner as before. 
 If now there be any error in the computed duration of the transit, the results of these two oper- 
 ations will be found unequal, since any change in the computed duration increases one, while 
 it diminishes the other, so that the mean between the two results (and in fact the mean of many 
 has been taken) must in all probability give very correctly the sun's horizontal parallax on the 
 day of the transit. Whence the sun's horizontal parallax, at the time of his mean distance from 
 the earth, may be found ; for (by Prop. XXIX.) as the sun's mean distance from the earth is to 
 his proportional distance at the transit, so is his horizontal parallax at that time to his mean hori- 
 zontal parallax. 
 
 In this manner the sun's mean horizontal parallax has been found, from comparing the tran- 
 sits of Yenus in 1761 and 1769, to be 8.65" or about 8f seconds. See Phil. Trans. Vol. LXII. 
 p. 611, and Ferguson's Astronomy, Chap. XXIil. <~ 
 
 SCHOL. The transits of Venus happen but very seldom ; the first that seems to have been 
 noticed was in the year 1639, by Mr. Hoirox and his friend Mr. Crabtree. With a view of en- 
 gaging the attention of future astronomers to the above method of determining the sun's parallax, 
 and thereby his real distance from the earth, Dr. Halley communicated a paper to the Royal 
 Society in the year 169", containing an account of the several years in which such a transit 
 would happen. He particularly mentioned those which would be seen in 1761 and 1769, pre- 
 suming that on those periods this important problem would be solved with great accuracy. No 
 other transit will happen until the vear 1874. 
 
 Except such transits as these, Venus exhibits the same appearance to us regularly every 
 eight years ; her conjunctions, elongations, and times of rising and setting, being nearly the same, 
 on the same days as before. 
 
 PROP. CXXXV1I. To find the distance of the sun from the earth. 
 
 In the triangle AHC, suppose H to be the sun. As the sine of 8f seconds, the horizontal Plate 10. 
 parallax of the sun AHC, is to radius, so is the semidiameter of the earth AC, which is found by Fl ^ - 1 ^' 
 mensuration to be 3985 English miles, to the number of semidiameters of the earth contained in 
 the distance of the sun from the earth. Hence the sun's distance from the earth is found to be 
 about 95,173,000 English miles ; for by log. we have 5.621914 (sine of 8".65) : 10.000000 : : 
 3.60042. (log. of 3985) : 95,173,000 miles. 
 
 PROP. CXXXV1II. To measure the distance of Mercury or Venus 
 from the sun. 
 
 Let S be the sun, E the earth, and M Mercury or Venus. Measure the angle SEM, and Plate 11. 
 observe accurately the time when this measure is taken. When Mercury has made one revo- Fl S- 3 - 
 lution, and arrives at the same point M, the earth will be in some other part of its orbit, 
 as Rj measure at that time the angle SRM, and observe the time when the measuie is 
 taken. 
 
 By these two observations the time in which the earth passes from E to R is known ; hence, 
 
393 OF ASTRONOMY. BOOK VII. PART I. 
 
 as 1 year is to the time employed in passing from E to R, so are S60 degrees to the arc ER ; 
 whence the arc ER, a id the angle ESR, are found. In the triangle ESR the sides SE, SR, (the 
 distance of the sun from the earth) being known, and also the contained angle RSE, let the 
 angles at the base SER, SRE, and the base RE be found. Then from the known angle SER 
 take away the angle SEM, which is also known, there will remain MER; and from the known 
 angle SRE take away the known angle SRM, there will remain MRE. The two angles MER, 
 MRE, being thus found, the third angle RME is also known; and the side RE is known. 
 Wherefore, the sine of the angle RME is to the side RE, as' the sine of the angle MRE is to the 
 side ME, or as the sine of the angle MER is to the side MR. In the triangle SUM, the sides 
 RS, RM, being thus found, the sum of the two sides RS, RM, is to their difference, as the tan- 
 gent of half the sum of the angles at the base RSM, RMS, is to the tangent of half their differ- 
 ence. To half the sum add half the difference, and the greater angle at M is found ; and from 
 half the sum take away half the difference, and the less angle at S is found. Whence, the sine 
 of the angle at M is to the side RS, or the sine of the angle at S is to the side RM, as the sine 
 of the angle at R is to the base SM, which is the distance required. 
 
 PROP. CXXXIX. To measure the distance of Mars from the sun. 
 
 Plate 11. Let S be the sun, E the earth, and M Mars. Measure the angle SEM ; when Mars has 
 '*>* made one revolution, observe the place of the earth in its orbit R, and measure the angle SRM. 
 
 Having found as before the arc ER, and the angle ESR, in the triangle ESR, in which the two 
 sides SE, SR, and the contained angle ESR, are known, let the angles at the base SER, SRE, 
 and the base RE, be found. If from the angle SEM (which has been observed) be taken SER, 
 there remains REM ; and if from the angle SRM (which has been observed) be taken SRE, 
 there remains ERM. Whence, in the triangle RME, the angles at R and E being found, the 
 third angle is known ; and the sine of the angle at M is to the side RE, as the sine of the angle 
 at E is to the side RM. Wherefore in the triangle SRM, the two sides of which, RS, RM, and 
 the contained angle at R, are known ; whence, as in the preceding Proposition, the two angles 
 at the base, S, M, and la'stly the base SM, which is the distance required, may be found. 
 
 PROP. CXL. To measure the distance of Jupiter or Saturn from the 
 sun by their satellites. 
 
 Plate 11. Let S be the sun, E the earth, and I Jupiter. First, observe the instant in which the satellite 
 F 'g- 5 - R disappears behind Jupiter, and the instant in which it again appears ; then, dividing the inter- 
 mediate time into two equal parts ; this will give the instant in which the earth E, Jupiter I, and 
 the satellite R, are in one right line BID. Next, observe the instant in which the satellite dis- 
 appears behind the shadow of Jupiter, and the instant in which it again appears ; and divide the 
 time between these instants into two equal parts to find the instant in which the satellite is in the 
 midst of the shadow, that is, in which the sun, Jupiter, and the satellite form a right line SIR. 
 Hence, the time taken up in passing from D to R is known ; whence, the time of tfie entire 
 revolution of the satellite is to 360 degrees as the time employed in passing from D to R is to 
 the arc DR. Thus the arc DR, and the angles RID, EIS, are found. Lastly, having taken an 
 observation of the angle 1ES, the other angle ESI is found ; and the side ES, the earth's distance 
 from the sun, is known ; whence, the sine of the angle EIS is to the side ES, as the sine of the 
 angle IES is to the side IS, the distance required. 
 
CHAP. IX. OF PARALLAXES, &c. 298 
 
 PROP. CXLI. To measure the distance of any planet from the sun. 
 
 Because the real distances of the planets from the sun are as their proportional distances ; 
 as the proportional distance of the earth from the sun is to the proportional distance of any 
 other planet from the sun, so is the real distance of the earth from the sun in mile*, to the real 
 distance of any other planet from the sun in miles. 
 
 Hence are found the distances of the planets from the sun in English miles. Mercury? 
 36,841,468; Venus, 68,891,486 ; Mars, 145,014,1485 Jupiter, 494,990,976; Saturn, 907,956,130: 
 and the Herschel, 1,800,000,000. 
 
 PROP. CXL1I. The horizontal parallax of any planet being given ; to 
 find its distance from the earth. 
 
 Let H be the planet, whose horizontal parallax AHC is known. The semidiameter of the p] ate ^ 
 earth AC being known, in the triangle CAH the sine of the angle AHC is to the side AC, as F'g- V- 
 the sine of the angle ffAC is to the side HC, the distance sought. 
 
 PROP. CXLIII. The distance of any planet being given, to measure its 
 real magnitude. 
 
 Let A be the earth and C the centre of any planet; and let the distance CA be known. Plate 11. 
 Suppose two right lines, AB, AD, drawn tangents to the planet, the angles CBA, CDA, are *>' ' 
 right angles ; therefore the square of AC is equal to the two squares of AB and EC together. 
 The same square of AC will also be equal to the two squares of AD and CD. And since the 
 square of the radius CB is equal to the square of the radius Cl), (on account of the spherical 
 figure of the planets) the square of the tangent AB is equal to the square of the tangent AD, and 
 the tangent AB to the tangent AD. Hence the two triangles ABC, ADC, are equal, and con- 
 sequently, the angles BAC, CAD, are equal. The angle BAD being measured bv a micrometer, 
 its half BAC is known ; whence, ill the triangle ABC, the nrie of the angle at B, which is a 
 right angle, is to the side AC, as the sine of the angle at A is to the side BC. The radius, and 
 consequently the diameter of the planet, being thus found ; because spheres are as the cubes of 
 their diameters, its magnitude is known by finding the cube of its diameter. 
 
 SPHOL. In Plate 15. Fig. 4, we have a view of the proportional magnitudes of the planets Plate 15. 
 Mercury, Venus, the Earth and Moon, Mars, Jupiter and Saturn, according to Ferguson ; with %' ' 
 the addition of Herschel. In proportion to these figures of the planets, the sun's diameter is 
 about two feet. 
 
 PROP. CXLIV. To find the periodical time of a planet. 
 
 Because, whilst any planet is performing its revolution the earth is carried forward in its 
 path, the planet, after one greatest elongation, must not only complete a revolution, but likewise 
 the whole angular space which the earth described in that time, before it arrives again at the 
 same elongation. Thus, before Venus can return to the same elongation, besides performing an 
 entire revolution in its orbit, (equal to 4 right angles) it must pass through as much more angular 
 space, as the earth has done in the mean time. Hence, as the angular motion of Venus is to the 
 angular motion of the earth in the time between the greatest elongation and its return, so is the 
 
OF ASTRONOMY. BOOK VII. PART 1. 
 
 periodical time of the earth to the periodical time of Venus. In this manner the periodical times 
 of all the planets may be found. 
 
 Or, observe when a planet is in any point of its orbit, and after any number of revolutions, 
 observe when it comes to the same point again; then divide that interval of time by the number 
 of revolutions, and you get the time of one revolution. The observations of ancient astronomers 
 are here very useful ; for as they have put down the places of the planets from their own obser- 
 vations, by comparing them with the places observed now, we take in a very great number of 
 revolutions, and, therefore, if we divide the interval of time by the number of revolutions, should 
 a small error be made in the whole time, it will affect so much less the time of one revolution. 
 The periodical times of the planets will be found in the table at the end of the chapter. 
 
 COR. Because the squares of the periodical times of the planets were found by Kepler to be 
 as the cubes of their distances, the periodical times of any two planets being known, and the 
 comparative or real distance of one of them from the sun being given, the distance of the other 
 may from this proportion be found. 
 
 PROP. CXLV. To find the mean velocities of the planets. 
 
 The periodical time of a planet being known, and also its diameter, and consequently its 
 circumference, (for the diameter of a circle is to its circumference nearly as 113 to 355,) its 
 mean velocity, or the velocity with which it would move if its motion were uniform, may be 
 thus found ; as the whole periodical time of the planet is to an h mr, so is the whole circumfer- 
 ence of its orbit to the angular space passed over in an hour. Thus it is found, that the mean 
 horary velocity of the earth is 68216.9 English miles. In like manner, the horary velocity of 
 the other planets may easily be found. 
 
 _ 
 
 COR. By comparing this proposition with Cor. Prop. CXVII. the velocity of light will be 
 found to be to the velocity of the earth in its orbit as 10632 to 1. 
 
 PROP. CXLVI. The planets revolve round their axes. 
 
 It is found by observation, that the earth revolves about its axis in 23h. 56' 4" mean solar 
 time ; Saturn in I2h. I >' i ; Jupiter in 9h. 56'; Mars in 24h. 40' ; Veuus in 23h. 20'. Tne sun 
 is found to revolve on his axis in 25d. lOh. 
 
 The time of Saturn's rotation is computed from the ratio of its-diameters, which Dr. Herschel 
 makes to be about 1 1 to 10. The time of the rotation of the other planets is not known ; nor 
 has it yet been determined whether they do revolve about their axes. 
 
 SCHOL. The following Table contains a synopsis of the distances, magnitudes, periods, &c. 
 of the several planets, according to the latest observations. 
 
o T 
 
 ? 
 
 
 
 
 
 w 
 
 g w 
 
 - 
 
 3 en 0. 05 01 
 
 -8- *>*.- 
 
 o a. 
 
 &! 
 
 to 
 - 
 
 O 
 *< 
 
 01 
 CO 
 
 01 
 
 O 
 
 r* 
 
 to 
 
 4k 1-1 
 to M 
 
 P 
 
 oo 
 
 4k 
 
 oo 
 
 Ol 
 00 
 
 O 
 tO 
 
 CA 
 
 w 
 
 O 
 
 (a 
 Ol 
 
 to to 
 O> l-p 00 
 
 0< ft --J 
 
 
 4k 
 O tfi> Ot 
 
 3 
 
 > *- 4- 
 to O W> 
 
 ' 
 
 *"* oo _ i< ci ^^ o 
 
 3 ^- 5* ^ 3* _ 3 
 
 3 .-t tO ' 
 _ 01 ^ O , 
 
 s 
 
 gto 
 
 to to to 
 
 to o eg 
 
 00 V 4i. 
 4k 0> to 
 
 O 
 
 tO 
 
 > 
 
 to 
 
 Ol 
 
 O 
 
 03 
 4k 
 
 C/> 
 
 c 
 
 Ul v- tO U> 
 
 - UD c/> 
 
 4k 
 
 o 
 
 -^ i to 
 tO i 00 
 
 1-1 4k 
 
 4k o> 
 
 Ol 
 
 C.? 
 
 00 ! O 4k 
 O>-W( 
 
 UiO>O4k 
 Ot>03 
 
 C t-0 - J 
 O'J -.. 
 
 4k 
 4k 
 
 01 
 
 Ol 
 
 en 
 vn 
 
 01 
 
 O 
 
 o> 
 tO 
 
 Co 
 
 4k 
 
 O> 
 
 4k 
 
 O 
 
 4k 
 
 O O 
 
 O5 O 
 
 4k i 
 
 O 
 
 4k 
 10 
 
 03 
 Co 
 
 tO 
 4k 
 O^ 
 G-J 
 O 
 
 OOUicO4k 
 
 " 
 
 Cl -C- r -O 
 
 4k -a K> 00 i OB Oi 
 O> tO 03 J^ ^ 
 Vi to 
 Oi 
 
 to 
 
 ooo-j 
 
 4k 
 
 lO 
 
 4k 
 O 
 
 Oi^jQ 
 
 C3 tO 
 OOo 
 to 5 
 
 es of the 
 lanets. 
 
 Tropical 
 
 of Aphelio 
 
 Motions of the 
 Aphelion in 
 100 yean. 
 
 
 Motions of 
 nodes in 100 
 
 "' 
 
 
 I rrf 
 
 
 
 00 
 )03 
 
 o<o o en 
 
 OO Oi to 
 
 \r -f * u 
 
 OO O O 
 O O J3 O 
 
 "o 
 
 O 
 O 
 
 O 
 O 
 
 Jn to k o to Oi to 
 
 tOOi4kOiOi00^4 
 
 1\x*>1^^ 
 
 OOOOOOO 
 OOOOOpJS 
 
 'oo'ooo'b'o 
 
 OOOOOOO 
 OOOOOOO 
 
 coi ta to 
 ~- 
 
 Oi 
 
 otto M 
 
 O to 
 
 4k 
 
 ota ** 
 
 O O 
 
 O O 
 
 to to i-ii i 
 o>ooioo 
 
 OiMOOOtOOO 
 
 -~4c.3ojooc,3- 
 
 OOOO 
 OOtOO 
 
 H- to 
 
 00<OOOO> 
 0, U 4k 
 
 l-'toOM0 
 
 to 
 
 01 
 
 en to 
 
 th 
 
 Mean dia 
 in Englis 
 
 ean distances from 
 the Sun, in round 
 numbers of miles. 
 
 ap- 
 
 t di- 
 rs as 
 seen rom 
 
 the Earth. 
 
 n 
 ren 
 
 - 
 
 b 
 
 oo 
 
 oo o to 
 to tn tj- 
 C 4k a 
 
 00 
 
 p"o" 
 CT" c 
 
 to 4k 
 
 5 4k 
 
 G3 
 
 Ol 0= 
 
 a\ to 
 
 oo 
 
 J S 
 
 3 g 
 
 o> oo o> 
 o oo a> 
 
 . l- to 
 
 ^ t3 
 
 - 
 
 Diur 
 roun 
 
 ions 
 to 
 
 O 
 a> 
 
 00 ( ^ O 
 
 CO Oi 
 ^-* 
 
 00 
 
 03 to 
 
 ^ oo 
 
 at 
 ing 
 1. 
 
 of 
 he 
 of 
 
jy sav* 
 
 BOOK VII. PART II. 
 
 OF THE CAUSES OF THE CELESTIAL MOTIONS AND OF OTHER 
 
 PHENOMENA. 
 
 CHAP. I. 
 
 Of the Cause of the Revolutions of the Heavenly Bodies in their Orlits. 
 
 PROP. CXLVII. The moon is retained in its orbit by a force which 
 impels it toward the centre of the earth. 
 
 Since (by Book II. Prop. I.) the moon, or any other planet, being put into motion, would 
 continually move on uniformly in a right line, there must be some force which draws it from its 
 rectilineal path. Whatever this force is, since it is found by observation that the moon by a 
 radius drawn to the earth's centre describes equal areas in equal times, it follows (from Book 
 II. Prop. LXXIII.) that it is impelled by that force toward the earth's centre. The earth 
 indeed is not at rest ; but because both the moon and earth revolve round the sun, the motion of 
 the moon with respect to the earth is the same as if the earth were at rest. 
 
 PRO I*. C XL VIII. The force which retains the moon in its orbit is, at 
 different distances from the earth, inversely as the squares of those distances. 
 
 The moon's orbit being an ellipse, which has the earth in one of its foci, the force which 
 retains it in its orbit must (by Book II. Prop. LXXXl.) in different parts of the orbit be 
 inversely as the squares of the distances from the earth. 
 
 PROP. CXL1X. The moon is retained in its orbit by a force which 
 carries it toward the earth with the same velocity with which a body, acted 
 upon by gravitation at the distance of the moon, would fall toward the earth. 
 
 Plate 11. ^et -^ER be the earth, PLV a part of the moon's orbit, LC an arc which the moon describes 
 Fig. 7. in its orbit in one minute of time. Since the moon describes its whole orbit in 27 days 7 hours 
 43 minutes, that is, in 39343 minutes, the length of the arc LC, which the moon describes in one 
 minute, is the -j^-j-j part of 360, or 33''. If the moon, setting out from L, were not impelled 
 toward the earth, it would move in the right line LB. Since therefore it moves in the arc LC, 
 there must be a force impelling it toward the earth's centre which draws it from the tangent LB, 
 so that, at the end of 1 minute, when it is arrived at C, it will have departed from the tangent 
 as far as BC, or LD ; or, because the moon describes the diagonal L (> in 1 minute, it would iu 
 the same time, by the projectile force, describe LB, and by the centripetal, LD. LD is then 
 
CHAP. I. CAUSE OF CELESTIAL MOTIONS. 297 
 
 the space through which the centripetal force makes the moon fall toward the earth in 1 minute ; 
 and LD is the versed sine of the arc LC, which is an arc of 33". Therefore the force which 
 impels the moon, would make it fall, in 1 minute of time, through the versed sine of an arc of 33". 
 Because AER, the earth's circumference, is found to measure 123249600 Paris feet, its semidiatn- 
 eter AT will be about 19615800 such feet. Since therefore the mean distance of the moon from 
 the earth is found to be 60 semidiameters of the earth, AT multiplied by 60 will give the length 
 of LT, a semidiameter of the moon's orbit, namely, 1176948000 feet. And as the radius is to 
 the versed sine of 33", so is LT to LD, or nearly as 1176948000 to 15 T V Paris feet, which is 
 nearly equal to 16 English feet, or 1 pole. Therefore, if the moon were to fall toward the 
 earth, the centripetal force which impels it toward the earth would make it fall 1 pole in the first 
 minute of its descent. But because (by Prop. CXLVIII.) this centripetal force is inversely as 
 the squares of the distances, a body which is at the distance of the moon, or 60 semidiameters of 
 the earth, will be attracted by a force as much less than that at the surface, as the square of 60, 
 or S600, is greater than the square of 1, or 1 ; that is, the force at the surface being 1, it will be 
 to the force at the distance of the moon, as 1 to -jjVff' and the velocities will have the same ratio. 
 But a body at the surface of the earth falls through 1 pole in a second of time ; that is, (by Book 
 II. Prop. XXVI.) through 3600 poles in a minute. Therefore, at the distance of the moon the 
 body would fall through 7 sV<r P ar * f this length* that is, through 1 pole in a minute. But it has 
 been shown that the moon, by its centripetal force, falls toward the earth 1 pole in a minute ; 
 therefore the moon is retained in its orbit by a force which moves it with the same velocity with, 
 which a body, acted upon by gravitation, and removed to the distance of the moon from the 
 earth, would fall toward the earth. 
 
 PROP. CL. The moon is retained in its orbit by the force of gravitation. 
 
 The force which retains the moon in its orbit agrees with gravitation in its direction (by 
 Prop. CXLV1I.) and in its degree of force; (by Prop. CXLIX.) it may therefore be concluded 
 to be the force of gavitation. 
 
 PROP. CLI. The primary planets are retained in their orbits by a force 
 which impels them toward the sun. 
 
 It is found by observation, that each of them, as they revolve in their respective orbits, des- 
 cribe, by a radius drawn to the sun, equal areas in equal times. Therefore (by Book II. Prop. 
 LXX11I.) the force which retains them in their orbit, impels them toward the sun. 
 
 PROP. CL1I. The forces which retain the primary planets in their 
 respective orbits are, at different distances from the sun, inversely as the 
 squares of those distances. 
 
 It appears from observation, that the squares of the periodical times of the planets are as the 
 oubes of their mean distances from the sun. For example, Saturn's periodical time being found 
 to be to Jupiter's about as 30 to 12, and the distance of the former from the sun, to that of the 
 latter nearly as 9 to 5 ; the squares of the times are 900 and 144 ; which are to one another 
 nearly as 729 to 125, the cubes of tiie distances. This proportion takes place in all the primary 
 
98 OF ASTRONOMY. BOOK VII. PART II. 
 
 planets ; hence (by Book II. Prop. LXXIX.) the force by which they are retained in their orbits 
 is inversely as the squares of the distances. 
 
 SCHOL. It has been remarked in the last chapter, that Kepler found by observation that the 
 squares of the periodical times of all the primary planets are as the cubes of their mean dis- 
 tances from the sun. Astronomers have since found, that the same law holds good in the secon- 
 daries with respect to their primaries. To Sir I. Newton we are indebted for an investigation 
 of this law on physical principles. He has demonstrated, that, in the present state of nature, 
 such a law was inevitable. 
 
 PROP. CLIII. The primary planets are retained in their orbits by the 
 force of gravitation. 
 
 The moon having been shown from the direction, and the law of its centripetal force, to he 
 retained in its orbit by gravitation ; since the primary planets are impelled toward the sun (by 
 Prop. CLI.) as the moon is toward the earth, and since their centripetal force acts \\ith respect 
 to the sun by the same law, by which the force which retains the ii'oon in its orbit acts with 
 respect to the earth, namely, that this force is inversely as the square of the distance of the plan- 
 et from the sun ; it may be concluded, as in the case of the moon, that they are retained in their 
 orbits by the force of gravitation. 
 
 This follows likewise from their moving in elliptical orbits, since it has been proved (Book 
 II. Prop. LXXXI.) that bodies revolving in such orbits have their centripetal forces inversely as 
 the squares of their distances from the centre about which they revolve. 
 
 PROP. CLIV. The satellites of Jupiter and Saturn are retained in their 
 respective orbits by the force of gravity. 
 
 They are observed to describe equal areas round the respective primaries in equal times, and 
 Consequently (by Book II. Prop. LXXIII.) are impelled toward them ; and the forces which re- 
 tain them in their orbits, are at different distances inversely as the squares of tho.-,e distances, 
 (by Book II. Prop. LXXXI.) because it has been observed that the squares of their periodical 
 times are as the cubes of their distances from their respective centres. Therefore the foice 
 which retains the satellites of Jupiter and Saturn in their orbits acts in the same manner, and 
 by the same law, as the force which retains the moon in its orbit acts with respect to the earth. 
 But all effects of the same sort are, without proof of the contrary, to be considered as pro- 
 duced by the same cause. Therefore the power which retains the satellites in their orbits is 
 gravitation. 
 
 PROP. CLV. The sun and any planet revolve round a common centre 
 of gravity, which remains at rest. 
 
 Plate 11. i*t S be the sun, and P any planet, mutually attracting each other. If neither of the two 
 
 Fig. a. bodies revolved in any orbit, they would move toward each other, and would meet at C, their 
 
 common centre of gravity; and during the approach of these two bodies, C, their common centre 
 
 of gravity, would be at rest, (by Book II. Prop. LI.) But il the body P has a projectile force 
 
 given to it in the direction Pt, and if this projectile force and its gravitation toward sS make it 
 
- 
 
 CHAP. I. CAUSE OF CELESTIAL MOTION. 399 
 
 describe an orbit round S, (by Book II. Prop. LXVIII.) such a projectile force will prevent the 
 body P from approaching to S, though it gravitates toward S. But if S has aot as great a 
 projectile force given to it at the same time in the opposite direction Ss, then because S con- 
 tinues to gravitate toward P, and there is no force which can prevent its approaching to P, it fol- 
 lows that S will approach to P, or as P revolves round S the" mutual gravitation of these two 
 bodies will diminish the distance SP. Now it appears (from Book II. Prop. LI.) that C, the 
 common centre of gravity, always divides this distance SP in the inverse ratio of S to P, or that 
 SC is always as much less than PC, as the quantity of matter in S is greater than the quantity 
 of matter in P ; consequently, since the quantity of matter in S and in P is always the same, 
 SC and PC have always the same ratio to one another. But as S approaches to P, SC decreas- 
 es. Therefore PC must decrease in the same ratio. But PC can decrease no otherwise than 
 either by the approach of P to C, or by the approach of C to P. But the projectile force pre- 
 vents P from approaching to C. Therefore C must approach to P. Thus it appears that, if P 
 has a projectile force given to it, and is made to revolve, unless S has an equal projectile force 
 given to it at the same time, the mutual gravitation of these two bodies toward one another will 
 put C, which is their common centre of gravity, in motion; contrary to Book II. Prop. LI. Cor. 
 Therefore as the planet P begins to move in the direction Pt, the sun S will likewise begin to 
 move in the direction Ss ; and C, their common centre of gravity, will continue at rest. And as 
 they tend mutually toward each other, or toward C, their common centre of gravity, their mo- 
 tions will not continue to be performed in right lii.es, but the planet P will revolve round C 
 in an orbit, of which PR is a part, and the sun S will revolve round C in an orbit, of which SQ 
 is a part. 
 
 PROP. CLVL The sun and any planet, whilst they mutually gravitate 
 toward each other, describe similar figures round their common centre of grav- 
 ity, and round each other. 
 
 Let S he the sun, P the earth, or any other planet, and C their common centre of gravity, pi ate u 
 about which (by the last Prop.) they revolve. To a spectator at P, who imagines the planet to be Fig. a. 
 at rest, the sun will appear to revolve about P, and the reverse at S. Because the common cen- 
 tre of gravity of the sun S, and any planet P, is always in a right line drawn from the sun to the 
 planet, if the planet moves through any small space from P to ;;, the line pC continued must pass 
 through the sun 5 and consequent^ the sun must have moved from S to s. Thus Pp, Ss, are 
 arcs described by the planet and sun in their respective orbits in the same time, and PC/7, Sos, 
 are areas described in the same time by the radii CS, CP. And because the vertical angles at 
 C are equal, and that SC is to PC as sC to pC, (for SP, sp, are both divided in C in the inverse 
 ratio of the quantities of matter in P and S) the areas PCj, SCs, are similar. In like manner, 
 any other parts of the two orbits described in the same time may be shown to be similar ; con- 
 sequently, the whole orbits are similar. 
 
 Again, when P has completed its revolution round C, or ^ or 1 of its orbit, it will appear to 
 a spectator at S, to whom S seems at rest, to Iwe completed its orbit, or ^ or A of its orbit 
 round S. And universally, the angular motion of the planet P about C, in any given time, will 
 be equal to its apparent angular motion about S, considered as at rest by a spectator at S. If 
 therefore the planet P in any given time ha> moved from P to p, in which -(by the last Prop.) 
 tlie sun S has moved from S to s, the angle PC;;, which is the measure of the planet's angular 
 
300 OF ASTRONOMY. BOOK VII. PART II. 
 
 motion about C, will be equal to the apparent angular motion round S. Let St be taken equal 
 to sp, and make the angle PS< equal to the angle PCp ; P will, by a radius drawn to S, apparent- 
 ly describe the area PSf, whilst, by a radius drawn to C, it is describing the area PC/>. Now, 
 because (as was before shown) SC is to PC, as *C to pC, (El. V. 18.) SC -f CP, or SP is to PC, 
 as sC -f Cp, or sp is to pC ; and sp is equal to St ; therefore PS is to PC, as S< is to pC, 
 Consequently, the two figures PCp, PSt, are similar. In like manner it may be shown, that 
 any other part, described in any given time, of the orbit the planet appears to move in round the 
 sun considered as at rest, will be similar to other parts, described in the same time, of the orbit 
 in which the planet moves round the common centre of gravity of the sun and the planet ; there- 
 fore the whole orbits are similar. And since the orbits which the sun, and the planet, describe 
 about their common centre have been proved to be similar, it follows, that the orbit which any 
 planet appears to describe round the sun, considered as at rest, is similar to the orbit which the 
 sun in the mean time describes round the common centre of gravity. 
 
 In like manner it might be proved, that the orbit which the sun S appears to describe round 
 the planet P, considered as at rest, is similar to either of the orbits which the planet and sun 
 describe about the centre of gravity. 
 
 COB. If the sun's apparent motion, seen from the earth, is an ellipse, having the earth in 
 one of its foci, the earth's apparent motion, seen from the sun, will be in a similar ellipse, hav- 
 ing the sun in one of its foci ; and if the sun and earth mutually gravitate toward each other, 
 they describe similar elliptic orbits about their common centre. 
 
 PROP. CLVII. The common centre of gravity of the sun and all th 
 planets is at rest, and is the centre of the solar system. 
 
 Since, from the mutual gravitation of the sun and any oae planet, they will revolve about 
 their common centre (by Prop. CLV.) the same must hold good with respect to the sun and all 
 the planets. Consequently, there must be some one point in the solar system which is its cen- 
 tre of gravity, and is at rest. 
 
 CHAP. II. 
 
 Of the Lunar Irregularities. 
 
 PROP. CLVIH. The nearer the moon is to its syzygies, the greater is 
 its velocity ; and the nearer it is to its quadratures, the slower it moves. 
 
 Plate 11. Let S represent the sun. T the earth, and LMNO the orbit of the moon ; let the moon be in 
 Fig. 8. one O f it s quadratures at L, and let the lines LS and TS be drawn. It is obvious, that the 
 tendency which the mooon has toward the sun is along the line LS, and that which the earth has, 
 is along the line TS ; let then the former of these be resolved into two others, the one along LA 
 parallel and equal to TS, the other from L to T, along the line LT. The former of these ten- 
 dencies being parallel and equal to that by which the earth tends along the line TS, alters not 
 the situation of the two bodies L and T with respect to each other, that is, it disturbs not the 
 motion of the body L ; but the other along LT increases its tendency toward T. And thi in- 
 
CHAP. II. OF THE LUNAR IRREGULARITIES. 30* 
 
 crease will he to the tendency the moon has to A, which is the same which the earth has to S, as 
 the distance LT to L \, or TS ; that is, the gravity of the moon toward the earth in the quad- 
 ratures is augmented hy the action of the sun, and that augmentation is to the tendency which 
 the earth has to the s in, as the length of the line LT, or tlie distance of the moon from the 
 earth, to TS, the distance of the earth from the sun. 
 
 Hence the greater the moon's distance is from the earth, the distance of the sun remaining 
 the same, the greater will this increase of the moon's gravity toward the earth be. But if the 
 distance of the moon from the earth remains the same, and the distance of the sun be augmented, 
 this additional ii. cease will be the less in the ratio of the cube of that distance. For, if TS be 
 increased while LT remains tl.e same, LT will be so much the less with respect to TS, that is, 
 the increase will be diminished in the ratio of the sun's distance ; but when TS, the distance of 
 the sun, is increased, the absolute force of the sun, and with it the above mentioned increase, 
 will be diminished also in proportion to the square of that distance ; consequently, taking in 
 both considerations, it will upon the whole be diminished in the ratio of the cube of that 
 distance. 
 
 Let now the moon be in one of its syzygies at M, then will the tendency it has to the sun, 
 more than that which the earth has, which is farther off at T, be to that which the .earth has, as 
 the difference of the squares of SM and ST is to the square of SM ; but the difference between 
 the squares of SM and ST has nearly the ratio to the square of SM, which twice MT, that is 
 MO, has to SM; because the difference between the squares of two numbers whose difference 
 is very small with respect to either of them (as the difference between SM and ST is with res- 
 pect to the distance of 5>) has little more than double the ratio to the square of the less number, 
 that the difference between the numbers themselves has to the less number. The tendency there- 
 fore which the moon, when at M, has to the sun, more than that which the earth has, is to that 
 which the earth has, nearly as MO, or twice TL, to SM, or because of the sun's great distance, 
 as twice LT to TS. Her tendency therefore to the earth is now diminished in that ratio; but 
 as was shown above, it was augmented in the quadratures in the ratio only of LT to TS. The 
 diminution here is therefore nearly double of the augmentation there. And whereas that aug- 
 mentation, when the distance of the sun remains the same, was shown to increase with the dis- 
 tance of the moon ; but when the distance of the moon remains the same, to decrease with the 
 cube of the sun's distance j this diminution being always nearly double of that, will do the 
 same. 
 
 When the moon is in the other syzygy at O, it is attracted toward the sun less than the 
 earth is by the difference of the squares of SO and ST; which, in effect, is the same thing as if 
 the earth were not attracted at all toward S, and the moon were attracted the contrary way;- 
 so that its tendency to the earth is here also diminished, as well as when it was at M, and al- 
 most in the same degree ; for, on account of the sun's great distance, the difference between 
 the squares of SO and ST is nearly the same as between ST and SM. 
 
 Or thus; the annual course of the moon round the sun being performed in the same time 
 that the earth's is, it ought to be retained in that course by the same force that the earth is; 
 whereas, when the moon comes to M, the action of the sun upon it is greater than it is upon the 
 earth, by the difference of the squares of SM and ST ; and when the moon is at O, it is less 
 than it ia upon the earth by the difference between the squares of ST and SO; so that in the 
 former case the moon is drawn too much toward the sun, and in the latter too little ; and there- 
 fore in both cases its tendency toward the earth is diminished, and almost in the same degree ; 
 
302 OF ASTRONOMY. BOOK VII. PAUT II. 
 
 because, as was observed above, the difference of the abovementionecl squares is nearly the same 
 in either case. 
 
 Next, let the moon be in the point of her orbit between the quadrature and the syzygy. 
 Then being nearer the sun than the earth is, she will be attracted with a stronger force ; let it 
 be expressed by iS produced to D,'till ID be of such a length, thatTS being put to express the 
 action of the sun upon the earth, ID may express the stronger force of the sun upon the moon ; 
 and let ID be resolved into two others, one of which let be la, equal and parallel to TS, then 
 will the other be aD, or its equal and parallel IG. This IG is the only disturbing force upon the 
 moon at L. the other Lrt being parallel to TS, affects the moon just as the sun does the earth; 
 and (bus alters not their situations with respect to each other. Let then this figure with the 
 Plate 11. line LG be transferred to fig. 9. This force LG may be resolved into LI and LH, the one a 
 Fl - 9 - tangent to the orbit of the moon, and the other a perpendicular thereto ; the former accelerates 
 the motion of the moon when going from the quadrature at O to the syzygy at B ; and will re- 
 tard it when going from B to R. 'I he other, when H falls upon TL produced, as in this figure, 
 diminishes the tendency of the moon toward the earth ; and when it falls between L and T, it 
 augments it. 
 
 Thus the nearer :he moon is to its syzygies, the greater will be its velocity ; and the nearer 
 it is to the quadratures, the slower it will move ; because one of the forces into which LG is 
 resolvable, accelerates its motion from the quadrature to the syzygies ; and retards it as much 
 from thence to the quadratures. 
 
 COR. Hence the moon in her monthly revolution is, by the action of the sun, alternately 
 accelerated and retarded. 
 
 PROP. CLIX. The moon describes equal areas in equal times only at 
 the syzygies and quadratures, and deviates from this law the farthest in the 
 octants. 
 
 Plate 11. The disturbing force being resolved into two others, one of them, at the quadratures, or syzy- 
 Fig. 9. gies, will be found to point from or toward T the centre of the earth directly, and therefore will 
 not hinder the moon from describing equal areas in equal times; the other likewise, in those 
 places will be found to tend toward the centre of the sun, and therefore neither of them will 
 prevent the moon there from describing equal areas in equal times, that is, will not at the quad- 
 ratures disturb the moon's motion at all. 
 
 But when the moon is in the octants, as at L, this force being resolved into two others, one 
 of them, as LH, will point directly to or from the centre of the earth, and therefore will increase 
 or diminish the moon's tendency toward the earth, but not hinder it from describing equal areas 
 in equal times. But the other, as LI, or HG, points neither toward the centre of the earth, nor 
 sun, and therefore, in the octants, prevents its describing equal areas in equal times. But this 
 being the mid-way between the quadrature and the syzygy, in both which places this disturbing 
 force doth not prevent the moon from describing equal areas in equal times, it follows, that 
 at the octants, this disturbing force will be greatest of all. 
 
 SCHOL. Hence it has always been found more difficult to obtain the moon's place in the 
 pctau ts, so as to agree with observation, than at the quadratures or syzjgies. 
 
CHAP. It. OF THE LUNAR IRREGULARITIES. 303 
 
 PROP. CLX. The orbit of the moon is more curved iu the quadratures, 
 and less in the syzygies, than it would lie if it were only attracted by the 
 earth. 
 
 For its motion (by Prop. CLVIII.) being accelerated thirtng its progress from the quadra- 
 tures to the syzygies, in the syzygies its motion will be quicker than it ought otherwise to be, 
 and therefore its centripetal force less than it would otherwise be. It will therefore at the syzy- 
 gies describe the portion of a larger curve, which consequently will be less curved than a smaller. 
 On tlie other hand, while the moon passes from the syzygies to the quadratures, its motion is 
 continually retarded, and therefore, at the quadratures, its motion will be slower than it would 
 otherwise be. At the quadratures, therefore, the moon will desciibe the portion of a lesser 
 curve, which therefore will be more curved than a larger curve. 
 
 O 
 
 PROP. CLX.I. When the earth is in its perihelion, the periodical time 
 of the moon will be the greatest ; when the earth is iu its aphelion, the peri- 
 odical time of the moon will be the least. 
 
 Since the irregularities explained in the three preceding Propositions proceed from the action 
 of the sun, it follows, that where the action of the sun is greatest, the irregularities arising from 
 it will be greatest too. But the nearer the earth is to the sun, the greater will be the action of 
 the sun upon the moon ; and the more the moon tends toward the sun, the less will it tend p . 
 toward the earth. When therefore the earth is at the perihelion A, and consequently at its Fig. 10. 
 least distance from the su'i S, the action of the sun upon the moon will be greatest, and destroy 
 more of its tendency toward the earth than at any other distance, as SE, SC, SD, &c. There- 
 fore when the earth is at the perihelion A, the moon will describe a larger orbit about the earth, 
 than when the earth is at any other distance from the sun, and consequently, her periodical 
 time will then be I he longest. 
 
 But the earth is at its perihelion in the winter, and consequently, then the moon will describe 
 the outermost circle about tiie earth, and her periodical :ime will be the longest; which agrees 
 with observation. For the same reason, when the earth is at its aphelion B, the tendency of the 
 moon toward the earth will be the greatest, and consequently, her periodical time the least. 
 And in this case, which will be in the summer, it will desciibe the innermost circle about the 
 earth. 
 
 PROP. CLXII. The line of the moon's apsis goes forward when the 
 moon is in syzygy, and backward when it is in quadrature; but it goes far- 
 ther forward than backward each time, so that at length it performs a revolu- 
 tion according to the order of the signs. 
 
 Since the moon describes an elliptical orbit CEDF about the earth S, placed in one of its foci, Plate It. 
 and since its centripetal force toward the earth, by means of the action of the sun (by Prop. **>' * 
 CLVHl.) is continually increasing, or decreasing, hut not equably ; that is, sometimes less, and 
 sometimes more, than in the inverse duplicate ratio of the distance of the moon from the earth, 
 therefore, the line of the moon's apsis AB will he continually going backward or forward ; that 
 is, the axis AB will not always He iu that situation, but go backward into the situation CD, or 
 
 *V % 0! 
 
 
OF ASTRONOMY. BOOK VII. PART II. 
 
 forward into the situation EF. Since however, taking one whole revolution of the moon about 
 the earth, the action of the sun more diminishes tlie tendency of the moon toward the earth than 
 it augments it, therefore the motion of the apses forward exceeds their motion backward. Upon 
 the whole, therefore, the apses of the moon's orbit go forward, or according to the order of the 
 signs. Their revolution is completed in about 9 years. 
 
 PROP. CLXIII. The eccentricity of the moon's orbit is varied in every 
 revolution of the moon, and is greatest when the moon is in syzygy, and least 
 when it is in quadrature ; and the orbit is most of all eccentrical when the line 
 of the apsis is in the syzygies, and least of all eccentrical when this line is in 
 the quadratures. 
 
 1 ' Because the moon describes an eccentrical orbit CEDE about the earth 8, and the action of 
 the sun upon it sometimes increases its tendency toward the earth, and sometimes diminishes it. 
 that is, makes its gravity toward the earth increase or decrease too fast ; if, while the moon 
 ascends from her lower apsis A, its gravity toward the earth decreases too fast, instead of des- 
 cribing the path DBF, and coming to the higher apsis at B, it will run out into a curve beyond 
 DBF ; that is, the orbit will become more eccentric, or farther from a circle. On the other hand, 
 if the moon is passing from her higher apsis B, to her lower A, and its gravity toward the earth, 
 by the action of the sun, increases too fast, it will approach nearer to the earth than the curve 
 CAE, and describe a curve within CAE, or a portion of an orbit less eccentric, or nearer to a 
 circle, than CEDF. And if we compare several revolutions of the moon together, we shall find 
 that when the line of the apsis is in the syzygies, the eccentricity will be the greatest of all, 
 because in that situation, the difference between the tendency which the moon has to the earth, 
 in one of the apses, and that which it has in the opposite one, is the greatest of all ; whereas, 
 when the line of the apsis is in the quadrature, this difference is the least, and therefore the lunar 
 eccentricity will be so too. 
 
 COR. When the gravity of the moon toward the earth decreases too fast, the eccentricity of 
 her orbit will increase ; and when her gravity toward the earth increases too fast, the eccentri- 
 city of her orbit will decrease ; and the orbit itself will approach nearer to a circle. 
 
 PROP. CLX1V. The line of the nodes moves backward, but not uni- 
 formly ; when it is in the syzygies it stands still, and moves fastest in the 
 quadratures. 
 
 Plate 11. When the line of the nodes is in the syzygies, as CD. the plane of the moon's orbit passes 
 *" ' through the centre of the sun S, as well as through that of the earth E ; whence, the disturbing 
 force acting in the direction of the line of the nodes, and consequently in the plane of the lunar 
 orbit, the moon is not drawn out of the plane of its orbit by the sun. But when the line of the 
 nodes is in any other situation, and the moon not in one of the nodes, it is continually drawn 
 out of the plane of its own orbit, on that side on which the sun lies. For instance, if the plane 
 of its orbit CGDF produced passes above the sun, the sun draws it downward ; if, on the con- 
 trary, the plane of its orbit produced passes below the sun, it draws it upward. Hence it fol- 
 lows, that when the line of the nodes is not in the syzygies, and the moon having passed either 
 of the nodes, has got out of the plane of the ecliptic ACBD, on either side of it, the action of 
 
CHAP. II. OF THE LUNAR IRREGULARITIES. 305 
 
 the sun occasions the moon to return back to the plane of the ecliptic sooner than it otherwise 
 would do. But where the moon enters that plane, there is the next node ; so that each node 
 does, as it were, come toward the moon ; and the nearer the line of the nodes is to the quadra- 
 tures, the greater is this effect, because, in that case, the sun is the farthest of all from the plane 
 of the lunar orbit produced. So that the line of the nodes goe backward the fastest of all, when 
 it is in the quadratures ; and not at all in the syzygies. 
 
 PROP. CLXV. The inclination of the lunar orbit is liable to change, 
 and is greatest when the nodes are in the syzygies, and least when they are 
 in the quadratures. 
 
 When the nodes are in the quadratures A, B, and the moon in its orbit AGBF has passed A, P' ate *' 
 and is approaching the syzygy which is next to the sun, the action of the sun upon the moon '^ 
 prevents its ascending so high, that is, departing so far from the plane of the ecliptic ADBC ; 
 whence the inclination of its orbit to the ecliptic will become less, and it will come to conjunc- 
 tion with the sun at H, making an angle with the ecliptic H YD, less than GAD. As the moon 
 goes on to the next quadrature B, the action of the sun upon the moon, in its descent toward the 
 node, hastens its descent, and thus, bringing it down to the ecliptic at K sooner than it would 
 otherwise arrive there, increases the inclination of the plane of its orbit as much as it was dimin- 
 ished in ascending from A to H. And for the same reason, while the moon passes from B to the 
 opposite syzygy F, the action of the sun decreases the inclination of its orbit, and increases it 
 again on its passage from thence to A, the next quadrature. 
 
 When the nodes are in the syzygies, C, D, the plane of the moon's orbit produced, passes 
 through the centre of the sun ; and consequently, not being affected by the action of the sun, its 
 inclination is neither increased nor diminished. 
 
 But while the nodes are passing from the syzygies C, B, to the quadratures A, B, the incli- 
 nation of the moon's orbit is diminished in every revolution of the moon ; and while they are 
 passing from thence to the syzygies, it is continually increasing. Suppose the nodes in the 
 octants at O and L, and the plane of AGBF, the orbit of the moon, so inclined to ADBC, the 
 ecliptic, that if produced it will pass above the sun S. When the moon is nearer the sun than 
 the earth is, it is attracted toward the sun more than the earth is ; and when farther off, the eartli 
 is attracted more than the moon is, that is, the moon is, as it were, attracted the other way. 
 Hence, whilst the moon is ascending from the ecliptic in passing from O to P, the disturbing 
 force being toward S, and the orbit above S, the mcon will not rise so high as P, and the inclina- 
 tion of its orbit will be diminished while it is passing over 90 degrees from the node O. In going 
 from a point below P to the next quadrature B, which is 45 degrees, the disturbing force being 
 still toward S, because the moon is as yet nearer the sun than the eartli is, and the moon now 
 descending toward the ecliptic, the attraction of the sun will hasten its descent, and therefore 
 cause it to move in a plane which will make with the plane of the ecliptic a larger angle than 
 before ; that is, in passing from P to B the inclination of the orbit is increased. But when the 
 moon has passed B, and is moving toward L, the disturbing force acting, in the plane of the 
 ecliptic, from the sun, and the moon still descending toward the ecliptic, the disturbing force, 
 attracting the moon upward, will retard its descent to the ecliptic, and cause it to move in a 
 plane which will make a less angle with the plane of the ecliptic than before ; that is, while it is 
 passing from B to the node L, the inclination of its orbit is diminished. Thus, while the moon 
 
306 OF ASTRONOMY. BOOK VII. PAHT II. 
 
 passes from O to L. the inclination of its orbit is diminished during three fourths of the passage. 
 In like manner, while the moon is ascending from L to I, because the disturbing force acts from 
 the sun, the inclination of its orbit is diminished ; and while it is descending from I to A, the 
 disturbing force still acting from the sun, the inclination is increased. But while it is still 
 descending from A to O, because the disturbing force acts toward the sun, the inclination is 
 diminished. Add to this, that while the moon passes from O to P, and from L to I, the disturb- 
 ing force is much greater than when it was passing from P to L, and from I to O, because the 
 difference between the distances of the moon and of the earth from the sun is greater in the 
 former case than in the latter. On the whole, therefore, while the nodes are between A and D, 
 B and F ; that is, while they are passing from syzygy to quadrature, the inclination of the lunai 
 orbit is diminished ; for, though the nodes have been supposed equally distant from the quadra- 
 ture and syzygy, it is obvious that the like effects must happen, though different in degree, when 
 they are nearer to the one than the other. 
 
 Next, let the nodes be in the octants I, P, between A and F, and B and G. While the moon 
 is ascending from the node I toward the quadrature A, the disturbing force acting from the sun, 
 it will be drawn upward, and the inclination of its orbit will be hereby increased. In ascending 
 from A to O, the disturbing force acting toward the sun, its ascent will be diminished, or the 
 inclination of its orbit lessened ; but in descending from O to the node P, the disturbing force 
 still acting toward the sun, it will be drawn downward, and consequently, the inclination of its 
 orbit will be increased. Thus, during one whole revolution of the moon in this position of its 
 nodes, the inclination of its orbit will be increased through three fourths of its passage. And 
 this will be true, as in the other case, when the nodes are not in the octants. Also, for the rea- 
 son mentioned in the other case, the force which increases the inclination of the orbit is, while it 
 acts, superior to that which diminishes it. While the nodes, therefore, are passing from the 
 quadratures to the syzygies, the inclination of the moon's orbit is increasing. From all which it 
 is manifest, that the inclination of the lunar orbit is the least when the line of the nodes is in 
 quadrature, and the moon in syzygy, and greatest when the line of the nodes is in syzygy. 
 
 PROP. CLXVI. The nodes of the moon are at rest, when the Hue of 
 the nodes is in syzygy ; they move in antecedentia, or from east to west, when 
 the line of the nodes is ia quadrature ; and also when it is between quadra- 
 ture and syzygy ; but their regress, in one revolution, is, in this case, less 
 than when the line of the nodes is in quadrature. 
 
 Plate 11. When the line of the nodes is in syzygy, because the disturbing force acts in the plane of the 
 'F'ff- 12. moon's orbit, it cannot change (he inclination of that plane to the ecliptic ; whence, the common 
 intersection of the two planes, or the line of the nodes, is immovable. If, whilst the line of the 
 nodes is in AB, the moon is passing from A through G to B, being constantly drawn toward the 
 plane of the ecliptic by the disturbing force, it will come to the plane sooner than it would have 
 done if no such force had acted upon it ; that is, before it has described 180, or is arrived at B. 
 In like manner, while the moon is passing from B to A, through F, being drawn tovvard the 
 plane of the ecliptic by the disturbing force, it will cross the ecliptic sooner than it would other- 
 wise have done, that is, before it arrives at A. Consequently, the nodes will have changed their 
 places, and moved in a contrary direction to the moon. In any other position of the line of the 
 nodes, the disturbing force will, for the same reason, cause the line of the nodes to move in ante- 
 
CHAP. 111. SPHEROIDICAL FORM OF THE EARTH. 30? 
 
 cedentia, though in a less degree ; because, whilst the moon is describing the greater part of its 
 orbit, it is drawn by the disturbing force (as was shown in the last Prop.) toward the ecliptic, 
 and consequently is made to cross the ecliptic sooner than it would otherwise have done, that is, 
 the nodes are, on the whole, in one revolution of the moon, made to move in a direction contra- 
 ry to that of the moon ; but this regress is less than when tire line of the nodes is in quadrature, 
 because, during part of the revolution in this oblique position of the line of the nodes, the nodes 
 move IK consequentia, or in the same direction with the moon, namely, whilst the disturbing 
 force (as was shown in the last Prop.) draws the moon from the plane of the ecliptic ; where- 
 as-, when the line of the nodes is in quadrature, they move in antecedentia during the whole 
 revolution. 
 
 SCHOL. I. The nodes perform one revolution, or pass through every part of the ecliptic, in 
 about, 19 years. 
 
 SCHOL. 2. All the irregularities of the moon are greater when the earth is in its perihelion, 
 than when it is in its aphelion, because the effect of the sun's action, whereby they are produced, 
 is inversely as the cube of its distance from the earth. They are also greater when the moon is 
 in conjunction with the sun, than in opposition, for the same reason; for the earth and moon, 
 taken together, are nearer the sun in the ( former situation of the moon, than they are in the 
 latter. 
 
 CHAP. III. 
 
 Of the Spheroidical Form of the Earth. 
 
 PROP. CLXVII. la the daily revolution of the earth round its axis, the 
 centrifugal force diminishes the weight of bodies more at the equator than in 
 any other place on the surface of the earth, in the duplicate ratio of the semi- 
 diameter to the cosine of the latitude of the place. 
 
 Let PEPe be the earth, PP the axis, Ee the equator. As the earth revolves upon its axis, pi ate 
 every place on its surface, except the two poles, describes a circle, the plane of which is perpen- P 'S- l 
 dicular to the axis, and the radius of which is the distance of that place from the axis. Thus, a 
 body placed at A will in one revolution of the earth describe a circle, the semidiametcr of which 
 will be AB, which, with the plane in which it lies, will be perpendicular to the axis PP. In like 
 manner, CE is the semidiameter of a circle described by the revolution of a place in the equator. 
 But CE is the semidiameter of the earth, and AB is the cosine of latitude of the place A ; for 
 AB is the sine of AP, the complement of AE, which is the latitude of the place. And a body 
 at E, revolving in a circle whose radius is CE, performs its revolution in the same time with a 
 body at A, revolving in a circle whose radius is AB. But where the periodical times are equal, 
 the centrifugal forces are as the radii, (by Book II. Prop. LXXV11.) Whence the body at E 
 has its centrifugal force as much greater than the body at A, as the radius UE is greater thati 
 the radius AB : and universally, the centrifugal force at the equator is to the centrifugal force 
 at anv other place on the surface of the earth, as the semidiameter of the earth to the cosine of 
 .the latitude of the place. 
 
308 OF ASTRONOMY. BOOK VII. PART IL 
 
 Moreover, if the centrifugal forces at E and A were equal, they would diminish the weights 
 of bodies unequally, on account of the different directions in which these forces act. The cen- 
 trifugal force at A, acting obliquely upon the force of gravitation toward C, can only diminish 
 this force by such a part of its action as is opposite to the direction of gravitation, that is, re- 
 solving Ab which may express the centrifugal force at A into Aa, ab, the part of the centrifugal 
 force which will act to diminish the gravity of the body at A, will be to the whole centrifugal 
 force at A, as Aa to Ab. Whereas at E, the whole centrifugal force, acting in direct opposi- 
 tion to the force of gravitation, will operate to diminish the. weight of a body at E. Hence the 
 force which acts to diminish the weight of a body, that is, the diminution at E is to the 
 same at A, as the whole centrifugal force Ab to the part Aa. But Ab is to Aa, as CE to 
 BA ; for, the triangles Aab and ABC being similar, At is to Aa, as AC or EC to BA. There- 
 fore, from the different directions in which the centrifugal forces act at E and A, the weight at 
 E is as much more diminished than at A, as EC, the semidiameter, is greater than AB, the cosine 
 of the latitude of the place A. 
 
 The centrifugal force then diminishes the force of gravitation in the ratio of EC to AB, both 
 because the centripetal force at E is greater than at A, and because it acts directly at E, but 
 obliquely at A. Therefore the centrifugal force diminishes the weight of a body at E, more than 
 at A, in the duplicate ratio of CE to BA, that isj as much more at E than at A, as the square of 
 CE is greater than the square of BA. 
 
 SCHOL. It is found by calculation from this Proposition, that gravity at the equator is di- 
 minished by the centrifugal force in the ratio of 288 to 289. 
 
 COR. 1. If the diurnal motion of the earth round its axis was about 17 times faster than it 
 is, the centrifugal force would, at the equator, be equal to the power of gravity, and all bodies 
 there would entirely lose their weight. But if the earth revolved still quicker than this, they 
 would all fly off. 
 
 COM. 2. Since a place in the equator describes a circle of 24933 miles in 24 hoars (Prop. 
 III. Cor.) it is evident that the velocity with which that place moves, is at the rate of about 17.3 
 miles per minute. The velocity in any parallel of latitude decreases in the proportion of the 
 cosine of latitude to radius. Thus, for the latitude of London, say, as Had. : Cos. 51 50' :: 
 velocity of the equator : velocity of London ; by logarithms, as 10.00000 i 9.794150 :: 1.232046 
 : 1.026196= 10.6 miles; that is, the city of London moves about the axis of the earth at the 
 rate of more than 10i miles in a minute of time. 
 
 CLXVIII. The earth is an oblate spheroid, elevated at the equator, and 
 depressed at the poles. 
 
 It has been found by observation, that a pendulum, shorter by 2.169 lines, is required to 
 vibrate seconds at the equator, than at the poles ; but (from Book II. Prop. XLIII. and XLI V.) 
 the lengths of pendulums vibrating in the same time are as the gravities at the places where 
 they vibrate; therefore the gravity at the poles is greater than at the equator. And it has been 
 found by Sir I. Newton, that this difference of gravity is so much greater than would arise from 
 the centrifugal force alone, that the ratio of the equatorial diameter of the earth to the polar 
 diameter, must be as 230 to 229, which makes the equatorial diameter exceed the polar bj 
 about 54 miles. 
 
 
CHAP. IV. PRECESSION OF THE EQUINOXES. 309 
 
 COR. 1. Hence bodies near the poles are heavier than the same bodies toward the equator; 
 (1.) Because they are nearer the earth's centre, where the whole force of the earth's attraction 
 is accumulated. (2.) Because their centrifugal force is less on account of their diurnal motion 
 being slower. For both these reasons, bodies carried from the poles toward the equator, gradual- 
 ly lose their weight. 
 
 Con. 2. The degrees of latitude upon the earth's surface arc longer at the poles than at 
 the equator. For an arc of a meridian near the poles is less curved than near the equator; that 
 is, it is an arc of a larger circle ; whence a degree measured upon that arc must be greater than 
 upon an arc of the same meridian at the equator. 
 
 COR. 3. The tendency of a heavy body, on any part of the surface of the earth between the 
 poles and the equator, is not directly toward the centre, but toward some point between 
 the centre and the equator. 
 
 SOHOL. The point toward which a body in any given place will tend may be determined. 
 
 For (by Prop. CLXV1I.) as radius EC is to the cosine of latitude of the place AB, so is the pi a te H. 
 centrifugal force at E to the centrifugal force at A in the direction A6. Produce, therefore, the Fi S >1 3. 
 line BA to b, till A6 has the same ratio to AC, as the quantity last found has to gravity upon 
 the surface of the earth. Complete the parallelogram A&Cc ,- the point sought will be c, and 
 the tendency of the body will be along Ac ; thus suppose the latitude of the place 51 46'. The 
 centrifugal force at the equator is found to be to (hat of gravity, as 1 to 289 ; hence, as radius 
 to the cosine of 51 46', so is 1 to 0.618, which is the centrifugal force at A. Consequently, the 
 centrifugal force at A is to the force of gravity, as 0.618 to 289 ; therefore, by the construction, 
 Ab or Cc is to AC in that ratio. The ratio of AC to Cc being thus found, as AC is to Cc, or as 
 289 is to 0.618, so will the sine of the angle of latitude ACc, or 51 46', be to 5' nearly, which is 
 the angle required, measuring the deviation of the line of direction of falling bodies at the given 
 latitude from a line drawn to tbe centre of the earth. 
 
 CHAP. IV. 
 
 
 
 Of the Precession of the Equinoxes. 
 
 DBF. LXIV. A Periodical Tear is the time in which the sun completes 
 its revolution through the ecliptic. 
 
 DBF. LXV. A Tropical Year is the time in which the sun completes its 
 revolution, setting out from any solstitial or equinoctial point, and returning 
 to the same. 
 
 PROP. CLXIX. The equinoctial points move in antecedentia, or go 
 backward from east to west, contrary to the order of the signs. 
 
 It is found from observation, that the equator and ecliptic do not always intersect each oth- 
 er in the same points, but that the points of intersection change their place, moving from east 
 to west, whilst the inclination of the planes remains the same. This motion is called the pre- 
 cession of the equinoxes, because it carries the equinoctial points in precedentia signa. 
 
310 OF ASTRONOMY. BOOK VH. PART II. 
 
 CLXX. The precession of the equinoxes makes the tropical year short- 
 er than the periodical year. 
 
 If, while the sun moves in the order of the signs, the equinoctial point moves in the contra- 
 ry direction, it is manifest, that the sun must arrive at the solstitial or equinoctial point from 
 which it set out, before it arrives at the same place in the zodiac, or must complete the tropical 
 year sooner than the periodical year. 
 
 The tropical year is observed to be 365 days, 5 hours, 49 minutes ; the periodical year, 365 
 days, 6 hours, 9 minutes, 12 seconds. 
 
 PROP. CLXXI. The precession of the equinoxes causes the poles of 
 the equator to describe a circle from east to west about the poles of the 
 ecliptic. 
 
 In this precession, the plane of the equator revolves from east to west, cutting the ecliptic, 
 which, with its axis, is at rest, in successive points. But while the plane of the equator is re- 
 volving, its axis must revolve with it the same way. And, since the plane of the equator is 
 always equally inclined to that of the ecliptic, the axis of the equator must always have the same 
 inclination to the axis of the ecliptic ; consequently; the poles of the equator will revolve round 
 the poles of the ecliptic, always preserving the same distance from each other; that is, the poles 
 of the equator will describe a circle about the poles of the ecliptic. 
 
 ESP. The precession of the equinoxes, and the revolution of the pole of the equator about 
 that of the ecliptic, may be thus represented on the celestial globe. Let the broad wooden hori- 
 zon represent the ecliptic ; place the axis of the globe perpendicular to the wooden circle ; the 
 ecliptic on the globe will then make an angle of 23 30' with the wooden horizon ; consequent- 
 ly, if the woodenhorizon represents the ecliptic, the circle which commonly represents the 
 ecliptic will m>?Wl!Btt!Seit the equator; and the two points in which this circle cuts the wood- 
 en horizon will repn'srnt the equinoctial points. If the globe, in this position, be turned slowly 
 round from east to west, these points of intersection will move round the same way, while the 
 inclination of the circle which now 'presents the equator to 1 that which represents the ecliptic 
 remains the same; whence the precession of the equinoxes is properly represented. Again, the 
 axis aml^oles (if the globe now representing those of the ecliptic, the axis and poles of the eclip- 
 tic, marked on the globe, will represent those of the equator; dnd in turning the globe round 
 from east to west, the points which represent the poles of the equator, will revolve the same way 
 round the poles of the gtoba-. which represent those of the ecliptic, and the axis of the supposed 
 equator will always, make the same angle with the plane of the supposed ecliptic. 
 
 / 
 
 PROP. CLXXII. The precession of the equinoxes is caused by the ac- 
 tion of the sun and moon on that excess of matter about the equatorial 
 parts of the earth, by which from a perfect sphere it becomes an oblate 
 spheroid. 
 
 Let ADCB Ioe the P 1 * 116 of the ccl 'P tic > s tlie sun > E the earth ' and AFBG a rin S encom- 
 passing the earth at any distance, as Saturn is encompassed by its ring. Let the half of this 
 ring AGB toward the sun be above the plane of the ecliptic, and the other half below it; then. 
 
CHAP. V. OF THE TIDES. 311 
 
 a line passing through A and B will be the line of the nodes of this ring. If it be supposed that 
 this ring moves round its centre E, the same way in which the moon moves round the earth, it 
 is obvious that every point of this ring will be acted upon by the disturbing force of the sun in 
 the same manner as the moon was shown to be acted upon in Prop. CLVIII. &c. Particularly, 
 tlie motion of the nodes of this ring, and consequently of the whole ring which moves with these 
 nodes, and its inclination to the plane in which its centre moves, will be aft'ected in the same 
 manner with the orbit of the moon ; whence, its nodes when in syzygies will stand still, and its 
 inclination will be greatest ; but in all other situations, the nodes will go backward, and fastest 
 of all when in the quadratures, at which time the inclination of the ring will be the least. This 
 will be the case, whatever be the thickness of the ring, or its distance from the centre. 
 
 If this ring be supposed to adhere to the earth, it is obvious that it will still have the motions 
 described above, and that in this situation, the earth itself must participate of these motions. 
 Now the earth being an oblate spheroid, having its equatorial diameter longer than that which 
 passes through its poles, this redundancy of matter, by which the form of the earth departs from 
 a perfect sphere, may be considered as a portion of the supposed ring, which receives from the 
 action of the sun tiie motions abovementioned, and communicates them to the earth. Hence the 
 equinoctial points, which are the nodes of the ring, when they are in syzygy, that is, at the 
 equinox, will stand still, and the inclination of the equator to the plane of the ecliptic will be the 
 greatest ; in all other situations they will go backward, and fastest when in quadrature at the 
 solstices 5 ana the inclination of the plane of the equator to that of the ecliptic is then the 
 least. 
 
 Con. Hence the axis of the earth, being perpendicular to the plane of the equator, changes 
 therewith its inclination to the plane of the ecliptic twice in every revolution of the earth about the 
 sun. For instance, it inci eases whilst the earth is moving from the solstitial to the equinoctial, 
 and diminishes as much in its passage from the equinoctial to the solstitial points ; which phe- 
 nomenon is called the Nutation of the Poles. 
 
 SOHOL. This precession of the equinoxes is found to be 50 seconds of a degree, every year, 
 westward or contrary to the sun's annual motion ; so that with respect to the fixed stars, the 
 equinoctial points fall backward SO degrees in 2160 years, whence the stars will appear to have 
 gone SO degrees forward, with respect to the signs of the ecliptic, which are reckoned from the 
 equinoctial point. Thus the stars which were formerly in Aries are now in Taurus, &c. This 
 period is completed in 5920 years. 
 
 CHAP. V. 
 
 Of tie Tides. 
 
 PROP. CLXXIII. The tides are caused by the attraction of the moon 
 and of the sun. 
 
 Let ApLn be the earth, and C its centre ; let the dotted circle PN represent a mass of wa- Plate II. 
 ter covering the surface of the earth ; let .M, m, be the moon ; S, s, the sun in different situa- Pli4te 14< 
 tions. Because the power of gravity diminishes as the squares of the distances increase, (by 
 
31S OF ASTRONOMY. BOOK VII. PAUT II. 
 
 Prop. CXLVIII.) the waters on the side of the earth A are more attracted bj the moon at M 
 than the central parts of the eartli C, and the central parts are more attracted than the waters 
 on the opposite side of the earth at L ; consequently (as was shown concerning the moon) the 
 waters on the side L will be as it were attracted from the centre of the earth, or will recede 
 from thence. Therefore, while the moon is at M, the waters will rise toward a and I on the 
 opposite sides of the earth A, L ; while, by the oblique attraction of the moon, the waters at P 
 and N will be depressed. 
 
 Or thus ; because (by Prop. CLV.) the moon and earth are continually revolving about their 
 common centre of gravity, suppose a; the points A, C, L, describing circles about this common 
 centre in the same periodical times, the forces required to retain them in these circles (as may 
 he inferred from Book II. Prop. LXXV.) will be to each other as their distances from the centre 
 aA, aC, aL. Consequently, the point L requires a greater force than C, and C than A, to retain 
 it in its orbit. Now these points are retained in their respective circles by the moon atM; 
 and consequently the point L, which is most remote, and therefore requires the greatest force, 
 is attracted the least, whilst A, the nearest point, is attracted the most. Thus, the water about 
 A being attracted too much, and that about L too little, both will have their gravity diminished 
 by the action of the moon, and will endeavour to leave the centre C ; while the water at P and 
 N, having their gravity increased by the same cause, will subside. Hence the form of the water 
 on the surface of the earth will become an oblong spheroid. 
 
 This oval of waters keeps pace with the moon in its monthly course round the earth; while 
 the earth, by its daily rotation about its axis, presents each part of its surface to the direct action 
 of the moon, twice each day, and thus produces two floods and two ebbs. But because the moon 
 is in the mean time passing from west to east in its orbit, it comes to the meridian of any place 
 later than it did the preceding day ; whence the two floods and ebbs require nearly 25 hours to 
 complete them. The tide is at the greatest height, not when the moon is in the meridian, but 
 some time afterward, because the force, by which the moon raises the tide, continues to act for 
 some time after it has passed the meridian. 
 
 As the moon thus raises the water in one place, and depresses it in another, the sun does the 
 same; but in a much less degree, on account of the small ratio of the semidiameter of the earth 
 to the distance of the sun ; for, as it was shown of the moon, that the force of the sun, by which 
 it disturbs its motion, is as the distance of the moon from the earth to that of the sun from the 
 same, so, in this case, the force of the sun to disturb the waters is as the semidiameter of the 
 earth to the distance of the sun, which ratio is very small. 
 
 COR. The moon being nearest to the eartli when in perigee, its attraction must then be 
 strongest, and the eS'ect or elevation of the waters, greatest. And the earth being in its perihe- 
 lion at the winter solstice, the sun's power to produce tides is greatest at that time. 
 
 PROP. CLXXIV. The tides are greatest at the new and full moons, 
 and least at the first and last quadratures ; and the highest of the former tides 
 are near the time of the equinoxes. 
 
 Plate 11. When the moon is in conjunction or opposition with the sun, as M, m, 8, the tides which each 
 
 Piff- 14. endeavours to raise are in the same place ; whence they are then greatest, and are called spring 
 
 tides. But, when the moon is in the first or last quarter, the sun being in the meridian when 
 
CHAP. V. OF THE TIDES. 313 
 
 the moon is in the horizon, as M, s, depresses the water where the moon raises it; whence the 
 tides are then least, and are called neap tides. On the full and new moons, which happen about 
 the equinoxes, when the luminaries are both in the equator or near it, the tides are the greatest 
 of all ; for first, the two eminences of water are at the greateat distance from the poles, and 
 hence the difference between ebb and flood is more sensible ; for if those eminences were at the 
 poles, it is obvious we should not perceive any tide at all ; secondly, the equatorial diameter of 
 the earth produced passes through the moon, which diameter is longer than any other, and con- 
 sequently there is greater disproportion between the distances of the zenith, centre and nadir, 
 from the centre of gravity of the earth and moon, in this situation than iu any other ; and thirdly, 
 the water rising higher in the open seas, rushes to the shores with greater force, where being 
 stopped, it rises higher still ; for it not only rises at the shores in proportion to the height it 
 rises to in the open seas, but also according to the velocity with which it flows from thence 
 against the shore. The spring tides, which happen a little before the vernal and after the autum- 
 nal equinox, are the greatest of all, because the sun is nearer the earth in the winter than in the 
 summer. 
 
 PROP. CLXXV. When the moon is in the northern hemisphere, it pro- 
 duces a greater tide while it is in the meridian above the horizon, than when 
 it is in the meridian below it ; when in the southern hemisphere, the reverse. 
 
 Let AFHD represent the earth, whose centre is T, and axis PO, the point P the north pole, p^^n. 
 and O the south pole, EQ the equator, FH a parallel to it on the south side, and KD another Fig. 15. 
 parallel to it on the north side. Let the fluid surrounding the earth form itself into an oblong 
 spheroid, whose longer axis HK produced, passes through the moon at L. The right lines TK, 
 or TH, drawn from the centre T, will represent the greatest height of the water in those places. 
 Then, supposing NM perpendicular to KH, TN or TM will denote the least height, and will 
 represent the height of the water in all parts of the globe through which the circle NM passes. 
 The right lines TE, TF, TH, TQ, TD, will show the height of the water in those respective 
 places E, F, H, Q, D. 
 
 Let us now consider some place in particular, which, by the diurnal motion of the earth, 
 describes the parallel KD. When that place is at K, the height of the water TK. is the greatest, 
 that is, it will be high water, and the moon L will be in the meridian. But afterward, when that 
 place comes to X, the height of the water will be the least, that is, it will be low water ; and 
 when the same place comes to D, it will be high water again. But because TK is greater than 
 TD, therefore, in the present case, when the moon is on the north side of the equator, or in the 
 northern signs, the height of the sea, or tide, will be greater when the. moon is in the part of the 
 meridian which is above the horizon, than when it is in the meridian, and below it. Hence it is 
 that the moon, when it is in the northern signs, makes the greatest tides on our side the equator 
 when it is above the earth. 
 
 Again, TH, on the south side the equator, is longer than TF ; and therefore, to a place that 
 describes the parallel FH, the greatest height of the water, when the moon is in the northern 
 signs, is when it is on that part of the meridian that is below the horizon of that place, and the 
 least tides when it is above the horizon. For the like reason, when the moon is in the southern 
 signs, the greatest tides on the other side of the equator will be when it is below our horizon, 
 and the least tides when it is above it. 
 
 40 
 
314 OF ASTRONOMY. HOOK VII. PART III. 
 
 COR. Hence it is evident, that when the moon is in the equator, the two tides are equally 
 high. For then the longer axis HK of the oblong spheroid coincides with the equator EQ, and 
 the two points of high water on any parallel, being equally distant from the highest points of the 
 tides, are consequently at the same height. 
 
 SCHOL. What hath been said of the tides must be understood upon supposition, that the 
 globe of the earth is covered entirely with water to a considerable depth ; bat continents which 
 stop the tide, straits between them, islands, and the shallowness of the sea in some places, which 
 are all impediments to the course of the water, cause many exceptions to what hath been above 
 laid down. These exceptions can only be explained from particular observations on the nature 
 of tides at different places. 
 
 BOOK VII. PART III. 
 
 OF THE FIXED STARS. 
 
 DEP. LXVII. Those bodies, which always appear in the heavens at the 
 same distance from each other, are called Fixed Stars ; because they do not 
 appear to have any proper motion of their own. 
 
 PROP. CLXXVI. The fixed stars are luminous bodies. 
 
 Because they appear as points of small magnitude when viewed through a telescope, they 
 must be at fuch immense distances, as to be invisible to the naked eve if they borrowed their 
 light; as is the case with respect to the satellites of Jupiter and Saturn, although they appear 
 of very distinguishable magnitude through a telescope. Besides, from the weakness of reflected 
 light, there can be no doubt but that the fixed stars shine with their own light. They are easily 
 known from the planets, by their twinkling. 
 
 SOHOL. The number of stars, visible at any one time, to the naked eye, is about 1000; but 
 Dr Herscl'el, by his magnificent improvements of the reflecting telescope, has discovered that the 
 whole number is great beyond ail conception. 
 
 PROP. CLXXVII. The fixed stars appear of different magnitudes. 
 
 The magnitudes of the fixed stars appear to be different from one another, which difference 
 ni&y arise either from u diversity in their real magnitudes, or distances; or from' both these 
 causes acting conjointly. The difference in the apparent magnitude of the stars is such as to 
 admit of their being divided into six classes, the largest being called stars of the first magnitude, 
 and the least, which are visible to the naked eye, stars of the sixth magnitude. Stars only visi- 
 ble by the help of glasses are called telescopic stars. 
 
 SCHOL. 1. It must not be inferred that all the stars of each class appear exactly of the same 
 magnitude ; there being great latitude given in this respect ; even those of the first magnitude 
 appear almost all different in lustre and size. There are also other stars of intermediate mag- 
 
BOOK VII. PART III. OF THE FIXED STARS. 315 
 
 riitudes, which, as astronomers cannot refer to any one class, they, therefore, place them between 
 two. Procyon, for instance, which Ptolemy makes of the first magnitude, and Tycho of the 
 second, Flamstead lays down as between the first and second. So that instead of 6 magnitudes, 
 we may say that there are almost as many orders of stars, as there are stars ; such considerable 
 varieties being observable in their magnitude, colour, and brightness. 
 
 SOHOL. 2. To the bare eye the stars appear of some sensible magnitude, owing to the glare 
 of light arising from the numberless reflections of the rays in coming to the eye; this leads us to 
 image in that the stars are much larger than they would appear, if we saw them only by the few 
 rays which come directly from them, so as to enter the eye without being intermixed with others. 
 
 EXP. Examine a fixed star of the first magnitude through a long and narrow tube ; which, 
 though it takes in as much of the sky as would hold a thousand such stars, scarcely renders that 
 one visible. 
 
 SCHOI.. 3. There seems but little reason to expect that the real magnitudes of the fixed stars 
 will ever be discovered with certainty; we must, therefore, be contented with an approxima- 
 tion, deduced from their parallax, (if this should ever be ascertained) and the quantity of light 
 they afford us compared with the sun. To this purpose, Dr. Herschel informs us that with a 
 magnifying power of 0450. and by means of his new micrometer, he found the apparent diame- 
 ter of * Lyiae to be 0.335". 
 
 Dr. Herschel's method of finding the annual parallax of the fixed stars is by observing how 
 the angle between two stars very near to each other varies in opposite parts of the year. The P [ ate il 
 following is the most simple case given by that great astronomer. Let G and E be two stars Fig. 16. 
 situated in a line with the earth at A, and supposed perpendicular to the diameter AB of the 
 earth's orbit, and when the earth is at B, obseive the angle GBE. Let P = the angle AGB, or 
 the annual parallax of G ; p the angle GBE found by observation ; M, in, the angles under 
 which the diameters of G and E appear, and draw GH, perpendicular to BG. Then p : P : : 
 
 GH : AB : : GE : AE : : (because M : m : : AE : AG) M m : M, hence P = -jj^ ^ the parallax 
 
 G. If G be a star of the first magnitude, and E one of the third, and p = 1", then P = 1|". 
 See Phil. Trans. Vol. LXXII. 
 
 This theory is only true upon the supposition that the stars are all of the sar.-e magnitude ; 
 and that a star of the second magnitude is at double the distance of one of the first, and so on. 
 These suppositions are certainly not founded on any analogy from the known and well estab- 
 lished principles of that system of bodies to which we belong. 
 
 SCHOL. 4. The ingenious observation of Kepler upon the magnitudes and distances of the 
 fixed stars, deserves to be introduced here, and the more so, as he was followed in the conjec- 
 ture by the great Dr. Halley. He observes, that there can be only 13 points upon the surface of 
 a sphere as far distant from each other as from the centre ; and supposing the nearest fixed stars 
 to be as far from each other as from the sun, he concludes that there can be only 13 stars of the 
 first magnitude. Hence at twice that distance from the sun there may be placed four times as 
 many, or 52. At three times that distance, nine times as many, or 117; and soon. These 
 numbers will give pretty nearly the number of stars of the first, second, third, &c. magnitudes. 
 Dr. Halley farther remarks, that if the number of stars be finite, and occupy only a part of space 
 the outward stars would be continually attracted to those within, and in time would unite into 
 one. But if the number be infinite, and they occupy an infinite space, all the parts would be 
 
 
316 
 
 OF ASTRONOMY. 
 
 BOOK VII. PART III. 
 
 nearly in equilibrio, ami consequently eacli fixed star being drawn in opposite directions would 
 keep its place, or move on till it had found an equilibrium. Phil. Trans. No. 564. See also the 
 introductory remarks of Dr. Herschel to a paper on the changes of the fixed stars. Phil. 
 Trans. 1796. 
 
 The fixed stars are divided into constellations, or 
 
 PROP. CLXXVIIL 
 
 systems of stars. 
 
 The ancients, that they might the better distinguish the stars with regard to their situation in 
 the heavens, divided them into several constellations, that is, systems of stars, each system con- 
 sisting of such as are near each other. And to distinguish these systems from one another, they 
 gave them the names of such men or things as they fancied the space they took up in the 
 heavens represented. To these, several new constellations have been added by modern astron- 
 
 omers. 
 
 SCHOL. The following table contains the names of the constellations, and the number of 
 stars observed in each by different astronomers. 
 
 The Ancient Conttcllations, 
 
 Ursa Minor 
 
 Ursa Major 
 
 Draco 
 
 Cepheus 
 
 Bootes, Jlrctophilax 
 
 Corona Borealis 
 
 Hercules, Engonasin 
 
 Lyra 
 
 Cygnus, Gallina 
 
 Cassiopea 
 
 Perseus 
 
 Auriga 
 
 Serpentarius, Ophiuchus 
 
 Serpens 
 
 Sagitta 
 
 Aquila, Vultur 
 
 Antinous 
 
 Delphinus 
 
 Equulus, Equi sectio 
 
 Pegasus, Equus 
 
 Andromeda 
 
 Triangulum 
 
 Aries 
 
 Taurus 
 
 Gemini 
 
 Cancer 
 
 Leo 
 
 Coma Berenices 
 
 The Little Bear 
 
 The Great Bear 
 
 The Dragon 
 
 Cepheus 
 
 Bootes 
 
 The Northern Crown 
 
 Hercules kneeling 
 
 The Harp 
 
 The Swan 
 
 The Lady in her Chair 
 
 Perseus 
 
 The Wagoner 
 
 Serpentarius 
 
 The Serpent 
 
 The Arrow 
 
 The Eagle 
 
 Antinous 
 
 The Dolphin 
 
 The Horse's Head 
 
 The Flying Horse 
 
 Andromeda 
 
 The Triangle 
 
 The Bam 
 
 The Bull 
 
 The Twins 
 
 The Crab 
 
 The Lion 5 
 
 Berenice's Hair, 5 
 
 Ptolemy. 
 
 8 
 
 Tycho. Haeliui. FlamMai!, 
 
 7 12 24 
 
 35 
 
 29 
 
 73 
 
 87 
 
 31 
 
 32 
 
 40 
 
 80 
 
 13 
 
 4 
 
 51 
 
 35 
 
 23 
 
 18 
 
 52 
 
 54 
 
 8 
 
 8 
 
 8 
 
 21 
 
 29 
 
 28 
 
 45 
 
 113 
 
 10 
 
 11 
 
 17 
 
 21 
 
 19 
 
 18 
 
 47 
 
 81 
 
 13 
 
 26 
 
 37 
 
 55 
 
 29 
 
 29 
 
 46 
 
 59 
 
 14 
 
 9 
 
 40 
 
 66 
 
 29 
 
 15 
 
 40 
 
 74 
 
 18 
 
 13 
 
 22 
 
 64 
 
 5 
 
 5 
 
 5 
 
 18 
 
 15 
 
 12 
 
 3 
 
 23 
 19 
 
 71 
 
 10 
 
 10 
 
 14 
 
 18 
 
 4 
 
 4 
 
 6 
 
 10 
 
 20 
 
 19 
 
 38 
 
 89 
 
 23 
 
 23 
 
 47 
 
 66 
 
 4 
 
 4 
 
 12 
 
 16 
 
 18 
 
 21 
 
 27 
 
 66 
 
 44 
 
 43 
 
 51 
 
 141 
 
 25 
 
 25 
 
 38 
 
 85 
 
 23 
 
 15 
 
 29 
 
 83 
 
 35 
 
 30 
 14 
 
 49 
 21 
 
 95 
 43 
 
BOOK VII. PART III. OF THE FIXED STARS. 317 
 
 The Ancient Constellations, Ptolemy. Tyclio. Hnelius. Flttnistcatl. 
 
 Virgo The Virgin 52 03 50 110 
 
 Libra, Chelae The Scales 17 10 20 51 
 
 Scorpius The Scorpion 24 10 20 44 
 
 Sagittarius The Archer 31 14 22 69 
 
 Capricornus The Goat 28 28 29 51 
 
 Aquarius The Water-bearer 45 41 47 108 
 
 Pisces The Fishes 38 SG 39 113 
 
 Cetus The Whale 22 21 45 97 
 
 Orion Orion 38 42 62 78 
 
 Eridanus, Fluviu- Eridanus, the River 34 10 27 84 
 
 Lepus The Hare 12 13 16 19 
 
 Canis major The Great Dog 29 13 21 31 
 
 Canis minor The Little Dog 2 2 13 14 
 
 Argo Navis The Ship 45 3 4 64 
 
 Hydra The Hydra 27 1!) 51 60 
 
 Crater The Cup . 7 I 10 31 
 
 Corvus The Crow 74 9 
 
 Centaurus The Centaur 37 .55 
 
 Lupus The Wolf 19 .-34 
 
 Ara The Altar 7 9 
 
 Corona Australia The Southern Crown 13 12. 
 
 Piscis Australia The Southern Fish 18 24 
 
 The >"ra> Southern Constellations. 
 
 Columba Noachi Noah's Dove 10 
 
 Robur Carolinum The Royal Oak 12 
 
 Grus The Crane 13 
 
 Phoenix The Phenix 13 
 
 Indus The Indian 12 
 
 Pavo The Peacock 14 
 
 Apus, Avis Indim The Bird of Paradise 1 1 
 
 Apis, Musca The Bee or Fly 4 
 
 Charnseleoa The Cameleon 10 
 
 Triangulum Australe The South Triangle 5 
 
 Piscis volans, Passer The Flying Fish 8 
 
 Dorado, Xiphias The Sword Fish 6 
 
 "Toucan The American Goose 9 
 
 fydrus The Water Snake 10 
 
 Hevelius's Constellations made out of the unformed Stars. 
 
 Hcrelius. Flatnstead. 
 
 L J nx The Lynx 19 44 
 
 Leo minor The Little Lion 53 
 
 Astenon and Cliara The Grey Hounds 23 25 
 
 Cerberus Cerberus" 4 
 
 Vulpecula & Anser The Fox and Goose 35 
 
S8 OF ASTRONOMY. BOOK VII. PART III. 
 
 Scutum Sobieski Sobieski's Shield 7 
 
 Lacerta The Lizard 10 16 
 
 Camelopardalis The Camelopard 32 58 
 
 Monoceros The Uriicora 19 31 
 
 Sextans The Sextant 11 41 
 
 SCHOL. 1. Stars not included in any constellation are called unformed stars. Besides the 
 names of the constellations, the ancient Greeks gave particular names to some single stars, or 
 small collections of stars ; thus, the cluster of small stars in the neck of the bull, was called the 
 Pleiades ; five stars in the bull's face, the Ifyades ; a bright star in the breast of Leo, the Lion'f 
 Heart ; and a large star between the knees of Bootes, Jlrcturus. 
 
 SCHOL 2. The constellations may be represented on two plane spheres projected on a great 
 circle, or on the convex surface of a solid sphere, as on the celestial globe, or most perfectly on 
 the concave surface of a hollow sphere. If the celestial globe be made use of, after rectifying it 
 to the time of the night, the stars may be found, by conceiving a line drawn from the centre of 
 the globe through any star in the heavens, and its representation upon the globe. Greek letters 
 have been added by Bayer to stars in the several constellations of his catalogue, (<* being affixed 
 to the largest star) by means of which any star may be easily found. 
 
 SCHOL. 3. Twelve of these constellations lie upon the ecliptic, including a space about 15 
 broad, called the Zodiac, within which all the planets move. The constellation Aries, about 
 2000 years ago, lay in the first sign of the ecliptic ; but on account of the precession of the equi- 
 noxes, it now lies in the second. Prop. CLXXIL Schol. In the foregoing table Jlntinous was 
 made out of the unformed stars near Jlquila ; and Coma Berenices out of the unformed stars 
 near the L.on's tail. They are both mentioned by Ptolemy, but as unformed stars. The con- 
 stellations as far as the triangle, with Coma Berenices, are northern; those after Pisces are 
 southern. 
 
 PROP. CLXXIX. The luminous part of the heavens, called the Milky 
 Way, consists of fixed stars too small to be seen by the naked eye. 
 
 This is found from observations made with telescopes. 
 
 In a paper on the Constructions of the Heavens, Dr. Herschel says, " it is very probable, that 
 the great stratum called the milky way is that in which the sun is placed, though perhaps not in 
 the centre of its thickness, but not far from the place where some smaller stratum branches from 
 it. Such a supposition will satisfactorily, and with great simplicity, account for all the phenom- 
 ena of the milky way, which, according to this hypothesis, is no other than the appearance of the 
 projection of the stars contained in this stratum, and its secondary branch." 
 
 In another paper on the same subject, he says, "thaj the milky way is a most extensive 
 stratum of stars of various sizes admits no longer of the least doubt ; and that our sun is actu- 
 ally one of the heavenly bodies belonging to it is as evident.'' 
 
 " We will now retreat to our own retired station in one of the planets attending a star in 
 the great combination with numberless others ; and in order to investigate what will be the 
 appearances from this contracted situation, let us begin with the naked eye. The stars of the 
 first magnitude, being in all probability the nearest, will furnish us with a step to begin our scale ; 
 setting olF, therefore, with the distance of Siriut or Arcturus, for instance, as unity, we will at 
 present suppose, that those of the second magnitude are at double, and those of the third at 
 
BOOK VII. PART III. OF THE FIXED STARS. 819 
 
 treble the distance, and so forth. Taking it, then, for granted, that a star of the seventh mag- 
 nitude is about seven times as far from us as one of the first, it follows that an observer, who is 
 enclosed in a globular cluster of stars, and not far from the centre, will never be able, with the 
 naked eye, to see to the end of it. For since, according to the aBove estimations, he can only 
 extend his view about seven times the distance of Sirius, it cannot be expected that his eyes 
 should reach the bordeis of a cluster, which has, perhaps, not less than fifty stars in depth every 
 where around him. The whole universe, therefore, to him, will be comprised in a set of constel- 
 lations, richly ornamented with scattered stars of all sizes. Or, if the united brightness of a 
 neighbouring cluster of stars should, in a remarkably clear night, reach his sight, it will put on 
 the appearance of a small, faint, nebulous cloud, not to be perceived without the greatest atten- 
 tion. Allowing him the use of a common telescope, he begins to suspect that all the milkiness 
 of the bright path which surrounds the sphere may be owing to stars. By increasing his power 
 of vision, he becomes certain, that the milky way is indeed no other than a collection of very 
 small stars, and the nebulae nothing but clusters of stars." 
 
 Dr Herschel then solves a general problem for computing the length of the visual ray ; that 
 of the telescope, which he uses, will reach to stars 49r times the distance of Sirius. Now (by 
 Prop. B. Cor. I. p. 323 ) Sirius cannot be nearer than 100,000 x 194,000,000 miles; therefore 
 Dr. Herschel's telescope will, at least, reach to 100,000 x 194,000,000 x 49" miles. And Dr. 
 Her&chel says, that in the most crowded part of the milky way, he has had fields of view that 
 contained no less than 568 stars, and these were continued for many minutes, so that, in a quar- 
 ter of an hour, he has seen 116,000 stars pass through the field view of a telescope of only 15' 
 aperture; and at another time, in 41 minutes, he saw 258,000 stars pass through the field of his 
 telescope. Every improvement in his telescopes has discovered stars not seen before, so that 
 there appear no bounds to their number, nor to the extent of the universe. See Phil. Trans. 
 Vol. LXXIV. and ixxvi. 
 
 SOHOL. 1. There are spots in the heavens, called Nebulai, some of which consist of clusters 
 of telescopic stars, others appear as luminous spots of different forms. The most considerable is 
 one in tlie mid-way between the two stars on the blade of Orion's sword, marked 6 by Bayer, 
 discovered in the year 1656 by Iluygens ; it contains only seven stars, and the other part is a 
 bright spot upon a dark ground, and appears like an opening into brighter regions beyond. Dr. 
 Halley and others have discovered nebulae in different parts of the heavens. In the Connoisstmce 
 des Temps for 1783 and 1*84, there is a catalogue of 103 nebulx observed by Messier and 
 Mechain. But to Dr. Herschel we are indebted for catalogues of 2000 nebulae and clusters of 
 stars which he himself has discovered. Some of them form a round compact system, others are 
 more irregular, of various forms, and some are long and narrow. The globular systems of stars 
 appear thicker in tlie middle, than they would do if the stars were all at equal distances from 
 each other ; they are therefore condensed toward the centre. That stars should be thus acci- 
 dentally disposed is too improbable a supposition to be admitted ; he supposes, therefore, that 
 they are brought together by their mutual attractions, and that the gradual condensation toward 
 the centre is a proof of a central power of such a kind. He observes also, that there are some 
 additional circumstances in the appearance of extended clusters and nebulas, that very much 
 favour the idea of a power lodged in the brightest part. For although tlie form of them be not 
 globular, it is plain that there is a tendency to sphericity. As the stars in tlie same nebulae 
 must be very nearly all at the same relative distances from us, and they appear nearly of the 
 same size, their real magnitudes must be nearly equal. Granting, therefore, that these nebula? 
 
ATS i Jfll Tii 
 
 ,120 OF ASTRONOMY. BOOK VII. PART III. 
 
 and clusters of stars are formed by mutual attraction, Dr. Herschel concludes that we may 
 judge of their relative age by the disposition of their component parts, those being the oldest 
 which are most compressed. He supposes, and indeed offers powerful arguments to prove, that 
 the milky way is the nebulse,'of which our sun is one of the component parts. See Phil. Trans. 
 Vol. LXXVI. and LXXIX. 
 
 SOHOL. -Z. Dr. Herschel has also discovered other phenomena in the heavens which he calls 
 nebulous stars ; that is, stars surrounded with a faint luminous atmosphere of large extent. 
 Those which have been thus styled by other astronomers, he says, ought not to have been so 
 called, for on examination they have proved to be either mere clusters of stars plainly to be dis- 
 tinguished by his large telescopes, or such nebulous appearances as might be occasioned by a 
 multitude of stars at a vast distance. The milky way consists entirely of stars ; and he says, 
 " I have been led on by degrees from the most evident congeries of stars to other groups in 
 which the lucid points were smaller, but still very plainly to be seen ; and from them to such 
 wherein they could but barely be suspected, until I arrived at last to spots in which no trace of 
 a star was to be discerned. But then the gradation to these latter were by such connected steps, 
 as left no room for doubt but that all these phenomena were equally occasioned by stars variously 
 dispersed in the immense expanse of the universe." 
 
 In the same paper is given an account of some nebulous stars, one of which is thus described; 
 " Nov. 13, 1790. A most singular phenomenon ! A star of the eighth magnitude, with a faint 
 luminous atmosphere of a circular form, and of about 3' in diameter. The star is perfectly in 
 the centre ? and the atmosphere is so diluted, faint and equal throughout, that there can be no 
 surmize of its consisting of stars, nor can there be a doubt of the evident connexion be- 
 tween the atmosphere and the star. Another star, not much less in brightness, and in 
 the same field of view with the above, was perfectly free from any such appearance." 
 Hence Dr. Herschel draws the following consequences: Granting the connexion between 
 the star and the surrounding nebulosity, if it consist of stars very remote which gives the 
 nebulous appearance, the central star, which is visible, must be immensely greater than the 
 rest ; or if the central star be no bigger than common, how extremely small and compressed 
 must be those other luminous points which occasion the nebulosity ? As, by the former supposi- 
 tion, the luminous central point must far exceed the standard of what we call a star ; so in the 
 latter, the shining matter about the centre will be too small to come under the same denomina- 
 tion ; we, therefore, either have a central body which is not a star, or a star winch is involved in 
 a shining fluid, of a nature totally unknown to ua. This last opinion Dr. Herschel adopts. 
 Light reflected from the star could not be seen at this distance. Besides, the outward parts are 
 nearly as bright as those near the star. Moreover, a cluster of stars will not so completely ac- 
 count for the milkinessor soft tint of the highest of these nebulae, as a self luminous fluid. " What 
 a field of novelty," says Dr. Herschel, " is here opened to our conceptions ! A shining fluid, of a 
 brightness sufficient to reach us from the regions of a star of the 8th, 9th, 10th, llth, 12th mag- 
 nitude, and of an extent so considerable as to take up 3, 4, 5, or 6 minutes in diameter." He 
 conjectures that this shining fluid may be composed of the light perpetually emitted from millions 
 of stars. See Phil. Trans. Vol. LXXXI. p. 1, on Nebulous Stars properly so called. 
 
 SCHOL. 3. New stars sometimes appear while others disappear. Several stars, mentioned 
 l>y ancient astronomers are not now to be found ; several are now visible to the naked eye, 
 which are not mentioned in the ancient catalogues ; and some stars have suddenly appeared, 
 and again, after a considerable interval, vanished; also a change of place has been observed in 
 some stars. 
 
VII. PART III. OF THE FIXED STARS. S21 
 
 The following are remarkable, and well authenticated examples. The first new star we have 
 an accurate account of, is that discovered by Cornelius Gemma, on Nov. 8, 1572, in the Chair 
 of Cassiopeia. It exceeded Sirius in brightness, and Jupiter in apparent magnitude. Tycho 
 Brahe observed it, and found that it had no sensible parallax. It gradually decayed, and after 
 16 mouths, disappeared. 
 
 On August 13, 1596, David Fabricius observed a new star in the JVecfc of the Whale, 25* 
 45' of Aries, with 15 54' south latitude. It disappeared after October in the same year ; was 
 discovered again in 1637. 
 
 In the year 1600, William Jansenus discovered a changeable star in the ueck of the Swan. It 
 was seen by Kepler, who wrote a treatise upon it, and determined its place to be 16 18' rsr, and 
 55 30' or 32' north latitude. Ricciolus saw it in 1616, 1621, and 1624, and is certain that it was 
 invisible from 1640 to 1650. M. Cassini saw it again in 1655 ; it increased till 1660 ; then 
 decreased ; and at the end of 1661, it disappeared. In November, 1665, it appeared again, and 
 disappeared in 1681. In 1715 it appeared, as it does at present, of the 6th magnitude. 
 
 In 1686 Kircher observed % in the Swan, to be a changeable star; and from 20 years' obser- 
 vations, the period of the return of the same phases was found to be 405 days; the variations of 
 its magnitude are subject to some irregularity. 
 
 In the year 1604, Kepler discovered a new star near the heel of Serpentarius, so very bril- 
 liant, that it exceeded every fixed star, and e"en Jupiter, in apparent magnitude. 
 
 Montanari discovered two stars in the Ship, marked /3 and y by Bayer, to be wanting. He 
 saw them in 1664, but lost them in 1668. He observed also that , in Medusa's Head, varied in. 
 its magnitude. 
 
 Mr. Goodricke has determined the periodical variation of Algol, or /3, in Medusa's Head 
 (observed by Montanari to be variable) to be about 2 days 21 hours. Its greatest brightness is 
 of the second magnitude, and least of the fourth. Phil. Trans. Vol. LXXIII. 
 
 SCHOL. 4. From an attentive examination of the stars with good telescopes, many of which 
 appear only single to the naked eye, are found to consist of two, three, or more stars. Dr 
 Maskelyne had observed * Herculis, to be a double star. Dr. Hornsby found it Bootis to be 
 double. Other astronomers had made similar discoveries. But Dr. Herschel, by his highly 
 improved telescopes has found about 700, of which not more than 42 had been observed before. 
 
 The following are a few of the most remarkable ; 
 
 * Herculis, Flam. 64, a beautiful double star ; the two stars very unequal ; the largest red, 
 and the smallest blue, inclining to green. 
 
 * Geminoruni, Flam. 66, double, a little unequal, both white; with a power of 146 their dis- 
 tance appears equal to the diameter of the smallest. 
 
 t Lyrce, Flam. 4 and 5, a double-double star ; at first sight it appears double at a considerable 
 distance, and by a little attention each will appear double; one set are equal, and both white ; 
 the other unequal, the largest white, and the smallest inclined to red. 
 
 ft Lyra, Flam. 10, quadruple, unequal, white; but three of them a little inclined to red. 
 
 A Orionis, Flam. 39, quadruple, or rather a double star, and has two more at a small dis- 
 tance; the double star considerably unequal, the largest white ; smallest, pale rose-colour. 
 
 ^ Herculis, Flam. 86, double, very unequal ; the small star is not visible with a power of 278, 
 but is seen very well with one of 460 ; the largest is inclined to a pale red, smallest, duskish. 
 * Lyra, Flam. 3, double, very unequal, the largest a fine brilliant white, the smallest dusky: 
 41 
 
333 OF ASTRONOMY. BOOK V1T. PART III. 
 
 it appears with a power of 227, Dr. Herschel measured the diameter of this fine star, and found 
 it to be 0".S553. 
 
 The examination of double stars with a telescope is a very excellent and ready method of 
 proving its powers. Dr. Herschel recommends the following method. The telescope, and the 
 observer, having been some time in the open air, adjust the focus of the telescope to some single 
 star of nearly the same magnitude, altitude, and colour of the star, to be examined ; attend to 
 all the phenomena of the adjusting star as it passes through the field of view ; whether it be 
 perfectly round, and well defined, or affected with little appendages playing about the edge, or 
 any other circumstances of the like kind. Such deceptions may be detected by turning the 
 object-glass a little in its cell, when these appendages will turn the same way. Thus you may 
 detect the imperfections of the instrument, and therefore will not be deceived when you come to 
 examine the double star. Phil. Trans. Vol. LXXH. and LXXV. 
 
 SCHOL. 5. The number of stars is unknown. The catalogue published by Bayer contains 
 1160; that by Flamstead, which includes many telescopic stars, contains 3000. But the most 
 complete catalogue is that published by the Rev. Mr. Wollaston, in 1789. 
 
 PROP. CLXXX. The longitude of the fixed stars increases, while their 
 latitude remains the same. 
 
 Because the vernal equinoctial point (by Prop. CLXXI.) moves westward, the distance 
 between any given star and that point, that is, its longitude, will increase. But since this change 
 is produced by the precession of the equinoxes, which is performed round the axis of the ecliptic, 
 this motion will make no change in the distance of the fixed star from the ecliptic, that is, in. 
 its latitude. 
 
 COR. Hence the constellations of the zodiac are to the east of those signs or arcs of the 
 zodiac which are called by the same names. The first part of the constellation Aries, by the 
 precession of the equinoxes, has gone so far to the east, since the names were first given to the 
 signs, that it is now 30 from the first degree of Aries in the line of the ecliptic. 
 
 DBF. LXVIII. The Jlnnual Parallax of a heavenly body is the change 
 of its apparent place, as it is viewed from the earth in its annual motion. 
 
 Plate 11. If ADBC be the orbit of the earth, S the sun, and A, B, the earth in opposite parts of its 
 ' orbit ; the change in ti.e apparent place of any body, as viewed from A and from B. is its annual 
 
 parallax. 
 
 [PROP. CLXXXI. The annual parallax of any heavenly body is propor- 
 tional to the angle which a diameter of the earth's orbit would subtend, if it 
 was viewed from that body. 
 
 If when the earth is at A, the fixed star E appears at or near the pole, and when the earth 
 has passed to the opposite point B, a different star F appears at or near the same pole, the star 
 E will have changed its place in respect of the pole ; for when the pole is at F, the star E, which 
 was at or near it before, is at the distance EF from it ; the apparent length of this distance EF 
 (by Def. LXVIII.) is the star's annual parallax. Now, if AB, the diameter of the earth's orbit, 
 
BOOK VII. PART III. OF THE FIXED STARS. v 
 
 was to be viewed from the star E, it would subtend the angle AEB ; but, because the axis of the 
 earth is always parallel to itself, AE aad BF, which coincide with the axis, are likewise parallel; 
 whence (El. I. 29 ) the angle EBF, subtended by EF, is equal to AEB, subtended by AB; and 
 AEB is the parallactic angle. 
 
 COR. The annual parallax of any heavenly body is inversely as its distance from the earthj; 
 for the angle AEB (by Book VI. Prop. LX1X.) is inversely as the distance of AB, the axis of 
 the earth's orbit. 
 
 PROP. A. If the distance of an object be greater than 100,000 times 
 the base, the angles at the stations will not sensibly differ from right angles ; 
 consequently the lines drawn from the object to the stations are, physically 
 speaking, parallel. 
 
 Suppose one of the angles to be 90. Then, since the most accurate instruments for the 
 mensuration of angles cannot be depended upon to less than 2", the tangent of which is to radi- 
 us as 1 to 103,132, if the base have a less ratio to the distance than this, that is, than about 1 to 
 100,000 (for the angle, the tangent of which is to radius in this ratio, is 2".06, or very little more 
 than two seconds) the angle at the other station will not sensibly differ from 90. See Button's 
 Logarithms. 
 
 SOHOL. It has been seen that Dr. Herschel depends upon his instruments for the accuracy 
 of measuring quantities much less than 2". 
 
 PROP. B. The parallax of an object, the distance of which is above 
 100,00i) times greater than that between the two stations of observation, is 
 insensible. v 
 
 If the object be at a greater distance from either station than 100,000 times the base, the an- 
 gle at one of the stations being 90, the angle at the other will be more than 89 59' 57.9', the 
 difference of which angle and 90, being scarcely more than 2", is too small to become sensible 
 by observation. 
 
 COR. 1. If the parallax of an object (observed with an instrument sufficiently exact to 
 measure an angle of 2") be insensible, the distance of it from either station cannot be less than 
 100,000 times the base, from the extremities of which it is observed. 
 
 SCHOL. It is to be remarked, that, though the distance of the object cannot be less than 
 100,000 times the base, yet it may be greater in any assignable ratio. 
 
 COR. 2. Line? drawn from any given points in a base, to an object, may be esteemed, in 
 practice, parallel, without any sensible error, if the distance of the object be more than 100,000 
 times the base. 
 
 COR 3. Rays, therefore, diverging from any point in the sun's disk upon the surface of the 
 earth, may be esteemed parallel, if their distance from each other do not exceed about 970 miles 
 at the earth's surface ; because 970 is to the distance of the earth from the sun in a proportion 
 of 1 to 100,000. 
 
OF ASTRONOMY. BOOK VII. PART III. 
 PROP. CLXXXII. The fixed stars have no sensible annual parallax. 
 
 Plate 11. When the place of the star E is observed by the best instruments from opposite points of the 
 
 Fig. 16. earth's orbit, its apparent place in the heavens remains the same, which could not be the case if 
 the angle of its parallax were so much as two seconds. 
 
 COR. 1. Hence it appears, that the fixed stars are so remote, that a diameter of the earth's 
 orbit bears no proportion to their distance, or (by Prop. CLXXXl.) that a diameter of the earth's 
 orbit, if viewed from one of the fixed stars, would appear as a point. 
 
 COR. 2. The distance of the stars must be greater than 100.000 times the base, from the ex- 
 tremities of which it is observed ; that is, greater than 100,000 times the diameter of the orbit of the 
 earth, or greater than 100,000 X 194,000,000. 
 
 COR. 3. Two planes drawn parallel to each other, and passing through the extremities of a 
 diameter of the orbit of the eartn, if produced, will appear to coincide with the same great circle 
 of the heavens ; because the diameter of the earth's orbit, when seen from the fixed stars, sub- 
 tends an angle less than 2''. In the same mancer, if a plane passing through the earth's centre, 
 be parallel to a plane drawn to the surface, these planes, when produced, apparently coincide 
 with the same great circle in the heavens. 
 
 COR. 4. The parallax of a fixed atar, being not more than 2", the sun, when viewed from 
 
 32' "." 1" 
 
 that star, would appear under an angle less than or less than , and therefore could 
 
 not be distinguished from a point. 
 
 SCHOL. Since bodies equal in magnitude and splendour to the sun, being placed at the dis- 
 tance of the fixed stars, would appear to us as the fixed stars now do, it may be supposed prob- 
 able, that the fixed stars are bodies similar to the sun, which is the centre of the solar system. 
 This being the case, the reason will appear, why a fixed star, when viewed through a telescope 
 magnifying 200 times, appears no other than a point. For the apparent diameter of the star be- 
 ing less than ^5. part of a second when magnified 200 times, will subtend an angle less than 
 2'' at the eye of a spectator, observing it in the telescope. 
 
 The parallax of the fixed star, when viewed from the opposite parts of the earth't orbit, is 
 here assumed 2', but it is probable that the parallax of the nearest star is much less, and con- 
 sequently the distance greater, in the same proportion, as the parallax is less. 
 
 PROP. CLXXXIIL The motion of the earth, and the progressive 
 motion of light, will make a fixed star, which has no sensible parallax, de- 
 viate fi-om its true place in the direction in which the earth moves. 
 
 Visc\7 ' ^ a star P asses through the zenith of any place when the earth is at A, it will (by last Prop.) 
 pass through the zenith of the same place when the earth is at B, the opposite extremity of the 
 earth's orbit. Consequently, such a star might be seen through a vertical telescope in the same 
 perpendicular at any point of the earth's orbit, if the motion of light from the star were in- 
 stantaneous. But the progressive motion of light will cause the star to deviate from the perpen- 
 dicular ; for, let the earth be moving from B to A, and let the velocity of light be to the veloci- 
 ty of the earth, as CA to BA, and let CB be the diagonal of the parallelogram formed from CA, 
 BA. Then the direction of a telescope, in order to see the star S when the earth is arrived at 
 
BOOK VII. PART III. OF THE FIXED STARS. 325 
 
 A, must be AH, parallel to BC. For, suppose BC to be a very long, slender telescope, 
 through which only one ray of light could pass at a time, or to be the axis of a larger telescope. 
 The star S cannot be seen through this telescope, but through a telescope perpendicular to B, if 
 the earth be stationary at B, and the progress of light instantaneous. But if the telescope in 
 the position BC were to continue in this position, and tolnove along with the earth to A, so as to 
 come into the situation AH, when the earth arrives at A, the star S might then be seen through it. 
 For, since the straight course of the ray is the line C A, in which it must always be, if it comes to 
 the eye without interruption; and since the ray cannot come directly along CB, the axis of the 
 telescope, and arrive at the eye in this axis, unless it is always in the axis ; that is, since the 
 ray, in order to come to the eye, must be always in the line CA, and also in the line CB, it must 
 be always in the common intersection of these two lines. Now C is the common intersection 
 when the earth is at B ; e is the common intersection when the earth is at E ; /, when it is at F ; 
 g, when it is at G ; and A, when at A ; the telescope, at each station, being successively in the 
 situations CB, EE, FF, GG, HA. Thus the common intersection descends down the line CA, 
 while the earth moves from B to A ; and, since the velocity of light is to that of the earth, as 
 C A to BA, a ray of light will likewise have descended down CA, while the earth was moving 
 from B to A. Therefore, in the whole motion of the telescope, the ray will have been in the 
 common intersection in the line CA, and the axis of the telescope, and consequently will have 
 passed along the axis of the telescope, and will come without interruption to the eye at A. 
 
 Thus it appears, that by the progressive motion of light, a ray which, coming from S, enters, 
 at C, a telescope in the situation CB, will arrive at the eye, when the telescope, carried along 
 BA with a velocity which is to that of a ray of light, as BA to C A, is come into the situation 
 HA ; and consequently, (Book VI. Prop. II.) the eye will see the star through the telescope in 
 the direction AH, the axis of the telescope ; that is, some point in the line AH produced will be 
 the apparent place of the star. Thus the star's apparent place has deviated from its true place 
 S in the direction BA, in which the earth was moving, so that if the motion of the earth is from 
 north to south, the star, which appeared in the zenith of the place when the earth was at B, will 
 appear to the southward of the zenith when the earth is arrived at A, and the reverse when the 
 earth is moving from south to north. 
 
 According to Bradley's observations, made on the star '/ in the constellation Dragon, this 
 star deviated southward from the zenith from Dec. till March, when it had departed from the 
 zenith 20". From that time till June its southern deviation decreased, after which it deviated 
 northward, and in September appeared about 20" toward the north of its station in June, from 
 which time till December it continued returning to its first situation. Thus the deviation of the 
 star was always in the direction of the earth's motion, and contrary to that of any deviation 
 which might be supposed to arise from the annual parallax of the star. But such a deviation 
 could not happen unless the earth moved, and the motion of light was progressive ; for if the 
 earth did not move, since the star is fixed, no alteration could be made in the apparent place of 
 the star by the progressive motion of its rays in a vertical direction ; and if the earth moves, 
 and the propagation of light were instantaneous, the earth's velocity would be nothing in respect 
 of the velocity of light, or BA with respect to CA would be nothing ; whence the angle ACB, 
 and its alternate angle CAH, would vanish, and AH would become coincident with AC, and 
 consequently the star would have no deviation from its true place. Hence we may conclude 
 from the deviation of the star above described, both that the earth moves, and, that the motion of 
 light is progressive. 
 
820 OF ASTRONOMY. BOOK V [I. PART III. 
 
 COR. From these observations it is found, that the velocity of star-light is such as carries it 
 through a space equal to the sun's distance from the earth in 8' 13". 
 
 SCHOL. Sir Isaac Newton has shown that the sun, by its attractive power, retains the plan- 
 ets belonging to our system in their orbits; he has likewise pointed out the method whereby the 
 quantity of matter contained in the sun may be accurately determined. Dr. Bradley has as- 
 signed the velocity of the solar light with a degree of precision exceeding our utmost expecta- 
 tion. Galileo and others have ascertained the rotation of the sun upon its axis, and determin- 
 ed the position of its equator. By means of the transit of Venus over the disk of the sun, 
 our mathematicians have calculated its distance from the earth ; its real diameter and magni- 
 tude; the density of the matter of which it is composed; and the laws of the fall of heavy bodies 
 on its surface. 
 
 In the year 1779, there was a spot on the sun, which was large enough to be seen by the 
 naked eye; it was divided into two parts, and must have been 50000 miles in diameter ; this 
 phenomenon may be accounted for, from some natural change of an atmosphere. For if such 
 of the fluids which enter into its composition be of a shining brilliancy, whilst others are 
 merely transparent, then any temporary cause which should remove the lucid fluid will permit 
 us to see the body of the sun through the transparent ones. If an observer were placed on the 
 moon, he could see the solid body of the eartli only in those places where the transparent fluids 
 of our atmosphere would permit him. In others, the opaque vapours would reflect the light "of 
 the sun without permitting his view to penetrate the surface of our globe. He would probably 
 find that our planet had occasionally some shining fluids in its atmosphere, such as the northern 
 lights. And there is good reason to believe, that all the planets emit light in some degree ; for 
 the illumination which remains on the moon in a total eclipse cannot be entirely ascribed to the 
 light which may reach it by the refraction of the earth's atmosphere. For, in some cases, as in 
 the eclipse of 1790, the focus of the sun's ravs refracted by the earth's atmosphere, must be 
 many thousand miles beyond the moon. 
 
 There are appearances also which denote a phosphoric quality in the atmosphere of Venus. 
 
 Dr. Herschel supposes, that the spots in the sun are mountains on its surface, which, con- 
 sidering the great attraction exerted by the sun upon bodies placed at its surface, and the slow 
 revolution it has about its axis, he thinks may be more than 300 miles high, and yet stand very 
 tirmly. He says, that in August, 1792, he examined the sun with several powers from 90 to 
 500. And it evidently appeared, that the black spots are the opaque ground or body of the sun ; 
 and that the luminous part is an atmosphere, which being intercepted or broken, gives us a 
 glimpse of the sun itself. 
 
 Hence he concludes, that the sun has a very extensive atmosphere, which consists of elastic 
 fluids that are more or less lucid and transparent ; and of which the lucid ones furnish us with 
 light. This atmosphere, he thinks, is not less than 1843, nor more than 2765 miles in height; 
 and he supposes that the density of the luminous solar clouds need not be exceedingly more 
 than that of our aurora borealis, in order to produce the eifects with which we are ac- 
 quainted. 
 
 The sun, then, appears to be a very eminent, large, and originally luminous body, and the 
 only one belonging to our system. Its similarity to the other globes of the solar system, with 
 regard to its solidity ; its atmosphere; its surface diversified with mountains and vallies; its 
 rotation on its axis j and the fall of heavy bodies OH its surface 5 leads us to suppose that it is 
 
BOOK VII. PART III. OF THE FIXED STARS. 337 
 
 most probably inhabited, like the rest of the planets, by beings whose organs are adapted to the 
 peculiar circumstances of that vast globe. 
 
 If it be objected, that from the effects produced at the distance of 97,000,000 miles, we may 
 infer, that every thing must be scorched up at its surface We reply, that there are many facts 
 in natural philosophy which show that heat is produced by the sun's rays only when they act on 
 a calorific medium ; they are the cause of the production of heat by uniting with the matter of 
 fire which is contained in the substances that are heated ; as the collision of the flint and 
 steel will inflame a magazine of gunpowder, by putting all the latent fire which it contains 
 into action. 
 
 On the tops of mountains of sufficient height, at the altitude where clouds can seldom reach 
 to shelter them from the direct rays of the sun, we always find regions of ice and snow. Now 
 if the solar rays themselves conveyed all the heat we find on this globe, it ought to be hottest 
 where their course is the least interrupted. Again ; our aeronauts all coufirm the coldness of 
 the upper regions of the atmosphere ; and since, therefore, even on our earth the heat of the 
 situation depends upon the readiness of the medium to yield to the impression of the solar rays, 
 we have only to admit, that on the sun itself the elastic fluids composing its atmosphere, 
 and the matter on its surface are of such a nature as not to be capable of any extensive affection 
 of its own rays ; and this seems to be proved by the copious emission of them ; for if the elastic 
 fluids of the atmosphere, or of the matter contained on the surface of the sun, were of such a 
 nature as to admit of an easy chemical combination with its rays, their emission would be very 
 much impeded. Another well known fact is, that the solar focus of the largest lens thrown into 
 the air, will occasion no sensible heat in the place where it has been kept for a considerable time, 
 although its power of exciting combustion, when proper bodies are exposed, should be sufficient to 
 fuse the most refractory substances. 
 
 It is by analogical reasoning that we consider the moon as inhabited. For it is a secondary 
 planet of considerable size, its surface is diversified, like that of the earth, with hills and val- 
 lies. Its situation, with respect to the sun, is much like that of the earth ; and by a rotation on 
 its axis, it enjoys an agreeable variety of seasons, and of day and night. To the moon, our 
 globe would appear a capital satellite, undergoing the same changes of illumination as the moon 
 does to the earth. The sun, planets, and the starry constellations of the heavens, will rise and 
 set there as they do here ; and heavy bodies will fall on the moon as they do on the earth. 
 There seems, then, only to be wanting, in order to complete the analogy, that it should be in- 
 habited like the earth. 
 
 It may be objected, that, in the moon, there are no large seas; and its atmosphere (the exist- 
 ence of which is doubted by many) is extremely rare, and unfit for the purposes of animal life ; 
 that its climates, its seasons, and the length of its days and nights, totally differ from ours ; 
 that without dense clouds, which the moon has not, there can be no rain, perhaps no rivers and 
 lakes. 
 
 In answer to this, it may be observed, that the very difference between the two planets 
 strengthens the argument. We find even on our own globe, that there is a most striking dis- 
 similarity in the situation of the creatures that live upon it. While man walks on the ground, 
 the birds fly in the air, and the fishes swim in the water. We cannot surely object to the con- 
 veniences afforded by the moon, if those that are to inhabit its regions are fitted to their condi- 
 tions as well as we on this globe of ours. The analogy already mentioned establishes a high 
 probability that the moon is inhabited. 
 
328 OF ASTRONOMY. BOOK VII. PART III. 
 
 Suppose, then, an inhabitant of the moon, who has not properly considered such analogical 
 reasonings as might induce him to surmise that our earth is inhabited, were to give it as his 
 opinion, that the use of that great body, which he sees in his neighbourhood, is to carry about his 
 little globe, in order that it may be properly exposed to the light of the sun, so as to enjoy an 
 agreeable and useful variety of illumination, as well as to give it light by reflection, when direct 
 light cannot be had. Should we not condemn his ignorance and want of reflection ? The earth, 
 it is true, performs those offices which have been named, for the inhabitants of the moon, but we 
 know that it also affords magnificent dwelling-places to numberless intelligent beings. 
 
 From experience, therefore, we affirm, that the performance of the most salutary offices to 
 inferior planets, is not inconsistent with the dignity of superior purposes ; and in consequence 
 of such analogical reasonings, assisted by telescopic views, which plainly favour the same opin- 
 ion, we do not hesitate to admit that the sun is richly stored with inhabitants. 
 
 This way of considering the sun, is of the utmost importance in its consequences. That 
 stars are suns can hardly admit of a doubt. Their immense distance would effectually exclude 
 ^them from our view, if their light were not of the solar kind. Besides, the analogy may be 
 traced much farther ; the sun turns on its axis ; so does the star Algol ; so do the stars called ft 
 Lyrae, ^Cephei, * Antinoi, Ceti, and many more, most probably all. Now from what other 
 cause can we, with so much probability, account for their periodical changes ? Again ; our sun's 
 spots are changeable so are the spots on the star Ceti. But if stars are suns, and suns are 
 inhabitable, we see at once what an extensive field for animation opens to our view. 
 
 It is true, that analogy may induce us to conclude, that since stars appear to be suns, and 
 suns, according to the common opinion, are bodies that serve to enlighten, warm, and sustain a 
 system of planets, we may have an idea of numberless globes that serve for the habitation of 
 living creature. But if these suns themselves are primary planets, we may see some thousands 
 of them with the naked eyes, and millions with the help of telescopes; and, at the same time, 
 the same analogical reasoning still remains in full force with regard to the planets which these 
 suns may support. 
 
 Moreover, from the observations of Dr. Herschel, on the compressed clusters of stars, it ap- 
 pears, that in many instances there cannot be space for the revolutions of a system of planets 
 and comets, and therefore it is highly probable that these suns are capital primary planets which 
 exist for themselves, and are connected together in one great system for mutual support. See 
 a very curious and valuable paper on the nature and construction of the sun and fixed stars, by 
 Dr. Herschel ; read to the Royal Society, Dec. 18, 1794. From this paper, the foregoing scho- 
 lium has been taken. See also Dr. Herschel's paper on the periodical star, a, Herculis ; with 
 remarks, tending to establish the rotatory motion of the stars on their axes. Phil. Trans v 1796. 
 

 
 APPENDIX TO THE ASTRONOMY, 
 
 CONTAINING SOLAR AND LUNAR TABLES, WITH THEIR EXPLANATION AND 
 
 USE, AND THE PROJECTION OF ECLIPSES, SELECTED FROM 
 
 "EWING'S PRACTICAL ASTRONOMY." 
 
 EXPLANATION OF THE TABLES. 
 
 Tables of the Mean Motions of Celestial Objects. 
 
 1 HE idea that the sun, moon, and stars, performed all their motions in circles, was, perhaps, one of 
 the first which men received concerning these very distant bodies. The regular returns of day and night, 
 of the seasons of the year, and of almost all things in the visible world, would serve to confirm it ; and 
 although it is well known that the orbits of the planets are not perfect circles, it is equally known that 
 they differ very little from circles ; and therefore modern astronomers retain the idea, and form tables of 
 the motions of the planets, as if their orbits were circles, and their motions always uniform, passing over 
 equal spaces in equal times Sur.l' ->-3 called Tables of Mean Motion; and the longitude of a planet 
 computed from such Tables for any gi . ~n time is called its mean longitude. 
 
 TABLE I. II. Contain the Sun's Mean Motion, and the Precession of the Equinoctial Points in Julian 
 
 fears. 
 
 The astronomical year is that space of time wherein the earth moves round the sun, or wherein the 
 sun seems to move round the whole ecliptic from any point, such as the vernal equinox, to the same again ; 
 which, according to the most accurate observations, consists of 365 natural days, 5 hours, 48 minutes, 54 
 Seconds, ; but in civil reckoning there are two kinds of years, common and bissextile. A common year 
 consists of 365 days, and the bissextile of 366. These'are called Julian years, from Julius Ciesar, who 
 introduced this method of computation. 
 
 Mr. Maylier makes the sun's mean motion in 365 days to be 11s. 29 45' 40".7. Admitting his num- 
 bers, we have, 
 
 For the 4th year, 
 Year. S. ' " S. ' " 
 
 1 11 29 45 40.7 To the 3d year 11 29 17 2.1 
 
 2 11 29 31 20.4 Add mo. for 1 year 11 29 4540.7 
 
 3 11 29 17 2.1 And for 1 day 59 8.3 
 4001 51.1 
 
 The 4th year = 001 51.1 
 
 That is, in four years the earth goes four times round the sun, and 1' 51" more. In the same manner, 
 the sun's mean motion may be found for any number of Julian years, as in the Table. 
 
 The yearly motion of the earth's or sun's apogee is found to be 1' 6" ; which being subtracted from 
 the sun's yearly motion in longitude, the remainder is the sun's mean anomaly for 1 year ; and for any 
 number of years the mean anomaly is found in the same manner as the mean longitude. 
 
 It has been found by observation, that the equinoctial points move backward, or contrary to the order 
 of the signs, at the rate of 50".S yearly, called the precession of the equinoxes ; and multiplying 50" .3 
 by 2, 3, 4, 5, &c. the numbers in Table II. are found. 
 
 42 
 
330 APPENDIX TO THE ASTRONOMY. 
 
 TABLE III. Contains the Sun's Mean Longitude and Anomaly, with the Obliquity of the Ecliptic, for 
 
 years current of the Christian JEra. 
 
 The table is composed in this manner : The mean longitude and anomaly of the sun with the obliquity 
 of the ecliptic, are found for the noon of the last day of Dec. 1760, which is accounted his mean longitude, 
 &c. for the succeeding year 1761.. To these numbers add the sun's mean motions for 20 years, taken 
 from Table I; the sums are the sun's mean longitude and anomaly for 1781. To these add the mean 
 motions lor 10 years, and the mean longitude and anomaly for 1791 are known. And for the following 
 >ears add the sun's mean motions in one year to his mean longitude and anomaly for the preceding 
 year continually ; remembering to add the motion of 1 day more for every bissextile year, until the num- 
 ber of years which the Table is to contain be completed. 
 
 The annual differences of the obliquity of the ecliptic are very small, and not always regular. It 
 appears by the Table that the difference is only about 25" in 60 years ; therefore the obliquity of the 
 ecliptic may be stated at 23 28' during the next fifty years without sensible error. 
 
 When the sun's mean longitude and anomaly are wanted for any year which is not in the Table, 
 take the mean motions for the intermediate years from Table I. and if the year is before 1761, subtract 
 them ; if after it, add them to the numbers for 1761, and the remainder or sum will be the mean longi- 
 tude and anomaly required. 
 
 In these additions 12 signs are to be rejected as often as they occur ; and in the subtractions 12 signs 
 must be borrowed when necessary. 
 
 TABLE IV. Contains the sun's mean motions for the days of the year, distributed into 12 calendar 
 months. At the noon of January 1st there is one astronomical day past, because it began at the noon 
 of December Slst ; and therefore the sun's mean motions in one day are put down. These being multi- 
 plied by 2, 3, 4, 5, &c. (allowance being made for the fractions in one day's motion) the products are the 
 numbers of the Table ; remembering to place them properly, viz. the motions for 31 days at the last day 
 of January ; for 32 days at the 1st day of February; and so on until the Slst of December. 
 
 TABLE V. Contains the sun's mean motions for hours, minutes, and seconds, which may be under- 
 stood from what has been said of his motions for days. 
 
 TABLE VI. Equations of the Sun's Centre. 
 
 The sun's mean ;ind trae longitudes differ more or less in every point of the earth's orbit except 
 two, viz. the aphelion and perihelion ; and these differences are called equations of the sun's centre. 
 One cause of the difference is, that the earth's orbit is an ellipse, in the periphery whereof the true longi- 
 tude is reckoned from the vernal equinox, and the mean longitude is reckoned from the same point in 
 the circumference of a circle, whose diameter is the greater axis of the ellipse. The circle and ellipse 
 coincide only in the aphelion and perihelion ; and therefore when the sun or earth is in either of these 
 points, the mean and true longitudes are the same and there is no equation. 
 
 Another cause of the difference between the mean and true longitudes arises from the unequal motion 
 of the earth in its orbit; for while the earth proceeds from its aphelion to its perihelion, its motion is 
 continually accelerated ; and from its perihelion to its aphelion, its motion is continually retarded ; and 
 this is true of every planet. 
 
 "When the sun appears in the earth's aphelion, his longitude is about Ss. 9?, and his anomaly is 0, be- 
 cause it is reckoned from that point ; and when he appears in the earth's perihelion, his longitude is 
 nearly 9s. 9, and his anomally is 6 signs. In the first six signs of anomaly, the equation found in the 
 Table is to he subtracted from the mean longitude ; in the other six signs it is to be added, and the 
 remainder or sum is the true longitude. 
 
 TABLE VII. Contains the Logarithms of the Earth's Distances from the Sun. 
 
 Because the earth's orbit is an ellipse, and the sun in one of its foci, the earth's distance from the 
 sun varies every moment ; for it is greatest in aphelion, and decreases as the earth proceeds from thence 
 toward the perihelion, where it is least. The several distances decrease in the first six signs of anom- 
 aly, ant! increase in the other six. "When the anomaly is 3 or 9 signs, the earth is at its mean distance 
 
EXPLANATION OF THE TABLES. 331 
 
 from the sun, expressed by 1000000. In all other points the distance is either greater or less than the 
 mean. 
 
 The earth's distance from the sun being calculated for every degree of anomaly, the logarithms of 
 these distances are contained in the Table, and are of use in computing the longitudes of the other planets. 
 
 TABLE VIII. The Sun's Declination to every Degree of his Longitude. 
 
 At the time of either equinox the sun is in the equinoctial circle, but at all other times he appears 
 at some distance from it, either north or south ; and this distance is called his declination. 
 
 At the vernal equinox the sun has no declination; but from that point his declination increases 
 northward, until he comes to the summer solitice, where it is greatest; and from thence it decreases 
 until the sun is at the autumnal equinox, when again it is nothing; and then changes its name from nortli 
 to south, and increases southward to the winter solstice, when again it is greatest; and from thence it 
 decreases until the sun appears in the vernal equinox. 
 
 Those points of the ecliptic, which are equally distant from the equinoxes or solstices, being also 
 equally distant from the equinoctial circle, have the same declination ; and therefore the declination being 
 calculated for every degree of the first quadrant of the ecliptic, answers for the whole ; for the begin- 
 ning of the sign Taurus, of Virgo, of Scorpio, and of Pisces, are all at the same distance from the equi- 
 noctial circle, and consequently these points have the same declination ; only in the two first the 
 declination is north, and iu the two last it is soutli ; and the same is true of all other points of the eclip- 
 tic which are equally distant from the equinoctial or solstitial points. 
 
 TABLE IX. Tlie Sun's Apparent Semidiameter and Hourly Motion. 
 
 When the earth is in its aphelion, the sun's diameter appears least, and his apparent motion slowest; 
 his diameter at that time is only 31' 34", and his hourly motion 2' 23". W'hile the sun moves appar- 
 ently from the aphelion to the perihelion, his diameter and hourly motion increase with his anomaly, 
 and are greatest in the perihelion ;'his diameter being then 32' 58", and his hourly motion 2' 33". The 
 difference between the aphelion and perihelion diameter is 64", and between the hourly motions 10". 
 
 The Table contains the sun's semidiameter and hourly motion to every 10 of mean anomaly ; which, 
 by taking proportional parts for the intermediate degrees, will serve to find them for any given anomaly. 
 They are frequently of use in Astronomy, especially in calculating eclipses. 
 
 TABLE X. Contains the equation of time for every degree of the sun's longitude ; and by using 
 proportion for the minutes and seconds, the equation may be found for any given longitude. For example, 
 let it be required to find the equation of time when the sun's longitude" is 6s. 10 50' SO". J)ns. 11m. 
 12s. 
 
 s. m. s. s. ' " s. 
 
 6 10 gives 10 57 As 1 : 18 : : 50 SO : 15 
 
 Gil 11 15 Add - 10 57 
 
 Diff. 18 True equation 11 12 
 
 TABLE XI. The Sun's Longitude every Day at JYoon. 
 
 The conveniency of having the sun's longitude nearly true for the noon of every day in the year 
 inclined me to insert this Table. The reason why it could not be made accurate is, a common year 
 of 365 days differs from the time wherein the sun seems to move round the whole ecliptic by almost 6 
 hours, or a day in four years ; which causes a difference in the sun's longitude at noon on the same days 
 of different years ; and therefore, to have made the Table more perfect, it must have been calculated for 
 four years, which was not thought expedient. 
 
 TABLES of the Moon's Mean Millions. 
 
 These Tables are made in the same manner as those of the sun or earth, the moon's period being 
 known ; but her motions are more in number than those of the earth ; for besides her mean longitude and 
 anomaly, the longitude of her node must be known, in order to calculate her place for any given time. 
 
 I 
 
832 APPENDIX TO THE ASTRONOMY. 
 
 The earth is one of the foci of the moon's orbit, and is the centre of her motion ; for the moon 
 revolves round the earth in the same manner as the earth moves round the sun. 
 
 The moon's period, or the time wherein she moves once round the earth, is 27 days, 7 hours, 43 min- 
 utes, 5 seconds ; and to find the moon's mean motion in a common year of 365 days, the proportion is, 
 As the moon's period 27d. 7h. 43m. 5s. 
 Is to her whole orbit or 360 ; 
 So is a common year of 365 days 
 To 13 revolutions, and 4s. 9 23' 5". 
 The 13 revolutions are rejected, and 4s. 9 23' 5" are taken for the moon's motion in 365 days. 
 
 For the Moon's Mean Motion in one day. 
 
 As the moon's period 27d. 7h. 43m. 5s. 
 
 Is to the whole circle or 360 ; 
 
 So is one day 
 
 To the moon's mean motion in 1 day 13* 10' 35". 
 
 Having the moon's mean motion in 1 year and 1 day, the Table of her mean longitudes for Julian 
 years is made by multiplying the motion for 1 year by 2, 3, 4, &c. and to the 4th adding the motion for 1 
 day, the products are the mean longitudes for these years ; and by_ multiplying the motion for I year by 
 the several numbers of years, and adding the motion in 1 day to every 4th or bissextile year, the Table 
 may be made to any extent. 
 
 To calculate the Moon's Mean Anomaly. 
 
 The moon's apogee moves once round her whole orbit in 8 years 309 days 8 hours 20 minutes ; or 
 (adding 2 days for leap years) in 3231 days 8 hours $.0 minutes. Then, 
 As 3231d. 8h. SOm. 
 Is to the whole circle of 360 ; 
 So is a common year of 365 days 
 To the mo. of the J 's apogee in 1 vear 40 3^ 50". 
 
 S ' " 
 
 From the 3) 's m. mot. in Ion. during 1 year 4 9 23 5 
 Substract the motion of her apogee in ditto 1 10 39 50 
 
 There remains the J 's mean anomaly in 1 year 2 28 43 15 
 To find the Moon's Mean Anomaly for one Day. 
 
 Divide the motion of her apogee in a year, viz. 40 39' 50" by 365, and the quotient 6' 41" is the 
 motion of the moon's apogee for 1 day. 
 
 O / II 
 
 From the moon's motion in longitude for 1 day 13 10 35 
 Subtract the motion of her apogee for ditto. 641 
 
 There remains the moon's mean anom. for 1 day 13 3 54 
 
 Having found the moon's mean anomaly for 1 year and 1 day, the several mean anomalies in the Table 
 are found in the same manner as the longitudes, viz. by multiplying the mean anomaly for 1 year by 2, 
 3, 4, Sec. and adding the motion for 1 day to every 4th or bissextile year. 
 
 To find the Mean Motion of the Moon's Node. 
 
 The moon's node moves backward round her whole orbit in 18 years 224 days 5 hours 5 therefore, for 
 its motion in one year, 
 
 As 18 years 224 days 5 hours 
 
 Is to the whole circle or 360, 
 
 So is a year of 365 days 
 
 To the motion of the moon's node in 1 year 19 19' 43". 
 
EXPLANATION OF THE TABLES. 333 
 
 For the motion of the moon's node in 1 day, divide 19 19' 43" by 565, and the quotient, which is 
 nearly 3' 11", is its motion for one day. 
 
 The motion of the moon's node for 1 year and 1 day being known, ils motion for any numher of years 
 is found in the same manner as those of mean longitude and anomaly, viz,, by multiplying one year's 
 motion by 2, 3, 4, &c. and adding the motion of a day to every fourth or bissextile year. 
 
 TABLE II. Containing the Moon's Mean Longitude and Anomaly, with the Longitude of her Node, for 
 
 current Fears. 
 
 To make the Table, find, either by calculation or a good observation, the moon's mean longitude and 
 anomaly, with the longitude of her node, on the noon of the last day of December preceding the year, 
 where the Table is to begin, which here is the year 1760, and these are the mean places for 1761, the 
 first in our Table. For 1781, the next year in the Table, take the mean motions for 20 years from Table 
 I. and add the longitude and anomaly to those for 1761, but subtract the motion of the node} the results 
 are the mean places for 1781 ; then take the mean motions for 10 years, and apply them in the same 
 manner to the numbers for 1781, and the mean places for 1791 will be known. 
 
 For the following years the Table is carried on by the continual addition of the mean longitude and 
 anomaly for one year, as also the motion for 1 day more every 4th or bissextile year, and subtracting the 
 motion of the node. 
 
 TABLE HI. Contains the Moon's Mean Motions for Days. 
 
 It begins with the mean motions for one day already found. These being multiplied by every number 
 of days from I to 365, place the products at the proper days of the several months, and the Table is made. 
 
 TABLE IV. Contains the Moon's Mean Motions for Hours, Minutes and Seconds. 
 
 The numbers are found by dividing the mean motions for 1 day by 24, the quotients are the mean 
 motions for an hour; and these again divided by 60, give the mean motions for 1 minute, &c. The 
 motions for the different numbers ofnours and minutes are found by multiplication. 
 
 The moon's motions are affected with many inequalities, and therefore many equations are necessary 
 to reduce her mean longitude to the true. The method of finding the arguments, of taking out the equa- 
 tions, and applying them, is described in Prob. VII. 
 
 The Tables for finding the moon's latitude, parallax, diameter, hourly motion in longitude and latitude, 
 at any given time, are sufficiently plain from their titles : hnd the method of applying them is described in 
 Prob. VIII XIII. 
 
 TABLES of the Mean Motion of the Moon from the Sun. 
 
 The moon's mean motion in a common year of 365 days is 4s. 9 23' 5", over and above 13 revolu- 
 tions ; and the sun's apparent mean motion in the same time is Ils. 29 45' 40". 
 
 8. ' " 
 
 From 4 9 23 5 
 
 Subtract 11 29 45 40 
 
 The remainder 4 9 37 25 is the moon's mean motion from the sun in a common year. 
 
 In the same manner, the moon's mean motion from the sun may be found for a day, an hour, or for 
 any number or parts of these. 
 
 Having the moon's mean motion from the sun in a year and a day, it may be found for any number of 
 years by multiplication ; remembering to add the motion for one day to every four ycaVs' motion for leap 
 year. 
 
 To make a Table of the moon's mean motion from the sun for current years of the Christian era, sub- 
 tract the sun's mean longitude from the moon's tor every year which the Table is to contain ; and the 
 remainders will be the numbers expressing the moon's mean motion from the sun ; as in the Table. 
 
 The table for months is made by subtracting the sun's mean longitude from the moon's on the last day 
 of the preceding month ; and the remainder is the moon's mean motion from the sun on the first day of 
 
 
334 APPENDIX TO THE ASTRONOMY. 
 
 the following month. The Tables for clays, hours, minutes, and seconds, are made in the same manner ; 
 and the method of using these Tables is described in Prob. XIV. 
 
 TABLE of Mean New Moons in March, with the Mean Anomalies of the Sun and Moon; and the Sun's 
 
 Mean Distance from the Moon's Node. 
 
 To make Table I. Calculate the time of mean new moon in March, for the year with which the 
 Table is to begin, by the rule, Prob. XIV. or by any other method. Calculate also the sun and moon's 
 mean anomalies for that time, and the sun's mean distance from the moon's node, from the Tables of 
 mean motion, and write them down in order. 
 
 For the following years. If the new moon falls after the llth day of March, add 12 lunations to the 
 time for the former year ; and if the next is a common year, subtract 365 days from the sum ; the remain- 
 der is the time of mean new moon in March the next year; but if the next be a leap year, subtract 366 
 days. If the mean new moon falls before the llth of March, add 13 mean lunations, and subtract 365 or 
 366 days from the sum, according as the next is a common or bissextile year. 
 
 Calculate also the sun and moon's mean anomalies, with the sun's mean distance from the moon's 
 node for 12 or 13 lunations, and add them to those of the former year ; and the numbers for the second 
 year in the Table will be known. Proceed in the same manner for every succeeding year until the Table 
 is completed. 
 
 TABLE II. Contains 13^ Mean Lunations, with the Anomalies, and the Sun's Mean Distance from the 
 
 Moon's Node. 
 
 The numbers are computed from the Tables of mean motion thus: Take the mean anomalies of the 
 sun and moon out of the Tables for 29 days 12 hours 44m. 3s. ; take also the mean motion of the sun, and 
 of the moon's node, out of their proper Tables for the same time ; their sum is the sun's mean distance 
 from the moon's node. Having found the numbers for one lunation, multiply them by 2, 3, 4, &c. until 
 there are 13 products, and divide the numbers for one lunation by 2 for the half lunation. 
 
 By these Fables the times of mean syzygies for any month of a given year, within the limits of Table 
 I. may be found. 
 
 The Tables of equations for reducing the mean to the true times of new and full moons, in the fol- 
 lowing pages, depend on the mean anomalies of the sun and moon, with the sun's mean distance from the 
 moon's node, and are expressed in time for the conveniency of calculation. The method of applying 
 them is given in Problem XVI. 
 
 Construction and Use of Logistical Logarithms. 
 
 Logistical Logarithms are artificial numbers deduced from common logarithms, and made for minutes 
 and seconds either of degrees or time, used in working proportions, wherein some or all of the given 
 terms arc sexagesimals. 
 
 The numbers which compose the Table of logistical logarithms are regulated by its extent. For if the 
 Table contains no more than the minutes and seconds in 1, or 3600", the logistical logarithm of 1" is the 
 logarithm of 3600", viz. 3.5563 ; and all the following numbers are the differences between 3.5563 and the 
 logarithms of the several numbers of seconds from 1 to 3600". Therefore to calculate the logistical loga- 
 rithm of any quantity less than 1, subtract the logarithm of the given number of seconds from 3.5563 ; 
 the remaind'er is the logistical logarithm required. 
 
 EXAM. Let it be required to find the logistical logarithm of 10', or 600". 
 From the common log. of 3600 = 3.5563 
 Subtract the common lox. of 600 =2.7781 
 
 There remains the logistical log. of 10' = 0.7782 
 
 But if the Table is to contain more than 1, such as in Dr. Maskelyne's Proportional Logarithms, 
 which extends to 5, or 10300"; then, in such a Table, the proportional or logistical logarithm of 1" is the 
 logarithm of 10800, viz. 4.0334 ; and all the rest are the differences between 4.0334 and the logarithms of 
 the several numbers of seconds from 1 to 10800". 
 
EXPLANATION OF THE TABLES. 335 
 
 EXAM. Let it be required to find the logistical or proportional log. of 10' or 600". 
 
 From the log. of 10800 = 4.0334 
 
 Subtract the log. of - 600 = 2.7781 
 
 Remains the prop. log. of 10' 1.2553 
 
 Hence there may be as many different systems of logistical or proportional logarithms as any one 
 chooses to assume -different extents of the Table. Our Table extends only from 1" to 1, which is suffi- 
 cient for common use. Logistical logarithms consist of four figures besides the index. 
 
 The numbers on the head of the Table are either degrees or minutes; and those in the left hand col- 
 umn, on the side, are minutes or seconds ; but if the numbers on the head be hours or minutes, those in 
 the left hand column, on the side, will be minutes or seconds of time. Therefore these numbers change 
 their denomination as occasion requires. The second line on the head of the Table are the numbers of 
 seconds in the minutes which stand over them. 
 
 To take the logistical logarithm of any number of minutes and seconds out of the Table, the given 
 number must be within the limits of the Table. 
 
 Find the minutes on the head of the Table, and the seconds in the left hand column on the side, and 
 under the minutes and opposite to the seconds stands the logistical logarithm required. 
 
 EXAMPLES. 
 
 / n 
 
 ^ 
 
 Giren numbers 40 Logist log. 1.9542 
 
 1 10 1.7112 
 
 7 40 8935 
 
 9 15 8120 
 
 50 792 
 
 50 37 739 
 
 A logistical logarithm being given, to find the number of minutes and seconds answering. 
 
 If the given logarithm be found in the Table, the minutes are on the head of the column and the 
 seconds on the side ; but if it be not found exactly in the Table, take the next greater than it, and the 
 minutes and seconds answering are found on the head and side of the table, as before. 
 
 EXAMPLES. 
 
 L. Log. Values. 
 
 i it 
 
 1.7112 1 !0 
 
 9823 6 15 
 
 6746 12 41 
 
 5175 18 13 
 
 . When the logistical logarithm of any number less than 3600 is required, find it, or the nearest less, 
 n the second line on the head of the Table; and below the number on the head, and opposite to its ex- 
 cess on the side, stands the logistical logarithm. Thus the logistical logarithm of 1276 is 4505. 
 
 Logistical logaiithms are used in finding a fourth proportional, when some or all of the given terms, 
 are minutes and seconds either of motion or time ; and the method of operation is the same as in work- 
 ing with common logarithms, that is, by adding the logarithms of the second and third terms, and sub- 
 tracting the logarithm of the first term from the sum, the remainder is the logarithm of the fourth term, or 
 answer. 
 
 EXAM. 1. Suppose the sun's hourly motion was 2' 27", and it were required to find bis motion in 
 47' 10". The answer is 1' 55''. 
 
 As GOm. 
 
 To 2' 27" 13890 
 
 So 47m. 10s. 1045 
 
 To 1' 55" 14935 
 
336 APPENDIX TO THE ASTRONOMY. 
 
 2. Suppose the hourly motion of the moon from the sun is 32' 8", and the moon is 58' 12" behind 
 the sun. In what time will the moon overtake the sun ? Jlns. Ih. 4Sm. 40s. 
 
 A". B. In such questions, when the fourth proportional would exceed 60m. which is beyond the limit 
 of the Table, take f , j, or i, of the third term, and multiply the fourth term, when found, by 2, 5, or 4. 
 
 As 32' 8" 2712 
 
 To 60m. 
 
 So 29' 6" 3143 
 
 To 54m. 20s. 431 
 
 2 
 
 Doubled is Ih. 48m. 40s. the answer. 
 
 In like manner, when two of the given terms exceed the limits of the Table, divide all the terms by 
 some number, such as 2, 3, or 4, and multiply the answer by the same number which divided the given 
 terms. Half of each term. 
 
 Suppose we have, As 76' 34" 38' 17" 1951 
 
 Is to 24h. 12h. 6990 
 
 So is 64' 20" sa' 10" 2707 
 
 To 20h. 10m. nearly lOh. 5m. 9697 
 
 Doubled 20h. 10. 7746 
 
 PROBLEMS, 
 
 Showing the Use and Application of the Tables, and Projection of Eclipses. 
 
 PROB. I. The Longitudes of two Places, and the Time at one of them being given, to find the correspond- 
 ing Time at the other. 
 
 Reduce the difference of the longitudes to time, by allowing at the rate of 15 to an hour, and add it to 
 the given time for a place toward the east; but subtract it for a place toward the west. 
 
 EXAM. 1. Suppose the time at Greenwich to be 6h. 7' 8" P. M. required the corresponding time at 
 Cambridge, in longitude 71 7' west. 
 
 h. ' " 
 
 Time at Greenwich 6 7 8 P. M. 
 
 Dili', long. 71 7' = 4 44 28 
 
 Time at Cambridge 1 22 40 P. M. 
 
 EXAM. 2. The time at Cambridge being 7h. 43' 57" A. M. required the corresponding time at Green- 
 wich. 
 
 h. ' " 
 
 Time at Cambridge 7 43 57 A. M. 
 
 Diff. long. 71 7' = + 4 44 28 
 
 Time at Greenwich 28 25 P. M. 
 
 EXAM. 3. The time at Hamburg, in longitude 10 38' E. being 6h. 30' 40" A. M. required the cor- 
 responding time at Cambridge, in longitude 71 7' W. 
 
 h. ' " 
 
 Time at Hamburgh 6 30 40 A. M. 
 
 Diff. long. 81 45' 5 27 
 
 Time at Cambridge 1 3 40 A. M. 
 
USE AND APPLICATION OF THE TABLES, &c. 337 
 
 PROB. II. To calculate the true Longitude of the Sun for any given Time and Place by the Tables. 
 
 When the given time is apparent, reduce it to mean time, and if the given place be not in the me- 
 ridian of the Tables, that is, Greenwich, reduce it to that meridian by Proh. I. and then write down the 
 years, months, days, hours, &c. under one another in a column on the left hand. 
 
 Take out of the Tables the sun's mean longitude and anomaly, answering to each part of the time, and 
 write them down on the right hand of the former numbers ; then add them as they stand in their several 
 columns, rejecting 12 signs, or any multiple thereof, and the sum will be the mean longitude of the sun 
 for the given time. 
 
 To reduce the mean longitude to the true. Enter the Table with the sun's mean anomaly, and take 
 out the equation of the centre, making proportion for the odd minutes, &c. and if the sign of the sun's 
 mean anamoly be at the head of the Table, subtract the equation from the mean longitude ; but if it be at 
 the foot of the Table, add it to the mean longitude, and the true longitude of the sun will be had. 
 
 EXAM. I. Required the sun's longitude 1796, Feb. 19, llh. 21' 50" in the forenoon, apparent time, 
 in the meridian of Edinburgh. 
 
 D. h. ' " 
 
 Given time February - - 18 23 21 SO 
 Edinburgh meridian W. + 12 25 
 
 Equation of time - + 14 14 
 
 Mean time at Greenwich - 18 23 43 
 
 1796 
 
 Feb. 18. bissex. 
 23h. 
 48' 
 
 9" - 10 29 9 10 7 19 40 51 
 
 Equation of the centre + 1 28 SO 
 
 M. Long. 
 
 M. Anom. 
 
 S. ' " 
 
 S. ' " 
 
 9 10 51 52 
 
 6 1 23 42 
 
 1 17 18 40 
 
 1 17 18 31 
 
 56 40 
 
 5G 40 
 
 1 58 
 
 1 58 
 
 Sun's true longitude 11 03740 
 
 2. Required the sun's longitude 1795, June 22, at noon, apparent time, at Greenwich, 
 
 h. ' " 
 
 Mean time - 000 
 
 Equation of time - - + 1 28 
 
 1 28 
 
 M. Lone;. M. Anom. 
 
 S. * " S. ' " 
 
 1795 - 9 10 7 2 6 39 58 
 
 June 22, - 5 20 S 1 1 5 20 30 SO 
 
 Oh. 1' - 2 2 
 
 28" - 1 1 
 
 3 38 6 11 21 10 31 
 
 Equation uf the centre + 17 23 
 
 Sun's true longitude 3 55 29 
 
 PROB. III. The Mean Anomaly of the Sun being given, to find the Logarithm of his Distance from, the 
 
 Earth. 
 
 With the mean anomaly of the sun take out the logarithm of his distance from the earth from the 
 Table. 43 
 
338 APPENDIX TO THE ASTRONOMY. 
 
 .EXAM. Let the sun's mean anomaly be 7s. 19 40' 51", what is the logarithm of his distance from the 
 earth ? 
 
 For 7s. 19 the log. is - 4.995224 
 
 For 40' 51" the part is + 69 
 
 Answer - 4.995293 
 
 PROB. IV. To find the Apparent Semidiameter, and also the Hourly Motion of the Sun for any given 
 
 Time. 
 
 Calculate tho sun's mean anomaly for the given time, and with the mean anomaly of the sun take out 
 his semkliameter and hourly motion From the Table. 
 
 EXAM. Required the sun's apparent semidiameter and hourly motion 1796, February 19th, llh. 21' 
 in the forenoon. 
 
 The sun's mean anomaly at that time is 7s. 19 42', which gives his semidiameter 16' 13" and his 
 hourly motion 2' 31". 
 
 PROB. V. The Longitude of the Sun, or any Point of the Ecliptic, being given, to find the correspond- 
 ing IJeelination. 
 
 With the given longitude take the declination out of the Table. 
 
 EXAM. Let the sun's longitude be 11s. 37' 40", what is his declination? Ans. 11" 15' 47" south. 
 To longitude 11s. the declination is li 29' 5" S. decreasing; the difference between this and the 
 next is 21' 12" ; Then, 
 
 ' " L.L. 
 
 As 1000 o ' " 
 
 To 21 12 4518 From 1 1 29 5 
 
 So is 37 40 2022 Subtract 13 18 
 
 To 13 18 6540 Declinat. 11 15 47 S. 
 
 PROB. VI. The Time of the Tear and the Sun's Declination being given, to find his Longitude ; or the 
 Declination of a Star, or of any Point of the Ecliptic, being given, to find its Longitude. 
 
 With the given declination take the longitude out of the same Table, as in the preceding problem. 
 Thus, take the declination in the Table nearest less than that given, and write down the corresponding 
 longitude; marking whether the declination be increasing or decreasing; then subtract the declination 
 found in the Table from the next greater] the remainder is the difference of declination for 1 of lon- 
 gitude. Subtract also the declination found in the Table from that given ; and then, as the difference 
 of declination for 1 of longitude is to 1, so is the difference between the given declination and that 
 found in the Table to a part, to be applied to the longitude already' found ; the result is the true 
 longitude. 
 
 EXAM. February 19, 1796, the sun'a declination is 1 1 15' 47" S. what is his longitude ? Jlns. 1 Is. 
 ffl 37' 39". 
 
 1st. To declin. 1 1 7' 53" S. the long, is 11s. 1, the declination decreasing. 
 
 o // 
 
 Cd. From the given declin. - 11 15 47 
 
 Subtract the next less in the Table - 11 7 53 
 
 Difference - 7 54 
 
 3d. Also from the next greater 11 29 5 
 
 Subtract the next less in the Table - 1 1 7 53 
 
 Diff. declin. for 1 long. 21 12 
 
USE AND APPLICATION OF THE TABLES, &c. 339 
 
 o / a 
 
 4th. As - 21 12, 
 
 To - - 100 
 
 So is - - - 7 54 
 
 To - - - 22 21 4287 
 
 To be subtracted. 
 
 s. a I II 
 
 From the longitude found 11 100 
 
 Subtract the prop, part 22 2 1 
 
 There remains the true longitude 11 37 39 
 
 PROB. VII. To calculate the true Longitude of the Moon for any given Time by the Tables. 
 
 Calculate the true longitude of the sun and his mean anomaly for the given time by Prob II. 
 
 From the Tables of the moon's mean motion, take out the mean longitude, anomaly, and ascending 
 node, for the given year; under which write down the mean motions for the month, day, hours, &c. 
 
 Add the numbers in the columns of the mean longitude and anomaly, rejecting 12 signs or any mul- 
 tiples thereof when they occur; but from the longitude of the node for the given year subtract the sum of 
 all the numbers below it, borrowing 12 signs when necessary; and thus the moon's mean longitude, mean, 
 anomaly, and the mean longitude of her node, will be obtained. 
 
 For the Arguments of the Equations, 
 
 The sun's mean anomaly is the first argument. Subtract the true longitude of the sun from the mean 
 longitude of the moon, the remainder is the mean distance of the moon from the sun, of which take the 
 double. 
 
 To and from the double distance of the moon from the sun add and subtract the first argument, the 
 sum and remainder are the second and third arguments. 
 
 Add and subtract the moon's mean anomaly to and from the double distance of the moon from the 
 sun for the fourth and fifth arguments. 
 
 To and from the fifth argument add and subtract the first argument, and the sixth and seventh argu- 
 ments are found. 
 
 Subtract the first argument fronfthe moon's mean anomaly, the remainder is the eighth argument. 
 
 Subtract (lie moon's mean anomaly from the moon's mean distance from the sun, the remainder is the 
 ninth argument. 
 
 Subtract the true longitude of the sun from the mean longitude of the moon's node, and the remainder 
 is the ten tli argument. 
 
 With these arguments take the several equations out of the Tables, and find their sum. 
 
 With the mean anomaly of the sun take out of the Tables the equations of the node and of the moon's 
 mean anomaly. Apply the equation to the longitude of the node, and its equated longitude will be 
 known. Correct the mean anomaly of the moon by its own equation, and also by the sum of the ten 
 former equations, and the correct anomaly of the moon will be had ; which is the eleventh argument, with 
 which take out of the Table the equation of the centre. Correct the mean longitude of the moon both by 
 the sum of the ten former equations and by the equation of the centre, and the equated longitude of the 
 moon will be known. 
 
 Subtract the sun's true longitude from the equated longitude of the moon, the remainder is the twelfth 
 argument, with which take the variation out of the Table, and apply it to the equated longitude of the 
 moon. 
 
 Subtract the correct longitude of the node from the longitude of the moon twelve times corrected, and 
 from double of the remainder subtract the moon's correct anamoly, the remainder is the 13th argument; 
 with which take the thirteenth equation from the Table and apply it to the moon's longitude last found, 
 which will give the moon's longitude in her orbit. 
 
 From the moon's longitude in her orbit subtract the correct longitude of the node, the remainder is the 
 fourteenth argument ; with which take out of the Table the reduction of the moon's place in her orbit to 
 the ecliptic. And, lastly, with the mean longitude of the moon's node take the equation of the equinoxes 
 
310 APPENDIX TO THE ASTRONOMY. 
 
 from the Table ; and apply both the reduction and the equation of the equinoxes to the moon's longitude 
 in her orbit, according to their signs, and the true ecliptic longitude of the moon will be found. 
 
 EXAM. It is required to find the moon's longitude 1795, June 22, at noon, apparent time in the meri- 
 dian of Greenwich. 
 
 The equation of time is + 1' 28", therefore the mean time is Oh. 1' 28". 
 Sun's longitude 3s. 55' 29". Mean anomaly, 1 Is. 21 10' 31". 
 
 Equations. 
 
 i) s m. jjonij. 
 
 
 46 
 
 1795 
 June 22. - 
 Oh. 1m. 
 28s. 
 
 Sum of 10 eqs. 
 
 Equat. centre 
 
 5 's equat. long. 
 Variation 
 
 13th 
 
 D 's orbit long. 
 Reduction 
 
 Equinoxes 
 "^ 's eclin. !ono-_ 
 
 s , o a 
 
 1 5 37 57 
 3 29 31 
 33 
 15 
 
 6 13 
 3 10 
 
 36 29 
 14 35 
 33 
 15 
 
 4 9 57 43 
 
 9 9 40 
 
 4 48 6 
 1 34 
 
 4 46 39 
 = Arg. 1 1th. 
 
 5 5 9 45 
 -f 19 
 
 9 23 
 
 51 52 
 3 18 
 
 5 5 28 45 
 4- 5 35 20 
 
 9 23 
 
 + 
 
 48 34 
 19 
 
 5 .11 4 5 
 
 + 20 48 
 
 9 24 
 
 7 34 
 
 5 11 24 53 
 + 46 
 
 5 11 25 39 
 
 6 43 
 
 5 11 18 56 
 
 15 
 
 .<; 11 1841 
 
 50 
 20 36 
 
 1 14 
 51 
 
 + 23 31 
 4 31 
 
 + 19 
 
 1 42 
 48 
 52 
 
 14 
 19 
 36 
 
 4 31 
 
 Arguments found. 
 
 Arguments found. 
 
 Arguments found. 
 
 D 
 O 
 
 3 from O 
 
 Doubled 
 O Anom. 
 
 2d = 
 Sd- 
 
 3> Anom. 
 4th - 
 
 o 5 9 45 
 
 3 55 29 
 
 6th = 
 7th 
 
 S. ' " 
 
 6 5 47 11 
 
 12th = . 
 
 3) 's long, equat. 
 ft 
 
 Doubled 
 j) 's Anom. 
 
 ISth = - 
 
 5 in orbit 
 ft 
 
 14th 
 
 s. ' 
 3 55 29 
 
 6 23 26 9 
 
 2 10 8 36 
 
 2 4 14 16 
 
 8th 
 
 9 23 51 52 
 11 21 10 31 
 
 5 11 4 5 
 4 46 39 
 
 4 8 28 32 
 11 21 10 31 
 
 10 2 41 21 
 2 4 14 16 
 9 23 51 52 
 
 1 10 17 26 
 
 3 29 39 3 
 
 9th 
 
 2 20 34 52 
 904 7 "?d. 
 
 4 17 18 1 
 
 4 10 32 24 
 
 
 4 8 28 32 
 9 23 51 52 
 
 ft 
 
 O 
 
 10th 
 
 4 26 27 18 
 
 4 48 3 
 . __3 55 29 
 
 5 11 25 39 
 
 2 2 20 24 
 
 5th = 
 
 n oQ KO <\A 
 
 
 6 14 36 40 
 11 21 10 31 
 
 D'seq 
 
 
 1 10 39 
 
 . long. - 51145 
 
 
USE AND APPLICATION OF THE TABLES, &c. 
 
 PROB. VIII. The Moon's Longitude, Anomaly, and Node, being given, to find her Latitude by the Tables. 
 
 Subtract the corrected longitude of the moon's node from the moon's longitude in her orbit, the 
 remainder is the first argument of latitude. 
 
 From the moon's longitude in her orbit subtract the sun's longitude ; and from double of the remain- 
 der subtract the first argument of latitude, the remainder is the second argument. 
 
 Subtract the moon's mean anomaly from the first argument of latitude, and the remainder is the 
 third argument. 
 
 From the third argument subtract the moon's mean anomaly for the fourth argument ; also subtract 
 the moon's mean anomaly from the fourth argument, and the remainder is the fifth argument. 
 
 With these arguments take out the numbers with their proper signs from the Tables, and take the 
 sum of those which are affirmative, and also the sum of the negative ; the difference of these sums, with 
 the sign of the greater, will be the latitude of the moon ; and will be north if marked with the sign +, 
 but south if marked with the sign . 
 
 EXAM. It is required to find the moon's latitude when her longitude in her orbit is 5s. 1 1 25' 39", 
 mean anomaly 9s. 23 51' 52", and node 4s. 46' 39". The sun's longitude is 3s. 55' 29". 
 
 Arguments found. 
 
 Arguments found. 
 
 Equations. 
 
 } 's orbit Ion. 
 $L 
 
 Arg. 1st = 
 
 3 ... 
 
 O - 
 
 Doubled 
 1st 
 
 s. ' " 
 5 11 25 39 
 4 36 39 
 
 d = 
 
 1st 
 
 J 's Anom. 
 
 3d = 
 4th = 
 5th 
 
 S. ' " 
 3 10 11 20 
 
 1 
 2 
 3 
 4 
 5 
 
 1 It 
 
 3 21 43 
 8 40 
 
 // 
 
 16 
 2 
 16 
 
 1 10 49 
 9 23 51 52 
 
 1 10 49 
 
 5 11 25 3V 
 3 35 29 
 
 3 16 57 8 
 9 23 51 52 
 
 3 3D 23 
 34 
 
 34 
 
 i*slat.is329'49"N. 
 
 2 10 SO 10 
 
 5 23 5 16 
 9 23 51 52 
 
 4 21 20 
 1 10 49 
 
 
 + 3 29 49 
 Am, The moor 
 
 7 29 13 24 
 
 PROB. IX. Tlie JMbon's Longitude with the Arguments for finding it, being knotvn, to find the Moon's 
 Equatorial Parallax or the Horizontal Parallax of the Moon for any Place on the Equator. 
 
 With the 5th, llth, and 12th arguments of longitude, take as many numbers from the Tables, with 
 their proper signs ; and from the sum of the affirmative numbers subtract the sum of the negative, the 
 remainder is the moon's equatorial parallax, or the horizontal parallax of the moon to a place on the 
 equator. 
 
 EXAM. Let the moon's longitude be 5s. 11 18' 41", and the arguments as below ; required her equa- 
 torial parallax. Jim. 56' 5". 
 
 5th Arg. 6 14 36 40 
 
 1 1th 9 24 7 34 
 
 12th 2 10 8 36 
 
 Moon's equatorial parallax 
 
 36 
 55 49 
 
 20 
 
 + 56 25 
 20 
 
 56 5 
 
312 
 
 APPENDIX TO THE ASTRONOMY. 
 
 PROB. X. The Equatorial Parallax of the Moon being given and the Latitude of a Place, to find the 
 Horizontal Parallax of the Moon at that Place ; and to reduce the Apparent Latitude of the Place 
 to the Centre of the Earth, supposing the Earth is an Oblate Speruid. 
 
 Enter the Table, with the given apparent latitude of the place on the side, and the equatorial parallax 
 at the top, and making proportion, if necessary, find the reduction of parallax j which subtracted from tke 
 equatorial parallax, leaves the horizontal parallax required. 
 
 With the apparent latitude of the place take the reduction of latitude out of the same Table ; which 
 being subtracted from the apparent latitude, leaves the latitude reduced. 
 
 N. B. The horizontal parallax and latitude thus found are what should be employed in computing 
 the moon's parallaxes in longitude, latitude, and declination ; which will now be performed by the com- 
 mon rules, founded on the supposition of the earth's being a sphere. 
 
 EXAM. Let the equatorial parallax be 56' 5", and the apparent latitude of the place 51 28' 40" : the 
 moon's horizontal parallax and the reduced latitude of the place are required. 
 
 Equatorial parallax 56 5 
 Reduction 9 
 
 Latitude 
 Reduction. 
 
 51 28 40 
 14 42 
 
 Horizontal parallax 5556 Reduced lat. 51 1358 
 
 PROB. XI. The Equatorial Parallax being given, to find the Moon's Horizontal Diameter. 
 Enter the Table, with the moon's equatorial parallax, and take out her horizontal diameter. 
 EXAM. The moon's equatorial parallax being 56' 5", the diameter is 30' 34". 
 
 .PROB. XII. To find the Moon's Hourly Motion in Longitude; her Longitude in her OMt, icith the 
 
 Arguments for finding it, being known. 
 
 With the 5th, llth, and 12th arguments of longitude, take out the numbers from the Tables, the sum 
 of these, regard being had to their signs, is the moon's hourly motion in longitude. 
 
 EXAM. Let the moon's longitude bs 5s. 11 25' 39", and the arguments as below ; required her hourly- 
 motion in longitude. 
 
 
 S. ' " 
 
 + , n 
 
 
 
 5th Arg. 
 
 6 14 36 40 
 
 4 
 
 
 nth 
 
 9 24 73-1 
 
 31 20 
 
 
 12th 
 
 2 10 8 3C 
 
 
 31 
 
 
 
 + 32 1 
 
 
 
 
 31 
 
 
 Answer 
 
 31 30 
 
 PROB. XIII. To find the. Moon's Hourly Motion in Latitude, the Arguments for finding her Latitude 
 
 being known. 
 
 With the first and second arguments of latitude, take two numbers out of the Tables, their sum, if 
 they have the same sign, or their difference, if they have contrary signs, is the moon's hourly motion in 
 latitude ; which tends to the north if it has the sign +, but to the south if it has the sign . 
 
 EXAM. It is required to find the moon's hourly motion in latitude, the arguments of latitude being. 
 
USE AND APPLICATION OF THE TABLES, &c. 343 
 
 s. ' 
 
 1st Arg. 1 10 39 
 
 2d 3 10 21 21 
 
 Answer 
 
 14 tending north. 
 
 PROB. XIV. To find the Time of the Mean Syxygies in any given Year and Month. 
 
 From the Tables of the mean motion of the moon from the sun take out the motions for the given year 
 and month ; add them, and subtract the sum from 12 signs; then, if the time of mean conjunction or new 
 moon be sought, the remainder, or the nearest less than it, being found in the Table of days, will give the 
 day of mean new moon ; and after subtracting the number found in the Table, the remainder or the next 
 less is to be found in the Table of hours, which will give the hour of the day when it is mean new moon ; 
 aud after another subtraction, the remainder is to be found in the Table of minutes for the minute of 
 mean new moon, and the nest remainder wdl give the seconds. But if the time of mean opposition or 
 full moon be sought, add 6 signs to the first remainder, and the sum being found in the Table of days, 
 will give the day of mean full moon ; the hours and minutes are to be found as before. 
 
 EXAM. 1. Required the time of mean new moon in January, K97. *ins. Jan. 27d. 7h. 40m. 21s. 
 
 S. ' " 
 
 1797 - 26 57 10 ' 
 
 12 
 
 Remains 11 3 2 50 
 
 27 days - - 10 29 9 
 
 3 53 50 
 7 hours - - 3 S3 20 
 
 20 30 
 40 minutes - - 20 19 5'" 
 
 21 seconds ... 10 55 
 
 In this operation the Tables give the moon's mean distance from the sun on the 1st day of January Os. 
 26 57' 10", which is subtracted from 12 signs to find how much the moon wants of a conjunction with 
 the sun ; the remainder shows that she wants 11s. 3 2' 50". In the Table of days it is found that she 
 moves over 10s. 29 9' in 27 days ; therefore the mean new moon happens on the 27th day of January. 
 
 Subtract 10s. 29 9' from 11s. 3 2' 50", there remains 3 53' 50", which in the Table of hours gives 7 
 hours; and subtracting 3 33' 20", there remains 20' 30", which gives 40 minutes; and by subtracting 
 again, there remains 10" 55'", and this gives 21 seconds. 
 
 2. It is required to find the time of mean full moon in January 1797. Jlns. Jan. 12d. 13h. 18m. 18s. 
 
 go i " 
 
 1797 26 57 10 
 
 12 
 
 11 3 2 50 
 Add 6000 
 
 5 3 2 50 
 12 days 4 26 17 20 
 
 6 45 30 
 13 hours 6 36 12 
 
 9 18 
 18 minutes 9 8 35'' 
 
 18 seconds 9 25 
 
APPENDIX TO THE ASTRONOMY. 
 
 By this problem the times of mean new and full moon may be calculated for every month in the year. 
 or any longer time, with very little trouble; for having found the time of mean conjunction and opposi- 
 tion in the month of January, add to these times a mean lunation, viz. 29d. 12h. 44m. 3s. continually, 
 rejecting the days in the month wherein the mean new or full moon is required, and the times will be 
 known. 
 
 KXAJI. Let it he required to find the mean new and full moons in every month of the year 1797. 
 
 Mean fu 
 
 11 Moon. 
 
 Mean 
 
 new Moon. 
 
 January 
 One lunation 
 
 February 
 March 
 April 
 May 
 June 
 July 
 August 
 September 
 October 
 November 
 December 
 
 D. h. in. ,. 
 
 12 13 18 18 
 + 29 12 44 3 
 
 January 
 One lunation 
 
 February 
 March 
 April 
 May 
 June 
 July 
 August 
 September 
 October 
 November 
 December 
 
 O. h. m. >. 
 
 27 7 40 21 
 4-29 12 44 3 
 
 11 2 2 21 
 
 25 -M a4 24 
 
 12 14 40 24 
 
 27 9 8 27 
 
 11 3 30 27 
 
 25 21 52 30 
 
 10 16 14 30 
 
 25 10 36 33 
 
 9 4 58 33 
 
 23 23 20 36 
 
 8 17 42 36 
 
 23 12 4 39 
 
 7 6 2b 39 
 
 32 48 42 
 
 5 19 10 4s 
 
 20 13 ^2 45 
 
 5 7 54 45 
 
 20 2 16 48 
 
 3 20 38 4b 
 
 18 15 51 
 
 3 9 22 5i 
 
 18 3 44 54 
 
 J\*ote. The mean and true syzygies never happen at the same time, except when the moon is in or 
 very near her apogee or perigee ; for in these points many of her inequalities either vanish or are very 
 small ; when the moon is in any other point of her orbit there is some interval of time between the mean 
 and true conjunctions or oppositions. The greatest is about 14 hours. 
 
 PROB. XV. The Time of Mean Conjunction or JVew Moon being given, to find the true Time. 
 
 Calculate the longitudes of both sun and moon for the time of mean conjunction ; (Prob. II. VII.) 
 and if they are equal to one another, the mean and true conjunctions happen at the same time ; but if 
 they differ, subtract the least from the greatest. Find the hourly motions of both sun and moon at the 
 time, (Prob. IV. XII.) and subtract the sun's hourly motion from the moon's, the remainder is the hourly 
 motion of the moon from the sun ; and then, 
 
 As the moon's hourly motion from the sun 
 Is to one hour or 60 minutes ; 
 
 So is the difference between the sun and moon's long. 
 To the time between the mean and true conjunction. 
 
 If the moon's longitude be less than the sun's, the interval is to be added to the time of mean conjunc- 
 tion ; but if it be greater, subtract the interval ; the sum or remainder is the time of true conjunction. 
 
 When it is required to find the time of true conjunction very accurately, the sun and moon's longitude 
 must be calculated again for the time found ; and their difference, if there be any, turned into time, and 
 applied to the time last found, will give the true time of conjunction. 
 
 If the difference be great, it will be necessary to renew the operation again. 
 
 EXA.M. It is required to find the time of true conjunction in January 1797; the time of mean con- 
 junction or new moon being January 27th, 7h. 40m. 21s. in the meridian of Greenwich. An$. Jan. 27d. 
 ISh. 40m. 
 
USE AND APPLICATION OF THE TABLES, &c. 345 
 
 The sun's longitude then is 10s. 8 28' 38", and his mean anomaly 6s. 28 3' 51". 
 For the Moon's Longitude. 
 
 ro. Long. 
 
 Apogee. 
 
 8 
 
 Arguments. 
 
 Equations. 
 
 s. ' " 
 1797 10 7 34 43 
 Jan. 27th 11 25 45 46 
 7h. S 50 35 
 40 m. 21 57 
 21s. 1 1 
 
 S. ' " 
 24 6 53 
 11 22 45 17 
 3 48 38 
 21 46 
 11 
 
 s. ' " 
 3 I 15 6 
 
 1 
 
 2 
 S 
 4 
 5 
 f 
 7 
 
 8 
 
 9 
 10 
 
 S. ' " 
 6 28 3 51 
 5 26 12 59 
 5 5 17 
 19 11 5& 
 11 7 7 6 
 6 5 10 14 
 4 8 2 32 
 5 22 58 54 
 11 8 1 49 
 4 21 19 40 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 
 Sum 
 
 + 
 / // 
 
 19 
 30 53 
 
 37 
 5 
 31 
 
 / // 
 
 5 15 
 
 4 
 38 
 
 11 
 59 
 
 1 25 47 
 56 
 S 
 
 10 7 33 12 
 10 equations + 25 18 
 
 21 2 45 
 10 25 
 
 2 29 48 18 
 4 25 
 
 10 7 58 30 
 Equat. of centre 297 
 
 20 52 20 
 +25 18 
 
 2 29 43 53 
 
 10 5 49 23 
 Variation 3 1 o 
 
 21 17 38 
 
 12 
 13 
 
 11 27 20 45 
 1 20 46 52 
 
 5 
 
 = llth arg. 
 
 + 32 25 
 77 
 
 + 25 18 
 
 7 7 
 
 10 5 46 8 
 13th 1 5 
 
 D 's orb. long. 10 5 45 3 
 O's long. 10 8 28 38 
 
 Short of conj. 
 
 2 43 35 
 
 Moon's hourly motion 
 Sun's ditto 
 
 Hourly motion 3 from O 
 As 27' 14" 
 To Ih. 
 So 2 43' 35" 
 To 6h. Om. 24s. 
 
 Mean conjunction January 
 
 Interval add 
 
 29 47 
 
 2 S3 
 
 27 14 
 
 D. li. m. s. 
 27 7 40 21 
 + 6 24 
 
 Jlns. New Moon January 27 13 40 45 
 
 The time of true opposition or full moon is found in the same manner as the time of conjunction. The 
 operation is to be continued until the difference of the sun and moon's longitude be 6 signs. 
 
 The rules here given for calculating the times of new and full moon are always good ; but the tedious- 
 ness of the operation has given occasion to the invention of other Tables, whereby the problem may be 
 solved with less trouble. The best of these were published among Dr. Halley's Astronomical Table's in 
 the year 1749, but not as his invention. The ingenious Mr. James Ferguson having adapted them to the 
 Gregorian style, and made some alterations in the arrangement, printed them in his Astronomy about 
 the year i760. Such of them as were judged proper for this work, are to be found in the Tables of mean 
 new moons, &c. extended to the year 1821. 
 
 These Tables contain the times of mean conjunction or new moon in March for every year, which is 
 made the first month with the mean anomalies of the sun and moon, and also the sun's mean distance 
 from the moon's node at the time, with equations for reducing the mean to the true time of conjunction 
 or opposition, whereby these times may be found for any month of a given year within the limits of the 
 Table. 
 
 44 
 
316 
 
 APPENDIX TO THE ASTRONOMY. 
 
 PROB. XVI. To calculate the Time of Neiv or Full Moon for any Month of u given Year by the Tables. 
 
 1st. For the Month of March. 
 
 Write down the time of mean new moon in March for the given year, with the mean anomalies of the 
 sun and moon, and also the sun's mean distance from the moon's ascending node, out of the Table. 
 
 When the time of Full Moon in March is required. 
 
 If the new moon happens before the 15th of the month, add half a lunation, with the anomalies, &c. 
 to the former numbers for new moon, the sum is the time of full moon ; but if it happens after the 15th, 
 subtract half a lunation with the anomalies from the numbers for new moon, and the time of mean, full 
 moon in March will be known. 
 
 2d. For any Month after March. 
 
 When the time of mean new or full moon is required in any month after March, take out the numbers 
 for March as before, and under them write down as many lunations with their anomalies as the given 
 month is after March; and by the sum of these, the time of mean new or full moon may be known, 
 together with the mean anomalies and the sun's distance from the moon's node ; which are the arguments 
 for finding the several equations, to reduce the time of the mean syzygies to the true. 
 
 With the sum of the days enter the Table of days, under the given month ; and opposite to that num- 
 ber in the left hand column is the day or the mean syzygies; but if the sum be less than any of those 
 under the given month, add a lunation with the anomalies to the former numbers, and then enter the 
 Table with the sum under the given month, and in the left hand column is the day of the month required. 
 
 The time of mean syzygy being known, to find the true by the Tables. 
 
 1. With the sun's mean anomaly enter the Table, and take out the first equation, (making proportion 
 for the odd minutes, &c.) and apply it to the time of the mean syzygy according to its sigo. 
 
 2. With the sun's mean anomaly take the equation of the moon's mean anomaly out of the Table, and 
 apply it according to its sign, and the moon's equated anomaly will be known. 
 
 3. With the moon's equated anomaly take the second equation out of the Table, which being applied 
 to the former time, according to its sign, the result will be the time of the syzygy very nearly. 
 
 4. Subtract the moon's equated anomaly from the sun's mean anomaly, and witli the remainder take 
 the third equation out of the Table, and apply it to the former equated time according to its sign. 
 
 Lastly, with the sun's mean distance from the moon's node take the fourth equation out of the Table, 
 and apply it to the last found equated time, according to its sign, and the result is the time of the true 
 syzygy. 
 
 EXAM. 1. Required the time of full moon in January, 1797. 
 
 
 O's anom. } >s anom. 
 
 O's dist. from J 
 
 1796, March 
 | lunation 
 
 Full in March 
 Add 10 lunations 
 
 1797, January 
 1st equation 
 
 3d equation 
 
 1) h. m. , 
 811 39 44 
 + 14 18 22 
 
 s. ' ' 
 8 8 3 47 
 14 S3 10 
 
 b. " 
 3 7 2 14 
 6 12 54 SO 
 
 8. ' " 
 8 26 47 
 15 20 7 
 
 25 G 1 4G 
 295 7 20 SO 
 
 8 22 36 57 
 921 3 14 
 
 9 19 56 44 
 8 18 10 4 
 
 8 1.1 46 54 
 10 6 42 20 
 
 12 IS 22 16 
 + ! 30 
 
 6 13 40 11 
 
 6 8 29 38 
 
 6 8 6 48 
 
 + 22 50 
 
 6 22 29 14 
 arg. 4th equat. 
 
 12 14 22 46 
 1 20 45 
 
 5 10 38 
 arg. 3d. equat. 
 
 C 8 29 So 
 arg. 2d. equal. 
 
 3d equation 
 4th equation 
 
 .0ns. Januarj 
 
 12 13 2 1 
 
 26 
 + 1 6 
 
 12 IS 2 41 
 
USE AND APPLICATION OF THE TABLES, &c. 
 
 2. It is required to find 
 
 the time of conj 
 
 unction, or new moon, in the 
 O's Anomaly.] 5 's Anomaly. 
 
 month of August, 1820. 
 O's dist. from 3's JJ. 
 
 1820, March 
 5th lima. 
 
 August 
 1st equat. 
 
 2d equat. 
 
 . h. m. s. 
 14 1 42 45 
 147 15 40 15 
 
 s. ' " 
 8 12 25 31 
 4 25 31 37 
 
 s. ' " 
 6 24 37 23 
 4952 
 
 S. ' " 
 11 19 35 34 
 5 3 21 10 
 
 S 17 23 
 2 31 47 
 
 1 7 57 8 
 -11 2 44 51 
 
 11 S 42 25 
 57 34 
 
 4 22 56 44 
 arg. 4th. equat. 
 
 8 14 51 IS 
 
 4 47 r 
 
 2 5 12 17 
 arg. 3d. equat. 
 
 11 2 44 51 
 arg. 2d. equat. 
 
 3d equat. 
 4th 
 
 .flns. Aug. 
 
 8 10 4 6 
 
 427 
 
 1 29 
 
 8 58 K 
 
 Note 1. These examples are wrought for the meridian of Greenwich ; but the answers may be reduc- 
 ed to the time under any other meridian by adding or subtracting the difference of longitude in time to or 
 from them, according as the place is situated to the east or west of Greenwich. 
 
 2. These Tables give the times of new and full moon with little trouble, and sufficiently true for com- 
 mon use ; being rarely above 1 or 2 minutes wide of the truth : but when it is required to find the 
 moment of conjunction or opposition accurately, calculate the longitudes of the sun and moon for the time 
 found by the tables ; and if they are the same, or differ 6 signs, according as the new or full moon is 
 required, the time is truly found ; but if not, take their difference. Find the hourly motion of the moon 
 from the sun ; and then, 
 
 As the hourly motion of the moon from the sun 
 Is to 60 minutes ; 
 
 So is the difference between the sun and moon's long. 
 To a number of minutes, &c. 
 which, applied to the time formerly found, will give the true time. 
 
 3. The precise moment of conjunction or opposition is seldom necessary except in calculating eclipses. 
 
 4. In the preceding examples the time of orbit conjunction or opposition is taken for the answer; that 
 is, the time when the moon's longitude in her orbit is the same with the sun's longitude in the ecliptic, is 
 taken for the time of conjunction ; and the time when the moon's orbit longitude differs 6 signs from the 
 sun's longitude, is taken for the time of opposition. 
 
 The answers are in mean time ; and if the equation of time be applied to them with a contrary sign 
 to that in the Table, the apparent time will be known. 
 
 To calculate more accurately the time of new moon in August, 1820. The answer by our Tables is 
 August 8d. 9h. 58m. 10s. The sun's longitude at this time is 4s. 16 9' 44",- and his mean anomaly Is. 7 
 23' 13". 
 
348 
 
 APPENDIX TO THE ASTRONOMY. 
 
 The calculation of the moon's longitude for the same time follows. 
 
 J's m. 
 
 Long. 
 
 5 's anom. 
 
 D's . 
 
 1820 
 August 8th 
 9h. 
 58m. 
 10s. 
 
 10 equations 
 Equat. centre 
 Variation 
 
 13th equation 
 
 5 's orb. long. 
 Sun's long. 
 
 Difference 
 
 S. ' " 
 3 19 18 47 
 18 43 26 
 4 56 28 
 31 50 
 5 
 
 S. ' " 
 11 1 fc 
 11 24 17 51 
 4 53 58 
 31 34 
 5 
 
 S. ' " 
 6 25 42 
 
 11 39 
 1 12 
 
 7 
 
 4 13 35 36 
 25 26 
 
 10 29 44 34 
 + 12 54 
 
 11 24 45 23 
 -f 5 28 
 
 4 13 10 10 
 + 314 
 
 10 29 57 28 
 25 2(i 
 
 11 24 50 51 
 
 4 16 11 14 
 
 + 1 
 
 10 29 32 2 
 arg. llth. 
 
 4 16 11 15 
 1 1 
 
 4 16 10 14 
 4 16 9 44 
 
 SO 
 
 Arguments. 
 
 Equations. 
 
 
 S. ' " 
 
 t 
 
 
 
 1 
 
 1 7 3 13 
 
 6 37 
 
 
 2 
 
 1 2 14 57 
 
 
 29 
 
 5 
 
 10 17 28 31 
 
 51 
 
 
 4 
 
 10 24 36 18 
 
 
 36 
 
 5 
 
 25 7 10 
 
 
 33 43 
 
 6 
 
 2 2 30 23 
 
 1 49 
 
 
 7 
 
 11 17 43 57 
 
 
 10 
 
 8 
 
 9 22 21 21 
 
 
 39 
 
 9 
 
 2 40 29 
 
 
 3 
 
 Hi 
 
 7 8 35 39 
 
 57 
 
 __ 
 
 
 
 
 T5 40 
 
 12 
 19 
 
 00 1 30 
 in 13 8 4fi 
 
 + 10 14 
 
 + 10 14 
 
 25 26 
 
 Moon's hourly motion 28 54 
 Sun's ditto. 2 24 
 
 Hourly motion 5 from 26 SO 
 
 Former time August 
 Interval subtract 
 
 As 26' 30" 
 To 60m. 
 So is 30" 
 
 To 1m. 7s. 
 
 B. k. m. s. 
 
 8 9 58 10 
 
 1 7 
 
 17243 
 
 True time of new moon 8 9 57 3, which is nearly the same with the former. 
 
 PROB. XVII. To find out the Number of Eclipses there are in any given Year, and in what Months 
 
 und Days they happen. 
 
 From the Table of the moon's mean motion take out the mean longitude of the moon's nodes for the 
 given year ; and then by the Table of the sun's longitude for every day of the year, find when the sun's 
 longitude will be nearly the same with that of the nodes, for at these times the eclipses must happen. The 
 months when the sun is in or near the places of the nodes may be called the node-months. In this inquiry 
 it may be proper to remember that the moon's nodes move backward about 1 38' every month. 
 
 To find the Days when Eclipses happen. 
 
 Calculate the times of the mean 
 distance from the moon's node ; and 
 
 syzygies in the node-months out of the Tables, and also the sun's 
 if the sun's distance from the node at the time of new moon be less 
 
USE AND APPLICATION OF THE TABLES, &c. 
 
 349 
 
 than 18 degrees, an eclipse of the sun may be expected ; and if the sun's distance from the node at the 
 time of full moon be less than 12 degrees, there may be an eclipse of the moon. We say there may be, 
 because we speak only of mean motions here. 
 
 EXAM. It is required to find the number of eclipses in the year 1796; and on what days they happen. 
 
 The mean longitude of the moon's north node, on the 1st of January 1796, is 3s. 20 35', and of the 
 south node 9s. 20 35'. The sun enters the sign Cancer about the 20th of June, and the sign Capricorn 
 about the 22d of December ; therefore the node-months are January, July, and December. 
 
 1. Calculate the time of mean new moon in January. 
 
 New moon. 
 
 O's dist. from 3>'s JJ. 
 
 1795, March 
 10 lunats. + 
 
 D. h. m. b. 
 20 2 51 6 
 295 7 20 SO 
 
 s. ' " 
 7 22 24 
 10 6 42 20 
 
 New } 1796 Jan. 9 10 11 38 5 29 6 20 sun eclipsed. 
 
 The sun's mean distance from the moon's north node being 5s. 29 6' 20", his distance from the soutli 
 node is only 55' 40"; and therefore the sun will be eclipsed to some place of the earth. 
 
 2. For the time of mean new moon in July. 
 
 New moon. 
 
 D. h. ' " 
 
 1796, March 8 11 39 44 
 
 4 lunations + Ii8 2 56 12 
 
 New 3> July 
 
 4 14 35 56 
 
 O's dist. from J's 
 
 s. 
 8 
 4 
 
 26 47 
 2 40 56 
 
 3 7 43 sun eclipsed. 
 
 S. For the time of mean full moon in December. 
 
 New moon. 
 
 Sun's dist. from Q. 
 
 )). h. m. s. 
 17%, March 8 1 1 39 44 
 \ lunation + 14 18 22 9. 
 
 S. ' " 
 8 26 47 
 + 15 20 7 
 
 Full 3> in March 23 6 1 46 
 9 lunations + 265 18 36 27 
 
 8 15 46 54 
 + 9626 
 
 Full D Dec. 14 38 13 
 
 5 21 49 moon eeli 
 
 The sun's distance from the moon's north node being 5s. 21 49', the moon's distance from her south 
 node is only 8 1 1', and therefore she is eclipsed. 
 
 4. For the time of mean new moon in December. 
 
 Add -| lunation to the numbers for full moon. 
 New moon. I Sun's dist. from SI. 
 
 
 
 i>. 
 
 h. 
 
 m. 
 
 s. 
 
 H. 
 
 O 
 
 / 
 
 / 
 
 
 Full i 
 
 Dec. 
 
 14 
 
 
 
 38 
 
 13 
 
 5 
 
 21 
 
 49 
 
 
 
 
 > lunation 
 
 + 14 
 
 18 
 
 22 
 
 2 
 
 + 
 
 15 
 
 20 
 
 7 
 
 
 New 
 
 D Dec. 
 
 28 
 
 19 
 
 
 
 15 
 
 6 
 
 7 
 
 9 
 
 7 sun 
 
 eclipsed. 
 
 We have found that there are four eclipses in the year 1796, viz. three of the sun and one of the moon. 
 The times of the mean conjunctions and oppositions are : 
 
350 
 
 APPENDIX TO THE ASTRONOMY. 
 
 Sun is eclipsed 
 
 And the moon is eclipsed 
 
 D. h. m. s. 
 
 Jan. 9 10 11 38 
 
 July 4 14 35 56 
 
 Dec. 28 19 15 
 
 . _14 38 13 
 
 PROB. XVIII. To calculate an Eclipse of the Moon. 
 
 1. Calculate the true time of opposition by Prob. XVI. and compute the longitude of the sun, and the 
 orbit longitude of the moon, for that time ; if these differ 6 signs, the time of opposition is truly found : 
 but if their difference be less or greater than 6 signs, mark the defect or excess, and find the moon's 
 hourly motion from the sun ; then, as the hourly motion of the moon from the snn is to one hour, so is the 
 defect or excess to the interval ; which being applied to the time formerly found, gives the true time of 
 opposition. This operation must be repeated until the longitude of the sun and the orbit longitude of the 
 moon differ precisely 6 signs. 
 
 2. Calculate the moon's latitude, horizontal parallax, and hourly motion, by the Tables ; take also 
 the sun and moon's seniidiamctcrs out of the Tables. 
 
 3. Add the sun's horizontal parallax (which is 9 seconds) to the moon's for the time, and from the 
 sum subtract the sun's semidiameter ; the remainder is the semidiameter of the earth's shadow ; to which 
 add the moon's semidiameter. and the sum is the semidiameter of the moon and earth's shadow ; from which 
 subtract the moon's latitude, and the remainder is called the parts deficient. If these be less than the 
 moon's diameter, the eclipse will be partial ; if they be equal to the moon's diameter, the eclipse will be 
 total without continuance ; and if they be greater than the moon's diameter, the eclipse will be total with 
 continuance. 
 
 EXAM. 1. It is required to calculate the eclipse of the moon, whicU will happen in December, 1797. 
 
 1. Calculate the time of opposition. 
 
 
 Sun's Anom. 
 
 Moon's Anom. 
 
 O'sdist. from SI. 
 
 1797, new D March 
 .', lunation 
 
 Full moon March 
 9 lunations 
 
 Full moon December 
 1 equation 
 
 2 equation 
 
 D. h. m. s. 
 27 9 12 24 
 14 18 22 2 
 
 S. ' " 
 8 26 25 59 
 14 33 10 
 
 s. ' " 
 2 12 39 19 
 6 12 54 SO 
 
 s. ' " 
 9 9 9 48 
 
 15 20 7 
 
 12 14 50 22 
 
 + 265 18 36 27 
 
 8 11 52 49 
 8 21 56 54 
 
 7 29 44 49 
 7 2-2 21 4 
 
 8 23 49 41 
 9626 
 
 S 9 26 49 
 1 52 47 
 
 5 3 49 43 
 3 21 23 13 
 
 3 22 5 53 
 42 40 
 
 5 29 51 47 
 arg. 4th equat. 
 Moon eclipsed. 
 
 3 7 34 2 
 + 8 53 8 
 
 1 12 26 30 
 arg. 3d. equat. 
 
 3 21 23 13 
 arg. 2d eq'uat. 
 
 G equation 
 4 ditto 
 
 Full moon December 
 
 3 16 27 10 
 
 3 11 
 
 1 
 
 3 16 23 58 
 
USE AND APPLICATION OF THE TABLES, &c. 
 
 3. Calculate the sun and moon's longitude for the time of full moon. 
 For the Sun's Longitude. 
 
 351 
 
 Sun's Long. 
 
 
 Sun 
 
 's Anom. 
 
 1797 
 December 3d 
 I6h. 
 23m. 
 58s. 
 
 Equation 
 Sun's longitude 
 
 S. 
 9 
 11 
 
 
 
 10 
 
 2 
 
 37 
 9 
 39 
 
 33 
 17 
 25 
 57 
 2 
 
 s 
 6 
 11 
 
 o 
 
 1 
 
 2 
 
 8 
 8 
 39 
 
 II 
 
 17 
 46 
 25 
 57 
 2 
 
 8 
 
 13 
 
 27 
 51 
 
 44 
 45 
 
 5 
 
 3 
 
 57 
 
 27 
 
 8 
 
 12 
 
 35 
 
 59 
 
 For the. Moon's Longitude. 
 
 Moon's m. 
 
 Long. 
 
 Moon's Anom 
 
 Moon's Q. 
 
 1797 
 Dec. 3d. 
 
 IGh. 
 23m. 
 58s. 
 
 10 equations 
 Equat. centre. 
 Tariation 
 
 13th equation 
 
 Moon's orb. Ion. 
 Sun's Ion. 
 
 Past opposition 
 
 S. ' " 
 10 7 34 43 
 4 26 45 
 8 47 S 
 12 37 
 32 
 
 S. o ' " 
 24 6 53 
 2 22 54 4 
 8 42 36 
 12 31 
 32 
 
 g O / ft 
 
 3 1 15 6 
 
 17 50 45 
 2 7 
 3 
 
 2 17 1 40 
 + 1 26 1 
 
 3 25 56 36 
 + 9 44 
 
 2 13 22 11 
 
 + 43 
 
 2 18 26 41 
 5 45 57 
 
 3 26 6 20 
 + 1 25 1 
 
 2 13 26 14 
 
 2 IS 40 44 
 + 6 
 
 3 27 31 21 
 arg. llth. 
 
 2 12 40 50 
 1 13 
 
 2 12 39 37 
 8 12 35 59 
 
 3 38 
 
 Arguments. 
 
 Equations. 
 
 
 S. 
 
 
 
 t 
 
 n 
 
 
 
 +, 
 
 ft 
 
 
 
 1 
 
 5 
 
 3 
 
 57 
 
 27 
 
 
 
 5 
 
 
 
 
 2 
 
 5 
 
 12 
 
 48 
 
 49 
 
 
 
 
 17 
 
 3 
 
 7 
 
 4 
 
 53 
 
 55 
 
 
 
 
 
 43 
 
 
 4 
 
 4 
 
 4 
 
 47 
 
 58 
 
 
 
 
 
 47 
 
 
 5 
 
 8 
 
 12 
 
 54 
 
 46 
 
 1 
 
 17 
 
 15 
 
 
 6 
 
 1 
 
 10 
 
 52 
 
 13 
 
 
 
 1 
 
 SO 
 
 
 
 
 3 
 
 8 
 
 57 
 
 19 
 
 
 
 
 
 46 
 
 
 8 
 
 10 
 
 2! 
 
 59 
 
 9 
 
 
 
 
 26 
 
 
 
 2 
 
 8 
 
 29 
 
 5 
 
 
 
 
 19 
 
 10 
 
 6 
 
 
 
 46 
 
 12 
 
 
 
 
 
 2 
 
 
 12 
 
 6 
 
 
 
 4 
 
 45 
 
 + 1 
 
 26 
 
 3 
 
 1 2 
 
 IS 
 
 8 
 
 
 
 57 
 
 51 
 
 
 
 1 
 
 2 
 
 
 
 1 1 
 
 20 
 
 q 
 
 OQ 
 
 
 
 
 
 i * 
 
 6 Ij 
 
 y 
 
 + 1 
 
 25 
 
 1 
 
 
 Moon's hourly motion - 
 Sun's ditto 
 
 Hourly motion 3 from Q - - 32 46 
 
 L.L. 
 
 As hourly motion J from Q 32' 46" 2627 
 To one hour or 60m. - - - o 
 
 12178 
 
 To 6m. 39$. - 
 
 - 9551 
 
352 
 
 APPENDIX TO THE ASTRONOMY. 
 
 Former time December 
 Interval subtract 
 
 Time of full moon 
 
 D. h. ' 
 
 3 16 23 58 
 6 39 
 
 3 16 17 19 
 
 Sun's longitude 
 Moon's ditto 
 Reduction 
 
 For the Moon's Latitude. 
 
 Arguments 
 
 
 
 S. ' " 
 
 I it 
 
 1 
 
 11 29 13 23 
 
 4 11 
 
 2 
 
 53 23 
 
 + 8 
 
 3 
 
 6 25 15 56 
 
 + 6 
 
 4 
 
 2 29 19 20 
 
 25 
 
 9 
 
 11 3 22 44 
 
 7 
 
 Moon's lat. S. 4 29 
 
 8 12 35 43 
 
 2 12 35 43 
 
 + 12 
 
 For the Moon's Horizontal Parallax. 
 
 \rffumits. 
 
 / 
 + 11 
 + 58 32 
 + 27 
 
 1 
 2 
 3 
 
 Equato 
 Reduct 
 
 8 12 54 46 
 3 27 31 21 
 6 4 45 
 
 ial Parallax 
 ion 
 
 59 10 
 9 
 
 Horizontal Parallax 
 
 59 1 
 
 For the time of Reduction. 
 
 L. L. 
 
 As moon's hourly motion from Q 32' 46" 2627 
 
 To Ih. or 60*. 
 
 So is reduction 12" 
 
 To time of reduction + 22" 
 
 Sun's semidiameter 
 Moon's semidiameter 
 Angle of the moon's path with the ecliptic 
 
 
 
 24771 
 
 22144 
 
 16 17 
 16 8 
 
 5 40'. 
 
 Note. In eclipses, the disk of the sun or moon is supposed to be divided into twelve equal parts 
 called digits, and each of these again divided into 60 equal parts or minutes. The magnitude of an 
 eclipse is estimated by the number of digits. 
 
 Calculation of the Eclipse. 
 
 > H 
 
 Moon's horizontal parallax 5_9 1 
 
 Sun's ditto '+ 9 
 
 Sum 59 10 
 
 Sun's semidiameter subtract - 16 17 
 
 Semidiameter of the earth's shadow 
 Moon's semidiameter add 
 
 Semid. of the moon and earth's shadow 
 Moon's latitude subtract 
 
 Remains the parts deficient 64 32 
 
 The eclipse will be total. 
 
USE AND APPLICATION OP THE TABLES, &c. 353 
 
 For the Digits eclipsed. 
 
 As moOn's semidiameter l& 8" 
 To 6 digits - .. 
 
 So parts deficient 54' 32" 
 
 To digits eclipsed 20 16' - 
 
 Mean time of orbit opposition Dec. 
 Time of reduction add and subtract 
 
 Middle of the eclipse S 16 17 41 
 
 Ecliptic opposition - - - S 16 16 57 
 
 To each of these add the equation of time + 9 22 
 
 Appar. time of the middle of the eclipse 3 16 27 3 
 of ecliptic opposition 3 16 26 19 
 
 To find the Scruples of Incidence, or that Part of the Moon's Path ivhich she passes over between the be- 
 ginning and middle of the Eclipse. 
 
 Reduce the sum of the semidiameters of the moon and earth's shadow to seconds, and also the moon's 
 latitude ; find their sum and difference ; and to the logarithm of their sum add the logarithm of their dif- 
 ference, and divide the sum by 2, the quotient is the logarithm of the scruples of incidence in seconds. 
 
 In this example the semidiameter of the moon and earth's shadow is 
 
 I H II 
 
 59 1 = 3541 
 Moon's latitude 4 29 = 269 Logarithms. 
 
 Sum 3810 3.5809450 
 
 Difference 3272 S.514813S 
 
 2)7.0957583 
 
 Scruples of incidence 3530 3.5478791 
 or 58' 50" 
 
 For the time of half duration. 
 
 ' " L. L. 
 
 As hourly mo. D from Q = 32 46 1'627 
 
 To 1 hour or 60 minutes 
 
 So scruples of incidence 58 50 (49' 25") 3096 
 
 To time of half duration Ih. 47' 42". (6 53 51) 469 
 
 o 
 
 1 47 42 
 
 Because the terms of this proportion are such, that the fourth term comes out too great for the Table 
 of logistical logarithms, take half of the third term ; and then double the fourth term for the answer^ 
 
 D. h. ' " 
 
 Apparent time of the middle of the eclipse 3 16 27 3 
 Tune of half duration, subtract and add zp 1 47 43 
 
 Beginning - - - 3 14 39 21 
 
 End - - . - 6 18 14 45 
 
 In total eclipses of the moon, she continues some time in total darkness ; to calculate this time, it is 
 
 45 
 
354 APPENDIX TO THE ASTRONOMY. 
 
 necessary to find the half length of that part of the moon's path which lies wholly within the earth's 
 shadow, which some call the scruples of total darkness. The operation is performed thus : 
 
 Subtract the moon's semidiameter from that of the earth's shadow ; reduce the remainder, and also the 
 moon's latitude, to seconds ; then find their sum and difference, and take the half sum of the logarithms 
 of the sum and difference, and the number answering is the scruples of total darkness. 
 
 In this example, 
 
 i it 
 
 The semidiameter of the earth's shadow is 42 55 
 The moon's diameter - 16 8 
 
 26 45 
 / n n 
 
 The difference is 26 45 = 1605 
 
 Moon's latitude 4 23 = 269 
 
 ^ Logarithm!. 
 
 Sum - 1874 S. 2727 696 
 
 Difference - - - 1336 3.1258065 
 
 2)6.3985761 
 
 Scruples - - - 1584 S. 1992880 
 
 Or - 26' 24". 
 
 For the time of half duration of total darkness. 
 
 ' L. L. 
 
 As hourly motion of moon from sun == 32 46 2627 
 To one hour or 60 - 
 
 So are scruples 6 24 3565 
 
 To time of half duration 48' 21" - 938 
 
 D. h. ' " 
 
 Apparent time of the middle of the eclipse 3 16 27 3 
 Half duration of total darkness subtract and add zp 48 21 
 
 Beginning of total darkness - - 3 15 38 42 
 
 End of ditto - - - 3 17 15 23 
 
 From the calculation it appears, that the moon will be eclipsed at London, 1797, December 3d. 
 
 Appar. time, 
 h. 
 
 Beginning of the eclipse - - 14 39 
 
 Beginning of total darkness 15 38 
 
 Ecliptic opposition - 16 26 
 
 Middle of the eclipse 
 
 End of total darkness - 17 15 
 
 End of the eclipse - 18 14 
 
 Digits eclipsed 20 16' from S. side of earth's shadow. 
 Duration of the eclipse 3h. 55'. 
 Duration of total darkness ih. 36'. 
 
 EXAM. 2. Calculation of an eclipse of the moon which will happen 1802, Sept. llth. 
 
 Mean time of orbit opposition, Sept. lid. lOh. 40'. At that time sun's longitude 5s. 18 25' 16"; 
 moon's longitude 11s. 18 25' 26", latitude north 37' 6" ascending. Reduction = 1' 40", time of reduc- 
 tion 2' 48". 
 
USE AND APPLICATION OF THE TABLES, &c. 355 
 
 / n 
 
 Moon's hourly motion - - 38 4 
 Sun's ditto - - - 2 26 
 
 Moon's hourly motion from sun - 35 38 
 
 O I ft 
 
 Moon's semidiameter - - -"- 016 43 
 
 Sun's ditto - - 15 57 
 
 Moon's horizontal parallax - - - 61 17 
 
 Angle of the moon's path with the ecliptic is 5 38 
 
 Calculation of the Eclipse. 
 
 Moon's horizontal parallax 
 Sun's ditto . 
 
 Sum ... - 61 26 
 
 Sun's semidiameter subtract - - 15 57 
 
 Semidiameter of earth's shadow - 45 29 
 
 Moon's semidiameter, add - - - -f- 16 43 
 
 Semidiameter of moon and earth's shadow 62 12 
 Moon's latitude, subtract ... 37 6 
 
 Parts deficient - - - - 25 6 
 
 For the digits eclipsed. 
 
 ' " L.L. 
 
 As moon's semidiameter - - 16 43 5550 
 
 To 6 digits 10000 
 
 So parts deficient - 25 6 3785 
 
 To digits - 9 8235 
 
 D. h, m. s. 
 
 Mean time of orbit opposition - - 11 10 40 
 Time of reduction, subtract and add to ttit zp 2 48 
 
 Middle of the eclipse 11 10 37 12 
 
 Ecliptic opposition - - 11 10 42 48 
 
 Equation of time add to each - - J- 3 31 
 
 Apparent middle of the eclipse 11 104043 
 ecliptic opposition - - 11 1046 19 
 
 For the scruples of incidence. 
 
 Semid. of the 5 and 's shadow 62 12 = 3732 
 Moon's latitude - - - 37 6 = 2226 
 
 Logarithms. 
 
 Sum . . . 5958 3.7751005 
 Difference 1506 3.1778250 
 
 2)6.9529255 
 
 Scruples 2905 3.4764627 
 
 Or 49' 45" 
 
 V. 
 
396 APPENDIX TO THE ASTRONOMY. 
 
 For the time of half duration. 
 
 ' " L. L. 
 
 As moon's hourly motion from sun 35 38 2263 
 
 To 1 hour or 60m. - 
 
 So scruples of inciil. 9 59 7789 
 
 To | of half duration - - 1648 5526 
 
 -ba> _J ** 
 
 Half duration - Ih. 24m. Os. 
 
 D. li. m. s. 
 
 Apparent middle of the eclipse 11 104043 
 
 Time of half duration, subt. and add - - q= I 24 
 
 Beginning ... 11 9 16 43 
 
 End 11 12 4 43 
 
 1802, September llth, the moon is eclipsed. 
 
 Appar. time. 
 
 h. m. 
 
 Beginning - -. 9 16 
 
 Middle - 10 40 
 
 Ecliptic opposition - 10 46 
 
 End - - - 12 4 
 
 Duration 2 hours 48 minutes. Digits eclipsed 9. 
 
 PROB. XIX. To describe a Figure representing a lunar Eclipse. (Plate XIV. Fig. 4 ) 
 
 EXAM. 1. It is required to project the eclipse, calculated in the 1st example of the last problem. 
 
 1. Draw the straight line AB for part of the ecliptic, in which take any point C, and draw CD at 
 right angles to AB, for the axis of the ecliptic; which is drawn downward, because the moon's latitude at 
 the beginning and middle of the eclipse is south. 
 
 2. Take the scmidiameter of the moon and earth's shadow, 59' ]'', from any scale, and with it, as a 
 radius, describe the circle ADB ; the moon's centre is in one point of the circumference of this circle at 
 the beginning; and in the opposite point of it, at the end of a total eclipse. 
 
 S. Take the semidiameter of the earth's shadow, 42' 55", from the same scale, and with it, from the 
 centre C, describe the circle KLM, to represent the earth's shadow. 
 
 4. Make CD the radius of a line of chords on the sector, and from it take the angle of the moon's 
 path with the ecliptic 5 40', and set it from D to E toward the left hand, (b'ecause the moon's latitude is 
 south ascending northward) and draw the line CE for the axis of the moon's orbit. 
 
 5. Take the moon's latitude 4' 29" from the scale, which set from C to F on the line CE ; and 
 through F draw the straight line PN at right angles to CE ; then the points N, F, and P are the places 
 of the moon's centre at the beginning, middle, and end of the eclipse. Take the moon's semidiameter 
 from the scale, and with it describe circles from the centres N, F, and P. These shall represent the 
 moon at these points. 
 
 To mark the hours on the Moon's Path. 
 
 The middle of the eclipse is December 3d. I6h. 27m. 3s. which, iu civil reckoning, is 4h. 27m. in the 
 morning of December 4th, marked on the moon's path by the point F. Now, for the place of four 
 hours, say, 
 
 As one hour 
 
 Is to the moon's hourly motion from the sun 32' 46" ; 
 
 So is 37m. 3s. 
 
 To 14' 46", the distance of 4h. from F. 
 
USE AND APPLICATION OF THE TABLES, &c. 357 
 
 Take 14' 46" from the scale, and set it from F toward the right hand for the place of 4 hours; then 
 take the moon's hourly motion from the sun 32' 46" and set it on'the moon's path from 4 to 3, and to 5, 
 and also from 5 to 6, and the hours are marked. Divide each of the hour distances into six equal parts, 
 and the place of the moon's centre will be known at every 10 minutes. 
 
 EXAM. 2. Required the projection of the eclipse cnlculated in the 2d example of the last problem. 
 (Plate XIV. Fig. 5.) 
 
 This is done in the same manner as the former ; only, because the moon's latitude is north, the semi- 
 circles ADB and KLM must be above the straight line AB ; and because the moon's latitude is ascend- 
 ing northward, the angle of her path with the ecliptic 5 38' must be set from D to K toward the right 
 hand. Draw CE for the axis of the moon's orbit, and thereon set the moon's latitude 37' 6" from C to 
 F ; and through F draw the straight line NP at right angles to CE ; and then the points N, F, and P, are 
 the places of the moon's centre at the beginning, middle, and end of the eclipse, as in the former example. 
 
 Take the moon's semidiameter 16' 4.3'' from the scale, and with it, for a radius, describe circles on the 
 centres N, F, and P, which will represent the moon's disk when she is at these points ; and then place, 
 the hours on the moon's path, as directed in the former example. 
 
 PREVIOUS to the Calculation and Projection of Solar Eclipses, it will be proper to premise some 
 things for explaining the terms used in the graphical delineation of them. 
 
 Although the earth is nearly a globe, yet when it is viewed from a very distant point, such as the 
 centre of the sun, it will appear as a Hat, round surface, like the moon, to us ; and this appearance is 
 called the earth's disk. 
 
 By the continual rotation of the earth round its axis, every point on its surface describes a "circle par- 
 allel to the equator ; and the circle described by any point or place is called the path of the vertex of 
 that place. 
 
 The representations of the paths of places on the earth's disk are not always of the same form, but 
 differ with the sun's longitude ; for when the sun is in either of the equinoctial points, the paths of all 
 places are represented by straight lines; and when the sun is in any other point of the ecliptic, they are 
 represented by ellipses, more or less eccentric according as the sun is nearer to, or farther from, the equi- 
 noctial circle. 
 
 In describing these paths on the earth's disk, respect must be had to the position of the axes of the 
 earth and ecliptic at the time. 
 
 When the sun is in any of the ascending signs Vf, ?, X, f , 8, II, the northern half of the earth's 
 axis lies to the right hand of the axis of the ecliptic, as seen from the sun ; but when he is in any of the 
 descending signs 25, &, "j;, A, m,, j , it lies to the left hand. When the sun is in either of the solsti- 
 tial points, the two axes coincide ; and when he is in either of the equinoctial points, they form the 
 greatest angle. 
 
 When the sun is in any of the northern signs, Aries, Taurus, &c. the north pole of the earth is in the 
 tipper or enlightened part of the earth's disk ; and when he is in any of the southern signs, Libra, Scorpio, 
 &c. the north pole of the earth is in the under or obscure part of the disk-. 
 
 The transverse diameter of any path is always at right angles to the earth's axis, and the conjugate 
 diameter coincides with or is a part of it. 
 
 PROB. XX. The Latitude of a Place, the Time of the Year, and the Sun's Longitude, being given, to 
 describe the Earth's Disk, and the Path of that Place thereon. 
 
 EXAM. It is required to describe the earth's disk, and the path of a place in latitude 51 32' N. 1803, 
 August 17th, at 8h. 19m. in the morning, apparent time. (Plate XV. Fig. 2.) 
 
 Sun's longitude 4s. 23 24' 45", distance from the solstice 53 2<f 45" ; declination N. 13 43' 43". 
 
 Draw the straight line AB to represent a part of the ecliptic, and let C be the point therein, which is 
 opposite to the sun at the time : from C draw CH at right angles to AB, and CH is the axis of the eclip- 
 tic, and II its pole. 
 
 From the centre C describe the semicircle AHB to represent the northern half of the earth's disk. 
 Make CA or CB the radius of the line of chords on the sector, and take the chord of 2S| degrees, which set 
 from H both ways to /and g, and draw the straight line /Ye 1 ; the north pole of the earth is always in 
 fhis line. 
 
338 APPENDIX TO THE ASTRONOMY. 
 
 Make/V the ratlins of the line of sines on the sector, and take the sine of 53 24' 45'', the sun's dis- 
 tance from the solstice, which set oft' from V to P toward the left hand, because the sun is in the sign 
 Virgo, and draw the straight line CP/t for the earth's axis ; then P is the north pole of the earth. Or the 
 angle contained between the earth's axis and that of the ecliptic may be found more accurately by calcu- 
 lation, thus ; 
 
 o / 
 
 As radius - S. 90 10. 
 
 Is to the sine of the sun's (list, from the solstice 53 25 9.9047106 
 
 So is the tan. of the distance of the poles 23 28 9.6376106 
 
 To the tun. of the angle contained by the axes 19 13 9.5423212 
 
 Now set off the chord of 19 IS' from II to h, and join CII, which will cut/g- in P, the place of the 
 north pole. 
 
 Make CA the radius of the line of chords on the sector ; take the chord of 38 28', the complement of 
 the latitude, and set it off from h, both ways, to e and n, where make marks. Again, take the chord of 
 the sun's declination 13 43' 43" from the sector, and set it off from the points e and n, both ways, to 
 D and F, and to M and G; then draw the lines DM, FG, cutting CP in 12 and 12 ; and the line 12 K 
 13 is the conjugate axis of the ellipse, which bisect in K, and through K draw the line 6 K 6 at right an- 
 gles to CP. 
 
 Make CA the radius of the line of sines on the sector, and take the sine of the co-latitude 38 28', and 
 set it off from K, both ways, to 6 and 6. These hours fall on the circumference of the disk, when the 
 sun is in either of the equinoctial points ; but at all other points they fall within it. The line 6 K 6 is 
 the transverse axis of the ellipse. 
 
 Make K 6 the radius of the line of sines on the sector, and from it take the sines of 15, 30, 45, 60", 
 and 75, and set them off from K, both ways, toward 6 and 6 ; and through these points draw lines paral- 
 lel to CP. Again, make K 12 the radius of the line of sines on the sector, and take from it the sines 
 of 75, 60, 45, 30, and 15, and set them off on the parallels, beginning with that next to 12 K 12 ; 
 and through these points draw a curve line which will be the path ot the given place. Mark the 
 hours on the path as per Fig. and divide them into halves, quarters, and smaller parts, at pleasure. 
 
 The 12 next to P marks midnight, when the sun's declination is north ; and that part of the path is 
 below the disk ; but the other 12 between K and C is noon or mid-day. The path touches the circum- 
 ference of the disk a little before 5, which is the time of sun-rising ; and again a little after 7, the time of 
 sun -setting. 
 
 In north latitude, when the sun's declination is south, the 12 nearest to P is mid-day, and the other is 
 midnight. 
 
 In the same manner, the parallel of latitude, or path of any place, may be described on the earth's 
 disk for any given time, and is used in representing the appearance of an eclipse of the sun. 
 
 PROB. XXI. To calculate an Eclipse of the Sun. 
 
 1. Calculate the time of mean conjunction, with the sun's distance from the iRoon's node, and there- 
 bv find if an eclipse will happen ; and if it do, calculate the true time of conjunction, and the sun and 
 iiioon's longitudes at that time. 
 
 2. Calculate the moon's latitude, horizontal parallax, horary motion, semidiameter, and angle of her 
 path with the ecliptic. 
 
 3. Calculate the sun's declination, hourly motion, and his distance from the solstice. 
 Having found these necessary requisites, write them down in order to be ready for use. 
 
 EXAM. The sun will be eclipsed in August, 1803. 
 The time of mean conjunction is Aug. 17d. 7h. 12' 24". 
 Sun's mean distance from the moon's node 6s. 2 8' 8". 
 And the true conjunction is Aug. 16d. 20h. SO' 34" mean time. 
 
 s. ' " 
 
 The sun's longitude then 4 23 25 3 
 
 The moon's orbit longitude 4 S3 28 49 
 
 Past conjunction 346 
 
USE AND APPLICATION OF THE TABLES, &c. 359 
 
 Sun's mean anomaly Is. 15 8' 15". 
 
 Moon's hourly motion 30 57 
 
 Sun's ditto subtract 2 24 
 
 Moon's hourly motion from sun 28 33 
 
 As moon's hourly motion from O 28' S3" 
 
 To 1 hour or 60 minutes 
 
 So is difference of longitudes 3' 46" 
 
 To interval T 54" 8796 
 
 D. h. ' " 
 
 From the time formerly found 16 20 30 34 
 
 Subtract the interval 7 54 
 
 Correct time of conjunction 16 20 22 40 
 
 Equation of time 3 48 
 
 Apparent time of conjuction 16 20 18 54 
 
 or 8h. 18' 54" in the morning of Aug. 17th. 
 
 For the Sun and Moon's places at the corrected time of Conjunction. 
 
 m. ' " m. s. " 
 
 As 60 : 2 24 :: 7 54 : 18 to be subtracted from Q' s lon g- 
 60 : 30 57 : : 7 54 : 4' 4" to be subtracted from the D 's long. 
 
 s. ' " 
 
 And then the sun's longitude is 4 23 24 45 
 The moon's ditto 4 23 24 45 
 
 2. The moon's horizontal parallax, or semi- > n 55 9 
 diameter of the earth's disk 5 
 
 3. Sun's distance from the nearest solstice 53 24 45 
 
 4. Sun's declination north - 13 43 43 
 
 5. Angle of the moon's path with the ecliptic 5 43 
 
 6. The moon's latitude south descending 46 
 
 7. The moon's hourly motion from the sun 28 33 
 
 8. The sun's semidiameter 15' 52" 
 
 9. The moon's semidiameter 15 14 
 
 10. The semidiameter of the penumbra, or sum ) ,. _ 
 of the semidiameters of sun and moon 5 
 
 In this example the sun's distance from the nearest solstice is found by subtracting 3 signs from his 
 longitude The angle of the moon's path with the ecliptic is taken out of the Table. 
 The method of hnding the other numbers has been taught formerly. 
 
 PROB. XXII. To describe a Figure representing a solar Eclipse, and to find its Magnitude, Beginning, 
 
 Middle, anil End, at London. (Plate XV. Fig. 2.) 
 
 EXAM. Project the solar eclipse, calculated in the last problem. 
 
 Draw the straight line AB to represent a part of the ecliptic ; take the semidiameter of the earth's 
 disk from a scale, and with it as radius describe the semicircle AHB for the northern half of the earth's 
 disk, and thereon describe the path of London, as has been taught in the preceding problem. 
 
 Make the semidiameter of the disk CA the radius of the line of chords on the sector, and take the 
 chord of 5 43', the angle of the moon's path, which set off on the circumference of the disk, from 
 H toward the right hand, to M, because the moon's latitude is south descending, and draw CM for the 
 axis of the moon's path. 
 
360 APPENDIX TO THE ASTRONOMY. 
 
 Set off' the moon's latitmie 46" south on MC, produced below AB to X ; and through X, at right an- 
 gles to MC, draw the line of the moon's path, which cuts AB on the left hand of X. The point X is the 
 time of conjunction by the Tables. 
 
 To find the place of 8 hours on the moon's path. 
 
 I-, lit 
 
 As 1 hour or GO minutes - 
 
 To the 3 's hourly motion from the O 28 ' 35" 3225 
 So is the time of conjunction after 8h. 18' 52" 5025 
 
 To the distance of 8 hours from X 8' 58" 8250 
 
 Take 8' 58" from the scale, and set it ofi' from X toward the left hand for the place of 8 hours on the 
 moon's path. 
 
 Take also from the scale 28' S3", the moon's hourly motion from the sun, and set it off both ways 
 from 8 to 9 and 7, and also from 7 to 6, &c. and the places of the hours on the moon's path will be 
 known. Divide each hour distance into 12 equal parts by dots, and the place of the moon's centre, or 
 rather the centre of the penumbra, will be known at every 5 minutes during the eclipse. 
 
 To find the middle of the Eclipse. 
 
 Apply one side of a square to the path of the moon or penumbra's centre, and move it backward and 
 forward until the other side of the square cuts the same hour and minute on the path of London and of 
 the penumbra's centre, which in this example is at y and u in 6 hours 32 minutes, the middle of the eclipse. 
 
 lake the sun's semidiameter 15' 52" from the scale, and with it describe a circle about the point y for 
 the sun's disk ; then take the moon's semidiameter 15' 4" from the scale, and with it describe a circle 
 about the point u for the moon's disk at tbe middle of the eclipse. The part of the sun's disk cut 
 ofi' by the moon's is the magnitude of the eclipse as it will appear at London. In this example it is 
 about 3 digits. 
 
 Lastly, take the semidiameter of the penumbra 30' 56" from the scale ; and setting one foot of the 
 compasses on the moon's path, and the other on the path of London, toward the left hand, carry that ex- 
 tent backward and forward until both points fall on the same hour and minute in each path, and that is 
 the beginning of the eclipse at London. With the same extent of the compasses, and one foot on each 
 path, carry them backward and forward toward the right hand ; and where both points fall on the same 
 time, that will be the end of the eclipse at London. These trials give, 
 
 h. m. 
 
 The beginning of the eclipse at - 5 50 
 
 The middle - - - 6 32 
 
 The end - 7 30 
 
 Duration Ih. 40m. Digits 3 50'. 
 
 JVote 1. The projection of a solar eclipse will exhibit the appearance of it more naturally, if some al- 
 terations be made in the preceding process, adapted to the supposition, that the observer is on the earth. 
 These alterations consist in drawing the axes of the earth and moon's' path on the side of the axis 
 of the ecliptic contrary to that which is required by the rule, and in numbering the hours in the op- 
 posite direction. For the relative position of the axes, as seen from the sun, is inverted with respect to 
 an observer of the sun on the earth. If a projection, made according to the preceding rule, be turned 
 over from right to left, it will appear, if visible through the paper, to correspond with one of this con- 
 struction. 
 
 Plate XV. Fig. 3, is an example, being the same eclipse projected in this manner, and requires no 
 farther explanation. 
 
 Note 2. The situation of the point on the sun's limb, with respect to a vertical and a horizontal di- 
 ameter,where an eclipse begins, may be easily determined by Projection. Thus, with the points in the 
 respective paths, where the centres of the sun and moon are at the beginning of the eclipse, as centres, 
 and their semidiameters as radii, describe circles, touching each other. Draw a line from the centre 
 (C Fig. 2 or 3, Plate XV.) through the centre of the sun, and it will give the vertical diameter; and a 
 diameter perpendicular to this will be horizontal. Then the line of chords on a sector being adapted to 
 (he semidiameter of the sun, the arc of the circumference, contained between the point of contact and an 
 extremity of one of these diameters, measured on the chord line, will give the required situation of the 
 point, where the eclipse begins. 
 
. 
 
 
 
 
 SOLAR AND LUtfAR TABLES. 
 
 
 TABLES OF THE SUN'S MEAN MOTIONS. 
 
 I. The Sun's Mean Longitude and 
 
 II. The Pre- 
 
 Anomaly in Julian I ears. 
 
 cession of the 
 
 Yean. 
 
 M. Longitude. 
 
 M. Anomaly. 
 
 Equinoctial 
 Points. 
 
 
 S. ' " 
 
 S. ' " 
 
 O 1 II 
 
 1 
 
 11 29 45 40 
 
 11 29 44 35 
 
 50.3 
 
 2 
 
 11 29 31 21 
 
 U 29 29 
 
 1 40.7 
 
 3 
 
 11 29 17 2 
 
 U 29 13 44 
 
 2 31 
 
 B 4 
 
 1 51 
 
 U 29 57 27 
 
 3 21.4 
 
 5 
 
 11 29 47 32 
 
 11 29 42 2 
 
 4 11.7 
 
 6 
 
 11 29 33 13 
 
 11 29 26 37 
 
 5 2.1 
 
 7 
 
 11 29 18 54 
 
 U 29 11 12 
 
 5 52.4 
 
 B 8 
 
 3 43 
 
 U 29 54 55 
 
 6 42.8 
 
 9 
 
 11 29 49 23 
 
 H 29 39 29 
 
 7 33.1 
 
 10 
 
 U 29 35 4 
 
 11 29 24 4 
 
 8 23.5 
 
 11 
 
 11 29 20 45 
 
 11 29 8 39 
 
 9 13.8 
 
 B 12 
 
 5 34 
 
 11 29 52 22 
 
 10 4.2 
 
 13 
 
 11 29 51 15 
 
 11 29 36 57 
 
 10 54.5 
 
 14 
 
 11 29 36 55 
 
 11 29 21 31 
 
 11 44.9 
 
 15 
 
 11 29 22 36 
 
 U 29 6 6 
 
 12 35.2 
 
 B 16 
 
 7 35 
 
 11 29 49 49 
 
 13 25.6 
 
 17 
 
 11 29 53 6 
 
 11 29 34 24 
 
 15 15.9 
 
 18 
 
 11 29 38 47 
 
 11 29 18 59 
 
 15 6.3 
 
 19 
 
 U 29 24 27 
 
 11 29 3 33 
 
 15 56.6 
 
 B 20 
 
 9 17 
 
 11 29 47 17 
 
 16 47 
 
 B 40 
 
 18 33 
 
 11 29 34 33 
 
 33 34 
 
 B 60 
 
 27 50 
 
 11 29 21 50 
 
 50 21 
 
 B 80 
 
 37 
 
 11 29 9 6 
 
 178 
 
 H 100 
 
 46 23 
 
 11 28 56 23 
 
 1 23 55 
 
 B 200 
 
 1 32 46 
 
 11 27 52 46 
 
 2 47 50 
 
 B 300 
 
 2 19 9 
 
 I 1 26 49 9 
 
 4 11 45 
 
 a 400 
 
 3 5 32 
 
 U 25 45 32 
 
 5 35 40 
 
 B 500 
 
 3 51 55 
 
 11 24 41 55 
 
 6 59 35 
 
 111. (S'ztn's Mean Longitude and 
 Anomaly in Years current. 
 
 Obliquity of the 
 Ecliptic 1st 
 
 
 January. 
 
 Yean. 
 
 M. Longitude. | . Anomaly. 
 
 
 A. D. 
 
 go/ //go/ // 
 
 a i a 
 
 1761 
 
 9 10 20 51 
 
 6 1 31 11 
 
 23 28 16 
 
 1781 
 
 9 10 30 7 
 
 6 1 18 27 
 
 23 28 9 
 
 1791 
 
 9 10 5 12 
 
 6 42 32 
 
 23 27 50 
 
 B 1792 
 
 9 10 50 1 
 
 6 1 26 15 
 
 23 27 48 
 
 1793 
 
 9 10 35 41 
 
 6 1 10 49 
 
 23 27 47 
 
 1794 
 
 9 10 21 22 
 
 6 55 24 
 
 23 27 48 
 
 1795 
 
 9 10 7 3 
 
 6 39 59 
 
 23 27 50 
 
 B 1796 
 
 9 10 51 52 
 
 6 1 23 42 
 
 23 27 52 
 
 1797 
 
 9 10 37 33 
 
 6 1 8 17 
 
 23 27 54 
 
 1798 
 
 9 10 23 13 
 
 6 52 51 
 
 23 27 55 
 
 1799 
 
 9 10 8 54 
 
 6 37 26 
 
 23 27 58 
 
 1800 
 
 9 9 54 35 
 
 6 22 1 
 
 23 28 
 
 1801 
 
 9 9 40 16 
 
 6 6 36 
 
 23 28 1 
 
 1802 
 
 9 9 25 56 
 
 5 29 51 10 
 
 23 28 
 
 1803 
 
 9 9 11 37 
 
 5 29 35 45 
 
 23 28 
 
 B 1804 
 
 9 9 56 26 
 
 6 19 28 
 
 23 27 58 
 
 1805 
 
 9 9 42 6 
 
 6042 
 
 23 27 55 
 
 1806 
 
 9 9 27 48 
 
 5 29 48 38 
 
 23 27 51 
 
 1807 
 
 9 9 13 29 
 
 5 29 33 13 
 
 23 27 48 
 
 B 1808 
 
 9 9 58 17 
 
 6 16 48 
 
 23 27 44 
 
 1809 
 
 9 9 43 57 
 
 6 1 31 
 
 23 27 42 
 
 1810 
 
 9 9 29 37 
 
 5 29 45 57 
 
 83 27 40 
 
 1811 
 
 9 9 15 17 
 
 5 29 30 32 
 
 23 27 39 
 
 B 1812 
 
 9 10 5 
 
 6 14 15 
 
 23 27 39 
 
 1813 
 
 9 9 45 45 
 
 5 29 58 49 
 
 23 27 40 
 
 1814 
 
 9 9 31 25 
 
 5 29 43 26 
 
 23 27 41 
 
 1815 
 
 9 9 17 5 
 
 5 29 27 58 
 
 23 27 43 
 
 B181G 
 
 9 10 1 53 
 
 6 11 41 
 
 23 27 46 
 
 1817 
 
 9 9 47 33 
 
 5 29 56 15 
 
 23 27 48 
 
 1818 
 
 9 9 S3 13 
 
 5 29 40 50 
 
 23 27 50 
 
 1819 
 
 9 9 18 53 
 
 5 29 25 24 
 
 23 27 52 
 
 B 1820 
 
 9 10 3 41 
 
 6097 
 
 23 27 52 
 
 1821 
 
 9 9 49 22 
 
 5 29 55 42 
 
 23 27 51 
 
 1841 
 
 9 9 58 39 
 
 5 29 42 59; 23 27 59 
 
 46J 
 
APPENDIX TO THE ASTRONOMY. 
 
 TABLE IV. The Sun's Mean Longitude and Anomaly for JllonHis and Days. 
 
 V 
 
 January. 
 
 V 
 
 February. 
 
 
 
 Sf 
 
 March. 
 
 Hfl 
 W 
 
 X 
 
 April. 
 
 3 
 
 
 ^ 
 
 
 ^ 
 
 
 '~*t 
 
 
 f 
 
 Longitude. 
 
 Anomaly. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 S. ' " 
 
 S. ' " 
 
 
 S. ' " 
 
 S. o ' " 
 
 
 S. ' " 
 
 S. ' " 
 
 
 S . ' " 
 
 S. ' " 
 
 l 
 
 59 8 
 
 59 8 
 
 1 
 
 1 1 32 27 
 
 1 1 32 21 
 
 1 
 
 1 29 8 20 
 
 1 29 8 9 
 
 1 
 
 2 29 41 38 
 
 2 29 41 22 
 
 2 
 
 1 58 17 
 
 1 58 17 
 
 2 
 
 1 2 31 35 
 
 1 2 31 29 
 
 2 
 
 2 7 28 
 
 2 7 17 
 
 c 
 
 3 40 46 
 
 3 40 30 
 
 3 
 
 2 57 25 
 
 2 57 24 
 
 3 
 
 1 3 30 43 
 
 1 3 30 37 
 
 5 
 
 2 1 6 36 
 
 2 1 6 25 
 
 { 
 
 3 1 39 55 
 
 3 1 39 38 
 
 4 
 
 3 56 33 
 
 3 56 32 
 
 4 
 
 1 4 29 52 
 
 1 4 29 46 
 
 4 
 
 2 2 5 45 
 
 2 2 5 34 
 
 4 
 
 3 2 39 3 
 
 3 2 38 46 
 
 5 
 
 4 55 42 
 
 4 55 41 
 
 5 
 
 1 5 29 
 
 1 5 28 53 
 
 5 
 
 2 3 4 53 
 
 2 3 4 42 
 
 ; 
 
 3 3 38 11 
 
 3 3 37 54 
 
 6 
 
 5 54 50 
 
 5 54 49 
 
 6 
 
 1 6 28 8 
 
 1 6 28 1 
 
 6 
 
 2441 
 
 2 4 3 49 
 
 6 
 
 3 4 37 20 
 
 3 4 37 3 
 
 7 
 
 6 53 58 
 
 6 53 57 
 
 7 
 
 1 7 27 17 
 
 1 7 27 10 
 
 7 
 
 2 5 3 10 
 
 2 5 2 58 
 
 7 
 
 3 5 36 28 
 
 3 5 36 11 
 
 8 
 
 7 53 7 
 
 7 53 6 
 
 8 
 
 1 8 26 25 
 
 1 8 26 17 
 
 8 
 
 2 6 2 18 
 
 2626 
 
 8 
 
 3 6 35 36 
 
 3 6 35 18 
 
 9 
 
 8 52 15 
 
 8 52 14 
 
 9 
 
 1 9 25 S3 
 
 1 9 25 26 
 
 9 
 
 2 7 1 26 
 
 2 7 1 14 
 
 c 
 
 3 7 34 45 
 
 3 7 34 27 
 
 10 
 
 9 51 2S 
 
 9 51 21 
 
 10 
 
 1 10 24 42 
 
 1 10 24 35 
 
 iO 
 
 2 8 35 
 
 2 8 22 
 
 to 
 
 3 8 33 53 
 
 3 8 33 35 
 
 11 
 
 10 50 32 
 
 10 50 30 
 
 11 
 
 1 11 23 50 
 
 1 11 23 42 
 
 11 
 
 2 8 59 43 
 
 2 8 59 30 
 
 11 
 
 3 9 33 1 
 
 3 9 32 43 
 
 12 
 
 11 49 40 
 
 11 49 38 
 
 12 
 
 1 12 22 58 
 
 1 12 22 50 
 
 12 
 
 2 9 58 51 
 
 2 9 58 38 
 
 12 
 
 3 10 32 10 
 
 3 10 31 52 
 
 13 
 
 12 48 48 
 
 12 48 46 
 
 IS 
 
 I 13 22 7 
 
 1 13 21 59 
 
 13 
 
 2 10 58 
 
 2 10 57 47 
 
 13 
 
 3 11 31 18 
 
 3 11 SO 59 
 
 14 
 
 13 47 57 
 
 13 47 54 
 
 14 
 
 1 14 21 15 
 
 1 14 21 7 
 
 14 
 
 2 11 57 8 
 
 2 11 56 55 
 
 14 
 
 3 12 30 26 
 
 3 12 SO 7 
 
 15 
 
 14 47 5 
 
 14 47 2 
 
 15 
 
 1 15 20 23 
 
 1 15 20 15 
 
 15 
 
 2 12 56 16 
 
 2 12 56 3 
 
 15 
 
 3 13 29 35 
 
 3 13 29 16 
 
 16 
 
 15 46 13 
 
 15 46 10 
 
 16 
 
 1 16 19 31 
 
 1 16 19 23 
 
 16 
 
 2 13 55 25 
 
 2 13 55 11 
 
 16 
 
 3 14 28 43 
 
 3 14 28 24 
 
 17 
 
 16 45 22 
 
 16 45 19 
 
 17 
 
 1 17 18 40 
 
 1 17 18 31 
 
 17 
 
 2 14 54 33 
 
 2 14 54 19 
 
 17 
 
 3 15 27 51 
 
 3 15 27 32 
 
 18 
 
 17 44 30 
 
 17 44 27 
 
 18 
 
 1 18 17 48 
 
 1 18 17 39 
 
 18 
 
 2 15 53 41 
 
 2 15 53 27 
 
 8 
 
 3 16 27 
 
 3 16 26 40 
 
 19 
 
 18 43 38 
 
 18 43 35 
 
 19 
 
 1 19 16 56 
 
 19 16 47 
 
 19 
 
 2 16 52 50 
 
 2 16 52 36 
 
 9 
 
 3 17 26 8 
 
 3 17 25 48 
 
 20 
 
 19 42 47 
 
 19 42 43 
 
 20 
 
 1 20 16 5 
 
 20 15 66 
 
 20 
 
 2 17 51 58 
 
 2 17 51 44 
 
 20 
 
 3 18 25 16 
 
 3 18 24 56 
 
 21 
 
 20 41 55 
 
 20 41 51 
 
 21 
 
 1 21 15 13 
 
 21 15 4 
 
 21 
 
 2 18 51 6 
 
 2 18 50 51 
 
 21 
 
 3 19 24 25 
 
 3 19 24 5 
 
 22 
 
 21 41 S 
 
 21 40 59 
 
 22 
 
 1 22 14 21 
 
 22 14 11 
 
 22 
 
 2 19 50 15 
 
 2 19 50 
 
 22 
 
 3 20 23 36 
 
 3 20 23 13 
 
 23 
 
 22 40 12 
 
 22 40 8 
 
 23 
 
 1 23 13 SO 
 
 23 13 20 
 
 23 
 
 2 20 49 23 
 
 2 20 49 8 
 
 23 
 
 3 21 22 41 
 
 3 21 22 21 
 
 24 
 
 23 39 20 
 
 O 23 39 16 
 
 24 
 
 1 24 12 38 
 
 24 12 28 
 
 24 
 
 2 21 48 3' 
 
 2 21 48 16 
 
 24 
 
 3 22 21 50 
 
 3 22 21 29 
 
 25 
 
 24 38 28 
 
 24 38 23 
 
 25 
 
 1 25 11 46 
 
 25 11 36 
 
 25 
 
 2 22 47 40 
 
 2 22 47 25 
 
 25 
 
 3 23 20 37 
 
 3 22 20 37 
 
 26 
 
 25 37 37 
 
 25 37 59 
 
 26 
 
 1 26 10 55 
 
 26 10 45 
 
 26 
 
 2 23 46 4b 
 
 2 23 46 33 
 
 26 
 
 3 24 20 6 
 
 3 24 19 45 
 
 27 
 
 26 36 45 
 
 26 36 40 
 
 27 
 
 1 27 10 S 
 
 27 9 52 
 
 27 
 
 2 24 45 56 
 
 2 24 45 40 
 
 
 3 25 19 15 
 
 3 25 18 54 
 
 28 
 
 27 35 53 
 
 27 35 48 
 
 28 
 
 1 28 9 11 
 
 1 28 9 
 
 28 
 
 2 25 45 5 
 
 2 25 44 53 
 
 28 
 
 3 26 18 23 
 
 3 26 18 2 
 
 -29 
 
 28 35 2 
 
 28 34 57 
 
 
 In the months Janua- 
 
 29 
 
 2 26 44 13 
 
 2 26 43 57 
 
 29 
 
 3 27 17 31 
 
 3 27 17 9 
 
 30 
 
 29 34 10 
 
 29 34 5 
 
 
 ry and February of bis- 
 
 30 
 
 2 27 43 21 
 
 2 27 43 5 
 
 30 
 
 3 28 16 40 
 
 3 28 16 18 
 
 
 
 
 
 sextile years take away 
 
 
 
 
 
 
 
 
 
 
 
 
 
 31 
 
 1 S3 18 
 
 1 33 12 
 
 
 one day from the time. 
 
 31 
 
 2 28 42 SO 
 
 2 28 42 14' 
 
 
 
 
SOLAR TABLES. 
 
 308 
 
 The Sun's Mean Longitude and Anomaly for .Months and Utnjs. 
 
 o 
 
 May: 
 
 w 
 
 V 
 
 June. 
 
 C 
 
 July. 
 
 O 
 
 fS 
 
 August.j 
 
 X 
 
 Longitude. 
 
 Anomaly. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 S. ' " 
 
 S. ' " 
 
 
 S. ' " 
 
 S. ' " 
 
 
 S. ' " 
 
 s. ' " 
 
 
 S. ' " 
 
 S. ' " 
 
 1 
 
 3 29 15 48 
 
 3 29 15 26 
 
 1 
 
 4 29 49 6 
 
 4 29 48 39 
 
 1 
 
 5 29 23 16 
 
 5 29 22 43 
 
 1 
 
 6 29 56 34 
 
 6 29 55 55 
 
 2 
 
 4 14 56 
 
 4 14 34 
 
 2 
 
 5 48 15 
 
 5 47 47 
 
 2 
 
 6 22 24 
 
 6 21 51 
 
 2 
 
 7 55 43 
 
 7 55 4 
 
 S 
 
 4 1 14 5 
 
 4 1 13 43 
 
 3 
 
 5 1 47 23 
 
 5 1 46 55 
 
 3 
 
 6 1 21 33 
 
 6 1 21 
 
 C 
 
 7 1 54 51 
 
 7 1 54 12 
 
 4 
 
 4 2 13 13 
 
 4 2 12 51 
 
 4 
 
 5 2 46 31 
 
 5 2 46 S 
 
 4 
 
 6 2 20 41 
 
 6 2 20 8 
 
 4 
 
 7 2 53 59 
 
 7 2 55 20 
 
 5 
 
 4 3 12 21 
 
 4 3 11 58 
 
 5 
 
 5 3 45 40 
 
 5 3 45 12 
 
 5 
 
 6 3 19 49 
 
 6 3 19 15 
 
 5 
 
 7 S 53 8 
 
 7 3 52 29 
 
 6 
 
 4 4 11 30 
 
 4 4 11 7 
 
 6 
 
 5 4 44 48 
 
 5 4 44 20 
 
 6 
 
 6 4 18 58 
 
 6 4 18 24 
 
 6 
 
 7 4 52 16 
 
 7 4 51 37 
 
 7 
 
 4 5 10 38 
 
 4 5 10 15 
 
 7 
 
 5 5 43 56 
 
 5 5 43 2b 
 
 7 
 
 6 5 18 6 
 
 6 5 17 32 
 
 7 
 
 7 5 51 24 
 
 7 5 50 44 
 
 8 
 
 4 6 9 46 
 
 4 6 9 23 
 
 8 
 
 5 6 43 5 
 
 5 6 42 37 
 
 8 
 
 6 6 17 14 
 
 6 16 40 
 
 8 
 
 7 6 50 33 
 
 7 6 49 54 
 
 9 
 
 4 7 8 55 
 
 4 T 8 32 
 
 9 
 
 5 7 42 13 
 
 5 7 41 44 
 
 9 
 
 6 7 16 23 
 
 6 7 15 49 
 
 9 
 
 7 7 49 41 
 
 7 7 49 1 
 
 iO 
 
 4883 
 
 4 8 7 40 
 
 10 
 
 5 8 41 21 
 
 5 8 40 52 
 
 10 
 
 6 8 15 31 
 
 6 8 14 56 
 
 10 
 
 7 8 48 49 
 
 7 8 48 9 
 
 11 
 
 4 9 7 11 
 
 4 9 6 47 
 
 11 
 
 5 9 40 30 
 
 5 9 40 1 
 
 il 
 
 6 9 14 39 
 
 6 9 14 4 
 
 11 
 
 7 9 47 58 
 
 7 9 47 18 
 
 1 
 
 4 10 6 20 
 
 4 10 5 56 
 
 12 
 
 5 10 39 38 
 
 5 10 39 8 
 
 12 
 
 6 10 13 48 
 
 6 10 13 13 
 
 12 
 
 7 10 47 6 
 
 7 10 46 25 
 
 13 
 
 4 11 5 28 
 
 4 11 5 4 
 
 13 
 
 5 11 38 46 
 
 5 11 38 16 
 
 IS 
 
 6 11 12 56 
 
 6 11 12 21 
 
 13 
 
 7 11 46 14 
 
 7 11 45 33 
 
 14 
 
 4 12 4 36 
 
 4 12 4 12 
 
 14 
 
 5 12 37 55 
 
 5 12 37 25 
 
 14 
 
 6 12 12 4 
 
 6 12 11 29 
 
 14 
 
 7 12 45 23 
 
 7 12 44 42 
 
 15 
 
 4 13 3 45 
 
 4 13 3 21 
 
 15 
 
 5 13 37 3 
 
 5 13 36 33 
 
 15 
 
 6 13 11 13 
 
 6 13 10 38 
 
 15 
 
 7 13 44 31 
 
 7 13 43 50 
 
 16 
 
 4 14 2 53 
 
 4 14 2 28 
 
 10 
 
 5 14 36 11 
 
 5 14 35 41 
 
 16 
 
 6 14 10 21 
 
 6 14 9 45 
 
 16 
 
 7 14 43 39 
 
 7 14 42 58 
 
 17 
 
 4 15 2 1 
 
 4 15 1 36 
 
 17 
 
 5 15 35 20 
 
 5 15 34 50 
 
 17 
 
 6 15 9 29 
 
 6 15 8 53 
 
 17 
 
 7 15 42 48 
 
 7 15 42 7 
 
 18 
 
 4 16 1 10 
 
 4 16 45 
 
 18 
 
 5 16 34 28 
 
 5 16 33 57 
 
 18 
 
 6 16 8 38 
 
 6 16 8 2 
 
 18 
 
 7 16 41 56 
 
 7 16 41 16 
 
 19 
 
 4 17 18 
 
 4 16 59 53 
 
 19 
 
 5 17 S3 36 
 
 5 17 33 5 
 
 19 
 
 6 17 7 46 
 
 6 17 7 10 
 
 19 
 
 7 17 41 4 
 
 7 17 40 22 
 
 20 
 
 4 17 59 26 
 
 4 17 59 1 
 
 20 
 
 5 18 32 45 
 
 5 18 32 14 
 
 20 
 
 6 18 6 54 
 
 6 18 6 18 
 
 20 
 
 7 18 40 13 
 
 7 18 39 31 
 
 21 
 
 4 18 58 35 
 
 4 18 58 10 
 
 21 
 
 5 19 31 53 
 
 5 19 31 32 
 
 21 
 
 6 19 6 S 
 
 6 19 5 26 
 
 21 
 
 7 19 39 21 
 
 7 19 38 39 
 
 22 
 
 4 19 57 43 
 
 4 19 57 17 
 
 22 
 
 5 20 31 1 
 
 5 20 30 30 
 
 22 
 
 6 20 5 11 
 
 6 20 4 34 
 
 22 
 
 7 20 38 29 
 
 7 20 37 47 
 
 23 
 
 4 20 56 51 
 
 4 20 56 25 
 
 23 
 
 5 21 30 10 
 
 5 21 29 39 
 
 23 
 
 6 21 410 
 
 6 21 3 42 
 
 23 
 
 7 21 37 38 
 
 7 21 30 56 
 
 24 
 
 4 21 56 
 
 4 21 55 34 
 
 24 
 
 5 22 29 18 
 
 5 22 28 46 
 
 24 
 
 6 22 3 28 
 
 6 22 2 51 
 
 24 
 
 7 22 36 46 
 
 7 22 36 4 
 
 25 
 
 4 22 55 8 
 
 4 22 54 42 
 
 25 
 
 5 23 28 26 
 
 5 23 27 54 
 
 25 
 
 6 23 2 36 
 
 6 23 1 59 
 
 25 
 
 - 23 35 54 
 
 7 23 35 11 
 
 20 
 
 4 23 54 16 
 
 4 23 53 50 
 
 26 
 
 5 24 27 35 
 
 5 24 27 3 
 
 26 
 
 6 24 1 44 
 
 6 24 1 7 
 
 26 
 
 7 24 35 3 
 
 7 24 34 20 
 
 -27 
 
 4 24 53 25 
 
 4 24 52 58 
 
 27 
 
 5 25 26 43 
 
 5 25 26 11 
 
 27 
 
 6 25 53 
 
 6 25 15 
 
 27 
 
 7 25 84 11 
 
 7 25 33 28 
 
 28 
 
 4 25 52 33 
 
 4 25 52 6 
 
 28 
 
 5 26 25 51 
 
 5 26 25 19 
 
 28 
 
 6 26 1 
 
 6 25 59 23 
 
 28 
 
 7 26 33 19 
 
 7 26 82 30 
 
 29 
 
 4 26 51 41 
 
 4 26 51 14 
 
 20 
 
 5 27 24 59 
 
 5 27 24 26 
 
 29 
 
 6 26 59 9 
 
 6 26 58 31 
 
 29 
 
 7 27 32 28 
 
 7 27 31 44 
 
 30 
 
 4 27 50 50 
 
 4 27 50 23 
 
 3(; 
 
 5 28 24 8 
 
 5 28 23 35 
 
 30 
 
 6 27 58 18 
 
 6 27 57 40 
 
 30 
 
 7 28 31 36 
 
 7 28 30 52 
 
 31 
 
 5 28 49 58 
 
 4 28 49 31 
 
 31 
 
 
 
 31 
 
 6 28 57 26 
 
 6 28 56 48 
 
 31 
 
 7 29 30 44 
 
 7 29 30 
 
381. 
 
 APPENDIX TO THE ASTRONOMY. 
 
 The Sun's Mean Longitude and Anomaly for Months anil Days. 
 
 C 
 
 September. 
 
 C 
 
 October. 
 
 C 
 
 .November. 
 
 C 
 
 December. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 Longitude. 
 
 Anomaly. 
 
 
 
 
 
 
 
 
 
 
 
 S. ' " 
 
 S. ' " 
 
 
 
 
 
 
 
 
 
 1 
 
 8 29 53 
 
 8 29 9 
 
 1 
 
 9043 
 
 9 3 13 
 
 i 
 
 10 37 21 
 
 10 36 26 
 
 1 
 
 11 11 31 
 
 11 10 30 
 
 2 
 
 8 1 29 1 
 
 8 1 28 17 
 
 2 
 
 9 1 3 11 
 
 9 1 2 21 
 
 2 
 
 10 1 36 29 
 
 10 1 35 34 
 
 2 
 
 11 1 10 39 
 
 11 1 9 38 
 
 S 
 
 8 2 28 9 
 
 8 2 27 25 
 
 3 
 
 9 2 2 19 
 
 9 2 I 29 
 
 3 
 
 10 2 35 38 
 
 10 2 S4 43 
 
 S 
 
 11 2 9 47 
 
 U 2 8 46 
 
 4 
 
 8 3 27 18 
 
 8 3 26 S3 
 
 4 
 
 9 3 1 28 
 
 9 3 38 
 
 4 
 
 10 3 34 46 
 
 10 S 33 50 
 
 4 
 
 11 3 8 56 
 
 11 3 7 55 
 
 5 
 
 8 4 26 26 
 
 8 4 25 41 
 
 5 
 
 9 4 36 
 
 9 3 59 46 
 
 5 
 
 10 4 S3 54 
 
 10 4 32 58 
 
 5 
 
 11 4 8 4 
 
 U 4 7 3 
 
 6 
 
 8 5 25 34 
 
 8 5 24 49! 
 
 6 
 
 9 4 59 44 
 
 9 4 58 54 
 
 6 
 
 10 5 33 3 
 
 10 5 32 7 
 
 6 
 
 U 5 7 12 
 
 11 5 6 11 
 
 7 
 
 8 6 24 43 
 
 8 6 23 58' 
 
 7 
 
 9 5 58 53 
 
 9 5 58 2 
 
 T 
 i 
 
 10 6 32 11 
 
 10 6 31 15 
 
 7 
 
 11 6 6 21 
 
 11 6 5 20 
 
 8 
 
 8 7 23 51 
 
 S 7 23 6 
 
 8 
 
 J 6 58 1 
 
 9 6 57 10 
 
 S 
 
 10 7 31 19 
 
 10 7 30 23 
 
 8 
 
 11 7 5 29 
 
 11 7 4 27 
 
 9 
 
 8 8 22 59 
 
 S 8 22 14 
 
 9 
 
 9 7 57 9 
 
 9 7 56 18 
 
 9 
 
 10 8 30 27 
 
 10 8 29 30 
 
 9 
 
 11 8 4 37 
 
 11 8 3 35 
 
 10 
 
 8 9 22 8 
 
 8 9 21 22 
 
 10 
 
 9 8 56 18 
 
 3 8 55 27 
 
 10 
 
 10 9 29 36 
 
 10 9 28 39 
 
 10 
 
 11 9 3 46 
 
 11 9 2 44 
 
 11 
 
 8 10 21 16 
 
 8 10 20 30 
 
 11 
 
 9 9 55 26 
 
 9 9 54 35 
 
 11 
 
 10 10 28 44 
 
 10 10 27 47 
 
 11 
 
 11 10 2 54 
 
 11 10 1 52 
 
 12 
 
 8 11 20 24 
 
 8 11 19 38 
 
 12 
 
 9 10 54 34 
 
 9 10 53 42 
 
 12 
 
 10 11 27 5a 
 
 10 11 26 5S 
 
 12 
 
 11 11 2 2 
 
 11 11 1 
 
 13 
 
 8 12 19 33 
 
 5 12 18 47 
 
 13 
 
 9 11 53 43 
 
 9 11 52 51 
 
 13 
 
 10 12 27 1 
 
 10 12 26 4 
 
 IS 
 
 11 12 1 11 
 
 11 12 8 
 
 14 
 
 8 13 18 41 
 
 8 13 17 55 
 
 14 
 
 9 12 52 51 
 
 9 12 51 59 
 
 14 
 
 10 13 26 <a 
 
 10 13 25 12 
 
 14 
 
 11 13 19 
 
 11 12 59 16 
 
 15 
 
 8 14 17 49 
 
 3 14 17 2 
 
 15 
 
 9 13 51 59 
 
 9 13 51 7 
 
 15 
 
 10 14 25 17 
 
 10 14 24 19 
 
 15 
 
 11 13 59 27 
 
 11 13 58 24 
 
 16 
 
 8 15 16 58 
 
 8 15 16 11 
 
 16 
 
 9 14 51 8 
 
 9 14 50 16 
 
 16 
 
 10 15 24 26 
 
 10 15 23 28 
 
 16 
 
 11 14 58 36 
 
 11 14 57 S3 
 
 17 
 
 8 16 16 6 
 
 8 16 15 19 
 
 17 
 
 9 15 50 If 
 
 9 15 49 24 
 
 17 
 
 10 16 23 34 
 
 10 16 22 36 
 
 17 
 
 1 1 15 57 44 
 
 U 15 56 41 
 
 18 
 
 8 17 15 14 
 
 8 17 14 27 
 
 18 
 
 9 16 49 24 
 
 9 16 48 31 
 
 18 
 
 10 17 22 42 
 
 10 17 21 44 
 
 18 
 
 U 16 56 52 
 
 U 16 55 48 
 
 19 
 
 8 18 14 23 
 
 8 18 13 35 
 
 19 
 
 J 17 48 33 
 
 9 17 47 40 
 
 19 
 
 10 18 21 50 
 
 10 18 20 52 
 
 19 
 
 11 17 56 1 
 
 11 17 54 57 
 
 20 
 
 8 19 13 31 
 
 8 19 12 4S 
 
 20 
 
 9 18 47 41 
 
 9 18 46 48 
 
 20 
 
 10 19 20 59 
 
 10 19 20 
 
 20 
 
 11 18 55 9 
 
 11 18 54 5 
 
 21 
 
 8 20 12 39 
 
 8 20 It 51 
 
 21 
 
 9 19 46 49 
 
 9 19 45 56 
 
 21 
 
 10 20 20 7 
 
 10 20 19 8 
 
 21 
 
 11 19 54 17 
 
 11 19 53 13 
 
 22 
 
 8 21 11 48 
 
 8 21 11 
 
 22 
 
 9 20 45 58 
 
 9 20 45 5 
 
 22 
 
 10 21 19 16 
 
 iO 21 18 17 
 
 22 
 
 11 20 53 26 
 
 11 20 52 22 
 
 23 
 
 8 22 10 56 
 
 8 22 10 8 
 
 23 
 
 9 21 45 6 
 
 9 21 44 IS 
 
 2S 
 
 10 22 18 24 
 
 10 22 17 25 
 
 23 
 
 U 21 52 34 
 
 11 21 51 29 
 
 24 
 
 8 23 10 4 
 
 8 23 9 16 
 
 24 
 
 9 22 44 14 
 
 9 2-2 43 20 
 
 24 
 
 TO 23 17 32 
 
 10 23 16 S3 
 
 24 
 
 11 22 51 42 
 
 11 22 50 87 
 
 25 
 
 8 24 9 13 
 
 8 24 8 25 
 
 25 
 
 9 23 43 23 
 
 9 23 42 29 
 
 5 
 
 10 24 16 41 
 
 10 24 15 41 
 
 25 
 
 11 23 50 51 
 
 11 23 49 46 
 
 26 
 
 8 25 8 21 
 
 8 25 7 32 
 
 26 
 
 9 24 42 31 
 
 9 24 41 37 
 
 26 
 
 10 25 15 49 
 
 10 25 14 49 
 
 26 
 
 11 24 49 59 
 
 U 24 48 54 
 
 27 
 
 8 26 7 29 
 
 8 26 6 40 
 
 57 
 
 9 25 41 39 
 
 9 25 40 45 
 
 27 
 
 10 26 14 57 
 
 10 26 13 57 
 
 27 
 
 11 25 49 7 
 
 11 25 48 2 
 
 28 
 
 8 27 6 38 
 
 8 27 5 49 
 
 28 
 
 9 26 40 48 
 
 9 26 39 54 
 
 28 
 
 10 27 14 6 
 
 10 27 13 6 
 
 28 
 
 11 26 48 16 
 
 11 26 47 11 
 
 29 
 
 8 28 5 46 
 
 8 28 4 57 
 
 29 
 
 9 27 39 56 
 
 9 27 39 2 
 
 29 
 
 10 28 13 14 
 
 10 28 12 14 
 
 29 
 
 11 27 47 24 
 
 11 27 46 18 
 
 30 
 
 8 29 4 54 
 
 8 29 4 5 
 
 30 
 
 9 28 39 4 
 
 9 28 38 9 
 
 30 
 
 10 29 12 22 
 
 10 29 11 22 
 
 30 
 
 11 28 46 32 
 
 11 28 45 26 
 
 
 
 
 31 
 
 9 29 38 13 
 
 9 29 37 18 
 
 
 
 
 31 
 
 11 29 43 41 
 
 U 29 44 35 
 
TABLE V. Of the Sun 1 * 
 Mean Longitude and Anom- 
 aly for Hours, Minutes, and 
 Seconds. 
 
 SOLAR TABLES. 
 
 TABLE VI. Equations of the Sun's Centre. 
 Argument. Sun's Mean Anomaly. 
 
 H 
 
 / // /;/ 
 
 H 
 
 O / '/ 
 
 / 
 
 II III jy 
 
 / 
 
 / // III 
 
 n 
 
 "' iv v 
 
 // 
 
 ii m jy 
 
 i 
 
 2 27 50 
 
 31 
 
 1 16 23 
 
 
 
 4 55 41 
 
 32 
 
 1 18 51 
 
 3 
 
 7 23 31 
 
 33 
 
 I 21 19 
 
 4 
 
 9 51 22 
 
 34 
 
 1 23 47 
 
 5 
 
 12 19 12 
 
 35 
 
 I 26 14 
 
 6 
 
 14 47 2 
 
 36 
 
 1 28 42 
 
 7 
 
 17 14 53 
 
 37 
 
 1 31 10 
 
 8 
 
 19 42 43 
 
 38 
 
 1 33 38 
 
 9 
 
 22 10 34 
 
 39 
 
 1 36 6 
 
 10 
 
 24 38 24 
 
 40 
 
 1 38 34 
 
 11 
 
 27 6 14 
 
 41 
 
 1 41 1 
 
 12 
 
 29 34 5 
 
 42 
 
 1 43 29 
 
 13 
 
 32 1 55 
 
 43 
 
 1 45 57 
 
 14 
 
 34 29 46 
 
 44 
 
 1 48 25 
 
 15 
 
 36 57 36 
 
 45 
 
 1 50 53 
 
 16 
 
 39 25 27 
 
 46 
 
 1 53 21 
 
 17 
 
 41 53 17 
 
 47 
 
 1 55 48 
 
 18 
 
 44 21 7 
 
 48 
 
 1 58 16 
 
 19 
 
 46 48 58 
 
 49 
 
 2 44 
 
 20 
 
 49 16 48 
 
 50 
 
 2 3 12 
 
 21 
 
 51 44 39 
 
 51 
 
 2 5 40 
 
 22 
 
 54 12 29 
 
 52 
 
 288 
 
 23 
 
 56 40 19 
 
 53 
 
 2 10 36 
 
 24 
 
 59 8 19 
 
 54 
 
 2 13 3 
 
 25 
 
 61 36 9 
 
 55 
 
 2 15 31 
 
 26 
 
 64 3 59 
 
 56 
 
 -2 17 59 
 
 27 
 
 66 SI 49 
 
 57 
 
 2 20 27 
 
 28 
 
 68 59 31 
 
 58 
 
 2 22 55 
 
 29 
 
 71 27 21 
 
 59 
 
 2 25 23 
 
 30 
 
 73 55 11 
 
 60 
 
 2 27 50 
 
 Signs. 1 V_2 3 4 5 
 
 O 
 
 
 
 1 II 
 
 000 
 
 1 II 
 
 56 47 
 
 O / tt 
 
 1 39 6 
 
 O I tl 
 
 \ 55 37 
 
 O 1 II ' /' 
 
 I 41 120 58 53 
 
 30 
 
 1 
 
 1 59 
 
 58 30 
 
 1 40 7 
 
 1 55 39 
 
 1 40 120 57 7 
 
 29 
 
 2 
 
 3 57 
 
 1 12 
 
 1 41 6 
 
 1 55 38 
 
 1 39 100 55 19 
 
 28 
 
 3 
 
 5 56 
 
 1 1 53 
 
 1 42 3 
 
 1 55 36 
 
 I 38 60 53 30 
 
 27 
 
 4 
 
 7 54 
 
 1 3 S3 
 
 I 42 59 
 
 1 55 31 
 
 1 37 
 
 51 40 
 
 26 
 
 5 
 
 9 52 
 
 1 5 12 
 
 I 43 52 
 
 1 55 24 
 
 1 35 52 
 
 49 49 
 
 25 
 
 6 
 
 11 50 
 
 1 6 50 
 
 I 44 44 
 
 1 55 15 
 
 1 34 43 
 
 47 57 
 
 24 
 
 7 
 
 13 48 
 
 1 8 27 
 
 I 45 34 
 
 1 55 3 
 
 1 33 32 
 
 46 5 
 
 23 
 
 8 
 
 15 46 
 
 1 10 2 
 
 1 46 22 
 
 1 54 50 
 
 1 32 19 
 
 44 11 
 
 22 
 
 9 
 
 17 43 
 
 I 11 36 
 
 I 47 8 
 
 1 54 35 
 
 1 31 4 
 
 42 16 
 
 21 
 
 10 
 
 19 40 
 
 1 13 9 
 
 I 47 53 
 
 1 54 17 
 
 1 29 47 
 
 40 21 
 
 20 
 
 11 
 
 21 37 
 
 1 14 41 
 
 I 48 35 
 
 I 53 57 
 
 I 28 29 
 
 38 25 
 
 19 
 
 12 
 
 23 33 
 
 1 16 11 
 
 I 49 15 
 
 1 53 36 
 
 1 27 9 
 
 36 28 
 
 18 
 
 IS 
 
 25 29 
 
 1 17 40 
 
 1 49 54 
 
 1 53 12 
 
 1 25 48 
 
 34 SO 
 
 17 
 
 14 
 
 27 25 
 
 1 19 8 
 
 1 50 30 
 
 1 52 46 
 
 1 24 25 
 
 32 32 
 
 16 
 
 15 
 
 29 20 
 
 I 34 
 
 1 51 5 
 
 1 52 18 
 
 L 23 
 
 30 33 
 
 15 
 
 16 
 
 31 15 
 
 1 21 59] 1 51 37 
 
 1 51 48 
 
 1 21 34 
 
 28 S3 
 
 14 
 
 17 
 
 33 9 
 
 1 23 22 
 
 1 52 f> 
 
 1 51 15 
 
 1 20 6 
 
 26 S3 
 
 13 
 
 18 
 
 35 2 
 
 1 24 44 
 
 1 52 36 
 
 I 50 41 
 
 1 18 36 
 
 24 33 
 
 12 
 
 19 
 
 86 55 
 
 1 26 5 
 
 1 53 3 
 
 I 50 5 
 
 1 17 5 
 
 22 32 
 
 U 
 
 20 
 
 38 47 
 
 1 27 24 
 
 1 53 27 
 
 1 49 26 
 
 1 15 33 
 
 20 30 
 
 10 
 
 21 
 
 40 39 
 
 1 28 41 
 
 1 53 50 
 
 1 48 46 
 
 I 13 59 
 
 18 28 
 
 9 
 
 22 
 
 42 30 
 
 1 29 57 
 
 1 54 10 
 
 1 48 3 
 
 1 12 24 
 
 16 26 
 
 8 
 
 23 
 
 44 20 
 
 1 31 11 
 
 1 54 28 
 
 1 47 19 
 
 1 10 47 
 
 14 24 
 
 7 
 
 24 
 
 46 9 
 
 L 32 24 
 
 1 54 44 
 
 1 46 32 
 
 199 
 
 12 21 
 
 6 
 
 25 
 
 47 57 
 
 1 33 35 
 
 1 54 58 
 
 1 45 44 
 
 1 7 29 
 
 10 18 
 
 5 
 
 -36 
 
 49 45 
 
 1 34 45 
 
 I 55 10 
 
 1 44 53 
 
 1 5 49 
 
 8 14 
 
 4 
 
 27 
 
 51 32 
 
 1 35 53 
 
 1 55 20 
 
 1 44 1 
 
 147 
 
 6 11 
 
 3 
 
 28 
 
 53 18 
 
 1 36 59 
 
 1 55 28 
 
 I 43 7 
 
 1 2 24 
 
 047 
 
 2 
 
 29 
 
 55 3 
 
 1 38 3 
 
 1 55 34 
 
 1 42 10 
 
 I 39 
 
 024 
 
 1 
 
 30 
 
 56 47 
 
 1 39 6 
 
 1 55 37 
 
 1 41 12 
 
 58 53 
 
 000 
 
 
 
 Signs + It +10 +9 +8 + 7+6 
 
 471 
 
Sfifi 
 
 APPENDIX TO THE ASTRONOMY. 
 
 TABLE VII. Logarithms of the Sun's Distancefrom the Earth. 
 
 Argument. Sun's Mean Anomaly. 
 
 Signs. 012345 
 
 O 
 
 
 
 
 
 
 
 o 
 
 
 
 5.007286 
 
 5.006347 
 
 5.003749 
 
 5.000124 
 
 4.996405 
 
 4.993620 
 
 SO 
 
 1 
 
 5.007285 
 
 5.006284 
 
 5.003640 
 
 4.999995 
 
 4.996292 
 
 4.993555 
 
 29 
 
 3 
 
 5.007282 
 
 5.006220 
 
 5.003531 
 
 4.999867 
 
 4.996180 
 
 4.993491 
 
 28 
 
 3 
 
 5.007277 
 
 5.006154 
 
 3.003420 
 
 4.999739 
 
 4.996069 
 
 4.993429 
 
 27 
 
 4 
 
 5.007269 
 
 5.006087 
 
 5.003307 
 
 4.999611 
 
 4.995959 
 
 4.993369 
 
 26 
 
 5 
 
 5.007260 
 
 5.006018 
 
 5.003194 
 
 4.999483 
 
 4.995850 
 
 4.993311 
 
 25 
 
 6 
 
 6.007249 
 
 5.005946 
 
 5.003080 
 
 4.999354 
 
 4.995742 
 
 4.993255 
 
 24 
 
 7 
 
 5.007235 
 
 5.005872 
 
 5.002965 
 
 4.999227 
 
 4.995636 
 
 4.993201 
 
 23 
 
 8 
 
 5.007218 
 
 S.005797 
 
 5.002849 
 
 4.999099 
 
 4.995531 
 
 4993150 
 
 22 
 
 9 
 
 5.007200 
 
 5.005720 
 
 5.002732 
 
 4.998971 
 
 4.995427 
 
 4.993102 
 
 21 
 
 10 
 
 5.007180 
 
 5.005642 
 
 5.002614 
 
 4.998844 
 
 4.995325 
 
 4.993055 
 
 20 
 
 11 
 
 5.007158 
 
 5.005562 
 
 5.002495 
 
 4.998717 
 
 4.995224 
 
 4.993009 
 
 19 
 
 12 
 
 5.007134 
 
 5.005480 
 
 5.002375 
 
 4.998590 
 
 4.995126 
 
 4.992966 
 
 18 
 
 13 
 
 5.007107 
 
 5.005397 
 
 5.002254 
 
 4.998463 
 
 4.995028 
 
 4.992926 
 
 17 
 
 14 
 
 5.007079 
 
 5.005312 
 
 5.002134 
 
 4.998336 
 
 4.994932 
 
 4.992888 
 
 16 
 
 15 
 
 5.007048 
 
 5.005225 
 
 5.002012 
 
 4.998210 
 
 4.994836 
 
 4.992852 
 
 15 
 
 16 
 
 5.007015 
 
 5.005136 
 
 5.001890 
 
 4.998084 
 
 4.994743 
 
 4.992818 
 
 14 
 
 17 
 
 5.006980 
 
 5.005047 
 
 5.001767 
 
 4 997960 
 
 4 994652 
 
 4.992786 
 
 13 
 
 18 
 
 5.006943 
 
 5.004956 
 
 5.001643 
 
 4.997837 
 
 4.994562 
 
 4.992757 
 
 12 
 
 19 
 
 5.006905 
 
 5.004863 
 
 5.001518 
 
 4.997714 
 
 4.994471 
 
 4.992731 
 
 11 
 
 20 
 
 5.006864 
 
 5.004768 
 
 5.501393 
 
 4.997591 
 
 4 994387 
 
 4.992706 
 
 10 
 
 21 
 
 5.006821 
 
 5.004672 
 
 5.001268 
 
 4.997468 
 
 4.994302 
 
 4.992683 
 
 9 
 
 22 
 
 5.006776 
 
 5.004575 
 
 5.001143 
 
 4.997347 
 
 4,994219 
 
 i.992663 
 
 8 
 
 23 
 
 5.006730 
 
 5.004477 
 
 5.001016 
 
 4.997226 
 
 4.994138 
 
 4.992646 
 
 7 
 
 24 
 
 5.006681 
 
 5.004377 
 
 5.000889 
 
 4.997106 
 
 4.994058 
 
 4.992631 
 
 6 
 
 25 
 
 5.00G630 
 
 5.004275 
 
 5.000762 
 
 4.996987 
 
 4.993980 
 
 4.992618 
 
 5 
 
 26 
 
 5.006577 
 
 5.004173 
 
 5.000635 
 
 4 996868 
 
 4.993904 
 
 4.992607 
 
 4 
 
 27 
 
 5.006522 
 
 5.004069 
 
 5 000508 
 
 4.996750 
 
 4.993831 
 
 4.992599 
 
 3 
 
 28 
 
 5.006466 
 
 5.003963 
 
 5.000380 
 
 4.996634 
 
 4.993759 
 
 4.992593 
 
 2 
 
 29 
 
 5.006408 
 
 5.003857 
 
 5.000252 
 
 4.996519 
 
 4.993688 
 
 4.992590 
 
 1 
 
 SO 
 
 5 006347 
 
 5.003749 
 
 5.000124 
 
 4 996105 
 
 4.993620 
 
 4.992589 
 
 
 
 Signs. 11 10 9 8 7 6 
 
SOLAR TABLES. 
 
 TABLE VIII. The Sun's Declination to every 
 Degree oj' his Longitude. 
 
 Argument. Sun's Longitude. 
 
 367 
 
 
 V .North. 
 
 1 North. 
 
 2 n North. 
 
 
 Deg. 
 
 6 =2= South. 
 
 T n^ South. 
 
 3 / South. 
 
 Deg. 
 
 
 O Decliu. 
 
 Q Declin. 
 
 O Declin. 
 
 
 
 O f ff 
 
 off 
 
 ' '/ 
 
 
 
 
 000 
 
 11 29 5 
 
 20 10 25 
 
 30 
 
 1 
 
 23 53 
 
 11 50 6 
 
 20 22 57 
 
 29 
 
 2 
 
 47 47 
 
 12 10 56 
 
 20 35 7 
 
 28 
 
 3 
 
 1 11 39 
 
 12 31 3*. 
 
 20 46 55 
 
 27 
 
 4 
 
 1 35 30 
 
 12 51 59 
 
 20 58 20 
 
 28 
 
 5 
 
 1 59 20 
 
 13 12 12 
 
 21 9 21 
 
 25 
 
 6 
 
 2 23 8 
 
 13 32 12 
 
 21 19 59 
 
 24 
 
 7 
 
 2 46 54 
 
 13 51 58 
 
 21 30 IS 
 
 23 
 
 8 
 
 3 10 37 
 
 14 11 30 
 
 21 40 3 
 
 22 
 
 9 
 
 3 34 17 
 
 14 SO 48 
 
 21 49 29 
 
 21 
 
 10 
 
 3 57 54 
 
 14 49 52 
 
 21 58 30 
 
 20 
 
 11 
 
 4 21 27 
 
 15 8 40 
 
 22 7 6 
 
 19 
 
 12 
 
 4 44 57 
 
 15 27 13 
 
 22 15 17 
 
 18 
 
 13 
 
 5 8 22 
 
 15 45 30 
 
 22 23 3 
 
 17 
 
 14 
 
 5 31 42 
 
 16 3 31 
 
 22 30 24 
 
 16 
 
 IS 
 
 5 54 57 
 
 16 21 16 
 
 22 37 18 
 
 15 
 
 16 
 
 6 IS 6 
 
 16 38 4i 
 
 22 43 47 
 
 14 
 
 17 
 
 6 41 9 
 
 16 55 55 
 
 22 49 50 
 
 13 
 
 18 
 
 746 
 
 17 12 48 
 
 22 55 27 
 
 12 
 
 19 
 
 7 26 57 
 
 17 29 23 
 
 23 38 
 
 11 
 
 20 
 
 7 49 41 
 
 17 45 40 
 
 23 5 22 
 
 10 
 
 21 
 
 8 12 17 
 
 18 1 38 
 
 23 9 39 
 
 9 
 
 22 
 
 8 3* 45 
 
 18 17 18 
 
 23 13 29 
 
 8 
 
 23 
 
 8 57 5 
 
 18 82 38 
 
 23 16 53 
 
 7 
 
 2* 
 
 9 19 17 
 
 18 47 S8 
 
 23 19 50 
 
 6 
 
 25 
 
 9 41 19 
 
 19 2 18 
 
 23 22 20 
 
 5 
 
 26 
 
 10 3 12 
 
 19 16 37 
 
 23 24 22 
 
 4 
 
 27 
 
 10 24 56 
 
 19 30 35 
 
 23 25 57 
 
 3 
 
 28 
 
 10 46 30 
 
 1M 44 13 
 
 23 27 5 
 
 2 
 
 29 
 
 11 7 53 
 
 19 57 30 
 
 23 27 46 
 
 1 
 
 30 
 
 11 29 5 
 
 20 10 25 
 
 23 28 
 
 
 
 D. 
 
 O Declin. 
 
 O Declin. 
 
 O Declin. 
 
 D. 
 
 
 11 K South. 
 
 10 .z South. 
 
 9 Vf iSouth. 
 
 
 
 5 to; North. 
 
 4 SI North. 
 
 8 35 North. 
 
 
 
 
 O O O O 
 
 CQ ! 35 
 
 
 
 g 
 
 ^ OI CO C.O CO 
 
 
 
 
 ^ 
 
 CO CO CO CS 
 
 
 
 
 3 
 
 3 
 
 - 01 o? 
 
 
 
 'O 
 
 S 
 
 
 CO 
 
 
 
 j 
 
 , ^ CO Oi C7> 
 
 
 R 
 
 
 fl 
 
 CD CO CO CO 
 
 
 5 
 
 
 1 
 
 ^ 1 . ^ . . T-l 
 
 
 g 
 
 
 
 
 
 1 
 
 ^-v 
 
 
 S 
 
 O "* G^ G> 
 * CO CO to CO 
 
 
 f 
 
 
 1 
 
 &! Ol O! 
 
 
 1 
 
 
 i 
 
 1 
 
 ^ T-C CO >O t-. 
 
 
 
 
 
 o 
 
 ^-i 1-^ r 1 t 
 
 
 g 
 
 
 
 e CD -o o 
 
 
 c 
 
 
 
 
 
 ?, 
 
 
 
 
 
 ^ & 
 
 
 ll 
 
 ~ 00 O O O 
 
 
 & rt 
 
 
 
 " Ci C^ CO CO 
 
 
 a g 
 
 
 i: 
 
 
 
 ?^j O 
 
 
 
 
 ^ CN CN (N C* 
 
 
 g ^ 
 
 CO 
 
 H 
 
 
 X 
 
 1 
 
 % i 
 
 aj 
 
 P J 
 a e 
 
 
 g / - co : o oo 
 
 If* *~ 
 
 | 1 cc -o <o ic 
 
 
 8; 
 
 
 Z 
 
 . >O CO (-s. CO 
 * O( CN C! CM 
 
 
 *s 
 
 
 
 
 
 
 
 i 
 
 Ol C! C3 (N 
 
 
 .?> 
 
 
 S 
 
 
 
 s 
 
 Ox 
 
 . 1 
 
 3 i 10 t, o co 
 
 Cft 
 
 ty> = 
 * to 
 
 
 1 
 
 - 10 0) 
 
 
 u t- 
 
 
 1 
 
 _ 10 W5 O CC 
 
 
 ^ ^ 
 
 
 i 
 
 1-1 r-r r. 
 
 
 O 5 
 
 
 S 
 
 . -^f t 10 10 
 
 
 
 
 h 
 
 - CV) C) CN Ci 
 
 
 * 
 
 
 I 
 
 < d 07 01 O 
 
 
 M 
 
 
 a 
 
 
 _ 
 
 5 
 
 
 | 
 
 * TT to JO O 
 
 "* 
 
 | 
 
 
 I 
 
 ^ 1O 'O *O 1O 
 
 
 H 
 
 
 i 
 
 i. CO Tf) CO Tf 
 
 
 
 
 i 
 
 
 
 
 
 
 w e <s 
 
 
 
 o 
 
 : g ! l^ Oi Ci 
 
 M 
 
 
 i 
 
 
 - 22^2 
 
 i 
 
 1 
 
 n 
 
 2 5. ' 
 
 if 
 
868 
 
 APPENDIX TO THE ASTRONOMY. 
 
 TABLE X. Of the Equation of Time fitted to each Degree of the Ecliptic. Place of the Jlpogee 3s. 9. 
 
 Argument. 
 
 Sun's Longitude. 
 
 c 
 
 r o 
 
 4 1 
 
 n 2 
 
 S 3 
 
 a 4 
 
 nj> 5 
 
 == 6 
 
 "I " 
 
 / 8 
 
 XJ 9 
 
 M* 1(1 
 
 **v i' ' 
 
 x n 
 
 ft 
 
 ' J ? 
 
 + 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 
 
 
 
 + 
 
 + 
 
 + 
 
 
 T). S. 
 
 m. s 
 
 m. s. 
 
 m. s. 
 
 m. s. 
 
 m. s. 
 
 m. s. 
 
 m. s- 
 
 m. s 
 
 m. s. 
 
 m. s. 
 
 s. 
 
 
 
 7 3G 
 
 I 9 
 
 3 51 
 
 1 IS 
 
 5 57 
 
 2 20 
 
 7 38 
 
 15 31 
 
 13 33 
 
 1 11 
 
 11 28 
 
 14 19 
 
 1 
 
 7 17 
 
 1 23 
 
 3 47 
 
 1 26 
 
 5 59 
 
 2 4 
 
 7 58 
 
 15 39 
 
 13 17 
 
 42 
 
 11 45 
 
 14 13 
 
 2 
 
 6 58 
 
 I 36 
 
 3 41 
 
 1 40 
 
 6 
 
 1 48 
 
 8 19 
 
 15 46 
 
 13 
 
 12 
 
 12 1 
 
 14 6 
 
 3 
 
 6 39 
 
 I 48 
 
 3 37 
 
 I 55 
 
 6 1 
 
 1 31 
 
 8 40 
 
 15 52 
 
 12 42 
 
 + 17 
 
 12 17 
 
 13 59 
 
 4 
 
 6 20 
 
 2 
 
 3 32 
 
 2 7 
 
 6 1 
 
 1 14 
 
 9 1 
 
 15 57 
 
 12 23 
 
 46 
 
 12 32 
 
 13 51 
 
 5 
 
 6 1 
 
 2 11 
 
 3 26 
 
 2 20 
 
 6 
 
 56 
 
 9 21 
 
 16 2 
 
 12 4 
 
 I 1C. 
 
 12 46 
 
 13 43 
 
 6 
 
 5 42 
 
 2 22 
 
 3 19 
 
 2 33 
 
 5 59 
 
 38 
 
 9 41 
 
 16 6 
 
 11 44 
 
 1 45 
 
 12 59 
 
 13 34 
 
 7- 
 4 
 
 5 24 
 
 2 32 
 
 3 12 
 
 2 45 
 
 5 57 
 
 20 
 
 10 1 
 
 16 9 
 
 LI 23 
 
 2 14 
 
 13 12 
 
 13 24 
 
 8 
 
 5 5 
 
 2 42 
 
 3 4 
 
 2 58 
 
 5 54 
 
 + 1 
 
 10 20 
 
 16 11 
 
 11 1 
 
 2 43 
 
 13 24 
 
 13 14 
 
 9 
 
 4 47 
 
 2 51 
 
 2 56 
 
 3 11 
 
 5 51 
 
 18 
 
 10 39 
 
 16 13 
 
 10 39 
 
 3 11 
 
 13 35 
 
 13 3 
 
 10 
 
 4 28 
 
 3 
 
 2 47 
 
 3 23 
 
 5 47 
 
 37 
 
 10 57 
 
 16 13 
 
 10 16 
 
 3 39 
 
 13 45 
 
 12 51 
 
 11 
 
 4 9 
 
 3 8 
 
 2 38 
 
 3 35 
 
 5 42 
 
 57 
 
 11 15 
 
 16 IS 
 
 9 53 
 
 4 7 
 
 13 54 
 
 12 39 
 
 12 
 
 3 50 
 
 3 16 
 
 -I 29 
 
 3 46 
 
 5 37 
 
 1 17 
 
 11 S3 
 
 16 12 
 
 9 29 
 
 4 35 
 
 14 2 
 
 12 27 
 
 13 
 
 3 32 
 
 3 23 
 
 2 19 
 
 3 58 
 
 5 31 
 
 1 38 
 
 11 51 
 
 16 10 
 
 9 5 
 
 5 2 
 
 14 9 
 
 12 14 
 
 14 
 
 3 13 
 
 3 30 
 
 2 8 
 
 4 9 
 
 5 24 
 
 1 58 
 
 12 8 
 
 16 7 
 
 8 40 
 
 5 29 
 
 14 16 
 
 12 
 
 15 
 
 2 55 
 
 3 36 
 
 1 57 
 
 4 19 
 
 5 17 
 
 2 19 
 
 12 25 
 
 16 4 
 
 8 14 
 
 5 56 
 
 14 22 
 
 11 46 
 
 16 
 
 2 37 
 
 3 41 
 
 1 46 
 
 4 29 
 
 5 9 
 
 2 40 
 
 12 41 
 
 16 
 
 7 48 
 
 6 22 
 
 14 27 
 
 11 31 
 
 17 
 
 2 19 
 
 3 46 
 
 1 35 
 
 4 39 
 
 5 1 
 
 3 1 
 
 12 57 
 
 15 55 
 
 7 22 
 
 6 48 
 
 14 31 
 
 11 16 
 
 18 
 
 2 1 
 
 3 50 
 
 1 23 
 
 4 48 
 
 4 52 
 
 3 22 
 
 13 12 
 
 15 49 
 
 6 55 
 
 7 13 
 
 14 35 
 
 11 1 
 
 19 
 
 t 43 
 
 3 53 
 
 t 11 
 
 4 57 
 
 4 43 
 
 3 44 
 
 13 27 
 
 15 42 
 
 6 28 
 
 7 37 
 
 14 38 
 
 10 46 
 
 20 
 
 1 26 
 
 3 56 
 
 59 
 
 5 5 
 
 4 33 
 
 4 5 
 
 13 42 
 
 15 35 
 
 6 
 
 8 1 
 
 14 40 
 
 10 30 
 
 21 
 
 1 9 
 
 3 58 
 
 46 
 
 5 13 
 
 4 22 
 
 4 26 
 
 13 56 
 
 15 26 
 
 5 32 
 
 8 24 
 
 14 41 
 
 10 14 
 
 22 
 
 52 
 
 1 
 
 34 
 
 5 20 
 
 4 11 
 
 4 47 
 
 14 9 
 
 15 17 
 
 5 4 
 
 8 47 
 
 14 42 
 
 9 58 
 
 23 
 
 36 
 
 4 1 
 
 21 
 
 5 27 
 
 3 50 
 
 5 9 
 
 14 21 
 
 15 7 
 
 4 36 
 
 9 9 
 
 14 41 
 
 9 41 
 
 24 
 
 20 
 
 4 1 
 
 _, 
 
 5 33 
 
 3 46 
 
 5 30 
 
 14 33 
 
 14 56 
 
 4 8 
 
 9 31 
 
 14 40 
 
 9 24 
 
 '.. 
 
 + 4 
 
 4 1 
 
 + 5 
 
 5 39 
 
 3 3S 
 
 5 32 
 
 14 44 
 
 14 44 
 
 3 39 
 
 9 53 
 
 14 39 
 
 9 6 
 
 26 
 
 11 
 
 4 
 
 19 
 
 5 44 
 
 3 19 
 
 6 13 
 
 14 53 
 
 14 31 
 
 S 10 
 
 10 14 
 
 14 37 
 
 8 48 
 
 27 
 
 26 
 
 3 59 
 
 31 
 
 5 48 
 
 3 4 
 
 6 35 
 
 15 5 
 
 14 17 
 
 2 41 
 
 10 34 
 
 14 34 
 
 8 30 
 
 28 
 
 40 
 
 3 57 
 
 46 
 
 5 52 
 
 2 50 
 
 6 56 
 
 15 14 
 
 14 3 
 
 2 11 
 
 10 53 
 
 14 30 
 
 8 12 
 
 29 
 
 53 
 
 3 54 
 
 59 
 
 5 55 
 
 2 35 
 
 7 17 
 
 15 23 
 
 13 48 
 
 1 41 
 
 11 11 
 
 14 25 
 
 7 54 
 
 30 
 
 1 9 
 
 3 51 
 
 1 13 
 
 5 57 
 
 2 20 
 
 7 28 
 
 15 31 
 
 13 33 
 
 1 11|11 28 
 
 14 19 
 
 7 36 
 
 The equations with + are to be added to the apparent time to have the mean time ; those with 
 are to he subtracted from the apparent for the mean time. 
 
SOLAR TABLES. 
 
 369 
 
 TABLE XI. The Sun's Longitude for every Day in the Fear, at JVoon. 
 
 c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \-_ 
 
 January . 
 
 February. 
 
 March. 
 
 April. 
 
 May. 
 
 June. 
 
 July. 
 
 August. 
 
 Sept. 
 
 October. 
 
 Novem. 
 
 Decem. 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S. ' 
 
 S. o ' 
 
 S. ' 
 
 S. ' 
 
 S. ' 
 
 S. ' 
 
 S. ' 
 
 S. ' 
 
 S. o ' 
 
 S. ' 
 
 S. ' 
 
 S. ' 
 
 \ 
 
 9 11 21 
 
 10 12 54 
 
 11 11 8 
 
 11 55 
 
 1 11 12 
 
 2 11 2 
 
 3 9 41 
 
 4 9 17 
 
 597 
 
 6 8 26 
 
 7 9 15 
 
 8 9 32 
 
 
 
 9 12 23 
 
 10 13 55 
 
 It 12 8 
 
 12 54 
 
 1 12 10 
 
 2 12 
 
 3 10 38 
 
 4 10 14 
 
 5 10 6 
 
 6 9 25 
 
 7 10 15 
 
 8 10 33 
 
 3 
 
 9 13 24 
 
 10 14 56 
 
 11 13 8 
 
 13 53 
 
 1 13 9 
 
 2 12 57 
 
 3 11 35 
 
 4 11 11 
 
 5 11 4 
 
 6 10 24 
 
 7 11 16 
 
 8 11 34 
 
 4 
 
 9 14 25 
 
 10 15 57 
 
 11 14 8 
 
 14 52 
 
 1 14 7 
 
 2 IS 54 
 
 S 12 32 
 
 4 12 9 
 
 5 12 2 
 
 6 11 24 
 
 7 12 16 
 
 8 12 35 
 
 5 
 
 9 15 26 
 
 10 16 57 
 
 11 15 8 
 
 15 51 
 
 1 15 5 
 
 2 14 52 
 
 3 13 30 
 
 4 13 6 
 
 5 13 
 
 6 12 23 
 
 r 13 16 
 
 8 13 36 
 
 6 
 
 9 16 27 
 
 10 17 5 r 
 
 11 16 8 
 
 16 50 
 
 1 16 3 
 
 2 15 49 
 
 3 14 27 
 
 4 14 4 
 
 5 13 59 
 
 6 13 22 
 
 7 14 16 
 
 8 14 37 
 
 ( 
 
 9 17 29 
 
 10 18 59 
 
 11 17 8 
 
 17 49 
 
 1 17 1 
 
 2 16 46 
 
 3 15 24 
 
 4 15 2 
 
 5 14 57 
 
 6 14 21 
 
 ' 15 17 
 
 8 15 S8 
 
 8 
 
 9 18 30 
 
 10 20 
 
 11 18 8 
 
 18 48 
 
 1 17 59 
 
 2 17 44 
 
 3 16 21 
 
 4 15 59 
 
 5 15 55 
 
 6 15 21 
 
 7 16 17 
 
 8 16 39 
 
 9 
 
 9 19 Si 
 
 10 21 
 
 11 19 8 
 
 19 47 
 
 1 18 56 
 
 2 18 41 
 
 3 17 18 
 
 4 16 57 
 
 j 16 53 
 
 6 16 20 
 
 7 17 17 
 
 8 17 40 
 
 iO 
 
 9 20 3: 
 
 10 22 1 
 
 11 20 8 
 
 20 45 
 
 1 19 54 
 
 2 19 39 
 
 3 18 15 
 
 4 17 54 
 
 5 17 52 
 
 6 17 19 
 
 7 18 18 
 
 8 18 41 
 
 11 
 
 9 21 S3 
 
 10 23 1 
 
 11 21 7 
 
 21 44 
 
 1 20 52 
 
 2 20 35 
 
 3 19 13 
 
 4 18 52 
 
 5 18 50 
 
 6 18 19 
 
 7 19 18 
 
 8 19 42 
 
 12 
 
 9 22 34 
 
 10 24 2 
 
 11 22 7 
 
 22 43 
 
 1 21 50 
 
 2 21 33 
 
 3 20 10 
 
 4 19 49 
 
 5 19 49 
 
 6 19 18 
 
 7 20 19 
 
 8 20 43 
 
 13 
 
 9 23 35 
 
 10 25 3 
 
 11 23 7 
 
 23 41 
 
 1 22 48 
 
 2 22 30 
 
 3 21 7 
 
 4 20 47 
 
 5 20 47 
 
 G 20 18 
 
 7 21 19 
 
 8 21 44 
 
 14 
 
 9 24 36 
 
 10 26 3 
 
 11 24 6 
 
 24 40 
 
 1 23 45 
 
 2 23 28 
 
 3 22 4 
 
 4 21 45 
 
 5 21 46 
 
 6 21 17 
 
 7 22 20 
 
 8 22 45 
 
 15 
 
 9 25 37 
 
 10 27 4 
 
 11 25 6 
 
 25 39 
 
 I 24 43 
 
 2 24 25 
 
 3 23 2 
 
 4 22 42 
 
 5 22 44 
 
 6 22 17 
 
 7 23 20 
 
 8 23 47 
 
 16 
 
 9 26 S9 
 
 10 28 4 
 
 11 26 6 
 
 26 87 
 
 1 25 41 
 
 2 25 22 
 
 3 23 59 
 
 4 23 40 
 
 5 23 43 
 
 6 23 17 
 
 7 24 21 
 
 8 24 48 
 
 17 
 
 9 27 39 
 
 10 29 4 
 
 11 27 5 
 
 27 36 
 
 1 26 39 
 
 2 26 19 
 
 3 24 56 
 
 4 24 38 
 
 5 24 42 
 
 6 24 16 
 
 7 25 21 
 
 8 25 49 
 
 18 
 
 9 28 41 
 
 11 5 
 
 11 28 5 
 
 28 34 
 
 1 27 36 
 
 2 27 17 
 
 3 25 53 
 
 4 25 36 
 
 5 25 40 
 
 6 25 16 
 
 7 26 22 
 
 8 26 50 
 
 19 
 
 9 29 42 
 
 1115 
 
 11 29 4 
 
 29 33 
 
 1 28 34 
 
 2 28 14 
 
 3 26 51 
 
 4 26 S3 
 
 5 26 39 
 
 6 26 16 
 
 T 27 23 
 
 8 27 51 
 
 20 
 
 10 43 
 
 11 2 6 
 
 004 
 
 1 31 
 
 1 29 32 
 
 2 29 11 
 
 3 27 48 
 
 4 27 31 
 
 5 27 38 
 
 6 27 16 
 
 7 28 23 
 
 8 28 52 
 
 21 
 
 10 1 44 
 
 11 3 6 
 
 013 
 
 1 1 30 
 
 2 29 
 
 308 
 
 3 28 45 
 
 4 28 29 
 
 5 28 37 
 
 6 28 15 
 
 7 29 24 
 
 8 29 54 
 
 22 
 
 10 2 45 
 
 11 4 6 
 
 023 
 
 1 2 28 
 
 2 1 27 
 
 316 
 
 3 29 43 
 
 4 29 27 
 
 5 29 35 
 
 6 29 15 
 
 8 25 
 
 9 55 
 
 23 
 
 10 3 46 
 
 1157 
 
 032 
 
 I 3 26 
 
 2 2 25 
 
 323 
 
 4 40 
 
 5 25 
 
 6 34 
 
 7 15 
 
 8 1 26 
 
 9 1 56 
 
 24 
 
 10 4 47 
 
 11 6 7 
 
 4 1 
 
 1 4 25 
 
 2 3 22 
 
 330 
 
 4 1 37 
 
 5 1 23 
 
 6 1 33 
 
 7 1 15 
 
 8 2 26 
 
 9 2 57 
 
 25 
 
 10 5 48 
 
 11 7 7 
 
 5 1 
 
 1 5 23 
 
 2 4 20 
 
 3 3 57 
 
 4 2 35 
 
 5 2 21 
 
 G 2 32 
 
 7 2 lo 
 
 8 3 27 
 
 9 3 58 
 
 26 
 
 10 6 49 
 
 1187 
 
 060 
 
 1 6 21 
 
 2 5 17 
 
 3 4 55 
 
 4 S 32 
 
 5 3 19 
 
 6 3 31 
 
 7 3 15 
 
 8 4 28 
 
 9 4 59 
 
 27 
 
 10 7 50 
 
 1198 
 
 6 59 
 
 1 7 20 
 
 2 6 15 
 
 3 5 52 
 
 4 4 29 
 
 5 4 17 
 
 6 4 SO 
 
 7 4 15 
 
 8 5 29 
 
 9.6 1 
 
 28 
 
 10 8 51 
 
 11 10 8 
 
 7 59 
 
 1 8 18 
 
 2 7 12 
 
 3 6 49 
 
 4 5 27 
 
 5 5 15 
 
 6 5 29 
 
 7 5 15 
 
 8 6 30 
 
 972 
 
 29 
 
 (0 9 52 
 
 
 8 58 
 
 1 9 16 
 
 2 8 10 
 
 3 7 46 
 
 4 6 24 
 
 5 6 13 
 
 6 6 28 
 
 7 6 15 
 
 8 7 31 
 
 983 
 
 3( 
 
 10 10 52 
 
 
 9 57 
 
 1 10 14 
 
 297 
 
 3 8 43 
 
 4 7 22 
 
 5 7 11 
 
 6 7 27 
 
 7 7 15 
 
 8 8 32 
 
 994 
 
 31 
 
 10 11 53 
 
 
 10 56 
 
 
 2 10 5 
 
 
 4 8 19 
 
 589 
 
 
 7 8 15 
 
 
 9 10 5 
 
370 
 
 APPENDIX TO THE ASTRONOMY. 
 
 LUNAR TABLES. 
 
 TABLE I. Of the Moon's Mean Motions in 
 Julian Fears. 
 
 Years. 
 
 3 's M. Long. 
 
 j) 's M. Ano. 
 
 3> 's ft Retro. 
 
 
 
 
 
 1 
 
 2 
 S 
 B 4 
 
 5 
 
 s. ' " 
 4 9 23 5 
 8 18 46 11 
 28 9 16 
 5 20 42 57 
 10 6 2 
 
 2 28 43 15 
 5 27 26 30 
 8 26 9 45 
 7 56 54 
 3 6 40 9 
 
 19 19 43 
 1 8 39 26 
 1 27 59 9 
 2 17 22 3 
 3 6 41 46 
 
 6 
 7 
 B 8 
 9 
 10 
 
 2 9 29 7 
 6 18 52 13 
 11 11 25 53 
 3 20 48 59 
 8 12 4 
 
 6 5 23 23 
 9 4 6 39 
 15 53 47 
 3 14 37 3 
 6 13 20 17 
 
 3 26 1 29 
 4 15 21 12 
 5 4 44 6 
 5 24 3 49 
 6 13 23 32 
 
 11 
 B 12 
 IS 
 14 
 15 
 
 9 35 9 
 5 2 8 50 
 9 11 31 55 
 1 20 55 1 
 6 18 6 
 
 9 12 3 32 
 23 50 17 
 3 22 33 56 
 6 21 17 11 
 9 20 26 
 
 7 2 43 15 
 7 22 6 9 
 8 11 25 52 
 9 45 35 
 9 20 5 18 
 
 B 16 
 17 
 
 18 
 19 
 B 20 
 
 10 22 51 46 
 3 2 14 52 
 7 11 37 57 
 11 21 1 2 
 4 13 S4 43 
 
 I 1 47 34 
 4 30 50 
 6 29 14 4 
 9 27 57 19 
 I 9 44 30 
 
 10 9 28 12 
 10 28 47 55 
 11 18 7 38 
 7 27 21 
 26 50 15 
 
 B 40 
 B 60 
 B 80 
 B 100 
 B 200 
 
 8 27 9 26 
 1 10 44 9 
 
 5 24 18 52 
 10 7 53 35 
 8 15 47 HI 
 
 2 19 28 56 
 3 29 13 24 
 5 8 57 52 
 6 18 42 20 
 1 7 24 40 
 
 1 23 40 30 
 2 20 30 45 
 3 17 21 
 4 14 11 15 
 8 28 22 13 
 
 B 300 
 B 400 
 B 500 
 
 6 23 40 4f> 
 5 1 34 20 
 3 9 27 5r 
 
 7 26 7 
 2 14 49 20 
 9 3 31 40 
 
 1 12 33 45 
 5 26 45 
 10 10 56 15 
 
 TABLE II. Tlie Moon's Mean Longitude and 
 Anomaly for Current Fears. 
 
 A. JD. 
 
 Mean Long. 
 
 Mean Aiiom. 
 
 Long. ft. 
 
 
 S. ' " 
 
 S. ' " 
 
 S. oi" 
 
 1761 
 
 7188 
 
 10 12 34 50 
 
 2 7 33 33 
 
 1781 
 
 11 14 42 54 
 
 11 22 19 18 
 
 1 10 43 18 
 
 1791 
 
 7 14 54 59 
 
 6 5 39 35 
 
 6 27 19 46 
 
 1792 
 
 7 28 40 
 
 9 17 26 44 
 
 6 7 56 52 
 
 1793 
 
 4 16 51 45 
 
 16 9 59 
 
 5 18 37 9 
 
 1794 
 
 8 26 14 51 
 
 3 14 53 14 
 
 4 29 17 26 
 
 1795 
 
 1 5 37 57 
 
 6 13 36 29 
 
 4 9 57 43 
 
 B 1796 
 
 5 28 11 37 
 
 9 25 23 38 
 
 3 20 34 49 
 
 1797 
 
 10 7 34 43 
 
 24 6 53 
 
 3 1 15 6 
 
 1798 
 
 2 16 57 48 
 
 S 22 50 8 
 
 2 11 55 23 
 
 1799 
 
 6 26 20 54 
 
 6 21 63 23 
 
 1 22 35 40 
 
 1800 
 
 11 5 44 
 
 9 20 16 38 
 
 1 3 15 57 
 
 1801 
 
 3 15 7 5 
 
 18 59 52 
 
 13 56 14 
 
 1802 
 
 7 24 30 11 
 
 3 17 43 7 
 
 11 24 36 31 
 
 1803 
 
 3 53 16 
 
 6 16 26 22 
 
 11 5 16 48 
 
 B 1804 
 
 4 26 26 57 
 
 9 28 13 31 
 
 10 15 53 54 
 
 1805 
 
 9 5 50 2 
 
 6 56 46 
 
 9 26 34 11 
 
 1806 
 
 1 15 13 8 
 
 3 25 40 1 
 
 9 7 14 28 
 
 1807 
 
 5 24 36 14 
 
 6 24 23 16 
 
 8 17 54 45 
 
 B 1808 
 
 10 17 9 54 
 
 10 6 10 25 
 
 7 28 31 51 
 
 1809 
 
 2 26 33 
 
 1 4 53 40 
 
 7 9 12 8 
 
 1810 
 
 7 5 56 5 
 
 4 3 36 55 
 
 6 19 52 25 
 
 1811 
 
 11 15 19 11 
 
 7 2 20 9 
 
 6 32 42 
 
 B 1812 
 
 4 7 52,52 
 
 10 14 7 18 
 
 5 11 9 48 
 
 1813 
 
 8 17 15 57 
 
 1 12 50 33 
 
 4 21 50 5 
 
 1814 
 
 26 39 3 
 
 4 11 33 48 
 
 4 2 30 22 
 
 1815 
 
 5628 
 
 7 10 17 3 
 
 3 13 10 39 
 
 B 1816 
 
 9 28 35 49 
 
 10 22 4 12 
 
 2 23 47 45 
 
 1817 
 
 2 7 58 55 
 
 1 20 47 27 
 
 2 4 28 2 
 
 1818 
 
 6 17 22 
 
 4 19 30 42 
 
 1 15 8 19 
 
 1819 
 
 10 26 45 6 
 
 7 18 13 57 
 
 25 48 36 
 
 B1820 
 
 3 19 18 47 
 
 11 1 6 
 
 6 25 42 
 
 1821 
 
 7 28 41 54 
 
 1 28 44 21 
 
 11 17 5 59 
 
 184! 
 
 12 16 37 
 
 3 8 28 51 10 20 15 44 
 
LUNAR TABLES. 
 
 371 
 
 1 
 
 a 
 
 
 . 
 
 . 00 CS S O 4 
 
 G) co CO rr 
 
 10 IO tO t- l~- 
 
 CO OO CS O O 
 
 C! Oi~eO rP 
 
 rr 10 10 to b- 
 
 ^ 
 
 
 eg 
 
 S CO rr I-H o) 
 
 
 c; co rr 10 
 
 -H O} CO 10 
 
 14 G* CO rP co 
 
 I-H G? CO rr 
 
 IO 
 
 
 _ 
 
 - O CO K O CO 
 
 to OS G? to OS 
 
 Gt 10 CO 10 
 
 CO i i rr b> H 
 
 rr K. o co to 
 
 O CO to CS 01 
 
 IO 
 
 
 o 
 
 ,_| rH Cl 01 
 
 G? O) CO CO CO 
 
 
 10 14 
 
 14 i Gl G< G? 
 
 eo co co CO rP 
 
 rr 
 
 
 s 
 
 o co CO CO CO eo 
 
 eo CO co CO co 
 
 CO co co co eo 
 
 
 
 rp rP rf rr rP 
 
 rr 
 
 
 
 t- rH IO CS CO 
 
 b- ' IO OS CO 
 
 (^ 10 OS co 
 
 > - 10 CS CO 
 
 b- in O G! 
 
 to S rp 00 G) 
 
 to 
 
 
 
 5 10 10 rr co eo 
 
 Gl G3 4 
 
 10 10 rr co co 
 
 CH < -H 
 
 10 10 rr co co 
 
 Gl Gl -H 
 
 IO 
 
 
 _>> 
 
 eo b- *i 10 Cs 
 
 CO t^ >4 1O OS 
 
 C3 to O rp CO 
 
 O! to O rp 00 
 
 l-H IO OS CO , 
 
 -H lo Cs CO b~ 
 
 o 
 
 M 
 
 09 
 
 C 
 
 - 10 10 
 
 
 CO CO rf rp rp 
 
 IO 10 
 
 i4 ,-, G? C! 
 
 co co CO rr rr 
 
 10 
 
 
 
 cs 
 
 3 
 c 
 
 o co to O eo to 
 
 OS O? IO 50 -H 
 G( 
 
 rP b, O oo to 
 
 14 O? 14 Gl 
 
 Cs O7 to OS G? 
 
 1-H -H G< 1-1 
 
 10 CO 14 rP b- 
 
 O O? 14 
 
 O CO to OS O! 
 14 Gl 01 
 
 IO 
 
 1 
 
 < 
 
 yi G) O! CO CO CO 
 
 rr rP >o 10 to 
 
 to to t~ b~ oo 
 
 CO CS OS OS O 
 
 O 14 T-I O O 
 
 -H 14 14 OI OI 
 
 eo 
 
 
 ^ 
 
 t 01 - GJ b. 0! 
 
 CO i-H rT G! 
 
 ITS co rP -4 
 
 IO G? CO *4 
 
 b, G) t^ P t~ 
 
 rT G? IO CO 
 
 C! b~ 01 b^ C) 
 
 rp 14 10 G7 
 
 b~ G! b^ OI b- 
 
 CO i4 rP Gl 10 
 
 1 to 
 
 CO 
 
 
 2 
 
 10 10 to to b- 
 
 V^ OO OS CS O 
 
 O i4 G? G? CO 
 
 co rp rr 10 to 
 
 <O N. K. 00 Cl 
 
 CS O O 14 T4 
 
 01 
 
 
 .2 
 
 - CO rr 10 r* 
 
 G? co rr 10 14 
 
 Gl co rp 10 
 
 1-1 G! co rp 10 
 
 14 G) CO rP 
 
 10 -H GI co rr 
 
 IO 
 
 
 
 
 O CO to O CO 
 
 to OS G} 10 OS 
 
 G? IO OO i4 IO 
 
 oo .- rP N. O 
 
 rP b~ O co to 
 
 OS CO to OS O 
 
 IO 
 
 
 53 
 
 01 01 
 
 14 G? 14 GJ 
 
 W T4 14 
 
 W i4 C< G3 
 
 14 14 G^ 
 
 CM 
 
 "H 
 
 
 .3* 
 
 aj OJ GJ co co rf 
 
 rr rp 10 10 to 
 
 tO b* b- 00 00 
 
 00 Cs OS O O 
 
 14 14 O O O 
 
 -" i4 O! Gl CO 
 
 CO 
 
 IJavs. 
 
 1-1 t co rr 10 
 
 as t- oo os o 
 
 14 o^ co rf LO 
 
 to t-- cc os o 
 
 r-i G! co rr 10 
 G5 "S O? O OJ 
 
 <J t~, CO OS O 
 OJ O! OI OJ CO 
 
 CO 
 
 
 
 cs 
 
 .. ,s OJ j-i co 
 - rr 10 14 G) 
 
 rr rf 10 10 to 
 CO rr 10 i i 
 
 t-~ b~- 00 OS O) 
 O! co rr 10 
 
 O -H O! CO 
 Gl CO rf 10 
 
 CO rH rp If; t^ 
 
 -H C! co rr 10 
 
 tO b~ 00 
 
 .- 01 
 
 
 . 
 
 I-H rr 00 14 rT 
 
 K. O CO K o 
 
 CO to O G) tO 
 
 OS O? IO CO G? 
 
 10 oo 14 rr b*. 
 
 14 rr K 
 
 
 "jj 
 
 rf rr rr 10 10 
 
 
 G? GO CO CO rP 
 
 rr rr !o 10 10 
 
 
 
 S 
 
 
 *-<O?O?OO? loC7OJC'G* 
 
 01 e o? G? 
 
 C3 GJ OJ O7 O? 
 
 co eo co 
 
 
 
 S b~ -4 IO OS OI 
 
 rr rf CO G} O? 
 
 to O rp 00 OJ 
 
 H 14 IO IO 
 
 to O rp oo OI 
 rr rr co 0} GJ 
 
 to o rr oo oi 
 
 T-l 14 10 10 
 
 to o rr oo c} 
 
 rr rr co Gi G! 
 
 to o rr 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 JL* 
 
 - rr OO G? to O 
 
 
 co b-. 14 10 os 
 
 co K, 14 rr 00 
 
 o; to o rr oo 
 
 01 to o 
 
 1? 
 
 
 
 4 01 
 
 O? O? CO CO CO 
 
 rr rr 10 10 10 
 
 14 14 -H 
 
 GJ O) CO CO CO 
 
 rp rp 10 
 
 3 
 
 
 
 00 -H rr b- O 
 
 CO to OS Gl 10 
 
 00 i4 rr K. O 
 
 rp b- O co to 
 
 O> GJ IO 00 i4 
 
 rr b~ O 
 
 h 
 
 c 
 
 01 14 01 G) 
 
 |4 G? i-* GJ 
 
 01 14 
 
 r4 o i- O) 
 
 14 T4 O7 14 
 
 G! GJ 
 
 .^ 
 
 
 
 
 
 
 
 
 O> 
 
 F* 
 
 : r* el OJ co eo 
 
 rf rp n< 10 10 
 
 tO to b- K. 00 
 
 00 00 O OS O 
 
 O 14 -H O 
 
 O -4 14 
 
 JH 
 
 
 
 
 
 - - 
 
 .-*.-- 
 
 
 
 
 
 rH tO -4 tO 14 
 
 to -H to to 
 
 1 tO 14 tO 
 
 O I-H to to 
 
 -H to b, ci 
 
 b- CJ t- 
 
 
 (2\ 
 
 t rr I-H 10 0! 
 
 co 14 rp O* 10 
 
 co rP 14 10 
 
 G? CO 14 rp 
 
 O 10 CO rr 
 
 rH IO Gl 
 
 
 
 
 
 
 
 
 
 
 c 
 
 00 CS OS O T4 
 
 4 O O! CO CO 
 
 rr 10 10 to to 
 
 b~ 00 CO OS CS 
 
 O O -H O! O? 
 
 CO CO rr 
 
 
 o 
 
 - eo rr 10 r4 oi 
 
 CO rp 10 rH 
 
 GJ CO rp 10 
 
 14 G? CO rr 10 
 
 14 G! co rr >o 
 
 rH Cl 
 
 
 J 
 
 o -H rr b- rH rP 
 
 K O CO b, O 
 
 CO to OS G} to 
 
 os o? 10 oo 14 
 
 10 oo -H rr b- 
 
 14 rr b. 
 
 
 s 
 
 rH GJ rH oi 
 
 G? 14 
 
 
 -i r-i 0! 1-1 
 
 e* cj 14 
 
 14 01 
 
 
 s 
 
 o3 o? oi oj co co 
 
 rp rp 10 10 to 
 
 tO to h. b- 00 
 
 00 OS CS OS O 
 
 1 1 
 
 14 -H C 
 
 rH t4 14 
 
 Days. 
 
 r- ex co rr 10 
 
 CD b- 00 Cs O 
 
 ~ 2 *" 2 2 
 
 to - oo as o 
 
 G? eo TP 10 
 
 C^ O? G? O> CJ 
 
 to f- oo 
 
 OI Gl GJ 
 
 
 
 
 
 __ _, rn CJ CO CO 
 
 rr rr 10 to to 
 
 t-- 00 00 OS O 
 
 O 1-H 14 O) CO 
 
 CO rr 10 10 to 
 
 1- 1- 00 00 CS 
 
 o 
 
 
 eg 
 
 - 14 oi co rr 10 
 
 I-H 0} CO rp 
 
 10 -H o? rp 
 
 10 i4 C-! CO 
 
 rr 10 rH oi 
 
 CO rr IO rH 
 
 CO 
 
 
 
 1 
 
 1 
 
 
 *J 
 
 - CO to O G? 10 
 
 OS GJ 'O 00 i4 
 
 rP 00 i4 rP h. 
 
 O rr b- O CO 
 
 to cs co to os 
 
 Gl IO 00 Gl IO 
 
 00 
 
 
 Q 
 
 I-H 14 
 
 |4 o? O! O! CO 
 
 co co rT i4 rr 
 
 IO IO IO 
 
 14 14 14 
 
 O! OI Gl CO CO 
 
 CO 
 
 
 
 
 o O O O C O 
 
 o o o o o 
 
 O O O O O 
 
 
 
 
 rH 
 
 
 
 .. rr CO OI to O 
 
 rr oo G^ to O 
 
 CO t- r~ IO OS 
 
 <^J l^- I-H IT) Os 
 
 eo b 10 OS 
 
 CO b~ rn IO OS 
 
 CO 
 
 
 
 ~ 10 rr "^ co co 
 
 O) -H T4 
 
 10 rr rr CO G< 
 
 G) 1-1 r4 10 
 
 10 rP rp CO ot 
 
 01 -i rH IO 
 
 1O 
 
 
 
 
 
 
 
 
 
 
 
 J_* 
 
 * CO b~ rH IO CS 
 
 CO b, 14 10 CS 
 
 G? to O rr 00 
 
 OI to O rr b- 
 
 i IO O CO b~ 
 
 ^4 IO OS CO tO 
 
 o 
 
 :A 
 
 s 
 
 14 14 rH 
 
 G) O< CO CO CO 
 
 rp rr 10 10 10 
 
 
 rP G! GJ CO CO 
 
 rr rP rr 10 vo 
 
 
 5 
 
 o 
 
 CO to OS G? IO 
 
 00 i i rr t^ O 
 
 CO to CS G? 10 
 
 OS G! 10 CO i4 
 
 rp b, O co CO 
 
 CS OJ >O OO rH 
 
 IO 
 
 
 c 
 
 rH OJ O7 
 
 14 ^H O! 1-H 
 
 d 14 i-H 
 
 G? I-H OI G} 
 
 14 l 1 C^ 
 
 Gl rH 
 
 rl 
 
 0) 
 
 * 
 
 (^ O O 14 14 O? 
 
 GJ co co eo rr 
 
 rp 10 10 to to 
 
 tO K b~ 00 00 
 
 O OS O O O 
 
 rH rH O O -H 
 
 - 
 
 
 
 
 ^ 10 o 10 o 10 
 
 o 10 o 10 o 
 
 10 O 1O O 1O 
 
 o 10 o 10 < 
 
 tO -H tO -H tO 
 
 rH tO tO 
 
 to 
 
 
 f^ 
 
 co 14 rr G? 10 
 
 co rr 14 10 
 
 e* co 14 rr 
 
 G} 10 00 rp 
 
 i-i 10 G< CO 
 
 I-H rP OI 10 CO 
 
 
 
 C 
 
 O i4 i-i G! O! 
 
 co rr rr 10 10 
 
 tO >-- K 00 CO 
 
 CS OS O 14 -H 
 
 OJ O? CO rr rr 
 
 IO IO to to b~ 
 
 00 
 
 
 . 
 
 * 14 CJ CO rP 10 
 
 rn Gl eo rr 
 
 10 I-H o co 
 
 rr 10 O) CO 
 
 rr 10 -H OI 
 
 co rr 10 14 
 
 GJ 
 
 
 Is 
 
 CO tO OS O) 10 
 
 OS OI IO 00 -H 
 
 rr oo -H rr fr- 
 
 O CO b, O CO 
 
 to Cs CO to OS 
 
 OJ IO CO GJ 10 
 
 00 
 
 
 *, 
 
 rH OI G) 
 
 1-H ,-. O! 14 
 
 G) 0? 14 
 
 -H C7 ,- 0* 
 
 i4 i- G! 
 
 rH Gl OI 
 
 
 
 3 
 
 ^0 -H rn G! 
 
 Gt eo co co rp 
 
 1 
 
 rr 10 IO to to 
 
 b>. b>- b- 00 00 
 
 CS OS O O O 
 
 -H rH O O 1-H 
 
 rH 
 
 Oays. 
 
 1-1 G? co rr !O 
 
 to i- oo os o 
 
 
 
 r^ G! CO rT IO 
 
 to b, oo cs o 
 
 I-H c* co rr 10 
 G) OI G! O> G! 
 
 to i- cc 01 o 
 
 O( OI Gl Gl CO 
 
 CO 
 
 s 
 o 
 
 5 
 I 
 
 u 
 
 
 
 
 c3 
 
 S 
 
 -c 
 &> 
 
 ts 
 a 
 >. 
 
 a 
 
 OS 
 1-5 
 
872 
 
 APPENDIX TO THE ASTRONOMY. 
 
 
 cS 
 
 c 
 < 
 
 ^ 1^. CO OO O} O 
 
 - O *-l C! Tf 
 
 >. CX >S C7! W "> 
 
 r- 1 1 1 
 
 o oo oo co oo oo 
 
 O ' O CO 
 >O i-l (N CO 
 
 00 W 'O 00 rH 
 rH 0! 0* CO 
 
 00 00 OO OO 00 
 
 CO ^ IO fj to 
 Tf >0 C! 
 
 * >^ rH Tf b- 
 V) <n T? rf If 
 
 00 00 00 00 00 
 
 t^ 1^ CO OO O 
 
 CO ^ >0 r- 
 
 O ro to O CO 
 IO IO IO 
 
 00 00 00 CTi O 
 
 O O " ex ex 
 
 CO Tf 'O rH 
 
 to cfi ex to o 
 
 rH ^ rH 
 C7) C7> CTj C7^ C7^ 
 
 CO Tf * IT, TO 
 C! CO Tf IO 
 
 CM 1O CO rH 10 
 
 ex eM ex co co 
 
 O O CT> O O 
 
 
 
 
 b 
 
 = 
 
 s 
 
 Anomaly. I 
 
 CM tO -T t- 
 * T O co C) 1-t 
 
 CM tO O Tf 00 
 ~ to !O 
 
 O to 00 CS IO CO 
 
 rH 1-1 O( 
 
 en to to >~ t~. t. 
 
 i 10 en TJ t^ 
 t-i >o >n Tf 
 
 Ol O 31 CO b, 
 r- i-H r-l Cl O< 
 
 <-i TC t^ O CO 
 
 c* o< 
 
 OD 00 O O C 
 
 rH 'O CJ CO t^ 
 t CO eN CM rH 
 
 rH 10 CJ CO K. 
 CO CO CO -^ -^ 
 
 to o> ex 10 co 
 
 rH C! rH 0) 
 
 O O -H rH O 
 
 r- IO O5 CO t^. 
 rH 1O IO Tf 
 
 r- 10 oo ex to 
 
 1O 1O IO 
 
 i-l TT b- rH rr 
 O? rH rH 
 
 O rH rH eX 
 
 V5 O CO 1-^ 
 
 co ex ex rn 
 o rf oo ex to 
 
 rH 1- rH eX <X 
 
 t^ O CO to O 
 
 eX rH eX rH 
 
 ex co co Tf Tf 
 
 rH IO C} CM tO 
 rn IO IO -f 
 
 O Tf ^ rH IO 
 CO CO CO Tf Tf 
 
 CX 10 CO rH Tf 
 rn m rH CM 
 
 IO IO 1O tO tO 
 
 
 
 M. Mutiua j . 
 
 V C7> Tf C~. T 
 
 t Tf ,_ >o CM 
 
 00 C7> O O rH 
 - Tf >--. 0! CO 
 
 ni >o CT ex >o 
 
 0) r-l 1-1 
 
 , to b- b, oo oo 
 
 01 TJ> CT) >O O 
 
 co r-i TT ex 
 w M co t 
 
 Tf 10 r- t 
 
 CO r- IO 00 rH 
 W r- <N CN 
 
 00 O O> O O 
 
 10 o m o TJ 
 
 CO rH Tf Ot 1O 
 Tf IO >O to to 
 
 CO Tf IO rH 
 rj* t- O Tf h, 
 
 rH I-H ex 
 
 rH rH O O O 
 rH -H 
 
 C 10 C 10 O 
 
 CO Tf ^H 
 
 t-~ 00 00 C7> O 
 
 ex co Tt 10 
 
 O co to O5 eo 
 
 rH eX -l 
 
 rH rH ex ex j 
 
 1O O 10 O m 
 eX CO rH Tf 
 
 o -H rn ex ex 
 ex co Tf 10 
 
 to c> ex 10 o) 
 
 rn eX rH G) 
 CO CO Tf Tf 10 
 
 O 10 o 10 o 
 
 CM 1O CO Tf 
 
 co co Tf 10 m 
 
 rH CM CO Tf IO 
 
 CM >O OO rH T|< 
 ex r- rH 
 
 10 to to K. t^ 
 
 
 Ua 
 
 fS. 
 
 rH CX CO Tf 10 
 
 <o t^ oo av c 
 
 rH 
 
 ex co rf ir 
 
 to 1^ CO Cl O 
 
 r- e>> co TT 10 
 ex ex ex o? ex 
 
 tc ^ oo o o 
 O! o? o e>! co 
 
 
 
 
 
 
 
 
 
 
 
 
 eg 
 
 s 
 S 
 
 .. 1- 00 00 CT> O 
 
 - CM CO Tf IO rH 
 
 _ Tf h- O CO K. 
 CX O! CO CO CO 
 
 o to to to to O 
 
 O rH G3 Gi 00 
 CX CO Tf 10 
 
 O CO tO O CO 
 
 TJ- TT Tf Tf 10 
 
 to to to (o o 
 
 ^r q- 10 to & 
 rH ex co ^ >o 
 
 to cr> ex 10 co 
 
 IO O 
 
 to to v, t~ i^ 
 
 t-- cc cr> o 
 rH ex co q- 
 
 ex m oo rH -)< 
 
 r- rH rH eX 
 t-~ >^ V^ h- t^ 
 
 O r- r- O) CO 
 rH eX CO Tf 
 
 00 Tf 1^ O 
 
 ex co co co TP 
 ^ K K K -, 
 
 CO Tf Tf IO tO 
 
 IO rH CM CO 
 
 CO ^ O CO to 
 Tf Tf >O T, IO 
 
 h- k^ K K K- 
 
 to 
 
 Tf 
 
 
 10 
 
 K 
 
 
 S 
 
 Anomaly. 
 
 en co t^. rH 10 
 * Tf Tf CO CO ex 
 
 rH IO O CO K 
 
 - 10 10 >o 
 
 O CO O O CO 
 CX r- rH 
 
 oj Tf >n 10 to to 
 
 CTJ CO 1-, r~ IO 
 rH rH IO 
 
 rH IO Ol CO tO 
 rH rH rH O! O? 
 
 to o o! 10 oo 
 
 CX M rH 
 
 to K K 00 00 
 
 o> ex to o T 
 rr -t co co ex 
 
 O Tf 00 ci to 
 
 CO CO CO ^< TJI 
 
 rH -^f h, C CO 
 rH eX rH CX 
 
 O) C~> G~i O O 
 
 rH T 1 
 
 00 CX to O ^ 1 
 rH rH IO 
 
 o Tf oo ex 10 
 
 IO IO 1O 
 
 to CT> <H to CD 
 
 rH rH 0! 
 
 rH r- O O O 
 
 co ex to o Tf 
 Tf Tf eo CO ex 
 
 C7 K IO 
 
 r- rn ex ex 
 
 ex o oo -< Tf 
 rn ex CM 
 
 rH r- ex ex v> 
 
 00 CM to O Tf 
 
 rH rH IO 
 
 OV CO t, rH Tf 
 CM CO CO Tf Tf> 
 
 t~ O CO tO C* 
 rH rH CX 
 
 CO Tf Tf Tf 10 
 
 00 
 
 Tf 
 
 CO 
 
 Tf 
 
 ex 
 ex 
 
 10 
 
 
 Pi 
 
 5 
 
 | 
 
 < 
 
 S 
 
 _ CO CO 00 CO O> 
 v CO 1-1 T W '0 
 
 O rH rH o? CM 
 - CJ CO t irj 
 
 Tf ^ O CO K 
 
 - w 
 aj >O >o 3 o w 
 
 Tf C7S TJ* Cft Tf 
 CO TJ" rH IO 
 
 CO rf tf 10 >O 
 
 rH O? CO Tf 10 
 
 o M to ci o? 
 
 rH <J( rt 
 
 K >. oo oo o 
 
 C3> TT Cl Tf Oi 
 eX OJ rH Tt 
 
 to K t^ oo oo 
 
 i- ex co rr 
 
 to c> ex 10 oo 
 
 rH eX rH eX 
 O5 O) O O i-i 
 
 ^r CTI TC a> ^ < 
 ex 10 co TT 
 
 O Oj O rH rH 
 
 10 ex co TP 
 
 rH IO 00 rH T< 
 eX rH rH 
 
 rH O O rH 1 
 
 o> Tf a> Tf 05 
 -i 10 CM oo 
 
 ex ex co Tf -r 
 o rH ex co 
 
 K r- Tf K O 
 
 ex rH ex CM 
 
 rH C? O? CO CO 
 
 Tf O5 TT Cl Tf 
 rH Tf CM 1O CO 
 
 10 m to to K 
 
 Tf IO rH CM 
 
 eo to O CO to 
 
 rH rn eX 
 
 TT Tf m 10 10 
 
 & 
 
 CO 
 
 CO 
 
 Oi 
 
 to 
 
 Da 
 
 y- 
 
 I-l CX CO Tf >O 
 
 tO 00 Cy> O 
 
 rn ex co T >o 
 
 <O t - 00 CT3 O 
 
 1 ex 
 
 *H CM CO T IO 
 
 ex GQ eM ti ex 
 
 to t^ a- as o 
 
 CM CM CM CM CO 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 eg 
 
 
 
 
 ^ 00 O C^ O 
 
 -l O? T>1 >O 
 
 _ o o> 'o oo -H 
 
 TJi If) 1O >O 
 
 o t f f Tf >O 
 
 rH CN CO CO T 
 
 rn c CO rj- 
 
 IO CO rH Tf >s- 
 
 rH ^H rH 
 
 10 10 10 10 10 
 
 Tf TS tO tO t- 
 10 rH eX CO 
 
 O * ^ C co 
 ex ex ex co co 
 
 to 'o 10 10 
 
 ICO <JU O> C O 
 *3i >o o co 
 to o co to ci 
 
 CO CO Tf -^ 
 IO IO lf> IO O 
 
 r rH CX r --J C'J 
 
 Tf 10 - ex 
 
 c; 10 as w 5 
 
 IO IO IO 
 
 in 10 to to 
 
 TT IO O tO t-^ 
 CO Tf IO rH 
 
 00 Tf 00 rH 
 rH rH rH CM 
 
 to to to to to 
 
 
 
 
 t 
 
 _^> 
 
 2 
 
 S 
 
 c 
 
 < 
 
 ^ O t 00 Cl tO 
 * 10 Tji CO CO 0* 
 
 if 00 CM O O 
 
 - 10 urj , 
 
 00 rH IO CO 
 
 " rH 1-1 
 
 CO CO < TJ- >* >0 
 
 O -rj 1 00 <N ; 
 W rH 1O 
 
 rf 00 O? !O O 
 
 rH O) CN O< 
 
 Tt K O CO O 
 
 W CX rH 
 
 >o to to t^ (^ 
 
 c TT oo o; to 
 10 ^- co co ex 
 
 '0 t- rn m O5 
 
 CO CO rf ^ M" 
 
 C5 eX IO CO rH 
 
 en rH c) ex 
 
 b- 00 00 OS C> 
 
 O ^ t~ rn IO 
 0! rH 10 
 
 CO t- rH IO 00 
 1O 
 
 ^" N. rH / 
 
 TH rH ex 
 
 -1 rH r-. 
 
 Cft CO t- . 
 
 Tf Tt- co co ex 
 
 CM to O Tf CO 
 
 rH rH CM CM CM 
 
 O co to a> ex 
 rH ex i-i 
 
 O O i-l rH CX 
 
 d co b, i 10 
 
 rH rH 10 
 
 CM to O Tf b- 
 
 CO CO Tf Tf Tf 
 
 IO CO rH Tf b- 
 
 CM rH CM 
 
 ex CM co co Tf 
 
 
 
 (^ 
 
 -. 
 
 _0 
 '-*J 
 
 o 
 3 
 
 s 
 
 00 CO 00 CO 00 
 < Tf r- IO IJ< 
 
 - CO CO TJ- Tf IO 
 0) CO rf 
 
 Cl W IO CO 1-1 
 
 c) c* oi 
 
 a - '.0 -f f 5 10 
 
 C7 OO CO CO CO 
 CO rH Tf CX 
 
 to to t, >- oo 
 
 IO C! CO 
 
 t 00 rH TT 1^ 
 -H r. 0) 
 
 CO tO ^ K >, 
 
 OO CO 00 CO 00 
 IO CO Tf rH 
 
 00 CJ) O O rH 
 
 r IT, r-. ex co 
 
 O CO 1-- O CO 
 
 ex oi 
 
 00 00 Oi Oi O 
 
 CO CO OT CO CO 
 10 O< CO rn 
 
 rH ex w eo TJ- 
 
 Tf IO rn eX 
 
 to O5 CO to C5 
 
 rH eX rH 0) 
 
 O O rH rH O 
 
 00 CO 00 CO 00 
 
 rr O? TJ c- 
 
 Tf 10 ir. to ^ 
 co Tf 10 - 
 
 O-) 10 00 C? 10 
 
 CX rn rH 
 
 e -H rn ex rx 
 
 CO 00 CO 00 CO 
 Tf rH 10 O? 
 
 ^ oo co cr> o 
 
 OJ CO Tf 10 r- 
 
 00 r- Tf K rH 
 C! rH CM CM 
 
 C? '-0 CO Tf Tf 
 
 
 I) 
 
 djs. 
 
 ^ co TT >o 
 
 to i, a; c~j cj 
 
 rn CX C-2 TT 10 
 
 to t~ co en o 
 
 r- 0-t CO TT '0 
 
 ex c ex es ex 
 
 to i - oc o o 
 ex o? CM CM co 
 
 
LUNAR TABLES. 
 
 873 
 
 September. 
 
 CS 
 
 'o 
 
 s 
 
 tO tO l~ CO CO 
 t rH Ol CO Tf IO 
 
 "0 so rH rr ~. 
 10 10 
 
 o 01 co co co 
 
 CT> O O CN 
 <N CO Tf IO 
 
 re K O CO 
 
 rH i-H i-l <N 0! 
 CO CO CO 00 CO 
 
 O CO CO Tf IO 
 
 ri C! CO TT 
 
 K O CO U3 SI 
 co to co c,o 
 
 CO CO CO CO CO 
 
 o to T >-- oo 
 
 >0 O! CO 
 
 C! O OS <N IO 
 Tf T)< ^1 IO 1O 
 
 CO CO CO CO CO 
 
 Ol Cj O O ii 
 Tt" IO rH O? CO 
 
 CO r* IO CO i-l 
 1O i-l 
 
 CO * Tf T) T)< 
 
 Ol Ol CO T}1 T}< 
 * 10 rH 01 
 
 rf K -H rr K 
 rH rH Ol O? 01 
 
 Tf Tf TJ< -<t rr 
 
 jj> 
 
 M 
 
 
 c 
 
 to O Tf 00 O! 
 
 * 01 rH 
 
 - 10 <n *, 
 
 10 10 10 
 
 o ^ 
 
 uj O O rH r-i O 
 
 to o Tt- co i-< 
 
 10 IO Tji CO CO 
 
 O T 00 <N 3 
 T- rl T~ O) <U 
 
 CO tO C35 5J IO 
 <N O5 
 
 O O 1-1 1-1 0? 
 
 to Oi CO t, 1-1 
 
 CS^ 1-1 !-< 
 
 O Tf 00 <N to 
 CO CO CO rf Tf 
 
 00 i-l Tf K O 
 
 1-1 tn 1-1 
 
 OJ CO CO CO ^ 
 
 if? c?r co t^ 
 
 10 tf & I.O CO 
 
 en co K 1-1 10 
 
 T)< 1O IO 
 
 CO to Ol CO tO 
 0) i-l i-l 
 
 't IO IO tO to 
 
 irs 01 CO b~ 1-1 
 
 GJ 1-1 -- 
 
 O CO K. T-i 'O 
 11 i-l W C< 
 
 CJ OJ 10 00 i-" 
 01 -i 0) 01 
 
 to >^ K co co 
 
 O CTJ CO b. T- 
 IO TT ^< CO CO 
 
 co 01 co o Tf 
 
 O! CO CO Tf Tf 
 
 Tf h^ O CO to 
 rH r. tX t 
 
 o> o> o o o 
 
 P\ 
 
 c 
 2 
 
 ^ 
 
 3 
 
 5? 
 
 ^ t~ O? * - f i-, 
 
 01 co ~ Tf 
 
 01. CD co Tf Tf 
 01 co Tf 
 
 10 CO rH Tf K 
 
 rH rH CM 
 
 _. - 
 
 O" t- o* t^ G! 
 <N 10 to rf 
 
 10 10 to t~ b- 
 
 IO i W CO 
 
 C Tf ^ O CO 
 0< 
 
 -< CS M CO 
 
 t^ tN t^ 0) b- 
 
 ri 10 O CO 
 
 00 00 O) O O 
 TJ- ir> O! 3 
 
 tO O> CO to O5 
 
 1- O! r- CM 
 
 eo co TJ< TJ- 10 
 
 CN t- W 1- O! 
 i -^ C} IO CO 
 
 -- - <N 0? CO 
 Tf IO i-l O* 
 
 >o o> o? >o 
 
 1-1 1-H 
 
 [ 10 to to -~ t- 
 
 t-. CO OO CO ^O 
 ^- i-l IO Ol 
 
 rf ^" IO IO to 
 CO rf >O rH 
 
 00 r- Tf 00 -1 
 
 O i-l Ol O! 
 
 l^ CO 00 C7> O 
 
 co 00 tn 00 co 
 
 CO rH Tf Ol 
 
 -, K. 00 OO C7> 
 
 0! CO Tf 10 
 
 n< K o eo *-~ 
 
 rH i-H Ol 
 
 O O rH rH rH 
 
 Da vs. 
 
 rH M CO Tf 10 
 
 to t, oo en o 
 
 O* CO 1- "0 
 
 to ^ 00 O> O 
 
 1-1 Ol CO T|i IO 
 Ol Ol O> OJ O 
 
 to tv. CO Ol O 
 
 O! Ol O! Ol CO 
 
 
 
 
 K 
 
 3 
 
 cj 
 
 1 
 
 to t, t 00 O> 
 
 t Tf 10 rH O* 
 
 tO O 00 to O 
 I-l rH CX CM Ol 
 
 O> O O ^- CT 
 CO IO rH (N 
 
 O? 10 O> (N 10 
 CO CO CO Tf Tt 
 
 CM CO ^< -<!f >O 
 CO rf 10 i 
 
 00 rH Tf 00 1-1 
 TJ- 10 10 10 
 
 to to t^ t^ oo 
 
 Ol CO Tf IO 
 
 * K O eo t, 
 
 1^ t 1-4 
 
 OJ C* OJ IN O5 
 
 CTi C7i O ri 
 i- Ol Tf >0 
 
 O CO to O CO 
 01 O! d Ol CO 
 
 01 Ol Ol O 0? 
 
 o; co co t TC 
 i- Ol CO i* 10 
 
 to o 01 10 oo 
 
 CO CO Tf Tf TJ- 
 
 Ol Ol Ol Ol O? 
 
 10 
 
 Ol 
 IO 
 
 01 
 
 Anomaly. 
 
 . CO l^ 10 Oi 
 * CO Ol Ol -i 
 
 O Tf 00 IN to 
 
 - >o to 10 
 o oj 10 co 01 10 
 
 Ol rH rH 
 r/} 00 Oi O*> O O 
 
 t 10 O> 
 10 10 TT CO 
 
 O CO ^ -H IO 
 H r- i-l <S N 
 
 co -< * ^ o 
 
 .1 (H O? 
 
 O -H 1-1 O O 
 
 co *^ 10 o> 
 
 CO (N O) i~ 
 
 CT) CO b~ i-H 10 
 O? CO CO Tt ^< 
 
 CO to O (53 
 
 -> CN 1-1 0^ 
 
 rf rt rH (?) G? 
 
 CO tO O 1 CO 
 >o 10 q" co 
 
 O> O! to O rf 
 
 Tf IO 1O 
 
 CO ^-t 00 i-l 
 05 ,-< 
 
 CO CO Tf ^ IO 
 
 01 to o *r oo 
 
 CO 01 O! rH 
 
 oo 01 co o >* 
 
 rH rH 01 01 
 
 Tf ^ O CO to 
 
 rH 01 rH CM 
 
 IO IO tO CO t 
 
 Ol tO O Tf CO 
 IO >O Tf CO 
 
 OO rH IO O> CO 
 Ol CO CO CO Tf 
 
 Ol Ol IO 00 rH 
 rH 1-1 Ol rH 
 
 k> CO 00 CO O} 
 
 0! 
 CO 
 
 - 
 
 Tf 
 
 3 
 
 05 
 
 ,^ 
 
 5 
 a 
 
 a 
 
 s 
 
 s 
 
 rH to i to 
 * O! "0 CO Tf 
 
 Tf Tf 10 CO tO 
 
 - co Tf "O rH 
 
 to o cj to 04 
 
 rH 01 rH 01 
 
 . O> O O O n 
 W) i 
 
 to to to 
 
 1O O< CO 
 
 K. K. 00 Oi Oi 
 W CO 1< 10 
 
 01 10 oo H 10 
 
 W " 1-H 
 
 1 O O I-l !-< 
 
 to to * 
 
 -H Tl< G3 IO CO 
 
 o o - c* 
 et co ** 10 
 
 00 < TC K. <-" 
 
 1-1 rs <N 
 
 - O) CO CO 
 
 to ^ o) ^ 
 
 Tt i-l IO O 
 
 CO CO Tf -rf U) 
 "> (N CO ^ >0 
 
 * K. O CO <O 
 -i 1-1 O) 
 
 Tf TC 10 10 >0 
 
 CH t^. Ol fr~ d 
 
 CO rH & Ol 
 
 to to t>. ^ co 
 
 r- (J< CO Tf 
 
 o co to o o 
 
 rH 01 1-1 
 
 tO to t~. b- CO 
 
 ts. Ol ^ Ol --. 
 IO CO Tf rH 
 
 00 O5 O O rH 
 10 Ol CO * 
 
 10 O Ol 10 00 
 01 01 
 
 CO CO O <3> O 
 
 01 
 1O 
 
 rH 
 
 >o 
 
 rH 
 (N 
 
 c 
 
 Days. 
 
 r- ( Ol CO Tf 10 
 
 to t^ oo o> o 
 
 Ol CO TT IO 
 
 to ^ oo o o 
 
 1-1 O! CO TT 10 
 
 01 01 01 01 01 
 
 to t- oo o> o 
 01 01 01 01 c> 
 
 rH 
 CO 
 
 
 
 
 . 
 
 3 
 
 eg 
 
 3 
 4 
 
 to >^ i^ oo Oi 
 
 ^ f-< Ol CO -^f O 
 
 co 1-1 Tf -, o 
 -3" T< TJI in 
 
 o O^ O Ol O^ Ci 
 
 O C -, (N 
 C* CO T >O 
 
 TC t, O CO to 
 IO IO 
 
 o> o> o o o 
 
 CO CO Tf ^- IO 
 
 1-1 C* CO -^ 
 
 O CO to O O) 
 - 1-1 0* 
 
 o o o o o 
 
 to to t- co co 
 
 10 1-1 O) CO 
 
 IO O) O* IO CO 
 O} d CO CO CO 
 
 o o o o o 
 
 O> O O rH Ol 
 Tji rH Ol CO 
 
 r- IO CO - 1 *t 
 
 TJ- TT ^r 10 10 
 
 o o o o o 
 
 Ol CO CO Tf 10 
 Tf 1O rH Ol 
 
 ^ Tf ^ 
 IO H 
 
 O ^ rH rH rH 
 
 IO 
 
 CO 
 
 n 
 
 rH 
 
 Anomaly. 
 
 O T 00 O! *O 
 - * CO 0< CN 
 
 o> co b, .1 m 
 
 " t >O IO 
 K O 5 h, O 
 
 0) -H 
 
 . h, t, 00 CO CT> 
 
 en 
 
 O Tf 00 GJ to 
 IO IO T 
 
 m co o o * 
 
 >- w-i (X <N 
 
 CO to Cl <}J 10 
 i Ot W 
 
 CT) O) O O * 
 
 O *! 00 i to 
 
 r co w o? 1-1 
 
 CO W to O T 
 O? CO CO Tj" Tf 
 
 00 i-l Tf l^ O 
 rH i C-) i-l 
 
 O -H 
 
 o -^ co CN to 
 
 1-1 O IO ^1< 
 
 CO <N '0 O> CO 
 TT IO IO IO 
 
 CO tO Ol O? to 
 
 <N 1-1 1-1 
 
 i W O? co CO 
 
 o t i^ i 10 
 
 t CO Ol Ol i-" 
 t-. ix >O O) CO 
 
 rH r- rH 01 
 O5 Ol "I CO r- 
 
 01 rH Ol Ol 
 CO * ^ >O IO 
 
 O CO t~ rH 1O 
 >O IO Tf 
 
 I-- Tf OD Ol 
 O! CO CO CO Tf 
 
 Tf ^ O CO tO 
 rH rH Ol 
 
 to tO K b- - 
 
 a> 
 eo 
 
 CO 
 
 Tf 
 
 0> 
 
 00 
 
 M. .vlotiun J . 
 
 ^ IO O O O IO 
 - *< "O (N CO 
 
 " D 5O K CO 00 
 -> CO rr 
 
 00 * b, O 
 <N IN Cl 
 
 . t>- 00 00 C} O5 
 
 WI 
 
 O 10 O 10 O 
 1-1 V C! 10 CO 
 
 O* C7 O O ** 
 IO CK CO -3 
 
 co t, O co to 
 
 r- w 
 
 O O 1-H ** 
 
 in o -< to 
 
 t 10 <N 
 O CO CO Tf 
 
 10 01 co 
 
 CD CO to O> <N 
 
 O) - 
 
 ^ i- 
 
 to i- tO 
 CO Ol 
 
 IO 10 tO CO K 
 Tf 10 y-l O? 
 
 <O 00 O >O OS 
 -i CN H OQ 
 
 Ol O) CO CO -t 
 
 tO r- tO r- tO 
 "O CO *t n 
 
 K. 00 O> O) O 
 CO Tf 10 Ol 
 
 rH Tf K - Tf 
 
 Ol " rH 
 
 Tf IO 10 to to 
 
 r tC rH tO r- 
 10 Ol CO r- 
 
 O rH Ol W CO 
 
 CO Tf IO r* 
 
 t^~ O CO -- O 
 
 01 I-H Ol Ol 
 
 tO K . CO 00 
 
 to 
 
 Tf 
 
 00 
 Ol 
 
 CO 
 
 o 
 
 .Jays. 
 
 CTCOTIO iot-,cosio 
 
 ^^ CN 00 TT JO 
 
 tO ^ CO C7> O 
 
 r- CM CO Tf IO 
 
 O> Ol Ol O! O> 
 
 to ^ oo 01 o 
 
 Oi O! O) O! CO 
 
 CO ' 
 
 -I 
 
 
 I 
 
 CO 
 
 sa 
 
 S 
 
374 
 
 APPENDIX TO THE ASTRONOMY. 
 
 
 CS 
 
 -*-j 
 O 
 
 
 
 .. rr 10 10 to <O 
 " O! CO * 10 
 
 . rr -, O CO b- 
 
 rr T 'o "> !o 
 b, l>- K K b- 
 
 t--. OO CO O O 
 1 CO Tf 
 
 O CO to O) CO 
 
 TH 
 
 oo oo oo oo oo 
 
 O Ol O! CO 
 I-H O! CO * 10 
 
 O O5 O? IO 00 
 r-l 1-1 W O) O 
 
 oo co oo oo oo 
 
 1 co rr 10 10 to 
 
 r-l Ol CO TP 
 
 OJ 1O 00 rH Tf 
 CO CO CO *!J* ^* 
 
 oo ac oo oo co 
 
 to t>. oo ci ci 
 
 10 r-i O! CO 
 
 K -H rr b, O 
 rf 10 10 10 
 
 co co oo co Ci 
 
 C O i i Ol Ol 
 
 1O rH Ol CO 
 
 eo N. o eo to 
 
 rH rH rH 
 
 Cl Ol Cl Cl Cl 
 
 co' 
 rT 
 
 Cl 
 Cl 
 
 rH 
 
 December. 
 
 _> 
 
 a 
 
 B 
 o 
 
 a 
 
 < 
 
 .. tO O "^ CO Ol 
 
 - TH i-c 10 m 
 
 (O O TT b- rH 
 TT IO IO IO 
 
 to O o* o d 
 01 01 rH 
 
 o5 rni ci 01 co co 
 
 to o Tt* fio tj^ 
 
 Tf TT CO Oi 0! 
 
 >O O CO K r-l 
 
 rH r- W 
 
 Gt 10 00 i-l Tf 
 
 rH tS il &! 
 
 q- T)< rf 10 10 
 
 tO O TT OO r-H 
 I-H rl IO 1O 
 
 10 O CO to O 
 O! O( O CO T)< 
 
 b. O CO to O 
 
 i-c O( 
 
 to to t~ t^ ^ 
 
 IO O CO t~ rH 
 
 ^r co eo 01 01 
 
 rf oo Ol to O 
 rr TT 10 10 
 
 01 10 oo rH m 
 
 r- Ol Ol 
 
 CO 00 O O) O 
 
 1O Cl CO b 1-1 
 rH 1O 1O 
 
 rr oo Ol >o Ci 
 
 1 1 1 1 1 1 
 
 CO rH rf K O 
 -H TH Ol 1-1 
 
 O 1- rH rH O 
 TH 1 * rH 
 
 IO CJl CO 1 rH 
 
 rr co co 01 01 
 
 CO ^ 1-4 IO Cl 
 
 O! Ol co co eo 
 eo to Ci 10 
 
 O! TH rH 
 O rH rH Ol Ol 
 
 <o 
 
 TH 
 
 CO 
 rr 
 
 00 
 Ol 
 
 w 
 
 
 
 
 B 
 O 
 
 "o 
 
 % 
 s 
 
 rr o 10 o >o 
 - co I-H rr d >o 
 
 10 to o b- t~ 
 
 * rH O! CO rr 
 
 TT K O CO tO 
 
 -H TH 01 
 
 ^ CO CO rT rr rr 
 
 o 10 o <o o 
 
 CO TJ 1 1-1 >O 
 
 00 O O O O 
 
 10 r-l CO Tf 
 
 Ol CO to O> O< 
 
 o; 1-1 
 
 10 10 to O b- 
 
 10 O 10 o 10 
 
 O CO TJ- 
 
 O) 0! CO CO 
 1O rH O) CO 
 
 IO C> O! IO 00 
 
 r-- 01 
 
 N. k^ oo co 01 
 
 o 10 o 10 o 
 01 10 co TJ< 
 
 rr Tf m to to 
 
 T >O rH Ol 
 
 rr CO rH Tf 
 
 Ol rH rH 
 
 Ol O O rH rH 
 
 IO O IO O 1O 
 rH 10 O? CO 
 
 IN. K oo ci en 
 
 CO VT "O rH 
 
 K O eo K O 
 
 Ol rH O! Ol 
 11 O O rH rH 
 
 c 10 o m o 
 1-1 rr Gl 10 CO 
 
 O O r- -H 01 
 
 eo rr 10 r-i 
 
 co to Ci eo to 
 
 H Ol TH Ol 
 
 01 01 01 <n eo 
 
 IO 
 
 CO 
 01 
 
 Cl 
 rT 
 
 Da 
 
 ys. 
 
 i Ol co rr '0 
 
 to l>. CO O O 
 
 1-1 Ol CO Tf IO 
 
 tO 1-, QO G5 O 
 
 rH ,-. , 0! 
 
 * O CO rf if; 
 0! 01 Ol 01 01 
 
 to t-~ oo Ci o 
 01 Ol Ol Ol co 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *J 
 
 o 
 
 s 
 
 .. IO 10 to N. N. 
 
 TH Gl CO ^ 
 
 O O! 10 00 1-1 
 
 . TH TH TH Ol 
 
 o to to to to 
 
 CO O O O O 
 IO r- CO Tf 
 
 Tf CO TH * t- 
 O! CO CO CO 
 
 to to to to to 
 
 *- (N 01 CO TJ- 
 10 rH Ol CO 
 
 O TJ< t, O CO 
 
 * Tj< TT IO IO 
 
 to to to to c o 
 
 TT 1O tO tO >. 
 rr W i-i 01 
 
 to o eo <o o 
 10 10 
 
 to to b, b, K 
 
 oo oo ci ci o 
 eo rr 10 O! 
 
 Ol 10 00 01 10 
 
 rH 1-1 rH Ol Ol 
 
 IN. t, s. I- K 
 
 rH r-t Ol CO CO 
 
 CO rr 10 1-1 
 
 00 rH rT CO i-l 
 
 o! eo co co TT 
 U -,-, -. K 
 
 
 Vovember. 
 
 Anomaly. I 
 
 k t-~ rH 1C Ol CO 
 
 - TH TH mm 
 
 Cl CO t^, O 1" 
 rr 10 10 
 
 , rT t- O TT b- 
 
 0* 0) TH 
 
 o rH TH 01 01 
 
 1-, r-c 10 O> CO 
 rf -^- CO O! O? 
 
 oo w to o * 
 
 i-l r-l <N 0) 
 
 O CO tO Cl (N 
 01 0< 
 
 co eo eo TT T) 
 
 IN. -t O Ql CO 
 1-Hr- 1C IO 
 
 00 Ol tO O CO 
 O? CO CO CO Tf 
 
 10 oo 1-1 Tf b 
 
 i-l 1-1 01 
 IO IO to tO to 
 
 ^ r- IO Oi CO 
 
 rr rr CO Ol Ol 
 
 ^ rH 10 O5 CO 
 
 rr 10 10 10 
 
 O CO to Cl CO 
 rH 01 rH 
 
 -, h- oo or o> 
 
 to o ^ oo 01 
 
 rH rH 10 10 
 
 t, rH IO 00 Ol 
 
 1-1 I-H rH Ol 
 
 tO Cl 01 10 CO 
 rH Ol rH Ol 
 
 Cl Cl C O 1-1 
 
 rH r- TH 
 
 to o rr co 01 
 
 rr rr CO 01 Ol 
 
 to O rr CO 
 O! eo co co rr 
 
 rH rf b- O co 
 
 01 TH rH 
 
 rn rH i-l 
 1 1 
 
 
 
 
 
 a 
 o 
 "w 
 
 o 
 
 s 
 s 
 
 .. rr Ol rr Cl rf 
 * CO IH rr W 
 
 CO 00 O5 CT> O 
 * TT >O 1-1 CO 
 
 oo -i >n eo rt 
 
 <N TH 0) 
 
 rf -H C G! CO eo 
 
 O TT O) T> O 
 >O CO ^ r* 
 
 o <s w o 
 
 TT 10 r- 
 
 Tf N. 1-H ^1 b 
 
 1-1 1-1 O? 
 Tf * IO IO IO 
 
 Tf OJ TJ O5 ** 
 
 10 01 co i-i 
 
 CO rf 10 >O <O 
 CO 'J" IO i-< 
 
 O CO to O CO 
 iH O! G* 
 
 tO tO k>. h. CO 
 
 02 ^r 01 rp 01 
 
 TJi Ot IO CO 
 
 to t^. t, oo 
 
 01 co rr 10 
 
 to o 01 10 a> 
 
 rH O! rH Ol 
 
 CO CO Cl Cl O 
 
 r-* 
 
 rr ci rr ci rr 
 
 rr -H 10 Ol 
 
 Cl O O rH Ol 
 rH CO rT 10 
 
 Ol IO 00 rH IO 
 
 Ol 1-1 1-1 
 
 O -H . o O 
 
 Ci rf Cl rr 01 
 
 CO rH ^ tjt IO 
 
 Ol e oo rr rr 
 
 rH Ol CO rT >O 
 
 00 i rr I- O 
 Ol Ol Ol 
 
 O rH TH 01 Ol 
 
 
 Da 
 
 ITS. 
 
 1-1 c^ O3 Tf >O 
 
 to ^ oo o> c 
 
 Ol CO TJi 10 
 
 t 1 ' 1 (--* 
 
 to t~ co ci o 
 
 ,H X r- 01 
 
 O! co rr IO 
 Ol Ol G O! O! 
 
 to t-, co 01 o 
 
 C! Ol Ol W CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 +j 
 O 
 
 s 
 
 10 o to c^ t-, 
 
 5 co rr >0 ri 
 
 _ o o o to 
 
 t.0 CO CO * -^ 
 
 * Tf Tf Tf Tf 
 
 CO Ci Ci O ^ 
 
 CO Tf 1-H 
 
 to Cl O? to O 
 T*> ^ 1O IO IO 
 
 * Tf Tf Tf TT 
 
 1-1 Ol CO CO ^1 
 Ol GO Tf IO 
 
 01 10 oo ~ 10 
 
 rH rH 
 
 10 10 >o >o 10 
 
 10 10 to to t- 
 
 rH Ol CO Tj< 1O 
 
 CD ** TP *~ O 
 rH Ol O! Ol CO 
 
 10 10 10 ir. >o 
 
 00 CO Cl C O 
 1-1 Ol rr 10 
 
 r b- c eo to 
 co co * rr rr 
 
 IO >O IO IO 1O 
 
 T-I o) o! co eo 
 
 rH Ol CO rr 
 
 O CO to Cl O! 
 10 IO 1O IO 
 
 10 10 10 10 to 
 
 rr 
 IO 
 
 IO 
 
 to 
 
 October. 
 
 
 
 j^ 
 
 3 
 
 s 
 
 o 
 a 
 < 
 
 ^ tO C5 CO tO O 
 - C? "-H 1-1 
 
 00 <N tO O -<f 
 
 ^r >o m 
 
 O (N 10 Oi O? 
 
 
 
 ^ r-i rt O 1-1 
 
 Tf 00 0! to C 
 
 1O Tf Tf CO CO 
 
 k- i-< IO O5 CO 
 
 1-1 r- -H O! 
 
 1O 00 r-l Tf b-. 
 
 1-1 " tN 
 
 H i-l W CO 
 
 t 00 Gl to O 
 d rH i-l 
 
 k -< 10 OS CO 
 Ol eo co co ^ 
 
 O CO <O O> Ol 
 
 Cl 1-1 Ol rH 
 CO Til ^1 t >O 
 
 rf 00 O! to O 
 
 10 rr rr co co 
 to o rr co 01 
 
 rr 10 >o 10 
 
 10 CO T? CO 
 
 0! 01 rH 
 
 IO to to K. K 
 
 ^r oo >*! to o 
 
 01 rH 
 
 to O rf 00 Ol 
 
 rH rH r- C! 
 
 rH rr >- O CO 
 rn 01 rH 0) 
 
 00 00 00 CJ Cl 
 
 rr CO * IO Cl 
 1O rr rr CO O! 
 
 IO Cl CO K. TH 
 
 O! Ol c? CO * 
 
 to ci 01 10 oo 
 
 rH rH 01 
 
 O O rH rH i 
 
 CO 
 O! 
 
 IO 
 
 rr 
 
 TH 
 
 o 
 
 
 r= 
 
 c 
 _o 
 
 '-*J 
 
 o 
 
 s 
 
 5- 
 
 * 00 CO 00 CO 00 
 
 ^ 10 co TT n 
 
 O O r-l rH O? 
 
 * TH CO Tf IO 
 
 o eo to 01 co 
 
 rt 0| ,- 
 
 K O O 1-1 
 
 '. : 00 co oo co 
 10 w co i- 
 
 O! CO Tf TT 10 
 
 i-H OJ CO Tf IO 
 
 to a> o? 10 OB 
 
 TH W rl O) 
 O? C* CO CO Tf 
 
 OO CO CO CO 00 
 Tt 1 Ol >O CO 
 
 10 to to -- co 
 
 Ol CO i*- 
 O! IO OO TH TT 
 
 0! -* rH 
 
 Tf IO 1O to to 
 
 CO OO CO 00 CO 
 rr rH 10 Ol 
 
 00 Cl Ol O i ' 
 IO rH CO Tf 
 
 K 1-1 rr ^ O 
 
 Ol rn Ol 01 
 
 to t^ IN. oo oo 
 
 CO CO 00 CO CO 
 CO rH rf Ol IO 
 
 Ol Cl CO t-O 
 10 1-1 Ol CO 
 
 CO K O CO to 
 rH rH Ol 
 
 Ol Cl O O C 
 
 co sc rf Cl rr 
 eo rr TH 10 
 
 rr 10 IO to to 
 
 rT 10 rH 01 
 
 Cl Ol to Cl d 
 
 01 TH 
 
 TH TH O TH 
 
 Ci 
 Ol 
 
 t, 
 
 CO 
 IO 
 
 Da 
 
 ys. 
 
 1- CO TT 10 
 
 to t>. oo 01 o 
 
 1 1 
 
 r-l Ol CO TJ" IO 
 
 to t~ oo o o 
 
 rH CM CO TT IO 
 
 G* 01 01 01 Ol 
 
 to t- oo 01 o 
 
 0! Ol Ol Ol eo 
 
 CO 
 
 i 
 
LUNAR TABLES. 
 
 375 
 
 TABLE IV. The Moon's Mean Motions for Hours, Minutes, 
 
 and Seconds. 
 
 
 M.Mo.j. 
 
 Anomaly. 
 
 Mot. ft. 
 
 
 M. Mo. > 
 
 Anomaly. 
 
 Mot. SI. 
 
 H. 
 
 a 1 II 
 
 1 II 
 
 / /' 
 
 
 
 
 
 / 
 
 t If lit 
 
 t it //' 
 
 " /// 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 32 56 
 
 32 39 
 
 8 
 
 31 
 
 17 1 10 
 
 16 52 3 
 
 4 6 
 
 2 
 
 1 5 53 
 
 1 5 20 
 
 16 
 
 3d 
 
 17 34 7 
 
 17 25 13 
 
 4 14 
 
 3 
 
 1 38 49 
 
 1 37 59 
 
 24 
 
 S3 
 
 18 7 3 
 
 17 58 
 
 4 22 
 
 4 
 
 2 It 46 
 
 2 10 39 
 
 32 
 
 34 
 
 18 40 
 
 18 30 31 
 
 4 30 
 
 5 
 
 2 44 42 
 
 2 43 18 
 
 40 
 
 35 
 
 19 12 55 
 
 19 3 12 
 
 4 37 
 
 6 
 
 3 17 39 
 
 3 15 59 
 
 48 
 
 36 
 
 19 45 53 
 
 19 35 50 
 
 4 47 
 
 7 
 
 3 50 35 
 
 3 48 38 
 
 56 
 
 37 
 
 20 18 48 
 
 20 8 SO 
 
 4 54 
 
 8 
 
 4 23 32 
 
 4 21 18 
 
 1 4 
 
 38 
 
 20 51 47 
 
 20 41 10 
 
 4 54 
 
 9 
 
 4 56 28 
 
 4 53 58 
 
 1 12 
 
 39 
 
 21 24 41 
 
 21 13 49 
 
 5 1 
 
 10 
 
 5 29 25 
 
 5 26 38 
 
 1 19 
 
 40 
 
 21 57 37 
 
 21 46 30 
 
 5 12 
 
 11 
 
 6 2 21 
 
 5 59 17 
 
 1 27 
 
 41 
 
 22 SO 36 
 
 22 19 12 
 
 5 25 
 
 12 
 
 6 35 18 
 
 6 31 57 
 
 1 35 
 
 42 
 
 23 3 SO 
 
 22 51 4fe 
 
 5 31 
 
 13 
 
 7 8 14 
 
 7 4 37 
 
 1 43 
 
 43 
 
 23 36 30 
 
 23 24 30 
 
 5 42 
 
 14 
 
 7 41 10 
 
 7 37 16 
 
 1 51 
 
 44 
 
 24 9 24 
 
 23 57 6 
 
 5 50 
 
 15 
 
 8 14 7 
 
 8 9 56 
 
 1 59 
 
 45 
 
 24 42 19 
 
 24 29 48 
 
 6 
 
 16 
 
 8 47 3 
 
 8 42 36 
 
 2 7 
 
 46 
 
 25 15 18 
 
 25 2 30 
 
 6 6 
 
 17 
 
 9 20 
 
 9 15 16 
 
 2 15 
 
 47 
 
 25 48 12 
 
 25 35 6 
 
 6 13 
 
 18 
 
 9 52 56 
 
 9 47 55 
 
 2 23 
 
 48 
 
 26 21 12 
 
 26 7 48 
 
 6 23 
 
 19 
 
 10 25 53 
 
 20 35 
 
 2 31 
 
 49 
 
 26 54 6 
 
 26 40 30 
 
 6 30 
 
 20 
 
 10 58 49 
 
 10 53 15 
 
 2 39 
 
 50 
 
 27 27 
 
 27 13 6 
 
 6 36 
 
 21 
 
 11 31 46 
 
 11 25 55 
 
 2 47 
 
 51 
 
 28 
 
 27 45 48 
 
 6 48 
 
 22 
 
 12 4 42 
 
 11 58 34 
 
 2 55 
 
 52 
 
 28 32 55 
 
 28 18 30 
 
 6 54 
 
 23 
 
 12 37 39 
 
 12 31 14 
 
 3 3 
 
 55 
 
 29 5 54 
 
 28 51 6 
 
 7 
 
 L l- 
 
 13 10 35 
 
 13 3 54 
 
 5 11 
 
 54 
 
 79 38 48 
 
 29 23 48 
 
 7 6 
 
 25 
 
 13 43 32 
 
 13 36 34 
 
 3 19 
 
 ;"*f> 
 
 30 11 48 
 
 29 56 24 
 
 7 18 
 
 2f 
 
 14 16 28 
 
 14 91" 
 
 3 26 
 
 56 
 
 30 44 42 
 
 30 29 6 
 
 7 24 
 
 27 
 
 14 49 24 
 
 14 41 53 
 
 3 34 
 
 57 
 
 31 17 36 
 
 31 1 48 
 
 7 SO 
 
 28 
 
 15 22 21 
 
 15 14 33 
 
 3 42 
 
 58 
 
 31 50 36 
 
 31 34 24 
 
 7 48 
 
 2 
 
 15 55 17 
 
 15 47 13 
 
 S 50 
 
 59 
 
 32 23 30 
 
 32 7 6 
 
 7 48 
 
 30 
 
 16 28 14 
 
 16 19 5H 
 
 3 58 
 
 60 
 
 32 56 39 
 
 32 39 48 
 
 7 56 
 
 TABLES OF THE MOON'S EQUATIONS. 
 I. Annual Equation of the Moon's Node. 
 Argument. Sun's Mean Anomaly. 
 
 S. 
 
 + o 
 
 + 1 
 
 + 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 o 
 
 
 
 
 
 4 SO 
 
 7 52 
 
 9 12 
 
 8 4 
 
 4 42 
 
 O 
 
 30 
 
 5 
 
 47 
 
 5 10 
 
 8 15 
 
 9 It 
 
 7 38 
 
 3 58 
 
 25 
 
 10 
 
 I S-J 
 
 5 48 
 
 8 34 
 
 9 6 
 
 7 9 
 
 3 16 
 
 20 
 
 15 
 
 2 19 
 
 6 23 
 
 8 49 
 
 8 56 
 
 6 37 
 
 2 26 
 
 15 
 
 20 
 
 3 4 
 
 6 56 
 
 9 1 
 
 8 43 
 
 6 1 
 
 1 38 
 
 10 
 
 25 
 
 3 48 
 
 7 26 
 
 3 8 
 
 8 25 
 
 5 23 
 
 49 
 
 5 
 
 30 
 
 4 30 
 
 7 52 
 
 9 12 
 
 8 4 
 
 4 42 
 
 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 -6 
 
 S. 
 
 Annual Equation of Moon's Mean Jlnomaly. 
 Argument. Sun's Mean Anomaly. 
 
 S. 
 
 + o 
 
 + 1 
 
 + 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 37 
 
 18 33 
 
 21 42 
 
 19 1 
 
 11 5 
 
 30 
 
 5 
 
 1 51 
 
 12 11 
 
 19 27 
 
 21 39 
 
 18 1 
 
 9 23 
 
 25 
 
 10 
 
 3 40 
 
 13 41 
 
 20 IS 
 
 21 27 
 
 16 53 
 
 7 36 
 
 20 
 
 15 
 
 5 29 
 
 15 4 
 
 20 49 
 
 21 5 
 
 15 37 
 
 5 45 
 
 15 
 
 20 
 
 7 15 
 
 16 21 
 
 21 16 
 
 20 34 
 
 14 13 
 
 3 52 
 
 10 
 
 25 
 
 8 58 
 
 17 31 
 
 21 3-1 
 
 19 52 
 
 12 42 
 
 1 56 
 
 5 
 
 30 
 
 10 37 
 
 18 33 
 
 21 42 
 
 19 1 
 
 11 5 
 
 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 S. 
 
 I. For the Moon's Longitude. 
 Argument. Sun's Mean Anomaly. 
 
 S. 
 
 + o 
 
 + 1 
 
 + 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 
 
 
 
 
 
 
 1 
 
 
 
 5 
 10 
 15 
 
 
 
 57 
 1 53 
 2 49 
 
 5 27 
 6 15 
 7 1 
 7 44 
 
 9 31 
 9 59 
 10 23 
 10 41 
 
 11 9 
 
 11 8 
 11 1 
 10 50 
 
 9 47 
 9 16 
 8 41 
 
 8 2 
 
 5 42 
 4 49 
 3 54 
 2 58 
 
 30 
 25 
 20 
 15 
 
 20 
 25 
 
 3 43 
 4 36 
 
 527 
 
 8 23 
 8 59 
 9 31 
 
 10 55 
 11 4 
 il 9 
 
 10 34 
 10 13 
 9 47 
 
 7 19 
 6 32 
 5 42 
 
 1 59 
 1 
 
 
 10 
 5 
 
 
 Is. 
 
 -t, 
 
 10 
 
 C) 
 
 
 
 ~'C 
 
 7 
 
 6 
 
 S. 
 
APPENDIX TO THE ASTRONOMY. 
 
 JI. For the 
 
 Longitude. 
 
 Arg. II. 2 2 from 
 + Arg. I. 
 
 III. For the Moon's 
 Longitude. 
 
 Arg. III. 2 3) from Q 
 - Arg. I. 
 
 S. 
 
 1 
 
 2 
 
 S. 
 
 
 s 
 
 
 
 __ i 
 
 2 
 
 S. 
 
 s. 
 
 + 6 
 
 + 7 
 
 + 8 
 
 S. 
 
 
 s. 
 
 + 6 
 
 + 7 
 
 + S 
 
 S. 
 
 
 
 
 
 
 
 28 
 
 48 
 
 30 
 
 
 
 
 
 
 38 
 
 1 5 
 
 O 
 
 30 
 
 5 
 
 5 
 
 32 
 
 51 
 
 25 
 
 
 5 
 
 7 
 
 43 
 
 1 8 
 
 25 
 
 10 
 
 10 
 
 36 
 
 52 
 
 20 
 
 
 10 
 
 13 
 
 4S 
 
 1 11 
 
 20 
 
 15 
 
 14 
 
 39 
 
 54 
 
 15 
 
 
 15 
 
 19 
 
 53 
 
 1 13 
 
 15 
 
 20 
 
 19 
 
 43 
 
 55 
 
 10 
 
 
 20 
 
 26 
 
 58 
 
 1 14 
 
 10 
 
 25 
 
 24 
 
 46 
 
 56 
 
 5 
 
 
 25 
 
 32 
 
 I 2 
 
 1 15 
 
 5 
 
 30 
 
 28 
 
 48 
 
 56 
 
 
 
 
 30 
 
 88 
 
 1 f- 
 
 1 15 
 
 
 
 S. 
 
 + 11 
 
 + 10 
 
 + 9 
 
 S. 
 
 
 S. 
 
 + 11 
 
 + 10 
 
 + 9 
 
 S. 
 
 s. 
 
 5 
 
 4 
 
 3 
 
 s. 
 
 
 s. 
 
 5 
 
 4 
 
 3 
 
 S 
 
 IV. For the Moon's Longitude. 
 Arg. IV. 2 3) from Q + J 's Mean Anomaly. 
 
 s. 
 
 -I- o 
 
 + 1 
 
 + 2 
 
 S. 
 
 s. 
 
 6 
 
 7 
 
 8 
 
 s. 
 
 O 
 
 
 
 ff 
 
 
 
 // 
 29 
 
 If 
 
 50 
 
 
 
 30 
 
 5 
 
 a 
 
 33 
 
 52 
 
 25 
 
 10 
 
 10 
 
 37 
 
 54 
 
 20 
 
 15 
 
 15 
 
 41 
 
 56 
 
 15 
 
 20 
 
 20 
 
 4% 
 
 57 
 
 10 
 
 25 
 
 24 
 
 47 
 
 57 
 
 5 
 
 30 
 
 29 
 
 50 
 
 58 
 
 
 
 S. 
 
 11 
 
 10 
 
 ._., Q 
 
 S. 
 
 . S - 
 
 + 5 
 
 + * 
 
 + S 
 
 s. 
 
 V. For the Moon's Longitude. Erection. 
 
 Argument V. 
 2 3) from Q j) 's Mean Anomaly. 
 
 s. 
 
 
 
 1 
 
 2 
 
 O 
 
 4 
 
 5 
 
 b. 
 
 
 
 
 
 
 
 39 44 
 
 1 9 11 
 
 1 20 28 
 
 1 10 12 
 
 40 45 
 
 O 
 
 30 
 
 1 
 
 1 23 
 
 40 56 
 
 1 9 53 
 
 1 20 29 
 
 1 9 30 
 
 39 31 
 
 29 
 
 2 
 
 2 46 
 
 42 7 
 
 1 10 34 
 
 1 20 28 
 
 1 8 46 
 
 38 16 
 
 2 
 
 3 
 
 049 
 
 43 18 
 
 1 11 14 
 
 1 20 25 
 
 181 
 
 37 
 
 27 
 
 4 
 
 5 32 
 
 44 27 
 
 1 11 52 
 
 1 20 21 
 
 1 7 15 
 
 35 45 
 
 26 
 
 5 
 
 6 55 
 
 45 36 
 
 1 12 29 
 
 1 20 16 
 
 1 6 28 
 
 34 27 
 
 25 
 
 6 
 
 8 17 
 
 46 45 
 
 1 13 5 
 
 1 20 9 
 
 1 5 40 
 
 33 10 
 
 24 
 
 7 
 
 9 4i 
 
 47 52 
 
 1 13 39 
 
 1 iiO 1 
 
 1 4 50 
 
 31 52 
 
 23 
 
 8 
 
 11 2 
 
 48 59 
 
 1 14 12 
 
 1 19 51 
 
 1 3 59 
 
 30 33 
 
 22 
 
 9 
 
 12 24 
 
 50 4 
 
 1 14 44 
 
 1 19 40 
 
 137 
 
 29 14 
 
 21 
 
 10 
 
 13 4f 
 
 51 9 
 
 1 15 15 
 
 1 19 27 
 
 1 2 13 
 
 27 5* 
 
 '20 
 
 11 
 
 15 8 
 
 52 13 
 
 1 15 44 
 
 1 19 13 
 
 1 1 19 
 
 26 34 
 
 19 
 
 12 
 
 16 30 
 
 :> 53 16 
 
 1 16 11 
 
 1 18 57 
 
 1 23 
 
 26 13 
 
 18 
 
 is 
 
 17 61 
 
 > 54 18 
 
 1 16 38 
 
 1 18 40 
 
 59 26 
 
 23 51 
 
 17 
 
 14 
 
 19 12 
 
 55 19 
 
 1 17 3 
 
 1 18 21 
 
 58 28 
 
 22 29 
 
 16 
 
 15 
 
 20 3i 
 
 56 19 
 
 1 17 26 
 
 1 18 1 
 
 57 29 
 
 21 7 
 
 15 
 
 16 
 
 21 5i 
 
 57 18 
 
 1 17 48 
 
 1 17 40 
 
 56 29 
 
 19 J4 
 
 14 
 
 17 
 
 23 1. 
 
 58 16 
 
 1 18 9 
 
 1 17 17 
 
 55 28 
 
 18 22 
 
 13 
 
 18 
 
 24 31 
 
 J 59 IS 
 
 1 18 29 
 
 1 16 53 
 
 54 26 
 
 16 58 
 
 12 
 
 19 
 
 2? 50 
 
 109 
 
 1 18 47 
 
 1 16 27 
 
 53 22 
 
 15 34 
 
 11 
 
 20 
 
 27 9 
 
 1 1 4 
 
 1 19 3 
 
 1 16 
 
 52 18 
 
 14 10 
 
 10 
 
 21 
 
 28 27 
 
 I 1 58 
 
 1 19 18 
 
 1 15 31 
 
 51 13 
 
 12 46 
 
 9 
 
 22 
 
 29 44 
 
 I 2 51 
 
 1 19 32 
 
 1 15 1 
 
 50 7 
 
 11 22 
 
 8 
 
 23 
 
 31 1 
 
 1 3 42 
 
 1 19 44 
 
 1 14 30 
 
 49 
 
 9 57 
 
 7 
 
 24 
 
 32 18 
 
 1 4 S3 
 
 1 19 55 
 
 1 IS 67 
 
 47 51 
 
 8 32 
 
 6 
 
 25 
 
 33 3* 
 
 1 5 2* 
 
 1 20 4 
 
 1 13 23 
 
 46 42 
 
 077 
 
 5 
 
 26 
 
 3i 49 
 
 1 6 10 
 
 1 20 12 
 
 1 12 47 
 
 45 33 
 
 5 42 
 
 4 
 
 27 
 
 36 4 
 
 1 6 57 
 
 1 20 18 
 
 1 12 1C 
 
 44 22 
 
 4 16 
 
 3 
 
 28 
 
 37 18 
 
 1 7 43 
 
 1 20 23 
 
 1 11 32 
 
 43 10 
 
 2 51 
 
 2 
 
 29 
 
 38 31 
 
 1 8 28 
 
 1 20 26 
 
 1 10 63 
 
 41 58 
 
 1 25 
 
 1 
 
 30 
 
 39 44 
 
 1 9 11 
 
 1 20 28 
 
 1 10 12 
 
 40 45 
 
 000 
 
 
 
 S. 
 
 + 11 
 
 + 10 
 
 + 
 
 + 8 
 
 + 7 
 
 + 6 
 
 S. 
 
LUNAR TABLES. 
 
 377 
 
 
 VI, For the ]> 's Longitude. 
 
 Argument. 
 Arg. V. + Arg. I. 
 
 VII. For tlie J 's Longitude. 
 
 Argument. 
 Arg. V. Arg. I. 
 
 s. 
 
 + 
 
 + 1 
 
 + * 
 
 S. 
 
 s. 
 
 6 
 
 7 
 
 8 
 
 S. 
 
 
 
 1 
 
 
 
 1 
 
 1 47 
 
 o 
 
 30 
 
 5 
 
 It 
 
 1 11 
 
 1 52 
 
 -25 
 
 10 
 
 21 
 
 I 19 
 
 I 56 
 
 20 
 
 15 
 
 32 
 
 1 27 
 
 I 59 
 
 15 
 
 20 
 
 42 
 
 1 35 
 
 2 -. 
 
 10 
 
 25 
 
 52 
 
 1 41 
 
 2 & 
 
 5 
 
 30 
 
 1 2 
 
 1 47 
 
 2 3 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 S. 
 
 S. 
 
 + 5 
 
 + 4 
 
 + 3 
 
 S. 
 
 s. 
 
 + o 
 
 + 1 
 
 + 2 
 
 S. 
 
 s. 
 
 6 
 
 7 
 
 8 
 
 S. 
 
 o 
 
 n 
 
 // 
 
 ft 
 
 O 
 
 
 
 
 
 23 
 
 40 
 
 30 
 
 5 
 
 4 
 
 27 
 
 42 
 
 25 
 
 10 
 
 8 
 
 SO 
 
 44 
 
 20 
 
 15 
 
 12 
 
 S3 
 
 45 
 
 15 
 
 20 
 
 16 
 
 36 
 
 40 
 
 10 
 
 25 
 
 20 
 
 38 
 
 46 
 
 5 
 
 30 
 
 23 
 
 40 
 
 47 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 S. 
 
 S. 
 
 + 5 
 
 + 4 
 
 + 3 
 
 S. 
 
 IX. For the Moon's Longitude. 
 
 Argument IX. 
 Mean Disk D from Q D 's Mean Anom. 
 
 S 
 
 
 
 1 
 
 =F2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 o 
 
 
 
 
 38 
 
 30 
 
 23 
 
 1 9 
 
 1 1 
 
 o 
 
 30 
 
 5 
 
 8 
 
 41 
 
 23 
 
 S3 
 
 1 12 
 
 54 
 
 25 
 
 10 
 
 16 
 
 42 
 
 16 
 
 42 
 
 1 14 
 
 45 
 
 20 
 
 15 
 
 23 
 
 41 
 
 7 
 
 51 
 
 1 13 
 
 35 
 
 15 
 
 20 
 
 29 
 
 39 
 
 + 3 
 
 58 
 
 I 11 
 
 23 
 
 10 
 
 25 
 
 34 
 
 35 
 
 13 
 
 1 5 
 
 1 7 
 
 12 
 
 5 
 
 30 
 
 38 
 
 30 
 
 23 
 
 1 9 
 
 1 1 
 
 
 
 
 
 S. 
 
 + 11 
 
 + 10 
 
 9 
 
 8 
 
 7 
 
 6 
 
 S. 
 
 VIII. For the 5 's Longitude. 
 
 Argument. 
 5 '9 Mean Anom. Arg. I. 
 
 S 
 
 + 
 
 + 1 
 
 + 2 
 
 S. 
 
 s. 
 
 6 
 
 7 
 
 8 
 
 S. 
 
 
 
 n 
 
 >t 
 
 rr 
 
 o 
 
 
 
 
 
 21 
 
 36 
 
 .0 
 
 5 
 
 4 
 
 24 
 
 38 
 
 25 
 
 10 
 
 7 
 
 27 
 
 40 
 
 20 
 
 15 
 
 11 
 
 30 
 
 41 
 
 15 
 
 20 
 
 14 
 
 32 
 
 41 
 
 10 
 
 25 
 
 18 
 
 34 
 
 42 
 
 5 
 
 30 
 
 21 
 
 36 
 
 42 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 S. 
 
 S. 
 
 + 5 
 
 + 4 
 
 + 3 
 
 s 
 
 X. For the Moon's Longitude. 
 
 Argument X. 
 Mean Long. J. True Long. Q. 
 
 S. 
 
 + 
 
 + 1 
 
 + 2 
 
 s. 
 
 S. 
 
 + 6 
 
 + 7 
 
 + |B 
 
 s. 
 
 O 
 
 a 
 
 / n 
 
 n 
 
 
 
 
 
 
 
 52 
 
 52 
 
 30 
 
 5 
 
 10 
 
 57 
 
 46 
 
 25 
 
 10 
 
 21 
 
 59 
 
 39 
 
 20 
 
 15 
 
 30 
 
 1 
 
 30 
 
 15 
 
 20 
 
 39 
 
 1 
 
 21 
 
 10 
 
 25 
 
 46 
 
 57 
 
 10 
 
 5 
 
 30 
 
 52 
 
 52 
 
 
 
 
 
 S. 
 
 11 
 
 .10 
 
 ~ 9 
 
 S. 
 
 S. 
 
 5 
 
 4 
 
 s!s. 
 
378 
 
 APPENDIX TO THE ASTRONOMY. 
 
 XL For the Moon's Longitude. Equation of the Centre. 
 Argument XI. Moon's Correct Anomaly. 
 
 
 s. 
 
 
 
 1 
 
 2 
 
 3 
 
 __ 4 
 
 5 
 
 8. 
 
 
 
 
 
 
 
 
 
 
 
 000 
 
 2 58 30 
 
 5 16 2! 
 
 6 17 38 
 
 5 38 46 
 
 3 20 56 
 
 30 
 
 1 
 
 6 11 
 
 3 3 59 
 
 5 19 48 
 
 6 18 2 
 
 5 35 41 
 
 3 14 57 
 
 29 
 
 2 
 
 12 22 
 
 3 9 24 
 
 5 23 10 
 
 6 18 18 
 
 5 32 28 
 
 3 8 54 
 
 28 
 
 3 
 
 18 32 
 
 3 14 46 
 
 5 26 26 
 
 6 18 28 
 
 5 29 8 
 
 3 2 47 
 
 27 
 
 4 
 
 24 42 
 
 3 20 5 
 
 5 29 38 
 
 6 18 32 
 
 5 25 4 
 
 2 56 36 
 
 26 
 
 5 
 
 30 52 
 
 3 25 22 
 
 5 32 43 
 
 6 18 28 
 
 5 22 10 
 
 2 50 21 
 
 25 
 
 6 
 
 37 2 
 
 3 30 35 
 
 5 35 43 
 
 6 18 17 
 
 5 18 30 
 
 2 44 2 
 
 24 
 
 7 
 
 43 11 
 
 3 35 45 
 
 5 38 38 
 
 6 17 59 
 
 5 14 44 
 
 2 37 40 
 
 23 
 
 8 
 
 49 19 
 
 3 40 51 
 
 5 41 27 
 
 6 17 34 
 
 5 10 52 
 
 2 31 14 
 
 22 
 
 9 
 
 55 26 
 
 3 45 54 
 
 5 44 11 
 
 6 17 3 
 
 5 6 53 
 
 2 24 45 
 
 21 
 
 10 
 
 I 1 33 
 
 3 50 54 
 
 5 46 48 
 
 6 16 24 
 
 5 2 48 
 
 2 18 13 
 
 20 
 
 11 
 
 1 7 39 
 
 3 55 50 
 
 5 49 20 
 
 6 15 38 
 
 4 58 37 
 
 2 11 37 
 
 19 
 
 12 
 
 1 13 44 
 
 4 42 
 
 5 51 45 
 
 6 14 45 
 
 4 54 19 
 
 2 4 59 
 
 18 
 
 13 
 
 1 19 47 
 
 4 5 30 
 
 5 54 5 
 
 6 13 45 
 
 4 49 55 
 
 1 58 18 
 
 17 
 
 14 
 
 1 25 50 
 
 4 10 15 
 
 5 56 19 
 
 5 12 38 
 
 4 45 25 
 
 I 51 34 
 
 16 
 
 15 
 
 I 31 51 
 
 4 14 56 
 
 5 58 27 
 
 S 11 24 
 
 4 40 50 
 
 1 44 48 
 
 15 
 
 16 
 
 1 37 51 
 
 4 19 33 
 
 6 28 
 
 5 10 3 
 
 4 36 8 
 
 1 37 59 
 
 14 
 
 17 
 
 1 43 49 
 
 4 24 5 
 
 6 2 24 
 
 G 8 34 
 
 4 31 20 
 
 1 31 8 
 
 13 
 
 18 
 
 1 49 45 
 
 4 28 34 
 
 6 4 13 
 
 G 6 59 
 
 4 26 27 
 
 1 24 16 
 
 12 
 
 19 
 
 I 55 41 
 
 4 32 58 
 
 6 5 56 
 
 6 5 16 
 
 4 21 29 
 
 1 17 21 
 
 11 
 
 20 
 
 2 1 34 
 
 4 37 18 
 
 6 7 32 
 
 6 3 27 
 
 4 16 24 
 
 I 10 25 
 
 10 
 
 21 
 
 2 7 25 
 
 4 41 33 
 
 692 
 
 6 1 SO 
 
 4 11 14 
 
 I 3 27 
 
 9 
 
 22 
 
 2 13 15 
 
 4 45 44 
 
 6 10 26 
 
 5 59 27 
 
 4 5 59 
 
 56 27 
 
 8 
 
 23 
 
 2 19 2 
 
 4 49 50 
 
 6 11 43 
 
 5 57 Hi 
 
 4 38 
 
 49 27 
 
 7 
 
 24 
 
 2 24 48 
 
 4 53 52 
 
 6 12 54 
 
 5 54 58 
 
 3 55 13 
 
 42 25 
 
 6 
 
 25 
 
 2 30 31 
 
 4 57 49 
 
 6 13 58 
 
 5 52 34 
 
 3 49 42 
 
 35 22 
 
 5 
 
 26 
 
 2 36 12 
 
 5 1 41 
 
 6 14 55 
 
 5 50 2 
 
 3 44 6 
 
 28 19 
 
 4 
 
 27 
 
 2 41 50 
 
 5 5 29 
 
 6 15 46 
 
 5 47 23 
 
 3 38 26 
 
 21 15 
 
 5 
 
 28 
 
 2 47 26 
 
 5911 
 
 6 16 SO 
 
 5 44 38 
 
 3 32 40 
 
 14 10 
 
 2 
 
 29 
 
 2 53 
 
 5 12 49 
 
 6 17 7 
 
 5 41 46 
 
 3 26 51 
 
 075 
 
 1 
 
 30 
 
 2 58 30 
 
 5 16 21 
 
 6 17 38 
 
 5 38 46 
 
 3 20 56 
 
 000 
 
 
 
 S; 
 
 + H 
 
 + 10 
 
 + 9 
 
 + 8 
 
 + 7 
 
 + 6 
 
 S. 
 
 
LUNAR TABLES. 
 
 ! 
 
 379 
 
 XII. For the J>'sLong. Variation, 
 
 Argument XII. 
 3 '& equaled Long. O '* true Long. 
 
 XIII. For the 
 
 Long. 
 
 Argument XIII. 
 
 Uoub. eq. itist. J from 
 J C. An. 
 
 XIV. For the J'sion*. Red. 
 Argument XIV. 
 
 Long. } in Orb. C. 
 Long. SI 
 
 s. 
 
 + 
 
 + 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 S. 
 
 Eq 
 
 I 
 
 s- 
 
 + 
 
 + 1 
 
 4-2 
 
 S. 
 
 o 
 
 
 
 1 
 2 
 
 5 
 4 
 
 
 
 
 
 
 O 
 
 30 
 29 
 
 28 
 27 
 26 
 
 S. 
 
 6 
 
 / // 
 
 
 7 
 15 
 22 
 29 
 
 7 
 
 8 
 
 S. 
 
 
 1 14 
 
 2 27 
 3 40 
 4 53 
 
 30 9 
 30 43 
 31 15 
 31 44 
 32 11 
 
 29 6 
 28 26 
 27 44 
 27 
 26 13 
 
 2 
 3 16 
 4 30 
 5 43 
 6 57 
 
 32 28 
 33 3 
 33 36 
 34 6 
 34 35 
 
 35 1 
 35 24 
 35 45 
 36 3 
 36 19 
 
 31 55 
 31 15 
 30 33 
 29 48 
 29 2 
 
 
 
 
 3 
 
 10 
 15 
 
 20 
 
 25 
 30 
 
 / // 
 42 
 48 
 54 
 () 59 
 1 4 
 
 / // 
 1 13 
 1 16 
 1 19 
 
 I 21 
 
 1 23 
 
 o 
 
 30 
 25 
 20 
 15 
 10 
 
 5 
 
 
 5 
 6 
 7 
 8 
 9 
 
 10 
 11 
 12 
 13 
 14 
 
 15 
 16 
 17 
 18 
 19 
 
 6 6 
 
 7 is 
 
 8 30 
 9 41 
 10 51 
 
 32 36 
 32 58 
 33 18 
 33 35 
 33 49 
 
 25 25 
 24 35 
 23 43 
 22 49 
 21 53 
 
 8 10 
 9 22 
 10 34 
 11 46 
 12 56 
 
 28 13 
 27 22 
 26 29 
 25 35 
 -24 38 
 
 25 
 24 
 23 
 22 
 21 
 
 i'O 
 19 
 18 
 17 
 16 
 
 15 
 
 14 
 13 
 12 
 11 
 
 35 
 42 
 
 I 9 
 I 13 
 
 1 :24 
 1 24 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 s. 
 
 12 
 13 9 
 14 16 
 15 23 
 16 28 
 
 34 1 
 34 H 
 34 18 
 34 22 
 34 23 
 
 20 56 
 19 5T 
 18 57 
 17 55 
 16 52 
 
 14 6 
 15 15 
 
 16 23 
 17 29 
 18 35 
 
 36 52 
 36 43 
 36 51 
 
 36 57 
 36 59 
 
 23 40 
 22 40 
 -21 39 
 20 35 
 19 30 
 
 18 24 
 17 17 
 16 8 
 14 58 
 13 47 
 
 12 36 
 11 23 
 10 9 
 8 54 
 
 7 39 
 
 S. 
 
 + 5 
 
 + 4 
 
 + 3 
 
 s. 
 
 nation of the Equinad 
 Points. 
 
 Argument. 
 rtean Long. Moon's J. 
 
 17 31 
 
 18 34 
 19 35 
 20 35 
 21 33 
 
 34 22 
 34 19 
 34 13 
 34 4 
 33 53 
 
 15 47 
 14 41 
 13 34 
 12 26 
 11 17 
 
 19 39 
 20 42 
 21 44 
 22 44 
 23 43 
 
 37 
 36 57 
 36 53 
 36 45 
 36 35 
 
 s. o 
 
 _1|_2 
 
 s. 
 
 20 
 21 
 
 22 
 23 
 24 
 
 25 
 26 
 
 27 
 28 
 29 
 
 22 30 
 23 24 
 24 17 
 25 8 
 25 57 
 
 33 39 
 
 33 23 
 33 4 
 32 43 
 32 19 
 
 10 8 
 8 57 
 7 45 
 6 34 
 5 21 
 
 4 8 
 2 54 
 1 41 
 
 hO 27 
 
 47 
 
 24 40 
 25 35 
 26 29 
 27 21 
 28 10 
 
 36 22 
 36 6 
 35 48 
 35 28 
 35 5 
 
 10 
 9 
 
 8 
 
 V 
 
 i 
 
 6 
 
 s. 
 
 4- c 
 
 + 7 
 
 + 8 
 
 s. 
 
 o 
 
 
 5 
 10 
 15 
 
 20 
 
 25 
 30 
 
 // 
 
 
 1 
 
 3 
 5 
 6 
 
 // 
 
 9 
 10 
 12 
 13 
 14 
 
 // 
 
 16 
 16 
 17 
 17 
 18 
 
 o 
 30 
 25 
 20 
 15 
 10 
 
 26 45 
 27 30 
 28 13 
 28 54 
 29 32 
 
 31 53 
 31 24 
 30 53 
 30 19 
 29 44 
 
 28 59 
 29 45 
 SO 29 
 31 10 
 31 50 
 
 34 39 
 34 11 
 33 41 
 33 8 
 32 32 
 
 6 24 
 5 7 
 3 51 
 2 34 
 1 17 
 
 5 
 4 
 
 2 
 1 
 
 8 
 9 
 
 15 
 16 
 
 18 
 18 
 
 5 
 
 
 3( 
 
 30 9 
 
 29 6 
 
 2 2 
 
 32 28 
 
 n 55 
 
 
 
 
 
 S. 
 
 4-n:4-10 
 
 + 9 
 
 S 
 
 S. 
 
 11 
 
 10 
 
 ^9 
 
 + 8 
 
 4-7 
 
 + 6 
 
 S. 
 
 S. 
 
 5 4 
 
 3 
 
 s. 
 
 s 
 
 
 
 1 
 
 2 
 
 S. 
 
 s. 
 
 6 
 
 7 
 
 8 
 
 S. 
 
 
 
 
 
 
 
 
 
 
 5 53 
 
 5 53 
 
 30 
 
 1 
 
 14 
 
 6 
 
 5 46 
 
 29 
 
 2 
 
 28 
 
 6 6 
 
 5 38 
 
 28 
 
 3 
 
 43 
 
 6 1 
 
 5 30 
 
 27 
 
 4 
 
 57 
 
 6 18 
 
 5 21 
 
 26 
 
 5 
 
 1 11 
 
 6 23 
 
 5 12 
 
 25 
 
 6 
 
 I 25 
 
 6 28 
 
 5 3 
 
 24 
 
 7 
 
 1 39 
 
 6 32 
 
 4 53 
 
 23 
 
 8 
 
 I 52 
 
 6 36 
 
 4 43 
 
 22 
 
 9 
 
 2 6 
 
 6 39 
 
 4 33 
 
 21 
 
 10 
 
 2 19 
 
 6 41 
 
 4 22 
 
 20 
 
 11 
 
 2 S3 
 
 6 44 
 
 4 11 
 
 19 
 
 12 
 
 2 46 
 
 6 45 
 
 4 
 
 18 
 
 13 
 
 2 59 
 
 6 47 
 
 3 48 
 
 17 
 
 14 
 
 3 11 
 
 6 47 
 
 3 36 
 
 16 
 
 15 
 
 3 2-4 
 
 6 48 
 
 3 24 
 
 15 
 
 16 
 
 3 30 
 
 6 47 
 
 3 11 
 
 14 
 
 17 
 
 3 48 
 
 6 47 
 
 2 59 
 
 13 
 
 18 
 
 4 
 
 6 45 
 
 2 46 
 
 12 
 
 19 
 
 4 11 
 
 6 44 
 
 2 38 
 
 11 
 
 20 
 
 4 22 
 
 6 41 
 
 2 19 
 
 10 
 
 21 
 
 4 33 
 
 6 39 
 
 2 6 
 
 9 
 
 22 
 
 4 43 
 
 6 36 
 
 1 52 
 
 8 
 
 23 
 
 4 53 
 
 6 32 
 
 1 39 
 
 7 
 
 24 
 
 5 3 
 
 6 28 
 
 1 25 
 
 6 
 
 25 
 
 5 12 
 
 6 23 
 
 1 11 
 
 5 
 
 26 
 
 5 21 
 
 6 18 
 
 57 
 
 4 
 
 27 
 
 5 30 
 
 6 12 
 
 43 
 
 S 
 
 28 
 
 5 38 
 
 6 6 
 
 28 
 
 2 
 
 29 
 
 5 46 
 
 6 
 
 1-1 
 
 1 
 
 
 
 5 53 
 
 5 53 
 
 
 
 
 
 S. 
 
 4-11 
 
 + 10 
 
 + 9 
 
 >. 
 
 s. 
 
 + 5 
 
 + 4 
 
 4-3 
 
 S. 
 
380 
 
 APPENDIX TO THE ASTRONOMY. 
 
 TABLES FOR FINDING THE MOON'S LATITUDE. 
 
 I. For the Moon's Latitude. 
 
 Argument I. 
 Long. D in Orbit Cor. Long. J 
 
 s. 
 
 + o 
 
 + I 
 
 -f 2 
 
 S. 
 
 s. 
 
 6 
 
 ___ <* 
 
 8 
 
 S. 
 
 o 
 
 o f ft 
 
 f II 
 
 a / ir 
 
 o 
 
 
 
 000 
 
 2 34 18 
 
 4 27 23 
 
 30 
 
 1 
 
 5 23 
 
 2 38 56 
 
 4 30 2 
 
 29 
 
 2 
 
 10 46 
 
 2 43 32 
 
 4 32 37 
 
 28 
 
 3 
 
 16 9 
 
 2 48 5 
 
 4 35 6 
 
 27 
 
 4 
 
 21 31 
 
 2 52 35 
 
 4 37 31 
 
 26 
 
 5 
 
 26 53 
 
 2 57 1 
 
 4 39 50 
 
 35 
 
 6 
 
 32 15 
 
 3 1 24 
 
 4 42 4 
 
 24 
 
 7 
 
 37 36 
 
 3 5 44 
 
 4 44 14 
 
 23 
 
 8 
 
 42 56 
 
 3 10 1 
 
 4 46 17 
 
 22 
 
 9 
 
 48 16 
 
 A 14 14 
 
 4 48 16 
 
 21 
 
 10 
 
 53 35 
 
 3 18 23 
 
 4 50 9 
 
 20 
 
 11 
 
 58 52 
 
 3 22 29 
 
 4 51 58 
 
 19 
 
 12 
 
 I 4 9 
 
 3 26 32 
 
 4 53 40 
 
 18 
 
 13 
 
 I 9 24 
 
 3 30 30 
 
 4 55 18 
 
 17 
 
 14 
 
 1 14 39 
 
 3 34 25 
 
 4 56 50 
 
 16 
 
 15 
 
 1 19 51 
 
 3 38 16 
 
 4 58 16 
 
 15 
 
 16 
 
 1 25 3 
 
 3 42 2 
 
 4 59 38 
 
 14 
 
 17 
 
 1 SO 13 
 
 3 45 45 
 
 5 53 
 
 13 
 
 18 
 
 1 35 21 
 
 3 49 24 
 
 523 
 
 12 
 
 19 
 
 1 40 27 
 
 3 52 58 
 
 538 
 
 11 
 
 20 
 
 I 45 32 
 
 3 56 28 
 
 547 
 
 10 
 
 21 
 
 1 50 35 
 
 3 59 54 
 
 550 
 
 9 
 
 22 
 
 I 55 35 
 
 4 3 16 
 
 5 5 48 
 
 8 
 
 23 
 
 2 34 
 
 4 6 33 
 
 5 6 31 
 
 7 
 
 24 
 
 2 5 30 
 
 4 9 45 
 
 577 
 
 6 
 
 25 
 
 2 10 25 
 
 4 12 53 
 
 5 7 38 
 
 5 
 
 26 
 
 2 15 16 
 
 4 15 57 
 
 584 
 
 4 
 
 27 
 
 2 20 6 
 
 4 18 55 
 
 5 8 23 
 
 3 
 
 28 
 
 2 24 52 
 
 4 21 49 
 
 5 8 38 
 
 2 
 
 29 
 
 2 29 36 
 
 4 24 38 
 
 5 8 46 
 
 1 
 
 30 
 
 -2 34 18 
 
 4 27 23 
 
 5 8 49 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 s. 
 
 S. 
 
 -f 5 
 
 + 4 
 
 + 3 
 
 s. 
 
 II. For the Moon's Latitude. 
 
 Argument II. 
 Double Dist. 5 in Orb. from Q Arg. I. 
 
 s. 
 
 ft 
 
 + 1 
 
 + 2 
 
 IS. 
 
 s. 
 
 6 
 
 7 
 
 8 
 
 S. 
 
 
 
 / // 
 
 / // 
 
 / / 
 
 
 
 
 
 
 
 4 24 
 
 7 3fr 
 
 30 
 
 1 
 
 9 
 
 4 32 
 
 7 42 
 
 29 
 
 3 
 
 18 
 
 4 40 
 
 7 46 
 
 28 
 
 S 
 
 28 
 
 4 48 
 
 7 51 
 
 27 
 
 4 
 
 37 
 
 4 55 
 
 7 55 
 
 26 
 
 5 
 
 46 
 
 3 3 
 
 7 59 
 
 25 
 
 6 
 
 55 
 
 5 11 
 
 8 3 
 
 24 
 
 7 
 
 I 4 
 
 5 18 
 
 8 6 
 
 23 
 
 8 
 
 I 14 
 
 5 25 
 
 8 10 
 
 22 
 
 9 
 
 1 23 
 
 5 32 
 
 8 13 
 
 21 
 
 10 
 
 1 32 
 
 5 40 
 
 8 16 
 
 20 
 
 11 
 
 1 41 
 
 5 47 
 
 8 20 
 
 19 
 
 12 
 
 I. 50 
 
 5 54 
 
 8 22 
 
 18 
 
 13 
 
 1 59 
 
 6 
 
 8 25 
 
 17 
 
 1 
 
 2 8 
 
 6 7 
 
 8 28 
 
 16 
 
 15 
 
 2 17 
 
 6 14 
 
 8 30 
 
 15 
 
 16 
 
 2 26 
 
 6 20 
 
 8 33 
 
 14 
 
 17 
 
 2 34 
 
 6 26 
 
 8 35 
 
 13 
 
 18 
 
 2 43 
 
 6 33 
 
 8 37 
 
 12 
 
 19 
 
 2 52 
 
 6 39 
 
 8 39 
 
 11 
 
 20 
 
 3 1 
 
 6 45 
 
 8 40 
 
 10 
 
 21 
 
 3 9 
 
 6 5] 
 
 8 42 
 
 9 
 
 22 
 
 3 18 
 
 6 56 
 
 8 43 
 
 8 
 
 23 
 
 3 26 
 
 7 2 
 
 8 44 
 
 7 
 
 24 
 
 3 35 
 
 7 7 
 
 8 45 
 
 6 
 
 25 
 
 3 43 
 
 7 13 
 
 8 46 
 
 5 
 
 26 
 
 3 52 
 
 7 18 
 
 8 47 
 
 4 
 
 27 
 
 4 
 
 7 23 
 
 ^ 48 
 
 3 
 
 28 
 
 4 8 
 
 - 28 
 
 8 48 
 
 2 
 
 29 
 
 4 16 
 
 7 33 
 
 8 48 
 
 1 
 
 30 
 
 4 24 
 
 7 38 
 
 8 48 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 S. 
 
 S. 
 
 + 5 
 
 + 4 
 
 f s 
 
 s. 
 
LUNAR TABLES. 
 
 381 
 
 III. For the Moon's Latitude. IV. For the Moon's Latitude. V. For the Moon's Latitude. 
 
 Argument. Argument. 
 
 Arg. I 3) 's Mean Anom. Arg. III. 3>'s Mean Anora. 
 
 S. 
 
 s. 
 
 
 + 6 
 
 1 
 
 -2 
 
 S. 
 s. 
 
 + ' 
 
 + 6 
 
 o 
 
 
 10 
 
 'f 
 
 
 
 3 
 
 ft 
 
 9 
 tl 
 
 If 
 
 15 
 
 16 
 
 17 
 18 
 
 o 
 
 JO 
 20 
 
 20 
 30 
 
 S. 
 
 6 
 9 
 
 13 
 
 15 
 
 '0 
 
 
 S. 
 
 + 11 
 
 5 
 
 + 10 
 
 + 9 
 
 s. 
 
 ~4 
 
 3 
 
 s. 
 
 s. 
 
 
 
 1 
 
 2 
 
 S. 
 
 s. 
 
 + 6 
 
 + 7 
 
 + 8 
 
 s. 
 
 o 
 
 so 
 
 20 
 
 o 
 
 
 10 
 
 a 
 
 
 4 
 
 ff 
 
 13 
 16 
 
 ff 
 
 22 
 24 
 
 20 
 ,30 
 
 9 
 13 
 
 + 11 
 
 19 
 
 22 
 
 25 
 25 
 
 + 9 
 
 10 
 
 
 8. 
 
 S. 
 
 S 
 
 + 10 
 
 s. 
 
 5 
 
 4 
 
 S 
 
 Argument* 
 Arg. IV. T) 's Mean Anom. 
 
 s. 
 s. 
 
 + 
 
 + 1 
 
 + 2 
 
 S. 
 S. 
 
 
 
 30 
 20 
 
 10 
 
 
 
 a. 
 
 6 
 
 7 
 
 8 
 
 o 
 
 
 10 
 
 20 
 30 
 
 S. 
 
 ff 
 
 
 
 3 
 
 tf 
 8 
 10 
 
 ff 
 
 14 
 15 
 
 5 
 8 
 
 12 
 14 
 
 16 
 16 
 
 11 
 
 10 
 
 _p 
 
 s. 
 
 + 5 
 
 + 4 
 
 + 3 
 
 s. 
 
 FOR THE MOON'S EQUATORIAL PARALLAX. 
 
 I. Argument V of Longitude, viz. 2 
 from O 5 's Mean Anomaly. 
 
 S. 
 
 
 
 1 
 
 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 
 
 n 
 
 n 
 
 ti 
 
 // 
 
 ff 
 
 ff 
 
 
 
 
 
 37 
 
 32 
 
 19 
 
 
 
 18 
 
 32 
 
 30 
 
 5 
 
 37 
 
 SO 
 
 16 
 
 S 
 
 21 
 
 34 
 
 25 
 
 10 
 
 36 
 
 28 
 
 IS 
 
 6 
 
 24 
 
 35 
 
 20 
 
 15 
 
 36 
 
 26 
 
 10 
 
 9 
 
 26 
 
 36 
 
 15 
 
 20 
 
 35 
 
 24 
 
 7 
 
 13 
 
 29 
 
 37 
 
 10 
 
 25 
 
 34 
 
 21 
 
 3 
 
 16 
 
 31 
 
 37 
 
 5 
 
 30 
 
 32 
 
 19 
 
 
 
 18 
 
 32 
 
 38 
 
 
 
 S. 1110 
 
 9 
 
 + 8 
 
 + 7 
 
 + c 
 
 S. 
 
 II. Argument XI of Longitude, viz. correct 
 Anomaly of 3 . 
 
 s 
 
 + o 
 
 + 1 
 
 + 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 . 
 
 
 
 
 5 
 10 
 15 
 
 20 
 
 25 
 30 
 
 
 
 
 
 
 
 
 
 30 
 25 
 20 
 15 
 10 
 
 5 
 
 
 54 13 
 54 14 
 54 15 
 54 18 
 5* 22 
 
 54 27 
 54 33 
 
 54 33 
 54 41 
 54 49 
 54 58 
 55 9 
 
 55 20 
 55 32 
 
 55 32 
 55 46 
 55 59 
 56 14 
 56 29 
 
 57 1 
 57 17 
 57 34 
 57 51 
 
 58 7 
 
 58 39 
 58 55 
 59 10 
 59 24 
 59 36 
 
 59 59 
 60 8 
 60 15 
 60 21 
 CO 25 
 
 56 45 
 
 .57 1 
 
 53 24 
 58 39 
 
 59 48 
 59 59 
 
 60 26 
 60 29 
 
 S. 
 
 + 11 
 
 + 10 
 
 + 9 
 
 + 8 
 
 + 7 
 
 + els. 
 
383 
 
 APPENDIX TO THE ASTRONOMY. 
 
 III. Argument XII. of Longitude, viz. ])' 
 equated Long. Q's true Long. 
 
 TABLE fur the reduction of lat. und 
 Hor. Par. for EUipticity. -^ 
 
 s 
 
 + 
 
 + 1 
 
 2 
 
 3 
 
 + 4 
 
 + 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 25 
 
 12 
 
 14 
 
 26 
 
 13 
 
 14 
 
 >0 
 
 5 
 
 25 
 
 8 
 
 17 
 
 25 
 
 8 
 
 18 
 
 '5 
 
 10 
 
 34 
 
 + 3 
 
 20 
 
 24 
 
 4 
 
 21 
 
 20 
 
 15 
 
 22 
 
 1 
 
 23 
 
 22 
 
 + 
 
 24 
 
 15 
 
 20 
 
 19 
 
 5 
 
 24 
 
 19 
 
 1 
 
 25 
 
 10 
 
 25 
 
 16 
 
 10 
 
 25 
 
 16 
 
 10 
 
 2 
 
 ; 
 
 30 
 
 12 
 
 14 
 
 26 
 
 13 
 
 14 
 
 2 
 
 o 
 
 S. 
 
 +11 
 
 +10 
 
 9 
 
 8 
 
 + 7 
 
 + 
 
 s. 
 
 TABLE for finding the Moon's Diameter. 
 Argument. Equatorial Parallax. 
 
 Equat. 1'ar. 
 
 ])'s Ui.tm. 
 
 quat. Par. 
 
 's Uiam. 
 
 i n 
 
 54 6 
 
 54 10 
 54 20 
 54 30 
 54 40 
 
 / // 
 
 29 26 
 29 31 
 
 29 37 
 29 42 
 29 48 
 
 / // 
 
 53 
 58 10 
 58 20 
 58 30 
 
 58 40 
 
 / // 
 
 31 37 
 31 42 
 31 47 
 31 53 
 31 58 
 
 54 50 
 55 
 55 10 
 55 20 
 55 30 
 
 29 53 
 29 58 
 30 4 
 30 9 
 30 15 
 
 58 50 
 59 
 59 10 
 59 20 
 59 30 
 
 32 4 
 32 9 
 32 15 
 32 20 
 32 26 
 
 55 40 
 55 50 
 56 
 56 10 
 56 20 
 
 30 20 
 30 26 
 SO 31 
 30 37 
 30 42 
 
 59 40 
 59 50 
 60 
 60 10 
 60 20 
 
 32 31 
 32 36 
 32 42 
 32 47 
 32 53 
 
 56 30 
 56 40 
 56 50 
 57 
 57 10 
 
 30 47 
 30 53 
 30 58 
 31 4 
 31 9 
 
 60 30 
 60 40 
 60 50 
 61 
 61 10 
 
 32 58 
 33 4 
 33 9 
 33 15 
 
 33 20 
 
 57 20 
 57 30 
 57 40 
 57 50 
 58 
 
 31 15 
 31 20 
 31 26 
 31 31 
 31 37 
 
 61 20 
 61 30 
 61 40 
 
 61 50 
 62 
 
 33 26 
 33 31 
 33 36 
 33 42 
 33 47 
 
 o 
 
 S 
 
 a 
 
 a s 
 
 (UT) 
 
 5 Z 
 
 a 2 
 - 
 
 II 
 
 II 
 
 15 
 
 23 
 
 , -a 
 
 II 
 
 I JJ 
 s -5 
 
 1.2 
 Jfto 
 
 1! 
 
 H 6 
 
 **M 
 
 O 
 
 
 deduct. 
 
 ied. 5 Hor.Par. 
 
 Lat. 
 
 )f Lat. 
 
 lorizontal Par. 
 
 
 
 53' 
 
 57' 
 
 61' 
 
 
 
 // 
 
 // 
 
 
 n 
 
 
 
 0. 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 2 
 
 047.9 
 
 0.0 
 
 0.0 
 
 0.0 
 
 4 
 
 1.35.5 
 
 0.1 
 
 0.1 
 
 1 
 
 6 
 
 2.22.7 
 
 0.1 
 
 0.1 
 
 0.1 
 
 8 
 
 5. 9.2 
 
 0.2 
 
 0.2 
 
 0.2 
 
 10 
 
 3.54.8 
 
 0-3 
 
 0.3 
 
 0.4 
 
 IT 
 
 4.39.3 
 
 ~a3 
 
 0.5 
 
 0.5 
 
 14 
 
 5.22.4 
 
 0.6 
 
 0.7 
 
 0.7 
 
 16 
 
 6. 3.9 
 
 0.8 
 
 0.9 
 
 O.S 
 
 18 
 
 6.43.7 
 
 1.0 
 
 1 1 
 
 1.2 
 
 20 
 
 7.21.5 
 
 1.2 
 
 1.5 
 
 1.4 
 
 22 
 
 7.57.2 
 
 "TT5 
 
 ~Mi 
 
 1.7 
 
 24 
 
 8.30.7 
 
 1.8 
 
 1. 
 
 2.0 
 
 26 
 
 9. 1.6 
 
 2.0 
 
 2.2 
 
 2.3 
 
 2b 
 
 9.29.9 
 
 2.3 
 
 2.5 
 
 2.7 
 
 30 
 
 955 4 
 
 2.7 
 
 2.9 
 
 3.1 
 
 32 
 
 10 18 1 
 
 o.U 
 
 3.: 
 
 ~374 
 
 34 
 
 10.37.8 
 
 3.3 
 
 3.(. 
 
 0.8 
 
 66 
 
 10.54.? 
 
 3.7 
 
 a.9 
 
 4.2 
 
 38" 
 
 11. 7.7 
 
 4.0 
 
 4.3 
 
 4.6 
 
 40 
 
 11 17.8 
 
 4.4 
 
 4.7 
 
 5.0 
 
 42 
 
 11.24.7 
 
 4.7 
 
 5.1 
 
 5.5 
 
 44 
 
 11.28.2 
 
 5.1 
 
 5.5 
 
 5.9 
 
 46 
 
 11.28.4 
 
 5.5 
 
 5.9 
 
 6.3 
 
 48 
 
 11.25 1 
 
 5.9 
 
 6.3 
 
 6.7 
 
 50 
 
 11.186 
 
 6.2 
 
 6,7 
 
 7.2 
 
 52 
 
 11. 8.8 
 
 ~67b 
 
 7.1 
 
 7.6 
 
 54 
 
 10.55.6 
 
 6.9 
 
 7.5 
 
 8.0 
 
 56 
 
 10.39.3 
 
 7.3 
 
 78 
 
 8.4 
 
 58 
 
 10.19.9 
 
 7.6 
 
 8.2 
 
 8.8 
 
 60 
 
 9.57.4 
 
 7.9 
 
 8.5 
 
 9.1 
 
 ba 
 
 9 32.0 
 
 8.3 
 
 8 9 
 
 9.5 
 
 64 
 
 9. 3.8 
 
 8.6 
 
 9.2 
 
 9.9 
 
 66 
 
 8.32.9 
 
 8.8 
 
 9.5 
 
 10.2 
 
 68 
 
 7.59.6 
 
 9.1 
 
 0.8 
 
 10.5 
 
 70 
 
 7.23.8 
 
 94 
 
 10.1 
 
 10.8 
 
 72 
 
 6.45 9 
 
 J.6 
 
 103 
 
 11.0 
 
 74 
 
 6 6.0 
 
 9.8 
 
 10.5 
 
 11.3 
 
 76 
 
 5.24.3 
 
 10.0 
 
 107 
 
 11.5 
 
 78 
 
 4.4 l.t 
 
 10.1 
 
 10.9 
 
 11.7 
 
 80 
 
 356.3 
 
 10.3 
 
 11.1 
 
 11.8 
 
 82 
 
 3.10.4 
 
 10.4 
 
 11.2 
 
 12.0 
 
 84 
 
 2.237 
 
 10.J 
 
 11.3 
 
 12.1 
 
 86 
 
 1.36.2 
 
 105 
 
 11.3 
 
 12.1 
 
 88 
 
 0.48.S 
 
 106 
 
 11.4 
 
 12.2 
 
 90 
 
 0. O.C 
 
 10.6 
 
 11.4 
 
 122 
 
LUNAR TABLED. 
 
 383 
 
 TABLES FOR FINDING THE MOON'S HOURLY MOTION IN LONGITUDE 
 
 AND LATITUDE. 
 
 I. For the Moon's Hourly Motion in Long. 
 
 Arg. V. of Long. viz. 2 J from O 3> 'a 
 Mean Anomaly. 
 
 s. 
 
 ( 
 
 1 
 
 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 D 
 
 n 
 
 n 
 
 II 
 
 // 
 
 f/ 
 
 ft 
 
 O 
 
 
 
 41 
 
 36 
 
 -}\ 
 
 
 
 21 
 
 37 
 
 30 
 
 s 
 
 41 
 
 34 
 
 18 
 
 3 
 
 24 
 
 38 
 
 25 
 
 10 
 
 41 
 
 32 
 
 15 
 
 7 
 
 27 
 
 40 
 
 ;o 
 
 1 5 
 
 40 
 
 30 
 
 11 
 
 10 
 
 30 
 
 41 
 
 i5 
 
 
 
 39 
 
 27 
 
 8 
 
 14 
 
 32 
 
 42 
 
 10 
 
 V '-, 
 
 38 
 
 24 
 
 4 
 
 17 
 
 35 
 
 42 
 
 5 
 
 30 
 
 36 
 
 21 
 
 
 
 21 
 
 37 
 
 43 
 
 
 
 S. 
 
 11 
 
 10 
 
 9 
 
 -f- 8 
 
 f " 
 
 f 6 
 
 S. 
 
 
 III. For the Moon's Hourly Motion in Longi 
 
 Arg. XII. of Long. viz. J's equat. 
 Long. Q's true Long. 
 
 S. 
 
 + o 
 
 1 
 
 2 
 
 3 
 
 =F 4 
 
 + 5 
 
 S. 
 
 
 
 
 
 
 
 
 
 
 
 40 
 
 19 
 
 21 
 
 40 
 
 20 
 
 21 
 
 30 
 
 5 
 
 39 
 
 13 
 
 26 
 
 39 
 
 13 
 
 27 
 
 25 
 
 10 
 
 37 
 
 + 6 
 
 31 
 
 37 
 
 7 
 
 32 
 
 20 
 
 15 
 
 34 
 
 1 
 
 35 
 
 34 
 
 + <> 
 
 36 
 
 15 
 
 20 
 
 30 
 
 8 
 
 38 
 
 30 
 
 7 
 
 39 
 
 10 
 
 25 
 
 25 
 
 15 
 
 3! 
 
 25 
 
 14 
 
 41 
 
 5 
 
 30 
 
 19 
 
 31 
 
 40 
 
 20 
 
 21 
 
 42 
 
 (.: 
 
 S. 
 
 + 11 
 
 10 
 
 9 
 
 8 
 
 =F 7 
 
 -I- & 
 
 S. 
 
 II. For the. Moon's Hourly Motion in Long. 
 
 Arg. XI. of Long. viz. the correct Anom- 
 aly of the J . 
 
 s. 
 
 + 
 
 + I 
 
 + 2 
 
 + 3 
 
 + 4 
 
 + 5 
 
 S. 
 
 
 
 
 
 
 
 
 
 
 
 29 35 
 
 29 57 
 
 31 2 
 
 32 42 
 
 34 36 
 
 36 10 
 
 30 
 
 5 
 
 29 35 
 
 30 5 
 
 31 17 
 
 33 
 
 34 54 
 
 36 21 
 
 25 
 
 10 
 
 29 37 
 
 30 15 
 
 31 32 
 
 33 19 
 
 35 11 
 
 36 30 
 
 20 
 
 15 
 
 29 -40 
 
 30 25 
 
 31 49 
 
 33 39 
 
 35 28 
 
 36 38 
 
 15 
 
 20 
 
 29 45 
 
 30 36 
 
 32 6 
 
 33 58 
 
 35 43 
 
 36 43 
 
 10 
 
 25 
 
 29 50 
 
 30 49 
 
 32 23 
 
 34 17 
 
 35 57 
 
 36 46 
 
 5 
 
 30 
 
 29 57 
 
 31 2 
 
 32 42 
 
 ?4 36 
 
 36 10 
 
 36 48 
 
 
 
 S. 
 
 + 11 
 
 + 10 
 
 + 9 
 
 + 8 
 
 + r 
 
 + 6 
 
 s. 
 
 I. For the Moon's Hourly II. For the Moon's Hourlij 
 Motion in Lat. Motion in Lat. 
 
 Argument I. of Lat. 
 
 Argument II. of Lat. 
 
 8. 
 
 + o 
 
 + 1 
 
 + 2 
 
 S. 
 
 S. 
 
 6 
 
 7 
 
 8 
 
 s. 
 
 
 
 
 
 
 
 
 2 58 
 
 2 34 
 
 1 29 
 
 30 
 
 5 
 
 2 57 
 
 2 26 
 
 1 15 
 
 25 
 
 10 
 
 2 55 
 
 2 16 
 
 1 1 
 
 20 
 
 15 
 
 2 52 
 
 2 6 
 
 46 
 
 15 
 
 20 
 
 2 47 
 
 1 54 
 
 31 
 
 10 
 
 25 
 
 2 41 
 
 1 4 
 
 15 
 
 5 
 
 30 
 
 2 34 
 
 1 29 
 
 
 
 
 
 S. 
 
 -f 11 
 
 + 10 
 
 + 9 
 
 s. 
 
 S. 
 
 5 
 
 4 
 
 3 
 
 s. 
 
 s. 
 
 + t 
 
 + 1 
 
 + 2 
 
 S. 
 
 s. 
 
 
 
 7 
 
 8 
 
 s. 
 
 o 
 
 ff 
 
 // 
 
 n 
 
 o 
 
 
 
 4 
 
 4 
 
 2 
 
 30 
 
 5 
 
 4 
 
 3 
 
 2 
 
 25 
 
 10 
 
 4 
 
 3 
 
 1 
 
 20 
 
 15 
 
 4 
 
 3 
 
 1 
 
 15 
 
 20 
 
 4 
 
 3 
 
 1 
 
 10 
 
 25 
 
 4 
 
 2 
 
 
 
 5 
 
 30 
 
 4 
 
 2 
 
 
 
 
 
 S. 
 
 + 11 
 
 + 10 
 
 + 9 
 
 S. 
 
 S. 
 
 5 
 
 4 
 
 3 
 
 s. 
 
384 
 
 APPENDIX TO THE ASTRONOMY. 
 
 Jingle of the vinibte Path of the Moon with the Ecliptic in 
 Eclipses. 
 
 Argument Long. J in her orbit Long. Q. 
 
 Tables of the Mean Motions of the Moon from tht 
 Sun. 
 
 
 Signs 6. 
 
 
 
 Horary Motion of the Moon from the Sun. 
 
 
 
 t 
 
 ; 
 
 
 / 
 
 ' 
 
 
 
 
 
 
 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 
 O 
 
 o / 
 
 O / 
 
 O / 
 
 ' 
 
 / 
 
 / 
 
 / 
 
 / 
 
 O / 
 
 O ' 
 
 O 
 
 
 
 5 47 
 
 5 4f. 
 
 5 45 
 
 5 44 
 
 5 43 
 
 5 42 
 
 5 41 
 
 5 41 
 
 5 40 
 
 5 39 
 
 30 
 
 e. 
 
 5 46 
 
 5 45 
 
 5 44 
 
 5 43 
 
 5 42 
 
 5 42 
 
 5 41 
 
 5 40 
 
 5 40 
 
 5 39 
 
 27 
 
 6 
 
 5 45 
 
 5 44 
 
 5 43 
 
 5 42 
 
 5 41 
 
 5 40 
 
 5 39 
 
 5 39 
 
 5 38 
 
 5 38 
 
 24 
 
 q 
 
 5 42 
 
 5 41 
 
 5 40 
 
 5 39 
 
 5 39 
 
 5 38 
 
 5 37 
 
 5 37 
 
 5 36 
 
 5 35 
 
 21 
 
 1" 
 
 5 39 
 
 5 38 
 
 5 37 
 
 5 36 
 
 5 35 
 
 5 35 
 
 5 34 
 
 5 33 
 
 5 33 
 
 5 32 
 
 18 
 
 15 
 
 5 35 
 
 5 34 
 
 5 33 
 
 5 32 
 
 5 31 
 
 5 31 
 
 5 30 
 
 5 29 
 
 5 29 
 
 5 28 
 
 16 
 
 
 11 Signs 5. 
 
 
 A.D. 
 
 3. 
 
 1761 
 
 9 20 47 17 
 
 1781 
 
 2 4 12 47 
 
 1791 
 
 10 4 49 47 
 
 B 1792 
 
 2 26 38 39 
 
 1793 
 
 7 6 16 4 
 
 1794 
 
 1 15 53 29 
 
 1795 
 
 3 25 30 54 
 
 B 1796 
 
 8 17 19 45 
 
 1797 
 
 26 57 10 
 
 1798 
 
 5 6 34 35 
 
 1799 
 
 9 16 12 
 
 1800 
 
 1 25 49 25 
 
 1801 
 
 6 5 26 49 
 
 1802 
 
 10 15 4 15 
 
 1803 
 
 2 24 41 39 
 
 B 1804 
 
 7 16 30 31 
 
 1805 
 
 11 26 7 56 
 
 1806 
 
 4 5 45 20 
 
 1807 
 
 8 15 22 45 
 
 B 1808 
 
 1 7 11 37 
 
 1809 
 
 5 16 49 3 
 
 1810 
 
 9 26 26 28 
 
 1811 
 
 2 6 3 54 
 
 B 1812 
 
 6 27 52 47 
 
 1813 
 
 11 7 30 12 
 
 1814 
 
 3 17 7 38 
 
 1815 
 
 7 26 45 3 
 
 B 1816 
 
 18 33 56 
 
 1817 
 
 4 28 11 22 
 
 1818 
 
 9 7 48 47 
 
 181< 
 
 ) 1 17 26 13 
 
 B 182( 
 
 ) 6 9 15 6 
 
 182' 
 
 I 10 18 52 32 
 
 Years complete. 
 
 
 CO'" 
 o 
 
 1 
 
 4 9 37 24 
 
 2 
 
 8 19 14 49 
 
 S 
 
 28 52 14 
 
 B 4 
 
 5 20 41 6 
 
 5 
 
 10 18 SO 
 
 6 
 
 2 9 55 54 
 
 7 
 
 6 19 33 19 
 
 B 8 
 
 11 11 22 10 
 
 9 
 
 3 20 59 36 
 
 10 
 
 8 37 
 
 11 
 
 10 14 24 
 
 B12 
 
 5 2 3 16 
 
 13 
 
 9 11 40 40 
 
 14 
 
 1 21 18 5 
 
 15 
 
 6 55 30 
 
 B 16 
 
 10 22 44 21 
 
 17 
 
 3 2 21 46 
 
 18 
 
 7 11 59 10 
 
 19 
 
 11 21 36 35 
 
 B20 
 
 4 13 25 26 
 
 Months. 
 
 
 S. ' 
 
 anuary 
 
 0000 
 
 February 
 
 17 54 48 
 
 March 
 
 11 29 15 16 
 
 April 
 May 
 
 17 10 3 
 22 53 23 
 
 June 
 
 1 10 48 11 
 
 July 
 
 1 16 31 22 
 
 August 
 
 2 4 26 20 
 
 Septem. 
 
 2 22 21 8 
 
 October 
 
 2 28 4 29 
 
 Novem. 
 
 3 15 59 16 
 
 Decem. 
 
 3 21 42 37 
 
 In months after February of bissextile years subtract 
 one day from the time found by the tables. 
 

 LUNAR TABLES. 
 
 385 
 
 TABLES of the Mean Motions of the Moon from 
 the Sun. 
 
 TABLET. Of Mean JVetc; Moons, %c. in March, from 
 1791 to 1821. 
 
 Days. 
 
 
 s. 
 
 1 
 
 12 11 27 
 
 2 
 
 24 22 53 
 
 3 
 
 1 6 34 20 
 
 4 
 
 1 18 45 47 
 
 5 
 
 2 57 13 
 
 6 
 
 2 13 8 40 
 
 7 
 
 2 25 20 7 
 
 & 
 
 3 7 31 33 
 
 9 
 
 3 19 43 
 
 10 
 
 4 1 54 27 
 
 11 
 
 4 14 5 53 
 
 12 
 
 4 26 17 20 
 
 13 
 
 5 8 28 47 
 
 14 
 
 5 20 40 13 
 
 15 
 
 6 2 15 40 
 
 16 
 
 6 15 3 7 
 
 17 
 
 6 27 14 33 
 
 18 
 
 7 9 26 
 
 19 
 
 7 21 37 27 
 
 20 
 
 8 3 48 53 
 
 21 
 
 8 16 20 
 
 22 
 
 8 28 11 47 
 
 23 
 
 9 10 23 13 
 
 24 
 
 9 22 34 40 
 
 25 
 
 10 4 46 7 
 
 26 
 
 10 16 57 33 
 
 27 
 
 10 29 9 
 
 28 
 
 U 11 20 27 
 
 29 
 
 11 23 31 53 
 
 30 
 
 5 43 20 1 
 
 31 
 
 17 54 47 
 
 Hours and Minutes. 
 
 i. 
 
 3 1 11 
 
 li. 
 
 O ' /> 
 
 n 
 
 n in tut 
 
 If 
 
 n 111 nil 
 
 1 
 
 30 29 
 
 31 
 
 15 44 48 
 
 2 
 
 1 57 
 
 32 
 
 16 15 17 
 
 3 
 
 1 31 26 
 
 33 
 
 16 45 45 
 
 4 
 
 2 1 54 
 
 34 
 
 17 16 13 
 
 5 
 
 2 32 23 
 
 35 
 
 17 46 42 
 
 6 
 
 3 2 52 
 
 .36 
 
 18 17 10 
 
 7 
 
 3 33 20 
 
 37 
 
 18 47 39 
 
 8 
 
 4 3 49 
 
 ?8 
 
 19 18 7 
 
 
 
 4 34 18 
 
 o9 
 
 19 48 36 
 
 10 
 
 5 4 46 
 
 40 
 
 20 19 5 
 
 11 
 
 5 35 15 
 
 41 
 
 20 49 33 
 
 12 
 
 6 5 43 
 
 .42 
 
 21 20 2 
 
 13 
 
 6 36 12 
 
 43 
 
 21 50 31 
 
 14 
 
 7 6 41 
 
 44 
 
 22 20 59 
 
 15 
 
 7 37 9 
 
 45 
 
 22 51 28 
 
 16 
 
 8 7 38 
 
 46 
 
 23 21 56 
 
 17 
 
 8 38 6 
 
 47 
 
 23 52 25 
 
 18 
 
 9 8 35 
 
 4! 
 
 24 22 54 
 
 19 
 
 9 39 4 
 
 49 
 
 24 53 22 
 
 20 
 
 10 9 32 
 
 50 
 
 25 23 51 
 
 21 
 
 10 40 1 
 
 !si 
 
 25 54 19 
 
 22 
 
 11 10 30 
 
 ;52 
 
 26 24 48 
 
 o< 
 
 11 40 59 
 
 :53 
 
 26 55 17 
 
 24 
 
 12 11 28 
 
 54 
 
 27 25 45 
 
 25 
 
 12 41 56 
 
 55 
 
 27 56 14 
 
 26 
 
 13 12 25 
 
 5C 
 
 28 26 43 
 
 27 
 
 13 42 54 
 
 57 
 
 28 57 12 
 
 28 
 
 14 13 22 
 
 58 
 
 29 27 40 
 
 g( 
 
 14 43 51 
 
 59 
 
 29 58 9 
 
 30 
 
 15 14 19 
 
 6C 
 
 30 28 37 
 
 A Lunation 
 Haifa Lunation 
 
 <! h. m. s. 
 29 12 44 3 
 14 18 22 2 
 
 
 
 
 
 Q's MeanlJist. 
 
 
 Mean Nc>v 3) 
 
 Sun's Mean 
 
 Moon's Mean 
 
 from the "]) 's 
 
 
 in March. 
 
 Anomaly. 
 
 Anomaly. 
 
 Node. 
 
 A.D. 
 
 <1. It. in. S. 
 
 S. ' " 
 
 S. ' " 
 
 / II 
 
 tt 
 
 i791 
 
 4 14 8 35 
 
 8 3 31 47 
 
 9 26 23 45 
 
 4 18 52 22 
 
 1702 
 
 22 11 41 15 
 
 8 21 53 59 
 
 9 2 52 
 
 5 27 35 24 
 
 1793 
 
 11 20 9 51 
 
 8 11 9 51 
 
 7 It 48 57 
 
 6 5 38 11 
 
 1794 
 
 30 18 2 32 
 
 8 29 32 3 
 
 6 17 26 4 
 
 7 14 21 IS 
 
 1795 
 
 20 2 51 8 
 
 8 18 47 55 
 
 4 27 14 9 
 
 7 22 24 
 
 1796 
 
 8 11 SO 44 
 
 8 8 3 47 
 
 3 7 2 14 
 
 8 26 47 
 
 1797 
 
 27 9 12 24 
 
 8 26 25 59 
 
 2 12 39 19 
 
 9 9 9 48 
 
 1798 
 
 16 18 1 1 
 
 8 15 41 51 
 
 22 27 25 
 
 9 17 12 35 
 
 1799 
 
 6 2 49 37 
 
 8 4 57 43 
 
 11 2 15 30 
 
 9 25 15 22 
 
 1800 
 
 25 22 17 
 
 8 23 19 55 
 
 10 7 52 36 
 
 11 3 58 23 
 
 1801 
 
 14 9 10 53 
 
 8 12 35 47 
 
 8 17 40 41 
 
 11 12 1 10 
 
 1 802 
 
 3 17 59 29 
 
 8 1 51 39 
 
 6 27 28 46 
 
 11 20 3 57 
 
 180S 
 
 22 15 32 9 
 
 8 20 13 51 
 
 6 3 3 52 
 
 28 46 58 
 
 1804 
 
 11 20 45 
 
 8 9 29 4S 
 
 4 12 53 57 
 
 1 6 49 45 
 
 1805 
 
 9 9 21 
 
 7 28 45 31 
 
 2 22 42 2 
 
 1 14 52 35 
 
 1806 
 
 19 6 42 1 
 
 8 17 7 43 
 
 I 28 19 8 
 
 2 23 35 36 
 
 1807 
 
 8 15 30 37 
 
 8 6 23 35 
 
 8 7 13 
 
 3 1 38 23 
 
 1808 
 
 26 13 3 17 
 
 8 24 45 47 
 
 11 13 44 19 
 
 4 10 21 24 
 
 1809 
 
 15 21 51 53 
 
 8 14 1 39 
 
 9 23 32 24 
 
 4 18 24 11 
 
 1810 
 
 5 6 40 29 
 
 8 3 17 31 
 
 8 3 20 29 
 
 4 26 26 58 
 
 1811 
 
 24 4 13 9 
 
 8 21 39 43 
 
 7 8 57 35 
 
 6 5 9 59 
 
 1812 
 
 12 13 1 45 
 
 8 10 55 35 
 
 5 18 45 40 
 
 6 13 12 46 
 
 1813 
 
 1 21 50 21 
 
 8 11 27 
 
 3 28 33 45 
 
 6 21 15 23 
 
 1814 
 
 20 19 23 1 
 
 8 18 33 3D 
 
 3 4 10 51 
 
 7 ,.9 58 24 
 
 1815 
 
 10 4 11 37 
 
 8 7 53 31 
 
 1 13 58 56 
 
 8 8 1 11 
 
 1816 
 
 28 1 44 17 
 
 8 26 15 43 
 
 19 36 2 
 
 9 16 44 12 
 
 1817 
 
 17 10 32 53 
 
 8 15 31 35 
 
 10 29 24 7 
 
 9 24 46 59 
 
 1818 
 
 6 19 21 29 
 
 8 4 47 27 
 
 9 9 12 12 
 
 10 2 49 46 
 
 1819 
 
 25 16 54 9 
 
 8 23 9 39 
 
 8 14 49 18 
 
 11 11 32 47 
 
 1820 
 
 14 1 42 45 
 
 8 12 25 31 
 
 6 24 37 23 
 
 11 19 35 34 
 
 1821 
 
 3 10 31 21 
 
 8 1 41 23 
 
 5 4 25 28 
 
 11 27 38 21 
 
380 
 
 APPENDIX TO THE ASTRONOMY. 
 
 ja 
 
 
 
 OC CTj C CN 
 CO CO rf rf rf 
 CO CO 00 CO to 
 
 GO rf IO tO t~ 
 
 rr rr rf -f Tf 
 
 CO CO CO CO CO 
 
 00 05 C i- G) 
 ^f TT 1O "O T) 
 
 CO CO O? fO CO 
 
 CO TP 1C tO I-- 
 
 IfJ T5 ITJ IO IO 
 
 CO CO CO CO CO 
 
 00 O O C! 
 >c. ir. to to to 
 CO CO CO CO CO 
 
 co rc m to 
 
 to to -0 to 
 CO CO CO CO 
 
 c 
 
 b- oo Oi s 
 
 r '.0 rf <0 -0 
 
 1-, 00 O> O 
 
 O! CO TT "O tO 
 
 I-- 30 Ci O 
 
 o-.' co rr .. ^: v- 
 
 - 
 
 CO CO V5 <O '.0 
 
 GO co co ^ : co 
 
 CO CO CO <0 CO 
 
 CO CO CO CO CO 
 
 co -r co co co 
 
 CO CO CO CO ro CO 
 
 o 
 
 1 
 
 y3 t^, ot Ci O 
 t^. t- I-- t~ 00 
 
 c?l O) Ol W Ol 
 
 i- CJ CO rf in 
 CO 00 CC 00 00 
 0! Ol 01 0) 01 
 
 CO > 00 Ci C 
 00 00 00 00 O) 
 W W <N G) Gl 
 
 ^- Ol CO * IO 
 
 c^ cr, oj c^ en 
 
 !O t-. JC. C. i; 
 
 c> o ir. 01 o 
 
 W O! r j' Ol 'O 
 
 - fM CO TJ U . '-^ 
 
 C C C C C 
 f. f.o CO co CO CO 
 
 s 
 
 _; i, v_ o C 
 
 O! co rr m 
 
 co t-- oo oi o 
 
 IO 1C C 'O O 
 
 *X CO rf ir. 
 
 r. >- v o> C 
 
 > 01 co rr m 
 
 
 Ol G-! C' CO (N 
 
 > O> Ol (M 
 
 C) O! G' G) 01 
 
 d O' O) G' O? 
 
 Ot OJ C' CM C-' 
 
 o) oo c> c; G> 
 
 1 
 
 in tO I-, CC O'j 
 
 01 0) C-J CN O! 
 
 00! M rf 
 
 <? Si 0! <*! 0! 
 'N d &I G) 
 
 m 3 t- oc CTJ 
 01 01 0! Gl 01 
 O! O! O! G$ O! 
 
 C C) C'. t 
 C' CO CO CO CO 
 
 .'. to t, oc Ci 
 
 CO CO CO CO CO 
 
 1C CN CO rf ir, 
 rf rf rr rf -r -f 
 
 1 
 
 in to t^ oo c? 
 
 00 00 CO 00 00 
 
 O - CM <o rf 
 Cl O5 Oi Ql CTJ 
 
 >o o t^ cc o 
 O) OD O3 w) O) 
 
 c; 01 co rr 
 
 C C C C 
 0' C! 01 O! C' 
 
 T. <i. t , JC J 
 
 O C O C 
 
 >' 01 r- 1 o> 01 
 
 C o) co TT 
 
 Ol O' O) C 1 O! 
 
 1 
 
 Tf in tO K 00 
 in in 'n in *r. 
 
 CR O C! CO 
 10 tO to <O <> 
 
 Tf "O CO >-, 00 
 
 to <o to o to 
 
 O O 1-1 O! CO 
 CO h~ I- I-. N. 
 
 rf 10 to K 00 
 
 1- 1. I. - (. 
 
 O) O T- O) CO rf 
 - OC OC 00 CO 00 
 
 _>> 
 
 3 
 
 jg 
 
 CO rf f> tO t- 
 
 Ol Ol Ol Ol Ol 
 
 00 Ol O W 
 OJ CO CO CO 
 
 CO rf m <O 1^ 
 CO CO CO CO CO 
 
 or ci o c! 
 co co rf rr rf 
 
 to rr in tr <- 
 
 rf rf rf rf t 
 
 oo cr. c 01 co 
 rf rf in in *? o 
 
 D 
 
 C 
 3 
 "- 
 
 CO rf in tO t^ 
 
 Ci CD Oi O5 Ci 
 
 00 CT> C - 1 (N 
 
 O> O) O O O 
 
 to t lo 'j: t^ 
 
 C C 
 
 co ;r> o 01 
 
 c c -< 
 
 CO rf >c to ^ 
 
 oo oi o 01 
 
 1 
 
 Ol CO rf m tO 
 O to tC tO (D 
 
 N. 00 Ol O 
 (O <> tO ^ b. 
 
 Ol CO rf in tO 
 
 t~ -- t~ t- t~ 
 
 b, co o o *- 
 
 K t~ t^ 00 00 
 
 O! CO rf 10 to 
 00 00 00 00 00 
 
 K. CO C7) C f " 
 JC 00 00 O- O> O4 
 
 |g 
 
 C- 
 
 01 co rr m to 
 co co co co co 
 
 K 00 O) O i 
 
 CO CO CO Tf TT 
 
 O 1 C?5 Tf V5 tO 
 rr Tf Tf Tf Tf 
 
 K oo O O 
 rf T}- rt 10 >n 
 
 O? CO rf 1C to 
 
 in in in n if. 
 
 >^ CC O5 C 
 in in m t. tr 
 
 t 
 
 01 co rf in 
 
 to t^ oo o-. o 
 
 1 O3 CO Tf 1O 
 
 to >-, oo o> o 
 
 C! co rf ir. 
 
 tc -. oc w c -< 
 
 " 
 
 
 
 
 
 
 ., - . ... 
 
 Hays. 
 
 . Gl CO rf m 
 
 3 I-- JO C2 C 
 
 01 CO Tf Ci 
 
 to t- oo o o 
 
 T- 01 
 
 - G-3 CO T "O 
 0! O! IN O> 01 
 
 O! O! CM O! CO CO 
 
 .1 
 
 8 
 
 '?u 
 
 *B 
 s 
 
 I 
 
 
 a I 
 
 12 
 
 J-S 
 
 _ rr on O! O C 
 t Ol rf in 
 
 rf 00 CO tO C 
 
 o? co in o? 
 
 CO rf 
 
 1 - 
 
 S 1 
 
 _ o o o o 
 
 . 0) 01 
 
 Ol Ol CO 
 
 o 
 
 u ' 
 
 rf M rr Ol 
 
 rf Ol rf 
 
 Ol rf 
 
 0! 
 
 "' -g 
 
 O - ON' CI CO 
 
 rf -rf in <0 to 
 
 b, 00 00 
 
 ir* 
 
 Ji 
 
 ^ 1-1 C3 CO f 5 
 
 CD K. 00 Ci O 
 
 _ o -- 
 
 
 
 
 O 1 r-l <N Ol 
 
 CO CO co rr "3- 
 
 in in to 
 
 o 
 
 9 
 
 * 
 
 
 
 CO 
 
 S ^* 
 
 .. Ci oo K to in 
 
 f co 01 o 
 
 en oo K 
 
 rr 
 
 rt 
 
 rf CO O! < 
 
 in rf co 01 
 
 in rf co 
 
 n 
 
 r. g 
 
 
 
 
 
 "c o 
 
 o <n b~ co Ci 
 
 rf o to Gl 00 
 
 CO Ci Wi 
 
 G^ 
 
 c c 
 
 Ol Ol 1-1 
 
 Ol Ol 
 
 * 
 
 " 
 
 1 
 
 . O ~+ Q* CO rf 
 
 in to to t^ oo 
 
 Ci O ^< 
 
 to 
 
 1 ^. 
 
 * r-. co in tn 
 
 co >n in T T 
 in "- co in * 
 
 CO Ol . . 
 CO in r* 
 
 O 
 
 J5 rt" 
 
 to 01 oo in f- 
 
 t^ rf O tO CO 
 
 o in 01 
 
 CO 
 
 ^ 
 
 " Ol CO 
 
 co rr in m 
 
 T Ol 
 
 CO 
 
 '= 5 
 
 o oo i^ to in 
 01 01 01 oj 01 
 
 rf CO OJ 1 
 O! Ol Ol Ol Ol 
 
 2 
 
 rf 
 
 ,= < 
 
 
 
 
 
 ' 
 
 ,J C * CO rf 
 
 in to t- oo Ci 
 
 O *H O 
 
 S 
 
 
 
 co to rj. 'N m 
 
 ^- 01 01 01 co 
 
 co O O 
 CO CO rf 
 
 01 
 
 5 
 
 
 
 
 
 3 
 
 c rf OO Ol to o 
 
 c rr (N in rr 
 
 rf 00 G4 to O 
 O3 in co 01 
 
 rf CO Ol 
 rf CO 
 
 Ol 
 0) 
 
 3 
 
 .a C) <-i rf ci in 
 
 rf ^ in 00 t~ 
 
 O CO - 
 
 CC 
 
 V 
 
 v- 
 
 Ci CT> oo oo -, 
 
 "3 0? "0 CC i rf 
 
 t--. O co to Ci 
 
 rf "I 4 in 
 
 0) in co 
 
 CO CO 1 
 
 2 
 
 
 j - co rr ,n 
 
 to i^ oo a-, o 
 
 Ol CO 
 
 -w 
 
LUNAK TABLES. 
 
 387 
 
 >> 
 
 1 
 
 S I 
 
 C 
 
 
 T 
 
 ! 
 
 Ci a; ; . to 10 "f 
 " 1 Of O Oi 
 
 CO CO i- O CTi 00 
 - . 01 0) 0) 
 
 i - <a 10 -T co CM 
 
 i- C O5 CO k-, CD 
 
 1O Tf CO O! O 
 
 
 1 
 
 IO 
 
 c.i . rr rr : i 
 s .-i in ot in m 10 c! 
 
 co co 'o GO of. o ci 
 
 CJT- tc; .^ IO -~ Tf 
 Tf Tf CO IO 
 
 - O T? CO 1-1 Ci 
 
 CO O CO CO 10 t^ 
 - Tf TJ- 
 
 00 SO 1O CO r-< O 
 
 CO OO 00 00 CO I- 
 Ci * O) Tf 
 
 00 O 10 CO C 
 
 O Tf CO C. O 
 
 C) Tf W "* 
 
 00 CD 10 CO i-< O 
 
 CD 
 
 
 o OCOOOOO 
 
 c o o c c o 
 
 o o o o o o 
 
 o ~ ~ o o o 
 
 o o o o c c 
 
 
 rf 
 
 rf Tf rf t>! .>- ef. 10 
 s i- O) en c'; rf rf 
 
 CO O) -^ O Cl OO *-, 
 - 0). 0) <N CM rn 
 
 oc i - TT , t-, co 
 
 rf Tf Tf TC '-0 CO 
 
 to 10 rr co o? i 
 
 <& t^ 00 00 O O* 
 
 0! 1-1 10 * CO 
 
 O Ol 00 t) 10 TJ- 
 
 OS 1-1 T- CO t, 10 
 - 1 ^f 0! 'T 
 
 CO 0) C CS X --D 
 10 10 10 
 
 co -1 t, o) m O 
 
 OJ CO 1 Tf rl 
 
 m Tf CN i- ci cc 
 10 10 10 10 rr n" 
 
 o c c c c o 
 
 - 
 
 co 
 
 i- of. c <- O co 
 
 10 10 Tf 
 
 . m 10 10 10 Tf TT Tf 
 co co co co co co co 
 
 co o) c.; co i- o 
 CO Of 10 CO 0! 
 
 -r Tf Tf co co co 
 
 CO CO CO CO CO CO 
 
 O 00 ,) CO IO 
 CO i-< IO O? 10 
 
 CO 0} O) r- C 
 CO CO CO CO CO CO 
 
 Tf 10 OJ C 
 
 ci o c? rr co 
 
 O Cl Ci 00 00 t, 
 CO CN O! O! 0< G! 
 
 O t^ IO O W re 
 10 C^ CO 10 
 
 so co 10 TT co tn 
 
 C~! W O! 0! G^ 
 
 00 
 
 el 
 
 0! O <-- Tf ti 
 CO OJ >0 Tf CJ. 
 
 r-l OJ CO CO Tf IO tO 
 
 - 01 of GJ CM 01. 01 o; 
 
 ~_i-i_- 
 
 00 00 tO CO t^ n 
 
 Tf 01 rf i 10 
 
 CO b- 00 00 Oi Ci 
 0) 01 Gl 01 ^0! 
 
 0! C O '-. OJ T 
 O 1O * i CO 
 
 O O O! C3 
 CO CO CO CO CO CO 
 
 t^ t- U_ Ci <J 00 
 IO CO . 
 
 <N CO CO CO Tf rf 
 CO CO CO CO CO CO 
 
 O O OC TT CC i 
 CO rf rf 10 10 
 
 rf Tf * Tf It 10 
 
 CO CO CO CO CO CO 
 
 en 
 
 F- 
 
 10 o Tf co a> o o 
 i TT _ co 10 r- Tf 
 
 to oo Ci o of. GO 10 
 
 "" Tf Tf Tf IO IO IO IO 
 
 * OO SO CO CTj CO 
 
 o co 10 T-I o) * 
 
 O 1 - 00 C i CS 
 10 if ; 10 
 
 ^ 00 00 t - Ci^ OI 
 10 r- (N CO TJ- 
 
 ci 10 o ^ oo a> 
 
 cr. -T oo i i o 
 
 TT IO IO 
 
 O G3 Tf 1O CO 
 
 o ~ <N i c cq 
 
 IO IO >O re TT CO 
 
 CD t^ 00 C-. O i- 
 
 
 
 o OOOCOCO 
 
 
 
 
 
 
 
 
 O h~ CO 01 00 CO O 
 5 CO ,-1 10 OI Tf 
 
 _ O 11 CO rf CO 00 Ci 
 
 C to co >; t- co 
 10 co 1-1 rr O! 
 
 CM Tf 1C t, & 
 
 Cj U". w 10 O G( 
 1O CO 1-1 ^ 1-1 IO 
 
 C 0! f >0 t^ 00 
 
 . t. o. 5i 'j; 
 o in c-! co 
 
 O CO 1O CD 00 
 
 Oi C. '-t rf Ci IO 
 Gl io O! 10 r-i rf 
 
 O C W CO 10 CO 
 
 ^ 
 
 (/ 
 
 o oooooc-o 
 
 ~ O O O O 
 
 o o o c o c 
 
 c c o c- 
 
 s c c- o 
 
 co 
 
 
 O .,) uO TT T. 
 
 1-- DO CJj O * Ot 
 
 CO ^P IO <X5 t^ 00 
 
 O"- w ^ G. CO -re 
 
 1-1 <yi yi c-' R? o? 
 
 T. 'O t^ GC C7* . 
 CI r>! i O' W 5 
 
 
 II 
 
 s. 
 
 * * 
 
 ** +5 
 "w 
 
 s I 
 
 s * 
 
 
 _ rj oo I-, CD 
 
 CO W - G-J Gt 
 
 "j TT co a< 
 l O! O-! r,i iyf 
 
 O Ci 00 1^ 1C 
 
 T, TT CO CM 1 
 
 C. Ci CO 1, CD 
 
 10 rf co G^ r-t O 
 
 
 o 
 
 .O "O IO O 
 TJ- 
 
 g - CO C CO <N 
 10 .0 
 
 ^ O? ^' G! n 
 
 rf u; as 
 
 10 q- n- 
 oo n" o io -H 
 
 Tf rr CO co CO 
 
 Oi in O5 
 
 CO t-i rt CO 
 
 U CO Cl rt O 
 0) G! ~ 
 
 IO <J3 CD IO GJ 
 i~ IO CO i-i 10 
 
 CO n -, CO 00 
 10 10 rf 
 
 n O O O 
 
 00 Gi CO O rf 
 G) CO -i rf 
 
 rf O 10 i- CO 
 rf rf CO CO G! 
 
 O O O O O 
 
 k- C CO CO Ci C 
 n 1O G! 10 
 
 CJ ^ CO 00 rf O 
 
 Cl i- 
 
 O C C C O O 
 
 to 
 
 -r 
 
 . C 0. <i, c O 
 aS =0 10 CO 
 
 C CI K '0 OJ 
 
 c CO CO CO CO CO 
 
 - 'O rf '- rf> rf 
 
 . c. _; c T- 
 
 CO CX CO 
 
 00 IO CO O K 
 G-l G-! O) G) n 
 
 CO ro rr5 r^ f^ 
 
 Ci Ci CD C C 
 TP IO ^- n 
 
 n- ci co co 
 
 CO CO CO f5 CO 
 
 b~ O Ci tC 00 
 rr CO 11 
 
 O t- CO O . 
 IO 10 io rf 
 
 CO C-! M G G 
 
 1- CO CO 10 G) 
 
 in co co 
 
 co O t- co o 
 rf Tf CO CO CO 
 
 GJ O) O> O) Gl 
 
 tc t ~ 10 o 'o 10 
 O( rf G) CO rf 
 
 CO Cl C: 10 i -, 
 
 C-' Gl i- 
 
 0) C- 1 G' G 1 OI OJ 
 
 K 
 
 CO 
 
 * 10 io 10 rt 'CO 
 
 g O O O O O 
 
 j; * t TJ- re rf 
 
 rr TT i. i- ; 
 
 <J) CO 1-1 CO 
 
 O C Oi O 00 
 
 n 1 < 
 
 If Tf Tf rf Tf 
 
 ^i <_ O t^ n 
 
 >r. T- c,; co rf 
 
 t, k^ CO '0 Tf 
 
 rf rf rf rf Tf 
 
 O O C} G< 
 rf CO C} 10 
 
 CO OJ i-l O CC 
 IO 
 
 rf -T t rf CO 
 
 i , a~> o c- Ci 
 
 G) -0 C! rf 
 
 (- 10 rf O! 
 10 in I.-. 10 10 
 
 CO rO CO CO 
 
 tO 00 T 10 O O 
 C! CO T rf rf CO 
 
 C) t 10 CO O 
 rf rf rf rf rf M 
 
 CO CO '0 CO CO CO 
 
 00 
 
 
 
 o O oo co o 
 
 i- 0} ctt 
 
 E" m K. O CO 
 CO CO CO Tf rf 
 
 _a co co co co co 
 
 ir. C5 1^ O ~. 
 
 C^ r- 10 G! 
 
 10 b a> o v 
 
 r? rf rf 1O IO 
 
 CO CO CO r.o CO 
 
 rr "- Gi 1 , Cl 
 
 CO Gi T 
 
 rf 10 K. cc C7l 
 in >o >o 10 >n 
 
 CO CO CO CO GO 
 
 , OC CO C,i 00 
 1-H O! W 1- 
 
 -< Gl CO rf 10 
 
 rf rf rf rf rf 
 
 C CO - _ .. 
 
 IO rf o 1O 
 
 CO CD K 00 CO 
 * rf -r T f 
 
 Oi 'O ^, CO u'. CO 
 
 oj 10 co rr 10 
 
 O Oi C C C O 
 rf rf rr -t rf -T 
 
 Oi 
 
 -4 
 
 O! Tj CD q- G! 
 w ^i U} CO n to 
 
 g CO CC O Tf ^ 
 
 js 'M 01 0? 0-1 O? 
 
 t, w. Ci i^ C( 
 O< G^ lO -j. 
 
 1O 00 r-H IO 
 OI G) cI CO CO 
 
 OJ <S C? tN GJ 
 
 rr co -C O O 
 rf n CO rf 
 
 oo tw m oo < 
 CO re rf Tt 10 
 
 G! <J1 G) 
 
 ct CO rf n 10 
 rf io 10 10 rf 
 
 rf K O CO CO 
 IO IO 
 
 O! GJ CO CO CO 
 
 c& rf C. C; 
 CO 7J 10 OO 
 
 Oi G) 10 K C 
 i G) 
 
 O co co co no 
 
 10 Cv. c". <_ u. o 
 CO CM rf 
 
 CO 10 00 C Gl 1O 
 
 CM Gl GJ CO CO CO 
 
 CO CO CO CO CO ro 
 
 c 
 
 ; 
 
 O 
 
 -/5 
 
 O 00 10 00 
 i CO 10 
 
 B" O rf 00 G? K 
 T^ 1-> 
 
 j= O O C O O 
 
 T Ci to CO 
 OI CO 'O n W 
 
 i- 10 oo n< oo 
 
 Crt G) CN CO CO 
 
 O O O O 
 
 Ci O! rf r- r. 
 CO IO i Gl 
 
 O! CO io Ci 
 rf rf 10 10 10 
 
 C O O C O 
 
 CO IO CO C <C 
 CO rf io 
 
 CO K CO O 
 
 r- - G^ 
 
 C CM G! t CD 
 
 rf 00 0! CO O 
 G! G! co co rr 
 
 i rT t t^ CD CN 
 10 Tf CO OI 1 
 
 Tf i-, - in Ci co 
 
 rf rf 10 -O 10 
 
 CC 
 
 
 CN CO rf 
 
 10 CD l^ 00 O) 
 
 . r- G) CO W 
 
 10 CO t- CC Oi 
 
 C n Gl CO rr 
 G) G) &l CJ G-) 
 
 .-. cu co w. c; 
 
 Gl O Gl Gl O> CO 
 
 
878 
 
 APPENDIX TO THE ASTRONOMY. 
 
 VI. The Second Equation of the Mean to the True 
 
 Argument. Moon's equated Anomaly. 
 
 VII. The third Equation 
 of the Mean to the 
 True Syzygy. 
 
 Arg. O' s Mean Anom. 
 Tl's Mean Anom. 
 
 VIII. Tlie.fourth Equation 
 of the .Mean to the 
 True Syzygy. 
 
 Q' s Mean 
 Dist. from j) ; s Q. 
 
 Argument. 
 
 
 b. o 
 
 1 
 
 ^ 
 
 3 
 
 4 
 
 
 
 
 
 
 h. in. s. 
 
 h. m. s. 
 
 ti. m. s. 
 
 IL. m. s. 
 
 h. m. a. 
 
 h. m. s. 
 
 O 
 
 
 
 000 
 
 5 12 48 
 
 8 47 8 
 
 9 46 44 
 
 8 8 59 
 
 4 34 33 
 
 30 
 
 1 
 
 10 58 
 
 5 21 56 
 
 8 51 45 
 
 9 45 3 
 
 8 3 1>. 
 
 4 26 1 
 
 29 
 
 2 
 
 21 56 
 
 5 30 57 
 
 8 56 10 
 
 9 45 12 
 
 7 57 23 
 
 4 17 25 
 
 28 
 
 S 
 
 32 54 
 
 5 39 5! 
 
 9 25 
 
 9 44 11 
 
 7 51 33 
 
 4 8 47 
 
 27 
 
 4 
 
 43 52 
 
 5 48 37 
 
 9 4 31 
 
 9 42 59 
 
 7 45 46 
 
 407 
 
 26 
 
 5 
 
 54 50 
 
 5 57 17 
 
 9 8 25 
 
 9 41 36 
 
 7 39 46 
 
 3 51 23 
 
 25 
 
 C 
 
 I 5 48 
 
 655! 
 
 9 12 9 
 
 9 40 S 
 
 7 33 36 
 
 3 42 32 
 
 24 
 
 7 
 
 1 16 46 
 
 6 14 19 
 
 9 15 43 
 
 9 38 19 
 
 7 27 22 
 
 3 33 38 
 
 23 
 
 8 
 
 1 27 44 
 
 6 22 41 
 
 9 19 5 
 
 9 36 24 
 
 7 21 2 
 
 3 24 42 
 
 22 
 
 9 
 
 1 38 40 
 
 6 30 57 
 
 9 22 14 
 
 9 34 18 
 
 7 14 30 
 
 3 15 44 
 
 21 
 
 10 
 
 1 49 33 
 
 6 39 4 
 
 9 25 12 
 
 9 32 1 
 
 7 7 50 
 
 3 6 45 
 
 20 
 
 11 
 
 2 23 
 
 6 47 
 
 9 27 54 
 
 9 29 3S 
 
 7 1 2 
 
 2 57 43 
 
 19 
 
 12 
 
 2 11 10 
 
 6 54 46 
 
 9 30 32 
 
 9 26 54 
 
 6 54 8 
 
 2 48 3D 
 
 18 
 
 13 
 
 2 21 54 
 
 7 2 24 
 
 9 32 58 
 
 9 24 4 
 
 6 47 9 
 
 2 39 34 
 
 17 
 
 14 
 
 2 32 34 
 
 7 9 52 
 
 9 35 12 
 
 9 21 S 
 
 6 40 6 
 
 2 30 28 
 
 16 
 
 15 
 
 2 43 9 
 
 7 17 9 
 
 9 37 14 
 
 9 17 51 
 
 6 32 56 
 
 2 21 19 
 
 15 
 
 16 
 
 2 53 38 
 
 7 24 19 
 
 9 39 8 
 
 9 14 28 
 
 6 25 40 
 
 2 12 8 
 
 14 
 
 17 
 
 343 
 
 7 31 18 
 
 9 40 51 
 
 9 10 54 
 
 6 18 18 
 
 2 2 53 
 
 13 
 
 18 
 
 3 14 24 
 
 7 38 9 
 
 9 42 21 
 
 979 
 
 6 10 49 
 
 1 53 36 
 
 12 
 
 19 
 
 3 24 42 
 
 7 44 51 
 
 9 43 42 
 
 9 3 IS 
 
 6 3 16 
 
 44 16 
 
 11 
 
 20 
 
 3 34 58 
 
 7 51 24 
 
 9 44 53 
 
 8 59 6 
 
 5 55 38 
 
 34 54 
 
 10 
 
 21 
 
 3 45 11 
 
 7 57 45 
 
 9 45 52 
 
 8 54 50 
 
 5 47 54 
 
 25 3.1 
 
 9 
 
 22 
 
 5 55 21 
 
 8 3 56 
 
 9 46 38 
 
 8 50 24 
 
 5 40 4 
 
 16 7 
 
 8 
 
 23 
 
 4 5 26 
 
 8 9 57 
 
 9 47 13 
 
 8 45 48 
 
 5 32 9 
 
 6 41 
 
 7 
 
 24 
 
 4 15 26 
 
 8 15 46 
 
 9 47 36 
 
 8 41 2 
 
 5 24 9 
 
 57 IS 
 
 6 
 
 25 
 
 4 25 20 
 
 8 21 24 
 
 9 47 49 
 
 8 36 6 
 
 5 16 5 
 
 ) 47 44 
 
 5 
 
 26 
 
 4 35 6 
 
 8 26 53 
 
 9 47 54 
 
 8 31 
 
 5 7 56 
 
 38 13 
 
 4 
 
 27 
 
 4 44 42 
 
 8 S3 11 
 
 9 47 46 
 
 S 25 44 
 
 4 59 42 
 
 28 41 
 
 S 
 
 28 
 
 4 54 11 
 
 8 37 19 
 
 9 47 S3 
 
 8 20 18 
 
 4 51 15 
 
 19 8 
 
 2 
 
 29 
 
 5 3 33 
 
 8 42 18 
 
 9 47 14 
 
 8 14 33 
 
 4 43 2 
 
 9 34 
 
 1 
 
 SO 
 
 5 12 48 
 
 8 47 8 
 
 9 46 44 
 
 3 8 59 
 
 4 S4 S3 
 
 
 
 C 
 
 
 S. 11 
 
 10 
 
 9 
 
 8 
 
 7 
 
 G 
 
 
 
 S. 
 
 1 
 
 2 
 
 
 
 S.4-6 
 
 + " 
 
 + 8 
 
 
 O 
 
 r 
 
 / // 
 
 / '/ 
 
 O 
 
 
 
 
 
 2 22 
 
 4 12 
 
 30 
 
 1 
 
 5 
 
 2 20 
 
 4 15 
 
 29 
 
 2 
 
 10 
 
 2 3(< 
 
 4 18 
 
 28 
 
 3 
 
 15 
 
 2 54 
 
 4 21 
 
 27 
 
 4 
 
 20 
 
 2 3b 
 
 4 24 
 
 20 
 
 it 
 
 25 
 
 2 42 
 
 4 27 
 
 25 
 
 6 
 
 30 
 
 2 46 
 
 4 SO 
 
 "2-i 
 
 7 
 
 35 
 
 2 50 
 
 4 32 
 
 25 
 
 8 
 
 40 
 
 2 54 
 
 4 34 
 
 i2 
 
 9 
 
 45 
 
 2 58 
 
 4 36 
 
 21 
 
 10 
 
 50 
 
 3 2 
 
 4 38 
 
 20 
 
 11 
 
 55 
 
 3 6 
 
 4 40 
 
 19 
 
 1 
 
 1 
 
 3 10 
 
 4 42 
 
 18 
 
 15 
 
 1 5 
 
 3 14 
 
 4 44 
 
 17 
 
 14 
 
 1 10 
 
 3 18 
 
 4 46 
 
 16 
 
 15 
 
 1 15 
 
 3 22 
 
 4 48 
 
 15 
 
 16 
 
 1 20 
 
 3 26 
 
 4 50 
 
 14 
 
 17 
 
 1 25 
 
 8 SO 
 
 4 51 
 
 13 
 
 18 
 
 1 SO 
 
 3 34 
 
 4 52 
 
 12 
 
 19 
 
 1 35 
 
 3 38 
 
 4 53 
 
 11 
 
 20 
 
 1 40 
 
 3 42 
 
 4 34 
 
 10 
 
 i\ 
 
 I 45 
 
 3 45 
 
 4 55 
 
 9 
 
 2-2 
 
 1 49 
 
 3 48 
 
 4 56 
 
 fc 
 
 23 
 
 1 52 
 
 3 51 
 
 4 57 
 
 4 
 
 24 
 
 1 56 
 
 3 54 
 
 4 57 
 
 e. 
 
 25 
 
 2 
 
 3 57 
 
 4 57 
 
 t 
 
 
 
 26 
 
 2 4 
 
 4 
 
 4 58 
 
 4 
 
 27 
 
 2 9 
 
 4 S 
 
 4 58 
 
 S 
 
 28 
 
 2 13 
 
 4 6 
 
 4 58 
 
 c 
 
 29 
 
 2 18 
 
 4 9 
 
 4 58 
 
 1 
 
 30 
 
 2 22 
 
 4 12 
 
 4 58 
 
 
 
 
 S 5 
 
 4 
 
 _SS. 
 
 
 
 S.-fll 
 
 + 10 
 
 + 9S. 
 
 
 
 + o 
 
 + 1 
 
 + 2 
 
 
 
 + 6 
 
 1 1 
 
 + 8 
 
 
 o 
 
 / // 
 
 f If 
 
 // 
 
 O 
 
 
 
 
 
 22 
 
 1 22 
 
 30 
 
 1 
 
 4 
 
 23 
 
 I 21 
 
 29 
 
 O 
 
 7 
 
 24 
 
 20 
 
 28 
 
 3 
 
 10 
 
 25 
 
 18 
 
 27 
 
 ^ 
 
 13 
 
 26 
 
 16 
 
 26 
 
 5 
 
 16 
 
 27 
 
 14 
 
 25 
 
 6 
 
 20 
 
 28 
 
 12 
 
 24 
 
 7 
 
 23 
 
 9 
 
 10 
 
 23 
 
 8 
 
 26 
 
 30 
 
 8 
 
 22 
 
 9 
 
 29 
 
 31 
 
 6 
 
 21 
 
 10 
 
 32 
 
 1 32 
 
 1 3 
 
 20 
 
 11 
 
 35 
 
 1 33 
 
 1 
 
 19 
 
 12 
 
 38 
 
 1 33 
 
 57 
 
 18 
 
 13 
 
 41 
 
 1 34 
 
 54 
 
 17 
 
 14 
 
 44 
 
 1 34 
 
 51 
 
 16 
 
 15 
 
 47 
 
 1 31 
 
 49 
 
 15 
 
 16 
 
 50 
 
 1 34 
 
 45 
 
 14 
 
 17 
 
 52 
 
 1 34 
 
 41 
 
 13 
 
 18 
 
 54 
 
 1 34 
 
 37 
 
 12 
 
 19 
 
 57 
 
 1 S3 
 
 34 
 
 11 
 
 20 
 
 1 
 
 1 33 
 
 31 
 
 10 
 
 21 
 
 1 2 
 
 1 32 
 
 28 
 
 9 
 
 22 
 
 1 5 
 
 1 31 
 
 25 
 
 8 
 
 23 
 
 1 8 
 
 1 30 
 
 22 
 
 7 
 
 24 
 
 1 10 
 
 1 29 
 
 19 
 
 6 
 
 25 
 
 1 12 
 
 1 28 
 
 16 
 
 f 
 
 26 
 
 1 14 
 
 1 27 
 
 13 
 
 4 
 
 27 
 
 1 16 
 
 1 26 
 
 10 
 
 o 
 
 28 
 
 I 18 
 
 1 25 
 
 G 
 
 2 
 
 29 
 
 1 20 
 
 1 24 
 
 3 
 
 1 
 
 30 
 
 1 22 
 
 1 22 
 
 
 
 
 
 
 S. 5 
 
 4 
 
 ss 
 
 
 
 S.-11 
 
 10 
 
 9S. 
 
 
LUNAR TABLES. 
 
 389 
 
 TABLE of Logistical Logarithms. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 
 
 
 60 
 
 120 
 
 180 
 
 240 
 
 300 
 
 360 
 
 420 
 
 
 
 
 60 
 
 120 
 
 180 
 
 240 
 
 300 
 
 360 
 
 420 
 
 
 
 00000 
 
 17781 
 
 14771 
 
 13010 
 
 11761 
 
 10792 
 
 1 0000 
 
 9331 
 
 30 
 
 20792 
 
 16021 
 
 13802 
 
 12341 
 
 11249 
 
 10378 
 
 J652 
 
 9031 
 
 1 
 
 35563 
 
 17710 
 
 14735 
 
 12986 
 
 11743 
 
 10777 
 
 9988 
 
 9320 
 
 31 
 
 20649 
 
 15973 
 
 13773 
 
 12320 
 
 11233 
 
 10365 
 
 9641 
 
 9021 
 
 2 
 
 32553 
 
 17 639 
 
 14699 
 
 1296i 
 
 11725 
 
 10763 
 
 9976 
 
 9310 
 
 32 
 
 20512 
 
 15925 
 
 1374 5 
 
 12300 
 
 11217 
 
 10352 
 
 9630 
 
 9012 
 
 3 
 
 30792 
 
 17570 
 
 14664 
 
 12939 
 
 11707 
 
 10749 
 
 9964 
 
 9300 
 
 33 
 
 20378 
 
 15878 
 
 13716 
 
 12279 
 
 11201 
 
 10339 
 
 9619 
 
 9002 
 
 4 
 
 29542 
 
 17501 
 
 14629 
 
 .2915 
 
 11689 
 
 10734 
 
 9952 
 
 9289 
 
 34 
 
 20248 
 
 15832 
 
 13688 
 
 12259 
 
 11186 
 
 10326 
 
 9608 
 
 8992 
 
 5 
 
 28573 
 
 17434 
 
 14594 
 
 12891 
 
 11671 
 
 10720 
 
 9040 
 
 9279 
 
 35 
 
 i0122 
 
 15786 
 
 13660 
 
 12239 
 
 11170 
 
 10313 
 
 9597 
 
 8983 
 
 6 
 
 27782 
 
 17368 
 
 14559 
 
 12868 
 
 11654 
 
 10706 
 
 9928 
 
 9209 
 
 36 
 
 20000 
 
 15740 
 
 13632 
 
 12218 
 
 11154 
 
 10300 
 
 9586 
 
 8973 
 
 7 
 
 27112 
 
 17302 
 
 14525 
 
 12845 
 
 11636 
 
 10692 
 
 9916 
 
 9259 
 
 37 
 
 19881 
 
 15695 
 
 13604 
 
 12198 
 
 11138 
 
 10287 
 
 9575 
 
 8964 
 
 8 
 
 26532 
 
 17238 
 
 14491 
 
 12821 
 
 11619 
 
 10678 
 
 9905 
 
 924 <- 
 
 38 
 
 19765 
 
 15651 
 
 13576 
 
 12178 
 
 11123 
 
 10274 
 
 9564 
 
 8954 
 
 9 
 
 26021 
 
 17175 
 
 14457 
 
 12798 
 
 11601 
 
 10663 
 
 9893 
 
 9238 
 
 39 
 
 19652 
 
 15607 
 
 13549 
 
 12159 
 
 11107 
 
 10261 
 
 9553 
 
 8945 
 
 10 
 
 25563 
 
 17112 
 
 14424 
 
 12775 
 
 11584 
 
 10649 
 
 9881 
 
 9228 
 
 40 
 
 19542 
 
 15563 
 
 13522 
 
 12139 
 
 11091 
 
 10248 
 
 9542 
 
 8935 
 
 11 
 
 25149 
 
 17050 
 
 14390 
 
 1275S 
 
 11566 
 
 10635 
 
 9869 
 
 9218 
 
 41 
 
 19435 
 
 15520 
 
 13495 
 
 12119 
 
 11076 
 
 10235 
 
 9532 
 
 8926 
 
 12 
 
 24771 
 
 16990 
 
 14357 
 
 12730 
 
 11549 
 
 10621 
 
 9858 
 
 9208 
 
 42 
 
 19331 
 
 15477 
 
 13468 
 
 12099 
 
 11061 
 
 10223 
 
 9521 
 
 8917 
 
 13 
 
 24424 
 
 16930 
 
 14325 
 
 12707 
 
 11532 
 
 10608 
 
 9846 
 
 9198 
 
 43 
 
 19228 
 
 15435 
 
 13441 
 
 12080 
 
 11045 
 
 10210 
 
 9510 
 
 8907 
 
 14 
 
 24102 
 
 16871 
 
 14292 
 
 12685 
 
 11515 
 
 10594 
 
 9834 
 
 9188 
 
 44 
 
 19128 
 
 15393 
 
 13415 
 
 12061 
 
 11030 
 
 10197 
 
 9499 
 
 8898 
 
 15 
 
 23802 
 
 16812 
 
 14260 
 
 12663 
 
 11498 
 
 10580 
 
 9823 
 
 9178 
 
 45 
 
 19031 
 
 15351 
 
 13388 
 
 12041 
 
 11015 
 
 10185 
 
 9488 
 
 8888 
 
 16 
 
 2S52S 
 
 16755 
 
 14228 
 
 12640 
 
 11481 
 
 10566 
 
 9811 
 
 9168 
 
 46 
 
 18935 
 
 15310 
 
 13362 
 
 12022 
 
 10999 
 
 10172 
 
 9478 
 
 8879 
 
 17 
 
 23259 
 
 16698 
 
 14196 
 
 12618 
 
 11464 
 
 10552 
 
 9800 
 
 9158 
 
 47 
 
 18842 
 
 15269 
 
 13336 
 
 12003 
 
 10984 
 
 10160 
 
 9467 
 
 8870 
 
 18 
 
 23010 
 
 16642 
 
 1416.- 
 
 12596 
 
 11447 
 
 10539 
 
 9788 
 
 9148 
 
 48 
 
 18751 
 
 15229 
 
 13310 
 
 11984 
 
 10969 
 
 10147 
 
 9456 
 
 8864 
 
 19 
 
 22775 
 
 16587 
 
 14133 
 
 12574 
 
 11430 
 
 10525 
 
 9777 
 
 9138 
 
 49 
 
 18661 
 
 15189 
 
 13284 
 
 11965 
 
 10954 
 
 10135 
 
 9446 
 
 8851 
 
 20 
 
 22553 
 
 16532 
 
 14102 
 
 12553 
 
 11413 
 
 !0512 
 
 9765 
 
 9128 
 
 50 
 
 18573 
 
 15149 
 
 13259 
 
 11946 
 
 10939 
 
 10122 
 
 9435 
 
 8842 
 
 21 
 
 22341 
 
 16478 
 
 14071 
 
 .2531 
 
 11397 
 
 10498 
 
 9754 
 
 9119 
 
 51 
 
 18487 
 
 15110 
 
 16233 
 
 11927 
 
 10924 
 
 10110 
 
 9425 
 
 8833 
 
 22 
 
 22139 
 
 16425 
 
 14040 
 
 12510 
 
 11380 
 
 10484 
 
 9742 
 
 9109 
 
 52 
 
 18403 
 
 15071 
 
 13208 
 
 11908 
 
 10909 
 
 10098 
 
 9414 
 
 8824 
 
 23 
 
 21946 
 
 16372 
 
 !4010 
 
 12488 
 
 11363 
 
 10471 
 
 9731 
 
 9099 
 
 53 
 
 18320 
 
 15032 
 
 13183 
 
 11889 
 
 10894 
 
 10085 
 
 9404 
 
 8814 
 
 24 
 
 21761 
 
 16320 
 
 1 397" 
 
 12467 
 
 11347 
 
 10458 
 
 9720 
 
 9089 
 
 54 
 
 18239 
 
 14994 
 
 13158 
 
 11871 
 
 10880 
 
 10073 
 
 9393 
 
 8805 
 
 25 
 
 21584 
 
 16-260 
 
 13949 
 
 12445 
 
 11331 
 
 10444 
 
 970 8 
 
 9079 
 
 55 
 
 18159 
 
 14956 
 
 13133 
 
 H852 
 
 10865 
 
 10061 
 
 9383 
 
 8796 
 
 26 
 
 '21413 
 
 16218 
 
 13919 
 
 12424 
 
 11314 
 
 10431 
 
 9697 
 
 9070 
 
 56 
 
 18081 
 
 14918 
 
 13108 
 
 11834 
 
 10850 
 
 10049 
 
 9372 
 
 8787 
 
 27 
 
 21249 
 
 16168 
 
 13890 
 
 12403 
 
 11298 
 
 10418 
 
 9686 
 
 9060 
 
 57 
 
 18004 
 
 14881 
 
 IS083 
 
 11816 
 
 10835 
 
 10036 
 
 9362 
 
 8778 
 
 28 
 
 21091 
 
 16118 
 
 13860 
 
 12382 
 
 1128, 
 
 10404 
 
 9675 
 
 9050 
 
 58 
 
 17929 
 
 14844 
 
 13059 
 
 11797 
 
 10821 
 
 10024 
 
 9351 
 
 8769 
 
 29 
 
 20939 
 
 16069 
 
 13831 
 
 12362 
 
 11266 
 
 10391 
 
 9664 
 
 904' 
 
 59 
 
 17855 
 
 14808 
 
 13034 
 
 11779 
 
 10806 
 
 10012 
 
 934 1 
 
 8760 
 
 30 
 
 20792 
 
 16021 
 
 13802 
 
 12341 
 
 11249 
 
 10378 
 
 9652 
 
 9031 
 
 60 
 
 17782 
 
 14771 
 
 13010 
 
 11761 
 
 10792 
 
 10000 
 
 9331 
 
 8752 
 
 53* 
 
300 
 
 APPENDIX TO THE ASTRONOMY. 
 
 TABLE of Logistical Logarithms. 
 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 
 8 
 
 y 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 
 480 
 
 540 
 
 GOO 
 
 060 
 
 720 
 
 780 
 
 840 
 
 900 
 
 960 
 
 
 480 
 
 540 
 
 600 
 
 660 
 
 720 
 
 780 
 
 84 C 
 
 900 
 
 960 
 
 
 
 8751 
 
 8259 
 
 7782 
 
 7368 
 
 6990 
 
 664a 
 
 6320 
 
 6021 
 
 5740 
 
 30 
 
 8487 
 
 8004 
 
 7570 
 
 7i7. r . 
 
 6812 
 
 6478 
 
 6168 
 
 5r'78 
 
 5607 
 
 1 
 
 4742 
 
 8231 
 
 T 774 
 
 7361 
 
 6984 
 
 6637 
 
 6315 
 
 6016 
 
 5736 
 
 31 
 
 8479 
 
 7997 
 
 7563 
 
 7168 
 
 6807 
 
 6473 
 
 6163 
 
 5871 
 
 6602 
 
 o 
 
 8733 
 
 8223 
 
 -707 
 
 7354 
 
 6978 
 
 6631 
 
 6310 
 
 6011 
 
 5731 
 
 32 
 
 8470 
 
 989 
 
 7556 
 
 7162 
 
 6801 
 
 6467 
 
 6158 
 
 5869 
 
 5598 
 
 s 
 
 8724 
 
 8215 
 
 "GO 
 
 7348 
 
 6972 
 
 6625 
 
 6305 
 
 6006 
 
 5727 
 
 33 
 
 8462 
 
 7981 
 
 7549 
 
 7156 
 
 6795 
 
 6462 
 
 6153 
 
 5864 
 
 5594 
 
 4 
 
 8715 
 
 8207 
 
 "753 
 
 7S41 
 
 0966 
 
 6620 
 
 6300 
 
 6001 
 
 5722 
 
 34 
 
 8453 
 
 7974 
 
 7542 
 
 7149 
 
 6789 
 
 0457 
 
 6143 
 
 5860 
 
 5589 
 
 5 
 
 8706 
 
 8199 
 
 -74.1 
 
 7335 
 
 6900 
 
 G6L4 
 
 0-294 
 
 6997 
 
 5718 
 
 35 
 
 8445 
 
 7966 
 
 7535 
 
 7143 
 
 6784 
 
 6451 
 
 6143 
 
 5855 
 
 5585 
 
 6 
 
 3697 
 
 8191 
 
 "738 
 
 7328 
 
 6954 
 
 6609 
 
 6289 
 
 5992 
 
 5713 
 
 36 
 
 8437 
 
 7959 
 
 7528 
 
 7137 
 
 6778 
 
 0-140 
 
 6138 
 
 58SO 
 
 5580 
 
 7 
 
 8688 
 
 8183 
 
 ^731 
 
 7322 
 
 6948 
 
 6603 
 
 6284 
 
 5987 
 
 5709 
 
 37 
 
 8428 
 
 7951 
 
 752S 
 
 7131 
 
 6772 
 
 6441 
 
 6133 
 
 5846 
 
 5576 
 
 8 
 
 8679 
 
 8175 
 
 7724 
 
 7315 
 
 6942 
 
 6598 
 
 6279 
 
 5982 
 
 5704 
 
 38 
 
 8420 
 
 7944 
 
 7515 
 
 7124 
 
 6766 
 
 6435 
 
 6128 
 
 5841 
 
 5572 
 
 9 
 
 8670 
 
 8167 
 
 7717 
 
 7309 
 
 6936 
 
 6592 
 
 6274 
 
 5977 
 
 5700 
 
 39 
 
 8411 
 
 7936 
 
 7508 
 
 7118 
 
 6761 
 
 6430 
 
 6123 
 
 5836 
 
 5567 
 
 10 
 
 8661 
 
 8159 
 
 7710 
 
 7302 
 
 6930 
 
 6587 
 
 6269 
 
 5973 
 
 5695 
 
 40 
 
 8403 
 
 7929 
 
 7501 
 
 7112 
 
 6755 
 
 6425 
 
 6118 
 
 5832 
 
 5563 
 
 11 
 
 8652 
 
 8152 
 
 7703 
 
 7296 
 
 6924 
 
 6581 
 
 6264 
 
 "968 
 
 5691 
 
 41 
 
 8395 
 
 7921 
 
 7494 
 
 7(06 
 
 6749 
 
 6420 
 
 6113 
 
 5827 
 
 5559 
 
 12 
 
 H64S 
 
 8144 
 
 7696 
 
 7289 
 
 6918 
 
 6576 
 
 6259 
 
 5963 
 
 5686 
 
 42 
 
 8386 
 
 7914 
 
 74 8P 
 
 7100 
 
 6743 
 
 6414 
 
 6108 
 
 5823 
 
 5554 
 
 f 
 
 8635 
 
 8136 
 
 7688 
 
 7283 
 
 6912 
 
 6570 
 
 6254 
 
 5958 
 
 5682 
 
 43 
 
 8378 
 
 7906 
 
 7481 
 
 7093 
 
 6738 
 
 6409 
 
 6103 
 
 5818 
 
 5550 
 
 14 
 
 8626 
 
 8128 
 
 7681 
 
 7276 
 
 6906 
 
 6565 
 
 6248 
 
 5954 
 
 5677 
 
 44 
 
 8370 
 
 7899 
 
 7474 
 
 7087 
 
 6732 
 
 6404 
 
 6099 
 
 5813 
 
 5546 
 
 ' t 
 
 8617 
 
 8120 
 
 7674 
 
 "270 
 
 6900 
 
 6559 
 
 6243 
 
 5949 
 
 5673 
 
 45 
 
 8SG1 
 
 7891 
 
 7467 
 
 7081 
 
 6726 
 
 6398 
 
 6094 
 
 5809 
 
 5541 
 
 16 
 
 8608 
 
 8112 
 
 7667 
 
 7264 
 
 6894 
 
 6554 
 
 6238 
 
 5944 
 
 5669 
 
 46 
 
 8353 
 
 7884 
 
 7461 
 
 7075 
 
 6721 
 
 6393 
 
 6089 
 
 5804 
 
 5537 
 
 17 
 
 8599 
 
 8104 
 
 7660 
 
 7257 
 
 6888 
 
 6548 
 
 6233 
 
 5939 
 
 5664 
 
 47 
 
 8345 
 
 7877 
 
 7454 
 
 706'.' 
 
 6715 
 
 6388 
 
 6084 
 
 5800 
 
 5533 
 
 18 
 
 8591 
 
 8097 
 
 7653 
 
 7251 
 
 6882 
 
 6543 
 
 6228 
 
 5935 
 
 5660 
 
 48 
 
 8337 
 
 7869 
 
 7447 
 
 7063 
 
 6706 
 
 6383 
 
 6079 
 
 5795 
 
 5528 
 
 19 
 
 8582 
 
 8089 
 
 7646 
 
 7244 
 
 6877 
 
 6538 
 
 6223 
 
 5930 
 
 5655 
 
 49 
 
 8328 
 
 7862 
 
 7441 
 
 7057 
 
 6704 
 
 6377 
 
 6074 
 
 5790 
 
 5524 
 
 20 
 
 8575 
 
 8081 
 
 7639 
 
 7238 
 
 6871 
 
 6532 
 
 6218 
 
 5925 
 
 5651 
 
 50 
 
 8320 
 
 7855 
 
 7434 
 
 7050 
 
 6698 
 
 6372 
 
 6069 
 
 5786 
 
 5520 
 
 21 
 
 8565 
 
 8073 
 
 7632 
 
 7232 
 
 6865 
 
 6527 
 
 6213 
 
 5920 
 
 5640 
 
 51 
 
 8312 
 
 7847 
 
 7427 
 
 7044 
 
 6692 
 
 6367 
 
 6064 
 
 5781 
 
 5516 
 
 22 
 
 8556 
 
 8066 
 
 7625 
 
 7225 
 
 6859 
 
 6521 
 
 6208 
 
 5916 
 
 5642 
 
 52 
 
 8304 
 
 7840 
 
 7421 
 
 7038 
 
 6687 
 
 636-2 
 
 605^' 
 
 -777 
 
 5511 
 
 23 
 
 8547 
 
 8058 
 
 7618 
 
 7219 
 
 6853 
 
 6516 
 
 6203 
 
 5911 
 
 5637 
 
 53 
 
 8296 
 
 78S2 
 
 7414 
 
 703? 
 
 6681 
 
 6357 
 
 6055 
 
 5772 
 
 5507 
 
 24 
 
 8539 
 
 8050 
 
 7611 
 
 7212 
 
 6847 
 
 6510 
 
 6198 
 
 5906 
 
 5633 
 
 54 
 
 8288 
 
 7825 
 
 7407 
 
 7026 
 
 6676 
 
 6351 
 
 6050 
 
 5768 
 
 5503 
 
 25 
 
 8530 
 
 8043 
 
 7604 
 
 7206 
 
 6841 
 
 6505 
 
 6193 
 
 5902 
 
 5629 
 
 55 
 
 8279 
 
 7818 
 
 7401 
 
 7020 
 
 6670 
 
 6346 
 
 6045 
 
 5763 
 
 5498 
 
 26 
 
 8522 
 
 8035 
 
 7597 
 
 7200 
 
 6836 
 
 6500 
 
 6188 
 
 5897 
 
 5624 
 
 56 
 
 8271 
 
 7811 
 
 7394 
 
 7014 
 
 6664 
 
 6341 
 
 6040 
 
 5758 
 
 5494 
 
 27 
 
 851* 
 
 8027 
 
 -590 
 
 7193 
 
 6830 
 
 6494 
 
 6183 
 
 5892 
 
 5620 
 
 57 
 
 8263 
 
 7803 
 
 7387 
 
 7008 
 
 6659 
 
 6336 
 
 6035 
 
 5754 
 
 5490 
 
 28 
 
 8504 
 
 S020 
 
 7583 
 
 7187 
 
 6824 
 
 6489 
 
 6178 
 
 5888 
 
 5615 
 
 58 
 
 8255 
 
 7796 
 
 7381 
 
 7002 
 
 0655 
 
 6331 
 
 6030 
 
 5749 
 
 5486 
 
 2S 
 
 8496 
 
 8012 
 
 7577 
 
 7181 
 
 68U 
 
 6484 
 
 6173 
 
 5883 
 
 5611 
 
 59 
 
 8247 
 
 7789 
 
 7374 
 
 6996 
 
 6648 
 
 6325 
 
 6025 
 
 574 r 
 
 5481 
 
 3C 
 
 8487 
 
 8004 
 
 757d 
 
 7175 
 
 6815 
 
 6478 
 
 6168 
 
 5878 
 
 5607 
 
 60 
 
 82397782 
 
 7368 
 
 6990 
 
 6642 
 
 6320 
 
 6021 
 
 5740 
 
 5477 
 

 LUNAR TABLES. 
 
 391 
 
 T ABLK of Logistical Logarithms. 
 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 II 
 
 
 18 
 
 19 
 
 20 I 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 
 1020 
 
 1080 
 
 1140 
 
 1200 
 
 1260 
 
 1320 
 
 1380 
 
 1440 
 
 1500 
 
 1020 
 
 1080 
 
 1140 
 
 1200 
 
 1260 
 
 1320 
 
 1380 
 
 1440 
 
 1500 
 
 
 
 5477 
 
 5229 
 
 4994 
 
 4771 
 
 4559435; 
 
 4164 
 
 3979 
 
 3802 
 
 30 
 
 5351 
 
 5110 
 
 4881 
 
 4664 
 
 4457 
 
 4260 
 
 4071 
 
 3890 
 
 3716 
 
 1 
 
 5473 
 
 5225 
 
 1990 
 
 4768 
 
 455614354 
 
 4161 
 
 3976 
 
 3799 
 
 31 
 
 5347 
 
 5106 
 
 4877 
 
 46GO 
 
 4454 
 
 4256 
 
 4068 
 
 3887 
 
 3713 
 
 2 
 
 5469 
 
 5221 
 
 4986 
 
 4764 
 
 45524351 
 
 4158 
 
 3973 
 
 3796 
 
 02 
 
 5343 
 
 5102 
 
 4874 
 
 4657 
 
 4450 
 
 4255 
 
 4065 
 
 3884 
 
 3710 
 
 3 
 
 5464 
 
 5217 
 
 4983 
 
 4760 
 
 45494347 
 
 4155 
 
 S970 
 
 3793 
 
 33 
 
 53S9 
 
 5098 
 
 4870 
 
 4653 
 
 4447 
 
 4250 
 
 4062 
 
 3881 
 
 3708 
 
 4 
 
 5460 
 
 5213 
 
 4979 
 
 4757 
 
 4546 
 
 4344 
 
 4152 
 
 3967 
 
 3791 
 
 34 
 
 5335 
 
 5094 
 
 4866 
 
 4650 
 
 4444 
 
 424 P 
 
 4059 
 
 3878 
 
 3705 
 
 5 
 
 5456 
 
 5209 
 
 4975 
 
 4753 
 
 4542 
 
 4341 
 
 4149 
 
 3964 
 
 3788 
 
 35 
 
 5331 
 
 5090 
 
 4863 
 
 4646 
 
 4440 
 
 4244 
 
 4055 
 
 3875 
 
 3702; 
 
 6 
 
 5452 
 
 5205 
 
 4971 
 
 4750 
 
 4539 
 
 4338 
 
 4145 
 
 3961 
 
 3785 
 
 36 
 
 5326 
 
 5086 
 
 4859 
 
 4G4S 
 
 4437 
 
 4240 
 
 4052 
 
 3872 
 
 3699 
 
 7 
 
 5447 
 
 5201 
 
 4967 
 
 4746 
 
 4535 
 
 4354 
 
 4142 
 
 3958 
 
 3782 
 
 37 
 
 5322 
 
 5082 
 
 4855 
 
 4639 
 
 4434 
 
 4237 
 
 4049 
 
 3869 
 
 3696 
 
 8 
 
 5443 
 
 5197 
 
 4964 
 
 4742 
 
 4532 
 
 4331 
 
 4139 
 
 3955 
 
 3779 
 
 38 
 
 5318 
 
 5079 
 
 4852 
 
 4636 
 
 4430 
 
 4234 
 
 4046 
 
 3866 
 
 3693 
 
 9 
 
 5439 
 
 5193 
 
 4960 
 
 4739 
 
 4528 
 
 4328 
 
 4136 
 
 3952 
 
 3776 
 
 39 
 
 *314 
 
 5075 
 
 4848 
 
 4632 
 
 4427 
 
 4231 
 
 4043 3863 
 
 3691 
 
 10 
 
 5435 
 
 5189 
 
 4956 
 
 4735 
 
 4525 
 
 4325 
 
 4133 
 
 3949 
 
 3773 
 
 40 
 
 5310 
 
 5071 
 
 4844 
 
 4629 
 
 4424 
 
 4228 
 
 4040 3860 
 
 3688 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11 
 
 5430 
 
 5185 
 
 4952 
 
 4732 
 
 4522 
 
 4321 
 
 4130 
 
 3946 
 
 3770 
 
 41 
 
 5306 
 
 5067 
 
 4841 
 
 4625 
 
 4420 
 
 4224 
 
 4037 3857 
 
 3685 
 
 12 
 
 5426 
 
 5181 
 
 4949 
 
 4728 
 
 4518 
 
 4318 
 
 4127 
 
 3943 
 
 3768 
 
 42 
 
 5302 
 
 5063 
 
 4837 
 
 4622 
 
 4417 
 
 4221 
 
 4034 
 
 3855 
 
 3682 
 
 13 
 
 5422 
 
 5177 
 
 4945 
 
 4724 
 
 4515 
 
 4315 
 
 4124 
 
 3940 
 
 3765 
 
 43 
 
 5298 
 
 5059 
 
 4833 
 
 4618 
 
 4414 
 
 4218 
 
 4031 
 
 3852 
 
 3679 
 
 14 
 
 5418 
 
 5173 
 
 4941 
 
 4721 
 
 4511 
 
 4311 
 
 4120 
 
 3937 
 
 3762 
 
 44 
 
 5294 
 
 5055 
 
 4830 
 
 4615 
 
 4410 
 
 4215 
 
 4028 
 
 3849 
 
 3677 
 
 15 
 
 5414 
 
 5169 
 
 4937 
 
 4717 
 
 4508 
 
 4308 
 
 4117 
 
 3934 
 
 3759 
 
 45 
 
 5290 
 
 5051 
 
 4826 
 
 4611 
 
 4407 
 
 4212 
 
 4025 
 
 3846 
 
 3674 
 
 16 
 
 5409 
 
 5165 
 
 4933 
 
 4714 
 
 4505 
 
 4305 
 
 4114 
 
 3931 
 
 3756 
 
 46 
 
 5285 
 
 5048 
 
 4822 
 
 4608 
 
 4404 
 
 4209 
 
 4022 
 
 3843 
 
 3671 
 
 17 
 
 5405 
 
 5161 
 
 4930 
 
 4710 
 
 4501 
 
 4302 
 
 4111 
 
 3928 
 
 3753 
 
 47 
 
 5281 
 
 5044 
 
 4819 
 
 4604 
 
 4400 
 
 4205 
 
 4019 
 
 3840 
 
 3668 
 
 18 
 
 5401 
 
 5157 
 
 496 
 
 4707 
 
 4498 
 
 4298 
 
 4108 
 
 3925 
 
 3750 
 
 48 
 
 5277 
 
 5040 
 
 4815 
 
 4601 
 
 4397 
 
 4202 
 
 4016 
 
 3837 
 
 3665 
 
 19 
 
 5397 
 
 5155 
 
 4922 
 
 4 70S 
 
 4494 
 
 4295 
 
 4105 
 
 3922 
 
 3747 
 
 49 
 
 5273 
 
 5036 
 
 4811 
 
 4597 
 
 4394 
 
 4199 
 
 4013 
 
 3834 
 
 3663 
 
 20 
 
 5393 
 
 5149 
 
 4918 
 
 4699 
 
 4491 
 
 4292 
 
 4102 
 
 3919 
 
 3745 
 
 50 
 
 5269 
 
 5032 
 
 4808 
 
 4594 
 
 4390 
 
 4196 
 
 4010 
 
 3831 
 
 3660 
 
 21 
 
 5389 
 
 5145 
 
 4915 
 
 4f>96 
 
 4488 
 
 4289 
 
 4099 
 
 3917 
 
 3742 
 
 51 
 
 5265 
 
 5028 
 
 4804 
 
 4590 
 
 4387 
 
 4193 
 
 4007 
 
 3828 
 
 3657 
 
 22 
 
 5 .84 
 
 S141 
 
 4911 
 
 4693 
 
 4484 
 
 4285 
 
 4096 
 
 3914 
 
 3739 
 
 52 
 
 5261 
 
 5025 
 
 4800 
 
 4587 
 
 4384 
 
 4189 
 
 4004 
 
 3825 
 
 3654 
 
 23 
 
 5380 
 
 5137 
 
 4907 
 
 4689 
 
 4481 
 
 4282 
 
 4092 
 
 3911 
 
 3736 
 
 53 
 
 5257 
 
 5021 
 
 4797 
 
 4584 
 
 4380 
 
 4186 
 
 4001 
 
 3822 
 
 3651 
 
 24 
 
 5376 
 
 5133 
 
 4903 
 
 4685 
 
 4477 
 
 4279 
 
 4089 
 
 3908 
 
 3733 
 
 54 
 
 5253 
 
 5017 
 
 4793 
 
 4580 
 
 4377 
 
 4183 
 
 3998 
 
 3820 
 
 3649 
 
 25 
 
 5372 
 
 5129 
 
 4900 
 
 4682 
 
 4474 
 
 4276 
 
 4086 
 
 3905 
 
 3730 
 
 55 
 
 5249 
 
 5013 
 
 4789 
 
 4577 
 
 4374 
 
 4180 
 
 3995 
 
 3817 
 
 3646 
 
 26 
 
 5368 
 
 5125 
 
 4896 
 
 4678 
 
 4471 
 
 4273 
 
 4083 
 
 3902 
 
 3727 
 
 56 
 
 5245 
 
 5009 
 
 4786 
 
 4573 
 
 4370 
 
 4177 
 
 3991 
 
 5814 
 
 3643 
 
 27 
 
 5364 
 
 5122 
 
 489C 
 
 4675 
 
 4467 
 
 4269 
 
 4080 
 
 3899 
 
 3725 
 
 57 
 
 5241 
 
 5005 
 
 4782 
 
 4570 
 
 4367 
 
 4174 
 
 3988 
 
 3811 
 
 3640 
 
 28 
 
 5359 
 
 oil*- 
 
 48by, 
 
 4671 
 
 4464 
 
 4266 
 
 4077 
 
 3896 
 
 3722 
 
 58 
 
 5237 
 
 5002 
 
 4778 
 
 4566 
 
 4364 
 
 4171 
 
 3985 
 
 3808 
 
 3637 
 
 29 
 
 5355 
 
 5114 
 
 488" 
 
 4668 
 
 4460 
 
 4263 
 
 4074 
 
 3893 
 
 3719 
 
 59 
 
 5233 
 
 4998 
 
 4775 
 
 4563 
 
 4361 
 
 4167 
 
 3982 
 
 3805 
 
 3635 
 
 3( 
 
 5351 
 
 5110 
 
 4881 
 
 4664 
 
 4457 
 
 4260 
 
 4071 
 
 3890 
 
 3716 
 
 60 
 
 5229|4994 
 
 4771 
 
 455914357 
 
 4164 
 
 3979 
 
 3802 
 
 3632 
 
393 
 
 APPENDIX TO THE ASTRONOMY. 
 
 TABLE of Logistical Logarithms. 
 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 
 1560 
 
 1620 
 
 1680 
 
 1740 
 
 1800 
 
 1860 
 
 1920 
 
 1980 
 
 2040 
 
 
 1560 1620 
 
 1680 
 
 1740 
 
 1800 
 
 1860 
 
 1920 
 
 '980 
 
 2040 
 
 
 
 3632 
 
 3468 
 
 3310 
 
 3158 
 
 3010 
 
 2868 
 
 2730 
 
 2596 
 
 2467 
 
 30 
 
 3549 
 
 3388 
 
 3233 
 
 308S 
 
 2939 
 
 2798 
 
 2663 
 
 2531 
 
 2403 
 
 1 
 
 3629 
 
 3465 
 
 3307 
 
 3155 
 
 3008 
 
 2866 
 
 2728 
 
 2594 
 
 2465 
 
 31 
 
 3546 
 
 3386 
 
 3231 
 
 3081 
 
 ^936 
 
 2?9f 
 
 266( 
 
 252! 
 
 2401 
 
 2 
 
 3626 
 
 3463 
 
 3305 
 
 3153 
 
 3005 
 
 2863 
 
 2725 
 
 2592 
 
 2462 
 
 32 
 
 3544 
 
 3383 3228 
 
 3078 
 
 2934 
 
 27!K 
 
 2658 
 
 2527 
 
 2399 
 
 3 
 
 3623 
 
 3460 
 
 3302 
 
 3150 
 
 3003 
 
 2861 
 
 2723 
 
 2590 
 
 2460 
 
 33 
 
 3541 
 
 3380 
 
 3i25 
 
 3076 
 
 2931 
 
 279x 
 
 65( 
 
 2525 
 
 2397 
 
 4 
 
 3621 
 
 3457 
 
 3300 
 
 3148 
 
 3001 
 
 2859 
 
 272l 
 
 2588 
 
 2458 
 
 34 
 
 353 
 
 3378 
 
 3223 
 
 3073 
 
 2929 
 
 2789 
 
 2654 
 
 352 
 
 2395 
 
 5 
 
 3618 
 
 3454 
 
 3297 
 
 3145 
 
 2998 
 
 2856 
 
 2719 
 
 2585 
 
 2456 
 
 35 
 
 3535 
 
 3375 
 
 3220 
 
 3071 
 
 2927 
 
 2787 
 
 2652 
 
 2520 
 
 5<J3 
 
 6 
 
 3615 
 
 3452 
 
 3294 
 
 314S 
 
 2996 
 
 2854 
 
 2716 
 
 258S 
 
 2454 
 
 36 
 
 3533 
 
 3372 
 
 3218 
 
 3069 
 
 2924 
 
 2786 
 
 -649 
 
 2518 
 
 2391 
 
 7 
 
 3612 
 
 3449 
 
 3292 
 
 3140 
 
 2993 
 
 2852 
 
 2714 
 
 2581 
 
 2452 
 
 37 
 
 3530 
 
 3370 
 
 3215 
 
 3066 
 
 2922 
 
 2782 
 
 -647 
 
 251C 
 
 2389 
 
 8 
 
 3610 
 
 3446 
 
 3289 
 
 3138 
 
 2991 
 
 2849 
 
 2712 
 
 2579 
 
 2450 
 
 38 
 
 8527 
 
 3367 
 
 3213 
 
 3064 
 
 2920 
 
 2730 
 
 xi645 
 
 2514 
 
 2387 
 
 9 
 
 3607 
 
 3444 
 
 3287 
 
 3135 
 
 2989 
 
 2847 
 
 2710 
 
 2577 
 
 2448 
 
 39 
 
 3525 
 
 3365 
 
 3210 
 
 3061 
 
 1917 
 
 2778 
 
 2643 
 
 2512 
 
 2384 
 
 10 
 
 3604 
 
 3441 
 
 3284 
 
 3133 
 
 2986 
 
 2845 
 
 2707 
 
 2574 
 
 2445 
 
 40 
 
 3522 
 
 3362 
 
 3208 
 
 3059 
 
 2915 
 
 2775 
 
 2640 
 
 2510 
 
 2382 
 
 11 
 
 360 1 
 
 3438 
 
 3282 
 
 3130 
 
 2984 
 
 2842 
 
 2705 
 
 2572 
 
 2443 
 
 41 
 
 3519 
 
 3359 
 
 3205 
 
 3056 
 
 2912 
 
 2773 
 
 2638 
 
 2507 
 
 2380 
 
 12 
 
 3598 
 
 3436 
 
 3279 
 
 3128 
 
 2981 
 
 2840 
 
 2703 
 
 2570 
 
 2441 
 
 42 
 
 35 16 
 
 3357 
 
 3203 
 
 3054 
 
 29 1C 
 
 2771 
 
 2636 
 
 2505 
 
 2378 
 
 13 
 
 >59(' 
 
 3433 
 
 3276 
 
 3125 
 
 2979 
 
 i838 
 
 2701 
 
 2568 
 
 24 3S 
 
 43 
 
 3514 
 
 SS.M 
 
 3200 
 
 3052 
 
 290fc 
 
 2769 
 
 2634 
 
 2503 
 
 2376 
 
 14 
 
 3593 
 
 3431 
 
 3274 
 
 3123 
 
 2977 
 
 2835 
 
 2698 
 
 2566 
 
 2437 
 
 44 
 
 3511 
 
 3351 
 
 3198 
 
 3049 
 
 2905 
 
 27' 6 
 
 2632 
 
 2501 
 
 2374 
 
 IS 
 
 3590 
 
 342 
 
 3271 
 
 3120 
 
 2974 
 
 2833 
 
 2696 
 
 2564 
 
 2435 
 
 45 
 
 3508 
 
 3349 
 
 3195 
 
 3047 
 
 2903 
 
 2764 
 
 2629 
 
 2499 
 
 2372 
 
 Ifi 
 
 3587 
 
 3425 
 
 326r 
 
 3118 
 
 2972 
 
 2831 
 
 2694 
 
 2561 
 
 2433 
 
 46 
 
 3506 
 
 3356 
 
 3193 
 
 3044 
 
 2901 
 
 '762 
 
 2627 
 
 2497 
 
 2370 
 
 17 
 
 3585 
 
 3423 
 
 3266 
 
 5115 
 
 2969 
 
 2828 
 
 <:692 
 
 2559 
 
 2431 
 
 47 
 
 3503 
 
 3344 
 
 3190 
 
 3042 
 
 3898 
 
 2760 
 
 2625 
 
 2494 
 
 2368 
 
 18 
 
 358S 
 
 3420 
 
 32<J4 
 
 3113 
 
 2967 
 
 2826 
 
 2689 
 
 2557 
 
 2429 
 
 48 
 
 3500 
 
 3341 
 
 3188 
 
 3039 
 
 ^896 
 
 2757 
 
 2623 
 
 2492 
 
 2366 
 
 19 
 
 3579 
 
 3417 
 
 3261 
 
 3110 
 
 2965 
 
 2824 
 
 2687 
 
 2555 
 
 2426 
 
 49 
 
 3497 
 
 3338 
 
 3185 
 
 3037 
 
 2894 
 
 2755 
 
 2621 
 
 2490 
 
 2364 
 
 20 
 
 3576 
 
 3415 
 
 3259 
 
 3108 
 
 2962 
 
 2821 
 
 -2685 
 
 2553 
 
 2424 
 
 50 
 
 3495 
 
 3336 
 
 3183 
 
 3034 
 
 2891 
 
 2753 
 
 2618 
 
 2488 
 
 2362 
 
 21 
 
 3574 
 
 3412 
 
 3256 
 
 3105 
 
 2960 
 
 2819 
 
 -2683 
 
 2531 
 
 2422 
 
 51 
 
 3492 
 
 3333 
 
 3180 
 
 303xJ 
 
 2889 
 
 2750 
 
 2616 
 
 2486 
 
 2359 
 
 -?2 
 
 3571 
 
 3409 
 
 3253 
 
 3103 
 
 2958 
 
 3817 
 
 2681 
 
 2548 
 
 2420 
 
 52 
 
 3489 
 
 3331 
 
 3178 
 
 3030 
 
 2887 
 
 -2748 
 
 2614 
 
 2484 
 
 2357 
 
 -23 
 
 3568 
 
 3407 
 
 3251 
 
 3101 
 
 2955 
 
 2815 
 
 2678 
 
 2546 
 
 2418 
 
 53 
 
 3487 
 
 3328 
 
 3175 
 
 3027 
 
 2884 
 
 2746 
 
 2612 
 
 2482 
 
 2355 
 
 M 
 
 3565 
 
 3404 
 
 3248 
 
 3098 
 
 2953 
 
 2812 
 
 2676 
 
 2544 
 
 2416 
 
 54 
 
 3484 
 
 3325 
 
 3!73 
 
 302^ 
 
 2882 
 
 2744 
 
 2610 
 
 2480 
 
 2353 
 
 25 
 
 3563 
 
 3401 
 
 3246 
 
 309G 
 
 2950 
 
 2810 
 
 2674 
 
 2542 
 
 2414 
 
 55 
 
 3481 
 
 3323 
 
 3170 
 
 3022 
 
 2880 
 
 2741 
 
 2607 
 
 2477 
 
 2351 
 
 26 
 
 35GO 
 
 3399 
 
 3243 
 
 3093 
 
 2948 
 
 2808 
 
 2672 
 
 2540 
 
 2412 
 
 56 
 
 3479 
 
 3320 
 
 3168 
 
 3020 
 
 2877 
 
 2739 
 
 2605 
 
 2475 
 
 2649 
 
 27 
 
 3557 
 
 3396 
 
 3241 
 
 3091 
 
 2946 
 
 2805 
 
 2669 
 
 2538 
 
 2410 
 
 57 
 
 3476 
 
 33l8 
 
 3165 
 
 3018 
 
 2875 
 
 2737 
 
 2603 
 
 2473 
 
 2347 
 
 28 
 
 3555 
 
 3393 
 
 3238 
 
 3088 
 
 2943 
 
 2803 
 
 2667 
 
 2535 
 
 2408 
 
 58 
 
 3473 
 
 3315 
 
 3163 
 
 3015 
 
 2873 
 
 27S. r 
 
 2601 
 
 2471 
 
 2345 
 
 29 
 
 3552 
 
 3391 
 
 3236 
 
 3086 
 
 2941 
 
 2801 
 
 2665 
 
 2533 
 
 2405 
 
 59 
 
 3471 
 
 3313 
 
 3160 
 
 3013 
 
 2870 
 
 2732 
 
 3599 
 
 2469 
 
 2343 
 
 30 
 
 3549 
 
 3388 
 
 3233 
 
 3083 
 
 2939 
 
 27982663 
 
 2531 
 
 2403 
 
 60 
 
 3468 
 
 3310 
 
 3158 
 
 3010 
 
 2868 
 
 2730 
 
 259 
 
 2467 
 
 2341 
 
LUNAR TABLES. 
 
 393 
 
 TABLE of Logistical Logarithms. 
 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 
 33 
 
 3G 
 
 37 
 
 38 
 
 39 
 
 4U 
 
 41 
 
 42 
 
 43 
 
 
 100 
 
 160 
 
 220 
 
 28. 
 
 234 
 
 2400 
 
 2460 
 
 3520 
 
 258 
 
 
 2100 
 
 2160 
 
 2220 
 
 2280 
 
 2340 
 
 2400 
 
 2460 
 
 2520 
 
 2580 
 
 
 
 341 
 
 2218 
 
 099 
 
 
 871 
 
 761 
 
 654 
 
 1549 
 
 1447 
 
 5o 
 
 2279 
 
 2159 
 
 2041 
 
 1927 
 
 1816 
 
 17"7 
 
 1601 
 
 1498 
 
 1397 
 
 1 
 
 2339 
 
 2216 
 
 -098 
 
 JH2 
 
 869 
 
 759 
 
 652 
 
 1547 
 
 1445 
 
 > ' 
 
 2277 
 
 2157 
 
 2039 
 
 1925 
 
 1814 
 
 170 
 
 1599 
 
 1496 
 
 ;395 
 
 2 
 
 2337 
 
 2214 
 
 2096 
 
 980 
 
 867 
 
 757 
 
 1650 
 
 1546 
 
 1443 
 
 32 
 
 2275 
 
 2155 
 
 2037 
 
 1 923 
 
 1812 
 
 1703 
 
 1598 
 
 1494 
 
 1393 
 
 g 
 
 2335 
 
 2212 
 
 2094 
 
 978 
 
 865 
 
 .755 
 
 648 
 
 1544 
 
 1442 
 
 33 
 
 2273 
 
 2153 
 
 2035 
 
 1921 
 
 1810 
 
 i roe 
 
 1596 
 
 1493 
 
 1392 
 
 4 
 
 2333 
 
 2210 
 
 2092 
 
 !)76 
 
 863 
 
 754 
 
 1647 
 
 1542 
 
 1440 
 
 34 
 
 2271 
 
 2151 
 
 2033 
 
 1919 
 
 1808 
 
 1700 
 
 1594 
 
 1491 
 
 1390 
 
 5 
 
 2331 
 
 2208 
 
 2090 
 
 1974 
 
 862 
 
 1752 
 
 645 
 
 1540 
 
 1438 
 
 .55 
 
 2269 
 
 2149 
 
 2032 
 
 1918 
 
 1806 
 
 1698 
 
 1592 
 
 1489 
 
 1388 
 
 6 
 
 2328 
 
 2206 
 
 2088 
 
 972 
 
 860 
 
 1750 
 
 1643 
 
 1539 
 
 1437 
 
 36 
 
 2267 
 
 2147 
 
 2030 
 
 1916 
 
 1805 
 
 1696 
 
 1591 
 
 1487 
 
 1387 
 
 7 
 
 2326 
 
 2204 
 
 20S(. 
 
 1970 
 
 1858 
 
 1748 
 
 1641 
 
 1537 
 
 1435 
 
 37 
 
 265 
 
 2145 
 
 2028 
 
 1914 
 
 1803 
 
 1694 
 
 589 
 
 1486 
 
 1385 
 
 8 
 
 2324 
 
 2202 
 
 2084 
 
 1968 
 
 1856 
 
 1746 
 
 1640 
 
 1535 
 
 1433 
 
 38 
 
 2263 
 
 2143 
 
 2026 
 
 1912 
 
 1801 
 
 1693 
 
 1587 
 
 1484 
 
 1383 
 
 c 
 
 2322 
 
 2200 
 
 208 
 
 1967 
 
 1854 
 
 1745 
 
 1638 
 
 1534 
 
 1432 
 
 39 
 
 2261 
 
 214i 
 
 2024 
 
 910 
 
 1799 
 
 1691 
 
 1585 
 
 1482 
 
 1382 
 
 10 
 
 2320 
 
 2198 
 
 2080 
 
 1965 
 
 1852 
 
 1743 
 
 1636 
 
 1532 
 
 1430 
 
 40 
 
 2259 
 
 2139 
 
 2022 
 
 1908 
 
 1797 
 
 1689 
 
 1584 
 
 1481 
 
 1380 
 
 11 
 
 2318 
 
 2196 
 
 2078 
 
 1963 
 
 850 
 
 1741 
 
 1634 
 
 1530 
 
 1428 
 
 41 
 
 2257 
 
 U37 
 
 2020 
 
 1906 
 
 1795 
 
 1687 
 
 1582 
 
 1479 
 
 1378 
 
 12 
 
 2316 
 
 2194 
 
 2076 
 
 1961 
 
 1849 
 
 1739 
 
 1633 
 
 1528 
 
 1427 
 
 42 
 
 2255 
 
 2135 
 
 2018 
 
 1904 
 
 1794 
 
 1636 
 
 15HO 
 
 1477 
 
 1377 
 
 i3 
 
 2314 
 
 2192 
 
 2074 
 
 19o9 
 
 1S47 
 
 1737 
 
 1631 
 
 1527 
 
 1425 
 
 43 
 
 2253 
 
 2133 
 
 2016 
 
 1 (03 
 
 1792 
 
 1684 
 
 1578 
 
 1476 
 
 1375 
 
 14 
 
 2312 
 
 2190 
 
 2072 
 
 1957 
 
 1845 
 
 1736 
 
 1629 
 
 1525 
 
 1423 
 
 44 
 
 2251 
 
 2131 
 
 2014 
 
 1901 
 
 1790 
 
 1682 
 
 1577 
 
 1474 
 
 1373 
 
 15 
 
 2310 
 
 21X8 
 
 2070 
 
 1955 
 
 1843 
 
 1734 
 
 1627 
 
 1523 
 
 1422 
 
 45 
 
 2249 
 
 2129 
 
 2019 
 
 1899 
 
 1788 
 
 1680 
 
 1575 
 
 1472 
 
 1372 
 
 16 
 
 2308 
 
 2186 
 
 2068 
 
 1953 
 
 1841 
 
 I '32 
 
 1026 
 
 1522 
 
 1420 
 
 4t 
 
 22-*7 
 
 2127 
 
 2010 
 
 1897 
 
 1786 
 
 678 
 
 1573 
 
 1470 
 
 1370 
 
 17 
 
 2306 
 
 2184 
 
 2066 
 
 1951 
 
 1839 
 
 1730 
 
 16Ss4 
 
 1520 
 
 1418 
 
 47 
 
 2245 
 
 2125 
 
 2009 
 
 1895 
 
 1785 
 
 1677 
 
 1571 
 
 1469 
 
 1368 
 
 18 
 
 2304 
 
 2182 
 
 2064 
 
 1950 
 
 1838 
 
 1728 
 
 1622 
 
 1518 
 
 1417 
 
 48 
 
 2243 
 
 2123 
 
 2007 
 
 1893 
 
 1783 
 
 1675 
 
 1570 
 
 1467 
 
 1367 
 
 19 
 
 2302 
 
 2180 
 
 '.062 
 
 1948 
 
 1836 
 
 1727 
 
 1620 
 
 1516 
 
 141n 
 
 49 
 
 2241 
 
 2121 
 
 2005 
 
 1891 
 
 1781 
 
 1673 
 
 1568 
 
 1465 
 
 1365 
 
 20 
 
 2300 
 
 2178 
 
 2061 
 
 1946 
 
 1834 
 
 1725 
 
 1619 
 
 1515 
 
 1413 
 
 50 
 
 2239 
 
 2119 
 
 2003 
 
 18S9 
 
 1779 
 
 1671 
 
 1566 
 
 1464 
 
 1363 
 
 21 
 
 2298 
 
 2176 
 
 2059 
 
 1944 
 
 1832 
 
 172f 
 
 1617 
 
 1513 
 
 1412 
 
 51 
 
 2-237 
 
 2117 
 
 2001 
 
 1888 
 
 1777 
 
 1670 
 
 1565 
 
 1462 
 
 1362 
 
 <J> 
 
 2296 
 
 21-4 
 
 2057 
 
 1942 
 
 1830 
 
 1721 
 
 1615 
 
 1511 
 
 14U 
 
 .V~ 
 
 2235 
 
 2115 
 
 1999 
 
 1886 
 
 1775 
 
 1668 
 
 156f 
 
 1460 
 
 1360 
 
 2o 
 
 s294 
 
 2172 
 
 2055 
 
 1940 
 
 1828 
 
 1719 
 
 1613 
 
 15H 
 
 140- 
 
 53 
 
 2233 
 
 2113 
 
 1997 
 
 1884 
 
 1774 
 
 1666 
 
 1561 
 
 1459 
 
 1359 
 
 24 
 
 2291 
 
 2170 
 
 2053 
 
 1938 
 
 18-27 
 
 1718 
 
 1612 
 
 1508 
 
 1407 
 
 54 
 
 2231 
 
 Mil 
 
 199. 
 
 1882 
 
 1772 
 
 1664 
 
 1559 
 
 1457 
 
 1357 
 
 25 
 
 2289 
 
 '1 i! 
 
 2051 
 
 1936 
 
 1825 
 
 1716 
 
 1610 
 
 I50f 
 
 1405 
 
 55 
 
 2229 
 
 2109 
 
 1993 
 
 1880 
 
 1770 
 
 1663 
 
 1558 
 
 1455 
 
 1355 
 
 26 
 
 2287 
 
 2167 
 
 204' 
 
 1934 
 
 1823 
 
 1714 
 
 1608 
 
 1504 
 
 1403 
 
 56 
 
 2227 
 
 2107 
 
 1991 
 
 1878 
 
 1768 
 
 1661 
 
 1556 
 
 1454 
 
 1354 
 
 2" 
 
 3285 
 
 216 
 
 2047 
 
 1933 
 
 182 
 
 1712 
 
 1606 
 
 1503 
 
 1402 
 
 57 
 
 2225 
 
 210* 
 
 1989 
 
 187f 
 
 1766 
 
 1659 
 
 1554 
 
 1452 
 
 1352 
 
 2 
 
 2283 
 
 216 
 
 2045 
 
 193 
 
 1816 
 
 1711 
 
 I'iOo 
 
 150 
 
 140f 
 
 T8 
 
 2223 
 
 2103 
 
 1987 
 
 1875 
 
 1765 
 
 1657 
 
 1552 
 
 1450 
 
 350 
 
 2 
 
 228 
 
 216 
 
 204, 
 
 192' 
 
 1817 
 
 1709 
 
 1603 
 
 1499 
 
 1398 
 
 <i 
 
 2220 
 
 2101 
 
 198f 
 
 1873 
 
 1763 
 
 1655 
 
 1551 
 
 1449 
 
 1349 
 
 SC 
 
 227 C 
 
 215 
 
 204 
 
 192' 
 
 I81f 
 
 1707 
 
 160 
 
 1498 
 
 1397 
 
 6( 
 
 2218 
 
 2099 
 
 1984 
 
 187 
 
 i76l 
 
 15' 
 
 1540 
 
 1447 
 
 '347 
 
 54J 
 
APPENDIX TO THE ASTRONOMY. 
 
 TABLE of Logistical Logarithms. 
 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 
 
 52 
 
 
 2640 
 
 2700 
 
 2760 
 
 2820 
 
 2880 
 
 -294C 
 
 3000 
 
 3060 
 
 3120 
 
 
 2640 
 
 2700 
 
 2760 
 
 2820 
 
 2880 
 
 '2940 
 
 300(- 
 
 3060 
 
 3120 
 
 
 
 1347 
 
 1249 
 
 1154 
 
 1061 
 
 969 
 
 880 
 
 792 
 
 706 
 
 621 
 
 
 1298 
 
 1201 
 
 1107 
 
 1015 
 
 924 
 
 835 
 
 749 
 
 663 
 
 580 
 
 1 
 
 1345 
 
 '248 
 
 1152 
 
 1059 
 
 968 
 
 878 
 
 790 
 
 704 
 
 620 
 
 31 
 
 1296 
 
 1200 
 
 1105 
 
 1013 
 
 923 
 
 834 
 
 747 
 
 662 
 
 579 
 
 2 
 
 1344 
 
 1246 
 
 1151 
 
 1057 
 
 96G 
 
 87' 
 
 789 
 
 703 
 
 619 
 
 32 
 
 1295 
 
 1198 
 
 1104 
 
 1012 
 
 921 
 
 833 
 
 74 C 
 
 661 
 
 577 
 
 3 
 
 1342 
 
 1245 
 
 1149 
 
 1056 
 
 963 
 
 875 
 
 787 
 
 702 
 
 617 
 
 
 1293 
 
 1197 
 
 1102 
 
 1010 
 
 920 
 
 831 
 
 744 
 
 659 
 
 576 
 
 4 
 
 1340 
 
 124.3 
 
 1148 
 
 1054 
 
 963 
 
 874 
 
 786 
 
 700 
 
 616 
 
 34 
 
 1291 
 
 1195 
 
 1101 
 
 1008 
 
 918 
 
 830 
 
 743 
 
 658 
 
 574 
 
 5 
 
 1339 
 
 1241 
 
 1146 
 
 1053 
 
 962 
 
 872 
 
 785 
 
 699 
 
 615 
 
 35 
 
 1290 
 
 1193 
 
 1099 
 
 1007 
 
 917 
 
 828 
 
 7-41 
 
 656 
 
 573 
 
 6 
 
 1337 
 
 1240 
 
 1145 
 
 1051 
 
 960 
 
 871 
 
 783 
 
 697 
 
 613 
 
 36 
 
 1288 
 
 1192 
 
 1098 
 
 1005 
 
 915 
 
 827 
 
 740 
 
 655 
 
 572 
 
 7 
 
 1335 
 
 1238 
 
 1143 
 
 1050 
 
 959 
 
 869 
 
 782 
 
 696 
 
 612 
 
 
 1287 
 
 1190 
 
 1096 
 
 1004 
 
 914 
 
 825 
 
 rss 
 
 654 
 
 570 
 
 g 
 
 1334 
 
 1237 
 
 1141 
 
 1048 
 
 957 
 
 868 
 
 780 
 
 694 
 
 610 
 
 38 
 
 1285 
 
 1189 
 
 1095 
 
 1002 
 
 912 
 
 824 
 
 
 
 569 
 
 9 
 
 1332 
 
 1235 
 
 1140 
 
 1047 
 
 956 
 
 866 
 
 779 
 
 693 
 
 609 
 
 39 
 
 1283 
 
 1187 
 
 1093 
 
 1001 
 
 911 
 
 822 
 
 736 
 
 651 
 
 568 
 
 10 
 
 1331 
 
 1233 
 
 1138 
 
 1045 
 
 954 
 
 865 
 
 777 
 
 692 
 
 608 
 
 40 
 
 1282 
 
 1186 
 
 1091 
 
 999 
 
 909 
 
 821 
 
 734 
 
 649 
 
 566 
 
 11 
 
 1329 
 
 1232 
 
 1137 
 
 1044 
 
 953 
 
 863 
 
 776 
 
 690 
 
 606 
 
 41 
 
 1280 
 
 1184 
 
 1090 
 
 998 
 
 908 
 
 819 
 
 733 
 
 648 
 
 565 
 
 lu 
 
 1327 
 
 1230 
 
 1135 
 
 1042 
 
 951 
 
 862 
 
 774 
 
 689 
 
 605 
 
 42 
 
 1278 
 
 1182 
 
 1088 
 
 996 
 
 906 
 
 818 
 
 7ol 
 
 647 
 
 563 
 
 13 
 
 1326 
 
 1229 
 
 1134 
 
 1041 
 
 950 
 
 860 
 
 773 
 
 687 
 
 60S 
 
 43 
 
 1277 
 
 1181 
 
 1087 
 
 995 
 
 905 
 
 816 
 
 730 
 
 645 
 
 562 
 
 14 
 
 1324 
 
 1227 
 
 1132 
 
 1039 
 
 948 
 
 859 
 
 772 
 
 686 
 
 602 
 
 ->4 
 
 1275 
 
 1179 
 
 1085 
 
 993 
 
 903 
 
 815 
 
 729 
 
 644 
 
 561 
 
 15 
 
 1322 
 
 1225 
 
 1130 
 
 1037 
 
 947 
 
 857 
 
 770 
 
 685 
 
 601 
 
 45 
 
 1274 
 
 1178 
 
 1084 
 
 992 
 
 902 
 
 814 
 
 727 
 
 642 
 
 559 
 
 16 
 
 1321 
 
 1224 
 
 1129 
 
 1036 
 
 945 
 
 856 
 
 769 
 
 683 
 
 599 
 
 46 
 
 1272 
 
 1176 
 
 1082 
 
 990 
 
 900 
 
 812 
 
 726 
 
 b41 
 
 558 
 
 17 
 
 1319 
 
 1222 
 
 1127 
 
 1034 
 
 944 
 
 855 
 
 767 
 
 682 
 
 598 
 
 47 
 
 1270 
 
 1174 
 
 1081 
 
 989 
 
 899 
 
 811 
 
 724 
 
 640 
 
 557 
 
 18 
 
 1317 
 
 1221 
 
 1126 
 
 1033 
 
 942 
 
 853 
 
 766 
 
 680 
 
 596 
 
 48 
 
 1269 
 
 UT d 
 7 O 
 
 1079 
 
 987 
 
 897 
 
 809 
 
 723 
 
 638 
 
 555 
 
 19 
 
 1316 
 
 1219 
 
 1124 
 
 1031 
 
 941 
 
 852 
 
 764 
 
 679 
 
 595 
 
 49 
 
 1267 
 
 1171 
 
 1078 
 
 986 
 
 896 
 
 808 
 
 721 
 
 637 
 
 554 
 
 20 
 
 1314 
 
 1217 
 
 1123 
 
 1030 
 
 939 
 
 850 
 
 763 
 
 678 
 
 594 
 
 50 
 
 1266 
 
 1170 
 
 1076 
 
 984 
 
 894 
 
 806 
 
 720 
 
 635 
 
 552 
 
 21 
 
 1313 
 
 1216 
 
 1121 
 
 1028 
 
 938 
 
 849 
 
 76-2 
 
 676 
 
 59S 
 
 51 
 
 1264 
 
 1168 
 
 1074 
 
 983 
 
 893 
 
 805 
 
 719 
 
 634 
 
 551 
 
 22 
 
 1311 
 
 1214 
 
 1119 
 
 1027 
 
 936 
 
 847 
 
 760 
 
 675 
 
 591 
 
 55 
 
 1262 
 
 1167 
 
 1073 
 
 981 
 
 891 
 
 803 
 
 717 
 
 633 
 
 550 
 
 23 
 
 1309 
 
 1213 
 
 1118 
 
 1025 
 
 935 
 
 846 
 
 759 
 
 673 
 
 590 
 
 53 
 
 1261 
 
 lies 
 
 1071 
 
 980 
 
 890 
 
 802 
 
 716 
 
 631 
 
 548 
 
 24 
 
 1308 
 
 1211 
 
 1116 
 
 1024 
 
 933 
 
 844 
 
 757 
 
 672 
 
 568 
 
 54 
 
 1259 
 
 1163 
 
 1070 
 
 9^8 
 
 888 
 
 801 
 
 714 
 
 630 
 
 547 
 
 25 
 
 1306 
 
 1209 
 
 1115 
 
 102i 
 
 932 
 
 843 
 
 756 
 
 670 
 
 587 
 
 5." 
 
 1257 
 
 1162 
 
 1068 
 
 977 
 
 887 
 
 799 
 
 713 
 
 t>28 
 
 546 
 
 26 
 
 1304 
 
 1208 
 
 1113 
 
 1021 
 
 930 
 
 841 
 
 754 
 
 669 
 
 585 
 
 56 
 
 1256 
 
 1160 
 
 1067 
 
 975 
 
 885 
 
 798 
 
 711 
 
 627 
 
 544 
 
 27 
 
 1303 
 
 1206 
 
 1112 
 
 1019 
 
 929 
 
 840 
 
 753 
 
 668 
 
 584 
 
 57 
 
 1254 
 
 1159 
 
 1065 
 
 974 
 
 884 
 
 796 
 
 710 
 
 626 
 
 543 
 
 28 
 
 1301 
 
 1205 
 
 1110 
 
 1018 
 
 927 
 
 838 
 
 751 
 
 666 
 
 583 
 
 58 
 
 1253 
 
 1157 
 
 1064 
 
 972 
 
 883 
 
 795 
 
 709 
 
 624 
 
 51] 
 
 29 
 
 1300 
 
 1203 
 
 1109 
 
 1016 
 
 926 
 
 837 
 
 750 
 
 665 
 
 581 
 
 59 
 
 1251 
 
 1156 
 
 1062 
 
 97T 
 
 8S< 
 
 793 
 
 707 
 
 623 
 
 540 
 
 30 
 
 1298 
 
 1201 
 
 1107 
 
 1015 
 
 924 
 
 835 
 
 749 
 
 663 
 
 580 
 
 60 
 
 1249 
 
 1154 
 
 1061 
 
 969 
 
 880 
 
 792 
 
 706 
 
 621 
 
 539 
 
TL 
 
 LUNAR. TABLES. 
 
 395 
 
 TABLE of Logistical Logarithms. 
 
 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 sa 11 1 sa 
 ^ . 
 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 j 
 
 .^r i 
 
 3240 
 
 3300 
 
 3360 
 
 3420 
 
 3480 
 
 3540; 
 
 
 418 
 
 339 
 
 26 i 
 
 185 
 
 110 
 
 36 
 
 
 416 
 
 337 
 
 260 
 
 1B4 
 
 109 
 
 35 
 
 
 415 
 
 336 
 
 258 
 
 182 
 
 107 
 
 34 
 
 
 414 
 
 335 
 
 257 
 
 181 
 
 106 
 
 33 
 
 412 
 
 333 
 
 256 
 
 180 
 
 105 
 
 31 
 
 411 
 
 SS2 
 
 255 
 
 179 
 
 104 
 
 30 
 
 410 
 
 331 
 
 253 
 
 177 
 
 103 
 
 29 
 
 408 
 
 329 
 
 252 
 
 176 
 
 101 
 
 28 
 
 407 
 
 328 
 
 251 
 
 175 
 
 100 
 
 27 
 
 406 
 
 327 
 
 250 
 
 174 
 
 99 
 
 25 
 
 UNIVERSITY OF CALIFORNIA LIBRARY 404 
 
 326 
 
 248 
 
 172 
 
 98 
 
 24 
 
 
 
 
 ,,., 
 
 V UC 
 
 <MX*U 
 
 **\j+r 
 
 o- 
 
 wj 
 
 11 
 
 100 
 
 403 
 
 324 
 
 247 
 
 171 
 
 96 
 
 23 
 
 
 12 
 
 522 
 
 442 
 
 362 
 
 284 
 
 208 
 
 132 
 
 58 
 
 4s> 
 
 482 
 
 402 
 
 323 
 
 246 
 
 170 
 
 95 
 
 22 
 
 
 13 
 
 521 
 
 440 
 
 361 
 
 283 
 
 206 
 
 131 
 
 57 
 
 43 
 
 480 
 
 400 
 
 322 
 
 244 
 
 169 
 
 94 
 
 21 
 
 
 14 
 
 520 
 
 439 
 
 359 
 
 282 
 
 205 
 
 130 
 
 56 
 
 44 
 
 479 
 
 399 
 
 320 
 
 243 
 
 167 
 
 93 
 
 19 
 
 
 15 
 
 518 
 
 438 
 
 358 
 
 280 
 
 204 
 
 129 
 
 55 
 
 45 
 
 478 
 
 398 
 
 319 
 
 242 
 
 166 
 
 91 
 
 18 
 
 
 16 
 
 517 
 
 436 
 
 357 
 
 279 
 
 202 
 
 127 
 
 53 
 
 46 
 
 476 
 
 396 
 
 318 
 
 241 
 
 165 
 
 90 
 
 17 
 
 
 17 
 
 516 
 
 435 
 
 356 
 
 278 
 
 201 
 
 126 
 
 52 
 
 47 
 
 475 
 
 395 
 
 316 
 
 239 
 
 163 
 
 89 
 
 16 
 
 
 18 
 
 514 
 
 434 
 
 354 
 
 276 
 
 200 
 
 125 
 
 51 
 
 48 
 
 474 
 
 394 
 
 315 
 
 238 
 
 162 
 
 88 
 
 15 
 
 
 19 
 
 513 
 
 432 
 
 353 
 
 275 
 
 199 
 
 124 
 
 50 
 
 49 
 
 472 
 
 392 
 
 314 
 
 237 
 
 161 
 
 87 
 
 13 
 
 
 20 
 
 512 
 
 431 
 
 352 
 
 274 
 
 197 
 
 122 
 
 49 
 
 50 
 
 471 
 
 391 
 
 313 
 
 235 
 
 160 
 
 85 
 
 12 
 
 
 2} 
 
 510 
 
 430 
 
 350 
 
 273 
 
 196 
 
 121 
 
 47 
 
 51 
 
 470 
 
 390 
 
 311 
 
 234 
 
 158 
 
 84 
 
 11 
 
 
 22 
 
 50 ( . 
 
 428 
 
 349 
 
 271 
 
 195 
 
 120 
 
 46 
 
 52 
 
 468 
 
 388 
 
 .TO 
 
 233 
 
 157 
 
 83 
 
 10 
 
 
 23 
 
 507 
 
 427 
 
 348 
 
 270 
 
 194 
 
 119 
 
 45 
 
 ~'3 
 
 467 
 
 387 
 
 309 
 
 232 
 
 156 
 
 82 
 
 8 
 
 
 24 
 
 506 
 
 426 
 
 346 
 
 269 
 
 192 
 
 117 
 
 44 
 
 54 
 
 466 
 
 38f 
 
 307 
 
 230 
 
 155 
 
 80 
 
 7 
 
 
 25 
 
 505 
 
 424 
 
 345 
 
 267 
 
 isn 
 
 116 
 
 42 
 
 55 
 
 464 
 
 384 
 
 306 
 
 229 
 
 153 
 
 79 
 
 6 
 
 
 26 
 
 SOS 
 
 423 
 
 344 
 
 266 
 
 J90 
 
 115 
 
 41 
 
 56 
 
 46- 
 
 383 
 
 305 
 
 228 
 
 152 
 
 "8 
 
 5 
 
 
 27 
 
 50C 
 
 422 
 
 342 
 
 265 
 
 189 
 
 114 
 
 40 
 
 57 
 
 462 
 
 382 
 
 504 
 
 227 
 
 151 
 
 77 
 
 4 
 
 
 28 
 
 501 
 
 420 
 
 341 
 
 264 
 
 187 
 
 112 
 
 39 
 
 58 
 
 46r 
 
 381 
 
 302 
 
 225 
 
 150 
 
 75 
 
 2 
 
 
 29 
 
 
 419 
 
 :40 
 
 262 
 
 186 
 
 111 
 
 38 
 
 59 
 
 45 
 
 379 
 
 301 
 
 224 
 
 148 
 
 74 
 
 1 
 
 
 30 
 
 498 
 
 418 
 
 339 
 
 261 
 
 185 
 
 110 
 
 36 GO 
 
 458 
 
 378 
 
 300 
 
 22J 
 
 147 
 
 73|