Phys. deptt 
 
 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Class 
 
 UNIVERSITY OF CALIFORNIA 
 
 LIBRARY 
 
 OF THE 
 
 DEPARTMENT OF PHYSICS 
 
 KM NQV.14iim 
 
 Accessions No. 6.L.<7.... Book No... / 
 

SCIENTIFIC MEMOIRS 
 
 EDITED BY 
 
 J. S. AMES, PH.D. 
 
 PROFESSOR OF PHYSICS IN JOHNS HOPKINS UNIVERSITY 
 
 IX. 
 THE LAWS OF GRAVITATION 
 
I 
 
THE 
 
 LAWS OF GRAVITATION 
 
 MEMOIKS BY NEWTON, BOUGUEK 
 AND CAVENDISH 
 
 TOGETHER WITH ABSTRACTS OF OTHER 
 IMPORTANT MEMOIRS 
 
 TRANSLATED AND EDITED BY 
 
 A. STANLEY MACKENZIE, PH.D. 
 
 PROFRSSOR OF PHYSICS IN BRYN MAWR COLLEGE 
 
 NEW YORK : CINCINNATI : CHICAGO 
 
 AMERICAN BOOK COMPANY 
 

 Copyright, 1900, by AMERICAN BOOK COMPANY. 
 w. P. i 
 
222544 
 
GENERAL CONTENTS 
 
 PAGE 
 
 Preface v 
 
 History of the subject before the appearance of Newton's Pnncipia. 1 
 
 Extracts from Newton's Principia and System of the World 9 
 
 Biographical sketch of Newton 19 
 
 Bouguer's The Figure of the Earth 23 
 
 Biographical sketch of Bouguer 44 
 
 The Bertier controversy 47 
 
 Account of Maskelyne's experiments on Schehallien 53 
 
 Cavendish's Experiments to determine the mean density of the Earth ... 59 
 
 Biographical sketch of Cavendish 107 
 
 Historical account of the experiments made since the time of Caven- 
 dish Ill 
 
 Table of results of experiments 143 
 
 Bibliography 145 
 
 Index.. 157 
 
PREFACE 
 
 IN preparing this volume, the ninth in the Scientific Me- 
 moirs series, the editor has had in mind the fact that the most 
 important of the memoirs here dealt with, that of Cavendish, 
 is frequently given for detailed study to young physicists in 
 order to train them in the art of reading for themselves period- 
 ical scientific literature. Certainly no better piece of work 
 could be used for the purpose, whether one considers the 
 intrinsic importance of the subject-matter, the keenness of 
 argument and the logical presentation in detail, or the use and 
 design of apparatus and the treatment of sources of error. 
 The main objections to Cavendish's work are those he himself 
 pointed out, and it is important to notice that, notwithstand- 
 ing all the advance in the refinement and manipulation of 
 apparatus which has been made during the century that has 
 elapsed since the date of Cavendish's experiment, his value for 
 the mean specific gravity of the earth, 5.448, must still be con- 
 sidered one of the most reliable, being not far from the latest 
 results of Poynting, Konig and Richarz and Krigar-Menzel, 
 Boys and Braun. 
 
 Believing that we in America devote insufficient time, if 
 any, to a study of Newton's great work, the editor has thought 
 it well to incorporate with the memoirs on the experimental 
 investigation of gravitational attraction the statements of New- 
 ton himself concerning that subject. 
 
 The laws of gravitation are embodied in the formula, 
 , r mm' 
 f =( *~f 
 
 which says that the attraction between two particles of matter 
 is directly proportional to the product of their masses, inverse- 
 ly proportional to the square of the distance between them, 
 and independent of the kind of matter and of the intervening 
 
PREFACE 
 
 medium. G is then a constant in nature, the Gravitation Con- 
 stant. It is more common perhaps to speak of the law, than 
 of the laws, of gravitation ; this has no doubt arisen from the 
 fact that they can be stated in a single mathematical formula. 
 The best evidence of the truth of these laws is indirect, for, 
 assuming them valid, astronomical measurements show that 
 they account for all the motions of the heavenly bodies. Such 
 measurements do not, however, enable us to find the numer- 
 ical value of G; for that purpose we must determine the at- 
 traction between two masses of known amount at a known 
 distance apart. It is with experiments of this character that 
 the present volume has to deal. As the masses used in such 
 experiments vary from a metal sphere of a few tenths of an 
 inch in diameter to a huge mountain mass, or to a shell of the 
 earth's crust 1250 ft. in thickness, and as the attraction has 
 been observed with such different instruments as the plumb- 
 line, the pendulum, the torsion balance, the pendulum bal- 
 ance and the beam balance, and yet the resulting value of G is 
 always about the same, we can regard these experiments as 
 constituting a further proof of Newton's laws, and the editor 
 has accordingly felt justified in using the title given. As- 
 suming the earth to be a sphere, the value of G is connected 
 with the value of the mean specific gravity of the earth, A, 
 by the equation 
 
 where g Is the acceleration due to gravity, and R the radius of 
 the earth ; and, accordingly, it is quite usual to state that the 
 aim of the above experiments is to find the mean density of 
 the earth. 
 
 The work on the attraction of mountain masses by the 
 French Academicians Bouguer and de la Condamine in Peru is 
 of very great importance, and is not known as it deserves to 
 be; almost all of their account of the work is therefore here 
 presented. It will be seen that they were the pioneers in two 
 of the methods which have been used for the measurement of 
 gravitational attraction ; and although, on account of imper- 
 fect instruments and unfavourable local conditions, their nu- 
 merical results are untrustworthy, they give the theory and 
 method of the experiments with great originality and clear- 
 ness. Such notes have been added to the memoirs as seemed 
 
 vi 
 
PREFACE 
 
 necessary to prevent the reader from wasting time over obscure 
 and inaccurate passages, and to suggest material for collateral 
 reading. 
 
 An eifort has been made to present along with the memoirs 
 a brief historical account of the various modes of experiment 
 used for finding the mean specific gravity of the earth, and a 
 table of results is added. As the literature on the subject 
 before the present century is not always easily obtainable, the 
 treatment of the matter for that period is given in compara- 
 tively greater detail. Believing that a bibliography contain- 
 ing every important reference to the subject is an essential 
 feature of a work of this kind, the editor has endeavoured to 
 make himself familiar with the whole of the very extensive 
 literature relating to it, and accordingly is fairly confident 
 that no important memoir has escaped his observation. From 
 the mass of material thus collected the bibliography given at 
 the end of the volume has been compiled. In order to keep 
 within the limits of space assigned, some references had to 
 be omitted, but they relate mainly to recent work, and it is 
 believed that they contain nothing of importance. 
 
 No effort has been made to deal with the mathematical side 
 of the subject ; accordingly the memoirs of Laplace, Legendre, 
 Ivory, etc., which deal with the finding of the mean specific 
 gravity of the earth by means of analytical methods are not 
 referred to ; but it is hoped that all the more important ex- 
 perimental investigations have been touched upon. 
 
 A. STANLEY MACKENZIE. 
 
 BKYN MAWR, October, 1899. 
 
 vii 
 
HISTORY OF THE SUBJECT 
 
 BEFORE THE 
 APPEARANCE OF NEWTON'S "PRINCIPIA" 
 
 DR. GILBERT'S contributions to the speculations on gravita- 
 tion are among the most important of the early writings on that 
 subject, although to Kepler also must credit be given for a 
 deep insight into its nature; the latter announces in his intro- 
 duction to the Astronomia Nova, published in 16Q9, his belief 
 in the perfect reciprocity of the action of gravitation, and in its 
 application to the whole material universe. Gilbert was led by 
 his researches on magnetism to the conclusion that the force of 
 gravity was due to the magnetic properties of the earth ; and 
 in 1600 announced [1*, I, 21] his opinion that bodies when re- 
 moved to a great distance from the earth would gradually lose 
 their motion downwards. The earliest proposals we find for 
 investigating whether such changes occur in the force of gravity 
 are in the works of Francis Bacon [2, Nov. Org. II, 36, and 
 Hist. Nat. I, 33]. He maintained that this force decreased 
 both inwards and outwards from the surface of the earth, and 
 suggested experiments to test his views. He would take two 
 clocks, one actuated by weights and the other by the compres- 
 sion of an iron spring, and regulate them so that they would 
 run at the same rate. The clock actuated by weights was then 
 to be placed at the top of some high steeple, and at the bottom 
 of a mine, and its rate at each place compared with that of the 
 other, which remained at the surface. There is no record .of 
 any trial of the experiment at that time. 
 
 After the founding of the Royal Society of London a stimu- 
 lus was given to experimenting upon this as upon many other 
 
 * The numbers in brackets refer to the Bibliography. 
 
 A 1 
 
ON 
 
 subjects. /&: |KiprJw,aS \e&([ 5be&>r<r that society by Dr. Power 
 [10, vol. 1, p. 133] on December 3, 1662, upon " Subterraneous 
 Experiments." A pound weight and 68 yards of thread 
 were put into one pan of a scale and counterpoised. The 
 weight was then lowered into a pit and attached by means of 
 the thread to the scale-pan held directly over the mouth of the 
 pit ; it was found to lose in weight by at least an ounce.* Three 
 weeks later Hooke [10, vol. 1, p. 163] made a report to the 
 society on some experiments he had performed at Westminster 
 Abbey. The report is worth reprinting, as giving some idea 
 of the method employed in such experiments, and of the state 
 of knowledge upon the subject at the time when Newton first 
 took it up. For it will be remembered that it was in 1665 that 
 Newton was led by his speculations on gravity to imagine that 
 since this action did not sensibly diminish with small changes 
 in height, it might perhaps extend to the moon, and be the 
 cause of that body's being retained in her orbit. Pursuing his 
 train of thought, he extended this explanation to the sun and 
 planets; and, taking into consideration Kepler's laws, it was 
 therefore necessary that the force must fall olf in the inverse 
 ratio of the square of the distance. When applying this law of 
 decrease from the earth to the moon, Newton used in deduc- 
 ing the length of the radius of the earth the rough estimate, 
 then current, of 60 miles to a degree of latitude, instead of 
 nearly 69^ ; and as a consequence the calculated motion of the 
 moon did not agree with the observed motion. He thereupon 
 laid aside for the time being any further thought upon the 
 matter. His attention was again called to the subject by a 
 letter from Hooke in 1679, and, Picard having in the meantime 
 measured the earth, Newton was able to apply the correct data 
 to the problem and to arrive at a beautiful agreement of the 
 calculated with the observed behaviour of the moon. From 
 that time date the wonderful researches which were the founda- 
 tion of the Principia. The following is Hooke's report : 
 
 "In prosecution of my Lord Verulam's experiment concern- 
 ing the decrease of gravity, the farther a body is removed be- 
 low the surface of the earth, I made trial, whether any such 
 difference in the weight of bodies could be found by their 
 
 * According to Le Sage [34], Descartes hud suggested a similar under- 
 taking twenty-five years earlier in a letter to Mcrsenne. 
 
 2 
 
THE LAWS OF GRAVITATION 
 
 nearer or farther removal from that surface upwards. To this 
 end I took a pair of exact scales and weights, and went to a 
 convenient place upon Westminster Abbey, where was a per- 
 pendicular height above the leads of a subjacent building, 
 which by measure I found threescore and eleven foot. Here 
 counterpoising a piece of iron (which weighed about 15 ounces 
 troy) and packthread enough to reach from the top to the bot- 
 tom, I found the counterpoise to be of troy- weight seventeen 
 ounces and thirty grains. Then letting down the iron by the 
 thread, till it almost touched the subjacent leads, I tried what 
 alteration there had happened to its weight, and found, that 
 the iron preponderated the former counterpoise somewhat more 
 than ten grains. Then drawing up the iron and thread with 
 all the diligence possibly I could, that it might neither get nor 
 lose any thing by touching the perpendicular wall, I found by 
 putting the iron and packthread again into its scale, that it 
 kept its last equilibrium; and therefore concluded, that it had 
 not received any sensible difference of weight from its nearness 
 to or distance from the earth. I repeated the trial in the same 
 place, but found, that it had not altered its equilibrium (as in 
 the first trial) neither at the bottom, nor after I had drawn it 
 up again ; which made me guess, that the first preponderating 
 of the scale was from the moisture of the air, or the like, that 
 had stuck to the string, and so made it heavier. In pursuance 
 of this experiment, I removed to another place of the Abbey, 
 that was just the same distance from the ground, that the for- 
 mer was from the leads; and upon repeating the trial there 
 with the former diligence, I found not any sensible alteration 
 of the equilibrium, either before or after I had drawn it up; 
 which farther confirmed me, that the first alteration proceeded 
 from some other accident, and not from the differing gravity of 
 the same body. 
 
 "I think therefore it were very desirable, from the determi- 
 nation of Dr. Power's trials, wherein he found such difference 
 of weight, that it were examined by such as have opportunity, 
 first, what difference there is in the density and pressure of 
 the air, and what of that condensation of gravity may be as- 
 cribed to the differing degrees of heat and cold at the top 
 and bottom, which may be easily tried with a common weather- 
 glass and a sealed-up thermometer ; for the thermometer will 
 shew what of the change is to be ascribed to heat and cold, 
 
 3 
 
MEMOIRS ON 
 
 and the weather-glass will shew the differing condensation. 
 Next, for the knowing, whether this alteration of gravity pro- 
 ceed from the density and gravity of the ambient air, it would 
 be requisite to make use of some very light body, extended 
 into large dimensions, such as a large globe of glass carefully 
 stopt, that no air may get in or out; for if the alteration 
 proceeded from the magnetical attraction of the parts of the 
 earth, the ball will lose but a sixteenth part of its weight (sup- 
 posing a lump of glass held the same proportion, that Dr. 
 Power found in brass); but if it proceed from the density of 
 the air, it may lose half, or perhaps more. Further, it were 
 very desireable, that the current of the air in that place were 
 observed, as Sir Robert Moray intimated the last day. Fourth- 
 ly, I think it were worth trial to counterpoise a light and heavy 
 body one against another above, and to carry down the scales 
 and them to the bottom, and observe what happens. Fifthly, 
 it were desireable, that trials were made, by the letting down of 
 other both heavier and lighter bodies, as lead, quicksilver, gold, 
 stones, wood, liquors, animal substances, and the like. Sixthly, 
 it were to be wished, that trial were made how that gravitation 
 does decrease with the descent of the body that is, by making 
 trial, how much the body grows lighter at every ten or twenty 
 foot distance. These trials, if accurately made, would afford 
 a great help to guess at the cause of this strange phaenom- 
 enon." 
 
 Dr. Power's experiment was repeated by Dr. Cotton, and an 
 account of his trials was given to the Society on June 1, 1664 
 [10, vol. 1, p. 433]. The weight was J lb., and the length of the 
 string 36 yards. A loss in weight of oz. was fonnd. 
 
 On September 1, 1664 [6, vol. 5, p. 307], we find a reference 
 to some experiments made at St. PauFs Cathedral by a com- 
 mittee of the Royal Society consisting of Sir R. Moray, Dr. 
 Wilkins, Dr. Goddard, Mr. Palmer, Mr. Hill and Mr. Hooke. 
 The results of these experiments were given to the Society on 
 September 14, 1664 [10, vol. 1, p. 466] ; the weight was 15 Ibs. 
 troy, the string about 200 ft. long, and the loss of weight 1 
 drachm. In a letter to Mr. Boyle [6, vol. 5, p. 536], dated 
 September 15th, Mr. Hooke gives more details, and remarks 
 that the balance was sensitive enough to be turned by a few 
 grains. He suggests the variation of the density of the air as 
 the cause of the loss in weight. Boyle [10, vol. 1, p. 470] pro- 
 
 4 
 
THE LAWS OF GRAVITATION 
 
 posed that Hooke's suggestion be tested by making the sus- 
 pended weight of a large glass ball loaded with mercury. 
 
 At a meeting of the Royal Society on March 14, 1665, Hooke 
 reported [10, vol. 2, p. 66, and 6, vol. 5, p. 544] that he had 
 tried Dr. Power's experiment at some wells near Epsom and 
 had found no loss in weight. Similar experiments were made 
 by Hooke at Ban stead Downs, in Surrey, and reported on 
 March 21, 1666 [10, vol. 2, p. 69, and 6, vol. 5, pp. 355 and 
 546]. The string was 330 ft. long, and the balance sensitive 
 to a grain, yet a pound shewed no change in weight when sus- 
 pended at the bottom of the well. He concludes that the 
 power of gravity cannot be magnetical, as Gilbert had sup- 
 posed. He says: "But in truth upon the consideration of the 
 nature of the theory, we may find, that supposing it true, that 
 all the constituent parts of the earth had a magnetical power, 
 the decrease of gravity would be almost a hundred times less 
 than a grain to a pound, at as great a depth as fifty fathom ; 
 for if we consider the proportion of the parts of the earth 
 placed upon one side beneath the stone, with the parts on the 
 other side above it, we may find the disproportion greater. 
 Unless we suppose the magnetism of the parts to act but at a 
 very little distance, which I think the experiments made in the 
 Abbey and St. Paul's will not allow of. If therefore there be 
 any such inequality of gravity, we must have some ways of 
 trial much more accurate than this of scales, of which I shall 
 propound two sorts," etc. It is interesting to notice that the 
 considerations upon which he makes his computations are prac- 
 tically those used by Airy in his Harton Colliery experiment. 
 
 On December 7, 1681 [10, vol. 4, p. 110], Hooke produced 
 before the society two pendulum-clocks adjusted to run at the 
 same rate. He proposed to put one at the top and the other 
 at the bottom of the monument on Pish Street Hill, and ob- 
 serve whether they would keep together. No notice of his 
 having tried the experiment has been found. This is the 
 method proposed by Bacon and used by Bouguer and many 
 others. 
 
 In 1682, Hooke read before the Royal Society "A Discourse 
 of the Nature of Comets" [4, pp. 149-191], in which he gives 
 his ideas on the subject of gravity (particularly on pages 170- 
 183). He considers gravity to be a universal principle, in- 
 herent in all matter, propagated by the same medium as that 
 
 5 
 
MEMOIRS ON THE LAWS OF GRAVITATION 
 
 by means of which light is conveyed, with unimaginable celer- 
 ity, to indefinitely great distances, and with a power varying 
 with the distance. He sums up his conceptions on gravitation 
 in nine propositions, which are of great interest, in that they 
 include many of the conceptions of Newton on this subject, 
 and yet were published four years before the Principia ap- 
 peared. 
 
PHILOSOPHIAE NATURALIS PRINCIPIA 
 MATHEMATICA 
 
 1st Edition, London, 1687. 2d Edition, Cambridge, 1713 (Cotes' Edition). 
 3d Edition, London, 1726 (Pemberton's Edition) 
 
 AND 
 
 DE MUNDI SYSTEMATE 
 
 London, 1727 
 BY 
 
 SIR ISAAC NEWTON 
 
 (Extracts taken from Davit's Edition of Motte's translation 
 3 volumes, London, 1803) 
 
CONTENTS 
 
 PAGE 
 
 On the attraction of spheres 9 
 
 Law of the distance 9 
 
 Law of the masses 13 
 
 Variation of gravity on the earth's surface 14 
 
 All attraction is mutual 15 
 
 Methods of showing the attraction between terrestrial bodies 17 
 
 Proof of its existence . . 17 
 
 Similar discussion for the case of celestial bodies 18 
 
 Final statements concerning tJie laws of gravitation 19 
 
 8 
 
THE MATHEMATICAL PRINCIPLES OF 
 NATURAL PHILOSOPHY 
 
 AND 
 
 SYSTEM OF THE WORLD 
 
 BY 
 
 SIR ISAAC NEWTON 
 
 BOOK I. PROPOSITION LXX1V. THEOREM XXXIV. 
 
 The same things supposed (if to the several points of a given 
 sphere there tend equal centripetal forces decreasing in a du- 
 plicate ratio of the distances from the points), I say, that a cor- 
 puscle situate without the, sphere is attracted with a force recip- 
 rocally proportional to the square of its distance from the 
 centre. 
 
 BOOK I. PROPOSITION LXXV. THEOREM XXXV. 
 
 If to the several points of a given sphere there tend equal cen- 
 tripetal forces decreasing in a duplicate ratio of the distances 
 from the points ; I say, that another similar sphere will be attract- 
 ed by it with a force reciprocally proportional to the square of the 
 distance of the centres. 
 
 For the attraction of every particle is reciprocally as the 
 square of its distance from the centre of the attracting sphere 
 (by prop. 74), and is therefore the same as if that whole at- 
 tracting force issued from one single corpuscle placed in the 
 centre of this sphere. But this attraction is as great as on the 
 other hand the attraction of the same corpuscle would be, if 
 that were itself attracted by the several particles of the attract- 
 ed sphere with the same force with which they are attracted by 
 
 9 
 
MEMOIRS ON 
 
 it. But that attraction of the corpuscle would be (by prop. 74) 
 reciprocally proportional to the square of its distance from the 
 centre of the sphere ; therefore the attraction of the sphere, 
 equal thereto, is also in the same ratio. Q. E. D. 
 
 Cor. 1. The attractions of spheres towards other homogene- 
 ous spheres are as the attracting spheres applied to the squares 
 of the distances of their centres from the centres of those 
 which they attract. 
 
 Cor. 2. The case is the same when the attracted sphere does 
 also attract. For the several points of the one attract the sev- 
 eral points of the other with the same force with which they 
 themselves are attracted by the others again ; and therefore 
 since in all attractions (by law 3) the attracted and attracting 
 point are both equally acted on, the force will be doubled by 
 their mutual attractions, the proportions remaining. 
 
 [Proposition LXXVI. proves the same thing for spheres made 
 np of homogeneous concentric layers.] 
 
 BOOK III. PROPOSITION V. THEOREM V. SCHOLIUM. 
 
 The force which retains the celestial bodies in their orbits 
 has been hitherto called centripetal force ; but it being now 
 made plain that it can be no other than a gravitating force, 
 we shall hereafter call it gravity. For the cause of that cen- 
 tripetal force which retains the moon in its orbit will extend 
 itself to all the planets. 
 
 BOOK III. PROPOSITION VI. THEOREM VI. 
 
 That all bodies gravitate towards every planet ; and that the 
 weights of bodies toiuards any the same planet, at equal distances 
 from the centre of the planet, are proportional to the quantities 
 of matter which they severally contain. 
 
 It has been, now of a long time, observed by others, that all 
 sorts of heavy bodies (allowance being made for the inequality 
 of retardation which they suffer from a small power of resist- 
 ance in the air) descend to the earth from equal heights in 
 equal times ; and that equality of times we. may distinguish to 
 a great accuracy, by the help of pendulums. I tried the thing 
 in gold, silver, lead, glass, sand, common salt, wood, water, 
 and wheat. I provided two wooden boxes, round and equal ; 
 
 10 
 
THE LAWS OF GRAVITATION 
 
 I filled the one with wood, and suspended an equal weight of 
 gold (as exactly as I could) in the centre of oscillation of the 
 other. The boxes hanging by equal threads of 11 feet made a 
 couple of pendulums perfectly equal in weight and figure, and 
 equally receiving the resistance of the air. And, placing the 
 one by the other, I observed them to play together forwards 
 and backwards, for a long time, with equal vibrations. . . . 
 and the like happened in the other bodies. By these experi- 
 ments, in bodies of the same weight, I could manifestly have 
 discovered a difference of matter less than the thousandth part 
 of the whole, had any such been. But, without all doubt, the 
 nature of gravity towards the planets is the same as towards 
 the earth. . . . Moreover, since the satellites of Jupiter per- 
 form their revolutions in times which observe the sesquiplicate 
 proportion of their distances from Jupiter's centre, their accel- 
 erative gravities towards Jupiter will be reciprocally as the 
 squares of their distances from Jupiter's centre that is, equal 
 at equal distances. And, therefore, these satellites, if sup- 
 posed to fall towards Jupiter from equal heights, would describe 
 equal spaces in equal times, in like manner as heavy bodies do 
 on our earth. ... If, at equal distances from the sun, any sat- 
 ellite, in proportion to the quantity of its matter, did gravitate 
 towards the sun with a force greater than Jupiter in propor- 
 tion to his, according to any given proportion, suppose of d to 
 e ; then the distance between the centres of the sun and of the 
 satellite's orbit would be always greater than the distance, be- 
 tween the centres of the sun and of Jupiter nearly in the sub- 
 duplicate of that proportion ; as by some computations I have 
 found. And if the satellite did gravitate to wards the sun 
 with a force, lesser in the proportion of e to d,, the distance of 
 the centre of the satellite's orbit from the sun would be less 
 than the distance of the centre of Jupiter from the sun in the 
 subduplicate of the same proportion. Therefore if, at equal 
 distances from the sun, the accelerative gravity of any satell- 
 ite towards the sun were greater or less than the accelerative 
 gravity of Jupiter towards the sun but by one y^Vo P arfc f the 
 whole gravity, the distance of the centre of the satellite's orbit 
 from the sun would be greater or less than the distance of Ju- 
 piter from the sun by one -g-gVo part of the whole distance 
 that is, by a fifth part of the distance of the utmost satellite 
 from the centre of Jupiter ; an eccentricity of the orbit which 
 
 11 
 
MEMOIRS ON 
 
 would be very sensible. But tbe orbits of the satellites are 
 concentric to Jupiter, and therefore the accelerative gravities 
 of Jupiter, and of all its satellites towards the sun, are equal 
 among themselves. . . . 
 
 But further ; the weights of all the parts of every planet 
 towards any other planet are one to another as the matter in 
 the several parts ; for if some parts did gravitate more, others 
 less, than for the quantity of their matter, then the whole 
 planet, according to the sort of parts with which it most 
 abounds, would gravitate more or less than in proportion to 
 the quantity of matter in the whole. Nor is it of any moment 
 whether these parts are external or internal ; for if, for exam- 
 ple, we should imagine the terrestrial bodies with us to be 
 raised up to the orb of the moon, to be there compared with 
 its body; if the weights of such bodies were to the weights of 
 the external parts of the moon as the quantities of matter in 
 the one and in the other respectively ; but to the weights of 
 the internal parts in a greater or less proportion, then likewise 
 the weights of those bodies would be to the weight of the 
 whole moon in a greater or less proportion ; against what we 
 have shewed above. 
 
 Cor. 1. Hence the weights of bodies do not depend upon 
 their forms and textures ; for if the weights could be altered 
 with the forms, they would be greater or less, according to the 
 variety of forms, in equal matter; altogether against experience. 
 
 Cor. 2. Universally, all bodies about the earth gravitate 
 towards the earth ; and the weights of all, at equal distances 
 from the earth's centre, are as the quantities of matter which 
 they severally contain. This is the quality of all bodies within 
 the reach of our experiments ; and therefore (by rule 3) to be 
 affirmed of all bodies whatsoever. . . . 
 
 Cor. 5. The power of gravity is of a different nature from the 
 power of magnetism ; for the magnetic attraction is not as the 
 matter attracted. Some bodies are attracted more by the 
 magnet ; others less ; most bodies not at all. The power of 
 magnetism in one and the same body may be increased and 
 diminished ; and is sometimes far stronger, for the quantity of 
 matter, than the power of gravity ; and in receding from the 
 magnet decreases not in the duplicate but almost in the tri- 
 plicate proportion of the distance, as nearly as I could judge 
 from some rude observations. 
 
 12 
 
THE L A W S F GRAVITATION 
 
 BOOK III. PROPOSITION VII. THEOREM VII. 
 
 That there is a power of gravity tending to all bodies, pro- 
 portioned to the^everal quantities of matter which' they contain.. 
 
 That all tbe/panets mutually gravitate one towards another, 
 we have proved before ; as well as that the force of gravity 
 towards every one of them, considered apart, is reciprocally as 
 the square of the distancfe of places from the centre of the 
 planets And thence (by prop. 69, book I, and its corollaries) 
 it follows, that the gravity tending towards all the planets is 
 proportional to the matter which they contain. 
 
 Moreover, since all the parts of any planet A gravitate to- 
 wards any other planet B ; and the gravity of every part is to 
 \\Q gravity of the whole as the matter of the part to the matter 
 tii the whole ; and (by law 3) to every action corresponds an 
 equal reaction ; therefore the planet B will, on the other hand, 
 gravitate towards all the parts of the planet A ; and its gravity 
 towards any one part will be to the gravity towards the whole 
 as the matter of the part to the matter of the whole. Q. E. D. 
 
 Cor. 1. Therefore the force of gravity towards any whole 
 planet arises from, and is compounded of, the forces of gravity 
 towards all its parts. Magnetic and electric attractions afford 
 us examples of this ; for all attraction towards the whole arises 
 from the attractions towards the several parts. The thing may 
 be easily understood in gravity, if we consider a greater planet 
 as formed of a number of lesser planets meeting together in 
 one globe ; for hence, it would appear that the force of the whole 
 must arise from the forces of the component parts. If it is 
 objected that, according to this law, all bodies with us must 
 mutually gravitate one towards another, I answer, that since 
 the gravitation towards these bodies is to the gravitation to- 
 wards the whole earth as these bodies are to the whole earth, 
 the gravitation towards them must be far less than to fall under 
 the observation of our senses. 
 
 Cor. 2. The force of gravity towards the several equal par- 
 ticles of any body is reciprocally as the square of the distance 
 of places from the particles ; as appears from cor. 3, prop. 74, 
 book I. 
 
 [ Under proposition X occurs the following important passage:] 
 However the planets have been formed while they were yet 
 in fluid masses, all the heavier matter subsided to the centre. 
 
 13 
 
MEMOIRS ON 
 
 Since, therefore, the common matter of our earth on the sur- 
 face thereof is about twice as heavy as water, and a little lower, 
 in mines, is found about three, or four, or even five times more 
 heavy, it is probable that the quantity of the whole matter of 
 the earth may be five or six times greater than if it consisted 
 all of water.* 
 
 [Under propositions XVIII. and XIX., Newton proves that 
 the axes of the planets are hss than the diameters drawn perpen- 
 dicular to the axes. He shows hoiv centrifugal force acts in 
 determining the form of the earth, and discusses the measurements 
 of terrestrial arcs known at that time; he deduces therefrom that 
 gravity will be leb'ened at the equator by ^- of itself ', and that 
 the earth will be higher at the equator than at the poles by 17.1 
 miles.] 
 
 BOOK III. PROPOSITION XX. PROBLEM IV. 
 
 To find and compare together the weights of bodies in the dif- 
 ferent regions of our earth. 
 
 Because the weights of the unequal legs of the canal of water 
 are equal ; and the weights of the parts proportional 
 to the whole legs, and alike situated 
 in them, are one to another as the 
 weights of the wholes, and therefore 
 equal betwixt themselves; the weights 
 of equal parts, and alike situated in 
 the legs, will be reciprocally as the 
 legs that is, reciprocally as 230 to 
 229. And the case is the same in 
 all homogeneous equal bodies alike 
 situated in the legs of the canal. 
 Their weights are reciprocally as the 
 legs that is, reciprocally as the dis- 
 tances of the bodies from the centre of the earth. Therefore, 
 if the bodies are situated in the uppermost parts of the canals, 
 or on the surface of the earth, their weights will be one to an- 
 other reciprocally as their distances from the centre. And, by 
 the same argument, the weights in all other places round the 
 whole surface of the earth are reciprocally as the distances of 
 
 * [ This was a wonderfully good guess on Newton's part, since tJie best of the 
 later determinations give about 5.5 for the mean specific gravity oftJie earth.} 
 
 14 
 
THE LAWS OF GRAVITATION 
 
 the places from the centre; and, therefore, in the hypothesis 
 of the earth's being a spheroid, are given in proportion. 
 
 [Newton then states that ( 'the lengths of pendulums vibrating 
 in equal times are as the forces of gravity"; he enumerates the 
 experiments on the periods of pendulums made at different parts 
 of the earth's surface, and tests his conclusions. 
 
 The folio iving remarks appear on pp. 20-25 of Motte's transla- 
 tion of the " tie Mundi Systemate" wherein Newton, after a 
 reference to his pendulum experiments, given on p. 11 of this 
 volume, says:} 
 
 Since the action of the centripetal force upon the bodies at- 
 tracted is, at equal distances, proportional to the quantities of 
 matter in those bodies, reason requires that it should be also 
 proportional to the quantity of matter in the body attracting. 
 
 For all action is mutual, and (by the third law of motion) 
 makes the bodies mutually to approach one to the other, and 
 therefore must be the same in both bodies. It is true that we 
 may consider one body as attracting, another as attracted; but 
 this distinction is more mathematical than natural. The at- 
 traction is really common of either to other, and therefore of 
 the same kind in both. 
 
 And hence it is that the attractive force is found in both. 
 The sun attracts Jupiter and the other planets; Jupiter at- 
 tracts its satellites ; and, for the same reason, the satellites act 
 as well one upon another as upon Jupiter, and all the planets 
 mutually one upon another. 
 
 And though the mutual actions of two planets may be dis- 
 tinguished and considered as two, by which each attracts the 
 other, yet, as those actions are intermediate, they do not make 
 but one operation between two terms. Two bodies may be 
 mutually attracted ea.ch to the other by the contraction of a 
 cord interposed. There is a double cause of action, to wit, the 
 disposition of both bodies, as well as a double action in so far 
 as the action is considered as upon two bodies ; but as betwixt 
 two bodies it is but one single one. It is not one action by 
 which the sun attracts Jupiter, and another by which Jupiter 
 attracts the sun; but it is one action by whicli the sun and 
 Jupiter mutually endeavour to approach each the other. By 
 the action with which the suii attracts Jupiter, Jupiter and 
 the sun endeavour to come nearer together (by the third law of 
 motion); and by the action with which Jupiter attracts the 
 
 15 
 
MEMOIRS ON 
 
 i 
 
 sun, likewise Jupiter aud the sun endeavour to come nearer to- 
 gether. But the sun is not attracted towards Jupiter by a 
 twofold action, nor Jupiter by a twofold action towards the 
 sun ; but it is one single intermediate action, by which both 
 approach nearer together. 
 
 Thus iron draws the loadstone as well as the loadstone 
 draws the iron ; for all iron in the neighbourhood of the load- 
 stone draws other iron. But the action betwixt the loadstone 
 and iron is single, and is considered as single by the philoso- 
 phers. The action of iron upon the loadstone is, indeed, the 
 action of the loadstone betwixt itself and the iron, by which 
 both endeavour to come nearer together ; and so it manifestly 
 appears, for if you remove the loadstone the whole force of the 
 iron almost ceases. 
 
 In this sense it is that we are to conceive one single action 
 to be exerted betwixt two planets, arising from the conspiring 
 natures of both; and this action standing in the same relation 
 to both, if it is proportional to the quantity of matter in the 
 one, it will be also proportional to the quantity of matter in 
 the other. 
 
 Perhaps it may be objected that, according to this phil- 
 osophy (prop. 74, book I), all bodies should mutually attract 
 one another, contrary to the evidence of experiments in ter- 
 restrial bodies ; but I answer that the experiments in terres- 
 trial bodies come to no account ; for the attraction of homo- 
 geneous spheres near their surfaces are (by prop. 72, book 
 I) as their diameters. Whence a sphere of one foot in diam- 
 eter, and of a like nature to the earth, would attract a small 
 body placed near its surface with a force 20,000,000 * times less 
 than the earth would do if placed near its surface ; but so 
 small a force could produce no sensible effect. If two such 
 spheres were distant but by one-quarter of an inch, they would 
 not, even in spaces void of resistance, come together by the 
 force of their mutual attraction in less than a month's time;f 
 
 * [If the sphere is one foot in diameter, thin number should be 40,000,000, 
 since the diameter of the earth is about 40, 000, 000 ft. But perhaps Newton 
 intended to say a sphere of one foot in radius. } 
 
 f [The time is very much less. On the assumption that each of the spheres is 
 one foot in diameter, Poynting (185, p. 10) finds the time to be about 320 sec- 
 onds. If, however, we take one foot as the radius of each sphere, Todhunter 
 (140, vol. I, p. 461) *lunox that the time is less than 250 seconds.} 
 
 16 
 
THE LAWS OF GRAVITATION 
 
 Fig.b 
 
 and less spheres will come together at a rate yet slower, viz., 
 in the proportion of their diameters. Nay, whole mountains 
 will not be sufficient to produce any sensible effect. A moun- 
 tain of an hemispherical figure, three miles high and six broad, 
 will not, by its attraction, draw the pendulum, two min- 
 utes * out of the true perpendicular; and it is only in the 
 great bodies of the planets that these forces are to be per- 
 ceived, unless we may reason about smaller bodies in manner 
 following.! 
 
 Let ABCD represent the globe of 
 the earth cut by any plane, AC, into 
 two parts, ACB and ACD. The part 
 ACB bearing upon the part ACD 
 presses it with its whole weight ; nor 
 can the part ACD sustain this press- 
 ure, and continue unmoved, if it is 
 not opposed by an equal contrary 
 pressure. And therefore the parts 
 equally press each other by their 
 weights that is, equally attract each 
 
 other, according to the third law of motion; and, if separated 
 and let go, would fall towards each other with velocities re- 
 ciprocally as the bodies. All which we may try and see in the 
 loadstone, whose attracted part does not propel the part at- 
 tracting, but is only stopped and sustained thereby. 
 
 Suppose now that ACB represents some small body on the 
 earth's surface ; then, because the mutual attractions of this 
 particle, and of the remaining part ACD of the earth towards 
 each other, are equal, but the attraction of the particle towards 
 the earth (or its weight) is as the matter of the particle (as we 
 have proved by the experiment of the pendulums), the at- 
 traction of the earth towards the particle will likewise be as 
 the matter of the particle; and therefore the attractive forces of 
 all terrestrial bodies will be as their several quantities of matter. 
 
 The forces (prop. 71, book I), which are as the matter in 
 
 * [Maskelyne (31) says with reference to this : "It will appear, by a very easy 
 calculation, that such a mountain would attract the plumb-line 1' 18" from the 
 perpendicular."] 
 
 \ [ This paragraph is of great importance, because in it Newton indicates 
 the methods of all ilie experiments yet made in order to measure gravitational 
 attraction in terrestrial bodies. ] 
 
 B 17 
 
MEMOIRS ON 
 
 terrestrial bodies of all forms, and therefore are not mutable 
 with the forms, must be found in all sorts of bodies whatsoever, 
 celestial as well as terrestrial, and t>e in all proportional to their 
 quantities of matter, because among all there is no difference 
 of substance, but of modes and forms only. But in celestial 
 bodies the same thing is likewise proved thus. We have shewn 
 that the action of the circumsolar force upon all the planets 
 (reduced to equal distances) is as the matter of the planets ; 
 that the action of the circumjovial force upon the satellites of 
 Jupiter observes the same law; and the same thing is to be said 
 of all the planets towards every planet; but thence it follows 
 (by prop. 69, book I) that their attractive forces are as their 
 several quantities of matter. 
 
 As the parts of the earth mutually attract one another, so 
 do those of all the planets. If Jupiter and its satellites were 
 brought together, and formed into one globe, without doubt 
 they would continue mutually to attract one another as before. 
 And, on the other hand, if the body of Jupiter was broken into 
 more globes, to be sure, these would no less attract one another 
 than they do the satellites now. From these attractions it is 
 that the bodies of the earth and all the planets effect a spheri- 
 cal figure, and their parts cohere, and are not dispersed through 
 the aether. But we have before proved that these forces arise 
 from the universal nature of matter (prop. 72, book I), and 
 that, therefore, the force of any whole globe is made up of the 
 several forces of all its parts. And from thence it follows (by 
 cor. 3, prop. 74) that the force of every particle decreases in 
 the duplicate proportion of the distance from that particle; 
 and (by prop. 73 and 75, book I) that the force of an entire 
 globe, reckoning from the surface outwards, decreases in the 
 duplicate, but, reckoning inwards, in the simple proportion of 
 the distances from the centres, if the matter of the globe be 
 uniform. And though the matter of the globe, reckoning from 
 the centre towards the surface, is not uniform (prop. 73, book 
 I), yet the decrease in the duplicate proportion of the distance 
 outwards would (by prop. 76, book I) take place, provided that 
 difformity is similar in places round about at equal distances 
 from the centre. And two such globes wil 1 (by the same prop- 
 osition) attract one the other with a force decreasing in the 
 duplicate proportion of the distance between their centres. 
 
 Wherefore the absolute force of every globe is as the quan- 
 
 18 
 
THE LAWS OF GRAVITATION 
 
 tity of matter which the globe contains ; but the motive force 
 by which every globe is attracted towards another, and which, 
 in terrestrial bodies, we commonly call their weight, is as the 
 content under the quantities of matter in both globes applied 
 to the square of the distance between their centres (by cor. 4, 
 prop. 76, book I), to which force the quantity of motion, by 
 which each globe in a given time will be carried towards the 
 other, is proportional. And the accelerative force, by which 
 every globe according to its quantity of matter is attracted 
 towards another, is as the quantity of matter in that other globe 
 applied to the square of the distance between the centres of 
 the two (by cor. 2, prop. 76, book I) ; to which force the ve- 
 locity by which the attracted globe will, in a given time, be 
 carried towards the other is proportional. And from these 
 principles well understood, it will be now easy to determine 
 the motions of the celestial bodies among themselves. 
 
 SIR ISAAC NEWTON was born at Woolsthorpe, near Grant- 
 ham, in Lincolnshire, in 1642. He was educated at the Grant- 
 ham grammar-school, entered Trinity College, Cambridge, in 
 1661, and received his degree four years later. He at once 
 began to make those magnificent discoveries in mathematics 
 and physics which have made his name immortal. In 1665 he 
 committed to writing his first discovery on fluxions, and shortly 
 afterward made the unsuccessful attempt, to which we have 
 already referred, to explain lunar and planetary motions. He 
 next turned his attention to the subject of optics ; his work in 
 that field includes the discovery of the unequal refrangibility 
 of differently coloured lights, the compositeness of white light 
 and chromatic aberration. Having erroneously concluded that 
 this aberration could not be rectified by a combination of lenses, 
 he turned his attention to reflectors for telescopes and made a 
 great advance in that direction. His name is also closely iden- 
 tified with the colours due to thin plates. From 1669 to 1701 
 he was Lucasian professor of mathematics at Cambridge. He 
 was elected to membership in the Royal Society in 1671, and 
 from 1703 until his death was its president ; he became a mem- 
 ber of the Paris Academy in 1699. The publication of his work 
 on Optics had caused some controversy, and such a lover of 
 peace was Newton, and so little did he care for the praise of 
 
 19 
 
MEMOIRS ON THE LAWS OF GRAVITATION 
 
 the world, that it was only at the earnest solicitation of Hal ley 
 that he was willing to give to the public the results of his won- 
 derful researches on central orbits, and universal gravitation ; 
 these included an explanation of the lunar inequalities, the 
 figure of the earth, the precession of the equinoxes and the 
 tides, and a method of comparing the masses of the heavenly 
 bodies. In 1669 he became a member of Parliament, in 1696 
 Warden of the Mint, and from 1699 until his death was Master 
 of the Mint. He gave much valuable aid in the recoinage of 
 the money and in questions of finance at this period. He was 
 knighted in 1705. During the latter years of his life much of 
 his time was devoted to his public duties. He died in 1727, 
 and was buried in Westminster Abbey. 
 
 20 
 
LA FIGURE DE LA TERRE 
 
 Determines par les Observations cle Messieurs Bouguer, et 
 de la Condamine, de 1'Academie Royale des Sciences, envoyes 
 par ordre du Roy au Perou, pour observer aux environs de 
 1'fCquateur. 
 
 Avec une Relation abregee de ce Voyage, qui contient la 
 description du Pays dans lequel les operations ont ete faites. 
 
 PAR M. BOUGUER 
 A Paris, 1749 
 
 Section 7, pp. 327-394 
 
 THE FIGURE OF THE EARTH 
 
 Determined by the observations of MM. Bouguer and de la 
 Condamine, of the Royal Academy of Sciences, sent to Peru 
 by order of the King to make observations near the equator. 
 
 With a brief account of their travels and a description of the 
 country in which the investigations were made. 
 
 BY PIERRE BOUGUER 
 Paris, 1749 
 
 Section 7 pp. 327-394 
 
 21 
 
CONTENTS OF SECTION VII 
 
 PAGE 
 
 Introduction 23 
 
 Chap. I. Experiments Made in Order to find the Length of the Seconds- 
 Pendulum 24 
 
 Description of Pendulum 24 
 
 Method of Observation (Omitted). 
 
 Observed Lengths of Seconds- Pendulum at Various Places 25 
 
 Corrections to be Made in the Observed Lengths 25 
 
 Corrected Lengths of Seconds-Pendulum at Various Places 27 
 
 Chap. II. Comparison of Attraction and Centrifugal Force (Omitted). 
 Chap. III. Remarks on the Diminution in Attraction at Different 
 
 Heights above Sea-level 27 
 
 Calculation of the Attraction Due to a Plateau 29 
 
 Deduction of the Mean Density of the Earth from Pend- 
 ulum Experiments. 32 
 
 Chap. IV. On the Deflection of the Plumb-line by a Mountain ... 33 
 
 Description of Mount Chimborazo 34 
 
 Its Deflection of the Plumb-line Calculated from the Theory. . 34 
 
 Various Ways Suggested for Showing the Deflection 35 
 
 Description of the Method Employed 38 
 
 Examination of the Attraction of Chimborazo 39 
 
 Meridian Altitudes at the First Station 40 
 
 Measurements Made to find the Relative Positions of the two 
 
 Stations 41 
 
 Meridian Altitudes at the Second Station (Omitted). 
 
 Corrected Meridian Altitudes at the Second Station 42 
 
 Calculations for the Observed Deflection of the Plumb-line. . . 42 
 
 Its Poor Agreement with that Calculated from Theory 43 
 
 Appendix (Omitted). 
 
SECTION VII. OP BOUGUER'S FIGURE 
 OF THE EARTH 
 
 ACCOUNT OF THE EXPERIMENTS OR OBSERVATIONS ON GRAV- 
 ITATION, WITH REMARKS ON THE CAUSES OF 
 THE FIGURE OF THE EARTH 
 
 1. HAVING discussed everything that bears on the earth con- 
 sidered as a geometrical body, it remains for us, before terminat- 
 ing this work, to verify the facts which give us some slight 
 knowledge of the interior conformation of this great mass con- 
 sidered as a physical body. . . . 
 
 2. The first question which presents itself on this matter is 
 a consideration of the part played in the flattening of the earth 
 by the attraction which compresses it from all sides, urging all 
 masses towards certain points. We know, since M. Eicher first 
 remarked it (in 1672 in Cayenne), that this force is not every- 
 where the same. It is greater towards the poles, and less to- 
 wards the equator. This agrees perfectly with the figure of the 
 earth, which appears to have yielded a little to the great press- 
 ure at the poles, and to be slightly elevated, on the contrary, 
 at the equator, where the compressing force was more feeble. 
 But does the effect correspond exactly to the cause upon which 
 we desire it to depend? Is the difference in attraction so great 
 that we can attribute to it all the inequality which exists, as 
 we have seen, between the two diameters of our globe ? To 
 answer this question it is necessary to determine, by exact ex- 
 periment, how much the attraction actually differs in different 
 parts of the earth. . . . We have two methods for observing 
 the change in attraction as we pass from one region to another ; 
 we have only to examine how much more quickly or more 
 slowly a pendulum of given length oscillates ; or else to find 
 the length of the pendulum whose time of vibration is exactly 
 
 23 
 
MEMOIRS ON 
 
 a second ; the differences which we shall find in the length of 
 this pendulum will determine the changes of the attraction as 
 we go from one region to another. 
 
 ACCOUNT OF THE EXPERIMENTS MADE FOR THE PURPOSE OF 
 DETERMINING THE LENGTH OF THE SECONDS-PENDULUM 
 
 3. My first experiments with the pendulum were made at- 
 Petit-Goave in the island of St. Domingue. They are reported 
 in the memoirs of the Academy for 1735 and 1736. . . . 
 
 4. The instrument which I almost always used, and which I 
 still use, is extremely simple. I make the pendulum always 
 exactly of the same length, and I compare its oscillations with 
 those of a clock which I regulate by daily observations. It is 
 not, properly speaking, by the different lengths of the pend- 
 ulum that I judge of the intensity of gravitation at different 
 places ; I judge of it only by the greater or less rapidity of the 
 oscillations, or by the number of oscillations made by the pend- 
 ulum in 24 hours. ... It appears to me to be much easier to 
 count the number of oscillations than to measure directly dif- 
 ferences of a few hundredths of a line* in the length of the 
 pendulum. 
 
 [Then follows an account of his pendulum. The bob was of 
 copper, composed of tzvo equal truncated cones joined at their 
 greater bases. The thread ivas a fibre of aloe, which is not af- 
 fected by the weather. The length was maintained constant by 
 having it always so that an iron rule just fitted in between the 
 clamp and the bob. The length of the equivalent simple pendulum 
 was 36 pouces, 7.015 lines. 
 
 Bouguer gives a description of a scale fixed behind the pend- 
 ulum, by means of which he could observe the decrement and the 
 time required by the pendulum to gain an oscillation on the 
 clock. ] 
 
 10. It is time to relate the experiments. ... I shall choose 
 one of those which I made on the rocky summit of Pichincha 
 [2434 toises above sea-level], in the month of August, 1737. The 
 
 * [72 ponces \ toise = 1.949 metres = 6.3945 ft, 12 lines = 1 pouce ] 
 
 24 
 
THE LAWS OF GRAVITATION 
 
 force of attraction Was feeble, not only because we were nearly 
 over the equator at this place, but also because we were at a 
 very great height above the surface of the sea. . . . 
 [Details of experiment.] 
 
 12. . . . We find in this way that the pendulum which beats 
 seconds at the equator, and in the highest accessible place on 
 the earth, is 36 polices 6.69 lines in length. I made other ex- 
 periments at the same place which agreed as exactly as possible 
 with this result. [One made by Don Antonio de Ulloa gave 36 
 pouces, 6.715 lines. We may take as the mean 36 pouces,^.7Q 
 lines. ] 
 
 13. I have found by the same proceedings and with the aid 
 of the same instruments, the length of the seconds-pendulum 
 at Quito [1466 toises above sea-level], to be 36 ponces, 6.82 or 
 6.83 lines. I have verified it at different times and in all sea- 
 sons of the year: at times of aphelion and perihelion, at the 
 equinoxes, and when the sun was at intermediate points; the 
 extreme results were 36 pouces, 6.79 lines and 6.85 lines, with 
 no differences which could not be attributed to the inevitable 
 errors of observation. . . . 
 
 [ The question of a possible yearly change is discussed. 
 
 Experiments were made with the same apparatus, in 1740,^ 
 I' Isle de I'Inca, 14' or 15' from the equator, and scarcely 40 toises 
 above sea-level. Bouguer regards this determination as that of 
 the true equinoctial pendulum.] 
 15. 
 
 Place 
 
 Length found by experiment 
 
 i 2434 toises absolute height. 
 Under the equator at ! 1466 " " 
 / Sea-level 
 
 36 pouce, 6.70 lines. 
 6.83 " 
 707 " 
 
 At Portobello 9 34' N latitude 
 
 7 16 " 
 
 At Petit-Goave, 18 27' " " 
 At Paris 
 
 7.33 " 
 
 8.58 ' 
 
 CORRECTIONS WHICH MUST BE APPLIED TO THE LENGTH OF 
 
 THE PENDULUM AS DETERMINED DIRECTLY FROM 
 
 THE EXPERIMENTS. 
 
 16. [Bouguer remarks that these corrections arise from changes 
 in tem/oerature and in the constitution of the atmosphere.'] The 
 
 35 
 
MEMOIRS ON 
 
 first cause does riot really change the length, it only makes it 
 appear different according as the measures we use are differ- 
 ently altered by heat or cold; but the other cause brings in a 
 real inequality,, since it produces nearly the same effect as if 
 the weight were greater or smaller. . . . 
 
 17. ... Since the temperature of Quito does not differ from 
 that of Paris in the middle of spring, we have only to refer all 
 our results to it. That is, without altering the lengths of the 
 pendulum found in these two cities, we have only to correct all 
 the others by increasing or diminishing them, according as the 
 metal rules we used were expanded by the heat or contracted 
 by the cold. [He concludes from his experiments that a change 
 of length of pendulum of .02 lines corresponds to a change of 
 temperature of 3R. Hence he had to add .075 lines to the length 
 found at sea-level, and subtract .05 lines from that found at 
 Pichincha.] 
 
 18. There is little more difficulty in finding the alteration in 
 the length of the pendulum caused by the medium in which 
 the experiments are- made. This medium, whether rare or dense, 
 has a certain weight, and that of the small mass of copper, of 
 which the bob of the pendulum is formed, is a little lessened 
 by it. The small mass tends to fall to the earth with only the 
 excess of its weight above that ot the air which surrounds it. 
 Thus our pendulums are acted on by a force a little less than if 
 we had performed the experiments in vacuo : and the length 
 of the seconds-pendulum, which we found directly from experi- 
 ment, is a little too short in the same proportion. 
 
 19. The use of the barometer enables us to find the ratio be- 
 tween the weight of mercury and of air in all the parts of the 
 atmosphere which are accessible. We observe how many feet 
 it is necessary to ascend or descend in order to change the 
 height of the mercury by a line. ... I have found in this way 
 that it was only necessary to express the first (the weight of 
 air) by unity, at the summit of Pichincha, if one expressed that 
 of copper by 11000. ... So I always found the seconds-pend- 
 ulum too small by y^^th part. To correct for this error 
 we must add .04 lines [at Pichincha ; .05 at Quito ; .06 at sea- 
 level.] . . . This is the first time that any one has taken ac- 
 count of this small correction which enters into the experi- 
 ments, but we cannot neglect it if we wish to attain the greatest 
 accuracy. . . . 
 
 26 
 
THE LAWS OF GRAVITATION 
 
 [Bouguer then proves that the time of vibration is not appreci- 
 ably affected by the resistance of the air.]* 
 
 22. Corrected lengths of the seconds-pendulum, or such as 
 they would be if the oscillations were made in vacuo. 
 
 Place. 
 
 Under the equator at ] 1466 
 
 ( Sea-level . 
 At Portobello, 9 34' N. latitude 
 At Petit-Goave, 18 27' " " . 
 At Paris . 
 
 2434 toises absolute height. 
 
 pouces,6.69 lines. 
 " 6.88 " 
 " 7.21 " 
 " 7.30 " 
 
 7.47 " 
 " 8.67 " 
 
 IT 
 
 COMPARISON OF ATTRACTION AND THE CENTRIFUGAL FORCE 
 
 WHICH BODIES ACQUIRE BY THE MOTION OF THE EARTH 
 
 ABOUT ITS AXIS, WITH REMARKS ON THE EFFECTS 
 
 OF THESE TWO FORCES. 
 
 [Bouguer finds that the primitive attraction (that is, the attrac- 
 tion the earth would have if it were at rest) is to the centrifugal 
 force as 288-JJ : 1. He gives a table showing the decrease in the 
 length of the seconds-pendulum at various latitudes, due to the 
 centrifugal force. The following headings will give an idea of 
 the matter contained in the rest of this chapter.] 
 
 The centrifugal force produced by the motion of the earth 
 about its axis is not sufficient to produce the observed differ- 
 ences in weight. 
 
 The primitive attraction does not tend towards a common 
 point as centre. 
 
 Ill 
 
 REMARKS ON THE DIMINUTION IN THE ATTRACTION AT DIF- 
 FERENT HEIGHTS ABOVE THE LEVEL OF THE SEA. 
 
 40. The experiments with the pendulum which we have 
 made at Quito and on the summit of Pichincha teach us that 
 
 * [See note on page 66] 
 
 27 
 
MEMOIRS ON 
 
 the attraction changes with the distance from the centre of the 
 earth. This force goes on diminishing as we ascend; I have 
 found the pendulum at Quito to be shorter than at sea-level by 
 .33 lines, or the y-gVr^ 1 P 111 ^ : Jin ^ in mounting to the summit of 
 Pichincha the pendulum is shortened again by .19 lines, and is 
 8TT tn P art shorter than at sea-level.* One cannot attribute 
 these differences to the centrifugal force, which, being greater 
 the higher we ascend, ought to diminish a little further the 
 primitive attraction. The centrifugal force is increased by the 
 height of the mountain by the -j-^j-yth part only, and as it is 
 itself but the ^-J^th part of the weight, it is clear that its new 
 increase corresponds to .001 lines only in the length of the 
 pendulum, and so does not sensibly contribute to the dimi- 
 nution of the other force. 
 
 41. If we compare the shortening which the pendulum re- 
 ceives with the height at which the experiment was made, we 
 see that the forces do not decrease in the simple inverse ratio 
 of the distances from the centre of the earth, but that they 
 follow rather the proportion of the square. Quito is 146(5 
 toises above sea-level, or 7^7 th of the radius of the earth ; but 
 it has been found that the attraction is less by a fraction much 
 more considerable namely, by a TTsr^h part, which is nearly 
 double ; this is not very far from the inverse ratio of the 
 square of the distance. . . . We have a second example in the 
 experiment made on Pichincha. The absolute height of this 
 mountain, which is 2434 toises above sea-level, is j^Vs^ 1 of the 
 radius of the earth. The diminution of the length of the pend- 
 ulum, or of the attraction, ought then to be the grjth part, 
 if it is to be in the inverse ratio of the square of the distance ; 
 but it was by no means so great in fact, only the g-J-^th part. 
 
 42. This diminution in attraction, as we go above sea-level, is 
 quite in conformity with what we otherwise know. We can 
 compare with the attraction here experimented upon that 
 which keeps the moon in its orbit, or which obliges it con- 
 tinually to perform a circle about us. These two forces aro 
 exactly in the inverse ratio of the squares of the distances 
 from the centre of the earth. We can make the same ex- 
 
 * [Pendulum observations were made at these and other places in Peru by dr 
 la Condamine also (8, pp. 70, 144, 162-169). For a complete biblior/raphy of 
 pendulum experiments, see that published by La Societe Fran$aise de Physique 
 (178, vol. 4).] 
 
 28 
 
THE LAWS OF GRAVITATION 
 
 amination with respect to the principal planets which have 
 several satellites, or with respect to" the sun, towards which 
 all the principal planets are attracted, and we shall always 
 find the law of the square. Why, then, do our experiments 
 constantly give a law not entirely in agreement with this? 
 Is it necessary to attribute the difference to some error on 
 our part ; or can it be that in the neighborhood of great masses 
 like the earth the law under consideration is observed in an 
 imperfect manner only ? 
 
 43. We shall find ourselves in a position to solve this diffi- 
 culty, perhaps, by remarking that the Cordilleras, on which we 
 were placed, form a kind of plateau, or, what in certain ways 
 amounts to the same thing, the surface of the earth is there 
 carried to a greater height or to a greater distance from the 
 centre. There is reason for believing that in this second 
 case the attraction would be a little greater ; for it is natural 
 to think that it depends upon the size of the attracting mass. 
 There are then two things to be considered in the case of 
 the experiments on the pendulum which I have reported. 
 These experiments were made 
 
 at a great height above the av- 
 erage surface of the earth, and 
 therefore the attraction ought 
 to be found a little less. But, 
 on the other hand, the group 
 of mountains on which Quito is 
 placed and on which Pichincha 
 rises, and all the other sum- 
 mits to which it acts as a 
 plinth, ought to produce nearly 
 the same effect as if the earth 
 at this place were larger or had 
 a greater radius. The attrac- 
 tion on this account ought to increase. Thus it depends on 
 a kind of chance, or, to speak more philosophically, it de- 
 pends on circumstances which we do not yet know, whether 
 the attraction at Quito will be equal to that at sea-level, or be 
 smaller or larger. 
 
 44. Suppose that the circle ADD represents the circum- 
 ference of the earth, of which C is the centre, and that Aa 
 is the amount by which Quito,, situated at a, is elevated 
 
 29 
 
 Fig. c. 
 
MEMOIRS ON 
 
 above sea -level. Imagine a new spherical shell of terrestrial 
 matter, occupying all the interval between the two concen- 
 tric surfaces ADD and add; or, which comes to the same 
 thing, imagine that the earth increases in radius, and that 
 Quito, without changing its position, remains at the level 
 of the sea, now supposed much higher. There is every reason 
 to think that the attraction at Quito would, as a consequence, 
 be found greater than it actually is at A or at D, in the 
 ratio of CA to Ca. It is necessary for that, however, to sup- 
 pose that the layer of earth enclosed between the two con- 
 centric surfaces is of the same density as all the rest ; for if 
 the density were different the increase would no longer be in 
 the same ratio. 
 
 45. Call r the radius, and A the density of the earth. Then 
 rA is the attraction at all the points A, D, etc., supposing 
 that the earth ends there. Call h the height Aa, which is 
 very small compared with r. Then the attraction at a is 
 less than at A, in the ratio of r 2 : (r-f 7*) 2 , or its diminution 
 will be as 2h:r; that is, if the attraction is rA at A, it is 
 (r 2A)A at a, and this supposes that the earth has CA only 
 for effective radius. But all this will be subject to change 
 if we add to our globe the layer Ac?D, whose density is 5. 
 This new spherical layer, if it had the same density as the 
 rest, would augment the attraction at the surface in the same 
 ratio as the radius of the earth became greater. The increase 
 would be in the ratio of r : r + U. 
 
 46. Thus the added layer would not only make up for the 
 decrease which the attraction actually suffers when we go 
 away from the earth, in rising by the height Ka h, but would 
 add a new amount to it, equal to half the diminution, since 
 it would make this attraction, which is actually r 2h at the 
 point , become r + Ji. It follows that the attraction which 
 the spherical layer can produce at its exterior surface at a 
 is expressed by 3A, or three times its thickness; but we must 
 multiply by the density S, because we suppose that the den- 
 sity of the layer and that of the earth as a whole are not 
 equal. 
 
 47. To recapitulate: When the earth has its radins, CA=:r, 
 the attraction at A is rA, and at the height h is (r 2&)A. 
 But when we add to the earth the spherical layer Ae?D, the 
 attraction at a becomes (r 2 
 
 30 
 
THE LAWS OF GRAVITATION 
 
 48. All that remains now to be remarked is that the Cordill- 
 eras of Peru, however great they may be, ought not to produce 
 the same effect as the spherical shell which we have assumed. 
 If the base EE of the Cordilleras were exactly double its 
 height, and this mass had the shape of the roof of a house of 
 indefinite length, then the Cordilleras would produce at a only 
 \ the effect of the entire spherical shell, as can be easily proved. 
 But there are further additions to be made in order to give a 
 more accurate idea of the Cordilleras of Peru. The base EE is 
 80 or 100 times greater than the height Aa, which augments 
 the effect in precisely the same ratio as the angle at a is 
 greater. This angle is only 90 when we find the effect ^ of 
 that which the whole spherical layer would produce, but on 
 account of the great width of the base of the Cordilleras the 
 angle is nearer 170, which doubles the effect. Moreover, the 
 Cordilleras do not terminate at the height of Quito in a single 
 summit like the ridge of a house ; it is, on the contrary, quite 
 10 or 12 leagues broad there. One can suppose then, without 
 fear of mistake, that the effect is the greatest which can be 
 produced by a chain of mountains. It is the J of that which a 
 spherical layer would produce, or f//3, and if we add to it the 
 attraction (r 2/z)A, which the globe ADD produces at a, we 
 shall have ( r 2A)A -f f/^* as the expression for the attraction 
 at Quito, when rA expresses that at sea-level. 
 
 49. The difference between the two is 2AA fAS, which 
 furnishes the subject of divers quite curious remarks. If the 
 matter of the Cordilleras were more compact than that of the 
 average of the whole earth, and their densities were as 4 : 3, 
 the difference 27iA |7i<J would become zero, and the attrac- 
 tion at Quito would be the same as at sea-level. If the density 
 B were still greater, our expression for the diminution would 
 change sign and become an increase, so that the pendulum 
 would be longer at Quito than at sea-level. But it is evident 
 
 * [This formula is independently foundby D' Alembert (\Z, vol. 6, pj9. 85-92), 
 by Young (51 and 95, vol. 2, p. 27), and by Poixson (65, vol. 1, pp. 492-6). 
 
 Under the form g*= g (l - | 7i + |^ it is known as "Dr. Young'- 
 
 Rule" where g } is the value of gravity at height h. and g is the value at the 
 sea-level. Faye (147) contends that the last term of the equation sliould be left 
 out ; and if Airy' 8 "flotation theory" (94), or Faye's compensation theory 
 (146-1), be true, there is no doubt thai this term requires correction.] 
 
 31 
 
MEMOIRS ON 
 
 that things are riot so. The difference in the length of the pend- 
 ulum is sufficiently great to let us see that the density of the 
 matter of which the Cordilleras is formed is much smaller than 
 that of the rest of the globe. 
 
 50. We have found by experiment a diminution of a -j-g^th 
 part in the length of the pendulum, or in the attraction, as we 
 go from the sea-level to Quito. So T ^ T corresponds to 27*A 
 %M, as compared with rA, which expresses the attraction at 
 
 sea-level ; that is, we have L- = a/ * A ~ % M . If we put - = 
 
 Idol rA r 
 
 which is the ratio of the height of Quito to the radius 
 
 of the earth, we shall have = -J x ?^ 1? Whence 
 
 lool <iZoi A 
 
 850 
 we deduce 3 = A, which tells us that the Cordilleras of 
 
 OiJtJ'J 
 
 Peru, in spite of all the minerals they contain, have less than J 
 the density of the interior of the earth.* 
 
 51. We admit that this determination may contain a few er- 
 rors on account of the large number of elements we had to em- 
 ploy in order to arrive at it. Nevertheless, if we once admit 
 that the attraction, when the other circumstances are the same, 
 follows exactly the direct ratio of the masses, we cannot doubt 
 that the Cordilleras of Peru have a density considerably less 
 than that of the rest of the globe. If we suppose A and 3 equal, 
 our expression for the difference of the attractions at Quito and 
 
 at sea-level would become -^- A ; which would make the differ- 
 
 2r 
 
 ence between the lengths of the pendulum 4 times too small, or 
 the attractions as the square roots, instead of the squares, 'of 
 the distances from the centre of the earth. The attraction at 
 Quito would be less than at sea-level by only the j-^Tj'th part, 
 and the pendulum would be really shorter by only 9 or 10 hun- 
 dred ths of a line, and in appearance by 2 or 3, on account of the 
 different constitution and temperature of the air. The differ- 
 ence of the lengths of the pendulum is certainly greater. Thus 
 it is necessary to admit that the earth is much more compact 
 
 * [To give Bouguer's result more accurately, the density of the earth is 4.7 
 times that of the Cordilleras. Saigey (74, p. 149) has made a recalculation of 
 these results, with the proper reduction to nacuo, and finds 4.25. He has done 
 the same for de la Condamine's pendultim experiments, with a result 4.50. 
 For Addendum, see p. 160.] 
 
THE LAWS OF GRAVITATION 
 
 below than above, and in the interior than at the surface. For 
 the soil of Quito is like that of all other countries ; it is a mixt- 
 ure of earth and stones, with some metallic constituents. . . . 
 Those physicists who imagined a great void in the middle of 
 the earth, and who would have us walk on a kind of very thin 
 crust, can think so no longer. We can make nearly the same 
 objection toWoodward's theory of great masses of water in the 
 interior. Bat let us continue to limit ourselves to the facts, or 
 to the only immediate deductions which we can draw from them. 
 These deductions are confirmed by the observations described 
 in the next chapter, which is in the form I gave it in Peru be- 
 fore forwarding it to France. 
 
 IV 
 
 MEMOIR ON ATTRACTION AND ON THE MANNER OF OBSERV- 
 ING WHETHER MOUNTAINS EXHIBIT IT (READ AT THE 
 ACADEMIE DES SCIENCES, IN OCTOBER, 1739) 
 
 52. It is very difficult not to accept attraction as a principle 
 of fact or of experience. The most rigid Cartesians, like all 
 other philosophers, cannot dispense with it in this sense. All 
 they can do is to reserve to themselves the right of explaining 
 it. ... Since all the planets circle about the sun, there must 
 necessarily be a force, I shall not say shoving them or drawing 
 them, but rather transporting them at each moment towards 
 this star. . . . Nothing prevents us from giving to this force 
 the name " attraction," and from trying to assign to it a phys- 
 ical cause. . . . 
 
 [Bouguer affirms that in establishing a new principle, it is not 
 only necessary to prove the insufficiency of all others, but their 
 impossibility also. ] 
 
 54. While waiting for all this to happen, it will contribute 
 to the perfection of physics if we examine more carefully into 
 attraction as a fact taught by experience. ... It appeared 
 to me that if all bodies act "at a distance," in proportion 
 to their mass, and according to the other laws which we know, 
 such enormous masses [the mountains of Peru] should pro- 
 duce a marked effect. I am well aware that they are very 
 small compared with the whole earth ; but one can approach 
 1000 or 2000 times nearer their centre, and if it is true that 
 c 33 
 
MEMOIRS ON 
 
 the attractions increase not only simply in the same ratio as 
 the distances diminish, but in the inverse ratio of their squares, 
 one ought to have a kind of compensation. 
 
 55. I shall content myself with justifying this in the case 
 of a single mountain called Chimborazo, the base of which 
 one is obliged to pass in going from the sea-side at Guaya- 
 quil to the more inhabited part of the province of Quito, 
 which is enclosed between the two chains of mountains here 
 formed by the Cordilleras, whose distance apart is 8 or 9 
 leagues. Chimborazo must be 3100 or 3200 toises above the 
 sea-level [he afterwards found it to be 3217 toises], and 1700 
 or 1800 above the level of the plateau. We know exactly the 
 relative heights of all the mountains we have seen, but not 
 having yet been able to compare any one with sea-level, we 
 are ignorant of their absolute heights. Chimborazo has ro.ots 
 which extend very far and become merged in those of the 
 other mountains, so that it is very difficult to determine the 
 true extent of its base. It must be more than 10,000 or 12,000 
 toises in diameter. But when we mount as high as possible, 
 to where the snow begins, which is 850 toises from the top 
 and renders the higher parts inaccessible, the mountain is still 
 more than 3500 toises in diameter. The top, instead of ter- 
 minating in a point, is rounded and blunt, and appears from 
 below to have a width of 300 or 400 toises. From these di- 
 mensions one can estimate its huge mass. In the present 
 investigation we need to know its height above ground only, 
 not above sea-level. Even so it must be 20,000,000,000 cubic 
 toises in volume. This is about the y;oir;iro"oroooth P ar t OI1 ^J 
 of the globe, and the effect of the attraction would be absol- 
 utely insensible, if one considered the quantity of matter 
 only. But as we can place ourselves at 1700 or 1800 toises 
 from the centre of gravity of the mountain, or 1900 times 
 nearer to it than to the centre of the earth, this proximity 
 ought to increase the effect about 3,600,000 times, and so 
 make it about 2000 times less than that which gravitation 
 produces, or the attraction caused by the whole mass of the 
 earth. This we get by employing only a rough calculation 
 and the lowest estimates. Calling the action of the moun- 
 tain 1, and that of the earth 2000, the direction of attraction 
 should be deflected from the vertical by about 1' 43". A 
 plumb-line which would be directed exactly to the centre of 
 
 34 
 
THE LAWS OF GRAVITATION 
 
 the earth, if its mass were exposed to the earth's attraction 
 a!6ne, ought then, on account of the action of the mountain, 
 to be inclined by this same quantity, which is, as we see, quite 
 considerable. 
 
 56. But how can we recognize this inclination; for all 
 gravitating bodies must be equally subject to it, and we 
 seem to lack a term of comparison? It would be useless to 
 have recourse to the level surfaces of the heaviest liquids, 
 since the attraction being equally altered with respect to 
 them, their surfaces, instead of being perfectly horizontal, 
 must suffer the same inclination. We see plainly, then, that, 
 in order to judge of the amount of this alteration, it will be 
 of no use to look just about us, we must seek another ver- 
 tical line far off which is subject to no action from the 
 mountain. But again, how are we to compare one vertical 
 with another ; or measure the angle which they make in 
 meeting towards the centre of the earth, and that with suf- 
 ficient accuracy? If while on the mountain, we observe with 
 the quadrant the height of a point far off, and then go to 
 that point and measure the height of the former place, it is 
 true that by the difference of these two heights we can judge 
 of the relative positions of the two vertical lines. But be- 
 sides that we must know the exact distance from one to the 
 other, it will be necessary also to suppose that the visual ray 
 is a straight line ; and it is not only certain that this is not 
 true, we know that it is subject by refraction to a very ir- 
 regular curvature. We cannot determine this curvature with 
 sufficient exactness to enable us to find the effect of the at- 
 traction. It seems to me, therefore, that we must seek in 
 the heavens a term of comparison. By this means, however, 
 we shall easily overcome every difficulty; and what a moment 
 ago seemed an impossibility becomes at once very simple. 
 
 57. We have but to station ourselves to the north or to 
 the south of a mountain, and as near as possible to its centre 
 of gravity, and observe the latitude. This observation can be 
 made with the greatest accuracy only by using a quadrant 
 or other equivalent instrument whose plumb-line will be de- 
 flected toward the mountain; this is the same as saying that 
 the zenith will recede from the mountain. Then we must 
 go east or west of this station to such, a distance that the at- 
 traction is negligible; and if we observe the latitude in this 
 
 35 
 
MEMOIRS ON 
 
 second place with the same care and with the same means 
 as in the first, it is evident that all the difference which we 
 shall observe will be due to attraction. In order to have 
 this second station precisely east or west of the first, we must 
 observe the azimuth of the sun at its rising or setting, by 
 finding its position with reference to some easily distinguished 
 point on the horizon ; in doing so we must often suppose the 
 latitude known; but the error we may make on this supposi- 
 tion will be of no consequence, and it will always be easy to 
 find two stations on the same parallel of latitude to within 
 3 or 4 sixtieths of a second. The latitude will be found pre- 
 cisely the same in the two places, if the vertical line has not 
 been altered in the first. Suppose, however, that without 
 seeking the latitude, we observe simply the meridian alti- 
 tudes of a star at the two stations ; the difference of these 
 two altitudes will indicate equally well the deflection of the 
 vertical line. It is evident that all the stars which pass the 
 meridian on the side of the apparent vertical line next to the 
 mountain will appear lower at the first station than at the 
 second ; for as the plumb-line approaches the mountain the 
 apparent zenith recedes from it and from these stars. It will 
 be quite the reverse with those stars which pass the meridian 
 on the other side of the apparent vertical line : they will ap- 
 pear higher at the first station.* 
 
 58. Instead of taking the stations both to the north or both 
 to the south, we could take them one to the north and the 
 other to the south, and exactly on the same meridian ; then 
 the effect of the attraction would be doubled, roughly speak- 
 ing, and we should find the sum of the contrary attractions. 
 The vertical line would be inclined in opposite directions at 
 the two stations; and the altitudes of stars which would be 
 increased in the one would be decreased in the other. The 
 physical effect being doubled would be more sensible, and 
 more susceptible of observation. If the two points were 
 equally distant from the centre of gravity of the mountain, 
 the action would be equal at both, and in order to get each 
 
 * \_This method of doubling the deflection caused by tlie mountain, by observ- 
 ing not one star, but at least two, one north and one south of the Rations, 
 is due to de la Condamine. See his account of the expedition (8, p. 68), Zach 
 (49) and Poynting (185, p. 14). This is the method actually employed by 
 Bouguer.] 
 
THE LAWS OF GRAVITATION 
 
 of them we should have merely to take half of the quantity 
 furnished hy the comparison of the observations. In other 
 cases the division would be a little more difficult; neverthe- 
 less it would be sufficient, as we shall shew later, to divide 
 the sum of the contrary attractions proportionally to the pro- 
 ducts of the quantity by which each station is more north or 
 more south, respectively, than the centre of gravity of the 
 mountain and the cube of the distance of the other station, 
 respectively, from the same centre. Thus we are under the 
 necessity of knowing the situation of each station with refer- 
 ence to the mountain; but we must know the distance from 
 one station to the other also, in order to determine geometric- 
 ally the difference of latitude between them. It is evident 
 that this difference must itself produce a change in the alti- 
 tude of each star, and we must know it before we can tell what 
 is the double effect of the attraction. To obtain the difference 
 in latitude of the two places, it would suffice ordinarily to 
 measure to the east or to the west of the mountain a base 
 directed nearly north and south, and to form on this base two 
 triangles which end at the two stations. 
 
 59. This way of making two observations from different 
 sides of the same mountain in order to render the effect of the 
 attraction more sensible, seems to me the more useful method, 
 as it depends less on the peculiarities of the places. We can 
 sometimes double the effect also by making the first observa- 
 tion at the north of one mountain and the second at the 
 south of another. If the two stations are not exactly on the 
 same east and west line, we have only to determine geometric- 
 ally their difference of latitude, and take account of it in the 
 comparison of the altitudes of the stars. 
 
 60. Finally, it is not only by observations made at the north 
 or at the south that we can discover whether mountains are 
 capable of acting "at a distance" ; it can be done also by ob- 
 servations made at the east or the west; but with this differ- 
 ence, that it will be no longer a question of observing latitude, 
 or of taking the meridian altitudes of stars; it will be only a 
 question of determining time exactly. It appears to me that 
 this last method would be often preferable to the preceding 
 ones, except that it requires two observers. Suppose that the 
 first of these is on the east side of a mountain, and the second 
 on the west side of another, or of the same, mountain. If each 
 
 37 
 
MEMOIRS ON 
 
 of them regulates carefully a chronometer by corresponding al- 
 titudes, it is evident that all these altitudes being altered by 
 the attraction which deflects the plumb-line, each chronome- 
 ter will be regulated as if the meridian were not exactly vertic- 
 al, but inclined below toward the mountain, and above away 
 from it. Let us suppose that the attraction amounts to a min- 
 ute of arc, and that the two mountains are on the equator ; 
 the first chronometer will denote midday 4 seconds of time 
 too soon, and the other 4 seconds too late. Thus, neglect- 
 ing the difference of longitude, which we could easily find by 
 measuring trigonometrically the distance of the two observ- 
 ers apart and reducing this distance to degrees and min- 
 utes, there would be a difference of 8 seconds, of time be- 
 tween the two chronometers. If the two mountains instead 
 of being on the equator were at latitude 60, each minute of 
 inclination which the attraction produced in a plumb-line 
 would produce 8 seconds of difference in the time of mid- 
 day, and therefore 16 seconds difference in the chronometers. 
 Finally, to judge of the attraction we need only know the exact 
 difference between the chronometers; and to find this, it would 
 always be sufficient to agree upon a signal, by fire or other- 
 wise ; and to observe at both stations the minute and second 
 of the instantaneous appearance of this signal. 
 
 61. I return to the first method because it appears to me to 
 be the simplest ; that is, suppose we station ourselves always 
 to the north or to the south of the mountain and confine our- 
 selves to observations of the latitude. It is evident that if. we 
 take at each station the meridian altitude of one star only, 
 we must know to the last degree of nicety the condition of the 
 quadrant we are using. There is no lack of methods for veri- 
 fying this instrument, but there is one which is extremely 
 valuable in the present instance, because, at the same time as 
 we work at verifying the quadrant, we are making the observ- 
 ations which decide the question at issue ; and in thus abridg- 
 ing the operations we avoid opportunities for errors. This 
 method is to take the meridian altitudes of an equal number 
 of stars toward the north and toward the south, and, provided 
 that the state of the instrument does not vary from one observ- 
 ation to another, it does not matter if it does change from 
 day to day. If it makes the altitudes of the stars on one side 
 the zenith too great, it will produce the same effect with re- 
 
 38 
 
THE LAWS OF GRAVITATION 
 
 spect to those on the other side. Thus the change will influ- 
 ence only the sum of the altitudes or the complements of the 
 altitudes, and will not alter the difference of the altitudes taken 
 on the different sides. The attraction, on the contrary, will not 
 alter the sum, but will change the difference; because at the 
 same time that it makes the stars on one side too high, it makes 
 those on the other side too low. It will always be easy to sep- 
 arate these two causes, and we shall not attribute to the one 
 that which arises from the other. To obtain at one stroke the 
 effect of attraction without being obliged to know the state of 
 the quadrant or the declinations of the stars, we need only ex- 
 amine whether the differences of the meridian altitudes taken 
 towards the north and towards the south are the same at the 
 two stations, or whether they are subject to a second differ- 
 ence. But it is necessary to remark that the altitudes being 
 increased on the one side while they are diminished on the 
 other, it is the half of this second difference which denotes 
 the physical effect of the attraction, both when this effect is 
 single and when it is double. In this latter case, it will be 
 necessary to divide the total effect in the ratio which the sep- 
 arate effects ought to have. 
 
 [Bouguer then proves this ratio to be that mentioned above (p. 
 37). He admits that some mountains might sliew less attraction 
 than that required by Neivtotis law (or even none), due to the 
 existence of great. cavities in the mass. He discusses the different 
 mountains in the neighbourhood of Quito, and for various reasons 
 decides upon Chimborazo as the one most suitable for the experi- 
 ment.] 
 
 EXAMINATION OF THE ATTRACTION OF CHIMBORAZO 
 
 65. I did not ascend this mountain alone as I did the pre- 
 ceding one. I had some time before communicated my design 
 and all my views to M. de la Condamine, and when on the 
 point of carrying them out I mentioned them to M. de Ulloa, 
 one of the two naval lieutenants who had assisted in the ob- 
 servations both of myself and of M. de la Condamine ever since 
 our arrival in the domains of His Catholic Majesty. These 
 gentlemen obligingly offered to accompany me, not only in the 
 preparatory examination, but also during the stay it was necess- 
 ary to make on the mountain side ; and as I knew it would be 
 to the advantage of the observations, I hastened to accept the 
 
MEMOIRS ON 
 
 offer. I had already thought that Chimborazo fulfilled ap- 
 proximately the necessary conditions : I knew that it was very 
 easy of access ; it could be seen from Quito, or rather from 
 Pichincha, from which it was 75,000 toises distant; and I had 
 already measured its height. ... On December 4th we estab- 
 lished ourselves on the south side of the mountain, at the bot- 
 tom of the snow line, 829 toises below the summit, but about 
 2400 above sea-level, and exactly 910 toises above the place at 
 Quito where I have always made my observations, and 344 
 toises above that part of Pichincha where there is a cross 
 which can be seen from all parts of the city, and where I passed 
 some days in March, 1737, in order to observe the astronomical 
 refraction. I shall not speak of the cold and the other discom- 
 forts we had to put up with ; snow covered our tent and all 
 the ground around as far as 800 or 900 toises below us, and we 
 lived in fear of being buried under its weight. It needed con- 
 tinual vigilance in order to avoid it. [M. de Ulloa fell ill, < and 
 had to descend the mountain on December 15th.] 
 
 [Left alone, Bouguer and de la Condamine observed the alti- 
 tudes of 10 stars, 4 on the south side and 6 on the north. Tlie 
 following are the altitudes as affected by the error of the instru- 
 ment and by refraction; they are the means of the readings of the 
 two observers.] 
 
 67. 
 
 On the north side : 
 Capella 
 
 Meridian altitudes at the first station 
 
 On 14th Dec. 
 
 On 15th Dec. 
 
 ' " 
 
 42 50 42| 
 
 59 53 55 
 66 18 15 
 69 20 
 72 33 42 
 
 32 58 10 
 
 38 58 52i 
 72 6 36| 
 75 9 12^ 
 
 o ' " 
 
 42 50 30 
 56 6 5 
 59 53 57 
 66 18 50 
 69 17 
 72 33 57 
 
 32 58 25 
 38 58 55 
 
 72 6 35 
 75 9 40 
 
 First head of Gemini . . 
 
 Second head of Gemini. . 
 
 Second horn of Aries 
 
 First horn of Aries 
 
 Aldebaran 
 
 On the south side : 
 Acarnar 
 
 Canopus .... . . 
 
 Tail of Cetus 
 
 Sirius 
 
 
 68. We observed the meridian altitude of the sun three 
 times. De la Condamine found it at the lower edge on De- 
 cember 15th to be 67 54' 26". This altitude, which is cor- 
 rected for the error of the instrument, but not for refraction 
 
 40 
 
THE LAWS OF GRAVITATION 
 
 and parallax, gives 1 2$' 53" south for the latitude of the 
 place where we were. I observed it on the 5th and 12th ; on 
 the 5th we had not yet regulated the chronometer nor traced 
 the meridian, and I found 1 30' 16". On the 12th I observed 
 the apparent altitude of the lower edge of the sun to be 
 68 5' 34" ; which gives 1 30' 6". 
 
 69. As soon as we were established on Chimborazo, I had sent 
 a tent about a league and a half to the west to a place called 
 I'Arenal to serve as the second station. [Bouguer then describes 
 the measurements made to find the exact position of the second 
 station; it ivas 3570 toises distant from the first, 174 toises lower, 
 and somewhat south of ivest of it. They began their observations 
 from the second station on Dec. 16th. Here they suffered more 
 from the wind and cold than at the more elevated station, as they 
 were more exposed to the prevailing east wind. It filled their 
 eyes with dust and continually threatened to overturn the tents.* 
 Tile screws of the quadrant could not be turned at night without 
 applying heat to them. Then follows a table of the altitudes as 
 observed. Since the second station was 505 toises, or 32", south 
 of the first, we must increase by 32" all the altitudes of stars ob- 
 served toward the north, and diminish the others by the same 
 amount. Moreover, since the second station was lower than the 
 first by 174 toises, the altitudes observed at the second station 
 must be diminished, to reduce them to the level of the first, on ac- 
 count of the excess of astronomical refraction. The following 
 are the corrected altitudes, and the differences for each star of 
 the mean determinations at the two stations :] 
 
 * [For de la Condamine's account of the experiments, see his Journal (8, 
 p. 69 and 8i, pt. 2, p. 146).] 
 
 41 
 
MEMOIRS ON 
 
 CORRECTED ALTITUDES AT THE SECOND STATION 
 
 Excess of altitudes 
 at the first station 
 over those at the 
 second, after the 
 latter have bee" 
 corrected 
 
 
 On 21st Dec. 
 
 On 22d Dec. 
 
 On the north side: 
 
 O / " 
 
 42 49 10 
 
 66 16 59 
 
 68 59 6 
 72 32 31! 
 
 32 56 53 
 
 38 57 16 
 
 72 4 47| 
 75 8 
 
 / 'I 
 
 42 49 15 
 
 59 53 23| 
 66 17 16 
 68 59 11 
 72 32 36! 
 
 32 56 28 
 
 38 57 36 
 72 4 50 
 
 75 8 7! 
 
 i n 
 
 1 24 
 
 1 25 
 1 10 
 1 6* 
 
 1 37! 
 1 28 
 
 1 48 
 1 22! 
 
 First head of Gemini... 
 Second head of Gemini. 
 Second horn of Aries. . . 
 First horn of Aries 
 Aldebaran 
 
 On the south side: 
 Acarnar 
 
 Canopus 
 
 Tail of Cetus . . 
 
 Sirius 
 
 
 [Bouguer considers the observations on the tail of Cetus and 
 the first horn of Aries as the best, but thinks it most legitimate 
 to take the mean of the altitudes of each star at each station and 
 to give equal weight to their differences. These differences are 
 given in the last column of the preceding table, and are, he main- 
 tains, too large and too uniform to be due to any defect in the ob- 
 servations. The averages of the excess of all the stars on the 
 north side and all those on the south side are now to be taken.} 
 
 74. ... They give about 1' 19" as the mean excess for the 
 north stars, and 1' 34" for the south. The second difference is 
 15". I leave it to my readers to say whether such a quantity 
 is sufficiently established by the means employed. My quad- 
 rant was 2.5 ft. in radius, and it must be remarked that any 
 errors which may exist in its graduation are of no importance 
 here ; since we have to do not with the altitudes themselves, 
 but with their differences. Suppose we admit the 15", it will 
 give 7".5 for the effect of attraction; it would be much greater 
 if we compared the tail of Cetus with the first horn of Aries. 
 However this is not the complete and absolute effect; for if 
 attraction really takes place, the mountain must have some 
 effect at the second station, which was about 4572 toises from 
 the centre of the mountain, and 61. 5 to the west of south. 
 At the first station we were nearly 16 west of south, and 1753 
 
 * [This is evidently a misprint for 1' 16".] 
 42 
 
THE LAWS OF GRAVITATION 
 
 toises distant. From these data we find that the effect at the 
 nearer station is to the effect we ought to find at the other as 
 1358 : 100, or as 13^ : 1 nearly. But since our observations 
 give only the difference of the two effects, we must increase 
 7". 5 by a 13th or 14th part of itself in order to have the total 
 effect [which makes it 8"]. 
 
 75. We must admit that this effect is very different from 
 what we had expected. But we know so little about the 
 earth's density, and on the other hand that of the mountain 
 may be so different from that which we have assumed it to be, 
 that there is no reason to be surprised at anything.* [It is a 
 tradition among the natives that Chimborazo is an extinct vol- 
 cano, and, if so, its density would be very hard to estimate. 
 Houyuer thinks it might be better to experiment on smaller and 
 denser mountains.] It is very probable that we shall find in 
 France or in England some hill of sufficient size, especially if 
 we double the effect ; and I shall be delighted if I find on my 
 return that the experiments that shall have been made either 
 confirm mine or throw new light on the matter. [At Riobamba 
 in Peru, December 30, 1738.] 
 
 [In an appendix Bouguer states that after a more thorough 
 survey of the Cordilleras he failed to find a more satisfactory 
 place at which to repeat his experiment. He suggests that the 
 converse effect be experimented upon : viz., the decrease in grav- 
 ity due to some deep canon among mountains. Assuming such 
 great cavities in Chimborazo as ivould make its real only half 
 its apparent volume, he finds his results would make it 6 or 
 7 times \ less dense than the earth; this he thinks not unreason- 
 
 * \J)e la Condamine also laid little stress on the numerical remit of the plumb- 
 line experiment ; for he says (8, p. 69), "ifwe can deduce from it nothing 
 decisively in favour of the Newtonian attraction, at least we find nothing con- 
 trary to that theory."] 
 
 f [In a critical analysis of all the experiments made to determine the density 
 of the earth up to that time, Saigey (74, p. 151 and in his "Petite Physique du 
 Globe," Paris, 1842, pt. 2, p. 151), in 1842, stated that Bouguer's calculations 
 were erroneous, because he confused the centre of attraction with tJie centre of 
 gravity of the mountain; he refers to a metliod, by means of which, using 
 Bouguer 's own mean result, he deduced the density of the earth to be 4.62 
 times that of tlie mountain ; but adds that if he had used only the results from 
 observations on the tail of Cetus and the first horn of Aries, tJie result would 
 have been 1.83, which is almost exactly the same as that found by Maskelyne by 
 the same method for the hill Schehallien.~\ 
 
 43 
 
MEMOIRS ON THE LAWS OF GRAVITATION 
 
 able. Bouguer further remarks that the distribution of density 
 in the earth may be such that the maximum attraction of the 
 earth is not at its surface but at some distance beneath it.] 
 
 [In connection with this work of Bouguer should be read a 
 paper (9) presented by him to the Academy on April 28, 1756, 
 on the possibility of detecting the deflection of a phimb-line due 
 to the ebb andfloiv of the tide. 
 
 Valuable accounts and discussions of Bouguer's work in Peril 
 are given by von Zach (43, 44, and 49), Schmidt (G4, vol. 2, 
 p. 475), Todhunter (140, chap. 12), Zanotti- Bianco (148, jo/. 
 2, pp. 122-25), and Poynting (185, pp. 10-14.] 
 
 PIERRE BOUGUER was born in 1698 at Croisic, Bretagne, and 
 was educated at the Jesuit College at Vannes. He succeeded 
 his father as professor of hydrography at Croisic in 1713, and 
 in 1730 accepted a similar position at Havre. His investiga- 
 tions concerning the intensity of light, embodied in a work, 
 Essai d'optique sur la gradation de la lumiere, published in 
 1729, led to his being elected a member of the Academy of 
 Sciences in 1731 ; he was promoted to the office of pensioned 
 astronomer in 1735. Along with two other members of the 
 Academy, MM. de la Condamine and Godin, he was sent to 
 Peru in 1735 to measure the length of an arc of the meridian 
 near the equator. Their labors there lasted ten years, and the 
 results of their observations were published in 1749 in La 
 Figure de la Terre, from which we have given the preceding ex- 
 tracts. For several years afterwards Bouguer was engaged in 
 a bitter controversy witli de la Condamine concerning their re- 
 spective shares in the Peruvian researches. In addition to his 
 works on photometry, he published several valuable treatises 
 on navigation, and various papers on atmospheric refraction 
 and other optical problems, and on mechanics. He died at 
 Paris in 1758. 
 
 44 
 
THE BEETIER CONTEOVEESY 
 
 45 
 
THE BERTIEB CONTROVERSY 
 
 IN June, 1769, there appeared in the Journal des Sciences et 
 des Beaux Arts a letter (11) by a M. Coultand, who signed 
 himself Former Professor of Physics at Turin. In it he de- 
 scribed some pendulum experiments made in the Alps of Savoy. 
 He claimed to have found that at a height of 1085 toises above 
 the base of the mountain the pendulum gained 28' in 2 months ; 
 20' 22" in 3 months at a height of 514 toises; and 15' 4" in 175 
 days at a height of 210 toises. So that it appeared as if the 
 attraction of gravitation increased with the distance from the 
 earth's centre, instead of behaving according to the Newtonian 
 law. A full account of the apparatus and observations was 
 added, and there seemed no reason why credence should not be 
 given to the results. The advocates of the Newtonian theory 
 felt called upon to account for this phenomenon consistent- 
 ly with their doctrine. D'Alembert (12 and 13, vol. 6, pp. 
 85-92) attacked the problem and found that the Newtonian 
 theory was adequate to explain the fact, provided the mean 
 density of the earth were about three eighths of that of the 
 mountain.* An abstract of Goultaud's alleged observations is 
 given by David (14). 
 
 In Dec., 1771, another letter appeared in the same journal 
 (15) signed by one Mercier, and addressed to Gessner, Pro- 
 fessor of Physics in the Univ. of Geneva. It described experi- 
 ments made in Valois similar to those of Coultaud and with 
 similar results, namely, that the attraction of gravitation is 
 directly proportional to the square of the distance. D'Alem- 
 bert then discussed the question again (13, vol. 6, pp. 93-98). 
 Further explanations on the Newtonian theory were forth- 
 
 * Compare the conclusion of Bonguer on p. 31 of this volume. 
 
 47' 
 
MEMOIRS ON 
 
 coming from Le Sage (16) and Lalande (17). Roiffe also dis- 
 cussed the experiments (18). 
 
 These results of Coultaud and Mercier seem, however, to 
 have been a cause of great exultation to a certain number of 
 scientists, especially ecclesiastics, who contended that the New- 
 tonians wished to take from them their Father, their God, in 
 asserting that bodies attract and move of themselves without 
 any Prime Mover. It is hard to believe that this feeling 
 existed so late as a century ago. One of the most active of 
 the opponents of the theory of Newton was Father Bertier, 
 de 1'Oratoire, who founded his 4th volume of Les Principes 
 Physiques on the above experiments. For several years a warm 
 discussion raged among French physicists over the question. 
 
 Le Sage, having had his suspicions aroused by some passage 
 in Mercier's letter, began a careful investigation into the gen- 
 uineness of the experiments both of Coultaud and Mercier. 
 He found them to be fabrications from beginning to end (19). 
 Le Sage does not mention whom he supposes to be the per- 
 petrators or instigators of the fraud. 
 
 A new impetus was given to the discussion by the publica- 
 tion of 2 letters (20) from Father Bertier describing experi- 
 ments with the balance similar to those performed by members 
 of the Royal Society of London a century before ; but Bertier 
 writes as if the idea were entirely a new one. The length of 
 the string used to suspend the weight from one arm of the 
 balance, after it had been counterpoised in the pan above, was 
 74 ft. In one case weights of 25 Ibs. were used, and when one 
 of them was suspended at the end of the string it lost in weight 
 1 ounce 3.5 drachms. Bertier concluded, much to his satisfac- 
 tion, that bodies weigh more the farther they are from the 
 centre of the earth. Roiffe followed with a paper (21) discuss- 
 ing the experiments made thus far and remarking that Bertier 
 had not taken account of the difference in the density of the 
 air at the two levels. Le Sage also criticised Bertier very 
 harshly (22). Repetitions of Bertier's experiment were made 
 by M. David, and Fathers Cotte and Bertier (23), the one with 
 a string of 170 ft. and weights of 1220 Ibs., the others with 
 a string of 45 ft. and weights of 150 Ibs. The one reported a 
 loss in weight of 1 oz., the others of 2 Ibs., in the same direc- 
 tion as indicated by Bertier's first experiment. An article by 
 David (24) in answer to Le Sage contains some scornful strict- 
 
 48 
 
THE LAWS OF GRAVITATION 
 
 ures on Newton and his principles which form amusing read- 
 ing at this late date. Rozier (25) criticised all the experi- 
 ments made on the laws of gravitation ; he refers to some 
 more made by Bertier (26) from which the latter concluded 
 that the loss in weight was proportional to the length of string 
 and to the weight. Rozier then announced the details of some 
 experiments of a similar kind made by himself, which gave 
 quite discordant results. David wrote another letter (27) with 
 more details of his experiments, but adding nothing of value. 
 Bertier followed with a similar letter (28). A committee of 
 the Academy of Dijon repeated the experiments with a sensit- 
 ive balance, and found (29) no change in weight except that 
 due to the different densities of the air at the higher and 
 lower levels. Some experiments on this same subject were 
 made by Achard (33) ; he found by using first a string and 
 then a brass chain with which to suspend the masses, that the 
 changes in weight of the suspended mass could be ascribed to 
 variations in the temperature and dampness of the air. Dolo- 
 mieu (30) made some experiments with a weight suspended in 
 a mine, similar to those of Dr. Power (page 2) ; his results 
 permitted no definite conclusions as to change in weight. In 
 connection with this work we might notice a valuable article 
 by Le Sage (34) on the history of the theory of gravitation and 
 the experiments made concerning it. He gives a brief account 
 of the views of Gilbert, Bacon, Kepler, Beaugrand, Fermat, 
 Pascal, Roberval, Descartes, and Gassendi ; finally he dis- 
 cusses Dolomieu's experiments and the possible variation in 
 density beneath the surface of the earth. 
 
 The controversy closes with the appearance of a letter (36) 
 from Bertier making an humble retraction of his statements 
 regarding the deductions to be drawn from his experiment ; he 
 admits that such experiments do not prove that bodies weigh 
 more as they are farther from the earth, but he declines to give 
 up his belief that such is the case. He again inveighs against 
 those who " by means of 101 different laws, which they make 
 God create to cover their ignorance, explain everything with a 
 facility that is truly delightful." 
 D 49 
 
THE SCHEHALLIEN EXPEEIMENT 
 
 51 
 
THE SCHEHALLTEN EXPEEIMENT 
 
 IN 1772, Maskelyne, the English Astronomer Royal, pro- 
 posed to the Royal Society (31) that the experiment of Bou- 
 guer on the attraction of a mountain be repeated in Great 
 Britain, as Bongner himself had suggested 30 years before. 
 Maskelyne had been informed of two places which might be 
 convenient for the purpose. One was near the confines of 
 Yorkshire and Lancashire, on the hill Whernside ; the other 
 in Cumberland, on the hill Helvellyn. The proposal was 
 favourably received by the Society, and Mr. Charles Mason was 
 sent to examine various hills in England and Scotland, and to 
 select the most suitable (32). Mason found that the two hills 
 referred to by Maskelyne were not suitable; and fixed upon 
 Schehallien in Perthshire as offering the best situation. At 
 the earnest solicitation of the Royal Society, Maskelyne him- 
 self undertook to make the necessary observations. He had at 
 his disposal a 10-foot zenith sector, and all his other instru- 
 ments were the best of their kind at the time. The work was 
 begun in the summer of 1774. The method of finding the de- 
 flection of the plumb-line due to the hill was exactly the second 
 of the methods described by Bouguer (page 36) ; he took read- 
 ings of the zenith-distances of certain stars at two stations, one 
 north and one south of the hill, and by this means doubled the 
 deflection of the plumb-line. Between June 30th and Septem- 
 ber 22d he took 1G9 star observations from the south station, 
 and 1G8 from the north station ; in all 337 observations on 43 
 stars. At the same time a very elaborate survey by triangula- 
 tion was made of the dimensions and form of the hill. This 
 was considered as made up of a very large number of prisms, 
 sufficient data for the determination of each of which were col- 
 lected during the survey. 
 
 53 
 
MEMOIRS ON 
 
 In his paper (32) describing the operations, Maskelyne cal- 
 culates, from 40 only out of the 337 observations, that the ap- 
 parent difference of latitude between the two stations is 54".6.* 
 The true difference of latitude is 43", leaving 11". 6 due to the 
 contrary attractions of the hill. 
 
 From a rough calculation, assuming the density of the 
 mountain to be the same as the mean density of the earth, 
 and that the law of attraction is that of the inverse square 
 of the distance, Maskelyne found that the attraction should be 
 twice that found by observation. Hence the mean density of 
 the earth is twice that of the hill. A more exact calculation 
 was promised for the future. Maskelyne draws two main con- 
 clusions: (1), that Schehallien has an attraction, and so, there- 
 fore, has every mountain ; (2), that the inverse square law of 
 the distance is confirmed ; for if the force were only a little 
 affected by the distance, the attraction of the hill would be 
 wholly insensible. 
 
 The survey of the hill and its environs was made during the 
 years 1774, 1775 and 1776. The calculation of the attraction 
 of the hill from these measurements was undertaken by Hut- 
 ton,! who employed several new and interesting methods. A 
 full account will be found in his paper (37 and 47, vol. 2, 
 pp. 1-68). Assuming that the density of the hill is the same 
 as the mean density of the earth, Hutton found that the attrac- 
 tion of the earth is to the sum of the contrary attractions of 
 the hill as 9933 : 1. Now Maskelyne had found the deflection 
 due to the contrary attractions of the hill to be 11". 6 ; whence 
 the attraction of the earth is to the sum of the attractions of 
 the hill as 1 : tan. 11". 6, or as 17781 : 1 ; or, allowing for the 
 
 * Von Zacli (49, App., pp. 686-692) has calculated the results from all 
 of the 337 observations, and finds for the apparent difference of latitude 
 54". 651, and for the deflection due to the contrary attractions of the hill 
 11". 632 ; which is in entire accord with Maskelyne's calculations. Saigey 
 (74, p. 153) also subjected the result to a test which was satisfactory. 
 Zanotti-Bmnco states (148, pt. 2, p. 134) that Saigey maintained that 
 Maskelyne did not choose his station at the most favourable part of the 
 hill-side, and that if he had done so he would have found the deflection 
 14" instead of 11". 6. 
 
 f For Button's own estimate of his share in the work, and for his con- 
 tempt for Cavendish's experiment, see bibl. No. 45. For a good account 
 of Hutton's method of calculation, see Zanotti-Bianco (148^, pt. 2, pp. 126- 
 32); see also Helmert (148, vol. 2, pp. 368-80). 
 
 54 
 
THE LAWS OF GRAVITATION 
 
 centrifugal force, as 17804 : 1 nearly. Hence the mean density 
 of the earth is to the density of the hill as 17804 : 9933, or as 
 9 : 5 nearly. Assuming the specific gravity of the hill to be 
 about 2.5, Hutton remarks that this would give 4.5 as the 
 mean specific gravity of the earth. Hutton revised this result 
 in his "Tracts" (47, vol. 2, p. 64); he takes the specific 
 gravity of the hill as 3, and hence the specific gravity of the 
 earth would be 5.4 nearly. 
 
 Playfair, with the aid of Lord Webb Seymour, made a care- 
 ful lithological survey of Schehallien, and published his re- 
 sults in 1811 (46 and 48). He found that the hill was made 
 up of two classes of rock, quartz of specific gravity 2.639876, 
 and micaceous rock, including calcareous, of specific gravity 
 2.81039. From two suppositions as to the distribution of these 
 two components in the interior of the hill, using Hutton's data 
 for the attraction, Playfair calculated the mean density of the 
 earth to be 4.55886 and 4.866997 respectively. Playfair con- 
 sidered the experiment on Schehallien so exact that he took the 
 mean of the above results, 4.713, as the best determination of 
 the mean density of the earth. 
 
 Hutton prefers to take 2.77 as the mean of Playfair's deter- 
 minations for the density of the hill, and the density of the 
 earth as f of 2.77, or 5 nearly. In a paper published in 1821 
 (52, 53 and 54), Hutton complains that his share in the Sche- 
 hallien experiment has always been underestimated ; he gives 
 a brief account of the observations, calculations and results, 
 and considers 5 as the most probable value of the mean density 
 of the earth. He shews that the Schehallien experiment could 
 not be made to give the same result as that of Cavendish, 5.48, 
 unless the deflection 11". 6 be diminished to about 10". 5 or 
 10". 4, which is manifestly too great an error to have been 
 committed by Maskelyne, considering the accuracy of the 
 observer and of the instruments, and the large number of ob- 
 servations made. Hutton suggests the repetition of the ex- 
 periment at one of the pyramids in Egypt. Some years later 
 Peters (80) made a calculation of the attraction of the Great 
 Pyramid. 
 
 For brief accounts of the Schehallien experiment and criti- 
 cisms upon it, reference should be made to Hutton (38 and 47, 
 vol. 2, pp. 69-77), von Zach (43, 44 and 49), Muncke (61, vol. 
 3, pp. 944-70), Schmidt (64, vol. 2, pp. 474-9), Meuabrea (71), 
 
 55 
 
MEMOIRS ON THE LAWS OF GRAVITATION 
 
 Schell (135), Todhunter (140, vol. 1, pp. 459-69). Zanotti- 
 Bianco (148, pt. 2, pp. 125-35) and Fresdorf (186, pp. 5-7). 
 
 Capt. Jacob has remarked (118 and 121) that by this method 
 we may measure the attraction of the mass of the mountain 
 above the surface,, yet we do not know how much ought to be 
 added or subtracted due to that below it. 
 
 Von Zach makes mention of several early astronomers who 
 assign anomalies in their geodetic measurements to the influence 
 of mountains on the plumb-lines of their instruments; the 
 reader is referred to von Zach, Humboldt (82, vol. 1, notes, 
 pp. 45-7) and Helmert (148, vol. 2, chap. 4), and to the ac- 
 count in this volume (p. 123) of the work of James and 
 Clarke. Von Zach himself made a very careful determination 
 in 1810, after the method used by Bouguer, of the attraction 
 of mount Mimet, near Marseilles. He found a deflection of 
 the plumb-line amounting to 2". He did not calculate the 
 density of the earth. His observations were published in book 
 form in 1814 (49). 
 
 For this work Maskelyne was presented by the Royal Society 
 with the Copley medal. At the presentation the President, Sir 
 John Pringle, delivered an address (35) on the attraction of 
 gravitation, giving a critical account of the state of the subject 
 before the time of Newton, as well as of its later developments. 
 
 56 
 
EXPERIMENTS TO DETERMINE THE 
 DENSITY OF THE EARTH 
 
 BY 
 
 HENRY CAVENDISH, ESQ., F.R.S. AND A.S. 
 Read June 21, 1798 
 
 (From the Philosophical Transactions of the Royal Society of London for tlie 
 year 1798, Part II. , pp. 469-526) 
 
 57 
 
CONTENTS 
 
 PAGK 
 
 Introduction 59 
 
 Description of the apparatus 61 
 
 Method of observing the deflection 64 
 
 " " " " time of vibration. 64 
 
 Effect of the resistance of the air 65 
 
 Account of the experiments 67 
 
 Testing for magnetic effects 68 
 
 Testing the elastic properties of the wire. 72 
 
 Further tests for magnetic effects - 75 
 
 Testing tJie effect of variation of temperature about the box 76 
 
 Final observations 80 
 
 On the tlieory of the experiment 88 
 
 Corrections to be made in the theory as first given 91 
 
 Effect of the variable position of the arm on the equations 97 
 
 When and how to apply the corrections 98 
 
 Table of results 99 
 
 Conclusion 99 
 
 Appendix : to find the attraction of the mahogany case on the balls 102 
 
 58 
 
EXPERIMENTS TO DETERMINE THE 
 DENSITY OF THE EARTH 
 
 BY 
 
 HENRY CAVENDISH, ESQ., F.R.S. AND A.S. 
 
 MANY years ago, the late Rev. John Miohell, of this society, 
 contrived a method of determining the density of the earth, hy 
 rendering sensible the attraction of small quantities of matter ; 
 but, as he was engaged in other pursuits, he did not complete 
 the apparatus till a short time before his death, and did not 
 live to make any experiments with it. After his death, the 
 apparatus came to the Rev. Francis John Hyde Wollaston, 
 Jacksonian Professor at Cambridge, who, not having conven- 
 iences for making experiments with it, in the manner he could 
 wish, was so good as to give it to me. 
 
 The apparatus is very simple ; it consists of a wooden arm, 6 
 feet long, made so as to unite great strength with little weight. 
 This arm is suspended in an horizontal position, by a slender 
 wire 40 inches long, and to each extremity is hung a leaden 
 ball, about 2 inches in diameter ; and the whole is inclosed in 
 a narrow wooden case, to defend it from the wind. 
 
 As no more force is required to make this arm turn round on 
 its centre, than what is necessary to twist the suspending wire, 
 it is plain, that if the wire is sufficiently slender, the most min- 
 ute force, such as the attraction of a leaden weight a few inches 
 in diameter, will be sufficient to draw the arm sensibly aside. 
 The weights which Mr. Michell intended to use were 8 inches 
 diameter. One of these was to be placed on one side the case, 
 opposite to one of the balls, and as near it as could conveniently 
 be done, and the other on the other side, opposite to the other 
 
 59 
 
MEMOIRS ON 
 
 ball, so that the attraction of both these weights would con- 
 spire in drawing the arm aside ; and, when its position, as af- 
 fected by these weights, was ascertained, the weights were to 
 be removed to the other side of the case, so as to draw the arm 
 the contrary way, and the position of the arm was to be again 
 determined ; and, consequently, half the difference of these po- 
 sitions would shew how much the arm was drawn aside by the 
 attraction of the weights. 
 
 In order to determine from hence the density of the earth, 
 it is necessary to ascertain what force is required to draw the 
 arm aside through a given space. This Mr. Michell intended to 
 do, by putting the arm in motion, and observing the time of its 
 vibrations, from which it may easily be computed.* 
 
 Mr. Michell had prepared two wooden stands, on which the 
 leaden weights were to be supported, and pushed forwards, till 
 they came almost in contact with the case ; but he seems to 
 have intended to move them by hand. 
 
 As the forcewith which the balls are attracted by these 
 weiglits is excessively minute, not more than 50)UO Vuuo ^ 
 their weight, it is plain, that a very minute disturbing force 
 will be sufficient to destroy the success of the experiment; and, 
 from the following experiments it will appear, that the disturb- 
 ing force most difficult to guard against, is that arising from 
 the variations of heat and cold ; for, if one side of the case is 
 warmer than the other, the air in contact with it will be rare- 
 fied, and, in consequence, will ascend, while that on the other 
 side will descend, and produce a current which will draw the 
 arm sensibly aside. f 
 
 * Mr. Coulomb has, in a variety of cases, used a contrivance of this kind 
 for trying small attractions ; but Mr. Michell informed me of his intention 
 of making this experiment, and of the method he intended to use, before 
 the publication of any of Mr. Coulomb's experiments. 
 
 f M. Cassini, in observing the variation compass placed by him in the 
 observatory (which was constructed so as to make very minute changes of 
 position visible, nnd in which the needle was suspended by a silk thread), 
 found that standing near the box, in order to observe, drew the needle 
 sensibly aside ; which I have no doubt was caused by this current of air 
 It must be observed, that his compass box was of metal, which transmits 
 heat faster than wood, and also was many i nches deep ; both which causes 
 served to increase the current of air. To diminish the effect of this cur- 
 rent, it is by all means advisable to make the box, in which the needle 
 plays, not much deeper than is necessnry to prevent the needle from strik- 
 ing against the top and bottom. 
 
 60 
 
THE LAWS OF GRAVITATION 
 
 As I was convinced of the necessity of guarding against this 
 source of error, I resolved to place the apparatus in a room 
 which should remain constantly shut, and to ohserve the mo- 
 tion of the arm from without, hy means of a telescope ; and to 
 suspend the leaden weights in such manner, that I could move 
 them without entering into the room. This difference in the 
 manner of observing, rendered it necessary to make some al- 
 teration in Mr. MichelFs apparatus ; and, as there were some 
 parts of it which I thought not so convenient as could be 
 wished, I chose to make the greatest part of it afresh. 
 
 Fig. 1 is a longitudinal vertical section through the instru- 
 ment, and the building in which it is placed. ABCDDCB- 
 AEFFE is the case ; x and x are the two balls, which are sus- 
 pended by the wires lix from the arm ghmh, which is itself 
 suspended by the slender wire gl. This arm consists of a 
 slender deal rod hnih, strengthened by a silver wire hyh', by 
 which means it is made strong enough to support the* balls, 
 though very light.* 
 
 The case is supported, and set horizontal, by four screws, 
 resting on posts fixed firmly into the ground ; two of them are 
 represented in the figure, by S and S ; the two others are not 
 represented, to avoid confusion. GG and GG are the end 
 walls of the building. W and W are the leaden weights ; 
 which are suspended by the copper rods RrPrR, and the 
 wooden bar rr, from the centre pin Pp. This pin passes 
 through a hole in the beam HH, perpendicularly over the cen- 
 tre of the instrument, and turns round in it, being prevented 
 from falling by the plate p. MM is a pulley, fastened to this 
 pin; and Mm, a cord wound round the pulley, and passing 
 through the end wall ; by which the observer may turn it 
 round, and thereby move the weights from one situation to the 
 other. 
 
 Fig. 2 is a plan of the instrument. AAAA is the case. SSSS, 
 the four screws for supporting it. hh, the arm and balls. W 
 and W, the weights. MM, the pulley for moving them. When 
 
 *Mr. Michell's rod was entirely of wood, and was much stronger and 
 stiffer than this, though not much heavier; but, as it had warped when it 
 came to me, I chose to make another, and preferred this form, partly as 
 being easier to construct and meeting with less resistance from the air, 
 and partly because, from its being of a less complicated form, I could more 
 easily compute how much it was attracted by the weights. 
 
 61 
 
MEMOIRS ON 
 
THE LAWS OF GRAVITATION 
 
 r' 
 
 the weights are in this position, both conspire in drawing the 
 arm in the direction 7AV; but, when they are removed to the 
 situation w and w, represented by the dotted lines, both con- 
 spire in drawing the arm in the contrary direction hw. These 
 weights are prevented from striking the instrument, by pieces 
 of wood, which stop them as soon as they come within of an 
 inch of the case. The pieces of wood are fastened to the wall 
 
 of the building ; and I find, that the weights may strike against 
 them with considerable force, without sensibly shaking the in- 
 strument. 
 
 In order to determine the situation of the arm, slips of ivory 
 are placed within the case, as near to each end of the arm as 
 can be done without danger of touching it, and are divided to 
 20ths of an inch. Another small slip of ivory is placed at each 
 and of the arm, serving as a vernier, and subdividing these 
 divisions into 5 parts ; so that the position of the arm may be 
 observed with ease to lOOths of an inch, and may be estimated 
 to less. These divisions are viewed, by means of the short 
 telescopes T and T (Fig. 1), through slits cut in the end of the 
 case, and stopped with glass ; they are enlightened by the lamps 
 L and L, with convex glasses, placed so as to throw the light 
 on the divisions ; no other light being admitted into the room. 
 
 The divisions on the slips of ivory run in the direction W0 
 (Fig. 2), so that, when the weights are placed in the positions 
 w and w, represented by the dotted circles, the arm is drawn 
 aside, in such direction as to make the index point to a higher 
 number on the slips of ivory ; for which reason, I call this the 
 positive position of the weights. 
 
 FK (Fig. 1) is a wooden rod, which, by means of an endless 
 screw, turns round the support to which the wire gl is fastened, 
 
MEMOIRS ON 
 
 and thereby enables the observer to turn round the wire, till 
 the arm settles in the middle of the case, without danger of 
 touching either side. The wire gl is fastened to its support at 
 top, and to the centre of the arm at bottom, by brass clips, in 
 which it is pinched by screws. 
 
 In these two figures, the different parts are drawn nearly in 
 the proper proportion to each other, and on a scale of one to 
 thirteen. 
 
 Before I proceed to the account of the experiments, it will 
 be proper to say something of the manner of observing. Sup- 
 pose the arm to be at rest, and its position to be observed, let 
 the weights be then moved, the arm will not only be drawn 
 aside thereby, but it will be made to vibrate, and its vibrations 
 will continue a great while ; so that, in order to determine how 
 much the arm is drawn aside, it is necessary to observe the ex- 
 treme points of the vibrations, and from thence to determine 
 the point which it would rest at if its motion was destroyed, or 
 the point of rest, as I shall call it. To do thia, I observe three 
 successive extreme points of a vibration, and take the mean 
 between the first and third of these points, as the extreme 
 point of vibration in one direction, and then assume the mean 
 between this and the second extreme, as the point of rest ; for, 
 as the vibrations are continually diminishing, it is evident, that 
 the mean between two extreme points will not give the true 
 point of rest. 
 
 It may be thought more exact, to observe many extreme 
 points of vibration, so as to find the point of rest by different 
 sets of three extremes, and to take the mean result ; but it 
 must be observed, that notwithstanding the pains taken to pre- 
 vent any disturbing force, the arm will seldom remain perfect- 
 ly at rest for an hour together; for which reason, it is best to 
 determine the point of rest, from observations made as soon 
 after the motion of the weights as possible. 
 
 The next thing to be determined is the time of vibration, 
 which I find in this manner: I observe the two extreme points 
 of a vibration, and also the times at which the arm arrives at two 
 given divisions between these extremes, taking care, as well as I 
 can guess, that these divisions shall be on different sides of the 
 middle point, and not very far from it. I then compute the 
 middle point of the vibration, and, by proportion, find the time 
 at which the arm comes to this middle point. I then, after a 
 
 64 
 
THE LAWS OF GRAVITATION 
 
 number of vibrations, repeat this operation, and divide the in- 
 terval of time, between the coming of the arm to these two 
 middle points, by the number of vibrations, which gives the 
 time of one vibration. The following example will explain 
 what is here said more clearly : 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle 
 of vibration 
 
 27.2 
 
 
 
 
 
 
 25 
 24 
 
 10 h 23' 4" | 
 57 f 
 
 
 
 10 h 23' 23" 
 
 22.1 
 
 
 
 
 246 
 
 
 27. 
 
 
 
 
 
 24.7 
 
 
 22.6 
 
 
 
 
 
 2475 
 
 
 26.8 
 
 
 
 
 
 24.8 
 
 
 23. 
 
 
 
 
 
 24.85 
 
 
 26.6 
 
 
 
 
 
 24.9 
 
 
 
 25 
 24 
 
 11 5 22 ) 
 
 6 48 J 
 
 
 
 11 5 22 
 
 23.4 
 
 
 
 
 
 The first column contains the extreme points of the vibra- 
 tions. The second, the intermediate divisions. The third, 
 the time at which the arm came to these divisions ; and the 
 fourth, the point of rest, which is thus found : the mean be- 
 tween the first and third extreme points is 27.1, and the mean 
 between this and the second extreme point is 24.6, which is 
 the point of rest, as found by the three first extremes. In like 
 manner, the point of rest found by the second, third, and 
 fourth extremes, is 24.7, and so on. The fifth column is the 
 time at which the arm came to the middle point of the vibra- 
 tion, which is thus found: the mean between 27.2 and 22.1 is 
 24.65, and is the middle point of the first vibration ; and, as 
 the arm came to 25 at 10 h 23' 4", and to 24 at 10 h 23' 57", we 
 find, by proportion, that it came to 24.65 at 10 h 23' 23". In 
 like manner, the arm came to the middle of the seventh vibra- 
 tion at ll h 5' 22"; and, therefore, six vibrations were performed 
 in 41' 59", or one vibration in 7' 0". 
 
 To judge of the propriety of this method, we must consider 
 in what manner the vibration is affected by the resistance of the 
 airland by the motion of the point of rest. 
 f/Let the arm, during the first vibration, move from D to B 
 (Fig. 3), and, during the second, from B to d ; Br7 being less 
 than DB, on account of the resistance. Bisect DB in M, and 
 E 65 
 
MEMOIRS ON 
 
 Bi7 in m, and bisect Mm in n, and let x be any point in the 
 vibration ; then, if the resistance is proportional to the square 
 of the velocity, the whole time of a vibration is very little al- 
 tered ; but, if T is taken to the time of one vibration, as the 
 diameter of a circle to its semi -circumference, the time of 
 
 TxDd 
 moving from B to n exceeds -J a vibration, by ^^ nearly ; 
 
 and the time of moving from B to m falls short of J a vibration, 
 
 D 6 M x 6 
 
 1 - > 3 - h ~t-t - j - -* 
 
 Fig. 3 
 
 by as much ; and the time of moving from B to x, in the sec- 
 ond vibration, exceeds that of moving from x to B, in the first, 
 
 . . 
 
 by - _ - , supposing Da to be bisected m 3 ; so that, 
 
 if a mean is taken, between the time of the first arrival of 
 the arm at x and its returning back to the same point, this 
 mean will be earlier* than the true time of its coming to B, by 
 
 The effect of motion in the point of rest is, that when the 
 arm is moving in the same direction as the point of rest, the 
 time of moving from one extreme point of vibration to the 
 other is increased, and it is diminished when they are moving 
 in contrary directions ; but, if the point of rest moves uni- 
 formly, the time of moving from one extreme to the middle 
 point of the vibration, will be equal to that of moving from 
 the middle point to the other extreme, and, moreover, the time 
 of two successive vibrations will be very little altered ; and, 
 therefore, the time of moving from the middle point of one 
 vibration to the middle point of the next, will also be very 
 little altered. 
 
 * [This word should be "later" as is observed by Todhunter (140, vol. 2, p. 
 165). For an elementary discussion of this kind of motion see Williamson and 
 Tarleton's " Treatise of Dynamics," ex. 13, 117. Poisson (65, vol. 1, pp. 
 353-361) and Menabrea (71) have given very elaborate analyses of the problem. 
 Cornu and Bailie (137, 141, 142, 143, and 157) proved in 1878 that the re- 
 sistance in the case under consideration is proportional to the first power of 
 the velocity.] 
 
 66 
 
THE LAWS OF GRAVITATION 
 
 It appears, therefore, that on account of the resistance of 
 the air, the time at which the arm comes to the middle point 
 of the vibration, is not exactly the mean between the times of 
 its coming to the extreme points, which causes some inaccur- 
 acy in my method of finding the time of a vibration. It must 
 be observed, however, that as the time of coming to the middle 
 point is before the middle of the vibration, both in the first 
 and last vibration, and in general is nearly equally so, the error 
 produced from this cause must be inconsiderable ; and, on the 
 whole, I see no method of finding the time of a vibration which 
 is liable to less objection. 
 
 The time of a vibration may be determined, either by previous 
 trials, or it may be done at each experiment, by ascertaining the 
 time of the vibrations which the arm is actually put into by 
 the motion of the weights ; but there is one advantage in the 
 latter method, namely, that if there should be any accidental 
 attraction, such as electricity, in the glass plates through which 
 the motion of the arm is seen, which should increase the force 
 necessary to draw the arm aside, it would also diminish the 
 time of vibration ; and, consequently, the error in the result 
 would be much less, when the force required to draw the arm 
 aside was deduced from experiments made at the time, than 
 when it was taken from previous experiments. 
 
 ACCOUNT OF THE EXPERIMENTS 
 
 In my first experiments, the wire by which the arm was sus- 
 pended was 39 inches long, and was of copper silvered, one 
 foot of which weighed 2 T 4 ^ grains ; its stiffness was such as to 
 make the arm perform a vibration in about 15 minutes. I im- 
 mediately found, indeed, that it was not stiff enough, as the 
 attraction of the weights drew the balls so much aside, as to 
 make them touch the sides of the case ; I, however, chose to 
 make some experiments with it, before I changed it. 
 
 In this trial, the rods by which the leaden weights were sus- 
 pended were of iron ; for, as I had taken care that there should 
 be nothing magnetical in the arm, it seemed of no signification 
 whether the rods were magnetical or not ; but, for greater se- 
 curity, I took off the leaden weights, and tried what effect the 
 rods would have by themselves. Now I find, by computation, 
 that the attraction of gravity of these rods on the balls, is to 
 
 67 
 
MEMOIRS ON 
 
 that of the weights, nearly as 17 to 2500 ; so that, as the at- 
 traction of the weights appeared, by the foregoing trial, to be 
 sufficient to draw the arm aside by about 15 divisions, the at- 
 traction of the rods alone should draw it aside about -fa of a 
 division ; and, therefore, the motion of the rods from one near 
 position to the other, should move it about ^ of a division. 
 
 The result of the experiment was, that for the first 15 min- 
 utes after the rods were removed from one near position to the 
 other, very little motion was produced in the arm, and hardly 
 more than ought to be produced by the action of gravity ; but 
 the motion then increased, so that, in about a quarter or half 
 an hour more, it was found to have moved } or 1J division, in 
 the same direction that it ought to have done by the action of 
 gravity. On returning the irons back to their former position, 
 the arm moved backward, in the same manner that it before 
 moved forward. 
 
 It must be observed, that the motion of the arm, in these ex- 
 periments, was hardly more than would sometimes take place 
 without any apparent cause; but yet, as in three experiments 
 which were made with these rods, the motion was constantly of 
 the same kind, though differing in quantity from J to 1 divis- 
 ion, there seems great reason to think that it was produced by 
 the rods. 
 
 As this effect seemed to me to be owing to magnetism, though 
 it was not such as I should have expected from that cause, I 
 changed the iron rods for copper, and tried them as before ; 
 the result was, that there still seemed to be some effect of the 
 same kind, but more irregular, so that I attributed it to some 
 accidental cause, and therefore hung on the leaden weights, and 
 proceeded with the experiments. 
 
 It must be observed, that the effect which seemed to be pro- 
 duced by moving the iron rods from one near position to the 
 other, was, at a medium, not more than one division ; whereas 
 the effect produced by moving the weight from the midway to 
 the near position, was about 15 divisions ; so that, if I had con- 
 tinued to use the iron rods, the error in the result caused there- 
 by, could hardly have exceeded -^ of the whole. 
 
 68 
 
THE LAWS OF GRAVITATION 
 
 EXPERIMENT I. AUG. 5 
 Weights in midway position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of mid. of 
 vibration 
 
 Difference 
 
 
 11.4 
 11.5 
 11.5 
 
 9 h 42' 0" 
 55 
 10 5 
 
 11.5 
 
 
 
 23.4 
 27.6 
 24.7 
 27,3 
 25.1 
 
 At 10 h 5', weights moved to positive position 
 
 25.82 
 26.07 
 26.1 
 
 At ll h 6', weights returned back to midway position 
 
 5. 
 
 
 
 
 
 
 
 11 
 12 
 
 48 } 
 1 30 f 
 
 
 
 o h r 13" 
 
 
 18.2 
 
 
 
 12 
 
 \ 
 
 14' 56" 
 
 
 12 
 11 
 
 16 29 ) 
 17 20 <i 
 
 
 
 16 9 
 
 
 6.6 
 
 
 
 11.92 
 
 
 
 14 36 
 
 
 11 
 12 
 
 30 24 ) 
 31 11 f 
 
 
 
 30 45 
 
 
 16.3 
 
 
 
 11.72 
 
 
 
 15 13 
 
 
 12 
 11 
 
 45 58 \ 
 
 47 4 f 
 
 
 
 45 58 
 
 
 7.7 
 
 
 
 
 
 
 Motion on moving from midway to pos. =14.32 
 
 pos. to midway = 14.1 
 Time of one vibration = 14' 55" 
 
 It must be observed, that in this experiment, the attraction 
 of the weights drew the arm from 11.5 to 25.8, so that, if no 
 contrivance had been used to prevent it, the momentum ac- 
 quired thereby would have carried it to near 40, and would, 
 therefore, have made the balls to strike against the case. To 
 prevent this, after the arm had moved near 15 divisions, I re- 
 turned the weights to the midway position, and let them re- 
 main there, till the arm came nearly to the extent of its vibra- 
 tion, and then again moved them to the positive position, 
 whereby the vibrations were so much diminished, that the balls 
 did not touch the sides ; and it was this which prevented my 
 observing the first extremity of the vibration. A like method 
 was used, when the weights were returned to the midway posi- 
 tion, and in the two following experiments. 
 
 69 
 
MEMOIRS ON 
 
 The vibrations, in moving the weights from the midway to 
 the positive position, were so small, that it was thought not 
 worth while to observe the time of the vibration. When the 
 weights were returned to the midway position, I determined 
 the time of the arm's coming to the middle point of each vibra- 
 tion, in order to see how nearly the times of the different 
 vibrations agreed together. In great part of the following ex- 
 periments, 1 contented myself with observing the time of its 
 coming to the middle point of only the first and last vibration. 
 
 EXPERIMENT II. AUG. 6 
 
 Weights in midway position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of mid. of 
 vibration 
 
 Difference 
 
 
 11 
 
 10 h 4' 0" 
 
 
 
 
 
 11 
 
 11 
 
 
 
 
 
 11 
 
 17 
 
 
 
 
 
 11 
 
 25 
 
 11. 
 
 
 
 Weights moved to positive position 
 
 29.3 
 
 24.1 
 
 30. 
 
 26.2 
 
 29.7 
 
 26.1) 
 
 28-7 
 
 27.1 
 
 28.4 
 
 26.87 
 
 27.57 
 28.02 
 28.12 
 28.05 
 27.85 
 27.82 
 
 Weights returned to midway position 
 
 6. 
 
 
 
 
 
 12 
 
 1 3 
 
 50 ) 
 
 
 13 
 
 4 
 
 34 \ 
 
 18.5 
 
 
 
 _^_ 
 
 
 
 13 
 
 18 
 
 29 ) 
 
 
 12 
 
 19 
 
 18 \ 
 
 6.5 
 
 
 
 
 
 
 
 11 
 
 33 
 
 48 ) 
 
 
 12 
 
 34 
 
 51 \ 
 
 15.2 
 
 
 
 
 
 
 
 13 
 
 45 
 
 8 { 
 
 
 12 
 
 46 
 
 22 J 
 
 7.1 
 
 
 
 
 
 
 
 11 
 
 2 3 
 
 48 ) 
 
 
 12 
 
 5 
 
 18 f 
 
 13.6 
 
 
 
 
 12.37 
 
 11.67 
 
 11. 
 
 10.75 
 
 l h 
 
 4' 
 
 1" 
 
 
 18 
 
 53 
 
 
 33 
 
 39 
 
 
 47 
 
 25 
 
 2 
 
 2 
 
 50 
 
 14 46 
 
 13 46 
 
 15 25 
 
 Motion of arm on moving weights from midway to pos. = 15.87 
 
 pos. to midway = 15.45 
 
 Time of one vibration =14' 42" 
 
 70 
 
THE LAWS OF GRAVITATION 
 
 EXPERIMENT III. AUG. 7 
 
 The weights being in the positive position, and the arm a little in motion 
 
 Exiiviue 
 IMHIIIS 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of mid. of 
 vibration 
 
 Difference 
 
 31.5 
 
 
 
 
 
 
 29 
 
 
 
 
 
 30.12 
 
 
 
 31 
 
 
 
 
 
 30.02 
 
 
 
 29.1 
 
 
 
 
 
 
 Weights moved to midway position 
 
 10 h 34' 55" 
 
 49 39 
 
 11 4 17 
 19 4 
 
 33 31 
 Mon 
 
 2 59 
 
 47 40 
 
 Motion of the arm on moving weights from pos. to mid.= 15.22 
 
 mid. to pos.=: 14.5 
 
 Time of one vibration, when in mid. position = 14' 39" 
 
 pos. position = 14' 54" 
 
 9. 
 
 
 
 
 
 14 
 
 10 h 34' 18" | 
 
 
 
 15 
 
 35 8 f 
 
 
 20.5 
 
 
 
 
 
 14.8 
 
 
 15 
 
 49 31 \ 
 
 
 
 14 
 
 50 27 f 
 
 
 9.2 
 
 
 
 
 
 14.07 
 
 
 14 
 
 11 5 7 ) 
 
 
 
 15 
 
 6 18 f 
 
 
 17.4 
 
 
 
 
 
 13.52 
 
 
 14 
 
 18 46 } 
 
 
 
 13 
 
 19 58 f 
 
 
 10.1 
 
 
 
 
 
 13.3 
 
 
 13 
 
 33 46 ) 
 
 
 
 14 
 
 35 26 f 
 
 
 15.6 
 
 
 
 
 Weights moved to positive p< 
 
 32. 
 
 
 
 
 
 28 
 
 2 48 ) 
 
 
 
 27 
 
 3 56 f 
 
 
 23,7 
 
 
 
 
 
 27.8 
 
 31.8 
 
 
 
 
 
 28.27 
 
 25.8 
 
 
 
 
 
 28.62 
 
 
 27 
 
 44 58 | 
 
 
 
 28 
 
 46 50 f 
 
 
 31.1 
 
 
 
 
 14' 44" 
 14 38 
 1447 
 1427 
 
 These experiments are sufficient to shew, that the attraction 
 of the weights on the balls is very sensible, and are also suf- 
 ficiently regular to determine the quantity of this attraction 
 pretty nearly, as the extreme results do not differ from each 
 other by more than -^ part. But there is a circumstance in 
 them, the reason of which does not readily appear, namely, 
 that the effect of the attraction seems to increase, for half an 
 
 71 
 
MEMOIRS ON 
 
 hour, or an hour, after the motion of the weights ; as it may 
 be observed, that in all three experiments, the mean position 
 kept increasing for that time, after moving the weights to the 
 positive position ; and kept decreasing, after moving them 
 from the positive to the midway position. 
 
 The first cause which occurred to me was, that possibly there 
 might be a want of elasticity, either in the suspending wire, or 
 something it was fastened to, which might make it yield more 
 to a given pressure, after a long continuance of that pressure, 
 than it did at first. 
 
 To put this to the trial, I moved the index so much, that the 
 arm, if not prevented by the sides of the case, would have stood 
 at about 50 divisions, so that, as it could not move farther than 
 to 35 divisions, it was kept in a position 15 divisions distant 
 from that which it would naturally have assumed from the 
 stiffness of the wire ; or, in other words, the wire was twisted 
 15 divisions. After having remained two or three hours in this 
 position, the index was moved back, so as to leave the arm at 
 liberty to assume its natural position. 
 
 It must be observed, that if a wire is twisted only a little 
 more than its elasticity admits of, then, instead of setting, as 
 it is called, or acquiring a permanent twist all at once, it sets 
 gradually, and, when it is left at liberty, it gradually loses part 
 of that set which it acquired ; so that if, in this experiment, 
 the wire, by having been kept twisted for two or three hours, 
 had gradually yielded to this pressure, or had begun to set, it 
 would gradually restore itself, when left at liberty, and the 
 point of rest would gradually move backwards ; but, though 
 the experiment was twice repeated, I could not perceive any 
 such effect. 
 
 The arm was next suspended by a stiffer wire. 
 
 EXPERIMENT IV. AUG. 12 
 Weights in midway position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of mid. of 
 vibration 
 
 Difference 
 
 
 21.6 
 
 9 h 30' 0" 
 
 
 
 
 
 21.5 
 
 52 
 
 
 
 
 
 21.5 
 
 10 13 
 
 21.5 
 
 
 
 72 
 
THE LAWS OF GRAVITATION 
 
 Weights moved from midway to positive position 
 
 27.2 
 
 
 
 
 
 
 22.1 
 
 
 
 
 
 24.6 
 
 
 
 27. 
 
 
 
 
 
 24.67 
 
 
 
 22.6 
 
 
 
 
 
 24.75 
 
 
 
 26.8 
 
 
 
 
 
 248 
 
 
 
 23.0 
 
 
 
 
 
 24.85 
 
 
 
 26.6 
 
 
 
 
 
 24.9 
 
 
 
 23.4 
 
 
 
 
 
 
 Weights moved to negative position 
 
 15. 
 
 
 
 
 
 
 
 17 
 19 
 
 19 25 ) 
 20 41 j 
 
 
 
 ' 10 h 20' 31'' 
 
 
 22.4 
 
 
 
 
 
 18.72 
 
 
 
 7' 0" 
 
 
 20 
 19 
 
 26 45 I 
 
 27 22 \ 
 
 
 
 27 31 
 
 
 15.1 
 
 
 
 
 
 18.52 
 
 
 
 6 57 
 
 
 19 
 20 
 
 35 1 ) 
 
 48 f 
 
 
 
 34 28 
 
 
 21.5 
 
 
 
 
 18.35 
 
 
 
 7 23 
 
 
 20 
 19 
 
 40 23 ) 
 
 41 18 j" 
 
 
 
 41 51 
 
 
 15.3 
 
 
 
 
 
 18.22 
 
 
 
 6 48 
 
 
 18 
 19 
 
 48 36 ) 
 49 24 j" 
 
 
 
 48 39 
 
 
 20.8 
 
 
 
 
 
 18.1 
 
 
 
 6 58 
 
 
 19 
 
 18 
 
 54 45 ) 
 55 45 J 
 
 
 
 55 37 
 
 
 15.5 
 
 
 
 
 
 
 Weights moved to positive position 
 
 31.3 
 
 
 
 
 
 
 
 25 
 23 
 
 11 10 25 I 
 11 3 J 
 
 
 
 11 10 40 
 
 
 17.1 
 
 
 
 
 
 24.02 
 
 
 
 7 3 
 
 
 22 
 23 
 
 17 6 ) 
 26 f 
 
 
 
 17 43 
 
 
 30.6 
 
 
 
 
 
 24.17 
 
 
 
 7 1 
 
 
 25 
 23 
 
 24 33 ) 
 
 25 17 f 
 
 
 
 24 44 
 
 
 18.4 
 
 
 
 
 
 2432 
 
 
 
 7 5 
 
 
 23 
 25 
 
 31 21 ) 
 32 9 \ 
 
 
 
 31 49 
 
 
 29.9 
 
 
 
 
 
 24.4 
 
 
 
 6 59 
 
 
 25 
 23 
 
 38 39 } 
 39 31 f 
 
 
 
 38 48 
 
 
 19.4 
 
 
 
 
 
 24.5 
 
 
 
 7 6 
 
 
 23 
 25 
 
 45 16 / 
 46 12 f 
 
 
 
 45 54 
 
 
 29.3 
 
 
 
 
 
 
 Moiion of arm on moving weights from midway to pos. = 3.1 
 
 pos. to neg. = 6.18 
 
 neg. to pos. 5.92 
 
 Time of one vibration in neg. position =1' 1" 
 
 pos. position =7' 3" 
 
 73 
 
MEMOIRS ON 
 
 EXPERIMENT V. AUG. 20 
 
 The weights being in the positive position, the arm was made to vibrate, by 
 moving the index 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of mid. of 
 vibration 
 
 Difference 
 
 29.6 
 21.1 
 29. 
 21.6 
 
 
 
 
 
 25.2 
 
 25.17 
 
 
 
 Weights moved to negative position 
 
 22.6 
 
 
 V 
 
 
 
 
 
 20 
 19 
 
 10 h 22' 47" ) 
 23 30 f 
 
 
 
 10 h 23' 11" 
 
 
 16.3 
 
 
 
 
 
 19.27 
 
 
 
 21.9 
 
 
 
 
 
 19.15 
 
 
 
 16.5 
 
 
 
 
 
 19.1 
 
 
 
 21.5 
 
 
 
 
 
 19.07 
 
 
 
 16.8 
 
 
 
 
 
 19.07 
 
 
 
 21.2 
 
 
 
 
 
 19.07 
 
 
 
 17.1 
 
 
 
 
 
 19.05 
 
 
 
 20.8 
 
 
 
 
 
 19.02 
 
 
 
 17.4 
 
 
 
 
 
 19.05 
 
 
 
 20.6 
 
 
 
 
 
 19.02 
 
 
 
 
 20 
 19 
 
 11 32 16 ) 
 
 33 58 j 
 
 
 
 11 33 53 
 
 
 17.5 
 
 
 
 
 
 18.97 
 
 
 
 7' 13" 
 
 
 19 
 20 
 
 41 16 ) 
 43 f 
 
 
 
 41 6 
 
 
 20.3 
 
 
 
 
 
 
 Weights moved to positive position 
 
 20.2 
 
 
 
 
 
 
 
 24 
 26 
 
 49 10 / 
 50 19 f 
 
 
 
 49 37 
 
 
 29/4 
 
 
 
 
 
 24.95 
 
 
 
 7 7 
 
 
 26 
 25 
 
 56 15 ) 
 
 47 f 
 
 
 
 \ 56 44 
 
 
 20.8 
 
 . 
 
 
 24 92 
 
 
 
 28.7 
 
 
 
 
 
 24.87 
 
 
 
 21.3 
 
 
 
 
 
 24.85 
 
 
 
 28.1 
 
 
 
 
 
 24.75 
 
 
 
 21.5 
 
 
 
 
 
 24.67 
 
 
 
 27.6 
 
 
 
 
 
 24.67 
 
 
 
 22. 
 
 
 
 
 
 24.7 
 
 
 
 
 24 
 25 
 
 45 48 | 
 46 43 f 
 
 . 
 
 46 21 
 
 
 27.2 
 
 
 
 
 
 24.7 
 
 
 
 7 1 
 
 
 25 
 24 
 
 53 11 ) 
 54 9 f 
 
 
 
 53 22 
 
 
 22.4 
 
 
 
 
 
 
 Motion of arm on moving weights from pos. to neg. = 5.9 
 
 neg. to pos. = 5.98 
 
 Time of one vibration, when weights are in neg. position = 7' 5" 
 
 pos. position = 7' 5" 
 
 74 
 
THE LAWS OF GRAVITATION 
 
 In the fourth experiment, the effect of the weights seemed 
 to increase on standing, in all three motions of the weights, 
 conformably to what was observed with the former wire ; but 
 in the last experiment the case was different ; for though, on 
 moving the weights from positive to negative, the effect seemed 
 to increase on standing, yet, on moving them from negative to 
 positive, it diminished. 
 
 My next trials were, to see whether this effect was owing to 
 magnetism. Now, as it happened, the case in which the arm 
 was inclosed, was placed nearly parallel to the magnetic east 
 and west, and therefore, if there was anything magnetic in the 
 balls and weights, the balls would acquire polarity from the 
 earth ; and the weights also, after having remained some time, 
 either in the positive or negative position, would acquire po- 
 larity in the same direction, and would attract the balls; but, 
 when the weights were moved to the contrary position, that 
 pole which before pointed to the north, would point to the 
 south, and would repel the ball it was approached to ; but yet, 
 as repelling one ball towards the south has the same effect on 
 the arm as attracting the other towards the north, this would 
 have no effect on the position of the arm. After some time, 
 however, the poles of the weight would be reversed, and would 
 begin to attract the balls, and would therefore produce the 
 same kind of effect as was actually observed. 
 
 To try whether this was the case, I detached the weights 
 from the upper part of the copper rods by which they were 
 suspended, but still retained the lower joint, namely, that 
 which passed through them ; I then fixed them in their posi- 
 tive position, in such manner, that they could turn round on 
 this joint, as a vertical axis. I also made an apparatus by 
 which I could turn them half way round, on these vertical axes, 
 without opening the door of the room. 
 
 Having suffered the apparatus to remain in this manner for a 
 day, I next morning observed the arm, and, having found it to 
 be stationary, turned the weights half way round on their axes, 
 but could not perceive any motion in the arm. Having suf- 
 fered the weights to remain in this position for about an hour, 
 I turned them back into their former position, but without its 
 having any effect on the arm. This experiment was repeated 
 on two other days, with the same result. 
 
 We may be sure, therefore, that the effect in question could 
 
 75 
 
MEMOIRS ON 
 
 not be produced by magnetism in the weights ; for, if it was, 
 turning them half round on their axes, would immediately have 
 changed their magnetic attraction into repulsion, and have 
 produced a motion in the arm. 
 
 As a further proof of this, I took off the leaden weights, and 
 in their room placed two 10-inch magnets ; the apparatus for 
 turning them round being left as it was, and the magnets being 
 placed horizontal, and pointing to the balls, and with their 
 north poles turned to the north ; but I could not find that 
 any alteration was produced in the place of the arm, by turn- 
 ing them half round ; which not only confirms the deduction 
 drawn from the former experiment, but also seems to shew, that 
 in the experiments with the iron rods, the effect produced could 
 not be owing to magnetism. 
 
 The next thing which suggested itself to me was, that pos- 
 sibly the effect might be owing to a difference of tempera- 
 ture between the weights and the case ; for it is evident, 
 that if the weights were much warmer than the case, they 
 would warm that side which was next to them, and produce a 
 current of air, which would make the balls approach nearer to 
 the weights. Though I thought it not likely that there should 
 be sufficient difference, between the heat of the weights and 
 case, to have any sensible effect, and though it seemed im- 
 probable that, in all the foregoing experiments, the weights 
 should happen to be warmer than the case, I resolved to ex- 
 amine into it, and for this purpose removed the apparatus used 
 in the last experiments, and supported the weights by the cop- 
 per rods, as before ; and, having placed them in the midway 
 position, I put a lamp under each, and placed a thermometer 
 with its ball close to the outside of the case, near that part 
 which one of the weights approached to in its positive position, 
 and in such manner that I conld distinguish the divisions by 
 the telescope. Having done this, I shut the door, and some 
 time after moved the weights to the positive position. At first, 
 the arm was drawn aside only in its usual manner ; but, in 
 half an hour, the effect was so much increased, that the arm 
 was drawn 14 divisions aside, instead of about three, as it 
 would otherwise have been, and the thermometer was raised 
 near 1.5 ; namely, from 61 to 62.5. On opening the door, 
 the weights were found to be no more heated, than just to pre- 
 vent their feeling cool to my fingers. 
 
 76 
 
THE LAWS OF GRAVITATON 
 
 As the effect of a difference of temperature appeared to be 
 so great, I bored a small hole in one of the weights, about 
 three-quarters of an inch deep, and inserted the ball of a small 
 thermometer, and then covered up the opening with cement. 
 Another small thermometer was placed with its ball close to 
 the case, and as near to that part to which the weight was 
 approached as could be done with safety ; the thermometers 
 being so placed, that when the weights were in the negative 
 position, both could be seen through one of the telescopes, by 
 means of light reflected from a concave mirror. 
 
 EXPERIMENT VI. SEPT. 6 
 
 Weights in midway position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Thermometer 
 
 in air 
 
 in weight 
 
 
 18.9 
 
 18.85 
 
 9 h 43' 
 
 10 3 
 
 18.85 
 
 55.5 
 
 
 13.1 
 18.4 
 13.4 
 missed 
 13.6 
 17.6 
 13.8 
 17.4 
 14.0 
 17.2 
 
 25.8 
 17.5 
 25.4 
 18.1 
 25.0 
 missed 
 24.7 
 19. 
 24.4 
 
 Weights moved to negative position 
 10 
 
 11 
 
 12 
 
 
 
 18 
 
 15.82 
 
 25 
 
 
 39 
 
 _ 
 
 46 
 
 15.65 
 
 53 
 
 15.65 
 
 
 
 15.65 
 
 7 
 
 15.65 
 
 14 
 
 
 
 55.5 
 
 55.5 
 
 55.5 
 
 55.8 
 
 55.8 
 
 Weights moved to positive position 
 
 
 23 
 
 
 
 
 30 
 
 21 
 
 .55 
 
 
 37 
 
 21 
 
 .6 
 
 
 44 
 
 21 
 
 05 
 
 
 51 
 
 
 
 
 
 5 
 
 
 
 
 12 
 
 21 
 
 .77 
 
 
 19 
 
 
 
 Motion of arm on moving weights from midway to = 3.03 
 
 -to +=5. 9 
 
 77 
 
MEMOIRS ON 
 
 EXPERIMENT VII. SEPT. 18 
 
 Weights in midway position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Thermometer 
 
 in air 
 
 in weight 
 
 
 19.4 
 19.4 
 
 8 h 30' 
 9 32 
 
 
 
 56.7 
 56.6 
 
 
 Weights moved to negative position 
 
 13.6 
 
 18.8 
 13.8 
 
 - 
 
 40 
 
 47 
 54 
 
 16.25 
 
 
 57.2 
 
 16.9 
 14. J 
 16.6 
 
 
 
 Eight extrer 
 
 10 58 
 11 5 
 12 
 
 ne points miss 
 15.62 
 
 ed 
 
 
 26.4 
 17.2 
 26.1 
 
 ' 
 
 Veights moved 
 20 
 28 
 35 
 
 to positive po& 
 21.72 
 
 ition 
 56.5 
 
 
 19.3 
 25.1 
 19.7 
 
 
 
 Four extren 
 10 
 17 
 24 
 
 le points miss 
 22.3 
 
 ed 
 
 
 Motion of arm on moving weights from midway to = 3.15 
 - to + = 6.1 
 
 EXPERIMENT VIII. SEPT. 23 
 in midway position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Thermometer 
 
 in air 
 
 in weight 
 
 
 19.3 
 19.2 
 
 9 h 46' 
 10 45 
 
 19.2 
 
 53.1 
 53.1 
 
 
 Weights moved to negative position 
 
 13.5 
 
 
 
 56 
 
 
 
 
 
 18.6 
 
 
 
 11 3 
 
 16.07 
 
 
 13.6 
 
 
 
 10 
 
 
 
 Four extreme points missed 
 
 17.4 
 
 
 
 44 
 
 
 
 14.1 
 
 
 
 51 
 
 15.7 
 
 
 17.2 
 
 
 
 58 
 
 
 
 
 
 53.6 
 
 f O * 
 
 53.6 
 
 78 
 
15.7 
 26.7 
 16.6 
 
 25.9 
 18.1 
 25.5 
 
 THE LAWS OF GRAVITATION 
 
 Weights moved to positive position 
 O h 1' 
 
 8 
 
 15 
 
 21.42 
 
 53.15 
 
 Two extreme points missed 
 36 
 
 43 
 
 50 
 
 21.9 
 
 Motion of arm on moving weights from midway to =3.13 
 
 -to +=5. 72 
 
 In these three experiments, the effect of the weight appeared 
 to increase from two to five tenths of a division, on standing 
 an hour ; and the thermometers shewed, that the weights were 
 three or five tenths of a degree warmer than the air close to the 
 case. In the two last experiments, I put a lamp into the room, 
 over night, in hopes of making the air warmer than the weights, 
 but without effect, as the heat of the weights exceeded that of 
 the air more in these two experiments than in the former. 
 
 On the evening of October 17, the weights being placed in 
 the midway position, lamps were put under them, in order to 
 warm them ; the door was then shut, and the lamps suffered to 
 burn out. The next morning it was found, on moving the 
 weights to the negative position, that they were 7. 5 warmer 
 than the air near the case. After they had continued an hour 
 in that position, they were found to have cooled 1.5, so as to 
 be only 6 warmer than the air. They were then moved to the 
 positive position ; and in both positions the arm was drawn 
 aside about four divisions more, after the weights had remained 
 an hour in that position, than it was at first. 
 
 May 22, 1798. The experiment was repeated in the same 
 manner, except that the lamps were made so as to burn only a 
 short time, and only two hours were suffered to elapse before 
 the weights were moved. The weights were now found to be 
 scarcely 2 warmer than the case ; and the arm was drawn 
 aside about two divisions more, after the weights had remained 
 an hour in the position they were moved to, than it was at first. 
 
 On May 23, the experiment was tried in the same manner, 
 except that the weights were cooled by laying ice on them; the 
 ice being confined in its place by tin plates, which, on moving 
 the weights, fell to the ground, so as not to be in the way. On 
 moving the weights to the negative position, they were found 
 
 79 
 
MEMOIRS ON 
 
 to be about 8 colder than the air, and their effect on the arm 
 seemed now to diminish on standing, instead of increasing, as 
 it did before ; as the arm was drawn aside about 2^ divisions 
 less, at the end of an hour after the motion of the weights, 
 than it was at first. 
 
 It seems sufficiently proved, therefore, that the effect in ques- 
 tion is produced, as above explained, by the difference of tem- 
 perature between the weights and case ; for in the 6th, 8th, and 
 9th* experiments, in which the weights were not much warmer 
 than the case, their effect increased but little on standing ; 
 whereas, it increased much, when they were much warmer than 
 the case, and -decreased much, when they were much cooler. 
 
 It must be observed, that in this apparatus, the box in which 
 the balls play is pretty deep, and the balls hang near the bot- 
 tom of it, which makes the effect of the current of air more 
 sensible than it would otherwise be, and is a defect which I in- 
 tend to rectify in some future experiments. 
 
 EXPERIMENT IX. APRIL 29 
 
 Weights in positive position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 34.7 
 
 
 
 
 [ 
 
 35. 
 
 
 
 
 
 34.84 
 
 
 34.65 
 
 
 
 
 V 
 
 "Weights moved to negative position 
 
 23.8 
 
 
 
 28 
 
 
 29 
 
 33.2 
 
 
 
 
 29 
 
 
 28 
 
 23.9 
 
 
 
 32. 
 
 
 
 24.15 
 
 
 
 31. 
 
 
 
 24.4 
 
 
 
 30.4 
 
 
 
 
 28 
 
 
 27 
 
 24.7 
 
 
 18' 29" \ 
 
 58 f 
 
 
 
 ll h 18' 43' 
 
 
 
 28.52 
 
 
 25 27 \ 
 
 57 J 
 
 
 
 25 40 
 
 
 28.25 
 
 
 
 
 28.01 
 
 
 
 
 27.82 
 
 
 
 
 27.63 
 
 
 
 
 27.55 
 
 > 
 
 
 
 27.47 
 
 
 7 4 ) 
 
 
 
 53 f 
 
 
 7 26 
 
 Motion of arm =6.32 
 Time of vibration = 6' 58" 
 
 * [This is evidently a misprint for 6th, 7th, and 8th.] 
 80 
 
THE LAWS OF GRAVITATION 
 
 EXPERIMENT X. MAY 5 
 
 Weights in positive position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 Difference 
 
 34.5 
 
 33.5 
 34.4 
 
 
 
 
 
 33.97 
 
 
 
 Weights moved to negative position 
 
 22.3 
 
 
 
 
 
 
 
 28 
 29 
 
 10 h 43' 42" I 
 44 6 J 
 
 
 
 10 h 43' 36" 
 
 
 33.2 
 
 
 
 
 
 27.82 
 
 . 
 
 r o" 
 
 
 28 
 
 27 
 
 50 33 ) 
 51 J 
 
 
 
 50 36 
 
 
 22.6 
 
 
 
 
 
 27.72 
 
 
 
 32.5 
 
 
 
 
 
 27.7 
 
 
 
 23.2 
 
 
 
 
 
 27.58 
 
 
 
 31.45 
 
 
 
 
 
 27.4. 
 
 
 
 23.5 
 
 
 
 
 
 27.28 
 
 
 
 
 27 
 28 
 
 11 25 20 ) 
 
 58 f 
 
 
 
 11 25 24 
 
 
 30.7 
 
 
 
 
 
 27.21 
 
 
 
 7 3 
 
 
 28 
 27 
 
 32 ) 
 32 40 J 
 
 
 
 32 27 
 
 
 23.95 
 
 
 
 
 
 27.21 
 
 
 
 6 56 
 
 
 27 
 
 39 19 ) 
 
 
 on .>> 
 
 
 
 28 
 
 40 2 i" 
 
 
 o &6 
 
 
 30.25 
 
 
 
 
 
 
 Motion of arm =6.15 
 
 Time of vibration = 6' 59" 
 
 EXPERIMENT XL MAY 6 
 
 Weights in positive position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 34.9 
 
 
 
 
 
 34.1 
 
 
 
 
 
 34.47 
 
 
 ' 34.8 
 
 
 
 
 
 34.49 
 
 
 34.25 
 
 
 
 
 
 Weights moved to negative position 
 
 23.3 
 
 
 
 
 
 
 28 
 29 
 
 9 h 59' 59" ) 
 
 10 27 f 
 
 
 
 10 h 0' 8" 
 
 33.3 
 
 
 
 
 
 28.42 
 
 
 
 29 
 
 6 52 ) 
 
 
 
 
 27 
 
 7 51 f 
 
 
 
 7 5 
 
 23.8 
 
 
 
 
 
 28.35 
 
 
 F 81 
 
MEMOIRS ON 
 
 32.5 
 
 
 
 
 
 28.3 
 
 
 24.4 
 
 
 
 
 
 missed 
 
 
 
 
 
 24.8 
 
 
 
 
 
 31.3 
 
 
 
 
 
 28.17 
 
 
 
 29 
 
 28 
 
 10 h 48' 37" ) 
 49 21 J 
 
 
 
 10 h 49' 8" 
 
 25.3 
 
 
 
 
 
 28.2 
 
 
 
 28 
 29 
 
 56 8 I 
 56 f 
 
 
 
 56 13 
 
 30.9 
 
 
 
 
 
 Motion of arm =6.07 
 
 Time of vibration = 7' 1" 
 
 In the three foregoing experiments, the index was purposely 
 moved so that, before the beginning of the experiment, the 
 balls rested as near the sides of the case as they could, without 
 danger of touching it ; for it must be observed, that when the 
 arm is at 35, they begin to touch. In the two following ex- 
 periments, the index was in its usual position. 
 
 EXPERIMENT XII. MAY 9 
 
 Weights* in negative position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 
 17.4 
 17.4 
 17.4 
 17.4 
 
 9 h 45' 0" 
 
 58 
 10 8 
 10 
 
 17.4 
 
 
 Weights moved to positive position 
 
 28.85 
 
 
 
 
 
 
 24 
 22 
 
 20 50 ) 
 21 46 f 
 
 
 
 10 h 20' 59" 
 
 18.4 
 
 
 
 
 
 23.49 
 
 
 28.3 
 
 
 
 
 
 23.57 
 
 
 19.3 
 
 
 
 
 
 23.67 
 
 
 27.8 
 
 
 
 
 
 23.72 
 
 
 20. 
 
 
 
 
 
 23.8 
 
 
 27.4 
 
 . 
 
 
 
 23.83 
 
 
 
 24 
 23 
 
 11 3 13 ) 
 54 \ 
 
 
 
 11 3 14 
 
 20.55 
 
 
 
 
 23.87 
 
 
 
 23 
 24 
 
 9 45 ) 
 10 28 f 
 
 
 
 10 18 
 
 27. 
 
 
 
 
 
 Motion of arm =6.09 
 
 Time of vibration = 7' 3" 
 
 82 
 
THE LAWS OF GRAVITATION 
 EXPERIMENT XIII. MAY 25 
 
 Weights in negative position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 16. 
 
 
 
 
 
 
 18.3 
 
 
 
 
 
 17.2 
 
 
 16.2 
 
 
 
 
 
 
 Weights moved 
 
 to positive position 
 
 29.6 
 
 
 
 
 
 
 
 25 
 24 
 
 10 h 22 
 
 22" I 
 45 [ 
 
 
 
 10 h 22' 56" 
 
 17.4 
 
 
 
 
 
 
 23.32 
 
 
 
 23 
 24 
 
 29 
 30 
 
 59 ) 
 23 f 
 
 
 
 30 3 
 
 28.9 
 
 
 
 
 
 
 23.4 
 
 
 
 24 
 23 
 
 36 
 
 37 
 
 58 \ 
 24 / 
 
 
 
 37 7 
 
 18.4 
 
 
 
 
 
 
 23.52 
 
 
 
 23 
 24 
 
 44 
 
 3 j. 
 31 f 
 
 
 
 44 14 
 
 28.4 
 
 
 
 
 
 
 23.62 
 
 
 19.3 
 
 
 
 
 
 
 23.7 
 
 
 27.8 
 
 
 
 
 
 
 23.7 
 
 
 
 24 
 23 
 
 11 5 
 6 
 
 26 ) 
 
 1 f 
 
 
 
 11 5 31 
 
 19.9 
 
 
 
 
 
 
 23.72 
 
 
 
 23 
 24 
 
 12 
 
 12 ) 
 50 \ 
 
 
 
 12 35 
 
 27.3 
 
 
 
 
 
 
 Weights moved 
 
 to negative position 
 
 13.5 
 
 
 
 
 
 
 21.8 
 
 
 
 
 
 
 17.75 
 
 
 
 18 
 17 
 
 37 
 38 
 
 &\ 
 
 
 
 37 39 
 
 13.9 
 
 
 
 
 
 
 17.67 
 
 
 
 17 
 
 18 
 
 44 
 45 
 
 *;} 
 
 
 
 44 45 
 
 21.1 
 
 
 
 
 
 
 17.62 
 
 
 14.4 
 
 
 
 
 
 
 17.6 
 
 
 20.5 
 
 
 
 
 
 
 17.52 
 
 
 14.7 
 
 
 
 
 
 
 17.47 
 
 
 20. 
 
 
 
 
 
 
 17.42 
 
 
 
 18 
 17 
 
 19 
 20 
 
 57 I 
 52 ] 
 
 
 
 20 24 
 
 15. 
 
 
 
 
 
 
 17.37 
 
 
 
 17 
 18 
 
 27 
 28 
 
 15 I 
 15 \ 
 
 
 
 27 30 
 
 19.5 
 
 
 
 
 
 
 Motion of the arm on moving 
 
 weights from to + = 6.12 
 
 
 + to -=5.97 
 
 Time of vibration at + 
 
 = 7' 6" 
 
 
 
 = 7' 7" 
 
 83 
 
MEMOIRS ON 
 
 EXPERIMENT XIV. MAY 26 
 
 Weights in negative position 
 
 Extreme 
 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibnition 
 
 
 16.1 
 16.1 
 16.1 
 16.1 
 
 9 h 18' 0" 
 
 24 
 46 
 49 
 
 16.1 
 
 
 Weights moved to positive position 
 
 27.7 
 
 
 
 
 
 
 
 23 
 22 
 
 10 
 1 
 
 46 \ 
 16 f 
 
 
 
 10" r i" 
 
 17.3 
 
 
 
 
 
 
 22.37 
 
 
 
 22 
 23 
 
 7 
 8 
 
 58 ) 
 
 27 \ 
 
 
 
 8 5 
 
 27.2 
 
 
 
 
 
 
 22.5 
 
 
 
 23 
 22 
 
 15 
 
 2 i 
 32 I 
 
 
 
 15 9 
 
 18.3 
 
 
 
 
 
 
 22.65 
 
 
 26.8 
 
 ' , 
 
 
 
 
 22.75 
 
 
 19.1 
 
 
 
 
 
 
 22.85 
 
 
 26.4 
 
 
 
 
 
 
 22.97 
 
 
 
 23 
 
 22 
 
 43 
 44 
 
 40 ) 
 22 \ 
 
 
 
 43 32 
 
 20. 
 
 
 
 
 
 
 23.15 
 
 
 
 22 
 23 
 
 49 
 50 
 
 53 ) 
 37 \ 
 
 
 
 50 41 
 
 26.2 
 
 
 
 
 
 
 Weights moved to negative position 
 
 12.4 
 
 
 
 
 
 
 
 16 
 17 
 
 11 7 
 
 8 
 
 53 ) 
 
 27 \ 
 
 
 
 11 8 25 
 
 21.5 
 
 
 
 
 
 
 17.02 
 
 
 
 17 
 16 
 
 15 
 16 
 
 30 I 
 3 \ 
 
 ;', 
 
 15 27 
 
 12.7 
 
 
 
 
 
 
 16.9 
 
 
 20.7 
 
 
 
 
 
 
 16.85 
 
 
 13.3 
 
 
 
 
 
 
 16.82 
 
 
 20. 
 
 
 
 
 
 
 16.72 
 
 
 13.6 
 
 
 
 
 
 
 16.67 
 
 
 
 16 
 17 
 
 50 
 51 
 
 33 ) 
 19 \ 
 
 
 
 50 58 
 
 19.5 
 
 
 
 
 
 
 16.65 
 
 
 
 17 
 16 
 
 57 
 58 
 
 53 ) 
 44 } 
 
 
 
 58 6 
 
 14. 
 
 
 
 
 
 
 Motion of arm by moving 
 
 weights from to + = 6.27 
 
 
 + to -=6.13 
 
 Time of vibration at + 
 
 = 7' 6" 
 
 - 
 
 = 7' 6" 
 
 84 
 
THE LAWS OF GRAVITATION 
 
 In the next experiment, the balls, before the motion of the 
 weights, were made to rest as near as possible to the sides of 
 the case, but on the contrary side from what they did in the 
 9th, 10th, and llth experiments. 
 
 EXPERIMENT XV. MAY 27 
 
 Weights in negative position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle ot 
 vibration 
 
 3.9 
 
 
 
 
 
 3.85 
 
 
 
 
 
 3.61 
 
 
 3.85 
 
 
 
 
 
 3.61 
 
 
 3.4 
 
 
 
 
 
 Weights moved to positive position 
 
 15.4 
 
 
 
 
 
 
 10 
 9 
 
 10 h 5' 59" ) 
 6 27 f 
 
 
 
 10 h 5' 56" 
 
 4.8 
 
 ' . 
 
 
 
 9.95 
 
 
 
 9 
 
 10 
 
 12 43 ) 
 13 11 f 
 
 
 
 13 5 
 
 14.8 
 
 
 
 
 
 10.07 
 
 
 
 10 
 9 
 
 20 24 ) 
 
 56 f 
 
 
 
 20 13 
 
 5.9 
 
 
 
 
 
 10.23 
 
 
 14.35 
 
 
 
 
 
 10.35 
 
 
 6.8 
 
 
 
 ' 
 
 10.46 
 
 
 13.9 
 
 
 
 
 
 10.52 
 
 
 
 11 
 10 
 
 48 30 \ 
 49 11 ] 
 
 
 
 48 42 
 
 7.5 
 
 
 
 
 
 10.6 
 
 
 
 10 
 
 11 
 
 55 26 ) 
 56 10 \ 
 
 
 
 55 48 
 
 13.5 
 
 
 
 
 
 Motion of the arm = 6.34 
 
 Time of vibration = 7' 7" 
 
 The two following experiments were made by Mr. Gilpin, 
 
 who was so good as to assist me on the occasion. 
 
 EXPERIMENT XVI. MAY 28 
 
 Weights in negative position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 
 rest 
 
 Time of middle of 
 vibration 
 
 22.55 
 
 
 
 
 
 8.4 
 
 
 
 
 
 15.09 
 
 
 21. 
 
 ' 
 
 
 
 14.9 
 
 
 9.2 
 
 
 
 
 
 85 
 
MEMOIRS ON 
 
 Weights moved to positive position 
 
 26.6 
 
 
 
 
 
 
 22 
 
 21 
 
 10 h 22' 53" ) 
 23 20 I 
 
 
 
 10 h 23' 15" 
 
 15.8 
 
 
 
 
 
 21. 
 
 
 
 20 
 21 
 
 30 7 | 
 36 f 
 
 
 
 30 30 
 
 25.8 
 
 
 
 
 
 21.05 
 
 
 
 22 
 21 
 
 37 23 I 
 55 \ 
 
 
 
 37 45 
 
 16.8 
 
 
 
 
 21.11 
 
 
 
 20 
 21 
 
 44 29 ) 
 45 4 f 
 
 
 
 45 1 
 
 25.05 
 
 
 
 
 
 21.11 
 
 
 
 22 
 21 
 
 51 54 ) 
 52 32 f 
 
 
 
 52 20 
 
 17.57 
 
 
 
 
 
 21.2 
 
 
 
 21 
 
 22 
 
 59 31 | 
 11 13 f 
 
 
 
 59 34 
 
 24.6 
 
 
 
 
 
 21.28 
 
 
 
 22 
 
 21 
 
 6 24 } 
 
 7 9 f 
 
 
 
 11 6 49 
 
 18.3 
 
 
 
 
 
 Motion of the arm = 6.1 
 
 Time of vibration =. 7' 16" 
 
 EXPEKIMENT XVII. MAY 30 
 
 Weights in negative position 
 
 Extreme 
 points 
 
 Divisions 
 
 Time 
 
 Point of 
 rest 
 
 Time of middle of 
 vibration 
 
 
 17.2 
 
 10 h 19' 0" 
 
 
 
 
 17.1 
 
 25 
 
 
 
 
 17.07 
 
 29 
 
 
 
 
 17.15 
 
 40 
 
 
 
 
 17.45 
 
 49 
 
 
 
 
 17.42 
 
 51 
 
 
 
 
 17.42 
 
 11 1 
 
 17.42 
 
 
 Weights moved to positive position 
 
 28.8 
 
 
 
 
 
 
 24 
 23 
 
 11 11 23 ) 
 49 J 
 
 
 
 ll h 11' 37" 
 
 18.1 
 
 
 
 
 
 23.2 
 
 
 
 22 
 23 
 
 18 13 I 
 43 \ 
 
 
 
 18 42 
 
 27.8 
 
 
 
 
 23.12 
 
 
 
 24 
 23 
 
 25 19 } 
 49 f 
 
 
 
 25 40 
 
 18.8 
 
 
 
 
 23.2 
 
 
 
 23 
 24 
 
 32 41 ) 
 33 13 \ 
 
 
 
 32 43 
 
 86 
 
THE LAWS OF GRAVITATION 
 
 27.38 
 
 
 
 
 
 23.31 
 
 
 
 24 
 23 
 
 ll h 39' 28" f 
 40 3 f 
 
 
 
 ll h 39' 44" 
 
 19.7 
 
 
 
 
 
 23.44 
 
 
 
 23 
 24 
 
 46 33 ) 
 
 47 11 J 
 
 
 
 46 46 
 
 27. 
 
 
 
 
 
 23.52 
 
 
 
 24 
 23 
 
 53 36 ) 
 54 17 f 
 
 
 
 53 48 
 
 20.4 
 
 
 
 
 
 23.57 
 
 
 
 23 
 24 
 
 34 ) 
 1 18 ( 
 
 
 
 6 55 
 
 26.5 
 
 
 
 
 
 23.55 
 
 
 
 24 
 23 
 
 7 34 ) 
 8 21 \ 
 
 
 
 7 50 
 
 20.8 
 
 
 
 
 
 23.59 
 
 
 
 23 
 24 
 
 14 30 ) 
 15 24 f 
 
 
 
 14 58 
 
 26.25 
 
 
 
 
 
 Weights moved to negative position 
 
 13.3 
 
 
 
 
 
 
 17 
 
 18 
 
 32 19 ) 
 
 48 f 
 
 
 
 32 44 
 
 22.4 
 
 
 
 
 17.95 
 
 
 
 18 
 17 
 
 39 46 > 
 40 19 J 
 
 
 
 39 44 
 
 13.7 
 
 
 
 
 
 17.85 
 
 
 
 17 
 
 18 
 
 46 26 ) 
 
 47 f 
 
 
 
 46 48 
 
 21.6 
 
 
 
 
 
 17.72 
 
 
 
 18 
 17 
 
 53 43 ) 
 54 20 f 
 
 
 
 53 50 
 
 14. 
 
 
 
 
 
 17.6 
 
 
 
 17 
 
 18 
 
 1 39 I 
 
 i 20 y 
 
 
 
 1 55 
 
 20.8 
 
 
 
 
 
 17.47 
 
 
 
 18 
 17 
 
 7 39 | 
 8 21 \ 
 
 
 
 7 59 
 
 14.3 
 
 
 
 
 
 17.37 
 
 
 
 17 
 
 14 54 ) 
 
 
 
 
 18 
 
 15 42 f 
 
 
 
 15 4 
 
 20.1 
 
 
 
 
 
 17.27 
 
 
 
 18 
 17 
 
 21 32 ) 
 22 22 J 
 
 
 
 22 5 
 
 14.6 
 
 
 
 
 
 Motion of the arm on moving weights from to + = 5.78 
 
 + to- =5.64 
 
 Time of vibration at 4- =T 2" 
 
 - =7' 3" 
 
 87 
 
MEMOIRS ON 
 
 OK THE METHOD OF COMPUTING THE DENSITY OF THE EARTH 
 FROM THESE EXPERIMENTS 
 
 I shall first compute this, on the supposition that the arm 
 and copper rods have no weight, and that the weights exert no 
 sensible attraction, except on the nearest ball ; and shall then 
 examine what corrections are necessary, on account of the arm 
 and rods, and some other small causes. 
 
 The first thing is, to find the force required to draw the arm 
 aside, which, as was before said, is to be determined by the time 
 of a vibration. 
 
 The distance of the centres of the two balls from each other 
 is 73.3 inches, and therefore the distance of each from the 
 centre of motion is 36.65, and the length of a pendulum vi- 
 brating seconds, in this climate, is 39.14; therefore, if the 
 stiffness of the wire by which the arm is suspended is such, 
 that the force which must be applied to each ball, in order to 
 draw the arm aside by the angle A, is to the weight of that 
 ball as the arch of A to the radius, the arm will vibrate in the 
 same time as a pendulum whose length is 36.65 inches, that is, in 
 
 y ' - seconds ; and therefore, if the stiffness of the wire is 
 
 such as to make it vibrate in N seconds, the force which must 
 be applied to each ball, in order to draw it aside by the angle 
 
 A, is to the weight of the ball as the arch of Ax^pX.y^ 
 
 -LAi O */ XT: 
 
 to the radius. But the ivory scale at the end of the arm is 
 38.3 inches from the centre of motion, and each division is -fa 
 of an inch, and therefore subtends an angle at the centre, whose 
 arch is T |-g ; and therefore the force which must be applied to 
 each ball, to draw the arm aside by one division, is to the weight 
 
 1 36.65 . 1 
 
 of the ball as to 1, or as to 1.* 
 
 * [Or thus: using the ordinary notation for the simple pendulum vibrating 
 through small arcs, if the force on each ball drawing the arm aside through an 
 
 arc subtending an angle of A were mg x ^ / , the arm would vibrate like a 
 
 /Of* pr 
 
 pendulum of the same length, and have a period of\J ~ seconds, because the 
 
 * oy.i4 
 
 period of a pendulum varies as the square root of its length. But the force varies 
 
 as - , ; therefore the force required to draw the arm through A with 
 (period} 1 ' 
 
THE LAWS OF GRAVITATION 
 
 The next thing is, to find the proportion which the attraction 
 of the weight on the ball hears to that of the earth thereon, sup- 
 posing the ball to be placed in the middle of the case, that is, 
 to be not nearer to one side than the other. When the weights 
 are approached to the balls, their centres are 8.85 inches from 
 the middle line of the case ; but, through inadvertence, the 
 distance, from each other, of the rods which support these 
 weights, was made equal to the distance of the centres of the 
 balls from each other, whereas it ought to have been somewhat 
 greater. In consequence of this, the centres of the weights are 
 not exactly opposite to those of the balls, when they are ap- 
 proached together; and the effect of the weights, in drawing 
 the arm aside, is less than it would otherwise have been, in the 
 
 o OK 
 
 triplicate ratio of ' to the chord of the an He whose sine is 
 
 8 85 36 ' 65 
 
 - , or in the triplicate ratio of the cosine of i this angle to 
 
 OD.OD 
 
 the radius, or in the ratio of .9779 to 1.* 
 
 period N" = mg x 
 
 arc of A c 
 
 36.65 
 X 8914 
 
 oa KK " on 1 A ^ ! - ^1^^ the force required to draw 
 
 OO.DO o". 14 
 
 the arm through 1 scale division with period N" 
 36.65 1 
 38.3 ' 20 36.65 
 = ^ X -^65- X 39Tl4^ N 
 
 1 36.65 1 
 
 = mg x 
 
 766N* 
 
 * [Let W be the position of 
 the "weight " of mass W, B 
 the position it was intend- 
 ed that it should have, and m 
 that of the ' ' ball " of mass m. 
 The distance mE, or WA, is 
 8.85 inches, and OW and Om 
 36.65 inches. Call WA and 
 Wm a and b respectively, and 
 G the gravitation constant. 
 Then it was intended that 
 the attraction to move the arm 
 
 should be - -2 ' m , but it is 
 
 G-W-m a 
 
 75 r ; a nd so is less than 
 
 b 2 b 
 
 was intended in the ratio of 
 to 1, or of Cos* ^ to 1.] 
 
 39.14" mgX 818 N 2 ] 
 
 89 
 
 Fig. d 
 
MEMOIRS ON 
 
 Each o.f the weights weighs 2,439,000 grams, and therefore is 
 equal in weight to 10.64 spherical feet of water ;* and therefore 
 its attraction on a particle placed at the centre of the ball, is to 
 the attraction of a spherical foot of water on an equal particle 
 
 (6 V 
 s-svj to 1. The 
 o.oO/ 
 
 mean diameter of the earth is 41,800,000 feet ;f and therefore, 
 if the mean density of the earth is to that of water as D to one, 
 the attraction of the leaden weight on the ball will be to that of 
 
 the earth thereon, as 10.64 x .9779 x (} to 41,800,000 D : : 
 
 \O.OO/ 
 
 1 : 8, 739,000 D.t 
 
 * [That is, iff equal to the weight of a sphere of water which can be inscribed in 
 a cube whose volume is 10.64 cu. ft., or we can express the volume of the sphere 
 by the number 10.64, when the unit of volume is that of a sphere of I foot in 
 
 diameter, that is, ofcu.ft The radius of a spherical foot of water is, accord- 
 ingly, 6 inches. Cavendish evidently uses Kirwaris estimate 0/253.35 grains 
 to the cu. in. of water. 
 
 The ensuing calculation can be stated thus : Call d and d' the densities of 
 water and of the earth respectively, m the mass of the ball, and G the gravita- 
 tion constant. The volume of the earth in spherical units is (41 800 OOO) 3 , and 
 itsraditisQxll 800 000 inches. 
 
 Gx 10.64 x 
 
 Attraction of weight on ball at 8.85 inches _ (8.85)- 
 
 x.9779 
 
 Attraction of earth on ball Gx (41_800 000 ) J x d' xwi 
 
 "~(6x41 800l)()0) r 
 
 .9779xl0.64x 
 
 f Y 
 
 \8.8,V 
 
 41 800 000 ^ 
 
 1 
 
 (I) 
 (2) 
 
 8 739 000 D ' 
 
 But we have already found (page 89) 
 Force required to draw the arm through 1 div. _ 1 
 Weight of ball '~818N ' 
 
 Dividing equation (1) by (2) we have 
 
 Attraction of weight on ball _____ _ 818 N 2 N 2 
 
 Force required to draw the arm through 1 div. ~ 8 739 GOOD ~ 10 683 D 
 = no. of div. through which the arm io drawn EEB div.] 
 
 f In strictness, we ought, instead of the mean diameter of the earth, to 
 take the diameter of that sphere whose attraction is equal to the force of 
 gravity in this climate; but the difference is not worth regarding. 
 
 \ [Ilutton has pointed out (54) that ihis number should be 8,740,000 ; but it 
 will not make any appreciable change in the value of D.] 
 
 90 
 
THE LAWS OF GRAVITATION 
 
 It is shewn, therefore, that the force which must be applied 
 to each ball, in order to draw the arm one division out of its 
 
 natural position, is xra of the weight of the ball ; and, if the 
 
 olo JN 
 
 mean density of the earth is to that of water as D to 1, the at- 
 
 traction of the weight on the ball is ^ of the weight of 
 
 o, 7 o",UUU L) 
 
 that ball ; and therefore the attraction will be able to draw the 
 ... . . . 818 N 2 N 2 
 
 arm out of its natural position by or * dlV18 ' 
 
 ions ; and therefore, if on moving the weights from the mid- 
 way to a near position the arm is found to move B divisions, or 
 if it moves 2 B divisions on moving the weights from one near 
 position to the other, it follows that the density of the earth, 
 
 _ . N 2 
 r 1S 
 
 We must now consider the corrections! which must be ap- 
 plied to this result ; first, for the effect which the resistance of 
 the arm to motion has on the time of the vibration : 2d, for the 
 attraction of the weights on the arm : 3d, for their attraction 
 on the farther ball : 4th, for the attraction of the copper rods 
 on the balls and arm : 5th, for the attraction of the case on the 
 balls and arm : and 6th, for the alteration of the attraction of 
 the weights on the balls, according to the position of the arm, 
 and the effect which that has on the time of vibration. None 
 of these corrections, indeed, except the last, are of much s4g- 
 nification, but they ought not entirely to be neglected. 
 
 As to the first, it must be considered, that during the vibra- 
 tions of the arm and balls, part of the force is spent in acceler- 
 ating the arm ; and therefore, in order to find the force re- 
 quired to draw them out of their natural position, we must 
 find the proportion which the forces spent in accelerating the 
 arm and balls bear to each other. 
 
 Let EDCMc (Fig. 4) be the arm. B and b the balls. Gs 
 the suspending wire. The arm consists of 4 parts ; first, a deal 
 rod Dcd, 73.3 inches long ; 2d, the silver wire DCW, weighing 170 
 grains ; 3d, the end pieces DE and ed, to which the ivory 
 
 * [This number should be 10,685. See last note.} 
 
 \ [For a discussion of these corrections, similar to that of Cavendish, but 
 with modern mathematical treatment, see Reich (67).] 
 
 91 
 
MEMOIRS ON 
 
 Fig. 4 
 
 vernier is fastened, each of which weighs 45 grains ; and 4th, 
 some brass work Cc, at the centre. The deal rod, when dry, 
 weighs 2320 grains, but when very damp, as it commonly was 
 during the experiments, weighs 2400 ; the transverse section is 
 of the shape represented in Fig. 5 ; the thick- 
 ness BA, and the dimensions of the part 
 Dl&ed, being the same in all parts ; but the 
 breadth B# diminishes gradually, from the 
 middle to the ends. The area of this sec- 
 tion is .33 of a square inch at the middle, and .146 at the 
 end; and therefore, if any point x (Fig. 4) is taken in cd, 
 
 1 ex . . , 2400 x. 33 
 
 and j is called x, this rod weighs -^ Q per inch at 
 
 
 , 
 cd 
 
 73.3 x. 238 
 
 ,, .,,. 2400X.146 2400 .33-. 184^ 
 
 the middle ; -- - at the end, and -r r x - 
 
 73.3 x. 238 
 
 73.3 
 
 .238 
 
 3320 -1848 
 
 at x; and therefore, as the weight of the wire is 
 
 7o.o 
 per inch, the deal rod and wire together may be con- 
 
 170 
 7373 
 
 ., - . , , 3490-1848 :r . . 
 
 side red as a rod whose weight at x= ^-^ - per inch. 
 
 I O.O 
 
 But the force required to accelerate any quantity of matter 
 placed at x, is proportional to x 3 ; that is, it is to the force re- 
 quired to accelerate the same quantity of matter placed at d as 
 x* to 1 ; and therefore, if cd is called /, and x is supposed to 
 flow, the fluxion of the force required to accelerate the deal 
 
 92 
 
THE LAWS OF GRAVITATION 
 
 x*lxx (3490 -1848*) . 
 rod and wire is proportional to - - - - i , the fluent 
 
 Y o. o 
 
 of which, generated while x flows from c to d, 
 
 = 1^3 X 
 
 so that the force required to accelerate each. half of the deal 
 rod and wire, is the same as is required to accelerate 350 grains 
 placed at d. 
 
 The resistance to motion of each of the pieces de, is equal to 
 that of 48 grains placed at d; as the distance of their centres of 
 gravity from C is 38 inches. The resistance of the brass work 
 at the centre may be disregarded ; and therefore the whole force 
 required to accelerate the arm, is the same as that required to 
 accelerate 398 grains placed at each of the points D and d. 
 
 Each of the balls weighs 11,262 grains, and they are placed 
 at the same distance from the centre as D and d; and there- 
 fore, the force required to accelerate the balls and arm to- 
 gether, is the same as if each ball weighed 11,660, and the arm 
 had no weight ; and therefore, supposing the time of a vibra- 
 tion to be given, the force required to draw the arm aside, is 
 greater than if the arm had no weight, in the proportion of 
 11,660 to 11,262, or of 1.0353 to 1. 
 
 To find the attraction of the weights on the arm, through d 
 draw the vertical plane dtvb perpendicular to Dd, and let w 
 be the centre of the weight, which, though not accurately in 
 this plane, may, without sensible error, be considered as placed 
 therein, and let b be the centre of the ball ; then wb is hori- 
 zontal andr=8.85, and db is vertical and=5.5; let wd=a, wb 
 
 dx 
 = b, and let -=-, or I x, =z; then the attraction of the weight 
 
 on a particle of matter at *, in the direction bw, is to its at- 
 traction on the same particle placed at b:: b 3 : (# 2 -f-zT)*, or is 
 
 b 3 
 proportional to - 3 , and the force of that attraction to 
 
 5 3 x (1 z) 
 move the arm, is proportional to a a ^, and the weight of 
 
 the deal rod and wire at the point x, was before said to be 
 
 3490 _ 1 848 z 1 642 + 1 848 z 
 
 = ; per inch; and therefore, if dx 
 
 <o.o to.o 
 
 flows, the fluxion of the power to move the arm 
 
 93 
 
73.3 
 
 MEMOIRS ON 
 ~r%^=*x (821+924 
 
 \ i 
 
 w*^ 
 
 ; which, as - = .08, 
 
 924 *' , r , fl 
 . The fluent of this 
 
 895 3 2 103 6 3 103 3 924 
 
 and the force with which the attraction of the weight, on the 
 nearest half of the deal rod and wire, tends to move the arm, 
 is proportional to this fluent generated while z flows from 
 to 1, that is, to 128 grains. 
 
 The force with which the attraction of the weight on the end 
 
 piece de tends to move the arm, is proportional to 47 x 3 [approxi- 
 mately], or 29 grains; and therefore the whole power of the 
 weight to move the arm, by means of its attraction on the near- 
 est part thereof, is equal to its attraction on 157 grains placed 
 
 157 
 at#, which is , or .0139 of its attraction on the ball.* 
 
 11/CD/O 
 
 It must be observed, that the effect of .the attraction of the 
 weight on the whole arm is rather less than this, as its attrac- 
 tion on the farther half draws it the contrary way ; but, as the 
 attraction on this is small, in comparison of its attraction on 
 the nearer half, it may be disregarded. 
 
 The attraction of the weight on the furthest ball, in the 
 direction btv, is to its attraction on the nearest ball :: wb 3 : wB 3 f- 
 ::. 0017:1; and therefore the effect of the attraction of the 
 weight on both balls, is to that of its attraction on the nearest 
 ball::. 9983 : 1. 
 
 * [A few minor misprints in Hie last two paragraphs in the original paper 
 have been corrected. A recalculation seems to give 142.5 instead ofl2S, and 
 28 instead of 29, grains; this would change the value ofD by 1 part in 1000.] 
 
 f [This is erroneously printed in the original as wd 3 : wD s .~\ 
 
 94 
 
THE LAWS OF GRAVITATION 
 
 To find the attraction of the copper rod on the nearest ball, 
 let 1) and w (Fig. 6) be the centres of the ball and weight, and 
 ea the perpendicular part of the copper rod, which consists of 
 two parts, ad and de. ad weighs 22,000 grains, and is 16 inches 
 long, and is nearly bisected by w. de weighs 41,000, and is 46 
 inches long, ivb is 8.85 inches, and is perpendicular to ew. 
 Now, the attraction of a line ew, 
 of uniform thickness, on b, in the 
 direction bw, is to that of the same 
 quantity of matter placed at w\\ bw 
 :eb; and therefore the attraction 
 of the part da equals that of 
 
 22,000 x wb 
 
 , or 16,300, placed at w; 
 
 db 
 and the 
 
 attraction of de equals 
 
 Fig. 6 
 
 that of 41,000x^x^-41,000 
 ed be 
 
 dw bw ,, 
 
 X yX-r-i i, or 2500, placed at the 
 
 ed bd 
 
 same point ; so that the attraction 
 
 of the perpendicular part of the w 
 
 copper rod on b, is to that of the 
 
 weight thereon, as 18,800 : 2,439,- 
 
 000, or as .00771 to 1. As for the 
 
 attraction of the inclined part of 
 
 the rod and wooden bar, marked 
 
 Pr and rr in Fig. 1, it may safely 
 
 be neglected, and so may the attraction of the whole rod on the 
 
 arm and farthest ball ; and therefore the attraction of the weight 
 
 and copper rod, on the arm and both balls together, exceeds the 
 
 attraction of the weight on the nearest ball, in the proportion 
 
 of .9983 + .0139 + . 0077 to one, or of 1.0199 to 1. 
 
 The next thing to be considered, is the attraction of the ma- 
 hogany case. Now it is evident, that when the arm stands at 
 the middle division, the attractions of the opposite sides of the 
 case balance each other, and have no power to draw the arm 
 either way. When the arm is removed from this division, it is 
 attracted a little towards the nearest side, so that the force re- 
 quired to draw the arm aside is rather less than it would other- 
 wise be ; but yet, if this force is proportional to the distance of 
 the arm from the middle division, it makes no error in the re- 
 
 95 
 
MEMOIRS ON 
 
 suit ; for, though the attraction will draw the arm aside more 
 than it would otherwise do, yet, as the accelerating force by 
 which the arm is made to vibrate is diminished in the same 
 proportion, the square of the time of a vibration will be in- 
 creased in the same proportion as the space by which the arm 
 is drawn aside, and therefore the result will be the same as if 
 the case exerted no attraction ; but, if the attraction of the 
 case is not proportional to the distance of the arm from the 
 middle point, the ratio in which the accelerating force is di- 
 minished is diiferent in different parts of the vibration, and 
 the square of the time of a vibration will not be increased in 
 the same proportion as the quantity by which the arm is drawn 
 aside, and therefore the result will be altered thereby. 
 
 On computation, I find that the force by which the attrac- 
 tion draws the arm from the centre is far from being propor- 
 tional to the distance, but the whole force is so small as not to 
 be worth regarding; for, in no position of the arm does the 
 attraction of the case on the balls exceed that of ^th of a spheric 
 inch of water, placed at the distance of one inch from the cen- 
 tre of -the balls ; and the attraction of the leaden weight equals 
 that of 10.6 spheric feet of water placed at 8.85 inches, or of 
 234 spheric inches placed at 1 inch distance ; so that the at- 
 traction of the case on the balls can in no position of the arm 
 exceed TT Yo of that of the weight. The computation is given 
 hi the Appendix. 
 
 It has been shown, therefore, that the force required to draw 
 the arm aside one division, is greater than it would be if the 
 arm had no weight, in the ratio of 1.0353 to 1, and therefore 
 
 7T-r Ta of the weight of the ball ; and moreover, the attraction 
 
 818 JN 
 
 of the weight and copper rod on the arm and both balls to- 
 gether, exceeds the attraction of the weight on the nearest ball, 
 
 1 0199 
 in the ratio of 1.0199 to 1, and therefore rr of the 
 
 weight of the ball ; consequently D is really equal to - 
 I 0199 N 2 N 2 i* 
 
 iMtead f 
 
 mer computation. It remains to be considered how much this 
 is affected by the position of the arm. 
 
 * [This should be 10,846 ; see note on pp. 90 and 91.] 
 96 
 
THE LAWS OF GRAVITATION 
 
 Suppose the weights to be approached to the balls ; let W 
 (Fig. 7) be the centre of one of the weights; let M be the cen- 
 tre of the nearest ball at its mean position, as when the arrn is 
 at 20 divisions ; let B be the point which it actually rests at ; 
 and let A be the point which it would rest at, if the weight was 
 removed ; consequently, AB is the space by which it is drawn 
 aside by means of the attraction ; and let M/3 be the space by 
 which it would be drawn aside, if the attraction on it was 
 
 Fig.l 
 
 the same as when it is at M. But the attraction at B is greater 
 than at M, in the proportion of WM 2 : WB 2 ; and therefore, 
 
 AT3 , ra WM 2 ... / t 2MB\ 
 
 AB = M/3 x ^gj = M/3 x \l + ^^ j , very nearly. 
 
 Let now the weights be moved to the contrary near position, 
 and let w be now the centre of the nearest weight, and b the point 
 
 of rest of the centre of the ball ; then M = Mft x M -f 
 
 and B&^M3xH-+2M3xl+; so that the 
 
 whole motion B& is greater than it would be if the attraction 
 on the ball was the same in all places as it is at M, in the ratio 
 
 of 14- to one ; and, therefore, does not depend sensibly 
 
 on the place of the arm, in either position of the weights, but 
 only on the quantity of its motion, by moving them. 
 
 This variation in the attraction of the weight, affects also 
 the time of vibration ; for, suppose the weights to be ap- 
 proached to the balls, let W be the centre of the nearest 
 weight ; let B and A represent the same things as before ; and 
 let x be the centre of the ball, at any point of its vibration ; let 
 AB represent the force with which the ball, when placed at B, 
 is drawn towards A by the stiffness of the wire ; then, as B is 
 the point of rest, the attraction of the weight thereon will also 
 equal AB ; and, when the ball is at x, the force with which it 
 is drawn towards A, by the stiffness of the wire, = Ao?, and that 
 with which it is drawn in the contrary direction, by the attrac- 
 
 WB 2 
 
 tion,=AB x ^-^ ; so that the actual force by which it is drawn 
 
 W x 
 G 97 
 
MEMOIRS ON 
 
 ABxWB 2 /, 2Bz\ 
 
 towards A=Az ^^ = AB + Rx ABx M +^^Jr=Bic 
 
 2B# x AB 
 
 near ty* ^ ^at the ac ^ ua ^ force with which 
 
 WR 
 
 the ball is drawn towards the middle point of the vibration, is 
 less than it would be if the weights were removed, in the ratio 
 
 2AB 
 of 1 TiTTr to one, and the square of the time of a vibration is 
 
 2AB 
 
 increased in the ratio of 1 to 1 w ^ ; which differs very little 
 
 from that of l + TT?r to 1, which is the ratio in which the 
 
 motion of the arm, by moving the weights from one near posi- 
 tion to the other, is increased. 
 
 The motion of the ball answering to one division of the arm 
 
 36 65* 
 = o? ~QO~Q 5 an d> ^ -M-^t is the motion of the ball answering 
 
 20 X uo.O 
 
 , .. . . MB 36.65<Z d 
 
 to d dmsions on the arm, _ = _-. = _ ; and 
 
 therefore, the time of vibration, and motion of the arm, must 
 
 rrected as follows : 
 
 If the time of vibration is determined by an experiment in 
 which the weights are in the near position, and the motion of 
 the arm, by moving the weights from the near to the midway 
 position, is d divisions, the observed time must be diminished 
 
 in the subduplicate ratio of 1 -^- to 1, that is, in the ratio 
 
 -LoO 
 
 of 1 -- * to 1; but, when it is determined by an experiment 
 185 
 
 in which the weights are in the midway position, no correction 
 must be applied. 
 
 To correct the motion of the arm caused by moving the 
 weights from a near to the midway position, or the reverse, 
 observe how much the position of the arm differs from 20 
 divisions, when the weights are in the near position ; let this 
 be n divisions, then, if the arm at that time is on the same side 
 of the division of 20 as the weight, the observed motion must be 
 
 * [This number, 36.65, Iwre, and again in the next line, is erroneously 
 printed in Cavendish's memoir as 36.35.] 
 f [In the original this is erroneously printed as mB.] 
 
THE LAWS OF GRAVITATION 
 
 diminished by the : part of the whole ; but, otherwise, it 
 
 loO 
 
 must be as much increased. 
 
 If the weights are moved from one near position to the other, 
 and the motion of the arm is 2d divisions, the observed motion 
 
 must be diminished by the : part of the whole. 
 
 185 
 
 If the weights are moved from one near position to the other, 
 and the time of vibration is determined while the weights are 
 in one of those positions, there is no need of correcting either 
 the motion of the arm, or the time of vibration.^/ 
 
 T 
 
 CONCLUSION 
 
 The following table contains the result of the experiments 
 
 Exper. 
 
 Mot. weight 
 
 Mot. arm 
 
 Do. corr. 
 
 Time vib. 
 
 Do. corr. 
 
 Density 
 
 1 j 
 
 m. to 4- 
 
 14.32 
 
 13.42 
 
 
 
 
 5.5 
 
 H 
 
 + to m. 
 
 14.1 
 
 13.17 
 
 14' 55" 
 
 
 
 5.61 
 
 oj 
 
 m. to + 
 
 15.87 
 
 14.69 
 
 
 
 
 
 4.88 
 
 2 i 
 
 + to m. 
 
 15.45 
 
 14.14 
 
 14 42 
 
 
 
 5.07 
 
 q5 
 
 + to m. 
 
 15.22 
 
 13.56 
 
 14 39 
 
 
 
 5.26 
 
 3 i 
 
 m. to + 
 
 14.5 
 
 13.28 
 
 14 54 
 
 
 
 5.55 
 
 I 
 
 m. to + 
 
 3.1 
 
 2.95 
 
 
 6' 54" 
 
 5.36 
 
 4\ 
 
 + to - 
 
 6.18 
 
 
 
 7 1 
 
 
 
 5.29 
 
 } 
 
 - to 4 
 
 5.92 
 
 
 
 7 3 
 
 
 
 5.58 
 
 KJ 
 
 + to - 
 
 5.9 
 
 
 
 7 5 
 
 
 
 5.65 
 
 5 i 
 
 - to + 
 
 5.98 
 
 
 
 7 5 
 
 
 
 5.57 
 
 j 
 
 m. to 
 
 3.03 
 
 2.9 1 
 
 
 
 5.53 
 
 6 i 
 
 - to + 
 
 5.9 
 
 5.71 
 
 7 A 
 
 
 5.62 
 
 7 i 
 
 m. to 
 
 - to + 
 
 3.15 
 6.1 
 
 3.03 1 
 5.9 f 
 
 i 1 
 
 by 
 
 6 57 
 
 5.29 
 5.44 
 
 
 m. to 
 
 3.13 
 
 3.00 | 
 
 mean 
 
 
 5.34 
 
 
 
 - to + 
 
 5.72 
 
 5.54J 
 
 
 
 5.79 
 
 9 
 
 + to - 
 
 6.32 
 
 
 
 6 58 
 
 
 
 5.1 
 
 10 
 
 + to - 
 
 6.15 
 
 , 
 
 6 59 
 
 
 
 5.27 
 
 11 
 
 -f -to 
 
 6.07 
 
 
 
 7 1 
 
 
 
 5.39 
 
 12 
 
 - to + 
 
 6.09 
 
 
 
 7 3 
 
 
 
 5.42 
 
 
 - to + 
 
 6.12 
 
 
 
 7 6 
 
 - 
 
 5.47 
 
 13 
 
 + to - 
 
 5.97 
 
 
 
 7 7 
 
 
 
 5.63 
 
 
 to + 
 
 6.27 
 
 
 
 7 6 
 
 
 
 5.34 
 
 14 
 
 + to - 
 
 6.13 
 
 
 
 7 6 
 
 
 
 5.46 
 
 15 
 
 - to + 
 
 6.34 
 
 
 
 7 7 
 
 
 
 5.3 
 
 16 
 
 - to + 
 
 6.1 
 
 
 
 7 16 
 
 
 
 5.75 
 
 17J 
 
 - to + 
 
 5.78 
 
 
 
 7 2 
 
 
 
 5.68 
 
 17 j 
 
 + to - 
 
 5.64 
 
 
 
 7 3 
 
 
 
 5.85 
 
 * [ The corrections neutralize each other, since they are the same for N 2 and 
 B, whose ratio enters into the expression for D.J 
 
MEMOIRS ON 
 
 From this table it appears, that though the experiments 
 agree pretty well together, yet the difference between them, 
 both in the quantity of motion of the arm and in the time of 
 vibration, is greater than can proceed merely from the error of 
 observation. As to the difference in the motion of the arm, it 
 may very well be accounted for, from the current of air pro- 
 duced by the difference of temperature ; but, whether this can 
 account for the difference in the time of vibration, is doubtful. 
 If the current of air was regular, and of the same swiftness in 
 all parts of the vibration of the ball, I think it could not ; but, 
 as there will most likely be much irregularity in the current, it 
 may very likely be sufficient to account for the difference. 
 
 By a mean of the experiments made with the wire first used, 
 the density of the earth comes out 5.48* times greater than 
 that of water ; and by a mean of those made with the latter 
 wire, it comes out the same ; and the extreme difference of the 
 results of the 23 observations made with this wire, is only .75 ; 
 so that the extreme results do not differ from the mean by more 
 than .38, or ^ of the whole, and therefore the density should 
 seem to be determined hereby, to great exactness. It, indeed, 
 may be objected, .that as the result appears to be influenced by 
 the current of air, or some other cause, the laws of which we 
 are not well acquainted with, this cause may perhaps act al- 
 ways, or commonly, in the same direction, and thereby make a 
 considerable error in the result. But yet, as the experiments 
 were tried in various weathers, and with considerable variety 
 in the difference of temperature of the weights and air, and 
 with the arm resting at different distances from the sides of 
 the case, it seems very unlikely that this cause should act so 
 uniformly in the same way, as to make the error of the mean 
 result nearly equal to the difference between this and the ex- 
 
 * [ This should be 5. 31. Had the third number in the column of densities been 
 5.88, instead of 4. 88, the average would have been as Cavendish gave it. But 
 Baity (79, p. 90) recalculated the densities from Cavendish's data, and found 
 4.88 to be correct. Curiously enough Cavendish made the same error in deduc- 
 ing the mean result of the whole number of experiments. It should be 5.448, not 
 5.48 (which would be had by putting 5.88 in place 0/4.88), with a probable error 
 of .033. The mean result of the last 23 observations is 5.48. The greatest dif- 
 ference of the single results from one another is . 97 ; and the extreme result 
 differs from the mean of all by .57, or fa of the whole. For an account of 
 Buttons reflections on Cavendish's "pretty and amusing little experiment " 
 see his paper (45 and 54) ; they are referred to in this volume on page 105.1 
 
 100 
 
T 11 E LA W S OF : GJi A V, IT /f I O N 
 
 treme ; and, therefore, it seems very unlikely that the density 
 of the earth should differ from 5.48 by so much as -fa of the 
 whole.* 
 
 Another objection, perhaps, may be made to these experi- 
 ments, namely, that it is uncertain, whether, in these small 
 distances, the force of gravity follows exactly the same law as 
 in greater distances. There is no reason, however, to think 
 that any irregularity of this kind takes place, until the bodies 
 come within the action of what is called the attraction of co- 
 hesion, and which, seems to extend only to very minute dis- 
 tances. With a view to see whether the result could be affected 
 by this attraction, I made the 9th, 10th, llth, and 15th experi- 
 ments, in which the balls were made to rest as close to the sides 
 of the case as they could ; but there is no difference to be de- 
 pended on, between the results under that circumstance, and 
 when the balls are placed in any other part of the case. 
 
 According to the experiments made by Dr. Maskelyne, on 
 the attraction of the hill Schehallien, the density of the earth 
 is 4 times that of water ; which differs rather more from the 
 preceding determination than I should have expected. But I 
 forbear entering into any consideration of which determination 
 is most to be depended on, till I have examined more carefully 
 how much the preceding determination is affected by irregular- 
 ities whose quantity I cannot measure. 
 
 * [See note on page 100.] 
 101 
 
APPENDIX 
 
 ON THE ATTRACTION OF THE MAHOGANY CASE ON THE BALLS 
 
 THE first thing is, to find the attraction of the rectangular 
 fc plane ck'pb (Fig 8) on the 
 
 point a, placed in the line 
 ac perpendicular to this 
 plane. 
 
 Let ac a, ck b, cb 
 
 x, and let 
 V 
 
 a 2 -h a* a 
 
 traction of the line bfl on 
 
 a, in the direction ab, = 
 
 - and therefore, if 
 *> ab x afi> 
 
 cb flows, the fluxion of the 
 attraction of the plane on the point a, in the direction cb, = 
 bx x bw bw 
 
 T X 
 
 = v 1 ', then the at- 
 
 Fig. 8 
 
 V* 1 + ^ 
 
 v w 
 
 :, the variable part of the fluent of which = log (v 
 
 v) 2 , and therefore the whole attractio 
 
 ck 4- ak 
 
 ion = log ( 
 -, . y ac 
 
 -^- ~\ so that the attraction of the plane, in the direction 
 bfi+afi) 
 
 cb, is found readily by logarithms, but I know no way of finding 
 its attraction in the direction ac, except by an infinite series.* 
 
 * \Playfair has given an expression in finite, terms for this attraction on pp. 
 225-8 of his paper in the Trans. Roy. Soc. Edin., vol. 6, 1812, pp. 187-243, 
 
 102 
 
MEMOIRS ON THE LAWS OF GRAVITATION 
 The two most convenient series I know, are the following : 
 First Series. Let = TT, and let A=arc whose tang, is TT, B 
 
 7T 3 7T 5 
 
 = A TT, C = B+-5~ , D = C -, etc. Then the attraction in the 
 o o 
 
 _ _ / , B?tf 2 3Cw 4 3-5Dw 6 \ * 
 direction ac= y'lw* x I A + -%~ + -%r + "2 T 4~- 6 ' . /' 
 
 For the second series, let A=arc whose tang. = , B=:A , 
 
 7T 7T 
 
 , D = C --=, etc. Then the attraction = a re. 90 
 
 7T J O7T 5 
 
 It must be observed, that the first series fails when IT is 
 greater than unity, and the second, when it is less ; but, if b is 
 taken equal to the least of the two lines ck and cb, there is no 
 case in which one or the other of them may not be used con- 
 veniently. 
 
 By the help of these series, I computed the following table : 
 
 
 .1962 
 
 .3714 
 
 .5145 
 
 .6248 
 
 .7071 
 
 .7808 
 
 .8575 
 
 .9285 
 
 .9815 
 
 i. 
 
 .1962 
 
 .00001 
 
 
 
 
 
 
 
 
 
 
 .3714 
 
 .00039 
 
 .00148 
 
 
 
 
 
 
 
 
 
 .5145 
 
 .00074 
 
 .00277 
 
 .00521 
 
 
 
 
 
 
 
 
 .6248 
 
 00110 
 
 .00406 
 
 .00778 
 
 .01183 
 
 
 
 
 
 
 
 .7071 
 
 00140 
 
 .00522 
 
 .01008 
 
 .01525 
 
 .02002 
 
 
 
 
 
 
 .7808 
 
 .00171 
 
 .00637 
 
 .01245 
 
 .01 896!. 02405 
 
 03247 
 
 
 
 
 
 .8575 
 
 00207 
 
 00772 
 
 .01522 
 
 02339.03116 
 
 .03964 
 
 .05057 
 
 
 
 
 .9285 
 
 00244 
 
 .00910 
 
 .01810 
 
 .02807. 03778!. 04867 
 
 06319 
 
 .08119 
 
 
 
 .9815 
 
 00271 
 
 .01019 
 
 .02084 
 
 .03193 
 
 .04868.05639 
 
 07478 
 
 09931 
 
 .12849 
 
 
 1. 
 
 00284 
 
 .01054 
 
 .02135 
 
 .03347 
 
 .045601.05975 
 
 .07978 
 
 .10789 
 
 .14632 
 
 .19612 
 
 ck 
 
 Find in this table, with the argument 7 at top, and the ar- 
 
 ctfc 
 
 cb 
 gurnent - in the left-hand column, the corresponding logarithm ; 
 
 entitled " Of the Solids of Greatest Attraction, or those which, among all the 
 8f>lirl& that have certain Properties, attract with the greatest Force in a given 
 Direction."] 
 
 * [In the last term of the series the coefficient D was omitted in the original.] 
 
 103 
 
MEMOIRS ON 
 
 then add together this logarithm, the logarithm of -j- t and the 
 
 cb afc 
 
 logarithm of ; the sum is the logarithm of the attraction. 
 
 To compute from hence the attraction of the case on the 
 A E ball, let the box DCBA (Fig. 
 
 1), in which the ball plaj^s, be 
 divided into two parts, by a 
 vertical section, perpendic- 
 ular to the length of the 
 case, and passing through 
 the centre of the ball ; and, 
 in Fig. 9, let the parallel- 
 epiped AEDEaMe be one of 
 these parts, ABDE being the 
 above-mentioned vertical sec- 
 tion ; let x be the centre of 
 the ball, and draw the par- 
 allelogram pnpmlx parallel to 
 and xgrp parallel to 
 and bisect /3S in c. 
 Now, the dimensions of the 
 box, on the inside, are Bft=1.75 ; BD=3.6; B/3=1.75; and 
 f)A 5 ; whence I find that, if xc and fix are taken as in the two 
 upper lines of the following table, the attractions of the differ- 
 ent parts are as set down below. 
 
 Fig. 9 
 
 Excess of attraction 
 
 Sum of these 
 Excess of attraction 
 
 xc 
 (3x 
 
 of T>drg above Bbrg 
 mdrp above nbrp 
 mesp above nasp 
 
 of Bbnp above Ddmd 
 Aanfl above JZemd 
 
 Whole attraction of the inside surface of the ) 
 half box. j" 
 
 .75 
 1.05 
 .2374 
 .2374 
 .3705 
 
 .5 
 
 1.3 
 .1614 
 .1614 
 .2516 
 
 .25 
 
 1.55 
 .0813 
 .0813 
 .1271 
 
 .8453 
 .5007 
 .4677 
 
 .5744 
 .3271 
 .3079 
 
 .2897 
 .1606 
 .1525 
 
 .1231 
 
 .0606 
 
 .0234 
 
 It appears, therefore, that the attraction of the box on x in- 
 creases faster than in proportion to the distance xc. 
 
 The specific gravity of the wood used in this case is .61, and 
 its thickness is f of an inch ; and therefore, if the attraction 
 of the outside surface of the box was the same as that of the 
 
 104 
 
THE LAWS OF GRAVITATION 
 
 inside, the whole attraction of the box on the ball, when ex 
 = .75, would be equal to 2 x.1231 x.61 xf cubic inches, or .201 
 spheric inches of water, placed at the distance of one inch from 
 the centre of the ball. In reality it can never be so great as 
 this, as the attraction of the outside surface is rather less than 
 that of the inside ; and, moreover, the distance of x from c can 
 never be quite so great as .75 of an inch, as the greatest motion 
 of the arm is only 1| inch. 
 
 Much has been written concerning the Cavendish experi- 
 ment ; the following references may be consulted to advantage. 
 
 Gilbert (40), in 1799, translated the greater part of Cavendish's 
 paper into German for his Annalen, adding many explanatory 
 notes. A few years later Brand es (42) gave a fresh mathemati- 
 cal analysis of the experiment, including the equations for the 
 time of swing of the torsion pendulum in the experiment pro- 
 posed by Muncke (see below). In 1815, the original paper of 
 Cavendish was translated entire into French by M. Chompre 
 (50). 
 
 In 1821, Hutton (54) recalculated the results of the experi- 
 ment after Cavendish's own formulae, and found, as he thought, 
 a "copious list of errata, some of which are large or import- 
 ant/' The mean of the first 6 experiments so corrected is 5.19, 
 and of the other 23 is 5.43 ; the mean of these two means is 
 5.31, which Hutton takes as the correct result given by the 
 Cavendish experiment. Baily states, however (79, pp. 92-96), 
 that Hutton himself had fallen into error, and that the com- 
 putations of. Cavendish are correct except in the one detail re- 
 ferred to on page 100 of this volume. Baily gives a very care- 
 ful criticism of the experiment on pp. 88-91 of his memoir. He 
 remarks that " Cavendish's object, in drawing up his memoir, 
 appears to have been more for the purpose of exhibiting a 
 specimen of what he considered to be an excellent method of 
 determining this important inquiry, than of deducing a result 
 that should lay claim to the full confidence of the scientific 
 world." Baily points out that the time was not determined 
 with due accuracy ; that the experiments were not arranged in 
 groups, in order to eliminate the error arising from the march 
 of the resting point; and that the distance between the weight 
 
 105 
 
MEMOIRS ON 
 
 and the ball was assumed constant. We shall see later from 
 the accounts of the investigations of Reich, Baily, Cornu and 
 Bailie, and Boys bow the errors in Cavendish's experiment have 
 been avoided. 
 
 Mnncke (61, vol. 3, pp. 940-70) has given an account of the 
 experiment and an admirable criticism of it, and compares the 
 result with that obtained by Maskelyne and Hufcton. He pro- 
 posed another method of using the torsion balance to find the 
 mean density of the earth ; he would find the time of vibration 
 with the masses first in the line of the balls and then in a 
 line at right angles to that direction. There would be no de- 
 flection to be measured. We have seen above that Brandes 
 gave the theory of this experiment. Investigations of this 
 nature have been made by Reich (83), Eotvos (192) and 
 Braun (193); for accounts of which see the latter part of this 
 volume. 
 
 A useful resume of Cavendish's paper was given by Schmidt 
 (64, vol. 2, pp. 481-7). He formed anew equations from which 
 to derive the value of the density of the earth, and found 5.52 
 using Cavendish's data. 
 
 Another mathematical investigation of the dynamical prob- 
 lem underlying the Cavendish experiment was made by Mena- 
 brea (71, 72 and 73), in 1840. It is a very elaborate analysis of 
 the whole problem. He examines the effect of the resistance 
 of the air on the time of vibration, and also shews how to find 
 the mass of the earth supposing that it is composed of spheroidal 
 layers of variable density. In Baily's memoir (79) is another 
 elaborate analysis, by Airy, of the mathematical theory of the 
 investigation. It treats especially of Baily's modification of 
 the Cavendish experiment (reproduced in Routh's Rigid Dy- 
 namics 1882, pt. 1, pp. 359-364). 
 
 An elementary treatment of the problem involved is given by 
 Gosselin (127), and from the formula he arrives at he derives 
 the value of the mean density of the earth as given by Caven- 
 dish's experiment, and gets 5.69. A similarly elementary treat- 
 ment by Babinet (132) gives 5.5. 
 
 An excellent account of Cavendish's work is given by Zanotti- 
 Bianco (148-J) and by Poynting (185, pp. 40-8) ; in the latter is 
 to be found a diagram showing the closeness of Cavendish's 
 separate results to the mean. 
 
 106 
 
THE LAWS OF GRAVITATION 
 
 CAVENDISH, son of Lord Charles Cavendish and a 
 nephew of the third Duke of Devonshire, was born at Nice in 
 1731 and died at London in 1810. He studied at Cambridge, 
 and becoming possessed, by the death of an uncle, of a large 
 fortune he devoted his life unostentatiously to private scientific 
 studies. Besides the investigation on gravitational attraction 
 here reprinted, he is remarkable for his researches in the field 
 of chemistry, and has been called the " Newton " of that sub- 
 ject. He worked on the constituents of the atmosphere and 
 on hydrogen ; he made the first synthesis of water, by burning 
 hydrogen in air, and found the density of hydrogen to be fa 
 (instead of -fa) of that of air. He determined the ratio of de- 
 phlogisticated to phlogisticated air to be about as 1 : 4. Caven- 
 dish also made many researches of great importance in the sub- 
 ject of electricity ; these have been collected and edited by 
 Clerk-Maxwell. Perhaps the most important of his electrical 
 investigations is that which proved that electrostatic attraction 
 takes place according to the law of the inverse square of the 
 distance. He is also the author of several papers on astronomi- 
 cal questions. Most of his writings are to be found in the 
 Philosophical Transactions of the period. 
 
 107 
 
HISTORICAL ACCOUNT OF THE EXPERI- 
 MENTS MADE SINCE THE TIME 
 OF CAVENDISH 
 
HISTORICAL ACCOUNT OF THE EXPERI- 
 MENTS MADE SINCE THE TIME 
 OF CAVENDISH 
 
 OARLINI. In 1821, Carlini, director of the Brera observatory 
 at Milan, made a series of experiments at the Hospice on Mt. 
 Cenis in the Alps, to determine the length of the seconds-pend- 
 ulum (55). He was led to do so from considering that the 
 Alps offered a favourable situation for a determination of the 
 mean density of the earth, and that no pendulum experiments 
 had been made there since the publication of the fictitious 
 ones of Coultaud and Mercier (see p. 47). Carlini compared 
 the time of vibration of a simple pendulum, made after the 
 general style of Borda's, with that of a standard clock whose 
 rate was noted daily. The height of the observing station was 
 1943 metres above sea-level, in latitude 45 14' 10". The cor- 
 rected length of the seconds -pendulum reduced to sea -level 
 was found to be 993.708 mm. Matthieu and Biot had found 
 the length of the "decimal" seconds-pendulum at Bordeaux, 
 in lat. 44 50' 25", to be 741.6151 mm. The calculated length 
 at Mt. Cenis would be 741.6421 mm. ; or, for the " sexagesimal " 
 seconds-pendulum (100000 decimal seconds = 86 400 sexages- 
 imal seconds) 993.498 mm. The difference between this length 
 and the observed length is .210 mm., which represents the at- 
 traction of the mountain on the pendulum. 
 
 The mountain is composed of schist, marble and gypsum, of 
 specific gravities 2.81, 2.86 and 2.32 respectively. Carlini took 
 the average of all three, 2.66, as the mean density of the hill. 
 Assuming that the hill was a segment of a sphere 1 geographical 
 mile in height and had a base of 11 miles in diameter, the at- 
 traction was calculated to be 5.0203, where 3 is the specific 
 gravity of the hill. With the same units the attraction of the 
 
 111 
 
MEMOIRS ON 
 
 earth is 14 394A, where A is the mean density of the earth.* 
 5.0203 .210 
 
 We cannot place very great confidence in this result, not only 
 on account of the fact that the extreme value of the length of 
 the seconds-pendulum varies from the mean by .032 mm. and 
 only 13 determinations were made ; but especially because the 
 size and density to be assigned to the mountain are largely a 
 matter of conjecture. 
 
 A resume of Carlini's paper was given by Saigey (56), and 
 by Schell (135). Excellent accounts of the experiment, with 
 criticisms of it, have been given by Zanotti-Bianco (148J, pt. 2, 
 pp. 136-45), Poynting (185,>pp. 22-4) and Fresdorf (186, pp. 
 8-11). 
 
 Sabine (58 and 82, notes, p. 47) remarks that Biot and Car- 
 lini had not properly reduced to vacuo the observed pendulum 
 lengths, and states that the corrected length of the seconds- 
 pendulum on Mt. Cenis is 39.0992 in. From the observations 
 made in the Formentera-Dunkirk survey he finds by interpo- 
 lation a pendulum length of 39.1154 in. for the latitude of the 
 Hospice. The difference between the observed and calculated 
 lengths is .0162 in. The difference calculated from the inverse- 
 square law is .0238 in. With Carlini's data and equations he 
 derives 4.77 for the value of A. 
 
 Schmidt (64, vol. 2, p. 480) gives a concise account of the 
 theory of the experiment, and remarks that Carlini made an 
 error in determining the attraction of a spherical segment. 
 Making the necessary correction he finds A = 4. 837, a result 
 not far from that of the Schehallien experiment. 
 
 In 1840, Giulio (70) also gave the true expression for the at- 
 traction of a spherical segment, and noted that several other 
 corrections must be made in Carlini's calculations ; the height 
 of the segment is 1.05 miles, iwstead of 1 mile ; the length of 
 the pendulum as determined by Biot must be corrected for an 
 error in the rule used to find the length, and for the altitude of 
 Bordeaux. Moreover, the reductions to vacuo had not been made 
 properly in either case. When all these corrections had been 
 applied to Carlini's results, Giulio found the value of A to be 4.95. 
 
 * This signification of A will be retained throughout the rest of the 
 volume. 
 
 112 
 
THE LAWS OF GRAVITATION 
 
 Saigey (74, p. 155) makes the observed pendulum length 
 corrected to vacuo 993.756 mm., and the calculated length 
 993.617 mm. With these numbers the value of A becomes 6.15. 
 
 Zanotti- Bianco (14% pt. 2, p. 136) mentions that Knopf 
 (149^) has compared the value of gravity as observed by Car- 
 lini on the top of the mountain with the value calculated for 
 the same place from observations made on the same parallel of 
 latitude, and found for A, 5.08. 
 
 AIRY, WHEWELL AND SHEEPSHANKS AT DOLCOATH MINE. 
 In 1826, Drobisch, in an appendix to a pamphlet on the figure 
 of the moon (57), suggested that experiments be made on the 
 change in the period of a pendulum when carried from the sur- 
 face of the earth to the bottom of a mine; he gave the theory 
 of the experiments and calculated the change resulting from 
 certain hypotheses. It is interesting to recall the fact that 
 Bacon proposed the same investigation two centuries earlier. 
 (See p. 1.) 
 
 At the very same time, unknown to Drobisch, experiments 
 of this nature were being tried in England by Airy and Whew- 
 ell at the copper mine of Dolcoath in Cornwall. Their meth- 
 od was to swing one invariable pendulum at the mouth of 
 the pit and compare its rate, by Rater's method of coincidences, 
 with that of a standard clock, and at the same time perform 
 the same operation upon another pendulum and another clock 
 at a depth of 1220 ft. in the mine. The pendulums were then 
 exchanged and the operations repeated. The greatest difficulty 
 experienced was that of comparing the rates of the two clocks. 
 The first series of experiments was abruptly stopped on account 
 of the damage received from fire by the lower pendulum. A 
 short account of the method was published in 1827 (60), and 
 Drobisch translated it for Poggendorff's Annalen (59), wherein 
 he gives also a more complete account of the theory and an ap- 
 plication of his equations to Airy's observations. Assuming 
 the mean density of the surface layer of the earth to be 2.587, 
 the experiments gave about 20 for the value of A. Drobisch 
 contends that the surface density should be taken to be 1/52, 
 considering how large an amount of the surface layer is water. 
 
 Two years later Airy and Whewell, assisted by Mr. Sheep- 
 shanks and others, attempted to repeat the experiments ; but 
 after overcoming various anomalies in the motions of the pend- 
 H 113 
 
MEMOIRS ON 
 
 ulums, the observations were stopped by a fall of rock in the 
 mine. The value of A found from this series was about 6. A 
 full account of the experiments was printed privately (62) in 
 1828, and Drobisch translated the pamphlet for the Annalen 
 (63). 
 
 REICH'S FIRST EXPERIMENT. In 1838, F. Reich, Professor 
 of Physics in the Bergakademie at Freiberg, published in book 
 form (67) the account of a series of experiments carried on by 
 him since 1835 to find, after the method of Cavendish, the 
 mean density of the earth. The adoption of the mirror and 
 scale method of measuring deflections seemed to him to prom- 
 ise a means of overcoming many of the difficulties against which 
 Cavendish had contended. The final observations were made 
 in the year 1837. 
 
 In order to avoid the effects due to irregularities of tempera- 
 ture, the apparatus was set up in a cellar room which was care- 
 fully closed up, and the observations made through a hole in 
 the door. The arm of the balance was 2.019 m. long, and its 
 moment of inertia was found after the manner used by Gauss 
 for a magnet. The average weight of each of the balls was 
 484.213 gr., and their distance below the arm was 77 cm. They 
 were composed of an alloy of about 90 parts tin, 10 parts bis- 
 muth, and a little lead. The attracting masses were of lead 
 45 kg. in weight and about 20 cm. in diameter, and hence 
 much smaller than those used ,by Cavendish. They were sus- 
 pended from pulleys running on rails parallel to the arm of the 
 balance, and could be quickly moved from the null to the at- 
 tracting positions. Only one mass was in the attracting posi- 
 tion at a time, on account of the fact that in every one of the 
 four attracting positions the distance from the mass to the ball 
 was slightly different ; whereas Cavendish used both masses at 
 once. The distance from mass to ball was measured at each 
 observation by means of a telescope moving along a horizontal 
 scale, and not once for all as was done by Cavendish. 
 
 After the suspended system was set up, Reich found a con- 
 tinual changing of the zero-point, which often lasted for 6 
 months. In his final observations this was not noticeable be- 
 cause of the length of time, 1^ years, which intervened between 
 the initial and final experiments. Accordingly the second 
 means of the elongations were found by him to be more con- 
 
 114 
 
THE LAWS OF GRAVITATION 
 
 stant than Cavendish found them. The following table will 
 show this, and also illustrate Reich's method of finding the 
 time of vibration : 
 
 Extremes 
 
 1st mean 
 
 2d mean 
 
 Time of passage 
 
 at 74. 5 
 
 at 75.5 
 
 95.2, 
 
 
 
 
 
 
 
 76.00 
 
 
 IV h 27' 36". 8 
 
 IV h 27' 30". 8 
 
 56.8 
 
 
 
 
 
 
 
 73.85 
 
 74.925 
 
 34 21 .2 
 
 34 29 .2 
 
 90.9 
 
 
 
 
 
 i 
 
 75.90 
 
 74.875 
 
 41 22 .4 
 
 41 12 .4 
 
 60.9 
 
 
 
 
 
 
 
 74.25 
 
 75.075 
 
 47 59 .2 
 
 48 10 .8 
 
 87.6' 
 
 
 
 
 
 Average or 3d mean 74.9583 
 Time of vibration determined from passage of 74.5 
 
 IV h 41' 22".4-IV h 27' 36". 8=13' 45". 6 
 47 59 .2- 34 21 .2=13 38 .0 
 Time of vibration determined from passage of 75.5 
 
 IV h 41' 12".4-IV h 27' 30".8=13' 41".6 
 
 48 10 .8- 
 
 34 29 .2=13 41 .6 
 
 aver. 13' 41". 6 
 
 By interpolation the time of a double vibration across 74.9583 
 is 13' 41". 708. This differs somewhat from Cavendish's method, 
 as will be seen by a reference to page 65. Reich considered it 
 a more accurate method than that of Cavendish, and remarks 
 that when he applied the hitter's method to the above observa- 
 tions he got results not very different, but on applying his own 
 method to Cavendish's observations he got somewhat different 
 results. Reich's method was adopted by Baily (79, pp. 44 and 
 47) in his experiments. 
 
 When in the above way at least three passages across the two 
 median points had been observed, Reich waited until the arm 
 was in its next extreme position and seemingly at rest and then 
 rapidly moved the attracting mass to its new position. It was 
 always so moved as to increase the swing. He assumed that 
 the motion was instantaneous, and used the last extreme of the 
 one series as the first of the next. Baily (79, p. 46) followed 
 him in this departure from the procedure of Cavendish. Cornu 
 and Bailie (142) have pointed out that this method leads to 
 error, and we shall refer to the matter again when describing 
 the results obtained by Baily (page 116). From each such set 
 of 4 extremes the resting-point and the time of vibration are 
 
 115 
 
MEMOIRS ON 
 
 found. From 2 such sets the deviation could be found, and 
 the mean density of the earth calculated. Reich did not, how- 
 ever, proceed in that way, but deduced one value of A from all 
 the observations of each day ; that is, he took the average of 
 all the deviations of that day for the final mean deviation, and 
 the average of all the times of vibration for the final mean time 
 of vibration, and from these deduced one value for the mean 
 density of the earth. 
 
 In applying corrections to the equations derived from a simpli- 
 fied form of the theory of the experiment, Reich followed Cav- 
 endish exactly. 57 observations were made, from which 14 de- 
 terminations of the value of A were deduced. The mean of all, 
 when corrected for the centrifugal force, was 5. 44 .0233, a re- 
 sult almost coinciding with Cavendish's. Reich admits at the 
 end of his paper that there were certain anomalies in the 
 motion of the beam which he could not account for. 
 
 A second series of 6 observations with iron masses 30 kg. in 
 weight and 20 cm. in diameter gave for A, 5.4522, which proves 
 that no disturbance could have arisen from magnetic action. 
 
 Valuable concise accounts of Reich's experiment are given 
 by Beaumont (66), Baily (68, 69 and 79, pp. 96-8), Schell 
 (135), Poynting (185, pp. 48-50) and Fresdorf (186, pp. 20-2). 
 
 BAILY. While Reich was making the investigations just re- 
 ferred to, a very comprehensive and elaborate series of experi- 
 ments upon almost the same plan was being carried on by the 
 English astronomer F. Baily. These experiments were under- 
 taken at the instance of the Royal Astronomical Society, and 
 in aid of them a grant of 500 was made by the British govern- 
 ment. The results were published in 1843 (79). They were 
 carried on in one of the rooms of Baily's residence, a one-story 
 house standing detached in a large garden. The apparatus was 
 almost the counterpart of that of Cavendish, except that the 
 balls were not suspended from the balance arm, but were 
 screwed directly on to its ends. The balance and its mahogany 
 case were, moreover, suspended from the ceiling, and the at- 
 tracting masses rested on the ends of a plank movable on a 
 pillar rising from the floor. As a protection against changes 
 of temperature this apparatus was then surrounded by a wooden 
 enclosure. The masses were of lead rather more that 12 in. in 
 diameter, weighing 380,469 Ibs. each. Torsion rods of deal and 
 
 116 
 
THE LAWS OF GRAVITATION 
 
 of brass, each about 77 in. long, were employed, and their 
 motion was observed by the mirror and scale method. Balls of 
 different materials and of various diameters were experimented 
 upon : viz., 1.5 in. platinum, 2 in. lead, 2 in. zinc, 2 in. glass, 
 2 in. ivory, 2.5 in. lead, and 2.5 in. hollow brass. The mode 
 of suspension was varied greatly, both single and double sus- 
 pension wires being used, and the material and distance apart 
 of the bifilar wires being frequently changed. The length of 
 the suspending wires was ordinarily about 60 in., and the time 
 of vibration varied from about 100 to 580 seconds. 
 
 The experiments were begun in Oct. 1838, and carried on 
 for 18 months, until about 1300 observations had been made ; 
 when, on account of the great discordance of the results, a 
 stop was made. Prof. Forbes suggested that these anomalies 
 might arise from radiation of heat, and advised the use of gilt 
 balls and, a gilt case. These changes were made, and the tor- 
 sion box also lined with thick flannel. They turned out to be 
 decided improvements, although some anomalies still existed, 
 and it is evident that the choice of a place for setting up the 
 apparatus was not a good one. 
 
 Baily adopted the method of Reich for reducing the time re- 
 quired to make the number of turning-points requisite for cal- 
 culating the deviation and period ; that is, the masses were 
 moved quickly from one near position to the other, and the last 
 turning-point of one series served for the first of the next. 
 Three new turning-points were observed at each position of the 
 masses, and each group of 4 was called an "experiment." 
 2153 such experiments were made during the years 1841-2. 
 The time of vibration was found for each experiment after the 
 method adopted by Reich. In deducing the mean density of 
 the earth from the observations Baily proceeded quite differ- 
 ently from Reich. There was always a slow motion of the zero- 
 point, and Baily, in order to take account of this, combined 
 the deflections and periods in threes. The difference between 
 the deflection of the 2d experiment and the average of the 1st 
 and 3d is twice the mean deviation. The average of the period 
 of the 2d experiment with the average of the 1st and 3d is the 
 mean period. From the mean deviation and mean period so 
 found a value of A is deduced. Another was then found from 
 comparing the 3d experiment with the 2d and 4th, and so on. 
 The mean of all the experiments gave for A, 5.6747.0038. Some 
 
 117 
 
MEMOIRS ON 
 
 of the experiments were made with the brass rod alone, without 
 any balls, the mean result for which was 5. 0666 .0038. 
 
 The mathematical analysis of the problem was given by 
 Airy, and is incorporated in Daily's paper (79, pp. 99-111); it 
 is also to be found in Routh's Rigid Dynamics, 1882, pt. 1, 
 pp. 359-64. 
 
 Baily published a condensed account of his work in several 
 journals (75, 76, 77, 78 and 80). A careful discussion of it is 
 given by Schell (135) and by Poynting (185, pp. 52-7). 
 
 In 1842, Saigey (74) wrote a full account of all the experi- 
 ments made before that date ; he gives his reasons for consider- 
 ing the pendulum method of finding A the least accurate, the 
 mountain method, somewhat better, and the torsion method the 
 best. He, finds great fault with the work of Baily, and con- 
 siders that his results are not so worthy of confidence as those 
 of Cavendish. Saigey contends that the anomalies observed 
 by Cavendish, Reich, and Baily cannot be accounted for by 
 radiation of heat, as Forbes suggested, because the balance 
 swings in an enclosure all points of which are at the same tem- 
 perature (thus begging the question); he confidently remarks 
 that these anomalies are caused by the passage of air into or 
 out of the case as the barometric pressure changes. The values 
 of A found by Baily increased from 5.61 to 5.77 as the density 
 of the balls used changed from 21.0 to 1.9 respectively; Saigey 
 thinks that this must arise from an error in calculating the 
 moment of inertia of the balance arm. He devises a graphical 
 method of making proper allowance for this supposed error, 
 and deduces as the final mean of all the experiments of Baily a 
 value 5.52, the extremes being 5.49 and 5.55. 
 
 Saigey made a new determination of A (74, vol. 12, p. 377), 
 from the difference 6". 86 of the astronomical and geodetical 
 latitudes of Evaux as calculated by Puissant. Applying the 
 method used by Hutton for Schehallien, and later by James 
 and Clarke for Arthur's Seat, he found the ratio of A to the 
 surface density of France to be 1.7. Assuming the latter den- 
 sity to be 2.5, the former becomes 4.25. 
 
 In 1847, Hearn tried to account for the anomalies in Baily's 
 results by assuming a magnetic action. He worked out the 
 theory (81) of such action, and found that it must be of a very 
 fluctuating nature and may be either positive or negative, and 
 even greater in magnitude than the force of gravitation. That" 
 
 118 
 
THE LAWS OF GRAVITATION 
 
 such a magnetic action does not really exist is to be deduced 
 from Reich's results with iron masses (see page 116). 
 
 Montigny offered to the Royal Academy of Belgium, in 1852, 
 a memoir in which he attributed the peculiarities in the be- 
 haviour of the torsion pendulum in the experiments of Caven- 
 dish and of Bciily to the rotation of the earth. Schaar (85), to 
 whom the memoir was referred by the Society, proved that the 
 rotation of the earth could not produce such effects, and the 
 memoir was not published. 
 
 It was Cornu and Bailie who first pointed out (142), in 1878, 
 the main error in Baily's method. It lies in his taking the 4th 
 reading of the turning-point of one series of experiments as the 
 1st of the next, as already explained. They shewed that the 
 rotation of the plank holding the masses could not be per- 
 formed rapidly enough to get the masses into the new position 
 before the arm had begun its return journey. They therefore 
 rejected the 1st of each series of 4 readings, and calculated A 
 from the other 3 in 10 cases taken at random from some of 
 Baily's most divergent values, and found 5.615 instead of 5.713. 
 Reducing Baily's final value in the same proportion they got 
 5.55. 
 
 A curious relation between density and temperature as pre- 
 sented in Baily's determinations was pointed out by Hicks 
 (166), in 1886. The mean density seems to fall with rise of 
 temperature. The most probable explanation of this is given 
 by Poynting (185, p. 56), who remarks that the experiments 
 with the light balls happened to be made in winter, and those 
 with the heavy balls in summer. Hicks also refers to several 
 slight corrections to be made in Airy's discussion of the theory 
 viz., for the air displaced by the attracting masses, for the 
 inertia of the air in which the balls move, and for expansion 
 with change of temperature. 
 
 REICH'S SECOND EXPERIMENT. Ten years after the appear- 
 ance of Baily's memoir, Reich published (83) an account of 
 some farther experiments with his apparatus. In the begin- 
 ning of his paper he pointed out that Baily's method of com- 
 bining the results of the separate experiments was better than 
 that used by himself. He proceeded to calculate the results 
 of his first experiments by Baily's method and found for A the 
 value 5.49.020. 
 
 119 
 
MEMOIRS ON 
 
 Being impressed with the anomalies in Baily's observations, 
 and especially with the variation of the final results with the 
 density of the balls, Reich determined to repeat his experi- 
 ments. His apparatus was set up this time in a second-story 
 room, and Baily's devices were employed in order to avoid the 
 effects of temperature changes. The only important change in 
 the arrangement of the apparatus was in the placing of the large 
 mass. It was now set in one of four depressions 90 apart in a 
 circular table revolving under the balance about a vertical axis 
 passing through the centre of one of the balls ; thus no correc- 
 tion was necessary for the attraction of the table and its sup- 
 ports upon the ball. The balls and masses were those used in 
 the first experiment. Three series of experiments were made 
 during the years 1847-50, one with a suspending wire of thin 
 copper, one with thick copper, and one with a bifilar iron sus- 
 pension. The final mean density of the earth was found to be 
 5.5832.0149. 
 
 In order to make a test of Hearn's explanation (see page 118) 
 of the peculiarities in Baily's results, Reicli made some further 
 experiments. He kept the North pole of a strong magnet near 
 the attracting lead mass for a whole day, and then suddenly 
 rotated the mass through 180 about a vertical axis; but no 
 effect was evident. Hence variations in the result are not due 
 to the magnetizing of the masses by the earth, or similar 
 causes. He then took off the tin balls and substituted success- 
 ively balls of bismuth and of iron. The values of A were re- 
 spectively 5.5233 and 5.6887 ; the largeness of the latter denotes 
 possibly a diamagnetic action of the lead mass ; but it shews 
 that under the original circumstances no measurable effect 
 could have arisen from magnetic action. 
 
 Prof. Forbes had suggested * to Reich that A could be found 
 from the period of the balance only, by noting the variation of 
 the time of vibration with the position of the attracting masses. 
 'Reich made some experiments of this nature by placing two 
 lead masses diametrically opposite to each other, first so that 
 the line joining them was perpendicular to the vertical plane 
 through the torsion arm, and next was in the plane. This 
 caused no deviation, but only a change in the time of swing of 
 
 * We have seen (page 106) that this method was suggested earlier, in 
 Gehler's PhysikaliscJie Worterbuch, and the equations given by Brandes (42). 
 
 120 
 
THE LAWS OF GRAVITATION 
 
 the balance. The value of A found in this way was 6.25, but 
 the apparatus was not well devised for the work. 
 
 Several abstracts of Reich's paper are to be found (84, 86, 
 87 and 185, pp. 50-2). 
 
 AIRY'S HARTON COLLIERY EXPERIMENT. We have already 
 referred to Airy's experiments in the Dolcoath mine in 1826-8. 
 In 1854, he again undertook to carry out investigations (100) 
 along the same lines, the introduction of the telegraph having 
 made easy the comparison of the clocks at the top and bottom 
 of the mine. He selected the Harton Colliery, near South 
 Shields, for the experiments, which were carried out by six ex- 
 perienced assistants of whom Mr. Dunkin was the chief. The 
 two stations were vertically above each other and 1256 ft. apart. 
 The apparatus was the best obtainable, and special precautions 
 were taken in order that the pendulum supports might be rigid. 
 
 Simultaneous observations of the two pendulums were kept 
 up night and day for a week ; then the pendulums were ex- 
 changed and observations taken for another week. Two more 
 exchanges were made, but the observations for them both were 
 made in one week. Each pendulum had six swings of nearly 
 4 hours each on every day of observation, and between success- 
 ive swings the clock rates were compared by telegraphic signals 
 given every 15 seconds by a journeyman clock. 
 
 The corrections and reductions were carried out by Airy in a 
 very elaborate manner. The results of the 1st and 3d series 
 agree very closely, as do those of the 2d and 4th, showing that 
 the pendulums had undergone no sensible change. By com- 
 paring the mean of the 1st and 3d series with the mean of the 
 2d and 4th, the ratio of the pendulum rates at the upper and 
 lower stations is obtained independently of the pendulums em- 
 ployed. The final result gave gravity at the lower station 
 greater than gravity at the upper by Trrrs^ 1 part, with an un- 
 certainty of g+oth P art f ^ ie i ncl 'ease ; or the acceleration of 
 the seconds-pendulum below is 2". 24 per day, with an uncer- 
 tainty of less than 0".01. 
 
 In order to calculate what this difference should be, suppose 
 the earth to be a sphere of radius r and mean density A, sur- 
 rounded by a spherical shell of thickness h and density S, then 
 
 , gravity below 2h 37/3 , 
 
 a simple analysis shews that 1-| (corn- 
 
 gravity above r rA 
 
 121 
 
MEMOIRS ON 
 
 pare p. 31). Airy gives a discussion of the effect of surface ir- 
 regularities ; it is shewn that, supposing the surface of the earth 
 near the mine to have no irregularities, the effect of those at dis- 
 tant parts of the earth may be neglected. He also assumes that 
 there is no sudden change of density just under the mine. He 
 proves that the effect of a plane of 3 miles in radius and of the 
 thickness of the sh'jll is ff of that of the whole shell, so that 
 only the neighbouring country need be surveyed. Since the 
 upper station is only 74 ft. above high water, it will be sufficient 
 to assume that any excess or defect of matter exists actually on 
 the surface. A careful survey of the environs of the mine was 
 made, and allowance made for each elevation and depression. 
 The general result is that the attraction of the regular shell of 
 
 matter is to be diminished by about ^-th part : - 
 
 gravity above 
 
 = 1.00012032-. 00017984 x-. Now from the pendulum ex- 
 
 periments Airy found = 1.00005185 . 00000019 ; 
 
 gravity above 
 
 hence ^=2. 6266 .0073. Prof. W. H. Miller found the aver- 
 o 
 
 age density of the rocks in the mine to be 2.50; hence Ar^6.566 
 .0182. 
 
 Airy had intended that the temperatures at the two stations 
 should be the same, but the temperature of the lower station 
 was 7. 13 F. higher than that of the upper. In a supplement- 
 ary paper (101) Airy makes a correction for this temperature 
 difference in two distinct ways, giving for the corrected A, 6.809 
 and 6.623 respectively. In this paper Stokes (102) investigates 
 the effect of the earth's rotation and ellipticity in modifying 
 the results of the Harton experiments. It was found to be 
 small, changing A from 6.566 to 6.565. 
 
 Airy published several preliminary notices of his work (88, 89 
 and 122), abstracts of which appeared in several journals (90, 
 91, 92, 98 and 111). Valuable resumes of the main paper are 
 also to be found (105, 107, 109, 112 and 119). 
 
 Haughton (106, 110, 113 and 116) gave a rough but simple 
 method of deducing A from Airy's figures, and arrived at 5.48 
 as the value of A. Knopf (149|) has severely criticized this 
 calculation. Another simple formula for the same purpose 
 was given by an anonymous writer (114). On the effect of 
 
 122 
 
THE LAWS OF GRAVITATION 
 
 great changes in density below the under station one should read 
 the paper by Jacob (118 and 121) already referred to. Schef- 
 fler (134) published in 1865, though it is dated 1856, the pro- 
 posal of an experiment similar to Airy's, but made no reference 
 to any earlier proposals of the same kind. Folie (136) calcul- 
 ated, in 1872, the attraction at the two stations in a manner 
 different from Airy's, by considering the shell as made up of 2 
 parts. Using Airy's data he arrived at 6.439 as the value of A. 
 Valuable summaries and criticisms of Airy's work are given 
 by Schell (135), Zanotti-Bianco (148J, pt. 2, pp. 146-60), Poynt- 
 ing (185, pp. 24-9) and Fresdorf (186$, pp. 13-7). 
 
 JAMES AND CLARKE. As a result of the calculations made 
 from the observations taken for the Ordnance Survey of Great 
 Britain and Ireland (104, 117, 120, 124, 125 and 126) by Lt. 
 Col. James, it was found that the plumb-line was considerably 
 deflected at several of the principal trigonometrical stations. 
 It was evident from the nature of the ground at the places 
 under consideration, that this deflection was due to irregulari- 
 ties of the surface. In order to study this action more care- 
 fully James decided to have the Schehallien experiment re- 
 peated at Arthur's Seat, near Edinburgh (103 and 125, pp. 572- 
 624). The observations were made during Sept. and Oct., 
 1855, with Airy's zenith -sector, on the summit of Arthur's 
 Seat (A), and at points near the meridian on the north (N) 
 and south (S) of that mountain, at about one-third of its al- 
 titude above the surrounding country. After corrections had 
 been applied, the results were as follows : 
 
 Station Astronomical lat.EEA Geodetical lat.=G A G 
 
 8 55 56' 26".69 55 56' 24".25 2".44 
 
 A 56 43 .69 56 38 .44 5 .25 
 
 N 57 9 .22 57 2 .71 6 .51 
 
 It will be noticed that even on the summit of the hill there 
 is an attraction of more than 5" toward the south, which can 
 not be due to the hill. Similarly, to the south of the hill the 
 attraction is not toward the north as we might expect. It is 
 evident that there is present some other attracting force, be- 
 sides that of Arthur's Seat, which appears to produce a general 
 deflection of 5" toward the south. 
 
 123 
 
MEMOIRS ON 
 
 Capt. Clarke, who made all the calculations, in order to find 
 the attraction according to Newton's law, used a modification of 
 the method of Hutton. He took account of all the surface 
 irregularities within a radius of about 24000 ft. The resulting 
 value for the ratio of the density of the rock composing the hill 
 to that of the whole earth was .5173 .0053. James investi- 
 gated the density of the rocks of Arthur's Seat and found it 
 to be on the average 2.75. This gives for A the value 5.316 
 .054. 
 
 In order to see whether the general deflection of 5" could be 
 accounted for by the presence of the hollow of the River Forth 
 to the north and the high land of the Pentland Hills to the 
 south, Clarke extended the calculated attraction to the borders 
 of Edinburghshire, some 13 miles away. He was able in this 
 way to account fora general deflection of 2".52, and he thought 
 that by carrying the calculations to Peeblesshire the whole 5" 
 might be accounted for. 
 
 Several abstracts of the original paper have been published 
 (108, 115 and 123). Poynting (185, pp. 19-22) has given a 
 valuable criticism of the work. 
 
 In connection with this investigation might be mentioned 
 the various writings on the subject of local attractions. Any 
 one wishing to become acquainted with this subject should read 
 Airy's account of his " flotation theory " (94 and 97), Faye's 
 account of his "compensation theory" (130, 146J and 147), 
 Pratt's papers (93, 96 and 99), Saigey (74), Struve (129), Pech- 
 mann (131), the treatises of Pratt (133), Clarke (149) and Hel- 
 mert (148, vol. 2). Many other references to papers by these 
 men as well as by Schubert, Peters, Keller, Bauernfeind and 
 others are to be found in the Roy. Soc. Cat. of Scientific papers 
 and in Gore's " A Bibliography of Geodesy" (174). See also 
 note on page 31 and remarks on page 56. We might here re- 
 call the determination of A by Saigey from local attraction (see 
 page 118). Pechmann (131) in the same way found in the Tyrol, 
 in 1864, two different values for A, 6.1311 .1557 and 6.352 
 .726, having assumed the density of the earth's crust to be 
 2.75. We shall refer later on to the determinations of Men- 
 denhall and Berget. 
 
 AND BAILLE. In 1873, Cornu and Bailie published 
 a short paper (137) stating that they had undertaken to repeat 
 
 124 
 
THE LAWS OF GRAVITATION 
 
 the Cavendish experiment under conditions as different as pos- 
 sible from those previously employed. They began by making 
 a thorough study of the torsion-balance in order to learn under 
 what conditions it would have the greatest precision and sensi- 
 tiveness. They found among other things that the resistance of 
 the air was proportional to the velocity (141, 142, 143 and 157). 
 
 The apparatus was set up in the cellar of the iScole Polytech- 
 nique. The arm of the balance was a small aluminium tube 
 50 cm. long, carrying on each end a copper ball 109 gr. in 
 weight. The suspension wire was of annealed silver 4.15 m. 
 long, and the time of vibration of the system 6' 38". The at- 
 tracting mass was mercury which could be aspirated from one 
 spherical iron vessel on one side of one of the copper balls to 
 another vessel similarly situated on the other side of the ball. 
 This method got rid of the disturbances arising from the move- 
 ment of the lead masses in the Cavendish form of the experi- 
 ment. The iron vessel was 12 cm. in diameter and the mer- 
 cury weighed 12 kg. Another great improvement was the 
 reduction of the dimensions of the apparatus to J of that used 
 by Cavendish, Reich and Baily, the time of oscillation and the 
 sensitiveness remaining the same. The motion of the arm was 
 registered electrically. 
 
 Two series of observations were made ; one in the summer 
 of 1872 gave A = 5. 56, and the other in the following winter 
 5.50. The difference was explained by a flexure of the torsion- 
 rod, and the former result was considered the better. 
 
 In a later report (142) they refer to some changes made in 
 their apparatus ; they increased the force of attraction by using 
 4 iron receivers, 2 on each side of each copper ball, and they 
 reduced the distance between the attracting bodies in the ratio 
 of -v/2 to 1. The time of vibration, 408", remained the same 
 within a few tenths of a second for more than a year. The new 
 value of A was 5.56. 
 
 We have already referred (page 119) to the fact that Cornu 
 and Bailie found out the error in the Baily experiments. 
 
 A final account of these experiments has not yet been pub- 
 lished. Abstracts of the papers cited are given by Poynting 
 (185, pp. 57-8) and by several journals (138 and 139). 
 
 JOLLY. In 1878, von Jolly of Munich published an account 
 (144 and 145) of the results of his study of the beam balance 
 
 125 
 
MEMOIRS ON 
 
 as an instrument for measuring gravitational attractions. He 
 discussed the sources of error in the balance readings and 
 methods of eliminating them. The variations due to tempera- 
 ture effects are very difficult to avoid, but by working in the 
 mornings only, and by covering the balance case with another 
 lined inside and out with silver paper, it was found to be pos- 
 sible to get quite concordant results. 
 
 Jolly applied the balance to test the Newtonian law of the 
 distance. Two extra scale pans were suspended by wires from 
 the ordinary scale pans of the balance and 5.29 m. below them. 
 The wires and lower scale pans were enclosed to prevent oscilla- 
 tions from air currents. Two kilogramme masses of pplished 
 nickel-plated brass were balanced against each other, first both 
 in the upper scale pans, and then one in the upper and the 
 other in the lower pan, in each case double weighings being made 
 after the manner of Gauss. The motion of the beam was noted 
 by the mirror and scale method, the mirror being fixed at the 
 middle of the beam and perpendicular to its length. If r is the 
 radius of the earth at sea-level, and h a height above it, then a 
 
 mass Qi at sea-level weighs Q 2 at h, where Q 2 =Qj (l - ) ap- 
 
 ,Q 2 1 000 000-1.5099 
 proximately. Jolly found by experiment ^-= , 
 
 V^j 
 
 Q 2 1 000 000-1.662 
 
 whereas the equation gives ^= TT^TTAA; ^ ne differ- 
 
 Vi 
 
 ence, .152 mg., Jolly thought, was due to local attractions. He 
 proposed to repeat the experiment at the top of a high tower, 
 and at the same time to find the mass of the earth by noting 
 the change in weight of one of the masses in the balance when 
 a large lead ball was brought beneath it. 
 
 The results of these experiments (153 and 154) were published 
 in 1881. The distance between the scale pans was now 21.005 
 m. The arm of the balance was 60 cm. long, and the maxi- 
 mum load 5 kg. Four hollow glass spheres of the same size 
 were made and in each of two 5 kg. of mercury were put, 
 and all were sealed up. Each scale pan had always one sphere 
 in it, and thus air corrections were avoided. An observation 
 was made as follows : first the mercury-filled spheres were bal- 
 anced in the upper pans, and then one in the upper pan was 
 balanced against the other in the lower. The change in weight 
 
 126 
 
THE LAWS OF GRAVITATION 
 
 observed was 31.686 mg.; whereas the change as calculated 
 from the formula should have been 33.059* mg. The differ- 
 ence is in the same direction as in the earlier experiment. 
 
 A sphere of radius .4975 m. and weight 5775.2 kg. was then 
 built up out of lead bars under the lower scale pan which 
 received the mercury-filled globe. The distance from the 
 centre of this sphere to that of the globe was then .5686 m. 
 The attraction of the sphere for the mercury-filled globe when 
 in the upper pan was neglected. 
 
 Observations were made exactly as before, and the change in 
 weight was 32.275 mg. The increase in weight due to the 
 presence of the lead is therefore .589 mg. Knowing the den- 
 sity of the lead to be 11.186, a simple calculation gives for the 
 mean density of the earth 5. 692 .068. 
 
 An account of these experiments is given by Helmert (148, 
 vol. 2, pp. 380-2), Zanotti- Bianco (148-|, vol. 2, pp. 175-82), 
 Wallentin (154J), Keller (167), Poynting (185, pp. 61-4) and 
 Fresdorf (186J, pp. 23-5). 
 
 MENDENHALL. In 1880, Prof. T. C. Mendenhall described 
 (150) a method of finding the period of a pendulum such that 
 a determination required 20 or 30 minutes only. At the begin- 
 ning and end of this time the pendulum throws a light trip- 
 hammer of wire which breaks a circuit and makes a record on 
 a chronograph on which a break-circuit clock is also marking. 
 The advantage of such an arrangement, in addition to the 
 short time required, is that the arc of vibration may be small 
 and will change very little. Mendenhall expressed a deter- 
 mination to find the variation of the acceleration due to gravity 
 on going from Tokio to the top of Mount Fujiyama. 
 
 A year later the results of these experiments were published 
 (151), having been made in Aug., 1880. An invariable pend- 
 ulum was used, made from a Kater's pendulum by removing 
 one ball and knife-edge. Its period at Tokio (barometer 30 in. 
 and temperature 23. 5 C.) was .999834 sec. On the top of 
 Fujiyama the barometer stood nearly stationary at 19.5 in. 
 during the observations, and the thermometer at 8. 5. After 
 approximate corrections were made for buoyancy, the time, 
 reduced to Tokio conditions, was 1.000336 sec. Assuming g at 
 
 * According to Helmert this should be 33.108 and according to Zanotti- 
 Bianco 33.053. 
 
 127 
 
MEMOIRS ON 
 
 Tokio to be 9.7984, as he had found in the previous year, it fol- 
 lows that at the summit of Fujiyama it is 9.7886. 
 
 No exact triangulation of the region had been made, but 
 Mendenhall assumed Fujiyama to be a cone 2.35 miles high 
 standing on a plain of considerable extent. The angle of the 
 cone was measured from photographs and found to be 138. 
 Fujiyama is an extinct volcano, said to have been made in a 
 single night, and hence its composition ought to be homogene- 
 ous. Its average density was taken as 2.12, but no great re- 
 liance can be placed on this number. Corrected for the differ- 
 ence in latitude, 19', between Tokio and Fujiyama, the time at 
 its base, supposing the hill taken away, would be .999847 sec. 
 The density of the earth, calculated from these data after the 
 manner of Carlini, was found to be 5.77. 
 
 Fresdorf (186^, pp. 11-13) describes fully the experiments 
 and -points out an error in Mendenhall's calculations ; the cor- 
 rected value for A is 5.667. Poynting (185, pp. 39-40) gives 
 -an abstract of the papers referred to. 
 
 STERNECK. Major von Sterneck has made several investiga- 
 tions of the variation of gravity beneath the earth's surface. 
 The earliest experiments (155) were made, in 1882, in the 
 Adalbert shaft of the silver mine at Pribram in Bohemia. The 
 method employed was to carry an invariable half-second pend- 
 ulum and a comparison clock from one station to another, and 
 find the period by the method of coincidences, the clock being 
 compared with a standard clock by carrying a pocket chron- 
 ometer from one to the other. The pendulum, of brass, was 
 a rod 24 cm. in length carrying a lens-shaped bob weighing 1 
 kg. The knife was of steel whose edge was so cut away that 
 it rested on a glass plate on two points only. The apparatus 
 was always enclosed in a glass case to prevent air currents. 
 The 3 stations at which observations were made were at the 
 surface, 516.0 m. and 972.5 m. below the surface respectively. 
 The respective periods at these stations were .5008550, .5008410 
 and .5008415 seconds, and the resulting values of A, found 
 from Airy's formula, were 6.28 and 5.01, the density of the 
 surface layer being taken as 2.75. It will be noticed that the 
 values of g at the two underground stations are practically the 
 same, and the results are unsatisfactory. 
 
 A year later (156) von Sterneck repeated his experiments at 
 
 128 
 
THE LAWS OF GRAVITATION 
 
 the same stations and at two additional ones. In order that 
 his observations might be independent of the rates of the clocks 
 used in finding the periods, Sterneck introduced an important 
 modification of the method adopted by Airy and by himself in 
 his earlier investigations. He made another pendulum similar 
 to the one described above ; one of these was always at the 
 surface station and the other at one of the underground sta- 
 tions, and their relative periods were compared by means of 
 electric signals sent simultaneously from a single clock. This 
 clock kept a circuit closed for half a second every other half 
 second and operated a relay with a strong current at each sta- 
 tion. The passage of the "tail" of the pendulum in front of 
 a scale was observed by means of a telescope, in the focal plane 
 of which was a shutter moved by the relay current every half 
 second, and at those instants only was the picture of the tail 
 of the pendulum allowed to pass to the eye through the tele- 
 scope. The time of a coincidence was when at one of these 
 flashes the tail appeared exactly at the middle of the scale ; the 
 time between two successive coincidences determines the period 
 of the pendulum. The observer at each of the two stations is 
 thus finding the period of his pendulum in terms of exactly the 
 same unit of time. When the observations were corrected, it 
 was found that the period at the highest underground station 
 was less than that at the next lower station, and the determin- 
 ation at the former station was consequently not used. The 
 values of A as determined from observations at the other sta- 
 tions were 5.71, 5.81 and 5.80, with a mean of 5.77. Helmert 
 (148, vol. 2, p. 499) has made a recalculation and finds that 
 these numbers should be 5.54, 5.71, 5.80 and 5.71 respectively. 
 Von Sterneck used his results at the surface and at these 
 underground stations to express g as a function of the depth. 
 Calling the value of g at the surface unity, and measuring r 
 from the centre of the earth and calling it equal to unity at 
 the surface, he deduced the following expression for the value 
 of g at any depth : 
 
 # = 2.6950 r-1.8087 r 2 +.1182 r\ 
 
 This would make g a maximum, 1.06, at r = .78. The density 
 would be expressed by the formula d= 15. 13 6 12.512 r, giving 
 15.136 for its value at the centre of the earth, and 2.624 at the 
 surface. These relations are at least suggestive if not con- 
 vincing. 
 
 i 129 
 
MEMOIRS ON 
 
 During the year 1883 von Sterneck used the same method 
 and apparatus to determine the variation in gravity for 3 sta- 
 tions above the earth's surface at Kronstadt. He found (158) 
 gravity greater at a higher point (Schlossberg) than at a lower 
 (Zwinger), and proved that neither the formula of Young (see 
 page 31) nor that of Faye and Ferrel for the reduction to sea- 
 level gave satisfactory results. 
 
 Twice in this year Sterneck made investigations at Krusna 
 hora in Bohemia. Here there was a mine with a horizontal 
 gallery 1000 m. long, and he wished to find the effect of the 
 overlying sheet of earth upon the value of gravity at various 
 points in the gallery. The same apparatus was used after some 
 improvements had been made. Observations were taken at the 
 mine mouth and at points 390 and 780 m. from the mouth, and 
 62 and 100 m. respectively below the surface of the ground. 
 The results shewed that gravity in the plateau increased with 
 the depth of the super-incumbent layer by the half of the 
 amount by which it would have changed in free space when 
 the distance from the centre of the earth was changed by the 
 same amount. Observations were made at 4 stations above 
 ground also at different elevations, and it was found that the 
 Faye-Ferrel rule accounted for the differences between them 
 much better than did the Bouguer-Young rule. 
 
 Further experiments (164) were made, in 1884, at Saghegy 
 in Hungary, and elsewhere, with results similar to those de- 
 scribed above. An important improvement was made in the 
 method of observing the coincidences. They were now ob- 
 served by the reflections of an electric spark from two mirrors, 
 one fixed on the pendulum stand, and the other attached to 
 the pendulum and when at rest parallel to the first. The spark 
 was made by the relay circuit every half second. 
 
 In 1885, Sterneck made a series of observations (165) at the 
 mouth and at 4 underground stations in the Himmelfahrt- 
 Fundgrube silver mine at Freiberg in Saxony. He was led to 
 do so by the publication of the results of some pendulum meas- 
 urements made there, in 1871, by Dr. C. Bruhns, who had found 
 that gravity decreased with the depth. Using Airy's formula, 
 von Sterneck found the following values for A at the 4 under- 
 ground stations in the order of their depth : 5.66, 6.66, 7.15 
 and 7.60, the density of the mine strata being 2.69. These re- 
 sults indicate an abnormal increase of gravity with depth. Von 
 
 130 
 
THE LAWS OF GRAVITATION 
 
 Sterneck noticed that in these experiments, as well as in those 
 made at Pribram, the increase in gravity is nearly proportional 
 to the increase in temperature. But although Hicks (166), as 
 we have seen (page 119), discovered a connection between the 
 values of A and the temperatures in Baily's experiments, and 
 Cornu and Bailie (page 125) got a larger result for A in summer 
 than in winter, we have no reason for looking upon the varia- 
 tions in temperature as an explanation of the anomalies under 
 consideration. An interesting criticism of von Sterneck's work 
 is given by Poynting (185, pp. 29-39). Short accounts of it 
 are given by Presdorf (186J, pp. 17-9) and Giinther (196 ^ vol. 
 1, p. 189). 
 
 WILSING. In 1887, J. Wilsing (170) made at Potsdam a de- 
 termination of the mean density of the earth by means of an 
 instrument which is called the pendulum balance, and is the 
 common beam balance turned through 90. It is practically a 
 pendulum made of a rod with balls at each end and a knife- 
 edge placed just above the centre of gravity. The instrument 
 used by Wilsing consisted of a drawn brass tube 1 m. long, 
 4.15 cm. in diameter and .16 cm. thick, strengthened near the 
 middle where the knife-edge is affixed. The knife-edge and 
 the bed on which it rested were of agate, and 6 cm. long. To 
 the ends were screwed the balls of brass weighing 540 gr. each, 
 and on the upper ball was a pin carrying discs which were used 
 for finding the moment of inertia and the position of the cen- 
 tre of gravity of the pendulum. Its motion was observed by 
 the telescope and scale method, a mirror being attached to the 
 side of the pendulum parallel to the knife-edge. The pend- 
 ulum was mounted on a massive pier in the basement of the 
 Astrophysical Observatory in Potsdam, and was protected from 
 air currents by a cloth-lined wooden covering. 
 
 The attracting masses were cast-iron cylinders each weighing 
 325 kg. They were so arranged on a continuous string passing 
 over pulleys that when one was opposite the lower brass ball on 
 one side of the pendulum the other was opposite the upper ball 
 on the other side. Their relative positions could be quickly 
 changed from without the room, so that the former mass came 
 opposite the upper ball and the latter mass opposite the lower ; 
 the deflection was now in the opposite direction from what it 
 was in the first case. 
 
 131 
 
MEMOIRS ON 
 
 The double deflection due to the change in position of the 
 masses, and the time of vibration are the quantities required 
 for the determination of A. The readings for these quantities 
 were made by the method of Baily, winch has been already de- 
 scribed. The time of vibration was determined first with the 
 discs on top of the upper ball, then with one removed and then 
 with still another removed. In this way the moment of inertia 
 was obtained. The theory of the instrument is complicated, 
 and for it reference must be made to the original paper. The 
 result obtained for A was 5. 594 .032. 
 
 In 1889, Wilsing published (172) an account of some further 
 observations with the same apparatus, some slight changes 
 having been made in it in the meantime. Extra precautions 
 were taken in order to avoid the effects of variations of tem- 
 perature. Experiments were made with the old balls, with new 
 lead balls, and with the pendulum rod alone. The mean re- 
 sult from these was 5. 588 .013 ; and the final average of all 
 his determinations 5.579.012. 
 
 A preliminary paper (163) was read by Wilsing before the 
 Berlin Academy, and also an extract (169) of his first paper. 
 A condensed translation of both papers was made by Prof. J. 
 H. Gore (171) for the Smithsonian Report for 1888, and a short 
 account of the work is given by Poynting (185, pp. 65-9) and 
 by Fresdorf (186, p. 28). 
 
 POYNTING. Prof. J. H. Poynting published in 1878 the 
 results (146) of a study of the beam balance. He found that 
 the sources of error were temperature changes producing con- 
 vection currents and unequal expansion of the arms, and the 
 necessity of frequently raising the knife-edges from the planes. 
 He tried to overcome the former difficulty by taking the same 
 precautions as those employed by users of the torsion balance ; 
 and he did away altogether with the raising of the beam be- 
 tween weighings, and when the weights had to be exchanged 
 held the pan fixed in a clamp. 
 
 The paper gives a description of his balance and illustrates 
 how it can be used, (I) to compare two weights, and (2) to find 
 the mean density of the earth. The motion of the beam was 
 observed by means of a telescope and scale, the mirror being 
 fixed at the centre of the beam. The deflection of the ray 
 could be multiplied by repeated reflections between tiiis mirror 
 
 132 
 
THE LAWS OF GRAVITATION 
 
 and another which was fixed and nearly parallel to the former. 
 The centre of oscillation was determined after the method of 
 Baily with the torsion balance. As a result of 11 observations 
 Prof. Poynting found the mean density of the earth to be 
 5.69.15. He felt justified, therefore, in proceeding to have a 
 more suitable balance constructed in order to make a more 
 careful determination of this quantity. 
 
 The investigation continued through many years, and the 
 results (180 and 185, pp. 71-156) were not published until 1891. 
 Many unforeseen difficulties arose during the progress of the 
 work, but by patience and skill Poynting was able to overcome 
 these difficulties and to begin to take observations in 1890. 
 The balance was of the large bullion type, 123 cm. long, and 
 made with extra rigidity by Oertling. It was set up in a base- 
 ment room at Mason College, Birmingham. The principle 
 upon which the experiment is based is as follows : two balls 
 of about the same mass are suspended from the two arms of 
 the balance. Beneath the balance is a turn-table carrying a 
 heavy spherical mass vertically under one of the balls. The 
 position of the beam is observed and the turn-table moved un- 
 til the mass is under the other ball and the position of the beam 
 again observed. The deflection measures twice the attraction 
 of the mass for the ball. The attraction of the mass for the 
 beam and wires, etc., is then eliminated by repeating these ob- 
 servations with the balls suspended at a different distance be- 
 low the arm, for then the attraction of the mass on the balance 
 remains the same, and we find the change in the attraction of 
 the mass for the ball with change of distance. The calculation 
 was complicated by the presence on the tnrn-table of another 
 mass as a counterpoise to the former one ; it was smaller than 
 that one and at a correspondingly greater distance from the 
 centre. It was used because certain anomalies could ,be ac- 
 counted for only on the supposition that the floor tilted when 
 the turn-table was rotated with the large mass only upon it. 
 
 Instead of the ordinary mirror fixed on the beam, Poynting 
 used the double-suspension mirror (see Darwin B. A. Rep., 
 1881). The riders were manipulated by mechanism from with- 
 out, and the observer was stationed in the room above, whence 
 he could make all changes and observations without opening 
 the balance room. The attracted and attracting masses were 
 made of an alloy of lead and antimony. The balls were gilded 
 
 133 
 
MEMOIRS ON 
 
 and weighed over 21 000 gr. each. The large mass weighed 
 150 000 gr., and the counterpoise about half as much. A first 
 set of observations gave A = 5. 52. The attracting bodies were 
 then all inverted in order to eliminate the effects of want of 
 symmetry in the position of the turn-table, and of homoge- 
 neity in the masses. A new set of observations gave A = 5.46. 
 The difference between the results of the two sets must have 
 been caused by a cavity or irregular distribution of density in 
 the large mass, and by other experiments Prof. Poynting found 
 that its centre of gravity was not at its centre of figure, but 
 was nearly at the place at which his gravitational experiments 
 would have suggested it to be. The mean result for A is taken 
 to be 5.4934, and for the gravitation constant, G, 6.6984xlO~ 8 . 
 
 Poynting remarks that the effects of convection currents are 
 greater in the beam balance than in the torsion balance, since 
 the motion of the former is in a vertical plane. He thinks 
 that a balance of greatly reduced dimensions would have been 
 preferable. The admirable way in which Prof. Poynting has 
 utilized the common balance for absolute measurements of 
 force caused the University of Cambridge to award him the 
 Adams Prize in 1893. 
 
 For a short account of this work see Wallentin (154J). 
 
 BERGET. In 1756, Bouguer read before the Academy of 
 Sciences the results (9) of some experiments made by him to 
 determine whether the plumb-line was affected by the tidal 
 motion of the ocean. He was not able to detect any such 
 effect. Towards the middle of last century Boscovitch pro- 
 posed (140, vol. 1, pp. 314 and 327) to place a long pendulum 
 in a very high tower by the edge of the sea, where the height 
 of the tide is very great, and to observe the deviation due to 
 the rise of the water, and thence to calculate the mean density 
 of the earth. Von Zach suggested (49, vol. 1, p. 17) a modification 
 of the experiment. Boscovitch also proposed the use of a reser- 
 voir after the manner about to be described, used by Berget. 
 In 1804, Robison, in his "Mechanical Philosophy"' vol. 1, page 
 339, points out that a very sensible effect on the value* of grav- 
 ity might be observed at Annapolis, Nova Scotia, due to the 
 very high tides there. The theory of this local influence is 
 given in Thomson and Tait's "Natural Philosophy " pt. II., 
 page 389. Struve (129) proposed to find A from observations 
 
 134 
 
THE LAWS OF GRAVITATION 
 
 on plumb-lines placed on each side of the Bristol Channel,.and 
 Keller (168) calculated the deflection of the plumb-line due 
 to the draining of Lake Fucino. In 1893, M. Berget utilized 
 this principle in order to find (181) the density of the earth. 
 
 He had the use of a lake of 32 hectares in area in the Com- 
 mune of Habay-la-neuve in Belgian Luxembourg. The level of 
 the lake could be lowered 1 m. in a few hours, and as quickly 
 regained. He could thus introduce under his instrument a 
 practically infinite plane of matter whose attraction could be 
 calculated and observed. The apparatus used to measure the 
 attraction was the hydrogen gravimeter such as Boussingault 
 and Mascart (Comp. Rend., vol. 95, pp. 126-8) used to find the 
 diurnal variation of gravity. The variation of the column of 
 mercury was observed by the interference fringes in vacuo be- 
 tween the surface of the mercury and the bottom of the tube, 
 which was worked optically plane. A first series of observa- 
 tions was made when the lake was lowered 50 cm., and another 
 when it was lowered 1 m. A change of 1 m. caused a displace- 
 ment of the mercury column of 1.26xlO~ 6 cm. The value of 
 the gravitation constant found was 6.80xlO~~ 8 , of A, 5.41 and 
 of the mass of the earth 5.85 x 10 27 gr. 
 
 M. Gouy remarks (182) that such a result would imply that 
 the temperature remained constant during hours to -g- o0 ^ 000 
 of a degree, which is impossible. Pavilion, with the greatest 
 care, was able to reach T -<ro"oo- f a degree only. So that the 
 result given by Berget can not be so accurate as he supposed. 
 
 For n, short account of the experiment see Fresdorf (186, 
 pp. 29-30). 
 
 BOYS. Prof. C. V. Boys read before the Royal Society, in 1889, 
 an important paper (175 and 176) on the best proportions and 
 design for the torsion balance as an instrument for finding the 
 gravitation constant. He shewed that the sensibility of the 
 apparatus, if the period of oscillation is always the same, is in- 
 dependent of the linear dimensions of the apparatus ; and re- 
 marked that the statements of Cornu on this point (page 125) 
 are not correct. There are great advantages to be gained by 
 reducing the dimensions of the apparatus of Cavendish 50 or 
 100 times ; the main one is that the possibility of variation of 
 temperature in the apparatus is enormously minimized. Then, 
 too, the case can be made cylindrical and corrections for its at- 
 
 135 
 
MEMOIRS ON 
 
 traction avoided. Until quartz fibres existed it would have 
 been impossible to have made this reduction in the dimensions 
 of the 'apparatus and retained the period of 5 to 10 minutes. 
 The introduction of this invaluable new means of suspension 
 is also due to Professor Boys. Another improvement in the 
 form of the apparatus devised by him is the suspending of the 
 small balls at different distances below the arm (the masses 
 must be at corresponding levels), so that each mass acts prac- 
 tically on one ball only. 
 
 Boys showed to the Society a balance of this design ; it had 
 an arm of only 13 mm. in length, was 18.7 times as sensitive as 
 that of Cavendish and behaved very satisfactorily. He pro- 
 posed to prepare a balance of this kind especially suitable for 
 absolute determinations and capable of determining the gravi- 
 tation constant to 1 part in 10 000. 
 
 An account of his completed work (187 and 189) was read in 
 1894. For the details of this beautiful experiment and the in- 
 genious way in which the apparatus was designed the original 
 paper must be consulted. The general design was that of his 
 earlier apparatus, but very great attention was given to the 
 minutest details, and especially to the arrangements for meas- 
 uring the dimensions. Some idea of the accuracy aimed at 
 may be got from considering that in order to obtain a result 
 correct to 1 in 10 000 it was necessary to measure the large 
 masses to 1 in 10 000, the times to 1 in 20 000, some lengths to 
 1 in 20 000 and angles to 1 in 10 000. The dimensions finally 
 used were, diameter of masses 2.25 and 4.25 in.; distance be- 
 tween masses in plan 4 and 6 in. ; distance between balls in 
 plan 1 in. ; diameter of balls .2 and .25 in. ; difference of level 
 between upper and lower balls 6 in. The masses were of lead 
 formed under great pressure, and the balls of gold. 
 
 The moment of inertia of the beam was determined by find- 
 ing the period when the balls were suspended from it, and 
 when they were taken away and a cylindrical body of silver, 
 equal in weight to the balls with their attachments, suspended 
 from the middle of the beam. The apparatus was enclosed by 
 a series of metallic screens to prevent temperature changes, 
 and outside of all was a double -walled wooden box with the 
 space between the walls filled with cotton- wool. The final result 
 for the gravitation constant was 6.6576 x 10~ 8 , and for A, 5.5270. 
 The last figure in each case has no significance, but Boys con- 
 
 136 
 
THE LAWS OF GRAVITATION 
 
 sidered that the next to the last could not be more than 2 in 
 error at the outside. He is still convinced that 1 part in 
 10 000 can be reached, but would increase the length of the 
 beam to 5 cm., since the disturbing moments due to convection 
 are proportional to the 5th power of the linear dimensions, not 
 to the 7th as he had originally supposed. An excellent resume 
 of the experiment is to be found in the lecture delivered by 
 Boys before the Royal Institution (188). 
 
 EOETVOES. A series of investigations upon gravitation is now 
 under way by Prof. R. von Eotvos of Budapest. He has pub- 
 lished a preliminary account (192) only of his experiments, but 
 they promise to be very elaborate and exhaustive. His paper 
 begins with a mathematical discussion of the space -variation 
 of gravity as deduced from the potential function. He investi- 
 gates the equipotential surface and the measurements necessary 
 to determine the principal radii of curvature, the variation of 
 gravity along the surface, and the variation perpendicular to the 
 surface. The latter has already been measured with the pend- 
 ulum, and by Jolly (144 and 145), Keller (152), Thiesen (179) 
 and others with the common balance. For the measurement 
 of the other quantities von Eotvos uses the torsion balance. 
 This he makes in two forms : the first is of the same general 
 type as that of Baily and is called the Krummungsvariometer, 
 since it is used to measure the difference of the reciprocals of 
 the principal radii of curvature ; the second is like that of 
 Boys in that one ball is on one end of the rod and the other 
 suspended 100 cm. below the other end by means of a wire, 
 and is called the Horizontalvariometer. The peculiarity of 
 these instruments is in the method devised for getting rid of 
 convection currents ; von Eotvos makes the case with double 
 walls of thin metal with an air-space of from 5 to 10 mm. So 
 steady is the motion that the balance can be used in any room 
 in the laboratory, and even in the free air at night. The period 
 is usually from 10 to 20 minutes, and the suspension wire is of 
 platinum of 100 to 150 cm. length. The rod swings in a flat 
 cylindrical box 40 cm. in diameter and 2 cm. deep. 
 
 Some investigations have been made of the variation of 
 gravity in the neighbourhood of the hill Saghberg, where von 
 Sterneck found great peculiarities.. Some preliminary deter- 
 minations have been made of the constant pf gravitation also, 
 
 137 
 
MEMOIRS ON 
 
 with a result 6.65xlO~ 8 . Von Eotvos speaks of the method 
 employed HS an entirely new one, but it is only a variation of 
 that already employed by Reich, and later by Dr. Braun, the 
 oscillation method. The instrument (the Krummungsvariom- 
 eter) is set up between two pillars of lead, and the time of vib- 
 ration observed both when the torsion rod is in the line joining 
 the pillars and when it is perpendicular to this line. The 
 paper is characterized by an almost total disregard of the work 
 already done in the field of gravitation. 
 
 Eotvos gives a description of two new instruments for use 
 in the study of gravitation. One he calls the Gravitationcom- 
 pensator ; in design it is similar to the others, but the arm 
 swings in a narrow tube. The tube is surrounded at each end 
 by the compensating masses having the balls at their centres ; 
 these masses are of the shape of a disc with two almost com- 
 plete quadrants taken away, just enough being left to hold the 
 remaining two quadrants together. By orienting these masses 
 any amount of compensating attraction required can be pro- 
 duced. The other instrument is called the Gravitationmulti- 
 plicator; underneath the torsion balance is a turn-table with 
 the attracting mass ; when the ball has reached its maximum 
 elongation in the direction of the mass, the latter is suddenly 
 moved to the opposite side, and so on. From the difference 
 of two successive elongations, and a knowledge of the damp- 
 ing, the amount of the attraction can be determined. This 
 is rather similar to a piece of apparatus proposed by Joly (177), 
 in 1890. 
 
 BRAtnsr. One of the latest and most elaborate determinations 
 of the mean density of the earth is that made by Dr. Carl 
 Braun, S.J., at Mariaschein in Bohemia (193). In its gen- 
 eral form the apparatus is like that employed by Reich in his 
 later experiments. Like Reich, too, he uses two distinct 
 methods of finding his results, the deflection and the oscilla- 
 tion methods. The experiments of Dr. Braun differ however 
 in several very important respects from those of Reich ; the 
 dimensions of the apparatus are much reduced, the masses are 
 suspended from wires and the deflection is determined differ- 
 ently ; but the respect in which it differs from all previous de- 
 terminations is in the fact that the torsion rod swings in a 
 partial vacuum of about 4 mm. of mercury. This was sug- 
 
 138 
 
THPJ LAWS OF GRAVITATION 
 
 gested by Faye (130) and by Boys, but no investigation of the 
 kind had ever been made. 
 
 The experiments were begun in 1887. In a corner of a liv- 
 ing-room a heavy stone slab was set into the stone wall ; on 
 this was a glass plate from which arose a brass tripod to carry 
 the suspension wire, 1 in. long, and the torsion-rod ; and on 
 the plate fitted airtight a conical glass cover within which a 
 vacuum could be made. The apparatus was so tight that the 
 pressure inside did not change in 4 years. Suspended from a 
 movable ring encircling the glass cover were the two masses, 
 about 42 cm. apart ; the masses used were two sets of spheres, 
 one of brass weighing 5 kg. each, and the other of iron, 112 mm. 
 in diameter, filled with mercury, and weighing about 9.15 kg. 
 each. The torsion rod was a triangle of copper wires and the 
 balls were suspended from its ends and lay in the same hori- 
 zontal plane 24.6 cm. apart. Each ball was of gilded brass and 
 weighed about 54 gr. In order to provide against temperature 
 changes, the whole apparatus was surrounded by metal screens, 
 cloth hangings and wooden enclosures. 
 
 The deflection method of observation is practically that of 
 Cavendish, but the position of the centre was found rather 
 differently. Dr. Braun observed the time of the passage in 
 each direction across the wire of the telescope of several scale 
 divisions near the centre, and took as the centre that point 
 with reference to which the time of oscillation was the same in 
 both directions. He found for the final corrected mean result 
 A = 5. 52962. 
 
 In the oscillation method the period was determined when 
 the masses were in the line joining the balls, and also when they 
 were in a line at right angles to that direction. The final cor- 
 rected mean result gave A = 5. 52920. The mean of all is 
 5.52945 ib. 0017. The extremes were 5.5094 and 5.5511. The 
 mean of all the results found in 1892 was 5.52770, and in 1894 
 was 5.53048. 
 
 The final result for the gravitation constant was 6.655213 
 
 xio- 8 . 
 
 In an appendix is given a further discussion of the correc- 
 tions to be made on account of damping. Dr. Braun found his 
 former estimate to be in error, and after examination gave as 
 the most probable final results, A = 5. 52700 .0014, and Gr 
 = (6.65816. 00168) x 10~ 8 . 
 
 139 
 
MEMOIRS ON 
 A concise account of the work is given in Nature (197). 
 
 KOENIG, RlCHARZ A^D KRIGAR - MENZEL. Ill 1884, Pro- 
 
 fessors A. Konig and F. Richarz proposed (159 and 160) to 
 determine the gravitation constant by a method which is a 
 modification of that used by Jolly (page 126). In the latter 
 experiment the lower set of scale pans was 21 m. beneath the 
 upper aud differences of temperature were unavoidable. The 
 improvement proposed was to have the sets of scale pans much 
 closer together, to measure the change of weight with height 
 after the manner of Jolly and then to insert between the upper 
 and lower pans a huge block of lead with holes in it for the 
 passage of the wires supporting the lower pans. A weighing 
 was made with two nearly equal masses, one in the right upper, 
 the other in the left lower pan ; then the former in the right 
 lower is balanced against the latter in the left upper pan. From 
 these weighings, taking account of the result of similar ob- 
 servations without the block, the value of 4 times the attrac- 
 tion of the block is determined, and from a comparison of this 
 result with the calculated attraction, the gravitation constant 
 can be determined. Professors Konig and Richarz seem to 
 have hit upon the same idea independently of each other. In 
 1881, Keller proposed (167) a somewhat similar modification of 
 the Jolly experiment. Professor A. M. Mayer suggested (161) 
 the use of mercury instead of lead for the attracting mass, but 
 Konig and Richarz replied (162) that Mayer had misunder- 
 stood the form of the experiment, and gave a lucid and simple 
 explanation of their method. 
 
 In 1893, appeared a report (183 and 184) on the observations 
 made to find the decrease of gravity with increase of height. 
 A description is given of the balance and of the improvements 
 introduced into it in order to overcome the liability to varia- 
 tion in its readings. The masses weighed against each other 
 were 1 kg. each, and the balance had a sensitiveness of 1 part 
 iirl 000 000. All exchange of weights was made automatically 
 without opening the covers. The apparatus was carefully sur- 
 rounded with metal screens to ward off temperature changes. 
 It was set up in a bastion of the citadel at Spandau, and in 
 consequence of the departure of Konig to accept a professor- 
 ship at Berlin, Dr. Krigar-Menzel assisted in the carrying on 
 of the research. The pans were 2.26 m. apart vertically. The 
 
 140 
 
THE LAWS OF GRAVITATION 
 
 change in gravity observed was .000006523 - - ; whereas the 
 
 sec. 2 ' 
 
 calculated value was .00000697. The difference is ascribed to 
 the local attraction of walls, etc. 
 
 The paper '198) embodying the final results was presented 
 to the Berlin Academy in Dec., 1897. A most exhaustive ex- 
 amination had been made of the possible sources of error, and 
 the devices for overcoming these difficulties were most ingen- 
 ious and elaborate. In the cases where the sources of error 
 could not be eliminated, as in the variations of temperature with 
 time and place, the effect is carefully considered and allowed 
 for. Observations were made continuously from Sept., 1890, to 
 Feb., 1896, and from the elegance of the method and the time 
 and care devoted to the working out of the result, this deter- 
 mination of the gravitation constant and the mean density of 
 the earth must be taken as one of the very best. 
 
 The block of lead weighed 100 000 kg., was 200 cm. high and 
 210 cm. square and was built up out of bars of lead 10 x 10 x30 
 cm. on the top of a massive pier. The amount of the settling 
 of the pier was measured and found to be not important, and 
 the shape of the block was not distorted by its own pressure. 
 The final value for G was (6.685 .011) 10~ 8 , and for A, 5.505 
 .009. 
 
 Professors Richarz and Krigar-Menzel published (191 and 
 199), in 1896, a condensed account of their work. Other ab- 
 stracts are also to be found (185, pp. 64-5, 186J, pp. 26-7, and 
 194). 
 
 NOTICES' In 1889, Dr. W. Laska of Prague proposed 
 (173) a method of finding the density of the earth. At the top 
 of a rod projecting above a pendulum is a lens which is so close 
 to a fixed plate of glass that Newton's rings are visible. A 
 hollow ball near the bob of the pendulum is then filled with 
 mercury and attracts the bob, bringing the lens nearer the 
 plate ; an observation of the movement of the Newton's rings 
 will measure the deflection of the bob. No further report has 
 been published. An account of his method is given by Gflnther 
 (196J, vol. 1, p. 197). 
 
 About the same time Professor Joly of Dublin suggested 
 (177) a resonance method for the same purpose. A pendulum in 
 a vacuous vessel has the same period as two massive ones kept 
 
 141 
 
MEMOIRS ON 
 
 going outside the vessel. The amplitude of the motion of the 
 inner pendulum due to a given number of swings of the outer 
 ones would give a measure of the constant of gravitation. 
 
 In 1895, Professor A. S. Mackenzie of Bryn Mawr College 
 published an account (190) of some experiments with the Boys' 
 form of torsion balance to determine whether the gravitational 
 properties of crystals vary with direction. No such variation 
 was found in the case of calc-spar, the crystal under investiga- 
 tion. He shewed further that the inverse square law holds 
 good in the neighbourhood of a crystal to one-fifth per cent. 
 
 Two years later appeared an account (196) of an investigation 
 by Professors Austin and Thwing of the University of Wiscon- 
 sin to determine whether gravitational attraction is indepen- 
 dent of the intervening medium, that is, whether there is a 
 gravitational permeability. No effect was found due to the 
 medium within the limits of error of the method. 
 
 At a meeting of the " Deutcher Naturforscher und Aerzte" 
 in Brunswick, in 1897, Professor Drtide read a paper (195) on 
 action at a distance, which contains a very valuable account of 
 the theory of gravitation, and should be consulted by any one 
 wishing to find a brief resume of that subject, and especially 
 for a discussion of the velocity of propagation of gravitation. 
 
 The latest work on the laws of gravitation is that of Profess- 
 ors Poynting and Gray (200) on the search for a directive 
 action of one quartz crystal on another. A small crystal was 
 suspended and its time of rotative vibration noted ; a large 
 crystal in the same horizontal plane was then rotated about 
 a vertical axis through its centre with a period either equal to, 
 or twice, that of the smaller crystal. If there were any directive 
 action the small crystal should be set in vibration by forced 
 oscillations; no such effect was found. 
 
 142 
 
THE LAWS OF GRAVITATOIN 
 
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THE LAWS OF GRAVITATION 
 
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THE LAWS OF GRAVITATION 
 
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MEMOIRS ON 
 
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 148 
 
THE LAWS OF GRAVITATION 
 
 75 1842 
 
 76 1842 
 
 F. Daily. 
 F. Baily. 
 
 77 1842 F. Baily. 
 
 78 1843 
 
 79 1843 
 
 80 1843 
 80 1845 
 
 F. Baily. 
 F. Baily. 
 
 A. G. 
 
 C. A. F. Peters 
 
 81 1847 G. W. Hearn. 
 
 82 1849 E. Sabine. 
 
 83 1852 F. Reich. 
 
 84 1852 - 
 
 85 1852 Schaar. 
 
 86 1853 
 
 87 1853 
 
 88 1855 
 
 89 1855 
 
 F. Reich. 
 
 G. B. Airy. 
 G. B. Airy. 
 
 90 1855 
 
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 (Report on Harton expts.). Mon. Not. Roy. 
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 149 
 
MEMOIRS ON 
 
 91 1855 
 
 92 1855 
 
 93 1855 J. H. Pratt. 
 
 94 1855 G. B. Airy. 
 
 95 1855 
 
 96 1856 
 
 97 1856 
 
 98 1856 
 
 99 1856 
 
 T. Young. 
 J. H. Pratt. 
 G. B. Airy. 
 
 J. H Pratt. 
 
 100 1856 G. B. Airy. 
 
 101 1856 G. B. Airy. 
 
 102 1856 G. G. Stokes. 
 
 103 1856 H. James and 
 
 Note sur les observations du pendule executees 
 dans les mines de Harton pour determiner la 
 densite moyennedela terre ; par M. Airy. Ann. 
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 Extrait du rapport presente a la 35 me seance 
 anuiversaire de laSociete Royale Astronomique 
 de Londres par le conseil de cette societe le 9 
 Fevrier, 1855. Arch, des Sc. Phys. et Nat., 29, 
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 On the attraction of the Himalaya mountains, 
 and of the elevated regions beyond them, upon 
 the plumb-line in India. Phil. Trans. Lond., 
 145,53-100. 
 
 On the computation of the effect of the attrac- 
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 parent astronomical latitude of stations in 
 geodetic surveys. Phil. Trans. Lond., 145, 
 101-4. 
 
 Miscellaneous works and life, by Peacock and 
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 (Same title as 93). Mon. Not. Roy. Astr. Soc., 
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 (Same title as 94). Mon. Not. Roy. Astr. Soc., 
 16, 42-43. 
 
 (Report on Harton expts.). Mon. Not. Roy. 
 Astr. Soc., 16, 104. 
 
 On the effect of local attraction upon the plumb- 
 line at stations on the English arc of the merid- 
 ian, between Dunnose and Burleigh Moor ; and 
 a method of computing its amount. Phil. Trans. 
 Lond., 146, 31-52, 
 
 A.ccount of pendulum experiments undertaken 
 in the Harton Colliery, for the purpose of de- 
 termining the mean density of the earth. Phil. 
 Trans. Lond., 146, 297-342. 
 Supplement to the "account of pendulum ex- 
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 being an account of experiments undertaken 
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 of the pendulum. Phil. Trans. Lond., 146, 
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 (Addendum to 101 ; on the effect of the earth's 
 rotation and ellipticity in modifying the numer- 
 ical results of the Harton experiment). Phil. 
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 A. R. Clarke. On the deflection of the plumb- 
 line at Arthur's Seat, and the mean specific 
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 150 
 
THE LAWS OF GRAVITATION 
 
 104 1856 H, James. 
 
 105 1856 
 
 106 1856 S. Haughton. 
 
 107 1856 G. B. Airy. 
 
 108 1856 H. James. 
 
 109 1856 G. B. Airy. 
 
 110 1856 
 
 111 1856 G. B. Airy. 
 
 112 1857 E. R. 
 
 113 1857 
 
 114 1857 
 
 115 1857 
 
 116 1857 
 
 117 1857 H. James. 
 
 118 1857 W. S. Jacob. 
 
 119 1857 G. B. Airy. 
 
 120 1857 H. James. 
 
 On the figure, dimensions and mean specific 
 gravity of the earth, as derived from the ord- 
 nance trigonometrical survey of Great Britain 
 and Ireland. Phil. Trans. Lond., 146, 607-26. 
 Ueber die in der Kohlengrube von Harton zur 
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 Account of the observations and computations 
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 of the deflection of the plumb-line at Arthur's 
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 (Same title as 101). Phil. Mag., [4], 12, 467-8. 
 Ueber die Dichtigkeit der Erde, hergeleitet aus 
 den Versuchen des Konigl. Astronomen (Hrn. 
 Airy) in der Kohlengrube Harton ; vonSr. Ehr- 
 wurd. Samuel Haughton, Fellow des Trinity 
 College in Dublin. Pogg. Ann., 99, 332^. 
 On the pendulum experiments lately made in 
 the Harton Colliery, for ascertaining the mean 
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 359-64. 
 
 Memoire sur les experiences enterprises dans la 
 mine de Harton pour determiner la densite 
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 Ueber die Dichtigkeit der Erde, hergeleitet aus 
 den Pendelbeobachtungen des Herrn Airy in 
 der Kohlengrube Harton von Herrn S. Haugh- 
 ton, Fellow am Trinity-College in Dublin. Zeit. 
 fur Math. u. Phys., 2, 68-70. 
 Ueber die Bestimmung der mittleren Dichtigkeit 
 der Erde. Zeit. fur Math. u. Phys.. 2, 128-30. 
 (Same title as 103). Proc. Roy. Soc. Edin., 3, 
 364-6. 
 
 (Notice of 106). Am. Journ. Sc., [2], 24, 158. 
 (Same title as 104). Phil. Mag. , [4], 1 3, 129-32. 
 On the causes of the great variation among the 
 different measures of the earth's mean density. 
 Phil. Mag., [4], 13, 525-8. 
 (Same title as 101). Proc. Roy. Soc. Lond., 8, 
 58-9. 
 
 (Same title as 104). Proc. Roy. Soc. Lond., 8, 
 111-6. 
 
 151 
 
MEMOIRS ON 
 
 121 1857 W. S. Jacob. (Same title as 118). Proc. Roy. Soc. Lend., 8, 
 
 295-9. 
 
 122 1858 G. B. Airy. (Same title as 111). Proc. Roy. Inst., 2, 17-22. 
 
 123 1858 (Same title as 103). Mon. Not. Roy. Astr. Soc., 
 
 18, 220. 
 
 124 1858 - (Same title as 104). Mon. Not. Roy. Astr. Soc., 
 
 18, 220-2. 
 
 125 1858 H. James and A. R. Clarke. Ordnance trigonometrical Survey 
 
 of Great Britain and Ireland. Account of the 
 observations and calculations of the principal 
 triangul.-ition ; and of the figure, dimensions and 
 mean specific gravity of the earth as derived 
 therefrom. 2 vol. London. 4 to . 
 
 126 1859 (Same title as 125). Mon. Not. Roy. Astr. Soc., 
 
 19, 194-9. 
 
 127 1859 P. F. J. Gosselin. Nouvelexamen sur la densite rnoyenne de la 
 
 terre. Mem. Acacl. Imp. de Mete, [2], 7, 469-85. 
 
 128* 1859-60 E. Sergent. Sulla densita della materia nell' intoruo del 
 globo, e sulla potenza della crosta terrestre. Atti 
 della. Soc. Ital. di 8c. Nat. Milano, 2, 169-175. 
 
 129 1861 O. Struve. Ueber einen von General Schubert an die Aka- 
 
 demie gerichteten Antrag betreffend die Rus- 
 sisch - Scandinavische Meridian - Gradmessung. 
 Bull. Acad. St. Petersb. phys. math, d., 3, 395- 
 424. 
 
 130 1863 H. A. E. A. Faye. Sur les instruments geodesiques et sur la 
 
 densite moyenne de la terre. Comp. Rend., >G, 
 557-66. 
 
 131 1864 E. Pechmann. Die Abweichung der Lothlinie bei astrono- 
 
 mischen Beobachtungsstationen und ihre Be- 
 rechnung als Erforderniss einer Gradmessung. 
 Denkschr. Acad. Wiss. Wien. math.-naturw. cl., 
 22, 41-88. 
 
 132 1864 J. Babinet. Note surle calcul de 1'experiencede Cavendish, 
 
 relative a la masse et a la densite moyenne de la 
 terre. Cosmos, 24, 543-5. 
 
 133 1865 J. H. Pratt. A treatise on attractions, Laplace's functions, 
 
 and the figure of the earth. 3 d . Ed n . Cambridge 
 and London. 8 V0 . 
 
 134 1865 H. Scheffler. Ueber die mittlere Dichtigkeit der Erde. Zeit. 
 
 fur Math. it. Phys., 1O, 224-7. 
 
 135 1869 A. Schell. Ueber die Bestimmung der mittleren Dichtigkeit 
 
 der Erde. Gottingen. 4 to . 
 
 136 1872 F. Folie. Sur le calcul de la densite moyenne de la terre, 
 
 d'apres les observations d'Airy. Bull. Acad. Roy. 
 Belg., [2], 33, 369-372 and 389-409. 
 
 137 1873 A. Cornu et J. B. Bailie. Determination nouvelle de la con- 
 
 stante de 1'attraction et de la densite moyenuetle 
 la terre. Comp. Rend., 76, 954-8. 
 
THE LAWS OF GRAVITATION 
 
 138* 1873 
 
 139 1873 
 
 145 
 146 
 
 1878 
 1879 
 
 (Notice of 137). Bull. Ilebd. de I'Assoc. Sclent, de 
 France., [1J, 12, 70. 
 
 A. Cornu and J. B. Bailie. Mutual determination of the con- 
 stant of attraction, and of the menu density of 
 the earth. Chemical New*, 27, 211. 
 
 140 1873 I. Todhunter. A history of the mathematical theories of at- 
 
 traction and i he figure of the earth, from the 
 time of Newton to that of Laplace. 2vol. Lou- 
 don. 8 V0 . 
 
 141 1878 A. Cornu et J. B. Bailie. Elude de la resistance de Fair dans 
 
 la balance de torsion. Comp. Rend., 86, 571-4. 
 
 142 1878 A. Cornu et J. B. Bailie. Sur la mesure de la densite moyenne 
 
 de la terre. Comp. Rend., 86, 699-702. 
 
 143 1878 A. Cornu et J. B. Bailie. Influence des termes proportioned an 
 
 carre des ecarts, dans le mouvemenl oscillatoire 
 dc la balance de torsion. Comp. Mend., 86, 
 1001-4. 
 
 144 1878 Ph. von Jolly. Die Anwendung der Waage auf Probleme der 
 
 Gravitation. Parti. Abh. Bay. Akad. Wiss.cl. 
 
 2, 13, Abth. 1, 157-176. 
 
 Ph. von Jolly. (Same title as 144). Wied. Ann., 5, 112-34. 
 J. H. Poynting. On a method of using the balance with great 
 
 delicacy, and on its employment to determine 
 
 the mean density of the earth. Proc. Roy. Soc. 
 
 Lond., 28, 2-35. 
 1880 H. A. E. A. Faye. Sur les variations seculaires de la figure 
 
 mathematique de la terre. Comp. Rend., 9O, 
 
 1185-91. 
 
 147 1880 H. A. E. A. Faye. Sur la reduction des observations du pen- 
 
 dule an niveau de la mer. Comp. Rend., 9O, 
 1443-6. 
 
 148 1880-4 F. R. Helmert. Die mathematischen und physikalischen 
 
 Theorieen der hohercn Geodasie. 2vol. Leip- 
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 148| 1880-5 O. Zanotti-Bianco. II problema meccanico della figuradella 
 
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 A. R. Clarke. Geodesy. Oxford. 8 V0 . 
 O. Knopf. Ueber die Methoden zur Bestimmung der mittleren 
 
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 T. C. Mendenhall. Determination of the acceleration due to the 
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 T. C. Mendenhall. On a determination of the force of gravity 
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 153 
 
 149 1880 
 149^* 1880 
 
 150 1880 
 
 151 1881 
 
 152 1881 F. Keller. 
 
MEMOIRS ON 
 
 154 1881 Ph. von Jolly. (Same title as 153). Wied. Ann., 14, 331-55. 
 154| 1882 J. G. Wallentin. Ueber die Methoden zur Bestimmung der 
 
 mittleren Dichte der Erde und eine neue dies- 
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 1, 212-7. 
 
 155 1882 R. von Sterneck. Untersucbungen ilber die Scbwere im In- 
 
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 156 1883 R. von Sterneck. Wiederbohmg der Untersuchungen tlber die 
 
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 Inst. Wien, 3, 59-94. 
 
 157 1883 J. B. Bailie. Sur la resistance de 1'air dans les mouvements 
 
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 158 1884 R. von Sterneck. Untersucbungen tlber die Scbwere auf der 
 
 Erde. Mitth. Mil.-Oeog. Inst. Wien, 4, 89-155. 
 
 159 1884 A. Kdnigand F. Richarz. Eine neue Metbode zur Bestimmung 
 
 der Gravitationsconstante. Sitzungsb. Akad. 
 Wiss. Berlin, 1203-5. 
 
 160 1885 A. Kouig and P. Ricbarz. (Same title as 159). Wied. Ann., 
 
 24, 664-8. 
 
 161 1885 A.M.Mayer. Methods of determining the density of tbe earth. 
 
 Nature, 31, 408-9. 
 
 162 1885 A. Konig and F. Ricbarz. Remarks on our method of deter- 
 
 mining tbe mean density of the earth. Nature, 
 31,484. 
 
 1J33 1885 J. Wilsing. Ueber die Anwendung des Pendels zur Bestim- 
 mung der mittleren Dicbtigkeit der Erde. Sitz- 
 ungsb. Akad. Wiss. Berlin, Hbbd. 1, 13-15. 
 
 164 1885 R. von Sterneck. Fortsetzung der Uutersuchungen ilber die 
 
 Sohwere auf der Erde. Mitth. Mil.-Geog. Inst. 
 Wien, 5, 77-105. 
 
 165 1886 R. von Sterneck. (Same title as 155). Mitth. Mil.-Geog. Inst. 
 
 Wien, 6, 97-119. 
 
 166 1886 W. M. Hicks. On some irregulariiies in tbe values of the 
 
 mean density of the earth, as determined by 
 Baily. Proc. Cam. Phil. 8oc., 5, pt. 2, 156-61. 
 
 167 1886 F. Keller. Sul metodo di Jolly per la determinazione della 
 
 densita media della terra. Atti Accad. Lincei. 
 Rend., [4], 2, 145-9. 
 
 168 1887 F. Keller. Sulla deviazione del filo a piombo prodotta dal 
 
 prosciugamento del Lago di Fucino. Atti Accad. 
 Lincei. Rend., [4], 3, 493-501. 
 
 169 1887 J. Wilsing. Mittheilung tlber die Resultale von Pendelbeo- 
 
 bachtungen zur Bestimmung der mittleren Dich- 
 tigkeit der Erde. Sitzungsb. Akad. Wiss. Berlin, 
 Hbbd. 1, 327-34. 
 
 170 1887 J. Wilsing. Bestimmung der mittleren Dicbtigkeit der Erde 
 
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 phys. Obs. Potsdam, 6, Stuck 2, 35-127. 
 154 
 
THE LAWS OF GRAVITATION 
 
 171 1888 J. H. Gore. Determination of the mean density of the earth 
 by means of a pendulum principle, by J. Wil- 
 sing, translated and condensed. Smithsonian 
 Rep. 1888. 635-46. 
 
 (Same title as 170). Publ. Astrophys. Obs. Pots- 
 dam, 6, Stuck 3, 133-91. 
 
 Ueber einen neuen Apparat zur Bestimmung 
 der Erddichte. Zeit. fur List. - Kunde., 9, 
 354-5. 
 
 A bibliography of geodesy. Washington. 4 to . 
 App. to U. S. Coast and Geod. Surv. Rep. for 
 1887. 
 
 On the Cavendish Experiment. Proc. Roy. Soc. 
 Lond., 46, 253-68. 
 C. V. Boys. (Same title as 175). Nature, 41, 155-9. 
 
 172 1889 J. Wilsing. 
 
 173 1889 W. Laska. 
 
 174 1889 J. H. Gore. 
 
 175 1889 C. V. Boys. 
 
 176 
 177 
 
 185 
 186 
 
 1889-90 
 1889-90 
 
 J. Joly. 
 
 178 1889-91 - 
 
 179 1890 Thiesen 
 
 180 1891 
 
 181 1893 A. Berget. 
 
 182 1893 Gouy. 
 
 183 1893 
 
 184 1894 
 
 1894 
 1894 
 
 187 1894 
 
 188 1894 
 
 (Report of meeting of Univ. Exptl. Assoc. 
 
 Dublin). Nature, 41, 256. 
 
 Collection de memoires relatifs a la physique, 
 
 publics par la Societe Francaise de Physique, 
 
 4 and 5. Paris. 8 V0 . 
 
 Determination de la variation de la pesanteur 
 
 avec la hauteur. Trav. et Mem. du Bur. Internat. 
 
 des Poids et Mes., 7, 3-32. 
 J. H. Poynting. On a determination of the mean density of 
 
 the earth and the gravitation constant by means 
 
 of the common balance. Phil. Trans. Lond., 
 
 [A], 182,565-656. 
 
 Determination experimentale de la constantede 
 
 Tattractiou universelle, ainsi que de la masse et 
 
 de la densite de la terre. Gomp. Rend., 116, 
 
 1501-3. 
 
 Sur la realisation des temperatures constantes. 
 
 Comp. Rend., 117, 96-7. 
 F. Richarz und O. Kri.uar-Menzel. Die Abnahme der Schwere 
 
 mil der Mohe beslimmt durch Wagungen. Sitz- 
 
 ungsb. Akad. Wiss. Berlin, 163-83. 
 F. Richarz und O. Krigar-Menzel. (Same title as 183). Wied. 
 
 Ann., 51, 559-83. 
 
 J. H. Poynting. The mean density of the earth. London. 8 V0 . 
 J. H. Poynting. A histciy of the methods of weighing the 
 
 earth. Proc. Birmingham Nat. Hist, and Phil. 
 
 Soc., 9, 1-23. 
 
 Die Methoden zur Bestimmung der mittleren 
 
 Dichte der Erde. Wiss. Beilage zum Jahresb. des 
 
 Gym. zu Weissenburg i. Elsass. 
 
 On the Newtonian constant of gravitation. Proc. 
 
 Roy. Soc. Lond., 56, 131-2. 
 
 (Same title as 187). Nature, 5O, 330-4, 366-8, 
 
 417-9 and 571. 
 155 
 
 
 186| 1894 G. Fresdorf. 
 
 C. V. Boys. 
 C. V. Boys. 
 
MEMOIRS ON THE LAWS OF GRAVITATION 
 
 189 1895 C. V. Boys. (Same title as 187). Phil. Trans. Lond., [A] 
 
 186, 1-72. 
 
 190 1895 A.S.Mackenzie. On the attractions of crystalline and isotropic 
 
 masses at small distances. Phy. Rev., 2, 321- 
 43. 
 
 191 1896 F. Richarz und O. Krigar-Menzel. Gravitationsconstante uud 
 
 mittlere Dichtigkeit der Ertle, bestimml durch 
 Wagungen. Sitzungsb. Akad. Wiss. Berlin, 
 1305-18. 
 
 192 1896 R. von Eotvos. Untersuclmngen liber Gravitation und Erd- 
 
 magmetismus. JVied. Ann., 59, 354-400. 
 
 193 1896 C. Brauu. Die Gravitationsconstante, die Masse und mitt- 
 
 lere Dichte der Erde nach einer neuen experi- 
 mentellen Bestimmung. Denkschr. Akad. Wiss. 
 Wien. math.-nalurw. cl., 64, 187-258c. 
 
 Iffi0- 1 QQA -7 The gravitation constant and the mean density 
 
 of the earth. Nature, 55, 296. 
 
 195 1897 P. Drude. Ueber Fernewirkungen. Wied. Ann., 62, i.- 
 
 xlix. 
 
 196 1897 L. W. Austin and C. B. Thwing. An experimental research on 
 
 gravitational permeability. Phy. Rev., 5, 294- 
 300. 
 
 196| 1897-9 S. Gunthcr. Handbuch der Geophysik. 2 vol. Stuttgart. 
 8 V0 . 
 
 197 1897 J. H. P(oynting). A new determination of ihe gravitation 
 
 constant and the mean density of the earth. 
 Nature, 56, 127-8. 
 
 198 1898 F. Richarz und O. Krigar-Menzel. Bestimmung der Gravita- 
 
 tionsconslante und mittlercn Dichtigkeit der 
 Erde durch Wauungen. Anhang Abh. Akad. 
 Wiss. Berlin, 1-196. 
 
 199 1898 F. Richarz und O. Krigar-Mcnzel. (Same title as 191). Wied. 
 
 Ann., 66, 177-193. 
 
 ^200 1899 J. H. Poynting and P. L. Gray. An experiment in search of 
 a directive action of one quartz crystal on an- 
 other. Phil. Trans. Lond., [A], 192, 245-56. 
 
 156 
 
INDEX 
 
 Aehard, 49. 
 
 Airy. 5, 106, 113. 118, 119, 121-124, 
 128-130; Theory of Cavendish Ex- 
 periment, 106, 118; Dolcoath Ex- 
 periments, 113; Harton Experi- 
 ments, 121. 
 
 Arthur's Seat, 118, 123, 124. 
 
 Attraction, Newton's Theorems on, 
 9 ; Newton's Error in Calculation 
 of, 16, 17 ; Primitive, 27 ; Of a 
 Plntenu, 29-32 ; Of a Spherical 
 Segment, Calculated by Newton, 
 17; by Carlini, Schmidt and Giu- 
 lio, 111, 112 ; Shown by Deflection 
 of Plumb-line, 33-43 ; Of Chirnbo- 
 razo, 34, 39 ; Of Schehallien, 43, 
 53-56 ; Due to Tides, 44, 134 ; Of 
 any Hill, Calculated by II niton, 
 54*; Of the Great Pyramid, 55; 
 Local, 56, 122-124, 126, 134, 135, 
 141 ; Of Mount Mimet, 56 ; Of 
 Mass Beneath Earth's Surface, 56, 
 
 122, 123; Of Arthur's Seat, 118, 
 
 123, 124 ; Of Evaux, 118 ; Of a 
 Cone, 428 ; Of an Infinite Plane, 
 135. 
 
 Austin and Thwing, 142. 
 
 B 
 
 Babinet, 106. 
 
 Bacon, 1, 2, 5. 49, 113 
 
 Baily, 100, 105, 106, 115-120, 125, 
 131-133, 137 ; Cavendish Experi- 
 ment Criticized by, 105 ; Error of, 
 Pointed out by Cornu and Bailie, 
 119 ; Anomalies in Results of, and 
 their Explanations, 118, 119. 
 
 Balance, Experiments with Beam, 
 2-5, 48, 49, 125, 132. 140 ; Experi- 
 ments with Torsion, 59-105, 114- 
 121, 124, 135, 137-139. 142; Mich- 
 ell Devised Torsion, 60; Experi- 
 ments with Pendulum, 131, 132. 
 
 Baucrnfeind, 124. 
 
 Beaumont, 116. 
 
 Benret, 124, 134, 135. 
 
 Bertier, 47-49. 
 
 Boscovitch, 134. 
 
 Bougner, 5, 21, 23-25, 27, 32, 33, 36, 
 39-44, 47, 53, 56, 130, 134; On 
 Tides, 44, 134 ; Life of, 44 ; First 
 to Take Account of Buoyancy of 
 Air, 26. 
 
 Boyle, 4. 
 
 Boys, 106, 135-137, 139, 142. 
 
 Brandes, 105, 106, 120; Theory of 
 Cavendish Experiment, 106; The- 
 ory of Oscillation Method, 105, 
 106, 120. 
 
 Braun, 106, 138, 139. 
 
 Carlini. 111-113, 128. 
 
 Cavendish, 54, 55, 59. 90, 91, 98, 100, 
 105-107, 114-116,118, 119, 125, 135, 
 136, 139 ; Error in Calculation of, 
 100, 105 ; Life of, 107. 
 
 Chimborazo, 22, 34. 39-41, 43. 
 
 Clarke, 56, 118, 123, 124. 
 
 Condamine, de la, 21, 28, 32, 36, 39- 
 41, 43, 44 ; Pendulum Experiments 
 of, 28 ; Method of, for Doubling 
 Deflection of Plumb-line, 36. 
 
 Cornu and Bailie, 66, 106, 115, 119, 
 124, 125, 131, 135. 
 
 Cotte, 48. 
 
 Cotton, 4. 
 
 Coulomb Balance, First Proposed by 
 Michell, 60. 
 
 Coultaud, 47, 48, 111. 
 
 I) 
 
 D'Alembert, 31. 47. 
 Damping. Method of Finding A, 
 138, 141, 142 ; Effect of, 139. 
 
 157 
 
INDEX 
 
 David, 47-49. 
 
 Deflection, of Arm of Torsion Bal- 
 ance, How Measured, by Caven- 
 dish. 64, 98 ; by Reich, 116, 119 ; 
 by Baily, 117, 119, 132, 133; by 
 Braun. '138 ; Affects the Period, 
 97 ; Error in Daily's Method of 
 Observing 119, 125 ; Multiplied 
 by Poynting, 132, 133. 
 
 Descartes, 2, 49 ; Suggested Method 
 of Measuring Gravity, 2. 
 
 Dimensions of Torsion Balance, Ef- 
 fects of, 125, 135, 137, 138. 
 
 Dolcoath, 113, 121. 
 
 Dolomieu, 49. 
 
 Drobisch, 113, 114. 
 
 Drude, 142. 
 
 E 
 
 EotvOs, 106, 137, 138. 
 
 Faye, 31, 124, 130, 139 ; Compensa- 
 tion Theory of. Correction of " Dr. 
 Young's Rule," 31, 124. 
 
 Ferrel, 130. 
 
 Flotation Theory, 31, 124. 
 
 Folie, 123. 
 
 Forbes, 117, 118, 120. 
 
 Forced Vibrations, J38, 141, 142. 
 
 Fresdorf, 56. 112, 116, 123, 127, 128, 
 131, 132, 135. 
 
 Fujiyama, 127, 128. 
 
 G 
 
 Gilbert, Dr., 1, 5,49. 
 
 Gilbert, L. W., 105. 
 
 Giulio, 112. , 
 
 Gore, 124, 132. 
 
 Gosselin. 106. 
 
 Gouy, 135. 
 
 Gravimeter, 135. 
 
 Gravitation, Early Conceptions of, 
 1, 49. 56 ; Early Experiments on, 
 by Members of Royal Society, 2-5 ; 
 As Explanation of Planetary Mo- 
 tion, by Newton, 2, 10-19 : Mag- 
 netic Theory of. 1, 4, 5. 12 ; Hooke's 
 Ideas Concerning, 5, 6 ; Compen- 
 sator. 138 ; Multiplicator, 138 ; Per- 
 meability, 142 ; Velocity of Prop- 
 agation of, 142. 
 
 Gravity, Proposed Experiment on, 
 by Bacon, 1 ; by Descartes, 2 ; 
 
 Decrease of, with Height, 27-33, 
 47-49, 111-113, 126-128, 130, 137, 
 140 ; Law of Increase of, with 
 Depth, 129, 130 ; Increase of, with 
 Temperature, 131 ; Mathematical 
 Discussion of, from Potential 137 
 
 Gray, 142. 
 
 Giinther, 131, 141. 
 
 H 
 
 Harton Colliery, 5, 121, 122 
 Haughton. 122. 
 Hearn, 118, 120. 
 Helmert, 54, 56, 124, 127, 129. 
 Hicks, 119, 131. 
 Hooke, 2, 4, 5. 
 Horizontalvariometer, 137. 
 Humboldt, 56. 
 
 Button, 54, 55, 90, 100, 105 106, 118, 
 124. 
 
 J 
 
 Jacob, 56, 123. 
 
 J;imes and Clarke, 56 118 123 124 
 Jolly, 125, 126, 137, 140. 
 Joly, 138, 141. 
 
 K 
 
 Keller, 124, 127, 135, 137, 140. 
 Kepler, 1, 2, 49. 
 Knopf, 113, 122. 
 Konig, 140. 
 
 Krigar-Menzel, 140, 141. 
 Krummungsvariometer, 137, 138. 
 
 Lalande, 48. 
 
 Laska, 141. 
 
 Law, of the Distance, 2, 9 29 47 
 101, 126, 142 ; Of the Masses, 13, 
 32; Of the Material, 12, 142; Of 
 the Medium, 142. 
 
 Lesage, 2, 48, 49. 
 
 M 
 
 Mackenzie, 142. 
 
 Magnetism. Gilbert's Explanation of 
 Gravitation by, 1,4, 5 ; Contrasted 
 with Gravitation, 12 ; Tests for 
 Effects of, by Cavendish, 67, 68 
 75, 76 ; by Reich, 116, 120 ; Sug- 
 gested by Hearn to Account for 
 Anomalies in Baily's Results, 118- 
 120. 
 
 158 
 
INDEX 
 
 Maskelyne, 17, 43, 53-56, 101, 106. 
 
 Mayer, 140. 
 
 Menabrea, 55, 66, 106. 
 
 Mendenhall, 124, 127, 128. 
 
 Mercier, 47, 48, 111. 
 
 Michel], 59, 60, 61. 
 
 Mine Experiments, 1, 2, 4, 5, 49, 
 
 113, 121, 128, 131. 
 Montigny, 119. 
 Muncke, 55, 105, 106. 
 
 N 
 
 Newton, 2, 6, 7, 9, 14-17, 19. 39, 43, 
 47-49, 56, 107, 110, 124, 126, 141 ; 
 Explanation of Planetary Motions 
 by, 2, 10 ; Pendulum Experiments 
 of, 11, 15 ; Guess as to Value of 
 A by, 14 ; Errors in Calculations 
 of, 16, 17 ; Indicates Methods of 
 Finding A, 17; Calculates Attrac- 
 tion of a Mountain, 17 ; Life of, 
 19; Attempts to Upset Theory of, 
 47. 
 
 P 
 
 Pechmann, 124. 
 
 Pendulum, Experiment with, Pro- 
 posed by Bacon, 1 ; by Hooke, 5 ; 
 Experiments with, by Newion, 
 10, 11, 15; by Bouguer, 24-33; 
 by Coultaud and Mercier, 47 ; by 
 Carlini, 111 ; by Airy, 113, 121 ; 
 by Mendenhall, 127 ; by Sterneck, 
 128-131 ; by Laska, 141 ; Correc- 
 tion for Buoyancy of Air on, First 
 Used, 26 ; Correction for, Due to 
 Resistance of Air, 27, 66 ; Methods 
 of Comparing One with Another, 
 113, 121, 128-130; Balance, 181. 
 
 Peters, 55, 124. 
 
 Playfair, 55, 102. 
 
 Plumb-line, Deflection of, Observed 
 at Chimborazo, 33-43 ; at Sche- 
 hallien, 43, 53-56; at Arthur's 
 Seat, 123; at Evaux, 118, 124; 
 in Tyrol, 124; Deflection of, Cal- 
 culated for Chimborazo, 34; how 
 to Observe, 35-39, 53 ; by Tides, 
 44, 134, 135. 
 
 Poisson, 31. 66 
 
 Power, 2-5, 49. 
 
 Poynting, 16, 36, 44, 106, 112, 116, 118, 
 119, 123-125, 127, 128, 131-134, 142. 
 
 Pratt, 124. 
 
 Pringle, 56. 
 
 Puissant, 118. 
 
 Pyramid, Attraction of the Great, 55. 
 
 R 
 
 Reich, 91, 106, 114-121, 125, 138. 
 
 Resistance of Air, Discussed by Bou- 
 guer, 27; by Cavendish, 65-67; 
 by Poisson, Menabrea, and Coruu 
 and Bailie, 66, 106, 125. 
 
 Richarz, 140, 141. 
 
 Robison, 134. 
 
 Roiffe, 48. 
 
 Royal Society, Experiments by 
 Members of, 2-6, 48. 
 
 Rozier, 49. 
 
 Sabine, 112. 
 
 Saigey, 32, 43, 54, 112, 113, 118, 124; 
 
 Correction of Peruvian Pendulum 
 
 Experiments by, 32, 43. 
 Schaar, 119. 
 Scheffler, 123. 
 Schehallien, 43, 53-55, 101, 112, 118, 
 
 123. 
 
 Schell, 56, 112, 116, 118, 123. 
 Schmidt, 44, 55, 106, 112. 
 Schubert, 124. 
 Sheepshanks, 113. 
 St. Paul's Cathedral, Experiments 
 
 at, 4, 5. 
 
 Sterneck, 128-131, 137. 
 Stokes, 122. 
 Struve, 124, 134. 
 
 Temperature, Effects of, on Torsion 
 Balance, Discussed by Cavendish, 
 60, 76-80; by Reich, 114; by 
 Baily, 116, 117; by Hicks and 
 Poynting, 119 ; by Boys, 135, 136 ; 
 by Eotvos, 137 ; by Braun, 139 ; 
 Change of A with, 125, 131 ; 
 Limit of Constancy of, 135. 
 
 Thiesen, 137. 
 
 Thomson and Tail, 134. 
 
 Tides, Action of, on Plumb-line, 44, 
 134. 
 
 Time of Vibration, How Found by 
 Cavendish, 64-67, 70 ; by Reich, 
 115, 119 ; by Baily, 117 ; by Men- 
 denhall, 127 ; As Affected by De- 
 flection, 97 ; As Affected by Con- 
 vection Currents, 80, 100, 134, 137 ; 
 A Found From, 105, 106, 120, 
 138, 139, 141. 
 
 Todhunter, 16, 44, 56, 66. 
 
 159 
 
INDEX 
 
 U 
 
 Ulloa, 25, 39, 40. 
 
 Vacuum, Experiment Made in, 138, 
 141. 
 
 W 
 
 Wallentin, 127, 134. 
 
 Westminster Abbey, Experiments 
 
 Made at, 2, 3, 5. 
 W he well, 113. 
 
 Wilsing, 131, 132. 
 
 Y 
 
 Young, Rule of, 31, 130. 
 
 Z 
 
 Zach, 36, 44, 54-56, 134 ; Maskelyne 
 Experiment Calculated by, 54 ; 
 Finds Attraction of Mount Mi- 
 met, 56. 
 
 Zanotti-Bianco, 44, 54, 56, 106, 112, 
 113, 123, 127. 
 
 ADDENDUM 
 
 Page 32. [Faye (146|) has calculated the diminution in the attraction ac- 
 cording to Ms formula (see note on p. 31), and finds it to be the ^^th part, 
 which is not far from that resulting from the experiment. His calculation can 
 also be stated in the following way : taking no account of the attraction of the 
 plateau, tJie observed pendulum lengtJis reduced to sea-level by Saigey are at 
 
 L'lsle de I'Inca . . 
 
 Quito 
 
 Difference 
 
 990.935 mm. 
 991.009 " 
 .074 " 
 
 which difference is of the order of the errors of the observations. See this vol- 
 ume, p. 130, Helmert (148, vol. 2, chap. 3), and Zanotti-Bianco (148%, pt. 1, 
 chap. 8, andpt. 2, p. 182).] 
 
 160 
 
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