^r 
 
 Mo 
 Division 
 Range 
 Shelf. 
 Received . 
 
 1870 
 

UNITY OF PURPOSE, 
 
 OR 
 
 RATIONAL ANALYSIS: 
 
 BEI 
 
 A TREATISE 
 
 DESIGNED 
 
 TO DISCLOSE PHYSICAL TRUTHS, AND TO DETECT 
 AND EXPOSE POPULAR ERRORS. 
 
 BY AUGUSTUS YOUNG. 
 
 " These are not, perhaps, very attractive speculations ; they disturb old and favorite 
 associations ; they serve to reduce many cherished traditions, much painfully acquired 
 knowledge, to obsolete lore ; but these things are so, and we must accustom ourselves to 
 regard them and their consequences without shrinking." 
 
 BOSTON: 
 
 PRINTED BY S. N. DICKINSON & CO. 
 1846. 
 
Entered according to Act of Congress, in the year 1846, by 
 
 AUGUSTUS YOUNG, 
 In the Clerk's Office of the District Court of the District of Vermont. 
 
CONTENTS. 
 
 [Although in the work, some of the respective Chapters are divided into, or are 
 composed of several Sections, such division will be disregarded in the Table of 
 Contents, the subject matters of the Chapters only being given.] 
 
 Page. 
 
 Introduction, 5 
 
 CHAPTER I. 
 On the Quadrature of the Circle, 27 
 
 CHAPTER II. 
 On the Law of Gravity, and the Popular Deduction and Promulgation of the 
 
 Supposed Law, 98 
 
 CHAPTER III. 
 
 Of the Proper Elements from which to determine the Laws of Force and 
 Motion, incident to the Heavenly Bodies, in their Eternal Rounds, 198 
 
 CHAPTER IV. 
 On the Hypothesis suggested by Kepler in respect to Elliptical Orbits, 217 
 
 CHAPTER V. 
 Outline of a Theory of the Solar System, 231 
 
 CHAPTER VI. 
 On the Tides, 246 
 
 CHAPTER VII. 
 An Investigation of the Theory of the Progressive Motion of Light, 261 
 
ERRATA. 
 
 Page 37, line 33, for may read might. 
 
 Page 48, line 35, before the word propose, read do not. 
 
 Page 48, line 42, for divisions, read divisors. 
 
 Page 58, line 43, before the word will, read when equated. 
 
 Page 60, line 19, before the word constantly, read when equated^ will. 
 
 Page 97, line 11, for space, read spaces. 
 
 Page 100, line 37, erase the word same commencing the line. 
 
 Page 107, line 14, for form, read force. 
 
 Page 116, line 33, erase the words the square root of. 
 
 Page 117, line 3, for minutes, read seconds, 
 
 Page 131, line 25, read the word we after the word it, at the end of the line. 
 
 Page 131, line 27. erase the word we commencing the line. 
 
 Page 210, line 6, for or, read is. 
 
 Page 216, line 10, for time, read line. 
 
 Page 224, line 30, for distance, read diameter. 
 
 Page 264, line 16, for sutor, read sector. 
 
 Page 286, line 17, for doctrines, read dictums. 
 
INTRODUCTION. 
 
 THE object of this work as declared in the title page, is to 
 elicit inquiry into, or reexamination of certain matters which the 
 world at large now receive implicitly, at the hands of others, as 
 scientific truths ; but which, perhaps, upon a more rigid and 
 scrupulous examination, divested of a too easy faith in matters 
 purely scientific, may be found to be but popular errors, which 
 should be eradicated for the benefit of physical science ; and 
 should the fact be disclosed that many matters which are now 
 almost universally esteemed sublime scientific truths, are but 
 dark and occult errors, the inquiry will naturally suggest itself, 
 whether it may not have been found necessary to clothe them in 
 a mathematical dress so wholly incomprehensible to the mass of 
 mankind, as to make it a hopeless task from want of leisure and 
 other facilities, to investigate the truth or falsehood promulgated 
 by the learned through the medium of what is so triumphantly 
 termed the higher branches of mathematics ; and hence, whether 
 mankind in general, have not been compelled to remain ignorant 
 of those supposed physical truths, except by a confidential faith 
 in those who profess to teach those things. And should my 
 labors in anywise serve to induce the learned to stoop a little 
 more to the necessities of the multitude, who lack leisure and 
 opportunity to acquaint themselves with all the modern devices 
 of mathematical science, my object will be fully accomplished. 
 
 I am aware that many will consider the subject matters of 
 which I treat, (namely, the cold, material laws of nature,) as 
 altogether grovelling in comparison with those flights of human 
 imagination, which flood the world in such profusion, and which 
 so liberally dispense to mankind the benefits of modern literature 
 by means of novels and tales of fiction ; and which are conceiv- 
 ed by many to be paramount to every thing intellectual beside : 
 but to fulfil a task so desirable to many, I confess I was never 
 
6 INTRODUCTION. 
 
 properly fitted and prepared; the preparation for it must, in 
 general, come through the medium of a collegiate education and 
 discipline. Besides, I have ever looked upon the universe as a 
 reality, and not as a fiction ; nor am I without hope, that eternal 
 truths may yet become as pleasing to a great portion of mankind 
 as is human error. Such is certainly the case to a very great 
 extent in the moral world, and it is certainly too manifest to be 
 overlooked or disregarded, that if a proper regard were had to 
 that unity of purpose in which the Christian religion is founded, 
 the world of mankind might be much more readily or easily 
 squared and quadrated into what is befitting a moral world, 
 than they could be equated to the same by means of Jewish 
 rites and ceremonies, or by heathen philosophy. It would 
 seem then to be but following out the principles of Christian 
 philsosophy, to take example from that great dispensation of 
 Divine truth in other departments of science, or in the natural 
 world, by endeavoring to adopt that unity of purpose which 
 alone seems calculated to lead to those truths which are to 
 benefit mankind. If such, then, were our foundation, whatever 
 errors and mistakes might occur in our attempts to extend its 
 benefits, we should not be left without a land-mark, a witness, 
 contained in such unity of purpose, whereby to correct those 
 errors. 
 
 Shall we not. then, in our researches for truth, endeavor to 
 follow out the true principles of Christian philosophy, with a 
 view to some unity of purpose, as being the most probable 
 condition upon which we may hope for success? 
 
 In my pursuit after truth, I shall no more be satisfied with 
 taking a middle flight, than Milton was in the sublimation of 
 fiction ; and as this must often bring me in contact with those 
 who have attempted the most daring flights in the scientific 
 world, (especially in the departments of mathematics and astron- 
 omy,) I shall speak of them and their doings, with a freedom 
 and boldness which I suppose to be justly due to the subject, 
 and to mankind ; nor will I, in this respect, ask from the world 
 other mercy than such as I show to others. 
 
 It may surprise the admiration which many have been taught 
 to entertain of Sir Isaac Newton, who by his flattering biog- 
 raphers, has been styled the creator of natural philosophy, ~ 
 that I should place Kepler on far higher ground as a philoso- 
 pher, so far, at least, as science has been benefited by their 
 researches or hypotheses. It may surprise many that I should 
 esteem Kepler as the father of physical astronomy ; and assert 
 that he has done more in laying the foundation of that sublime 
 science in those eternal truths upon which its superstructure 
 
INTRODUCTION. 7 
 
 should be erected, than all others ; and that, had a like sagacity 
 succeeded him in those who attempted its superstructure, its 
 strength and beauty would have been grand indeed in compari- 
 son with its present deformity. And although I shall treat of 
 Kepler's elliptical hypothesis with something more than doubts, 
 nevertheless, if it were an error in him, it could not have been 
 the cause of errors committed by others in respect to the laws of 
 force and motion, by which the phenomena of the heavenly 
 bodies can only be explained. 
 
 Sir Isaac Newton lived in a day of problems, in a time of 
 great anxiety in respect to the science of astronomy, which had 
 recently been ushered forth to the world, and somewhat advanced 
 by the labors and researches of Copernicus, Tycho, Kepler and 
 Galileo ; and hence, the world then required some master spirit 
 to collect the disjointed parts, and to raise the superstructure to 
 their implicit admiration and content. Newton also lived in a 
 country where science and the arts were better patronized by 
 national pride than in any other country at the time ; and he 
 seems to have been fully able to avail himself of all advantages 
 which came in his way ; for, if we are to believe his admiring 
 biographers, he was often enabled to keep his own discoveries a 
 profound secret until they were disclosed to the world by others, 
 although he never failed to assort his claim as the discoverer, and 
 to have his claim allowed by his country ; as in the case of Kep- 
 ler, Liebnitz and Flamstead ; although modern disclosures in 
 respect to the treatment of Flamstead, have caused even high 
 toned Englishmen to declare, that the name of Newton was no 
 longer to be revered. And whether the doubt expressed in respect 
 to Sir Richard Arkwright, may not with equal propriety be ap- 
 plied to Sir Isaac Newton, is for a candid world to determine, 
 namely, " some esteem this remarkable man as a genius of the 
 first order, gifted with an extraordinary power of invention ; 
 others, as an artful contriver, who understood how to appropriate 
 to himself the discoveries of others." 
 
 But the biographer of Newton was certainly acquainted with 
 what would induce implicit faith, and flatter a nation's vanity, 
 when he saw fit to insert the pitiful story about the fall of the 
 apple from the apple-tree, when Newton \vas sitting alone under 
 the tree ; and what Sir Isaac Newton asked himself on that 
 occasion ; and also what he said to his pet dog Diamond, on a 
 certain occasion ; how the fall of the apple suggested to the 
 mind of Sir Isaac, the hypothesis of the universal gravity of 
 inert matter; how Sir Isaac Newton devised a method for 
 ascertaining whether gravity extended from the earth to the moon, 
 and what would be the law of gravity in respect to distance ; 
 
8 INTRODUCTION. 
 
 and how his ecstatic agony, on supposing his conjecture about 
 to be realized by proof, became so great as to deprive him of the 
 power to finish out the problem and final result, which was 
 thenceforth to become the great physical law of the universe ; 
 and was therefore compelled to submit the finishing stroke to 
 another person. 
 
 But such stuff cannot well be accounted for upon any other 
 principle than that of a design to clothe Sir Isaac Newton with 
 something superhuman in the eyes of the world, and make him 
 an object of adoration ; and thereby to induce a belief, that such 
 wonders could have been performed by no other. And such 
 efforts have had their effect ; Newton has been immortalized, 
 and national pride has been gratified, however true science may 
 have been retarded in consequence. 
 
 The world has generally supposed that Newton first disclosed 
 the idea of attraction among the heavenly bodies ; and that 
 such attraction was the cause of the tides ; notwithstanding, Kep- 
 ler, some forty years previous to the advent of Newton's apple, 
 had not only fully declared the principle to the world, but had 
 done it in such a way and manner, as not to lead mankind into 
 those atheistical and abhorrent notions, which are taught in the 
 Newtonian philosophy in respect to the hypothesis. Kepler had 
 declared that the tides were caused by the attraction of the moon ; 
 that the earth and moon mutually attract each other ; and that, 
 in . consequence, they would come together, and that too, at a 
 point as much nearer the earth, as the earth is greater than the 
 moon, were they not prevented by the motion of the moon in its 
 orbit. And a better development or declaration for the future 
 progress of science, could not have been made. It was in nowise 
 calculated to lead into error ; it did not assume any particular 
 hypothesis in respect to universal gravity, or attempt to declare 
 its laws in respect to distance, but left them for future investiga- 
 tion, which long ere this, should have been both rational and 
 conclusive. 
 
 It is said of Newton, that, after receiving the hint from the 
 falling apple, setting out with the law of Kepler, namely, " that 
 the squares of the times of revolution of the different planets, are 
 proportioned to the cubes of their distances from the sun," he 
 found by calculation, that the force of solar attraction or gravity, 
 decreases proportionally to the square of the distance. 
 
 But I propose to show that if he set out with Kepler's said 
 law, he very soon lost sight of it ; as he did also of the whole 
 spirit and meaning of Kepler's olher law, which teaches that the 
 motion of a planet in an eccentric orbit, varies as the distance 
 of the planet from the sun varies ; which last great law of Kepler, 
 
INTRODUCTION. 9 
 
 Newton must have wholly misunderstood or misapprehended ; 
 else the error aforesaid could not have been committed ; for I 
 shall endeavor to prove, that neither Sir Isaac, nor any other per- 
 son, ever found the force of solar gravity to decrease propor- 
 tionally to the square of the distance. For. if such be the case, 
 Kepler's second law should be so far altered from the manner in 
 which he declared it, as to assert that the motion of a system of 
 planets must vary proportionally to each other, as their respective 
 distances vary from the sun, thereby wholly overthrowing the 
 great law of Kepler, with which Newton set out, as it would 
 cause the squares of the periods of a system of planets to be as 
 the fourth powers of their mean distances from the sun, as may 
 be readily ascertained by the most obtuse intellect. For a fact 
 more easily ascertained scarcely exists, than that the mean motions 
 of a system of planets are inversely as the square roots of their 
 mean distances from the sun, which is the only possible condition 
 in which the squares of the periods can be as the cubes of their 
 mean distances from the sun. 
 
 Another fact as simple, is, that if the mean motions of a sys- 
 tem of planets were inversely as their mean distances from the 
 sun, and also, if the motion of each individual planet varies 
 inversely as its distance varies during its period or revolu- 
 tion, then, indeed, the force of solar attraction would be inversely 
 as the square of the distance ; and the periods would be as the 
 squares of their mean distances from the sun ; and upon no 
 other condition whatever could the force of solar attraction be 
 inversely as the square of the distance. 
 
 It is also said by Newton's biographer, that having discovered 
 the law of gravity of the planets toward the sun, he endeavored 
 to apply it to the moon, by comparing the deflection of the moon 
 from a tangent to its orbit, with the law of falling bodies near 
 the earth ; and that, after many years of doubt, he at length 
 found that the attractive power of the earth varied inversely as 
 the square of the distance from its centre to the moon ; and that 
 after two years more, spent in penetrating the consequences of 
 this discovery, his immortal Principia came forth, of which it is 
 said, that not more than two or three of his contemporaries were 
 capable of understanding it, and that more than fifty years elapsed 
 before the great physical truth which it contained, (namely, his 
 law of gravity,) was thoroughly understood by scientific men 
 generally. Nor, indeed, was this accomplished, until a recon- 
 ciliation to the law was brought about by Clairaut, by one of 
 the strangest devices of which ingenuity, grown desperate, is ca- 
 pable, and which has produced a kind of Gordian knot in the 
 2 
 
10 INTRODUCTION. 
 
 science of astronomy, which we had much better cut than to 
 pother much in attempting to untie it. 
 
 But perhaps a better commentary can scarcely be made upon 
 our inclination to adore something which we know not, than that 
 contained in a prospectus, issued in the city of New York, by a 
 respectable publisher, this year, (1846) for publishing in this 
 country, for the first time, Newton's Principia, in which the 
 publisher remarks as follows : 
 
 " It is a singular fact that this remarkable work, the Principia 
 of Sir Isaac Newton, the foundation, at once, of his world- 
 wide fame, and the grandest monument of human intellect, 
 should have remained comparatively unknown in this country 
 to the present day. Singular, because the name of the immor- 
 tal author is familiar as a household word ; himself honored, 
 revered, looked up to as one of the demi-gods of by-gone 
 times," &c. " But the Principia has hitherto been inaccessible to 
 popular, use. A few copies in Latin, and occasionally one in 
 English, were and are to be found in some of our large libra- 
 ries, and in the possession of some loving disciple of the great 
 master. A dead language, however, in the one case, and an 
 enormous price in both, particularly in the ease of the English 
 edition, have thus far opposed very sufficient obstacles to any 
 general circulation of the work," &c. 
 
 And such is the blind idolatry of a world, thus worshipping 
 a fellow-mortal, which it may, at some future time, wish to make 
 the scape-goat of its errors. 
 
 But of the law of gravity, I propose to treat in its proper place ;. 
 remarking here, however, that, notwithstanding Sir Isaac New- 
 ton, Bernoulli, and others, have strangely endeavored to equate 
 and adjust the respective attractive powers of the sun and moon, 
 by calculating their respective influences in producing the tides, 
 in which Newton estimates the influence of the moon to that of 
 the sun as 3.5 to 1 ; Bernoulli as 2.5 to 1, and M. Thouroud 
 far less than either, nevertheless, the earth revolves farther in 
 its orbit in one day, about the sun, than the moon does about 
 the earth, in an entire revolution ; which circumstance ought to 
 be accounted for in some rational and proper manner. 
 
 But Sir Isaac Newton having determined his law of gravity, 
 physical consequents must necessarily follow or flow from it ; as 
 in the case of the motion of the moon's apogee, for instance ; and 
 although Sir Isaac, by his law of gravity, had furnished but just 
 one fourth of the requisite quantity, or only sufficient to produce 
 just one half of the actual or observed motion, yet -Clairaut, at 
 length, found out a way to make that answer. 
 
 So also in respect to the precession of the equinoxes, it is still 
 
INTRODUCTION. 11 
 
 insisted that Sir Isaac Newton assigned the true physical cause, 
 namely : the attraction of the sun upon the surplus matter of 
 the earth, over and above its greatest inscribed sphere ; although 
 Mr. Vince says, " it is acknowledged that Newton fell into an 
 error in his investigations as to the amount of the effect." But 
 this acknowledgment seems rather to have arisen from the ne- 
 cessity that the celebrated astronomer, Dr. Bradley, was under, of 
 having a place to stand on, in respect to his discovery of the nuta- 
 tion of the poles of the earth. And in respect to the theory of the 
 tides, I think Sir Isaac Newton has lost nothing from La Place's 
 modification of the cause and effect, as he has left it still more ab- 
 surd and inexplicable, if .possible, than he found it. 
 
 As my principal purpose is to treat of, or inquire into the 
 nature and effect of the more direct physical causes by which 
 the worlds are more immediately governed and controlled in their 
 eternal rounds, and as Sir Isaac Newton's theory of light does 
 not so essentially interfere with or affect those laws, I shall omit 
 to treat of that department in his philosophy, as that may be 
 better borne with, however absurd it may be, than his theory and 
 law of gravity, till a better philosophy shall correct the absurdi- 
 ties. But not so in respect to his theory of universal gravity, (as 
 by him taught to a too confiding world,) which, like some nox- 
 ious plant or vine, has interwoven itself into the whole fabric of 
 astronomy, disjoining its parts, hiding its beauties, and affecting 
 the operations of all its physical laws. 
 
 This is the theory which, while it asserts that all gross or tan- 
 gible matter in the universe, is inert, dead, and regardless, asserts 
 also, that all matter is equally endued with an innate or inherent 
 principle for the attraction of other matter ; and hence, that inert 
 matter is made to possess innate powers and innate tendencies. 
 This is the theory which requires the heavenly bodies to be pro- 
 jected in the dkection of a tangent to the orbit in which they 
 are to revolve; not that the planet is to pass in the direction 
 in which it is projected ; the projection being only one of the 
 resulting forces, the other being inherent in inert matter, con- 
 stantly operating, though at a vast distance, and that also 
 through space, and which must forever after, take the whole 
 charge of the body so projected, and safely guid-e it through all 
 the disturbing forces, during its eternal rounds. This is the the- 
 ory which has converted the whole universe into a system of 
 disturbing forces, and the whole corps of astronomers into an 
 exploring party, whose business it is to search out and regulate 
 disturbances. And it would necessarily be so, if the theory were 
 true; for the sun would then so disturb the original projectile 
 force, that the planet could not pass- at all in the line of projec- 
 
12 INTRODUCTION. 
 
 tion, and the planets, in turn, would disturb their neighbors, to- 
 gether with every stranger that came into their vicinity. Never- 
 theless, the disturbances are mutual, and not at all prevented by 
 the inertia of matter. 
 
 But, to me, this seems quite too fortuitous, if not absolutely 
 atheistical ; and, therefore, I must leave it to others to suppose 
 that a universe, governed and controlled by no other power than 
 that arising from disturbing forces, would operate better than one 
 governed and controlled by immutable laws of order and regu- 
 larity. Nevertheless, I do not accuse Sir Isaac Newton of en- 
 tertaining atheistical views or notions, for he even wrote upon the 
 prophecies of Daniel, and upon the Apocalypse, and were he 
 now living, would probably be the best expounder extant, of 
 their mysteries. 
 
 And finally, it is this theory, which, from the time of its pro- 
 mulgation, has served (with the world) " to make the worse 
 appear the better reason, to perplex and dash maturest counsels," 
 whereby astronomers have appeared to lose sight of first prin- 
 ciples, and rather implicitly to follow the persuasive sugges- 
 tions of a file leader, with a zeal that would have been far better 
 employed in the ways of truth, and in its development to the 
 world. 
 
 And now, if it should be thought that I have been somewhat 
 severe in my remarks, it ought also to be understood that I have 
 much idolatry to contend with, and even an apparent determina- 
 tion neither to behold nor acknowledge any error in anything 
 Sir Isaac Newton may have presumed to teach, promulgate or 
 find ; and hence we so often find the naked allegation, even in 
 the highest authorities, as if it were " confirmation strong as 
 proofs of holy writ," that Sir Isaac Newton found, &c. 
 
 But perhaps what I have said may only serve to show that my 
 faith in the infallibility of Sir Isaac Newton, is not so great, as 
 that of some others ; and if a further expression of such want 
 of faith should be required, I will here declare, that I have yet 
 to learn of a single beneficial discovery or principle, origin- 
 ally promulgated by Sir Isaac Newton, in anywise beneficial 
 to physical astronomy. 
 
 But I will not attempt to justify raillery, or irreverence upon 
 the ground that others before me have used such weapons. 
 Nevertheless, Sir Richard Phillips, (who, perhaps, has made 
 rather a feeble attack upon the stronghold, and who, perhaps, 
 should have adduced more efficient evidence to justify his prov- 
 ocation,) in speaking of the Newtonian method of deducing the 
 law of gravity from a consideration of the deflection of the moon 
 from a tangent to its orbit, speaks in this wise : 
 
 " How gross the imposture, then, of founding a system on a 
 
INTRODUCTION. 13 
 
 fictitious fall of sixteen feet per minute ! " &c. " How juggling, 
 too, to compare a paper fall at the moon, with a real fall at the 
 earlh ; and how delusive, even under that juggle, to take a minute 
 when even the juggle itself would hold for no other time ! " &c. 
 " But the wonder, after all, is far less, that a system-maker should 
 be seduced to make the assertion, than that mathematicians, and 
 men of moral worth should, from 1687 to 1831, have been mis- 
 taken enough to believe and teach such nonsense with solemn 
 emphasis." 
 
 The fact is, that Sir Richard, and also thousands of others, 
 perceived that there was discrepancy and error in the Newtonian 
 promulgation of the law of gravity; and strange indeed have 
 been the devices and attempts to correct or set aside the error ; 
 as also to reconcile the law with observed phenomena, in lieu of 
 reinvestigating the Newtonian method, and thereby, at once, 
 detecting the error which Sir Isaac committed. 
 
 Bat to me, Sir Isaac Newton's thoughts often appear to have 
 been extremely crude and unanalyzed, and in them he often 
 forgot his own philosophy, such as it was ; as, for instance, when 
 he wished to be wiser than Galileo, in respect to the direction a 
 body would take in its fall from a state of rest towards the earth, 
 he states that in some few hundred feet fall, it would advance 
 some half inch eastward, in consequence of the diurnal motion of 
 the earth ; and some modern experimenters, (believing such to be 
 the law,) have conceived that they have detected the half inch 
 advance ; but a more whimsical error could not have been ad- 
 vanced, if we suppose the law of gravity to be an omnipresent 
 law. But we are too ready to transgress against reason, if we 
 suppose the law to be with us. 
 
 Unity of purpose in the Divine Architect of the universe, is the 
 hypothesis in which I would involve all philosophy in respect to 
 cause and effect ; nor would I be too hasty in adopting conclu- 
 sions which might savor of fortuity, instead of being the result 
 of general and immutable laws. And whether some philoso- 
 phers of high repute, since the days of Copernicus, Galileo, and 
 Kepler, have sufficiently discarded fortuity or chance from being 
 co-workers in the operations of nature, is a subject worthy of 
 consideration ; and if not, it would not be a wonder if, thereby, 
 the universe had been thrown a little out of gear ; and if it 
 should, in consequence, require much equating to enable it to 
 fulfil what would seem to be its proper operations or destination. 
 
 The Copernican system of astronomy was not only the result 
 of great sagacity, at a time when much ignorance prevailed 
 upon the science of astronomy ; but, also, of a boldness border- 
 ing on temerity, in venturing to declare this theory to a world 
 
14 INTRODUCTION. 
 
 full of superstition and bigotry, and wedded, as the world al- 
 ways is, to those notions which have long been cherished by the 
 multitude as matters of implicit faith. 
 
 Galileo did excellent service to the science of astronomy, not 
 only in his discoveries by aid of the telescope, but in his other 
 philosophical researches; and especially, in ascertaining and 
 determining the law of falling bodies, near the earth. 
 
 But Kepler seems to have been the first who entertained 
 rational notions of physical astronomy ; at least, the first who 
 obtained any knowledge of those immutable laws, based upon 
 or founded in the immutability of numbers, by which the solar 
 system is governed in its harmonious movements ; which laws, 
 so by him discovered, from their great utility to the science of 
 astronomy, are, in great justice to their discoverer, called Kepler's 
 laws. And happy would the circumstance have been for the 
 science of astronomy, had those who succeeded those master- 
 builders, and essayed to erect the superstructure, been careful to 
 square their work by the foundation so skillfully and admirably 
 laid. 
 
 Those master-builders did not assume to build upon hypo- 
 thesis alone, but upon well-ascertained and demonstrated facts. 
 Thus Kepler, in announcing to the world the law of gravity, 
 did not attempt to announce a specific law, by which matter 
 operates upon matter, by means of an equal, innate, or inher- 
 ent gravity ; nor did he attempt to promulgate a law, by 
 which matter operates upon matter, through distance or space ; 
 and hence his announcement was not calculated to mislead 
 mankind in respect to a naked hypothesis, nor to cause them 
 to err by an erroneous, pretended demonstration, either of which, 
 to be sustained, would require the laws of nature to give way, 
 or at least to subscribe liberally to their support, to the neg- 
 lect of those matters more directly submitted to their charge and 
 control. 
 
 But in respect to Newton's broad hypothesis, a*s to the 
 equal innate gravity of all matter, may it not too strongly 
 imply a necessity, in matter itself, to possess certain innate, 
 inherent, or independent qualities, sufficient fo? its own govern- 
 ment ? Besides, if the law of force be an inherent and inde- 
 pendent quality or principle of matter, it may be hard to under- 
 stand why motion should not also be an inherent quality or 
 principle of matter. But if the Newtonian hypothesis be not 
 true, then what has been supposed to be different densities of 
 the planets may be accounted for upon the principle, that uni- 
 form laws operate upon matter alike, under like circumstances ; 
 which, manifestly, could not be the case upon the Newtonian 
 
INTRODUCTION. 15 
 
 % 
 
 theory, by which the worlds were launched on their never* 
 ending way by one original impulse, thence to be left wholly 
 dependent upon their equal inherent gravity for their future regu- 
 lation and control, notwithstanding the infinity of disturbing 
 forces forever operating from this supposed power of innate 
 gravity. 
 
 But what is certainly of vast importance to the science of 
 astronomy is, whether the force of gravity, which retains the 
 planets in their orbits, operates inversely as the distance, or in- 
 versely as the square of the distance, or in some other ratio. 
 
 It is quite certain that, attempting to bring the Newtonian 
 hypothesis, of the force of attraction or gravitation, to agree or 
 correspond with facts obtained from observation, has cost the 
 learned much trouble in exploring for equations, by which to 
 reconcile and harmonize the small deviations from the law, as 
 prescribed or promulgated. 
 
 To be sure Clairaut, Euler and others protested for a long 
 time against the law, as promulgated by Newton ; for that the 
 cause would not be adequate to the effects produced by the 
 law, as in case of the motion of the moon's apogee ; and, 
 for a time, the moon seemed well nigh divorced from all obliga- 
 tion to this earth, until Clairaut, by a device so fraught with 
 ingenuity that few, perhaps, will volunteer to examine its truth 
 or error, declared that he had reconciled the Newtonian law 
 of gravity with the motions of the moon. This seemed to be 
 the last great struggle. Opposition, in all quarters, appeared to 
 give way, and philosophy seemed not only compelled, but 
 rather inclined, to rest from opposition and strife, and to content 
 itself with exploring for disturbing forces and equations, by 
 which the Newtonian law of gravity might be justified, however 
 the harmony of nature might be torn in pieces to effect the pur- 
 pose. And notwithstanding many since have felt in their 
 minds to rebel against so unjust and arbitrary a law, as in 
 case of Lord Monboddo and Sir Richard Phillips, as well as 
 thousands of students who have been disciplined under this 
 law, yet no effectual force has ever been organized to resist 
 it. While, on the contrary, the superstructure has, for near two 
 centuries, been erecting upon the Newtonian foundation ; in 
 which the builders have, figuratively 'at least, made use of 
 the calculus for stones, and fluxions for mortar, until their lan- 
 guage, either spoken or written, has become so confounded, 
 that the world at large have long since ceased to have any un- 
 derstanding of it whatever; while the builders, (to use a figure 
 of M. Bailly,) like vultures over a battle-field, soar one above 
 another by such awful flights, that one can scarce discern anoth- 
 
16 INTRODUCTION. 
 
 er ; while mortals, who tread the earth beneath, can scarcely dis- 
 cern the lowest of them. 
 
 But, instead of contending against the sufficiency of the law 
 of gravity, as given by Newton, or endeavoring to reconcile the 
 law thus given, with facts which utterly deny its truth, had 
 Clairaut, Euler, and others, taken the trouble to reexamine the 
 evidence from which Newton deduced his supposed demonstra- 
 tion, they would readily have discovered where and how he 
 committed his error, namely, in assigning the square of the 
 distance to the rate of force, in lieu of assigning or referring it 
 to the amount of the fall of the body, in accordance with the law 
 of falling bodies near the surface of the earth. This one cor- 
 rection in the premises might have led to a further and better 
 consideration of the law of force or gravity, and eventually to a 
 true development of its principles, as applied to the control of a 
 revolving body ; by which they would have discovered that the 
 moon is entitled to just four times the force allowed to it by 
 Newton's law of gravity, and, consequently, to twice the motion 
 which his force would give. 
 
 But another great error in Newton's calculations (and which 
 will be considered in its proper place) was, in supposing that 
 the deflection of the moon from a tangent to its orbit, in a given 
 time, would be just equal to the fall of a body from a state of 
 rest, at the distance of the moon, in the same time. 
 
 Neither the limits nor design of this work permit me, were I 
 disposed, to enter the lists with philosophers, (either ancient or 
 modern,) who have theorized upon the phenomena of motion, 
 with much curious speculation, even to the doubting of its 
 existence. For the same reason I shall not go into the inquiry, 
 whether, according to Newton's theory, matter, by nature, is inert, 
 and will not move until urged by some adequate force ; and 
 that, from the same cause, inertia, when once put in motion, 
 it would, if not counteracted or disturbed by some other force, 
 forever continue to move in a right line, with a uniform velocity. 
 
 But, perhaps, after all, there may be something too imaginary 
 in such a theory, to justify its being considered a self-evident 
 proposition. For on recurring to the laws of nature, so far as 
 we are able to investigate them, we find them so contrived as 
 often to forbid what, 'to our fancies or imaginations, might 
 occur, if the laws of nature were otherwise than they are. We 
 know of no such thing in nature as that of a body moving, 
 either by a uniform motion or in a straight line ; for all matter 
 not only moves in curves, but with constantly varying veloc- 
 ities. Nor is the dilemma to be avoided, by reference to 
 a body falling toward the centre of the earth ; for the motion 
 
INTRODUCTION, 17 
 
 is not only accelerated, but the curve which it represents in its 
 fall has even perplexed La Place, and other great mathemati- 
 cians, to describe it. Hence it might seem, that those who have 
 assumed as a lemma, or self-evident proposition, that a uniform 
 motion, in a straight line, is the natural state or condition of the 
 motion of matter, have permitted their imaginations to devise 
 quite a different method from that which the laws of nature any 
 where permit. 
 
 Similar to the foregoing, is the case of those who fancy that 
 the natural state of matter is a state of rest ; and allege, that of 
 the truth of this proposition, also, we have abundant proof. We 
 know of no matter that is not constantly in motion, and that, 
 too, in accordance with what appears to us to be eternal and 
 immutable laws. To be sure, a stone, or other ponderous body, 
 lying upon the surface of the earth, appears to us to be at rest; 
 but who does not know that it is the tendency of such a body to 
 be in rr otion that causes it to remain in such apparent state of 
 rest, r.nd that its gravity is as much exerted upon the body 
 where it lies, as if it were actually falling towards the earth ? 
 
 So, also, the assumption appears to have been gratuitous and 
 wholly uncalled for, that the heavenly bodies must necessarily 
 have been projected, as with a view to motion, in straight lines ; 
 but that, by the power possessed by inert matter, they were 
 wholly prevented from moving in the direction in which they 
 were projected ; and this upon the assumed principle, that the 
 forces which produce motion in a curve are resulting forces, 
 that is, that the motion in a curve is the result of two forces 
 momentarily applied ; or that the straight line which the body 
 would pursue, were it not deflected from the same, arises from 
 the inertia of the body in a state of motion, being the result of 
 force long since expended ; and that the deflecting force is con- 
 stantly or momentarily operating upon the inertia of the body, 
 as though it were in a state of rest. But, upon the principle of 
 resulting forces, it may be hard to conceive why the motion 
 should not be continually or momentarily increased ; especially, 
 if the circle be what many have conceived it to be, namely, a 
 polygon consisting of an infinite number of sides ; in which 
 case, the deflecting force would be exerted by many different 
 impulses, and, consequently, the rate of motion would be 
 constantly changing. Nevertheless, the motion in a circle 
 remains as uniform as in a straight line ; and Lord Monboddo 
 says, that in the days of Aristotle the line of a circle was con- 
 sidered as simple as that of a straight line ; and in accordance 
 with this idea, it is certain, that the only rational conclusion to 
 be deduced from the attempts of those who would ascertain the 
 3 
 
18 INTRODUCTION. 
 
 periphery of a circle, by a redaction of its inscribed and circum- 
 scribed polygons, is, that the angles of the polygons, by becom- 
 ing more and more obtuse, at length wholly vanish in the 
 line of the circle ; and, hence, that where a tangent touches the 
 periphery of the circle, it forms no angle whatever with the line 
 of the circle. 
 
 But we nowhere discover, in the mechanism of the heavens, 
 that force is exerted by impulses ; and it would seem that the 
 composition and resolution of forces, as taught upon the par- 
 allelogram or otherwise, instead of teaching or inducing such 
 notions of force and motion, as being applicable to the heavenly 
 bodies, would have simply taught that motion, produced by 
 impulses, is as the square root of the force applied. But how the 
 eternal and immutable laws of nature would have been, had 
 they been otherwise than they are, I know not ; nor is my imag- 
 ination boundless enough to comprehend things so vast, and give 
 the information to mankind ; and should I attempt it, I should, 
 probably, find myself in " endless mazes lost." 
 
 In reference to the science of astronomy, which, from its 
 nature, requires, perhaps more than any other science, the aid of 
 mathematics, or rather the application of numbers in the de- 
 velopment of its truths, may it not be questioned, whether 
 a greater regard should not have been had to the edification of 
 those, whose lack of leisure and of other facilities, too often de- 
 prive them of the requisite advantages for investigating the more 
 abstruse sciences, and which might be afforded them, were the 
 learned inclined to stoop a little more to their necessities in that 
 respect ? 
 
 Thus the mathematical languages, if I may so speak, in 
 which the science of astronomy is now generally taught, are 
 become absolutely terrific to the great mass of mankind ; by 
 which means much of the science is to them but little if any 
 better than Rosicrusian mysteries ; which fact has a tendency to 
 cause their knowledge of the science to become less, in propor- 
 tion as the science is supposed to become advanced. For I will 
 ask, what proportion of a community can be expected to obtain 
 a knowledge of algebra, fluxions, or the calculus, or an infinity 
 of abstruse geometrical diagrams, with the view of obtaining 
 a knowledge of the science of astronomy, whereby they may 
 become able to distinguish for themselves the beauty of truth 
 from the deformity of error ? 
 
 But happy, however, may it have been for some, that they 
 were thus enabled to obscure their philosophy, even to such a 
 degree that the learned themselves were unable to penetrate or 
 dispel the dark fogs and mists that pervaded it. And it is 
 
INTRODUCTION. 
 
 19 
 
 almost to be feared, that the libraries of the present day will 
 share the fate of the Alexandrian, from so universal an ignorance 
 of their contents, or from our not being able to understand and 
 appreciate them. 
 
 Neither Galileo, nor Kepler, knew aught of the differential 
 or the integral calculus, or of fluxions ; nor had Des Cartes 
 applied algebra to astronomical investigations. Kepler made 
 use of the then popular method, by the use of the Arabic 
 characters ; and that, too, even without the assistance of log- 
 arithms. But since the days of Kepler, I know not of a 
 single discovery in physical astronomy which deserves the 
 name. 
 
 May it not, then, be possible, that less rigid or thorough 
 mathematical investigations have obtained, in consequence of the 
 adoption of other methods than would have been used, if the 
 popular or Arabic character had been retained in all mathemat- 
 ical investigations ? for of the use of it, all must have some know- 
 ledge who have obtained the rudiments of an education. 
 
 And may it not even now be questioned, whether the ordin- 
 ary mathematics, by popular notation, is not as susceptible 
 of demonstrating the most abstruse problems or propositions, 
 and of investigating as rigidly, as clearly, and with as much 
 scrutiny, the arcana appertaining to mathematical science, as 
 any of the substitutes extant? This method is, certainly, the only 
 one with which the mass of mankind will probably become 
 acquainted ; and, hence, to adopt into the sciences such a method 
 to the extent of its powers, would certainly bring science vastly 
 more within the reach or understanding of the mass of man- 
 kind, than it is at present ; and vastly more would be gained to 
 science by the general diffusion of knowledge, than perhaps 
 would be lost by a total abandonment of all other methods. 
 
 These being my views, I shall, in the present work, con- 
 fine myself, as much as possible, to the common or popular 
 method in mathematics: whether it be in attempting to disclose 
 new principles in the economy or philosophy of numbers ; or in 
 applying them to the investigation of the laws of nature, or of 
 scientific truth ; making use of as few additional symbols or 
 characters for the expression of abstract ideas, (in addition to the 
 natural numbers, or popular notation,) as the subject will admit. 
 
 It is not, perhaps, yet known to what extent the economy of 
 numbers, (even by the common or popular method,) may be 
 carried, in their development of the human mind in its upward 
 progress; which, from being, in their simplest application, per- 
 haps, the most abstract and simple of all our ideas, may also be 
 extended into the most intricate, abstruse and confused. For 
 
20 INTRODUCTION. 
 
 although unity, or individuality, is, perhaps, as distinct a concep- 
 tion as the mind entertains, yet, in proportion as unities or in- 
 dividualities are increased or multiplied, they become more and 
 more indistinct or incomprehensible to the mind ; so that few, 
 perhaps, have very clear conceptions of a million. 
 
 In the economy of numbers, as in other things contrived by 
 infinite wisdom for the benefit of man, there is a fitness of things 
 fully commensurate with the human understanding; and to a 
 knowledge of which, to the utmost of human capability, he 
 seems to be invited and urged by every consideration worthy of 
 his unbounded capacity, when fully developed. 
 
 Although it is not my purpose, in this place, to treat particu- 
 larly of the economy of numbers, it may, nevertheless, not be 
 wholly unprofitable to commence with it, and refer to some of the 
 qualities of unity or 1, and their peculiar application to the in- 
 vestigation of the immutable laws of nature. 
 
 Unity, then, or number 1, seems wonderfully calculated or 
 adapted, as a numerical standard, for all quantities capable or 
 susceptible of numerical value or admeasurement ; as of bulk 
 or magnitude, weight, space, time, force, motion, distance, &c.; 
 for, however great or small the quantity, it may be denoted by 
 unity, which is at the same time susceptible of indefinite math- 
 ematical or numerical divisibility. But what peculiarly dis- 
 tinguishes unity from all other numerical quantities, and makes 
 it the standard numerical quantity by which to compare all 
 others, is, that its powers and roots neither augment nor diminish 
 it. This, together with the constant reciprocity which exists in 
 the elements of the heavenly bodies, as connected with the laws 
 by which they are governed, whether it be in reference to the 
 elements of an individual planet of a system, or of all the several 
 planets of a system when compared with each other, makes it 
 of the first importance, in facilitating our operations and in 
 arriving at satisfactory results,, that unity be adopted as the stand- 
 ard measure of all numerical quantities or values. This being the 
 case, by always regarding unity as the radical centre of all nu- 
 merical quantities, the laws which govern the planetary world 
 will be found to be based upon much more uniform and simple 
 principles than they have in general been supposed to be r all 
 harmonizing and agreeing with the Keplerian law, that the 
 squares of the periods are as the cubes of the mean distances 
 from the sun. Thus, in any system of bodies revolving about 
 a central force, as in case of the primary planets about the sun,, 
 or of satellites about a primary planet, either of the revolving, 
 bodies may be taken at the mean distance of unity or 1, from 
 the central force, by which to compare the elements of all the 
 
I 
 
 INTRODUCTION. 21 
 
 other revolving bodies, their respective distances being pro- 
 portioned to that the distance of which is taken at unity. And in 
 such case, each and every element of that planet which is adopt- 
 ed as the standard, or the distance of which is assumed at 
 unity, , must also be assumed at unity, in order that the Kep- 
 lerian law may be fulfilled. 
 
 In respect to whatever of theoretic matter I may introduce, as 
 supposed deductions and conclusions drawn by way of analogy 
 from known facts and laws, I shall feel no anxiety in sustaining 
 the same, further than it shall be sustained by legitimate evi- 
 dence ; as in case of my theory in respect to the identity between 
 magnetism and the attraction of gravitation by which the heav- 
 enly bodies are controlled, of which I have conceived many 
 evidences to exist in the phenomena of nature. In support of 
 this theory, I shall adduce the evidence arising from the differ- 
 ent powers of attraction of the different planets, in proportion to 
 their respective bulks or magnitudes ; and in this case, I shall at- 
 tempt to show that the phenomena arises from a general law, 
 (not fortuitous) deducible from, the identity of gravitation with 
 magnetic attraction. 
 
 I shall call in the aid of the phenomena of the tides, arising from 
 the attractive power of the moon ; and shall, on this principle, 
 endeavor to account for many of the phenomena in respect to 
 the tides, in a very different way and manner from those com- 
 monly adopted respecting them ; and show why equal tides can 
 exist on opposite sides of the earth at the same time ; why the 
 tide-waters of broad oceans pile high upon the shores of broad 
 continents, while they are comparatively small on the shores of 
 the narrow parts of those continents, and still smaller in the 
 midst of broad oceans, where only small islands exist, &c. ; and 
 will endeavor to account, from the same simple cause, for the 
 reason why high tide is not as far behind the moon in the midst 
 of broad oceans, as upon the shores of broad continents ; why 
 large inland seas produce little or no perceptible tides, &c. 
 
 In support of the identity of gravitation and magnetism, to 
 some extent, at least, I propose to call to my aid the nice and 
 critical observations of Dr. Bradley, (Astronomer Royal to Geo. 
 II,) made upon some of the fixed stars, with the view of ascer- 
 taining whether they had any sensible parallax ; in which 
 he finally deduced the hypothesis of the aberration of light, 
 which, in astronomy, has become technical in its use and signi- 
 fication, denoting a small apparent motion of ihe fixed stars, oc- 
 casioned by the progressive motion of light and the earth's an- 
 nual motion in its orbit. And although the hypothesis has, 
 since his announcement of it, been elaborated into a sys- 
 
22 INTRODUCTION. 
 
 tern or branch of philosophy, by the aid and assistance of the 
 higher branches of mathematics, and its supposed consequences 
 or effects introduced into astronomical observations and calcula- 
 tions, I must declare my full dissent to the correctness of the 
 conclusion or hypothesis, believing the same to be in nowise 
 justified by the premises from which it is drawn. 
 
 In these introductory remarks I will refer to but one other 
 subject connected with my theoretic speculation, as collateral 
 or rather presumptive evidence in favor of that unity of pur- 
 pose, and fitness of things, which all will readily ascribe to the 
 Divine Architect of the universe. I refer to the subject of Geol- 
 ogy, a subject which, though recent as a science, or subject of 
 investigation, abounds with that multiplicity of facts and phe- 
 nomena which has already caused theories in respect to it, to 
 succeed each other in rather rapid succession, from the circum- 
 stance of a few newly discovered facts overturning the last 
 formed theory. 
 
 But the Huttonian, or Volcanic theory, that the surface of the 
 earth is but the cooled shell of what was once an entire mass 
 of fused, or melted matter, and that all the supposed catastro- 
 phes that have happened to the earth, in producing the appear- 
 ances and phenomena on and near its surface, as also the increase 
 of the temperature of heat as we descend into the earth, &c., which 
 are conceived to be the evidences of such hypothesis, is, in the 
 opinion of many, so deeply laid as not to be easily overturned. 
 But, to me, this hypothesis seems altogether too fortuitous, both 
 in its origin and consequences, to obtain credence for a moment, 
 from one who believes creation to be sustained by immutable 
 and uniform laws; and seems far better calculated to furnish 
 food for the marvellous, than to lead mankind to very high no- 
 tions in respect to the wisdom, power, and goodness of their 
 Creator. It would seem also, as if there would exist no necessity 
 for such an hypothesis, which so illy accounts for the phenomena 
 to which it is assigned, if due regard were had, or proper consider- 
 ation given, to what were, until of late, occult laws or operations 
 of nature, but which, in modern times, have been deduced 
 and drawn forth by the sagacity of man, not as unprofitable 
 knowledge, but such as ministers to his safety, his support and 
 his happiness. I refer to the modus operandi of the law of elec- 
 tricity, in the experiment by Dr. Franklin with the string of a 
 kite ; to galvanism, in the experiment by Galvani with the leg of 
 a frog ; and to electro-magnetism, arid the experiments by Oer- 
 sted, of no less sagacious intellect. 
 
 Those operations, or laws of nature, together with that of 
 magnetism, (and, perhaps, the operation of light,) are, in general, 
 
INTRODUCTION. 23 
 
 considered by philosophers so far identical, as, at least, to be 
 modifications of one great and universal law, the constant effect 
 of whose silent operations, are more apparent under the form of 
 gravitation or attraction, than under any other ; and that this 
 universal law, in all its various modifications and operations, 
 acts as a messenger of Omnipresence, as well in the government 
 and control of inanimate and unorganized matter, as in that 
 which is organized and animate, and, therefore, dependent upon 
 final causes. We see the operation of this law or principle, 
 in gravity or attraction, in controlling the heavenly bodies in 
 their motions. We see its operation in magnetism or mag- 
 netic attraction, as denoted by a common magnet or loadstone ; 
 presenting that wonderful phenomenon exhibited in the polarity 
 of a common compass needle, accompanied by the paradoxi- 
 cal fact or phenomenon, that what is termed the north pole of 
 the needle, is attracted by the south, and repelled by the north 
 pole of another needle ; which paradoxical fact 1 propose to no- 
 tice hereafter. Here then we discover (in the absence of any 
 more appropriate word by which to explain the occult opera- 
 tion) the magnetic fluid operating in different directions, namely, 
 towards the opposite poles of the earth, (except in case of the 
 variation or declination of the needle,) and this upon all parts 
 of the surface of the earth ; while at the poles of the earth, it 
 can have but one direction, as denoted by the dip of the needle. 
 
 If, then, we find this all-pervading, all-powerful, but occult 
 law, thus operating throughout the surface of the earth, (and 
 analogy would indicate the same in respect to other planets,) if 
 we find it operating in the movement and control of the vast 
 bodies which compose the solar system ; if we find it, under other 
 modifications or appearances, as those of electricity, galvanism, 
 or electro-magnetism, (when subjected to the control of man, 
 and made to subserve his purposes and curiosity,) operating by 
 its own immutable laws upon a small scale, as that of the 
 electric, the galvanic or electro-magnetic battery, or of the tele- 
 graph, is it to be supposed, or presumed, that those operations 
 are only incident to human contrivance ? or, rather, on the con- 
 trary, are not those laws, or principles, designed for similar but 
 vast and continued operations in the control of the universe, 
 which, otherwise, must be left to fortuity or chance ? For it is 
 inconceivable, how an immutable law can be subjected to the 
 performance of operations not in accordance with its common 
 sphere of action. 
 
 I would ask, then, if what are termed the positive and nega- 
 tive fluids (for here I will use no other expression to denote 
 them) of this all powerful agent, (under whatever modification 
 
24 INTRODUCTION. 
 
 it may operate,) meet in concentration at the centre of the earth, 
 whether it would not far better account for all the phenomena 
 upon and near the surface of the earth, than does the present 
 hypothesis of internal fires, as also for an infinite variety of 
 other phenomena, which the present hypothesis in nowise ac- 
 counts for, as that of the independent coal formation, for ex- 
 ample ? 
 
 Besides, upon the theory which I propose, there would be 
 nothing fortuitous ; all would result from the operation of con- 
 tinued laws ; there would be no apprehension that the internal 
 fires would become extinct ; or that the crust might give way, 
 and produce catastrophes, such as the imagination alone can 
 conceive. 
 
 As no excuse is pleaded for certain courses which I see prop- 
 er to take, or pursue, in the work, hence none need be granted ; 
 as, for instance, in case of the investigation of the law of 
 gravity, from a consideration of the law of falling bodies; in 
 which case the text on which I comment, (generally taken from 
 Maclaurin, Vince or Rees,) may not only appear to some to be 
 too often and too liberally quoted, but the commentaries may seem 
 to savor too much of repetition, all tending to prove one and the 
 same thing, namely, that the force of gravity varies inversely 
 as the distance varies. It is but few subjects, however, of which 
 I treat, and those are fundamental and important; and the 
 course I have pursued, I believe, has ever been esteemed laud- 
 able, in regard to important truths. They should be kept before 
 the people, until their importance shall be duly considered and 
 appreciated ; they should even be taught by way of inculcation : 
 and if error is to be removed, to make room for the truth, the 
 task is doubled. We should " try every art," like a bird that 
 " tries to tempt her new fledged offspring to the skies ; " nor be 
 satisfied with some formal intimation, believing the world to be 
 under obligation to hold or regard it as truth, or perish. And 
 although we may be inclined to adore whatever human devices 
 may have established, nevertheless, truth is too catholic in its na- 
 ture to be established by convention. 
 
 It is not expected that the human race will be born prodigies, 
 or even possessing much intuitive knowledge ; nor is it ration- 
 ally to be supposed, that a precocity will soon attach to human 
 nature, whereby infancy shall be the wisest portion of our lives ; 
 and whenever such has been the case, and any have become the 
 oracles at which we might knock, or to whom we might apply for 
 wisdom, we have beheld the sad reality, to use tire language of 
 Job, that "wisdom has died with them." No philosopher has 
 yet been sagacious enough, (or has yet had opportunity,) to 
 
INTRODUCTION. 
 
 25 
 
 detect and seize the ethereal spark and blow it into a flame ; to 
 lay hold of the evanescent thread in the tissue, and to unravel 
 and draw it forth, for the edification and benefit of man, 
 as in case of some of the other great laws of nature, long 
 hidden and occult, as those of electricity, galvanism, electro- 
 magnetism, &c. 
 
 Nevertheless, those prodigies always seem inclined to devel- 
 ope or disclose their numerical results, by means of legitimate 
 numbers ; and, hence, to the mass of mankind, they are, in 
 some respects, equally as beneficial and far more intelligible, than 
 many who have conceived their own craniums to be the very 
 archives of wisdom. 
 
 Since, then, mankind must plod on, not only in the acquiring 
 of knowledge from experience, but also from the stock which has 
 been accumulated as the general store, it is scarcely a mark of 
 wisdom, or philosophy, for those who have been placed as file 
 leaders, in the great march of human improvement, (or who 
 have been regarded as such,) to lead off so far in advance, that 
 their position can only be distinguished by the reflection of their 
 names from the sky ; and they hence become no guide whatever 
 to the multitude. 
 
 But as physical astronomy ought to have its foundation based 
 on truth, and as the involution and evolution of all the strange 
 curves in which modern astronomy compels the planetary 
 system to wander, are subjected, for the accuracy of their in- 
 flexure, to the curve of the circle, it is certainly desirable that 
 the circle, in order that it may be a true standard, should be 
 mathematically round. And in order to furnish such standard, 
 I shall proceed to an investigation of the quadrature of the cir- 
 cle, prior to a direct investigation of the laws of force and mo- 
 tion. 
 
 In my investigation of the quadrature, (as in all of my other 
 investigations,) I have endeavored to render the process as 
 simple and easy as possible, carefully avoiding any method 
 that may not at once be made familiar to any one having the 
 rudiments of an education. Hence I have adopted the letter p, 
 to express or denote generally the word polygon ; and this from 
 the circumstance that a figure at the right of the letter, may 
 readily denote the number of sides of the polygon, whenever it 
 may be necessary to give the number of sides. And this, to- 
 gether with a few denotations of polygonal quantities, expressed 
 by letters of the alphabet, with one or two signs definitely ex- 
 plained, is all the analytical machinery which I propose to adopt 
 or use. For I have yet to learn, that the beauty of truth depends 
 upon its robes, however flaunting or party-colored they may be. 
 4 
 
26 INTRODUCTION. 
 
 Hence, it does not require a volume of fluxions, and the cal- 
 culus, to explain the quadrature to mankind, or give them ra- 
 tional notions of the curve of the humble circle ; for all agree, 
 that those methods will not disclose the ratio between the diam- 
 eter and circumference. Archimedes, in his sagacity, sought 
 out the true method ; but, having fallen into an error in the oper- 
 ation, he missed the mark, and thence sought to detect the error 
 to the day of his death ; but, having failed, the error has yet 
 remained, and has at length been extended to a prodigious 
 length of tail. 
 
 But had Archimedes adopted unity as the diameter of the 
 circle, and geometrized upon lines in lieu of areas, he would at 
 once have discovered wherein his error lay. And hence, in 
 respect to the circle, we have a notable instance of the tenacity 
 of mankind, in adhering to and sustaining error, (however wide 
 from the truth,) if they imagine the law to be with them ; as in 
 the case, also, of Dr. Bradley's theory of the Progressive Motion 
 of Light in Time, which theory has placed the sun some 
 eight and one-fourth minutes of time out of its apparent place, 
 which theory has been elaborated by fluxions and the calcu- 
 lus, until no one is justified in believing his own eyes ; when, 
 perhaps, the error of the hypothesis may be readily shown by 
 the simplest diagram, and also that the Dr. drew conclusions 
 directly the reverse of the phenomena from which he deduced 
 them. 
 
 But an instance of this kind, not less remarkable, perhaps, 
 than any other, is that in respect to the Newtonian law of grav- 
 ity, the demonstration of which, perhaps, many suppose to be 
 a result obtained through the medium of fluxions, the calculus, 
 the problem of the three bodies, or by some mystical operation 
 of corpuscles or physical atoms upon each other, &c., when 
 even the Newtonian method, on which he labored for years, was 
 so extremely simple, that any school-boy may reexamine it, and 
 detect the error which Newton made. And such is the ease 
 with which all physical or mathematical truths may be demon- 
 strated, whilst error often requires a strange and uncommon dress 
 to hide its deformities. But 
 
 " Patron or intercessor, none appeared." 
 
 " And I looked, and there was none to help ; and I wondered 
 that there was none to uphold." 
 
UNITY OF PURPOSE, ETC. 
 
 CHAPTER I. 
 
 On the Quadrature of the Circle. 
 
 SECTION FIRST. 
 
 To find a rational proportion between the diameter and cir- 
 cumference of the circle, which might be definitely expressed in 
 some given power, or root, of numbers, has long been a desid- 
 eratum with mankind, as we learn not only from tradition and 
 the history of geometry and mathematics, but, also, from every 
 elementary work upon geometry and mathematics, in which we 
 readily discover that a very great portion of all the geometric 
 problems, propositions and demonstrations, have had in view 
 to ascertain the true ratio between the diameter and circum- 
 ference of the circle, as though it were the one thing needful in 
 science. 
 
 Nor has this longing for a paramount truth been without 
 its beneficial effects on science, as thereby rectilinear geometry 
 has been developed to a vast extent ; but its failure to develope 
 the true quadrature, has elicited expressions like the following, 
 even from modest authors, namely : " If there existed a ra- 
 tional proportion, that is, a proportion to be expressed in 
 whole numbers, of the surface of a circle to a square surface, 
 there would be, at the same time, a rational proportion between 
 the diameter and the circumference. But, from geometrical 
 reasons, no rational proportion of the diameter to the circum- 
 ference is possible ; it can be expressed only by approximation. 
 
28 ON THE QUADRATURE 
 
 However, the proportion thus obtained is quite as correct as is 
 necessary for any purpose in the applied mathematics." But, 
 however modest the above allegations are, compared with those 
 of some authors, I must dissent, not only from what the author 
 alleges as truths, but also from the consequence and effect which 
 such kind of teaching is calculated to have on mankind. 
 
 Why should the author speak of a ratio expressed in whole 
 numbers, when even the ratio between the linear measures of 
 the square cannot be so expressed, as that between the diagonal 
 and circumference, but must be expressed in the second power 
 of numbers ? And even though the ratio of the diameter of the 
 circle to its circumference cannot be definitely expressed short of 
 Ihe third power, or some power higher than the second, it 
 will be no less a definite expression of the true ratio, than 
 though it could be expressed in whole numbers in the first pow- 
 er. Nor was the author, perhaps, aware, how much more bene- 
 ficial to science the true ratio would be, though it were only 
 capable of being definitely expressed in some higher power than 
 the first or second. 
 
 Thus the expression of the author is fallacious. But when 
 the author alleges that, " from geometrical reasons, no rational 
 proportion of the diameter to the circumference is possible," I 
 think the allegation altogether too bold to aid science in its pro- 
 gress ; and it is an allegation certainly uncalled for, by those who 
 are in search of truth. It is sufficient that geometry, in the 
 hands of those who have furnished it with its reasons, has not 
 been able to disclose to mortals the true ratio between the diam- 
 eter and circumference of the circle. 
 
 But the same author, with all his wonted modesty, has gone 
 much further, and made use of allegations far more reprehensi- 
 ble, when he alleges that the ratio, obtained by a supposed ap- 
 proximation, is quite as accurate as is necessary for any purpose 
 in the applied mathematics. I have yet to learn that funda- 
 mental errors are as good as the truth, let such doctrine come 
 from what source it may ; and I predict that the popular ratio 
 will, sooner or later, be found not quite as accurate as is neces- 
 sary. But the supposition is, that geometry has not only furnished 
 reasons why no definite proportion, or ratio, can exist between 
 the diameter and circumference of the circle, but that it has been 
 able to direct mankind to an error so near the truth, that it is 
 equally as good, or beneficial, as truth itself. Thus geometry 
 itself, so full of rectitude and truth, must be bent and warped 
 from its legitimate station, and made subservient to human 
 error. 
 
OF THE CIRCLE. 29 
 
 Nevertheless, the Pythagorean theorem, or 47th problem of 
 the 1st Book of Euclid, with its infinity of mathematical ap- 
 plications, (as conclusive to the mind, as is the geometric 
 demonstration from which they are deduced,) yet remains, a 
 boon to science ; thus for example, that the diagonal and one side 
 of the square, may have a definite ratio expressed in the second 
 power of numbers. That, if any given surface be conceived to 
 be enlarged or diminished, (still retaining the same form,) the 
 area or surface will be enlarged or diminished, in a duplicate 
 ratio to that of any given line or measure of such area. That 
 when the diameter of the circle is unity, the area is equal to one- 
 fourth of the periphery ; and that the area of any circumscribing 
 polygon of such circle is equal to one-fourth of the perimeter of 
 such polygon, and that the area of the circumscribed square is 
 equal to the diameter of such circle. That, if the diagonal of a 
 square be 2, or twice unity, the area of such square is equal to 
 the diameter of its circumscribed circle. That, as the area of 
 the circumscribed square of a circle is double that of the area of 
 the inscribed square of such circle, so the area of the circum- 
 scribed circle, of a given square, is double the area of the in- 
 scribed circle of the same square, for the proportions of their 
 respective diameters are the same ; and that one-fourth of the 
 perimeter of any regular polygon, multiplied by the diameter of 
 the inscribed circle of such polygon, gives the area of such poly- 
 gon ; and that the same holds in respect to the circle. 
 
 These, together with the infinity of corollaries and conse- 
 quents, which flow from legitimate geometric demonstrations, 
 are, doubtless, sufficient to enable mankind to ascertain the true 
 quadrature, by the aid of right reason. 
 
 Thus, a very great part of our labor, on this portion of the sub- 
 ject, must consist in bringing together and comparing demonstrat- 
 ed principles, and mathematically arranging them, in such man- 
 ner as to enable us to arrive at the required result. This, perhaps, 
 may be done, if we take proper heed to those very appropriate 
 remarks of M. Legendre, in which he alleges, that " by deducing 
 consequences from one or more propositions, we may be led 
 back to some proposition already proved ; and that an indubita- 
 ble proof of their certainty is, that however we combine them 
 together, provided only our reasoning be correct, the results we 
 obtain are always perfectly accurate." Such language differs 
 widely, however, from that actual abomination to science, to 
 which the same author stoops, when he alleges, that iu the prob- 
 lem of the quadrature is now degraded to one of those idle 
 questions, about which no one possessing the least tincture of 
 
30 ON THE QUADRATURE 
 
 geometrical science, will spend any portion of his time ; that 
 although the true ratio of the diameter to the circumference has 
 not been obtained, yet, that no advantage whatever would be 
 derived from the true ratio, over the approximate ratio, &c. ; " 
 thus boldly alleging that error is equally beneficial with the truth. 
 And I will here ask of science, (and perhaps the disciples of Co- 
 pernicus can answer,) if any known distance from the truth 
 can well be ascertained, if we know not where the truth lies ? 
 
 The idea or conception of invoking the aid of geometry, for 
 the purposes of a true induction into a polygonal progression, 
 whereby the ratio between the diameter and circumference of 
 the circle might be obtained, is of high antiquity, as such was 
 the method adopted by Archimedes. And it is generally ac- 
 knowledged, that no essential improvement has yet been made 
 either in his process or result, otherwise than by means of certain 
 devices for facilitating what is supposed to be a true approxima- 
 tion towards the truth; and it seems to have been taken for 
 granted, that Archimedes was satisfied with his method, and the 
 results arising therefrom. But, in my view, it is wholly unrea- 
 sonable to suppose that Archimedes, after having written a treat- 
 ise upon the quadrature, if he had been satisfied with the prin- 
 ciples upon which he had proceeded, or with the results arrived at, 
 would, at the age of seventy-five years, have been poking in the 
 sand, with the view to catch a vision of the true principles of 
 the quadrature. He, doubtless, perceived that his method of 
 deducing a progression from his geometric formula, would ne- 
 cessarily lead to discrepancy and error, and that there existed, 
 somewhere in his device, an assumption based upon false and 
 erroneous premises. But we do not learn that he attempted to 
 degrade the question of the quadrature, or to call those fools who 
 might attempt to improve upon what he had done, (as if wisdom 
 was to die with him,) as has been the case with some others ; 
 nor do we find that he was bold enough to deny the existence of 
 what he did not comprehend, or to allege, as a teacher of science, 
 that error is equally beneficial with truth. 
 
 I shall not pretend or allege, that the popular error in respect 
 to the quadrature, arises from any defect in rectilinear geometry, 
 as that is perfect in its legitimate sphere ; nevertheless, all ex- 
 perience has shown that the equation of curves, as well as of 
 the laws of force and motion operating in planetary orbits, is 
 properly transferred to the transcendentalism of logarithms : Nor 
 has the necessity of a logarithmic equation been overlooked in 
 attempts to develop the true quadrature ; and, indeed, the most 
 approved method of extending what is conceived to be an ap- 
 
OF THE CIRCLE. 31 
 
 proximation, is based upon a logarithmic series. But I am con- 
 strained to allege, that in lieu of adapting a logarithmic series 
 to a true equation of the circle, they have only adapted it to 
 those errors incident to their supposed approximation, and which, 
 of course, only serves to confirm the error. For they have wholly 
 disregarded the varying capacities of polygonal measures, other- 
 wise than that arising from the presumption that the perimetrical 
 capacities were constantly increasing from the trigon or square, 
 to the periphery of the circle. And consequently they have given 
 the periphery of the circle a greater capacity than it actually 
 possesses ; for, if there be other extraordinary capacities to be 
 consulted or considered, than those of the perimeters only, in a 
 polygonal progression from the trigon (or from the square) to 
 the circle, it must be manifest, that the final capacity of the peri- 
 meter of the circle, must depend upon a proper equation of the 
 capacities of polygonal measures, throughout the whole progres- 
 sion. If, however, we attempt an approximation by means of a 
 logarithmic series, it is certainly of importance that such series 
 should be adopted as will lead to righ: conclusions ; nor should 
 some solitary coincidence in the application of numbers, (sup- 
 posed to have some analogy,) be sufficient to establish the cor- 
 rectness of such logarithmic series ; as may, perhaps, have been 
 the case in reference to a series carried on by an alternate divis- 
 ion by the odd numbers, upon the supposition that they are 
 equally applicable to the deflection of a revolving body from a 
 tangent to its orbit, as to the rectilinear fall of bodies near the 
 earth ; and, consequently, that they had a like application to the 
 curve of the circle ; and thus, trusting to fallacious appearances 
 or unfounded analogies, the truth has been permitted to suffer. For, 
 notwithstanding the law of the rectilinear fall of bodies near the 
 earth has so beautiful and simple a numerical expression, by 
 which law the sum of the spaces passed over in as many equal 
 moments from the commencement of the fall, is the square of the 
 whole number of moments, (or times ;) and also, by which the 
 square of the sum of all the moments (or times) will be the 
 sum of all the odd numbers in their order, up to the number of 
 moments, (or times,) yet, I am quite apprehensive that those 
 beautiful coincidences have been intruded into those investiga- 
 tions with which they have no particular concern, and have in- 
 duced conclusions highly detrimental to science. It is not my 
 intention, however, to reject legitimate evidence, in whatever 
 form it may be offered ; whether it be direct and positive, or 
 only presumptive and corroborative, as by the reductio ad 
 absoirdum ; or such as shall arise from the discrepancies and 
 
32 ON THE QUADRATURE 
 
 errors of the popular methods ; and I shall even attempt to in- 
 duce assent to the truth of my conclusion, from the beauty and 
 harmony of its analogies and applications, and its coincidences 
 with all the great laws of nature, which are the subjects of nu- 
 merical or mathematical calculation. For, to determine the true 
 measures of the circle, and to know from indubitable proof that 
 such determination is right, is, perhaps, all that can rationally be 
 required ; nor are we to be debarred from any legitimate evi- 
 dence that may arise to sustain us in our knowledge of the 
 truth. 
 
 My purpose is, to prove to the satisfaction of the world, that 
 the circumference of the circle, whose diameter is unity or 1, is 
 3.1748020, or the third or cube root of 32, and hence that the 
 area, or one fourth of the circumference is .7937005, or the cube 
 root of .5, in lieu of the popular series, .78539816339, &c., 
 &c. ; thus making the cube or third power of the diameter of 
 the circle, twice the third power of one fourth of the circumfer- 
 ence ; and, consequently, that the ratio of the diameter to the 
 circumference is commensurable in the third powers and roots 
 of definite numbers. 
 
 It is agreed that the surfaces of solids are as their bulks : 
 That the product of one sixth of the surface of any regular solid 
 by the diameter of its inscribed sphere, gives the bulk or con- 
 tents: That the bulks of solids increase or decrease in a tripli- 
 cate ratio, to that of any given linear measure of the solid : 
 That the surface of a solid increases or decreases by a duplicate 
 ratio to that of its linear measure : And also, that the ratio of 
 the increase of surface to that of bulk, is as that of the diame- 
 ter of the orbits to the periods of the planets. 
 
 It has been well demonstrated, that the bulk of the sphere is two 
 thirds that of its circumscribing cylinder ; hence, by my determi- 
 nation, the bulk of the prime sphere, or sphere whose diameter 
 is unity, is the third root of the fraction .148148, ad infini- 
 tum ; and the third power of its surface is 32. These pro- 
 portions between the sphere, cylinder and cube certainly pre- 
 sent many beautiful coincidences and reciprocities, not only 
 in the economy of numbers, but in the perfect adaptation and 
 fitness of things in respect to surfaces and solids, or their ra- 
 tional admeasurement, which was anciently thought so desir- 
 able for the furtherance of science, as to be earnestly attended to. 
 Nevertheless, the beauty, harmony and reciprocity in the ap- 
 plication, is but presumptive evidence, and is not to be consid- 
 ered sufficient, for satisfying. the abstract geometer or mathema- 
 tician, which is to be done. But as it is my intention to be tried 
 
OF THE CIRCLE. 33 
 
 by my peers, I shall address myself to them, in support of my 
 cause, in language as intelligible to all as I am capable of using, 
 or as circumstances will admit, scrupulously avoiding those meth- 
 ods of investigation which, to a great portion of mankind, are 
 dark and mysterious, and absolutely repulsive and terrific, like 
 the methods by fluxions and the calculus. For it is my in- 
 tention to test the question whether the great mass of mankind 
 are capable of judging for themselves in respect to physical 
 science, or whether they are forever to depend upon the few pro- 
 foundly learned, for an implicit faith instead of actual knowl- 
 edge. And this being my purpose, perhaps none will expect 
 me to give heed to those methods of investigation which have 
 been so little understood, save by the profoundly learned ; and 
 which, although carried to such an awful height as to be beyond 
 the ken of mortals in general, have never, as yet, perhaps, dis- 
 closed or developed a single physical truth worthy of science. 
 The method which I have adopted for comparing polygonal 
 quantities, with a view to some unity of purpose, varies some- 
 what, in many respects, from the popular methods ; but nev- 
 ertheless, retains a sufficient similarity to enable us to make 
 all proper comparisons between the methods and results ar- 
 rived at. And as it is my intention to avail myself, in behalf 
 of my views, of the discrepancies and errors incident to the 
 popular methods and results ; and to show that they are 
 wholly inadequate to equate the varying capacities of polygonal 
 measures, it will be my endeavor, so far as I may refer to the 
 popular methods, and the results arising therefrom, with a view 
 to a just comparison, faithfully to translate the same as nearly 
 to some unity of purpose, as the idiom or nature of the case will 
 admit; and to give full credit to the authors, in the use of the 
 general appellation of popular methods or popular determination. 
 It is often alleged by authors, (and perhaps with too much 
 emphasis or technicality,) that the circle, or any given polygon, 
 is properly the area or surface; but that, in the common ac- 
 ceptation, the periphery, or perimeter alone is called the circle, 
 or polygon ; and I must confess, that, for most purposes, I am 
 inclined to the common acceptation ; for the reason that it is the 
 linear quantities which must determine the numeral amount of 
 area or surface, and not the converse. The ratio between a line 
 which forms a perfect circle, and its greatest rectilinear extension, 
 (called the diameter,) is what has been so long and so anxiously 
 sought; and if such ratio be obtained, we shall have no difficulty 
 in assigning the ratio of the area to that of the square, whose area 
 is called unity ; whether such area of the square be merely con- 
 ventional, as being most convenient, as a standard, or whether 
 5 
 
34 ON THE QUADRATURE 
 
 it is founded in the fitness of things ; and which, from necessity, 
 has induced its adoption as the standard. And it certainly is 
 more proper to speak of the varying capacities of linear meas- 
 ures for bounding area or surface, than to speak of the capacities 
 of the areas or surfaces themselves. 
 
 We very properly speak of the circle, the square, the octagon, 
 &c., without regard to their numerical dimensions; and such 
 should always be the technical consideration of all regular 
 polygons ; namely, that a regular polygon, having any given 
 number of sides, is properly one and the same polygon, what- 
 ever may be its numerical dimensions. But as a polygon must 
 often be compared with the same polygon, possessing different 
 numerical dimensions, it will often be the most convenient 
 mode of comparison, to consider the same polygons, when ex- 
 pressed by different numerical dimensions, as being like poly- 
 gons ; that is, one is like the other. Thus, in the case of the prime 
 circle, or circle whose diameter is 1, the same polygon (of what- 
 ever number of sides) may be inscribed within, or circumscribed 
 about the circle ; and the inscribed polygon may be said to be 
 like, or the same polygon with, the one which circumscribes 
 the circle. And the importance of considering the identity or 
 sameness of a regular polygon to depend upon the number of 
 its sides, and not at all upon its numerical dimensions, in 
 our investigations of the quadrature, is, perhaps, much greater 
 than might generally be supposed ; and hence I shall en- 
 deavor to enforce the necessity of such consideration in those 
 places where a different consideration must necessarily lead to 
 discrepancy and error. Thus, calling the circle whose diame- 
 ter is unity, the prime circle, any circumscribing polygon with its 
 like inscribed polygon, forms a prime polygon, and is properly 
 to be considered as one and the same polygon, possessing differ- 
 ent numerical dimensions ; the circumscribed, being called the 
 major polygon, and the inscribed being called the minor poly- 
 gon. And in what numerical point or position the perimeters 
 of a major and minor polygon shall actually coincide, and there- 
 by form the periphery of the prime circle, is the question to be 
 determined. That such periphery exists, is a self-evident pro- 
 position ; nor is it in any way material, whether we conceive it 
 to be composed of the perfect coincidence of a major and minor 
 perimeter, or whether we conceive it to lie between the utmost 
 major and minor perimeters ; it is sufficient for us that it is but 
 a boundary, the thickness of which we can no more comprehend 
 than that of the infinite divisibility of matter. Nevertheless, as 
 we have all along conceived the major and minor perimeters to 
 be progressing towards the periphery of the circle, perhaps our 
 
OF THE CIRCLE. 35 
 
 utmost conception must be, that the periphery of the prime circle 
 lies between the perimeters of the utmost major and minor poly- 
 gons. And hence the utmost of our researches in respect to the 
 quadrature, can extend no further than that of finding the utmost 
 means, (either equal or proportional,) between two approxima- 
 ting quantities or numeral positions. This, however, is quite 
 sufficient ; for geometric points and lines, having no thickness, 
 the utmost mean between two approximating points or lines is 
 but the actual coincidence of those points or lines ; and all must 
 necessarily lie in, and be expressed by the same root or power. 
 It may, nevertheless, in the course of the work, often be more in 
 accordance with our habits of thinking, to conceive certain coin- 
 cidences to operate as a merger or extinguishment of a part of 
 the coinciding quantities, as in the case of diametric quantities, 
 for instance ; for whilst each polygon of sides is conceived to 
 have two diametric quantities, (namely, the diameters of its in- 
 scribed and circumscribed circles,) the circle is generally con- 
 ceived to have but one diametric quantity. 
 
 Nevertheless, it may often be necessary, by way of compari- 
 son and explanation, to consider the circle in the nature of a 
 polygon of sides, and possessing like quantities as that of an 
 inscribed and circumscribed diameter ; and in case the prime 
 circle is to be considered as the utmost polygon, partaking, as 
 well of the properties of the minor as of the major polygons, it 
 may also, for the purposes of comparison, be conceived to have 
 a major and minor perimeter, and a major and minor area ; in 
 which case, the major area will be numerically equal to one 
 fourth of the major perimeter, and one fourth of the minor perim- 
 eter will be conceived to be equal to a minor area of twice the 
 number of sides ; for such is numerically the utmost of our con- 
 ception in respect to a minor perimeter and its accompanying 
 area. And hence, so long as we retain the conception of a 
 minor perimeter, so long we retain the conception of a minor 
 area of twice the sides. 
 
 Now to obviate the difficulty in obtaining a satisfactory result 
 to the mind, arising from such inevitable conceptions, as well as 
 to determine the numerical points or position in which the major 
 and minor perimeter of a prime polygon actually coincide and 
 form the periphery of the prime circle, it will be necessary, lest 
 we mistake the ratio of progression of the minor perimeter to 
 that of the major, that we so prepare our work as to enable us 
 measurably to comprehend the end from the beginning, by the 
 adoption of such a course as will furnish a train of such absolute 
 and conclusive coincidences of the ratios of progression of 
 polygonal quantities, as inevitably direct to, and conclusively 
 
36 ON THE QUADRATURE 
 
 determine the numerical point or position in which the major 
 and minor perimeters of the utmost prime polygon must actually 
 coincide. And as the numeral value of the minor perimeter 
 of any prime polygon, is, to the major perimeter, as the diameter 
 of the inscribed circle of a like polygon is to the diameter of the 
 circumscribed circle of such polygon, I have sought to obtain 
 such trains of absolute coincidences, (and consequently the coin- 
 cidence of the major and minor perimeter of the utmost prime 
 polygon,) by a comparison of the class of prime polygons with 
 that class of polygons whose respective areas are 0.5, or half of 
 unity, which class of polygons, in contradistinction from the 
 prime polygons, I have thought proper to denominate primal poly- 
 gons; and the four denotations by means of letters of the alpha- 
 bet, by which such comparison is mostly made, I shall call prime 
 denotations ; and inasmuch as a full comparison must necessarily 
 extend to polygons of different numerical dimensions, from those 
 of the prime and primal, other denotations may be appended, 
 denominated auxiliary denotations ; all of which will be suffi- 
 ciently explained for use, in the appropriate place. 
 
 Perhaps it will not be inappropriate here, to give some intro- 
 ductory hints at what will appear far more conspicuous in the 
 work, namely : the admirable adaptation of the prime and pri- 
 mal polygons to an investigation of the quadrature ; not only 
 from the great number of actual coincidences so readily obtain- 
 ed in the ratios of progression of polygonal quantities, which so 
 satisfactorily dictate the numeral position of the final coinciden- 
 ces sought ; but from the manner in which those progressive 
 ratios unfold to us the varied and revolving capacities of poly- 
 gonal measures, while performing the quadrant of an orbit, (or, 
 if we please, the two octants of the quadrant,) thereby constantly 
 suggesting both the necessity and the manner of equating those 
 revolving capacities. 
 
 Thus the diameter of the prime circle is the inscribed diame- 
 ter of all major polygons, and the circumscribed diameter of all 
 minor polygons ; making unity an important point in this par- 
 ticular, but mainly from the fact that the prime square possesses, 
 in and of itself, important coincidences situated in unity, thereby 
 making the square the proper polygonal standard. Thus the 
 diameter of the inscribed circle of the square, and one fourth of 
 the perimeter of the same square, are always numerally equal ; 
 and when situated in unity, each is necessarily equal to the area; 
 each of which is then numerically equal to the diameter of the 
 circumscribing circle of the primal square, or square whose area 
 is half of unity, which circumstance becomes important in our 
 investigations ; while one fourth of the perimeter of the minor 
 
OF THE CIRCLE. 37 
 
 square is equal to the diameter of the inscribed circle of the pri- 
 mal square, and is also equal to the area of the minor octagon ; 
 thus making the square (namely, the polygon denoting unity 
 amongst the polygons) emphatically the standard from which 
 ratio and proportion are properly conceived to emanate, and with 
 which they must of necessity be compared. But in all prime po- 
 lygons below the square, (namely, of more sides than the square,) 
 which we are able specifically to investigate, one fourth of the pe- 
 rimeter is less than the diameter of the circumscribing circle of a 
 like primal polygon ; and by the popular methods this variation 
 is increased in the whole distance from the square to the circle ; 
 thereby making the capacity of the perimeter of the circle too 
 great, and that of the diameter too small. 
 
 Now this variance between one fourth of the perimeter of a 
 major polygon and the diameter of the circumscribing circle of 
 a like primal polygon, which is proportionally the same between 
 one fourth of the perimeter of a minor polygon and the diameter 
 of the inscribed circle of a like primal polygon, (which variance 
 is nothing in the square and circle, and is at its maximum in the 
 octagon,) is always proportionally equal to twice its equation ; 
 and to denote such equation, a sign or symbol will be adopted 
 and explained. Again, a comparison of the prime and primal 
 polygons is well calculated to disclose certain facts and phenom- 
 ena in polygonal economy, which the popular methods seem not 
 to have heeded or to have made any account of, and among 
 others, the transposition of polygonal quantities in unity, or in 
 the prime square, while passing from the trigon to polygons sit- 
 uated below the square, which phenomenon certainly deserves 
 consideration, and will receive it, by special reference, in due time 
 and in the proper place. 
 
 And it is possible that a notice and consideration of this 
 transposition of polygonal quantities in unity, or the prime 
 squ archly" have suggested some defect in the popular mode 
 of approximating by means of a logarithmic series; that is, the 
 method of obtaining constant dividends and divisors, and a 
 disregard of such transposition, may have afforded some of those 
 geometrical reasons which are said to exist, why " no ra- 
 tional proportion of the diameter to the circumference of the 
 circle is possible." This transposition of polygonal quantities 
 in the major square, is necessarily such, that, for the obtaining 
 of one fourth of the major octagon, the dividend and divisor may 
 always be obtained from the same nominal terms of the square, 
 as from the trigon, in deducing one fourth of the perimeter of the 
 major hexagon ; out as in any polygon below the square, the 
 relative situations of the terms are transposed from what they are 
 
38 4 ON THE QUADRATURE 
 
 in the trigon, it may, at least, be questioned, whether there 
 should not be some transposition of terms in polygons be- 
 low the square, from which the requisite dividends and divi- 
 sors should be obtained. Nevertheless, such transposition has 
 not been heeded in the popular methods. It will, however, be 
 quite manifest, that dividends and divisors obtained from any 
 given polygon below the square, for deducing one fourth of the 
 circumference of a major polygon of twice the sides, may be 
 obtained of like nominal quantities of the square for deducing 
 one fourth of the perimeter of the major octagon ; but not from 
 the trigon, in deducing one fourth of the perimeter of the major 
 hexagon. The consequence then becomes inevitable, that if we 
 proceed to obtain our constant dividends and divisors from no- 
 minal terms, the same as or like those we use in obtaining one 
 fourth of the perimeter of the major hexagon and octagon, the 
 divisor becomes too large for the dividend, and consequently 
 the quotient will be too small ; and the error will continue to 
 increase, by infinitesimal quantities, to the final determination, 
 producing some strange discrepancies near the final determina- 
 tion ; as that of forming premature coincidences and -again 
 breaking up and dispersing them ; while other quantities, equally 
 entitled to coincidence, are not permitted at all to coincide ; 
 and in the final result, those quantities which, from their ratios 
 of progression, seemed all along destined to form one general 
 coincidence in the circle, are scattered into various and diverse 
 coincidences ; and not a single term or polygonal quantity is 
 permitted to remain in that numeral point or position which 
 seemed destined, from every law of polygonal progression, to be 
 the point of coincidence for all the progressive quantities. 
 Thus one fourth of the periphery of the prime circle will be 
 .7853981, &c., and the diameter of the circle whose area is 0.5, 
 will necessarily be expressed by .797892, &c., and the three 
 points of equation which should have directed to the true place, 
 are caused to coincide in .791620 ; namely : .002080 below the 
 true place. 
 
 Because the product of one fourth of the perimeter of any 
 given polygon X the diameter of the inscribed circle of such 
 polygon gives the area of such polygon, consequently, for the 
 same cause, the area and one fourth of the perimeter of any 
 given polygon, the diameter of whose inscribed circle is unity, 
 are numerically equal ; and hence it is equally well, and vastly 
 more convenient in our investigations, to adopt one fourth of the 
 perimeter as the standard circumferential measure, in lieu of the 
 entire perimeter. 
 
 I shall plead no excuse for the desultory manner in which I 
 
OF THE CIRCLE. 39 
 
 shall treat the subject of the quadrature ; it being my design 
 solely to furnish the evidence, both in quantity and quality, to 
 satisfy the candid inquirer of the correctness of my solution of 
 this important problem. Hence, after setting forth an epitome 
 of the prime table, as a table of reference, and giving some ex- 
 planation of the few signs, characters, and denotations used in 
 the work, I shall proceed to present my evidence in such a way 
 and manner as I may conceive will elicit the most thorough and 
 thoughtful investigation from the inquirer. And conceiving the 
 whole subject to be wrapt up in consequents or corollaries, flow- 
 ing directly from well demonstrated principles which need not to 
 be repeated, the bare suggestion of which, by way of allegation, 
 being, in most cases, a sufficient reference to the demonstrated 
 principles from which they emanate or flow, at least by the aid of 
 a little reflection, and that correct reasoning of which Legendre 
 speaks, I shall not, in general, do more than to allege such 
 demonstrated truths, and such consequences as they inevitably 
 dictate. And in this respect, I may sometimes become quite 
 prolific in trains of such correlative matter as I may conceive 
 will throw light upon the subject ; for it is my intention to fur- 
 nish indubitable proof of the correctness of my determination, 
 by showing " that however we combine our evidence, provided 
 only our reasoning be correct, the results we obtain will always 
 be perfectly accurate." The limits, however, assigned to the 
 work, will admit but a very small portion of the evidence 
 which I have examined in favor of my determination ; nor am 
 I confident that the evidence which I shall offer, will be a judi- 
 cious selection from that which I have examined. 
 
 I am aware of the ordeal which the popular determination 
 of the quadrature of the circle has passed ; and of the fortified 
 position in which it now rests, as in the citadel of the world's 
 last hope ; and that, were it not for the eternity and immutability 
 of the principles of ratio and proportion, the popular error 
 must, long ere this, have been converted into a mathematical 
 truth. And it having been thoroughly tested by fluxions and the 
 calculus, (those methods which appear to me far better calculat- 
 ed for the equation of human errors, than for disclosing mathe- 
 matical truths,) the learned have in general, become satisfied 
 that the force of nature can no further go ; and hence, the ques- 
 tion is sure to be put to the modest inquirer, in respect to a true 
 solution of the quadrature, " whether he has examined the sub- 
 ject by means of the fluxions and the calculus;" and still we 
 find it alleged in Rees' Cyclopaedia that the hopes of mathema- 
 ticians, which were revived in respect to a true solution of the 
 quadrature, upon the invention of fluxions and the calculus, 
 
40 ON THE QUADRATURE 
 
 were, at length, abandoned in despair, after exhausting every 
 possible device by means of those methods, but without success; 
 and hence a true solution is now declared to be impracticable ; 
 and geometry is charged with having furnished reasons which 
 justify such declarations; but I have certainly understood legiti- 
 mate geometry quite differently. But the problem of the quadra- 
 ture ought not to be settled by convention, although I am aware 
 that important questions in respect to science have too often 
 been settled in that way ; as in case of the theory of universal 
 gravity ; and in case of the laws by which gravity operates in 
 respect to distance. 
 
 If, however, the popular determination of the quadrature, and 
 also the Newtonian theory and law of gravity, are forever to 
 remain as the world's best hope in respect to a sure foundation 
 upon which to base their scientific researches, then, to be sure, 
 astronomers must have an onerous task to perform, in which 
 one equation only serves to beget other supposed variances and 
 aberrations, and so on, ad infinitum ; in which case, the toils 
 and labors of astronomers will become perplexing, and, like the 
 labors of Sysyphus, never-ending. 
 
 As a matter of reference, I have thought proper to close this 
 section with an epitome of two tables ; the first, or prime table, 
 being formed by constantly multiplying unity, and each suc- 
 cessive product obtained, by the sixth root of .2 in ascending, or 
 by the sixth root of .5 in descending. Hence, if the natural 
 numbers are made exponents to the terms, both in ascending 
 and in descending, any two terms denoted by the same number, 
 will be reciprocals of each other ; the term denoted by 2., in the 
 descending column, would be the same which I have so con- 
 stantly denoted by , in the course of the work, and is, accord- 
 ing to my idea, the area of the circle whose diameter is unity. 
 And in accordance with such conclusion, if the diameter of the 
 sphere be taken in any term of the table, or if the diameter of 
 the circle be taken in any term of the table, the third power or 
 cube of the surface of such sphere, or the third power or cube of 
 the circumference of such sphere, or of such circle, will be ex- 
 pressed in terms of the table, if the terms are sufficiently ex- 
 tended, as they may be at pleasure. And the third power or 
 cube of the surface of the circumscribing cylinder of such 
 sphere, may be found expressed in terms of the second table, in 
 case the terms be sufficiently extended. 
 
OF THE CIRCLE. 
 
 41 
 
 FKlMh 
 1. 
 
 TAUt-JK. 
 1. 
 
 J1,UU1N. 
 
 .16666 
 
 U lAJDJUJBj. 
 
 6. 
 
 .890898 
 
 1.122462 
 
 .209986 
 
 4.762203 
 
 .793700 
 
 1.259921 
 
 .264566 
 
 3.779763 
 
 .707106 
 
 1.414213 
 
 .3333 
 
 3. 
 
 .629960 
 
 1.587401 
 
 .419973 
 
 2.381101 
 
 .561231 
 
 1.781786 
 
 .529133 
 
 1.889881 
 
 .5 
 
 2. 
 
 .666 
 
 1.5 
 
 .445449 
 
 2.244924 
 
 .839946 
 
 1.190550 
 
 .396850 
 
 2.519840 
 
 1.058265 
 
 .944940 
 
 .353553 
 
 2.828426 
 
 1.333 
 
 .75 
 
 .314980 
 
 3.174802 
 
 1.679888 
 
 .595274 
 
 .280615 
 
 3.563572 
 
 2.116531 
 
 .472467 
 
 .25 
 
 4. 
 
 2.666 
 
 .375 
 
 .222724 
 
 4.489848 
 
 3.359777 
 
 .297637 
 
 .198425 
 
 5.039684 
 
 4.233062 
 
 .236233 
 
 .176776 
 
 5.656852 
 
 5.333 
 
 .1875 
 
 .157490 
 
 6.349602 
 
 
 
 .140307 
 
 7.127144 
 
 
 
 .125 
 
 8. 
 
 
 
 .111361 
 
 8.979696 
 
 
 
 .099212 
 
 10.079360 
 
 
 
 .088388 
 
 11.313704 
 
 
 
 .078764 
 
 12.699204 
 
 
 
 .070153 
 
 14.254288 
 
 
 
 .0625 
 
 16. 
 
 
 
 .055680 
 
 17.959392 
 
 
 
 .049606 
 
 20.158720 
 
 
 
 .044194 
 
 22.627408 
 
 
 
 .039373. 
 
 25.398416 
 
 
 
 .035076 
 
 28.508576 
 
 
 
 .03125 
 
 32. 
 
 
 
 It will be seen that in the second table, the increase and de- 
 crease from 6, (or surface of the prime cube,) and its reciprocal, 
 is by a duplicate ratio to that in the prime table. If the diame- 
 ter and surface of a sphere be found in the prime table, its bulk 
 will be found in the second table, in case the tables be ex- 
 tended so as to give the required terms. 
 
 So if the diameter and bulk of a cylinder be contained in the 
 prime table, its surface will be contained in a term of the second 
 table, &c. 
 
 The above tables are only intended as examples, as they may 
 be extended at pleasure ; similar tables may also be formed, 
 applicable to the measures of the circle. 
 6 
 
42 ON THE QUADRATURE 
 
 SECTION SECOND. 
 
 Eternal wisdom has so contrived and adapted the economy 
 of numbers to the human capacity and the fitness of things, in all 
 those physical operations which are subject to ratio and propor- 
 tion, or to a numerical value, that it would seem wholly unne- 
 cessary for us to be groping in darkness in respect to those 
 things, if we but take heed to that unity of purpose in which they 
 appear to have been contrived. 
 
 But as my purpose now is to treat of polygonal economy, as 
 applicable to the economy of numbers, I will at once suggest 
 a fitness of things which has placed all proper investigation, 
 in respect to polygonal progression and the quadrature of the 
 circle, within very narrow limits, and which is, therefore, not 
 incomprehensible to any inquirer. 
 
 It is well agreed that the square, the diameter of whose in- 
 scribed circle is assumed at unity, properly denotes unity, 
 amongst the regular polygons, (and it is only of regular poly- 
 gons which I treat,) and that the right angle is properly termed 
 the angle of unity amongst angles ; and as all ratio and propor- 
 tion emanates from, and has a direct reference to or connection 
 with unity, hence the square, which denotes unity in so many 
 respects, namely, the diameter of whose inscribed circle is 1., 
 whose area is 1., and whose side is equal to the diameter of its 
 inscribed circle, and in which the diameter of its inscribed circle 
 is equal to the diameter of the circumscribed circle of the square, 
 whose area is .5, or half of unity, is admirably calculated for 
 a standard, by which to compare the quantities of tne inscribed 
 and circumscribed polygons of the circle, whose diameter is as- 
 sumed at unity. 
 
 Hence, in my direct investigation of the quadrature, by the 
 aid of symbols or denotations, I shall confine the subject to two 
 classes of polygons, called prime and primal ; the prime class 
 consisting of the inscribed and circumscribed polygons of the 
 circle, whose diameter is 1. the inscribed being called minor 
 polygons, and the circumscribed being called major polygons ; 
 the other class being that whose respective areas are .5, or 
 half of unity, which polygons of the last class I shall call primal 
 polygons. 
 
 In the comparison of the proper quantities of these- two 
 classes, the proper quantities of a primal polygon is its inscribed 
 
OF THE CIRCLE. 43 
 
 and circumscribed diameters ; namely, the diameter of its cir- 
 cumscribed circle, which will be denoted by B ; and the diame- 
 ter of its inscribed circle, which will be denoted by D. 
 
 The proper quantity of a minor polygon is one fourth of its 
 perimeter, which will be denoted by C ; and the proper quantity 
 of a major polygon is also one fourth of its perimeter, which 
 will be denoted by M. 
 
 These four letters are the prime denotations for investigating 
 the quadrature ; and should other letters or denotations be add- 
 ed, for the purpose of exemplification, they are to be esteemed 
 as only secondary or auxiliary. 
 
 It will be perceived, that in the prime denotations for investi- 
 gating the quadrature, I have scrupulously avoided any denota- 
 tion for area ; and this I have done in accordance with what 
 I have before stated, namely, that linear quantities are the 
 only quantities with which we have anything to do, in the inves- 
 tigation of the quadrature. The only question to be solved, 
 the only mystery in the case, is that of the ratio between di- 
 ameter and circumference, wholly regardless of area, otherwise 
 than as a consequent, depending on the ratio between the lines 
 which measure or bound area. Hence, if we obtain the true ratio 
 between the diameter and circumference, there will be no question 
 in respect to the true assignment of area. 
 
 The circle, then, or any regular polygon, is properly its diam- 
 etric and circumferential quantities, and not its area. 
 
 Again : that lines are the proper measures of surfaces or areas, 
 and not areas of lines, is too manifest to require comment, not- 
 withstanding the popular attempts to obtain the quadrature seem 
 rather adverse to my notions ; nevertheless, the ill success at- 
 tending that method, has not proved the legitimacy of the 
 method. Legitimate geometry does not geometrize upon areas, 
 but upon lines. 
 
 Geometry, however, seeks for three or more lines, from which 
 to deduce ratio and proportion ; but it never seeks to proportion 
 lines by means of areas, by a disregard of those lines. 
 
 Thus the business of legitimate geometry is, to proportion 
 lines to each other ; and in this it does not at all regard the 
 varied and varying capacities of polygonal perimeters for cir- 
 cumscribing area. Nevertheless, whatever may be the varied 
 capacities of polygonal perimeters for circumscribing area, in 
 proportion to their respective diametric quantities, or however 
 different that of the octagon, or of the polygon of sixteen sides, 
 may be from that of the square, we shall find that what may 
 properly be termed the capacity of the perimeter, in proportion 
 
44 ON THE QUADRATURE 
 
 to the diametric quantity, is the same in the circle as in the 
 square. 
 
 Now it has not wholly escaped the ken of mathematicians 
 and geometers, that the capacity of the perimeter of the octagon, 
 for circumscribing area, is greater, in proportion to its two diam- 
 etric quantities, than that of any other polygon ; nevertheless, 
 no advantage or benefit seems to have been taken of the fact, 
 although it is of such vast importance in a consideration of the 
 quadrature ; for, while attempting a geometrical induction into a 
 true polygonal progression, the method has been to geometrize 
 upon areas in lieu of lines, thus losing all benefits to be derived 
 from the varying capacities of polygonal measures. 
 
 But areas need not be even named or known, in a polygonal 
 progression, except in the primal class, where they silently keep 
 station in .5. 
 
 Such, then, being the case in respect to the capacities of poly- 
 gonal measures for bounding area, geometry will readily dictate 
 a polygonal progression from the major octagon to the prime 
 circle, on true principles of ratio and proportion, and in a way 
 and manner that cannot well be misunderstood ; nor need we 
 resort to other diagrams than such as are to be found in Legen- 
 dre, and other works on geometry. It may be well here, how- 
 ever, to remark, that the areas of polygons are, properly speak- 
 ing, proportioned to the diameters of their circumscribed cir- 
 cles. 
 
 I will now proceed to set forth the signs, characters and de- 
 notations to be used or employed, as a kind of analytical machin- 
 ery, in deducing the quadrature, with the requisite explanations 
 for reading the same, those denoting multiplication, division, 
 equality and proportion being used in the common, or popular 
 manner. 
 
 Numerical, or numeral quantity, may properly be conceived 
 to be that which is contained between and some given nume- 
 ral point) (or position,) as that of .5, or the square of .5, for ex- 
 ample ; or, numeral quantity may be contained between two 
 points, (or positions,) expressed or understood ; as that con- 
 tained between .5 and 1. for example ; and such will be the 
 technical meaning given to the word point, or points, when re- 
 ferring to boundaries of numerical quantities. Thus, we may 
 speak of a root or power of a given point, or that a given point 
 is a mean between two other given points, &c. 
 
 The word mean, when used to denote a point between two 
 other points, when not represented to be an equal mean, is always 
 to be understood as denoting proportional quantity ; thus 2. is a 
 
OF THE CIRCLE. 45 
 
 mean between 4. and 1. ; so 2. is the farther of two means from 
 8. to 1., or the farther of three means from 16. to 1., &c. 
 
 ^ denotes the first term of the prime table below unity ; %, the 
 second term, and so on. 
 
 ^ This sign, with a figure under it, is used to denote any 
 given number of terms of the prime table, at which one given 
 point may be situated either above or below another given point ; 
 thus, for example, 2. is ^ above 1., &c. 
 
 p denotes, generally, a polygon, and may be read polygon. 
 
 b denotes, generally, the diameter of the circumscribed circle 
 of any given polygon, except when the area is .5 ; in which case 
 it will be denoted by B. 
 
 d denotes, generally, the diameter of the inscribed circle of 
 any given polygon, except when the area is .5 ; in which case it 
 will be denoted by D. 
 
 m is used to denote, generally, one fourth of the perimeter of 
 a given polygon, except in case of a major polygon, namely, 
 when the diameter of the inscribed circle is 1. ; in which case it 
 will be denoted by M. 
 
 A figure placed at the right of JP, or of any letter denoting 
 polygonal quantity, signifies or denotes the number of sides of 
 the given polygon; thus p 3 denotes the trigon, p\ the square, 
 and so on. So b 8 or B 8 denotes the octagon, &c. 
 
 V is used to denote one side of a major polygon. 
 
 W is used to denote one side of a minor polygon ; and a 
 figure at the side of V or W, denotes the number of sides of the 
 given polygon. 
 
 What I shall term a series of polygons, is one commencing 
 with the square or with a polygon, the number of whose sides is 
 an odd number, and proceeding ad infinitum, by a constant du- 
 plication of the number of sides. 
 
 The term polygonal progression, is used to denote the order in 
 which the prime denotations progress through the major and 
 minor polygons, from the square to the circle. 
 
 A may be used to denote the area of a minor polygon. 
 
 I will now proceed to set forth a train of facts, which, in the 
 course of the work, may become important; the truth of them 
 has been so often and so well demonstrated, and they are in and 
 of themselves so manifest, as may readily be proved by any 
 inquirer, that it would be supererogation to do more than to 
 name or allege them; and in doing this I shall not attempt 
 method, but shall proceed in a very desultory manner, premis- 
 ing, however, that the point denoted by ^, (which is the cube 
 root of .5,) so constantly occurs as a mean between quantities 
 denoted by two polygonal denotations, which become proper- 
 
40 ON THE QUADRATURE 
 
 tional extremes to ^, that I have thought proper to use the word 
 adverse, in reference to such extremes ; as that they are adverse, 
 or one is the adverse of the other. 
 
 And first in the train of facts, I will speak of certain coin- 
 cidences and reciprocities, whereby we shall find that we need 
 not wander a great way from unity in our investigations of the 
 quadrature. 
 
 It is understood that the area of any given major p is M 
 of the same p; also that C of any given minor p is = the area 
 of a minor p of twice the sides. 
 
 The reciprocal of M of any given p is == d of a like p, whose 
 m is 1. ; and the reciprocal of C of any p is = b of a like p, 
 whose m is 1. ; and the reciprocal of the area of any minor p is 
 = b of a jo of half the sides, whose m is 1. 
 
 When d of any p is the reciprocal of M of a like p, the area 
 of such p is = d; and when b of any p is the reciprocal of the 
 area of a like minor p, the area of such p is = b. 
 
 Whenever a p is placed in connection with, or is compared 
 with a p of half the sides, it must always be understood to 
 be a p of twice the sides of another p ; thus neither the jt?4, nor 
 any p of an odd number of sides, can have a p of half the 
 sides. * 
 
 Thus we say b of any p whose area is = b, is a mean between 
 b of a p of half the sides, whose area is = b, and d of such p of 
 half the sides, whose area is such d; hence, such will con- 
 tinue in a series, from the p 8 to the circle. 
 
 So B of any p is a mean between B and D of a p of half the 
 sides; which order of progression will also continue from the j^S 
 to the circle. 
 
 Because d, when m is 1., is the reciprocal of M, hence, when 
 M is in ^, d will be the cube root of 2., or ^ above M. So also, 
 if C be in ^, b will be the cube root of 2, or T above C. 
 
 But ^ will be found to be an extremely important point, in 
 respect to polygonal progression by means of the prime denota- 
 tions, as also of auxiliary denotations, however they be multi- 
 plied or increased in numbers ; and as well in accordance with 
 the popular method as with my own, ^ is. me farther of two 
 means from M to D ; hence, when M is in ^, D is also. So ^ is 
 the farther of two means from C, to B of twice the sides ; so 
 that when C is in ^, B of twice the sides is also. 
 
 This, then, presents a consequence, which, perhaps, had better 
 be noticed here, namely, if M and C coincide in ^, thereby form- 
 ing M of the prime circle, B and D must necessarily coincide 
 in ^, thereby forming D or B of the circle ; in which case, B of 
 twice the sides could not there exist in reality. And it will be 
 
OF THE CIRCLE. 47 
 
 fully shown that if prime or auxiliary denotations, however nu- 
 merous they may become, have their final place in , ; that all 
 the denotations (except those which coincide above 1.,) will also 
 coincide and have their final place in ^ ; in which point all de- 
 notations coinciding below 1. will be merged, or they will be ex- 
 tinguished in the denotations of the circle. 
 
 When M and D are in ^, half the square of d, when m is 1, 
 is also in ^. So when C is in ^, half the square of &, when m 
 is 1, is also in ^. And when m of any given p is = B of a 
 like PJ the area of such p is half the square of b of a like p, 
 whose m is 1. So, also, when m of any given p is = D of a 
 like j9, the area of such p is = half the square of d of a like p 
 whose m is 1. 
 
 And it may be here remarked, that the easiest and simplest 
 method of obtaining M 8 from terms of the p 4, would be the 
 same by the popular method, as by my own, namely, by di- 
 viding the product of M 4 by C 4, by half the square of d 8 
 when m 8 is 1. But it is quite certain, that such process will 
 not avail any farther than that of finding M 8, as it is deter- 
 mined or shown to be otherwise, even by the popular method 
 and determination ; for in the popular determination, B by D -f- 
 half the square of 6, when m is 1, gives M, that is, accord- 
 ing to the final coincidences deduced by the popular method. 
 
 Hence, a method which will determine M 8, is not sure to de- 
 termine M of any p farther advanced in the series. 
 
 Any given linear measure of a given /?, is always a mean be- 
 tween the area of such p, (whatever may be its dimensions.) and 
 the point in which the area and a like linear measure of a like 
 p y are = ; thus m of any given j, is a mean between the area of 
 such 7?, and M of a like p, for the reason that if area be conceived 
 to be either increased or diminished, (still retaining the same 
 form,) area will flow by a duplicate ratio to that of any linear 
 measure of such area. 
 
 In any given jo, C is to D as M is to B, hence, C is to M 
 as D is to B. But in the p 3 the relative positions of C and D, 
 and of M and B are different from what they are in any given 
 p below the p 4, or between the p 4 and circle. Thus, in the p 
 3, M is above B, and C is above D, but in any p between the 
 p 4 and circle, B is above M, and D above C ; while in the p 4, 
 B and M are =, and D and C are =. 
 
 Hence, if we take our dividend and divisor for obtaining M 6, 
 from terms of the p 3, a like process applied to the /; 4 will give 
 M 8. But such process in obtaining a dividend and divisor 
 from terms of a major p of more sides than the p 4, will not be 
 likely to give M of &p of twice the sides. Nevertheless, any 
 
48 ON THE QUADRATURE 
 
 process applied to a p other than the p 4, that will give M of a p 
 of twice the sides, will, when applied to the p 4, give M 8. 
 
 In the series commencing with the p 3, M 6 is M of the first 
 secondary p obtained in the series ; and in the series commenc- 
 ing with the p 4, M 8 is M of the first secondary p obtained in 
 the series ; and M 6 and M 8 may both be obtained by means 
 of the popular dividends and divisors ; and in either case, M of 
 the secondary p is below a mean between M and C of the p of 
 half the sides ; and the difference between a mean between M 
 and C of the p 3 and M 6, is greater than that between a mean 
 between M and C of the p 4 and M 8. 
 
 But, by any system, the difference between M and C of a jt?, 
 and M of a p of twice the sides, eventually becomes extinguish- 
 ed ; for by the popular method, it must be extinguished in the 
 circle ; and few things can be made more manifest than that 
 such difference is extinguished in ,,. Neither will it be con- 
 tended that M is eventually above a mean between M and C of 
 &p of half the sides. 
 
 But this remarkable fact should not be disregarded, that while 
 we find M in ^ a mean between M and C of a p of half the 
 sides, we also find, that if a supposed progression be farther con- 
 tinued, (and for want of a proper equation of polygonal capaci- 
 ties, such may be the case,) M will forever after continue to be 
 a mean between M and C of a p of half the sides. 
 
 Hence, if we assume a mean between M and C to be the 
 mean place of M of twice the sides, and institute a polygonal 
 progression upon such assumption, we shall see how much de- 
 termined a mean between M and C is to coincide with the other 
 polygonal quantities in ^. 
 
 By such process then, the true and mean place of M will be 
 found to coincide in ^,- and will remain so ever after ; hence, 
 if they have not coincided before arriving at ^, then after arriv- 
 ing at ^, the law of progression must be changed, in order that 
 the progression may be continued farther. 
 
 But I propose to show that M becomes a mean between M 
 and C of a p of half the sides, before we arrive at ^; and 
 hence, that the above process is only given by way of the re- 
 ductio ad absurdum. 
 
 But as it was manifest, that the difference between M, and 
 M and C of a p of half of the sides, must eventually be extin- 
 guished, the popular methods have essayed to do it by a suc- 
 cession of dividends, and corresponding divisions ; in which 
 case, each successive dividend and divisor must depend upon 
 the correctness or the incorrectness of the preceding processes. 
 
OF THE CIRCLE. 49 
 
 Hence, if the process be erroneous, the result must be so too, 
 and the final quantities will not be found in their true places. 
 
 Now, with a view to some further examination of the popu- 
 lar mode, by means of dividends and divisors, I will simplify 
 it by translating it to a correspondence with some unity of pur- 
 pose. 
 
 Thus, the popular method essays to obtain M of the prime 
 circle, by a constant division of the product of M by A, (which 
 is but the square of C,) by half the sum of A and C, namely, 
 by an equal mean between A and C, supposing that such 
 process would serve, from the p 4 to the circle, to give M of a p 
 of twice the sides. 
 
 But it will be my business to show that the course pursued 
 conducts truly no farther than the rugged octagon, that p 
 whose perimeter has transcendental capacities. 
 
 Precisely the same continued and final results will be obtain- 
 ed as in the above process, if we divide the product of M by D, 
 or of C by B, by half the sum of B and D. Hence, we find 
 that B and D need not be excluded from participating in a 
 polygonal progression, with a view to obtain the quadrature. 
 
 So upon the same principle, (if correct,) m of any given /?, 
 (of whatever dimensions,) -r- by half the sum of d and b of such 
 /?, would give M of a p of twice the sides. 
 
 But, perhaps, the simplest of these methods is that in which 
 B and D are mostly used, as by dividing the product of M by 
 D, or C by B, by half the sum of B and D, and by such pro- 
 cess we certainly obtain M 6 and M 8. 
 
 Now the product of M by C of the p 4 -f- by half such M 
 and C, gives M 8, so the product of M by C of the p 3 -f- by 
 half of such M and C, gives M 6; but such process cannot avail 
 farther; for M by C of the/? 8 -H by half such M and C, gives 
 .783100, or even less than the popular M of the prime circle ; 
 which intimates very strongly, the extraordinary capacity of M 
 8. And yet nothing is more manifest, than that in the final 
 result, the product of M by C -^ by half the sum of such M and 
 C, gives M, or M of twice the sides, if we so choose to con- 
 sider it ; and if such progression were conducted on the popular 
 method, the final M would be far below that of the popular final 
 M, for the disproportion between the dividend and divisor would 
 be even greater than by the popular method. Hence, from the 
 p 8 to the circle, there is required an equation of perimetric ca- 
 pacity. 
 
 Various, indeed, (however erroneous the process and results,) 
 may be the methods of conducting a progression by means of a 
 constant succession of dividends and divisors. Thus, the pro- 
 7 
 
50 ON THE QUADRATURE 
 
 duct of M by D, or of C by B ~- by half the sum of C and 
 M, will give the same continued and final results, as if the pro- 
 duct of B by D be -4- by half the sum of such B and D, and 
 neither of these progressions, if thus conducted, will give the 
 termini below , ; for when M and D coincide in ^, (as they 
 must by any method,) the divisor, namely, half the sum of M 
 and C, cannot be above ^, unless we conceive C to be above M. 
 
 Now in any p between the p 4 and circle, A by B ~ by half 
 of A and B will give less, than when M by D is -r- by half of 
 M and D, for the reason that an equal mean between A and 
 B, is greater than an equal mean between M and D, while a 
 mean proportional between A and B is the same as that between 
 M and D. Nevertheless, the only point of coincidence of A 
 and B, or of M and D, is in ,, by any system. 
 
 Again, the product of C by D -H by half of A and D, is 
 identical in result, to that when M by D is -4- by half of C 
 and B. 
 
 Now if we denote the difference between C 8 and D 8, or 
 between M 8 and B 8, by J_, then the product of C 8 by D 
 8 is J_ above the square of C 8, and half the sum of A 8 
 and D 8 will consequently be J above the popular divisor, 
 namely, half of A 8 and C 8, and hence, the quotient will be 
 J above the popular quotient, that is, by such process, M 16 
 would be above the popular M 16, and it is manifest that such 
 process would dimmish the difference between M and B, or be- 
 tween C and D, while the popular mode serves to increase the 
 difference between C and D, or between M and B, or to 
 enlarge the quantity denoted by J_, or cause it to increase from 
 the p 8 to the circle. 
 
 Whether we -f- the product of C by D by half of A and D, 
 or the product of M by D by half of C and B, the divisor is 
 always to a mean between M and D, as such mean is to 
 the quotient. Hence, where the divisor becomes the adverse 
 of D, the quotient will be in ^, for a mean between M and 
 D, is a mean between the divisor and quotient, and ,, is the 
 farther of two means from D to a mean between M and D. 
 
 If then a progression must be carried on by means of con- 
 stant dividends and divisors, (which is at least problematical,) 
 there seems no good reason why diametric quantities may not 
 share in the progression, upon any system whatever. And 
 as the divisor is to the square root of the dividend, as such root 
 is to the quotient, hence, if M and D of any given p below the 
 p 4, being -i- by half of such M and D, will give M of a p of 
 twice the sides, then our conceptions at a certain stage of the 
 process will be right ; namely, when M and D coincide in ^r 
 
OF THE CIRCLE. 51 
 
 in which case an equal and a rational mean between them would 
 be the same ; and no further progress could be made ; and similar 
 may be shown to be the results of other progressions. 
 
 Nevertheless, all agree that the difference between the di- 
 visor, and B of the p of twice the sides of that from which the 
 dividend is obtained, may be eventually equated to _^_ ; or that 
 they may eventually coincide. But I have sufficiently treated of 
 polygonal progressions, by means of constant or successive divi- 
 dends and divisors. 
 
 SECTION THIRD. VJJ 
 
 Inasmuch as the point denoted by , assumes so much con- 
 sequence, in a consideration and comparison of the prime and 
 primal polygons, it may not be improper to note some few phe- 
 nomena in respect to it. 
 
 Thus, when m of any given p is made the adverse of D of a 
 like j9, the area of such given p is ,. Hence, if m of a given 
 p and D of a like p be adverse, the product of one area by the 
 other is half of ^ ; but if m of a p 4 and b of a p 4 be adverse, 
 the product of one area by the other is one fourth of ^ ; while if 
 m of one circle and b of another circle be adverse, the product of 
 one area by the other is half of ,. 
 
 If ^ be the farther of two means from one given point to 
 another, the point farthest from ^ is half the square of the 
 reciprocal of the nearest point ; hence M is always half the 
 square of the reciprocal of D, A of B, and C of B of twice the 
 sides. 
 
 The adverse of M is always T below e?, when m is 1. So the 
 adverse of C is always "4" below 6, when m is 1. So also, when 
 D of a p is the adverse of M of a like JP, the area of such p is 
 T below -d. So when b of a p is the adverse of A of a like p, 
 the area of such p is T below b. 
 
 The area of any given p is half the square of a mean between 
 such area and 2. ; hence M is half the square of a mean between 
 such M and 2. 
 
 If m of one p and d of a like p be =, the product of one 
 area by the other is the fourth power of such m or d. 
 
 Twice the area of any given p is the square of b of an equal 
 square, with that of the given p ; hence, b of an equal square, 
 with that of any given M, is = m of a like p with that denoted 
 by M", whose area is 2. 
 
 M and D coincide in ^ J hence, when D is in ^ m is the 
 square of d, and the area is the third power or cube of d. 
 Hence m of such p is the third root of the square of the area. 
 
52 ON THE QUADRATURE 
 
 If two points be adverse, the reciprocal of either is ' above 
 the other. 
 
 The product of B by D of any p is half of b of a like p 
 whose m is 1. 
 
 The square root of C is T above a mean between C and 
 ,2, ; so the square root of M is T above a mean between M 
 and 3- 
 
 If b of one circle and m of another, b of one p 4 and m of 
 another, and b of one p 3 and m of another be = in one and 
 the same point, the product of one area of the circle by the 
 other, will be double the product of one area of the p 4 by the 
 other, and quadruple the product of one area of the p 3 by the 
 other. 
 
 The square root of A is to C as the square root of M is 
 to M. 
 
 Twice the square of C4is the equal reciprocal of M4; so 
 twice the square of C 8 is the equal reciprocal of M 8. But twice 
 the square of C of a p of more sides than the p 8, is more than 
 the equal reciprocal of M. 
 
 Hence M 4, plus twice the square of C 4, is ; and M 8, plus 
 twice the square of C 8, is 2. But in any p of more sides than 
 the p 8, M plus twice the square of C, is more than 2. 
 
 b of a major polygon is the reciprocal of d of a like minor 
 polygon ; and when M is in ^, b is ^ above B of a like j9, and 
 b of a like minor p is then ^ above C. 
 
 The square root of b of any major/? is b of a p of twice the 
 sides, whose area is = the area of the given prime p. 
 
 The square root of d of any minor p is d of a like p, whose 
 area is = C of the given minor p. 
 
 D is the reciprocal of m of a like /?, whose area is 2. 
 
 The product of m by b of a p 3, is ^ above the area ; the pro- 
 duct of m by & of a jf?4, is T above the area ; and the product of 
 m by b of a circle is equal the area. 
 
 If m of the jf?4 be = M of any given p, the square of b of 
 such p 4 is the reciprocal of half the square of b of the given p 
 whose m is 1. 
 
 D of any given p is the reciprocal of b of a p 4, whose area is 
 = M of the given p. 
 
 A is to C as the square root of A is to the square root of M ; 
 hence, C is to M of twice the sides as the square root of such C 
 is to the square root of M of twice the sides. 
 
 M by C of the p 4 is = m 4 when the area is .5 ; and when M 
 and C of a p 4 are adverse, M by C is = m of a p of twice the 
 sides, whose area is .5. 
 
OF THE CIRCLE. 53 
 
 But M by C of any p between the p 4 and the p whose M and 
 C are adverse, is less than m of a like p whose area is .5. 
 
 Half of the higher of two reciprocal points is the reciprocal 
 of twice the lower point. 
 
 A mean between M and C of the p 4 is half of a mean be- 
 tween b and d of the p 4, whose area is 2., and the same with the p 
 whose M and C are adverse, and each of them is half of b of a 
 p of twice the sides, whose area is 2 ; hence, ^ above ^ is a 
 mean between b and d of a like p of that whose M and C are 
 adverse, whose area is 2. Consequently, ^ above ^ is b of a 
 like p to that whose M is in ^, whose area is 2. ; and, conse- 
 quently, if M of a jo be below ^, a mean between M and C 
 will be less than half of a mean between b and d of a like j9, 
 whose area is 2. 
 
 If the area of any given p is the third power, or cube, of any 
 given point, such area is a mean between d of such p and m 
 of a like jo, whose area is = such point. 
 
 So the third power of D of any given p is to .5, as such D is 
 to M of a like p. 
 
 So the third power of B of any given p is to .5, as such B is 
 to A of a like p. 
 
 Hence, the third power of d of a p, of whatever dimensions, 
 is to the area of such p as such d is to M of a like p. 
 
 So the third power of b of a p, of whatever dimensions, is to 
 the area of such p as such b is to A of a like p. 
 
 So also, the third power of m of any given p is to the area of 
 such p as such m is to the reciprocal of M of a like p. 
 
 The product of the area of any given p, of whatever dimen- 
 sions, by M of the prime circle, is = the product of the area of 
 the circumscribing circle of the given p, by A of a like p to the 
 given p. 
 
 When d of any given p is ^, the cube of the area of such p 
 is one fourth of the cube, or third power, of M of a like p. 
 
 So the third or cube root of the square of M of any given p 
 is m of a like jo, when the area is the third root of such M. 
 
 The area of any given p is to the square of d of such j9, as M 
 of a like p is to 1. 
 
 So the area of any given p is to the square of m of such p as 
 d when m is 1., is to 1. 
 
 When the area of a JP is 1., the product of d by b is = b of a 
 like jo, whose m is 1. 
 
 Such, then, are a few of the facts, corollaries, and conse- 
 quents, flowing from that vast store-house of demonstrated prin- 
 ciples, which the world has accumulated. 
 
 And would the prescribed limits of the work permit, much 
 
54 ON THE QUADRATURE 
 
 curious evidence might be deduced, in favor of my determination 
 of the quadrature, from a consideration of the sides of polygons, 
 (which I denote by V,) in which the sum of the squares of all 
 the several sides of a regular polygon, is a mean between the 
 square of one side and the square of the entire perimeter. 
 Thus, when V of the p 4 is 1., the area is 1., or = ^4 when w4 
 is 1. When V 8 is 1., the area is = four times d 8 when m 8 is 
 1. ; and when V16 is 1., the area is = sixteen times ^16 when 
 m 16 is 1., &c. 
 
 In such investigation, as also in a consideration of the sum of 
 all the powers of certain polygonal quantities, we should dis- 
 cover the inclination of the measures of the p 8 to be governed 
 by equal means, in lieu of proportional means. Thus, the sum 
 of all the powers of the square of C 8 is = d 8 when mS is 1. ; 
 the sum of all the powers of A 8 is twice d 8 when m 8 is 1. ; so 
 the sum of all the powers of M 8 is four times d 8 when m 8 is 
 1., or four times the reciprocal of M 8, &c. Nevertheless, the 
 rugged octagon, when unfolded, presents great mathematical 
 beauties ; and number 8, amongst numerical quantities, is cer- 
 tainly a conspicuous number, in its constant application in the 
 development of physical laws by the aid of numbers. I will 
 here suggest, (before proceeding further with explanatory mat- 
 ter,) a simple polygonal progression, based upon the principle, 
 that any given point, either above or below ^ is half the square 
 of "4^ above a mean between such point and ^ ; and hence, the 
 progression will be alike applicable to all polygons, from the jt?3 
 to the circle, and is designed to approximate I) and B to a coin- 
 cidence. 
 
 B of any given p is half the square of T above & of a like ;?, 
 whose area is T below the adverse of B of the given p ; and D 
 of any given p is half the square of T above d of a like jo, 
 whose area is ^ below the adverse of D of the given p. 
 
 ^ is a mean between B 4 and ^, and B 4 is half the square of 
 ^ above ^ ; and when b 4 is ^ the area of such p4= is T below 
 the adverse of B 4. 
 
 So D 4 is ^, and is half the square of T above a mean be- 
 tween D 4 and ^ ; and when d 4 is a mean between D 4 and 
 ^, the area of such p 4 is ^ below i, or is T below the adverse 
 ofD4. 
 
 Hence, such progression, by an infinite series, will extinguish 
 the mean between B and ^, upon the one hand, and between D 
 and , upon the other ; nor can the progression proceed further. 
 And it is manifest that, at the end of the progression, either in 
 respect to B or D, that the area will be .5, or f below B or D, 
 namely, T below ^. 
 
OF THE CIRCLE. 55 
 
 From the fact, that the product of the perimeter of any regu- 
 lar jf?, by one fourth of the diameter of the inscribed circle of 
 such polygon, gives the area of such p, it follows as a corollary, 
 that if the square root of such perimeter be taken as the hypoth- 
 enuse of a right angled triangle of two equal legs, or sides, that 
 the square of the distance from the centre of the base to the 
 right angle, (namely, the square of the altitude,) will be = the 
 area of the given p. Let such perimeter, then, be the perimeter 
 of a major polygon ; in which case, the square of the altitude 
 will always be = M of the given p. 
 
 In such case, the base will always be = m of a like j9, whose 
 area is 4 ; one of the equal sides will be = m of a like />, 
 whose area is 2 ; and the altitude will be = m of a like p, 
 whose area is 1. ; and the square of the altitude will be = M 
 of a like p; one of the equal sides will be = b of a p 4, of equal 
 area with that of the given 7?, or will be a mean between 2. and 
 M of the given p ; and when M of the given p is ^, b of an 
 equal p 4 will then be the reciprocal of ^. 
 
 1 is used to denote the difference between M 8 and B 8, or 
 between C 8 and D ; which difference is extinguished, or made 
 JL in the circle. If, then, A be a diminishing quantity from the p 8 
 to the circle, it ought to be so denoted in the course of the pro- 
 gression ; thus, if in the p 16 it shall be but a proportional half, 
 for instance, of what it is in the p 8, and so on, the same 
 should appear, as thus, i, J, J, &c. ; for if the diminishing quan- 
 tity be constantly denoted by ^ we shall be inclined to conceive 
 the same difference to exist between M and B or C and D, 
 when the same is wholly extinguished, as existed in the p 8. 
 
 And, in fact, when M and C become adverse, although they 
 then are in actual coincidence, we cannot well get rid of the 
 conception, that M is J. above ^ and C JL below ^ ; for as we 
 still retain in our conceptions the order of the progression, we 
 still conceive the terms as being separate, and their distance as 
 being the same as previously. 
 
 SECTION FOURTH. 
 
 It will not, probably, be doubted or denied, that the order in 
 which the four prime denotations progress, from the p 4 to the 
 prime circle, or to the coincidence of C and M and the coinci- 
 dence of D and B, is as follows, namely, C : D : : M : B, or 
 C : M : : D : B ; for such is the popular progression and final 
 determination. The difference between M and B and between 
 C and D being in the p 4, but greater in the p 8 than in the 
 p 6. 
 
56 ON THE QUADRATURE 
 
 Neither will it be denied that, by the popular method, the act- 
 ual difference between M and B and between C and D, is in- 
 creased from the p 4 to the circle ; so that the difference is even 
 greater (saying nothing about ratio,) in the circle than it is in 
 the p 8. 
 
 Neither will it be denied, that the progressive ratios of M and 
 D may coincide in ^, and that they cannot, upon any system or 
 principle, coincide in any other point or position ; and that the 
 same will be the case in respect to A and B, or C, and B of twice 
 the sides, in case they are permitted to progress thus far ; for 
 such is manifestly the law of progression, by any system or 
 method. 
 
 So it is manifest, also, that by the general law of progression, 
 when the progressive ratios of M and D coincide in ^, if C and 
 B do not then coincide with M and D, they will be adverse ; 
 so also, if M and C of a p of half the sides of that whose M 
 and D are in ^/, are not also in ,,, they must necessarily be 
 adverse. 
 
 Hence, when M and D coincide in ^ if a like or a similar dif- 
 ference still prevails between M and B and between C and D 
 as in the p 8, the formula for the four prime denotations, for the 
 p whose M and D are in ^ and the p of half the sides, will be 
 thus : C : D C : D M : B : M : B. But if, when M and D coin- 
 cide in 3> me difference between M and B and between C and 
 D is JL, or is equated to JL, then all the above eight denotations 
 are in ^ ; or, if we choose so to consider it, those denotations 
 belonging to the p whose M and D apparently coincide in ^ 
 are extinguished, by the coincidence of those of half the sides, 
 which may be shown to be universally the case in respect to all 
 denotations (of whatever kind) which apparently coincide in 
 ^; and hence, we may either consider them as vanishing, or 
 becoming extinguished by like denotations of half the sides, in 
 the point of apparent coincidence ; or we may consider them as 
 still existing in coincidence with those of half the sides. 
 
 And to show that such phenomena must take place, (confining 
 our investigations wholly to major, minor and primal polygons,) 
 one side of the minor p 4, when properly equated and adjusted, 
 is as well the base of two adjoining sides of the major p 8, as 
 of two adjoining sides of the minor p 8; and the like is true, in 
 respect to the p 8 and p 16, and so on, ad infinitum, until one side 
 of a minor p is = two sides of either a major or minor p of 
 twice the sides ; or, rather, until all numeral difference between 
 them is extinguished. 
 
 The application, thus far, in cases in which one and two sides 
 have been considered, has been general, and is adapted to any 
 
OF THE CIRCLE. 57 
 
 series of polygons; but, in order to avail ourselves of the full 
 benefits of the phenomena, we must confine ourselves to that 
 series of polygons which commences with the p 4 ; for the 
 reason, that in such a series M or C of any /?, (except that of the 
 p 4,) is composed of an even number of sides ; and hence, from 
 the p 8 onward to the circle, the phenomena apply equally as 
 well to M or to C, as to one or two sides, which is not the case 
 with any other series than that commencing with the p 4. 
 Hence, the commencement of our process is properly with the 
 p 8 ; and especially, when we wish to show the progress of a 
 polygonal progression downward, towards M of the prime 
 circle. 
 
 Thus C of the p 8 may be said to span both C and also M of 
 the p 16, and so on, until C becomes = C, as also M, of a p of 
 twice the sides, which can only occur in the circle ; and the con- 
 sequence necessarily is, that in either JP, C is = M, which may 
 be called the fourfold chord of the periphery. 
 
 Hence M, in descending, must necessarily become = M, 
 as also = C of a p of twice the sides ; and the only ques- 
 tion is, in what numerical point or position shall this fourfold 
 chord be formed or situated ? If it be in ^ tnen B and D final 
 are in ^ ; if it be above ^, then D and B final will be below %, ; 
 and if it be below ^, then D and B final will be above 3 j f r 
 if it be not in ^, then ^ wil l De tne farther of two means from 
 the coincidence of M and C to that of B and D. 
 
 If M and C coincide in ^, the difference between M and B 
 and between C and D will have become equated to _2_. If M 
 and C coincide below ,,, such difference will remain, or has 
 been increased or diminished from what it was in the p 8, ac- 
 cordingly as it shall be in the circle. And if M and C shall 
 coincide above ^, then such difference will have been extin- 
 guished, and will have again increased in the opposite direction. 
 And certainly I will not contend in favor of the last position, but 
 it is my business to contend against the second. 
 
 This fact, alone, then, would seem to present an insurmount- 
 able difficulty in the way of the popular mode, namely, when 
 the progressive ratios of M and D coincide in ^ ; if we attempt 
 to extend the progression farther, M and B will in future pro- 
 gress by the same ratio, and C and D will also progress by the 
 same ratio, which is contrary to the popular law of progression, 
 in which the difference between M and B, as also between C 
 and D is constantly increasing from the p 8 to the circle. And 
 certainly the very law of progression itself, ought not to be re- 
 quired to give way, under any circumstances. Besides, M and 
 B are properly regarded as descending quantities ; and two 
 8 
 
58 ON THE QUADRATURE 
 
 separate quantities, descending by the same ratio, must necessa- 
 rily converge. 
 
 Hence, it is impossible that the progression can be extended 
 after M and D coincide, without an alteration of the law of pro- 
 gression. If, then, the law of progression is not altered in ^ 
 then M and C, and B and D coincide in ^. 
 
 We perceive, that C and C and M of a p of twice the sides, 
 (in the series commencing with the p 4,) preserve their relative 
 situations from the p 8 and p 16 to the circle ; and D of the p 
 whose M and C are adverse, and D and B of the p of twice the 
 sides, (namely, the p whose M and D are in ^,) have the same 
 relative situations, in respect to each other, as C of a p whose 
 M and C are adverse, and C and M of the p of twice the sides, 
 have. 
 
 I have said, that from the p 8 to the circle the same relative 
 situation, between C, and C and M of a.p of twice the sides, is 
 constantly preserved ; and it may be of consequence to know, 
 whether from the p 8 to the circle the same relative situation is 
 constantly preserved between C and M of a p and C and M of 
 ap of twice the sides. 
 
 A mean between C and M is to a mean between D and B, as 
 C is to D or as M is to B ; or a mean between M 8 and C 8 is 
 _1 below a mean between B 8 and D 8 ; hence, if M 16 be a 
 mean between M 8 and C 8, it is JL below a mean between B 8 
 and D 8. 
 
 Now M 8 is a mean between M 4 and %_ below D 4, and M 8 
 is also a mean between C 4 and JL below B 4 ; and if M 16 be 
 a mean between M 8 and C 8, then the quantity denoted by Ji 
 must be constantly halved. Now M 16 is to a mean between 
 M 8 and C 8 as it is to a mean between C 8 and M 8, for they 
 are one point, and so on. So M 16 is to a mean between M 8 
 and JL, below D 8, as it is to a mean between C 8 and J_ below 
 B 8, for they are one and the same point. 
 
 If, then, M 16 is a mean between M 8 and JL below D 8 or 
 C 8, and JL below B 8, we then have the terms of a progression 
 from the p\ to the circle, in which, from the pS to the circle, the 
 same relative position will constantly hold between M and C of 
 any p and M and C of a p of twice the sides ; and it would 
 seem, in reality, as if such progression was corroborated by the 
 popular determination. 
 
 If, then, the p 4 be called the primitive p of the series, and all 
 the other polygons of the series be called secondary polygons, 
 then, while C 16 is a mean between M 16 aad C 8, M 16 will 
 be a mean between M 8 and C 8, and so on to the circle. 
 
 But to proceed with the progression on the foregoing ; M 8 is 
 
OF THE CIRCLE. 59 
 
 a mean between M 4 and A below D 4, and between C 4 and 
 2_ below B 4, (for they are one and the same.) M 16 is a mean be- 
 tween M 8, and !_ below D 8, and between C 8 and J_ below B 8. 
 M 32 is a mean between M 16, and below D 16, and between 
 C 16, and J below B 16, and so on. 
 
 Such being the case it is manifest that all the four denotations 
 must coincide together, which must of necessity be in ^. Or, 
 on the same principle, we have this progression : A 8 is to M 8 
 as M 8 is to L below B 4 ; A 16 is to M 16, as M 16 is to 1 be- 
 low B 8, and so on. Or thus ; M 4 is to M 8, as M 8 is to A be- 
 low D 4, M 8 is to M 16, as M 16 is to .1 below D 8, and so on. 
 
 Thus A and M would coincide with B of half the sides, and 
 M and D would coincide with M of twice the sides. And this 
 entirely comports with the law of progression. So again, C 4 is 
 to M 8, as M 8 is to 1 below B 4 ; C 8 is to M 16, as M 16 is to 
 _1 below B 8, and so on. 
 
 But the popular methods instead of equating JL to JL, have 
 only equated it to Ji, and that by placing M and C final, f be- 
 low their proper place. 
 
 But a great number of progressions of the kind, and on simi- 
 lar principles, may be instituted all of which will find their 
 termini in ^ : for m of any given p when the area is .5, namely, 
 M by D, or C by B, of any given p -r- by ^ gives m of a like 
 p, when the area is ^. 
 
 M of any given jo, is to b when the area is 1, as C is to d 
 when the area is 1, and the proportional difference between M 
 4, and b 4, when the area is 1, is less in the p 4, than in any p 
 between the p 4 and circle ; but this proportional difference is the 
 same in the p 4 and the circle. Hence the linear capacities are 
 the same in the p 4 as in the circle. The difference between M 
 and 6, when the area is 1, or between C and d, when the area is 
 1, is greater in the p 8 than in any p of the series commencing 
 with the p 4, and is denoted by 1_ ; hence the p 8 is the proper 
 p from which to commence our equation of polygonal capacities, 
 whether it be mathematically or geometrically performed. 
 
 M 8 is J_ below a mean between M 4 and I) 4, or C 4 and B 4, 
 and M 16 is J below a mean between M 8 and D 8, or C 8 and B 
 8, and so on. And by such progression, it is manifest that all 
 the terms coincide in ^. 
 
 Now no one will deny but that M 8 is _L below the square 
 root of m 4, when the area of such p 4 is .5 ; nevertheless, the 
 popular method has placed M of the prime circle, J below the 
 square root of m of the circle whose area is .5, from which it is 
 manifest that by such method, no progress has been made from 
 the p 16 to the circle, but that there has even been a retrograde 
 
60 ON THE QUADRATURE 
 
 from the truth, occasioned by an increase of the constant divisor 
 in an incorrect proportion to that of the constant dividend. 
 
 I have said it might be important to know avhether from the 
 p 8 to the circle, M and C, and M and C of a p of twice the 
 sides, constantly retained the same relative position in respect to 
 each other. It is quite certain that they do not from the p 4 to 
 the circle. Then draw the circle whose diameter is assumed at 
 unity ) and from any point in the periphery, draw one side of the 
 perimeter of the major/? 8, (namely, one side of M 8,) as a tan- 
 gent, and such tangent may be said to be the tangent to one side 
 of C 8, and also to two sides of M 16, as also two sides of 
 C 16, one side of M 16 being merged in such tangent, which 
 side of M 16 so merged, is the tangent of two sides of M 32, of 
 two sides of C 32, and of one side of C 16, and so on, ad in- 
 ftnitum, until the tangent is extinguished, namely, until M and 
 C coincide with M and C of twice the sides, or until M and C 
 have extinguished M and C of twice the sides, thus determining 
 the fact, that from the p 8 to the circle, M and C and M and C 
 of twice the sides constantly retain the same relative situation to 
 each other. And no fact can be made more manifest, than that 
 when M is in ^ M and C of half the sides are adverse, and D of 
 the p whose M is in *, is = M ; C of such p is a mean between 
 M of such p, and C of half the sides. B of such p is a mean 
 between M or D of such p, and M of half the sides. 
 
 What then must be the relative situation of M 16 in respect 
 to M 8 and C 8 ? Let legitimate geometry answer the question. 
 
 Hence, M 16 is a mean between C 8 and J_ below B 8, or 
 is a mean between M 8 and _L below D 8. Or M 16 is J below 
 a mean between M 8 and D 8 or J below a mean between C 
 8 and B 8, So M 32 is J below a mean between M 16 and D 
 16, or C 16 and B 16, and so on, ad infinitum, until M finally 
 coincides with M and D of half the sides and with C and B of 
 half the sides. And as before shown, such coincidence is in ,,. 
 
 We hence obtain or avail ourselves of the true equation of 
 polygonal measures ; but to attempt this without either unity of 
 purpose, or legitimate geometry, would be a hopeless task. 
 
 Hence if any final denotation is in ^, they all become final 
 in *, ; in which case also, if any denotation appears to be in ^ 
 it is there extinguished by a like denotation of half the sides. 
 And it might be a pleasant exercise in mathematics, to multiply 
 auxiliary denotations of other polygonal quantities ; as for in- 
 stance, a mean between C and D, which is also a mean between 
 A and B and a mean between M and D, which is also a mean 
 between C and B, and is the square root of m of a like p whose 
 area is 5, or half the square of the reciprocal of A, or of C, or 
 of M and C ; all of which, by the laws of polygonal progression, 
 
OF THE CIRCLE. 61 
 
 are seen to progress, or flow into, and to coincide in ,,. But I 
 forbear to enter upon it further. It may, however, be suggested, 
 that if we signify ihe p whose M and D appear to coincide in ^, 
 by placing figure 2 at the right of its denotations, and figure 1 at 
 the right of the denotations of the p whose M and C appear to be 
 adverse, then when M 2 and D 2 appear to coincide in ,, C 1 
 and M 1 actually coincide in ^, in which case M 1 will appear 
 to assume the identical place of C 1, when M 1 and C 1 are 
 conceived to be adverse ; and if the progression be conceived to 
 be still continued (for we may conceive a continuance of the 
 progression until we can conceive quantity to be extinguished 
 by constantly halving it) M and B will thence be conceived to 
 ascend, and C and D to descend, contrary to the order or law of 
 polygonal progression, which process, or imaginary process, (for 
 it is but imaginary,) will present many curious coincidences and 
 adverse reciprocities by way of the reductio ad absurdum, which 
 I will not here enter upon. 
 
 But I will cease to multiply evidence of this description, as- 
 suring the world that the store house is yet full, and by no means 
 exhausted by the pittance which I have drawn from it. 
 
 Z may denote a mean between M and D or C and B of any 
 given p, which is at the same time the square root of m of a like 
 p whose area is .5. Then, from the p 4 to the circle, Z and M 
 of a p of twice the sides, approximate each other, by any system. 
 Then, Z 4 is 1 above M 8, Z 8 is above M 16, Z 16 is above M 
 32, and so on, till Z is = M of twice the sides. 
 
 Z is the farther of three means from M to ,. Hence, Z can- 
 not be = M of twice the sides below ,,, for in that case M would 
 be below M of twice the sides, which is impossible. 
 
 If from the p 8 to the circle, a mean between M and C is M 
 of twice the sides, a progression on this principle will be identical 
 with that by which we say that Z 4 is _1 above M 8, Z 8 is above 
 M 16, &c. And then, from the p 8 onward, D is to a mean 
 between D and C, as a mean between M and I) is to M of twice 
 the sides, or as Z is to M of twice the sides, or as Z is to a mean 
 between M and C of the same p, which last also applies to the 
 1*4 
 
 Again, when Z becomes a mean between M and C (= M of 
 twice the sides) such mean between M and C (or M of twice the 
 sides) is a mean between M and D, or C and B, or B and D. 
 Then M is = B, and C is = D ; and M of twice the sides is also 
 = B of such p of twice the sides : thus in two intermediate poly- 
 gons, M is = B, and when M is = B, C is =D. 
 
 Because M8 is a mean between M4 and j below D 4, it is 
 a mean between C 4 and JL below B 4. Because M 16 is a 
 
62 ON THE QUADRATURE 
 
 mean between M 8 and J_ below D 8, it is a mean between C 8 
 and Ji below B 8, and so on. 
 
 So when M becomes a mean between M and D, or C and B 
 of half the sides, it is also a mean between M and C of half the 
 sides. When M is given, we conceive M of twice the sides, 
 even though there be no difference between them. 
 
 Hence, when M and D coincide, we conceive M of twice the 
 sides, even though it be = such M in ^ as a mean between M 
 and D in ,. 
 
 In order. that M may be a mean between M and D of half the 
 sides, it must be the farther of three means from M of half the 
 sides, to %,. 
 
 M below ^, cannot be a mean between M and D of half the 
 sides, for in that case, M of half the sides would be the lower 
 M, for M cannot be above M of half the sides. 
 
 By no system is M ever above a mean between M and D of 
 half the sides, that is, a mean between M and C is never above 
 a mean between M and D. 
 
 Whenever M is a mean between M and D of half the sides, 
 it is also a mean between M and C of half the sides, in which 
 case C and D are necessarily =. So when there is no difference 
 between M and B, or C and D, a mean between M and C of 
 such p is M of twice the sides. 
 
 By the popular method, M does not become a mean between 
 M and C of half the sides, except in the circle, and hence, M is 
 always, (from M 16 to the circle,) below a mean between M 
 and I), or C and B. But the fact is too manifest to be doubted, 
 that M in ^ is a mean between M and C of half the sides. 
 
 When M is in ,, Z is necessarily in ^, being then the farther 
 of three means from M to %,. 
 
 Z is above M of twice the sides, until they become =. 
 
 In ^ M is = Z of the same p. Below ^, M cannot be = Z 
 of any p. Hence M cannot be = Z of half the sides, except 
 when Z and M of the same p are = and such must be the case 
 in the point in which M coincides with M of twice the sides. 
 Such point is in ,. 
 
 The progression may also be carried on thus : The square root 
 of m 4, when the area is .5, is J_ above M 8. The square root of 
 m 8, when the area is .5, is above M 16, &c., until the square 
 root of m, when the area is .5, is = M of twice the sides. 
 
 Now it is manifest that the square root of w, when the area is 
 .5, cannot be = M of twice the sides, in any point above ,,, for 
 it is only in ^ that it becomes = M of a like p, for a mean between 
 M and D, (which is the square root of m when the area is .5,) 
 is above until M and D coincide in . 
 
OF THE CIRCLE. 63 
 
 It is equally manifest, that the square root of m when the area 
 is .5, cannot be = M of twice the sides below ,,, for the reason 
 that the square root of m, when the area is .5, is always the far- 
 ther of three means from M to ^ ; hence, for the square root of 
 m, when the area is .5, to be = M of twice the sides, in any 
 point below ^, would require M to be below M of twice the 
 sides, which would be a phenomenon that no one would expect. 
 
 Hence it is determined in the most conclusive manner, that ,, 
 is the point in which we must conceive the square root of m, 
 when the area is .5, to become = M of twice the sides ; while 
 in the same point, it is = M of the same p. 
 
 Hence, ^ is the very point in which M apparently becomes 
 = M of twice the sides, which is a phenomenon that apparently 
 must occur in the formation of the periphery of the circle. Nev- 
 ertheless, such M of twice the sides does not actually exist, it 
 being but the conception of M of twice the sides of that M which 
 has no sides, or the number of which (if number it may be called) 
 is infinite. And should any require corroborative evidence in 
 favor of my determination that M of the prime circle is in ^, 
 let them apply to legitimate geometry. 
 
 The equation of the capacities of polygonal measures, properly 
 commences with the p 8, in which p there is the greatest differ- 
 ence between M and B, or C and D, and this is plainly dictated 
 by geometry. And in reference to the varying capacities of po- 
 lygonal measures, the jo4, or unit polygon, is the perfect standard, 
 the true place of any one of its measures always being in the 
 mean place. But not so in respect to polygons situated between 
 the p 4 and its inscribed circle ; in which the linear capacities are 
 variable, the true places varying from the mean, and hence requir- 
 ing a constant equation, the variance being greater in the p 8 than 
 in any other p. 
 
 Thus the variance of the true place of m 8 from the mean place, 
 is proportionally twice as great in respect to d8, as it is in respect 
 to b 8, and d 8 is proportionally as much above its mean place, as 
 m 8 is below, while b 8 is always in its mean place in respect to 
 area, as well in the p 8 as in all polygons from the p 4 to the circle ; 
 for it must be recollected that these varying capacities have regard 
 or reference, to the admeasurment of area. 
 
 Nevertheless, the linear capacities of the circle are the same as 
 those of the p 4 ; and hence, in a progression from the p 4 to the 
 circle, those varying capacities must be regarded in order to avoid 
 constant and final error. 
 
 This must be done by geometrizing wholly by means of lines, 
 (which is the only legitimate geometry,) without any regard 
 whatever to areas, in the course of our progression ; for we may 
 
64 ON THE QUADRATURE 
 
 be assured, that when our lines are properly adjusted and pro- 
 portioned, that the areas will be so too. But it is too manifest, 
 that if we attempt to progress by means of areas instead of lines, 
 that we shall wholly fail to equate the varying linear capacities of 
 polygons situated between the j4 and its inscribed circle. 
 
 I have said that if one final denotation be in ^, (however the 
 denotations be multiplied,) then all the denotations will necessa- 
 rily coincide together in ^ ; but if any final denotation be not in 
 ^, then the final denotations will be scattered, some being above 
 %,, and some below, but none in ,. 
 
 And for the purpose of exhibiting the phenomenon, I will still 
 increase the number of denotations expressed by capital letters. 
 
 E may denote half the square of the reciprocal of A, of any 
 given p. F may denote half the square of the reciprocal of C 
 of any given p; namely, half the square of b of a p whose 
 m is 1, and G may denote half the square of the reciprocal of 
 M; namely, half the square of d of a p whose m is 1. Now 
 half the square of the reciprocal of C of any given p is = the 
 area of a like p whose m is = B of a like p ; and in the p 8 
 such area, or half the square of the reciprocal of C 8, is an equal 
 mean between M 4 and C 4, and is the proper divisor for ob- 
 taining M 8, when the dividend is the product of M 4 by C 4, 
 and by such, or a similar divisor does the popular method essay 
 to carry on a progression ; but it will not succeed beyond obtain- 
 ing M 8. So also, half the square of the reciprocal of M is = 
 the area of a like p when m is = D of a like p. 
 
 B then of any p is a mean between M and half the square of 
 the reciprocal of C of a like p, and D of any p is a mean be- 
 tween M and half the square of the reciprocal of M of such p. 
 
 B of any given p is the farther of three means from half the 
 square of the reciprocal of A of a like p to ^ and D of any 
 given p is the farther of three means from half the square of the 
 reciprocal of M to ,,. 
 
 Hence, a mean between half the square of the reciprocal of C 
 and ^, is the adverse of C, and a mean between half the square 
 of the reciprocal of M and ^, is the adverse of M. 
 
 But I will not further multiply terms, as any one can multiply 
 them at pleasure, and discover a vast variety of polygonal coin- 
 cidences and reciprocities, very many of which I have examined, 
 but have not here even hinted at. 
 
 Let then any denotation which I have signified by a capital 
 letter be final in ^. 
 
 Let A then, be considered final in ^ j tnen B is in ,,, for A 
 and B coincide in ^ ; then C of half the sides is final in , ; 
 so M is final in the point in which C is final ; D coincides with 
 
OF THE CIRCLE. 65 
 
 M of a like p in ^ ; so then D is final in ^ Z is final in ^, 
 because it is a mean between M and D, or C and B. A being 
 in ,2^, hence E is in ^, for I) is half the square of the reciprocal 
 of A ; and the like may be said in respect to C and F, or of M 
 and G ; or thus if d a denote the point in which d and the 
 area of any given p are =, which is always the reciprocal of M, 
 or is d when m is 1 ; if b a denote the point in which b and the 
 area of any given p are =, which is the reciprocal of A of a 
 like p, then D is the farther of three means from half the square 
 of d a to ^ ; hence when D is in ^, half the square of d a is in 
 2, ; when A and B are in ^, half the square of b a is in , ; and 
 when C is in ^, half the square of b a of twice the sides is in 2,. 
 Hence, if any denotation be final in ^, there is then a general 
 coincidence of all the special denotations in ^ excepting d a and 
 b a, which are then reciprocals of ^, and is the diameter of an 
 orbit in which the period is .5. I might have here added another 
 very important denotation, signifying a mean between C and D, 
 (as Z does between M and D,) which denotation may be ex- 
 pressed by Y, which is always the farther of three means from A 
 to ,,, and the square of which is a mean between .5 and A, of a 
 like p ; hence, when A is in ^, Y is in ^, at the same time 
 such Y being a mean between C and D, which are also in ^; 
 so Z in ^ is a mean between M and D when they coincide in %,. 
 
 Hence ^ seems to be emphatically the goal of the spirit land, 
 and deserves far more consideration than the narrow limits which 
 I have assigned to this work will justify ; it being my design to 
 select a few of the dictums and corollaries from amongst those 
 which I have examined and considered, and those too which are 
 least complex, or may be most easily understood. 
 
 But in respect to that focal point denoted by ^, at which all 
 polygonal denotations centre and find their home, it is possible 
 that a little farther hovering around it, by way of " closer and 
 closer contact," may not be inappropriate at the close of this 
 chapter. 
 
 Now if the difference between M and B or C and D be con 
 stantly denoted by _L, (as would be the case by the popular meth- 
 od, save that JL becomes enlarged from the p 8 to the circle, by 
 the popular mode,) there will be no difficulty in determining the 
 relative situation of the denotations of any /?, whenever any two 
 denotations coincide, or those of a p of half the sides, or of 
 twice the sides, &c. For in such case, conception will always 
 place B 1 above M, and D 1 above C. So F will be 1 above 
 M, and G will be JL above A, by conception ; and such is the 
 conception in the finale, or in the circle. Nevertheless, the popu- 
 lar determination furnishes as ample proof as can be desired 
 
66 ON THE QUADRATURE 
 
 that the progressive ratios of M and D, of A and B, of C and 
 B of twice the sides, of D and Z, of B and Y, &c. &c., have their 
 point of coincidence in ,,. 
 
 Let M and D then coincide in ^, and we conceive B to be L 
 above M, and D JL above C ; hence, when M and D are in , B 
 and C are adverse. But as C is = A of twice the sides, hence, 
 C of the p whose M is in ^, is a mean between ^, and C of 
 half the sides ; that is, C of half the sides will be conceived to 
 be J. below ^, and as D is J_ above C, hence, C of the p whose 
 M is in ^ is = D of a p of half the sides, (and here I will 
 make the allegation, that D can never be above C of twice the 
 sides, nor can M be below B of twice the sides.) 
 
 Thus we find D to be 1 below ^ an d as tL tne farther 
 of two means from M to D, hence M 1 is conceived to be 1. 
 above ^, or the adverse of C 1. 
 
 Hence, M and D of the p 2 is in ^, and B and C of the same 
 p are adverse, and M and C of the p 1, are adverse ; and hence it 
 is easy to assign all the relative denotations of these two poly- 
 gons. 
 
 In such case, as we still conceive the difference between B and 
 M, or D and C, to be the same as in the p 8, (namely, denoted 
 by J_) we necessarily conceive M and C of the p 2 to be sepa- 
 rated by the same amount as M and B, or C and D. 
 
 Hence, if we suppose such difference to be reality, in lieu of 
 mere conception, then the progression must be carried on, ad in- 
 finitum ; and it is quite manifest that such fancied progression 
 would have a termini nearly (though not quite) in the two points 
 of the popular termini ; not quite, I say, because, from the co- 
 incidence of M and D in ^, the popular law of progression must 
 be altered from what it has previously been, as is quite manifest, 
 for after M and D coincide, C and D must necessarily thereafter 
 progress by the same ratio ; so also must M and B, &LC. ; and a 
 mean between M and C thenceforward will be M of twice the 
 sides ; besides, if M and D descend by the same ratio, they will 
 converge towards each other, and hence, on this account alone, 
 will place the conceived termini of the final M above the popu- 
 lar place. This is a portion of the subject that may be exempli- 
 fied to great extent, but I forbear. 
 
 Y is the constant mean between C and D, and is the farther 
 of three means from A to ^ ; hence, when Y is in ^, A is also 
 in ^, and consequently, B is in %,. 
 
 Z is the constant mean between M and D, and between C and 
 B, and when M and D are in ^, Z is a mean between them ; 
 or when Z is in %, f M and D, as also the area of such major p 
 are in %,. Thus when Z is in ^, it is = M, or the area of such 
 major;?; and is = D of a like;?. 
 
OF THE CIRCLE. 67 
 
 So when Y is in ,, it is = A and B of such jo, for Y is the 
 farther of three means from A to ^ ; and A is = C of half the 
 sides until A and C become =, namely, until they become = M. 
 
 Hence A in ^ is = C, for Y cannot be = C of half the sides ; 
 and when Y is = C, it is also = D, for Y is a mean between C 
 and D. Hence C in , is = M. 
 
 We shall all agree that if the polygonal denotations expressed 
 by capital letters, be considered in the nature of flowing quantities, 
 from the^4, or rather from the pS, preserving the same relative 
 situation to each other, (viz. those descending and also those as- 
 cending,) until they commence forming coincidences in ^, that 
 Y and Z will coincide at the same time (if not at the same 
 place) that M and C do. 
 
 We shall all agree that the square of Z is always a mean be- 
 tween .5 and M, and that the square of Y is always a mean 
 between .5 and A ; consequently, when the square of Y becomes 
 a mean between .5 and C, C is = M ; and if such be in ^, then 
 all the denotations will coincide together in ^. Hence the place 
 of the coincidence of Y and Z is important. Y commences 
 from A of the p 8 ; and Z, from B of the p 8, and the popular 
 conclusion is, that they coincide below ^, that A, C and M 
 coincide below ^, that B and D coincide above ^ an d E, F 
 and G | above ^. So the popular conclusion is, that the deno- 
 tations on their passage to their final coincidences, often salute 
 each other and pass on ; thus M, Z, D and G coincide in ^, but 
 each is destined to pass on in order to form other coincidences. 
 So also M, and B of twice the sides, (according to the popular 
 method, will coincide f above ^, or the adverse of the popular 
 coincidence of M and C, with others formed, and again dispersed 
 in the vicinity of ^. But, by the popular mode, not a single 
 final coincidence is in ,. Nevertheless, if Y and Z were to co- 
 incide in ^, all the other denotations would coincide in ,. Hence, 
 if either denotation be final in ^, all the denotations would 
 coincide in ^ J in which case the capacities of the linear meas- 
 ures of the circle will be the same as those of the p4, and that 
 they are so, is certainly sustained by too much evidence, both 
 general and special, to be doubted or denied. 
 
 Our principal object is, to ascertain the point in which C 
 shall become = A, or when C shall become M, viz. when all 
 surrounding area shall be excluded from A, of the circle, by the 
 periphery ; or perhaps, after the manner of Leibnitz, that nu- 
 meral position in which the curve of the circle shall osculate all 
 the angles contained as well in the perimeter of the circumscribed, 
 as of the inscribed polygon. That is, when a mean between Y 
 and Z, (which, as in other cases, is always Y of twice the sides,) 
 
DO ON THE QUADRATURE 
 
 becomes extinguished by the coincidence of Y and Z, the square 
 of either Y or Z will be a mean between .5 and the osculating 
 point, or point in which C becomes = M, or is merged in M. If 
 then Y final be in ^, its square is a mean between .5 and A, 
 which is then = M ; its square is also a mean between .5 and C, 
 which is then = M. 
 
 A polygonal progression, then, does not (as has been sup- 
 posed,) depend upon an alternate succession of equal and rational 
 means ; but, on the contrary, from the p 8 onward to the circle, 
 namely, throughout the whole progression through the secondary 
 polygons of the series, it proceeds wholly by rational means; 
 in which, what may be called a denotation of twice the sides of 
 its fellows is always a mean between them. Thus, b a 8 is a 
 mean between b a 4 and d a 4 ; b a 16 is a mean between b a 8 
 and d a 8, and so on, ad infinitum. So b a 8 is = b 4, when m 4 
 is 1, and so on to a coincidence in the third or cube root of 2. 
 So a mean between E and G is E of twice the sides, unto a 
 coincidence of E and G ; a mean between B and 1) is B of twice 
 the sides, ad infinitum, and the like is the case in respect to Y 
 and Z. 
 
 Such being the premises, the conclusion need not be mis- 
 taken. 
 
 As a kind of synopsis to the quadrature, (upon which too 
 much evidence cannot well be adduced,) it may not be improper 
 to offer it in a dress or version somewhat new. 
 
 Thus b a may denote the point in which b and the area of 
 any given p are =; da may denote the point in which d and 
 the area of a p are = ; M a may denote the point in which m 
 and the area of a p are = ; (which point has heretofore been 
 denoted by M.) C denotes a mean between M a and A. 
 
 A mean between b a and da of any given p is = b a of a 
 p of twice the number of sides ; hence, a mean between b a and 
 da of any given j9, is the reciprocal of a mean between A and 
 M a of a like p. 
 
 B is the farther of three means from E, or half the square of 
 b a to 2,, an d D is the farther of three means from G, or half 
 the square of d a to %, j Y is the farther of three means from A 
 to 3, and is a mean between C and D ; and Z is the farther 
 of three means from M a to ^, and is a mean between M a 
 and D. 
 
 Thus, a mean between d a and b a is b a of a p of twice the 
 sides, until such mean between d a and b a no longer exists, or 
 until b a is merged and extinguished in da of the circle, 
 namely, until there is no longer da or b a of a p of sides. 
 
 When d a and b a coincide, it is manifest that A and C be- 
 
OF THE CIRCLE, 69 
 
 come merged and extinguished in Ma of the circle ; for A is 
 the reciprocal of b a, and C is the reciprocal of a mean be- 
 tween b a and d a. And the point in which b a becomes merged 
 in D , is the reciprocal of the point in which A and C become 
 merged in M a by any system. 
 
 A mean between E and G is = E of twice the sides, until 
 E becomes merged in G. A mean between B and D is = B 
 of twice the sides, until B is merged in D. And a mean 
 between Y and Z is = Y of twice the sides, until Y is merged 
 in Z. 
 
 Hence, we may very properly consider, that on arriving at the 
 quantities and measures of the circle, all means between similar 
 denotations are extinguished, or do not exist. Hence, that in the 
 quantities or measures of the circle, there is neither b a, E, B, 
 A, C, or Y, as all those (while they exist) denote polygons of 
 sides. 
 
 Nevertheless, in the final event, the only proper quantities and 
 measures of the circle, for which we are seeking, are those ex- 
 pressed by M a and D ; and the progressive ratios of M a and D 
 are found, by any system, to coincide in ^. 
 
 The object of research, however, is to ascertain the point in 
 which b a becomes extinguished in d a, for in the reciprocal of 
 such point A becomes merged or extinguished in Ma; and in a 
 like p, namely, in the circle, B becomes extinguished in D. Y in 
 Z, and E in G. 
 
 So long as b a exists, we necessarily conceive of a mean be- 
 tween b a and d a ; and the like is the case in respect to E and 
 G, B and D, A and M a, or Y and Z. Hence, we may as well 
 conceive that d a coincides with b a of twice the sides, as with 
 b a of a like number of sides with da; for d a forms a coin- 
 cidence as much with one as with the other. Hence, if we say 
 that da coincides with b a of twice the sides, it is also a coin- 
 cidence with b a of the same number of sides with da, (if we 
 may speak of number where there is no number,) and the 
 like is true in respect to D and B, of Ma and A, of G and E, 
 or Z and Y. 
 
 Now A 8 is JL below half the square of d a 8, and M a 8 is 
 JL below half the square ofba of twice the sides ; and such is, 
 by the popular determination, conceived to be the relative situa- 
 tion of A and M a, onward to the circle, in which case such JL is 
 supposed constantly to increase in value. 
 
 We have retained, then, as measures or quantities of the 
 circle, (after the extinguishment of E, B, C, A and Y,) those 
 of Ma, D, Z and G ; and, by the popular method of considering 
 the subject, as well as by any other method that may be de- 
 
70 ON THE QUADRATURE 
 
 vised, those four proper measures, or quantities, of the circle, 
 coincide together in ,. That is, all the proper measures of 
 the circle, situated below unity, necessarily coincide together 
 in ^; hence, if either proper measure of the circle is in ^ 
 the others are also. 
 
 But again, C, or a mean between A and M a, coincides with 
 B of twice the sides of such Ma; and we also say that D coin- 
 cides with B of twice the sides. But, by the popular method, 
 one of these coincidences would be in ^, while the other would 
 be i above ^. So M a, D, Z and G, (the proper measures of 
 the circle,) coincide in ^, and A, B, Y and E, (extinguishable 
 quantities,) are conceived to coincide together in ^. But A, B, 
 Y and E are as well means as principals ; and hence, M a, D, Z 
 and G will coincide in ^ ; A, B, Y and E will coincide in ^ ; 
 and a mean between A and M a, a mean between B and D, a 
 mean between E and G, and a mean between Y and Z, coin- 
 cide together in ,; and this upon the simple principles of po- 
 lygonal progression, which all may readily examine, and of the 
 truth of which all may satisfy themselves ; and such phenomena 
 cannot exist, unless ^ be the point in which the proper measures 
 of the circle coincide, and in which all other polygonal quanti- 
 ties cease to exist. 
 
 Nevertheless, when we consider ^ as the point in which a 
 mean between A and M a, between B and D, between E and G, 
 and between Y and Z coincide, we necessarily, in our concep- 
 tion, transpose the terms of half the sides, and imagine those 
 which have previously been above %, to nave passed below ^, 
 and those which have been previously below ^ to nave passed 
 above ^ ; which necessarily arises from our conception, that 
 there is a progression so long as we retain the terms denoting 
 polygons of sides. 
 
 Thus, if we conceive such means to coincide in ^ that is, if 
 we conceive C, as a mean between A and M a, to be in ^, Ma 
 will be conceived to occupy the identical place that A does, 
 when Ma is in ^ ; and A will appear to occupy the place of 
 Ma of half the sides of Ma in ^ ; B will be in the place of C, 
 when M a is in ^ ; and D will be in the place of B, when Ma 
 is in ^, &c. But this is but conception, founded upon that 
 which still conceives the progression to be continued ; which, 
 if true, would place M a below A or C, B below D, E below 
 G, and Z below Y ; and such a phenomena no one expects 
 to see. 
 
 When Ma and D are in ^, such Ma or D is half the square 
 of da; and when A and B are in 3> such A or B will be half 
 the square of b a. 
 
OP THE CIRCLE. 71 
 
 And if we conceive C to be in ^, namely, a mean between 
 A and M a, such C will be conceived to be half the square of a 
 mean between d a and b a ; in which case, such d a will be 
 conceived to be above b a, being an event which cannot occur. 
 
 Let us suppose ^ to be the termini of an infinite series, for 
 the progressive ratios of M a and D, and of A and B; B 4 and 
 M4 are in 1, D 4 in ^, and A 4 in ^ If we progress, by 
 successive means, between M a 4 and ,,, and between D 4 and 
 ,2,, the termini will be in ^; or if we proceed, by successive 
 means, between B 4 and , and between A 4 and ^, the termini 
 will also be in %,. But in such case, we conceive the last step 
 in the progression, from D4 to ^, to be one step in ad- 
 vance of a like number of steps, in the progression from A 4 to 
 3- That is, in the coincidence of D with ^, we conceive A to 
 be one step behind or below ^, which step we conceive to be 
 JL below ^. And hence we conceive, that when D (and conse- 
 quently M a) is in ^, B of the same number of sides is J_ above 
 , ; being thus transported, by our conception, from the fact, 
 that in proceeding from M a 4, or from B 4, we have progressed 
 or travelled in the same, or in the identical footsteps in both 
 cases, and also by the same number of footsteps. And in such 
 progression, from M a 4 to ^and from D 4 to ^, or from B 4 
 to , and from A 4 to ^, the results will be the same, whether 
 we progress by a constant mean or by the farther of two, the 
 farther of three, and so on, ad infinitum, until numbers or means 
 are exhausted, or to an actual coincidence. Nevertheless, the 
 most natural method, and at the same time that affording the most 
 ready explanation and evidence, is probably that proceeding by 
 the farther of two means. 
 
 In such case, (as by any other progression,) when the progres- 
 sive ratios of M a and D coincide in ,,, we conceive A to be 
 below ,, and B J_ above ^ 5 nevertheless, we have passed or 
 progressed by the same steps, in respect to B as to M a, from 
 their coincidence in the p 4, and hence the farther of two means 
 arrives as soon in ^ in one case as in the other. 
 
 And to aid us in a proper conception of the matter, the econ- 
 omy of numbers, and that of quantities, are so curiously adjusted 
 to each other, that we need not remain in much doubt ; for if we 
 conceive another step of the progression in respect to A and B, 
 we shall find A and B in their vanishing points according to the 
 popular determination, that is, in the points in which they 
 become merged or extinguished in M a and D of the circle ; for 
 by such additional step, we conceive A to be f below ^, and B 
 i above , or in the popular M a and D of the circle. 
 
 The utmost then, which can be claimed, is, that when M a 
 
72 ON THE QUADRATURE 
 
 and D coincide, the very next step in the progression, applied to 
 A and B, extinguishes them. And the same remark applies to 
 Y and to E, for by such progression, when M a and D are in ^, 
 (and consequently Z and G,) the very next step in the progres- 
 sion, applied to Y or to E, places them in Z and G final, accord- 
 ing to the popular notion of their final positions. 
 
 And because G and Z are in , when M a and D are contained 
 in it, and because E and Y coincide with A and B in ,,, and 
 are extinguished in the same p as are A and B in our present 
 consideration of the subject, we may dispense with G and Z, as 
 also with E and Y, and merely consider C as a mean between 
 A and M a. Thus in the foregoing example of a progression 
 from M a or B of the p 4, or from A and from D of the p 4, if 
 we adopt a simple mean in lieu of the farther of two means, 
 then when M a and D are in ^ if we conceive A and B to 
 take another step in the progression, we shall conceive it to place 
 A JL below ^j and B ^ above ,. But if, when M a is in 3/> the 
 quantity denoted by i_ is already extinguished, except in the 
 imagination or conception, then A and B are in ^ when M a 
 and D are ; in which case, such A and B of twice the sides will 
 only be in their respective vanishing means. 
 
 When either B or D coincide with a mean between B and D, 
 it is but the coincidence of B and D, or rather the extinguish- 
 ment of B as well as that of a mean between B and D ; notwith- 
 standing we may still conceive both B and such mean to exist; 
 and the like is true in respect to A, and M a, &c., in which case, 
 even such extinguished mean will be conceived to be B or A of 
 twice the sides ; for from the p 8 onward, B is but a mean between 
 B and D of half the sides, and A is but a mean between A and 
 M a of half the sides ; and when those means are extinguished, 
 M a and D only remain as proper quantities of the circle. 
 Hence, as C is but a mean between A and M a, when such mean 
 is extinguished, C is also extinguished, or is merged in M a. 
 
 Thus the extinguishment of A and B is the extinguishment 
 of flowing polygonal quantities, or rather of quantities denoting 
 polygons of sides ; and we must conceive them to flow until 
 those quantities which necessarily denote polygons of sides 
 are extinguished, and until we can have no conception of a 
 farther progression, or until any farther attempt at a progression 
 will apparently cause a retrograde movement. 
 
 Now the progressive ratios of A and B coincide in ^, by any 
 system, and if ^ be the point of their extinguishment, it is then 
 the point of a general coincidence ; and it is also, clear, that we 
 cannot conceive an extension of the progression beyond the 
 coincidence of A and B, for an attempt at a farther progression 
 
OP THE CIRCLE. 73 
 
 gives an apparent retrograde movement, and transposition of 
 terms. 
 
 Thus, if A and B coincide in ^, we conceive M a of a like 
 p to be L below , (going upon the ground that from the p 8,' 
 or even from the p 4, M a shall be constantly denoted by J_ be- 
 low B.) But A in ^, as in any other case, is a mean between 
 A and M a of half the sides ; or, as has all along been consider- 
 ed, is = C of half the sides. 
 
 C in ^, then, denotes a p of half the sides of that whose A is in 
 ,2,, and C is a mean between A and M a of the p which it de- 
 notes, and C in ^ denotes a p whose M a is A below ^ and 
 whose A is A above ^, (occupying the identical places. of Ma 
 and C of the p of half the sides of that whose M a and D are 
 
 in.) 
 
 Thus where A and B coincide, (or appear to,) we cannot con- 
 ceive of a p of twice the sides, for polygonal quantities denoting 
 sides, are then extinguished, and if we conceive a p of half the 
 sides, (namely, the p whose C is conceived to be in ,,,) we find 
 A to be JL above ,,, namely, ^_ above A of twice the sides ; 
 which phenomenon will not be required by any one. 
 
 Thus the final or extinguished mean between A and M a, 
 and the final or extinguished mean between B and D coincide 
 in ^ ; and until such apparent coincidence, we may conceive a 
 continuance of the progression. 
 
 Then considering C to denote simply a mean between A and 
 M a, when C is conceived to be in ,,, it is but the point of its 
 extinguishment ; namely, the point of extinguishment between 
 A and M a. Hence, in a general consideration of the quantity 
 denoted by C, its more proper consideration is that of simply a 
 mean between A and M a ; nevertheless, its consequence is 
 vastly important when considered as one of the prime linear 
 quantities. 
 
 Our utmost researches then, are not only extended to extin- 
 guished means in which certain approximating quantities coin- 
 cide, but also to the general coincidence of those extinguished 
 means ; which general coincidence is in ^, by any rational pro- 
 gression that may be devised. And then, and then only, do we 
 find the capacities of polygonal measures to again become the 
 same as they were in the p 4, and hence to be capable of a ra- 
 tional application. 
 
 I have before remarked upon the allegation, which we often 
 find repeated by eminent geometers and mathematicians, that a 
 polygon is, properly speaking, the area, and not its linear meas- 
 ures, as is often supposed ; but I have been rather inclined to 
 adopt the vulgar, or common acceptation, believing it to be 
 10 
 
74 ON THE QUADRATURE 
 
 best suited to the subject, if but one is to be adopted. Never- 
 theless, with a view to a full consideration of the subject, the 
 areas should be considered in connection with their linear meas- 
 ures ; at least so far that, whenever linear measures of polygons 
 are to be compared and proportioned to each other, each linear 
 measure shall denote or express the amount of area of the poly- 
 gon whose linear measure is given, or of a polygon of half the 
 sides, or of twice the sides ; thus B or D always denotes the area 
 to be .5 ; M a or M denotes either the area, or one-fourth of the 
 circumference of a major polygon ; while C denotes one fourth 
 of the circumference of a minor polygon, and also the area of a 
 minor polygon of twice the sides. 
 
 Hence, by the use of M, C, B and D we have as full and as 
 proper an expression or denotation of the areas of the major 
 and minor polygons, as also of all primal polygons, at least from 
 the octagon, (or polygon from which our equation commences,) 
 as though the areas themselves were expressed. 
 
 But the converse of this could not occur, and more especially 
 in respect to diametric quantities of polygons ; for each polygon 
 of sides, while it has but one perimeter, has two diametric quan- 
 tities, which must be considered in our investigations ; hence, 
 while by means of the linear measures we may denote the quan- 
 tities of all necessary areas, we cannot, by means of the areas, 
 denote all the necessary linear measures. 
 
 And to show that a direct use of areas may be dispensed 
 with, and that we may conduct our progression wholly by linear 
 quantities, it is only necessary to allege the fact, that a mean be- 
 tween A and B of any given jo, is also a mean between C and 
 D of the same p ; and hence, when A and B shall coincide, it 
 will be as well in the mean point between C and D, as between 
 A and B. So also, when C and D coincide, it must be in the 
 mean point between A and B of the same p ; and hence we 
 may, in general, adopt C and D in our progression, to the exclu- 
 sion of A and B. 
 
 So a mean between M (or M a) and D is also a mean be- 
 tween C and B of the same p ; hence, when M and D coincide, 
 it as well in the mean point between C and B, as between M 
 and D ; hence, when C and B shall coincide, it must, of neces- 
 sity, be in the mean point between M and D ; hence, in our pro- 
 gression we may, in general, adopt M and D, to the neglect of C 
 and B, regarding, however, the important fact, that C is to D 
 of any given p as M is to B. 
 
 Then the farther of three means from M to ^ (which is a mean 
 between M and D,) will be the mean place of M of twice the 
 sides ; and the farther of three means from C to ^, (which is a 
 
OF THE CIRCLE. 75 
 
 mean between C and D of twice the sides,) will be the mean 
 place of C, or of D of twice the sides. 
 
 Then, when M is in ^, it will be = the mean place of M of 
 twice the sides ; and when C is in ^, it will be = the mean 
 place of C of twice the sides. The consequence is, that when 
 A is in ^, such A is then = the mean place of C of the 
 same p. 
 
 It is, then, equally well to speak of the true and mean places 
 of C and of M, as of A, or of the area of a major p; never- 
 theless, if we adopt M a in lieu of M, we then make the ap- 
 plication of the true and mean places, as well to the area as 
 to M. 
 
 But, in respect to minor polygons, C is = A of twice the 
 sides ; nevertheless, the farther of three means from A is the 
 mean place of A of twice the sides, and the farther of three 
 means from C is the mean place of C of twice the sides, until 
 the progression is ended. 
 
 Thus the true and mean place of A 8 is the same, while the 
 true place of M a 8 is j_ below the mean place ; so also, the true 
 place of C 8, or of A 16, is J below the mean place. 
 
 Y, then, may denote the mean place of C and of D, and Z 
 may denote the mean place of Ma of twice the sides, or of B 
 of twice the sides ; thus B 8 is as well in its mean place as in 
 its true place ; and such construction, in respect to the true and 
 mean places of M a, B, C and D are not only in perfect accord- 
 ance with the law of polygonal progression, but the same is 
 even corroborated by the popular method. 
 
 Thus C is, proportionally, half as far below the mean place of 
 C or D, as Ma is below the mean place of Ma or B ; and C is 
 proportionally as far too low, in respect to the mean place of D, 
 as in respect to the mean place of B. 
 
 Now, while the true place of C 8 is a mean between the true 
 place of A 8 and the true place of M a 8, so the mean place of C 8 
 is a mean between the mean place of A 8 and the mean place of 
 MaS] and the like principle holds throughout the progression. 
 Or thus, the mean place of C is a mean between the mean place 
 of A and the mean place of M a, or of B. 
 
 When A is in ,, such A is in the mean place of C of the 
 same p ; hence such A (as the mean place of C) is a mean be- 
 tween the mean place of such A and the mean place of M a, 
 or B. 
 
 So A in ^ is a mean between its mean place and the mean 
 place of M a, or B ; A in ^ is in the mean place of C, or of D ; 
 and C in ^ is in the mean place of C, or of D of twice the sides. 
 And the like is true of B in ^ as in the case of A ; so M a and 
 
76 ON THE QUADRATURE 
 
 D in %, are i n the mean place of C and B of the same p, or of 
 M a and B of a p of twice the sides. 
 
 Thus infinite wisdom, which has furnished geometry and ma- 
 thematics with the best of reasons, but which never intended 
 that we should survey a field or measure and determine the con- 
 tents of an area, without taking into the account both the cir- 
 cumferential and diametric measures of such area, has, in the 
 present case, as if to invite us to quadrate our ways at once, 
 and not to be eternally attempting to equate them to some sup- 
 posed, but erroneous, rule or law, so contrived and adapted 
 the evidence, as not only to show us the necessity of an equation 
 of the varying capacities of polygonal measures, as we progress 
 from the square to its inscribed circle, but also to dictate, in 
 divers ways, the manner of conducting the same. The neces- 
 sity of this equation can only be detected, by a comparison of 
 the perimetric and diametric quantities of polygons ; nor can the 
 equation be made but by some law of progression, by which 
 those quantities are constantly compared with each other. 
 
 C is to D as M a is to B ; and this is the form in which the 
 progression proceeds, from the trigon to the circle ; nevertheless, 
 in the p 4 the relative situations of C and D are changed, as is 
 also the case in respect to M a and B ; C and D being equal in 
 the p 4, and M a and B being equal in the p 4. But the p 8 pre- 
 sents the variance between C and D, or between M a and B, 
 whichuis denoted by _L, and which requires to be equated to .. 
 in the circle, in lieu of being increased in value as we progress 
 from the p 8 to the circle, as is the case by the popular determi- 
 nation. 
 
 It may not, however, be surprising, that the mind should not 
 readily assent to an equation of the difference existing between 
 M a and B (and consequently between C and D,} of the p 8 to 
 JL, on arriving at ^, but rather that it should conceive the same 
 ratio to exist, as we progress onward from the p 8 ; and hence, 
 when the progressive ratio of M a is in ^, that the same ratio ex- 
 ists between M a and B as existed in the/* 8 ; and hence, the actual 
 numerical difference between M a and B, when M a is in ^, 
 would be but very little less than it is in the p 8. And such 
 would be the case, were toe capacities of the linear measures of 
 all the polygons situated between the p 8 and circle, the same 
 as they are in the p 8 ; but not so if, from the p 8 towards the 
 circle, the capacities of the measures of polygons are again ap- 
 proximating to what they were in the p 4, from whence we com- 
 menced our progression. 
 
 But having found in the p 8 (or in the first step in the progres- 
 sion,) a difference between M a and B, or between C and D r 
 
OF THE CIRCLE. 77 
 
 and finding the law to be universal, that M a is to B as C is to 
 D, and, consequently, that C is to M a as D is to B ; the mind 
 which is imbued with the least tincture of ratio and proportion 
 will naturally attach to any sign, denoting such difference in the 
 p 8, the conception that it is a sign denoting ratio and proportion, 
 and that, as such, it will accompany the polygonal progression, 
 and that it will constantly be denoted by the same character or 
 sign as is used for the p 8, which, if it were simply to denote 
 ratio, would only become extinguished in ; whereas, by a pro- 
 per equation of the capacities of polygonal quantities, it necessa- 
 rily becomes extinguished in the circle. 
 
 The consequence of such natural or rational conception of the 
 mind is, that when M a and D (which are in advance of their 
 mean places from the p 8 onward) coincide in ^, or in their 
 mean places, B is still conceived to be J_ above M a, and C 
 1_ below D ; .thus conceiving the difference of A between C and 
 B, notwithstanding they actually coincide with Ma and D 
 
 itt& 
 
 Such conception, however, is not unprofitable, as it enables us 
 to give (in a formula, if we please) the relative situations of the 
 polygonal quantities and measures, not only of the p whose M a 
 and D coincide in ^, but also of the p of half the sides, or of 
 twice sides ; for it will be recollected that the circle is composed 
 of a p of half the sides and a p of twice the sides, as well from 
 the minor as from the major polygons ; which phenomenon is 
 the result of a demonstration, the truth of which is readily per- 
 ceived. 
 
 It may not be inappropriate here, to suggest something farther 
 in reference to the measures and quantities of the p 8, and of the 
 important positions which they hold in polygonal economy. 
 Thus, the true places of A and of B of the p 8, are also their 
 mean places, and A 8 is the square of B 8. And while B 8 is 
 a mean between B of any given p, and the square root of A of 
 such p ; or between D of any given p and the square root of M a 
 of such j?, A 8 is a mean between B and the square root of Y of 
 any given p, viz. between B of any given p, and the square of 
 the mean place of C or D of such p> or of the mean place of A 
 of a p of twice the sides. 
 
 Hence, if B of a p be in ^ A 8 is the mean between such B 
 and the square of the mean place of C or of D, or of A of twice 
 the sides. 
 
 But B in ^ is = A, by any system, as also by the law of 
 polygonal progression, and is a mean between C and D of the 
 same p, or is in the mean place of C or D of the same p, and 
 is consequently in the mean place of A of twice the sides ; but, 
 
78 ON THE QUADRATURE 
 
 as I have before remarked, the place of A of twice the sides is 
 nowhere to be found. 
 
 So we say that A in , is = C of half the sides ; consequent- 
 ly, C in , is = the mean place of C or of D of twice the sides, 
 and A in ^ is = the mean place of A of twice the sides. And 
 because M # in ^ is = Z, hence, M a in ^ is = the mean place 
 of M a, or B of twice the sides. The consequence is, that if we 
 progress upon the mean place of M a and B, and upon the mean 
 place of C and D, they will eventually coincide with the true 
 places, and with each other. As the mean place of M a or of B 
 is a mean between M a and D, or between C and B, namely, 
 the farther of three means from M a to ^, hence, if our progres- 
 sive steps are upon the successive mean places of M a or B, we 
 can only arrive in , by an infinite series ; and as a mean be- 
 tween M a and D is also a mean between C and B, perhaps no 
 one will contend that an infinite series will cause a coincidence 
 of M a and D sooner than it will the coincidence of C and B, 
 or, that an infinite series of progressive steps will sooner ex- 
 tinguish a mean between M a and D, than between C and B. 
 
 Hence, C and B must coincide at the same time, and in the 
 same point with M a and D. And as a mean between A and B 
 is also a mean between C and D, an infinite approximation will 
 cause A and B and C and D to coincide at the same time, and 
 in the same point ; and in respect to the coincidence of A and 
 B, or of M a and D, the same is in %,. 
 
 But M a, D, G and Z finally abide, and denote the area and 
 linear measures of the circle ; and when they coincide together 
 in ^, (which occurs in accordance with any law of polygonal 
 progression,) the true and mean places of M a coincide in ^. 
 
 A, B, E and Y always denote polygons of sides, and hence 
 are extinguishable quantities, or extinguish able means ; for while 
 they are considered in the nature of quantities, they are also con- 
 sidered as mean proportionals, and, whether as quantities or 
 means, they are conceived to be progressing towards their final 
 extinguishment, or to a merger in the abiding quantities ; and 
 hence, when they cease to exist as quantities, they also cease to 
 exist as means ; and by the law of progression, A, B, E and Y 
 must apparently coincide in ^ ; and if either one of them is ex- 
 tinguished in ,,, they all are ; we nevertheless conceive them to 
 exist (both as quantities and means) in the point of their extin- 
 guishment ; and hence we reject the conception of their final ex- 
 tinguishment, even in the abiding quantities which finally denote 
 the circle ; nor can such conception be relieved, except by the 
 reductio ad absurdum. 
 
OF THE CIRCLE. 79 
 
 
 SECTION FIFTH. APPLICATION. 
 
 In making a short application of my determination of the 
 measures of the circle to that of solids, I shall go no farther than 
 a short comparison with the prime sphere, namely, the sphere 
 whose diameter is unity, and its circumscribed cylinder and 
 prisms, amongst which I place the cube ; nor need I go farther 
 than to compare the prime sphere with its circumscribing cylin- 
 der and cube. 
 
 What is often termed the volume or contents of a solid, I shall 
 call by the more appropriate name of bulk. 
 
 The surfaces of regular solids are as their bulks. 
 
 The bulk of the sphere is demonstrated to be two-thirds of the 
 bulk of its circumscribing cylinder ; and the surface of a sphere 
 is = to the lateral surface of its circumscribing cylinder. 
 
 The surface of the prime sphere, or of its circumscribing cyl- 
 inder, or any circumscribing prism, is six times the bulk of the 
 given solid. 
 
 The surface of a regular solid flows by a duplicate ratio, and 
 the bulk by a triplicate ratio to that of any given linear measure 
 of such solid. 
 
 The circumscribing cube of the prime sphere properly de- 
 notes unity amongst the solids, as the square does amongst the 
 polygons. 
 
 The circumference of the prime sphere is = the surface; 
 hence, if the diameter of a sphere be expressed in a term of the 
 prime table, any given power of the circumference will be ex- 
 pressed in terms of the table, as also of the surface, 
 but not of the bulk. Nevertheless, tables may be readily made 
 upon the same ratio between terms of the table, as that of the 
 prime table, in which the bulks of spheres may be expressed 
 when their surfaces are expressed in the prime table, and in 
 which the surfaces of cylinders and their bulks may also be ex- 
 pressed ; and hence the whole may be thrown into rational and 
 tabular form. 
 
 It is not my intention here (like Montucla) to enter into the 
 sports of mathematics, nor to make sport of those things which 
 I do not comprehend ; but I have before suggested that number 
 8, which has much to do with the octagon and its infinity of 
 equal means, was not destitute of beauties ; it is the sum of all 
 the powers of .8888, ad infinitum, which is the square root of the 
 revolving series .790123456790, ad infinitum, which series does 
 not appear to have a drop of octagonal blood in it. And if either 
 series be multiplied or divided by any natural number or definite 
 
80 ON THE QUADRATURE 
 
 quantity, the result will be a continuous, uniform, or revolving 
 series, by which an infinity of equal means may be produced as 
 exponential tables. 
 
 These revolving series are deduced from what may be termed 
 the equivalents of the natural numbers in their order, of which I 
 will give one or two examples in connection with a suggestion in 
 respect to the law of falling bodies. And although I may here- 
 after give a more extended numerical formula in respect to the 
 law of falling bodies, than the one I here give, and one more 
 fully showing the economy and adaptation of numbers to our 
 every necessity, yet, perhaps one of the simplest formulas that 
 may be given for the purpose is the following : 
 
 11111111 
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
 
 Thus if we take the natural numbers in their order, to the 
 number fifteen, and divide them into periods for extracting their 
 square root, making the figure 1, at the left, the first period, the 
 square root will come out in the constant figure 1, over the last 
 figure in each period, namely, over each odd number, making 
 eight figures in such root, and the sum of all the odd numbers 
 thus far will be 64, or the square of the sum of all the figures in 
 the root, and so on, ad infinitum, le't the natural numbers be ex- 
 tended ever so far. And in the extraction of such root, the re- 
 mainder will constantly come out in terms of the even numbers 
 in their order, as 2, 4, 6, 8, 10, 12, 14, &c. Thus while each 
 figure of the root denotes one moment of time, each odd num- 
 ber denotes the number of spaces passed over in each successive 
 moment, and the successive even numbers denote the final ve- 
 locity of the body at the end of each successive moment ; and 
 the total spaces passed over at the end of each successive mo- 
 ment, will be the square of the number of moments the body 
 has been falling. 
 
 Hence, the whole space passed over is the square of the whole 
 time of the fall ; and as the force which urges the body in its 
 fall is constant and uniform with the time, hence, in a falling 
 body, the whole space passed over from the commencement of 
 the fall, is as well the square of the whole force, as of the time, 
 which is too important a matter to be overlooked in our investi- 
 gations of the laws of force and motion, although I apprehend 
 it has not always been sufficiently regarded. 
 
 This, however, is one of the numerical beauties of numbers, 
 applicable to our wants, but we should not seek to extend it be- 
 yond its legitimate application. 
 
 But to show that this is a mere numerical dress of the law of 
 falling bodies, I will present the natural numbers in their order, 
 
OP THE CIRCLE. 81 
 
 or what is equivalent thereto, in another dress or form, and which 
 may be extended ad infinitum, and which will be found to have 
 some application to the measures of the circle, and of the round 
 bodies. 
 
 Thus, if we take the natural numbers in their order to any ex- 
 tent, and multiply the same by 2, and then divide the product by 
 2, we obtain a revolving series, ad infinitum, which may be 
 said to be equivalent to the natural numbers in their order; which 
 series will come out 1, 2, 3, 4, 5, 6, 7, 9, 0,1, 2, 3, 4, 5, &c., in 
 which the figure 8 will never occur. 
 
 If we then divide such series into periods for the extraction of 
 the square root, making figure 1, at the left, one period, the root 
 will continue to be expressed by figure 1, over the last figure in 
 each period, ad infinitum, and the remainder will constantly come 
 out in terms of the even numbers, or what is equivajent thereto, 
 namely, 2, 4, 6, *9, 1, 3, 5, 8, 0, 2, 4, 6, 9, 1, 3, &c., which equiv- 
 alent of the even numbers is produced in the same manner as 
 that of all the natural numbers. So also an equivalent to the 
 odd numbers is produced in the same manner, and will revolve 
 thus : 1, 3, 5, 8, 0, 2, 4, 6, 9, 1, 3, 5, 8, 0, 2, &c. Which equiva- 
 lents of the natural numbers in their order, when applied to the 
 measures of the circle and of the round bodies, are not only use- 
 ful, but their application is extremely beautiful ; but they do not 
 so well apply as formulas to express the law of falling bodies 
 near the earth. 
 
 But to return to a farther application of the circle. When the 
 diameter of the inscribed sphere of the cylinder is = d of the 
 circle whose m is 1, the bulk of such cylinder is the square of the 
 diameter of the inscribed sphere. T-ie circumference is four, and 
 the surface will be six times the diameter of its inscribed sphere. 
 But the proper diameter of the cylinder or cube, is the diameter 
 of its inscribed sphere. If the diameter of the cylinder be the 
 sixth root of six, its surface is six, and its bulk is = its diameter. 
 
 But this is the general law : wher the diameter of the sphere, 
 cylinder or cube is the reciprocal of ' e bulk of a like prime solid, 
 the bulk is the square of the diameter. Hence, when the diam- 
 eter of the sphere is 1.889881, or th?rd root of 6.75, the bulk is 
 the square of the diameter, and the \rface is six times the diam- 
 eter, and the circumference is 6. So when the diameter of 
 such solid is the third root of the reciprocal of the bulk of a like 
 prime solid, the bulk is 1. The reciprocal of the bulk of the 
 prime sphere is three times the square of M of the prime circle, 
 or is three times three halves of M 01 the prime circle, and the 
 square of such reciprocal of the bulk of the prime sphere, is the 
 third root of the square of M 3. 
 11 
 
82 ON THE QUADRATURE 
 
 The square root of the reciprocal of the bulk of a prime solid 
 is the point of equality between the bulk and diameter, and the 
 third root of the square of such point of equality is the diameter 
 of a like solid whose bulk is 1 ; and such point of equality in 
 the prism of three lateral sides is == B 6. 
 
 .888, ad infinitum, is six times the third power of the bulk of 
 the prime sphere, and is = the area of the circle whose m is "jp 
 above .666, ad infinitum^ and is three halves of the bulk of the 
 prism of three lateral sides, whose diameter is = d 3, when M 3 
 is 1, &c. 
 
 The reciprocal of the bulk of any prime solid is = the bulk of 
 the circumscribing cube of a like solid whose bulk is 1. But I 
 will not pursue the subject farther ; the object being simply to 
 show the reciprocities and coincidences, and the mathematical 
 beauties in which solids seem inclined to clothe themselves. 
 
 The father of true physical philosophy has .shown that the 
 squares of the periods of the planets are as the cubes of their 
 mean distances from the sun ; and wishing to be justified or 
 corroborated in my determination of the quadrature, by that great 
 physical law in regard to force and motion, I will here make a 
 reference of the consequences of my determination, to the two 
 great Keplerean laws ; and, in such reference, shall (for the pres- 
 ent purposes) consider orbits of different dimensions as being 
 circles of different dimensions ; premising that while the circle 
 whose diameter is unity is the prime circle, the orbit whose ra- 
 dius is unity is properly the prime orbit ; and this for reasons 
 that will readily be recognized. And I will here remark, that in 
 general we have as little to do with the area of an orbit in the 
 investigation of the laws of force and motion, as we have with 
 the areas of polygons in the investigation of the measures of the 
 circle. 
 
 Unity is the standard measure for anything susceptible of nu- 
 merical admeasurement, or of ratio and proportion ; hence, when 
 the radius of an orbit, or the mean distance of the planet is 1, the 
 period is 1 ; else the square of the period could not be as the 
 cube of the distance ; so when radius is 1, the force of gravity 
 must be called 1, for such is the only condition upon which 
 we can proportion the rate of gravity to different distances from 
 the same attracting power. So also, at the distance 1, the mean 
 rate of motion is 1, as also the rate of convergency of the orbit, 
 and especially the time of the period. 
 
 I will here assume the fact, that if the diameter of the inscribed 
 circle of any given polygon be = to the square of the circum- 
 ference of a like major polygon, the circumference of such poly- 
 gon will be the third power of the circumference of such major 
 
OP THE CIRCLE. 83 
 
 polygon ; and the same will hold good in respect to the circle, or 
 orbit. 
 
 Hence, when the diameter of an orbit is = the square of the 
 circumference of the circle whose diameter is 1, the circumfer- 
 ence of such orbit will be 32, or one fourth of the circumference 
 will be 8. 
 
 Or thus : when the radius of an orbit is four times the square 
 of the circumference of the prime circle, the square of the cir- 
 cumference of such orbit will be the third power of radius (the 
 mean distance) of such orbit ; and such necessarily occurs when 
 the circumference of the orbit is 256, in which case, the period 
 will be = the circumference. 
 
 Or the version may be thus : if the circumference of ah 
 orbit be the product of the third power of the circumference of 
 the prime circle, by 8, the period will be = the circumference- 
 for when the diameter of an orbit is 8, the period is = the diam- 
 eter, by any system of the circle. 
 
 So when the diameter of an orbit is 8, the circumference is 
 twice the square of the circumference of the square (or p 4) of 
 equal area with that of the prime circle. 
 
 The third power of the circumference of an orbit is 32 times 
 the third power of the diameter, and 256 times the third power 
 of the radius ; so when the circumference is 256, the square of 
 the circumference is the cube of radius. 
 
 So when the circumference is 64, the period will be 32, and 
 when the circumference is 16, the period will be 4, and so on. 
 So when the circumference is 4, the period is one eighth of the 
 circumference ; viz., the period will be .5. 
 
 Thus, when the radius of an orbit is the third root of any 
 given point, the period of the planet is the second, or square root 
 of the same point. And hence we may conclude that this great 
 law is founded in unity of purpose. 
 
 Suppose then a planet to revolve in a centric orbit, namely 
 with the sun in the centre, and of course having a uniform mo- 
 tion : Then if the force of gravity be inversely as the distance, 
 (as I allege it to be,) the rate of motion will be the square root 
 of the force applied ; but if the force of gravity varies inversely 
 as the square of the distance, as taught by Sir Isaac Newton, 
 then the rate of motion will be the fourth root of the force appli- 
 ed ; as may readily be seen by almost any one who will bestow 
 a little consideration on the subject. The circumferences of 
 circles are as their diameters, or as their radii ; hence, if Kepler, 
 could have supposed that his great law, as by him declared, 
 could have been mistaken or perverted, he might have declared 
 anew, that the third power of one eighth of the circumference 
 
84 ON THE QUADRATURE 
 
 is half the square of the period ; or that the square of twice 
 the period is the third power of one fourth of the circumference, 
 &c.j thus comparing time with circumference, in lieu of di- 
 ameter; and the law thus declared would have been equally 
 true and equally well received. For then, as now, when 
 the circumference is 4, the period will be .0, or equal to one- 
 eighth of the circumference ; and still, when the diameter is 8, 
 the period will be 8, and when the period is 4, it will be = one- 
 fourth of the circumference, or the square root of the circumfer- 
 ence, &c. 
 
 When the period is .5, or the square root of .25, the radius is 
 necessarily the cube root of .25, or the square of 3> an d the di- 
 ameter must be the reciprocal of ^ by any system ; else the 
 Keplerean system must fail. 
 
 This statement, however, will not quadrate with the popular 
 measures of the circle ; and it will be seen and acknowledged, 
 that in applying the popular measures of the circle to the laws 
 of planetary force and motion, it requires a great deal of calculi 
 to chink up with. According to this, the third power of one- 
 fourth of the circumference of an orbit when the diameter is 1, 
 is equal to the period of a planet in an orbit whose circumference 
 is 4; but not so says the popular method. 
 
 By any system, the rate of motion, and the period of a planet 
 have a determinate ratio to the diameter of the orbit ; but accord- 
 ing to the popular method, they have no determinate ratio to 
 the circumferential measure of the orbit; that is, according to 
 the popular notions, neither the motion nor time of the period, 
 have a determinate ratio to that of the line or path in which the 
 planet actually moves or revolves. 
 
 By any system, when the circumference is 4, or when the area 
 is = the diameter of the orbit, the period is the third power of 
 the diameter of an orbit whose area is .5, hence, the period of 
 any planet of a system is = the third power of the diameter of 
 another orbit of the system whose area is =. one-eighth of the 
 circumference of the given orbit. Thus, if the circumference of 
 an orbit be 256, the period is = the third power of the diameter 
 of an orbit wh se area is 32. 
 
 This application may also be made to orbits in connection 
 with polygons, namely, the reciprocal of M of any given poly- 
 gon is the diameter of an orbit in which the period is the third 
 power or cube of the diameter of the circumscribed circle of a 
 like polygon whose area is .5. 
 
 If the limits assigned to this work permitted, it would by no 
 means be an unpleasant task to contemplate the relationship of 
 the family of curves ; and wherein the parabola (that curve of 
 all others except that of the circle, the most useful and impor- 
 
OF THE CIRCLE. 85 
 
 tant to be understood,) conspires with the circle in the admeas- 
 urement of solids, &c., and serves to corroborate my determina- 
 tion of the quadrature. 
 
 It is entirely beyond my sagacity to determine how the ellipse 
 (or elongated circle) has been elaborated into notice and conse- 
 quence far beyond its merits, otherwise than by way of mathe- 
 matical or geometrical discipline to the mind ; and I am equally 
 uninformed why the science termed the science of conic sec- 
 tions, (from wh^ch the modest circle is generally excluded,) has 
 been regarded paramount to any other department ; of which 
 science it is said, by those in high places, that " it forms one of 
 the most important parts of mathematics, being distinguished 
 for elegance, and demonstrating, with surprising simplicity and 
 beauty, and in the most harmonious connection, the different 
 laws, according to which the Creator has made worlds to re- 
 volve," &c. " that few branches of mathematics delight a 
 youthful mind so much as conic sections ; that the emotions 
 which the pupil manifests when they unfold to him the great 
 laws of the universe, might be called natural piety ; and that in 
 teaching this science to young people, the descriptive method 
 (by diagrams) ought always to be connected with the analytic 
 method," &c. 
 
 I am aware that the ellipse, the parabola, and the hyperbola, 
 may be presented in a train of diagrams which in their appear- 
 ance are very imposing ; and that they may serve in connection 
 with the importance which is generally supposed to be attached 
 to their explication, to induce a species of faith or belief, which 
 some would call natural piety. But I have yet to learn that 
 this science discloses the laws according" to which the Creator 
 has made worlds to revolve ; or that, as has also been admir- 
 ingly expressed, in comparison with conic sections, in view 
 of the grandeur and beauty which they unfold to the mind in 
 the mathematical world, all that has preceded this part is but the 
 alphabet of the science of mathematics, &c. 
 
 It is true that the strong grounds thus far, of the Newtonian 
 philosophy, is in the belief that he disclosed, by their aid, the 
 law according to 'which the Creator has made worlds to revolve; 
 and so tenacious have been the sustainers of Newton's fame, 
 that the cry of robbery was raised when Bernoulli made the 
 allegation without giving Newton the credit, that "we now know 
 that the heavenly bodies perform their revolutions by conic sec- 
 tions ;" thereby inducing the world to attach vast importance to 
 what may, at some future day, be shown to be wholly delusive. 
 
 It is somewhat curious to contemplate the means and the in- 
 genuity which have been practised, to induce the natural piety 
 
86 ON THE QUADRATURE 
 
 conceived to flow from a contemplation of conic sections ; and 
 it is certain that those articles of faith have been fortified by 
 great care and caution, and by a continual inculcation ; so much 
 so, that not a word must be suggested, in our works on astron- 
 omy, that it would be possible for a planet to revolve in a circle, 
 whether centric or eccentric ; nor that the ellipse can be cut from 
 any other solid than the cone. Hence, the natural minds of 
 thousands have been so continually plied with the elliptical or- 
 bit, and with the magic qualities of the cone and of conic sec- 
 tions, that at length they have yielded, or, at least, rendered pas- 
 sive obedience. And it would not be wonderful if something 
 strange should come over the mind, when it is brought to assent 
 to the Newtonian hypothesis, that the Creator has caused worlds 
 to revolve in ellipses, however eccentric. 
 
 Sir Isaac Newton settled many knotty questons by his favor- 
 ite expression, of Neque novit natura limitem, which, perhaps, 
 may have various renderings, for it will as well apply to error as 
 to truth ; for when we have set out on the road to error, we can 
 find no bounds or limits to our career ; and such is, doubtless, 
 the case in respect to the hypothesis of elliptical orbits, as I 
 shall attempt to prove, an error which commenced with Kep- 
 ler, while making calculations upon Tycho's observations ; an 
 error which, although not depending upon any physical necessity 
 or law, was seized upon by Newton ; and he has given it such 
 importance, as nearly to obliterate the two great laws of Kepler, 
 and especially that which declares that the squares of the periods 
 are as the cubes of the mean distances, the modern version 
 of which is, that " the squares of the periods of the planets are 
 as the cubes of their semi major axes." 
 
 If such then be the law, namely, that a planet may revolve 
 in an orbit, however eccentric, even to extreme eccentricity, 
 and that the square of the period is the cube of the semi major 
 axis, it is manifest, that during the period of such planet, a 
 vastly greater amount of the force of attraction will be applied 
 to it, than to a planet revolving in a circle, (whether centric or 
 eccentric,) whose diameter is = to the major axis of the ellipse ; 
 while the planet revolving in the ellipse would pass over less 
 than two-thirds of the space of that revolving in the circle, in 
 the same time or period, and a far less proportion of area. 
 Hence, the popular notion makes the time of the period depend 
 wholly upon the major axis of the orbit, without regard to 
 amount of space passed over, or the amount of force expended 
 during the period. And such have been the doctrines and teach- 
 ings which have induced what is termed natural piety, in those 
 who have been induced to conceive that such things could be. 
 
OF THE CIRCLE. 87 
 
 If it shall be found, then, that a planet cannot revolve in an 
 orbit of extreme ellipticity, if we retrace the road which led to 
 such error, we shall find that a planet cannot revolve in an el- 
 liptical orbit, however small the ellipticity. 
 
 I will, however, in conclusion of this reference, suggest a dic- 
 tum or two, based upon the operation of the Keplerean law, 
 namely, that the square of the period is the cube of the distance, 
 and that the radius vector passes over equal areas in equal times ; 
 vconsequently, that the motion of the planet, at any given point of 
 the orbit, varies inversely as the distance varies, which laws, in 
 fact, are one and the same, operating in unity of purpose, although 
 the same has generally been resolved into two separate or dis- 
 tinct laws. This law, then, taken in connection, presupposes its 
 operation in an eccentric orbit, or one in which the central force 
 is not situated in the centre of the orbit. 
 
 This subject, however, has been dark, mysterious and inexpli- 
 cable, in consequence of the Newtonian error, in assigning the 
 law ; and notwithstanding it had become much the fashion im- 
 plicitly to avow that to be true which it was said Sir Isaac New- 
 ton had found, nevertheless, for many years after the promulga- 
 tion of the Newtonian law of gravity, the truth had not become 
 so far entangled in the meshes of error, as to place it wholly 
 beyond the power of redemption, until Clairaut effectually accom- 
 plished it, in his supposed accounting for the motion of the 
 moon's apogee, upon the principle of the Newtonian law of 
 gravity ; for, previous to that time, it had not been deemed high 
 treason for men in high places to speak disparagingly of the law. 
 
 But still the square of the period is the cube of the mean 
 distance, and still the motion varies inversely as the distance 
 varies, and consequently the line of the radius vector passes over 
 = ' areas in equal times ; and, consequently, the simplicity in the 
 laws of force and motion, by which the heavenly bodies revolve, 
 should not have been rendered thus complicated and inexplica- 
 ble. For the above phenomena once being established, directly 
 and conclusively determine the law of gravity, by which the phe- 
 nomena are caused or produced ; that is, the law is directly deduc- 
 ible from its effects ; and why it should not have been done, is 
 somewhat unaccountable. 
 
 I allege, then, that the intensity of gravity is inversely as the 
 distance, and, consequently, that the mean motion of a planet is 
 the square root of the mean force ; (not that the mean motion is 
 the fourth root of the mean force, as Sir Isaac Newton would 
 have it.) 
 
 Then, as the motion of a planet in an eccentric orbit varies 
 inversely as the distance varies, hence the motion, at any given 
 
88 ON THE QUADRATURE 
 
 point of the orbit, will be the square of what the motion of a 
 planet would be, in a corresponding point of a like orbit, whose 
 mean distance is the square root of the mean distance of the 
 given orbit; and this for the reason, that the mean motion of a 
 planet is inversely as the square root of the mean distance ; 
 else the squares of the periods of a system of planets cannot 
 be as the cubes of their mean distances from the gravitating 
 power. 
 
 Hence, if the mean distance be 4, and the perihelion distance- 
 1, the actual motion, at any given point of the orbit, is the square 
 of what the actual motion of a planet, revolving in a centric or- 
 bit, would be, at just half the distance of such given point. 
 Thus, if the mean distance be 4, the mean motion will be .5 ; or 
 the square of the motion of a planet, revolving around the same 
 central force in a centric orbit, at the distance 2, or at half the 
 mean distance of the given orbit. And if the perihelion distance 
 of such orbit be 1, the motion at the perihelion would be 2, or 
 the square of the motion of a planet, revolving in a centric orbit 
 about the same central force, at the distance .5, or half the dis- 
 tance of the given planet at perihelion, and the like would be the 
 case in any point of the orbit. 
 
 And such will be found to be some- of the necessary and inev- 
 itable results of the law of gravity, and they are in perfect ac- 
 cordance with the great phenomena observed by Kepler, namely, 
 that the squares of the periods are as the cubes of the distances ; 
 that the motion varies inversely as the distance varies ; and, con- 
 sequently, that the radius vector passes over equal areas in equal 
 times, and that none of these phenomena can occur, except the 
 force of gravity be inversely as the distance. 
 
 SECTION SIXTH. 
 
 Infallibility is certainly a rare quality to acquire during the 
 whole period of one's life, and but few may expect to be charged 
 or credited with it ; and I certainly have no desire that the world 
 should enter upon their books any such charges or credits to me ; 
 and should I be charged with an attempt to mar the record of 
 the world in respect to the infallibility of Sir Isaac Newton and 
 some others, for whom the votaries of science have long claimed 
 the entire admiration of the world, I have only to remark that 
 such admiration has savored quite too much of idolatry to sub- 
 serve the best interests of science ; and I have never yet believed 
 that the only proper way to extend or perfect science in the eyes 
 of a wondering or admiring world, is, by inducing a belief in the 
 infallibility of some one. of its votaries who may have essayed, to 
 
OP THE CIRCLE. O 
 
 lay its broad foundations. And having honestly conceived that 
 certain principles promulgated by Sir Isaac Newton as funda- 
 mental, (and so received and taught b^ the world,) are wholly 
 erroneous, and that long before this the errors would have 
 been detected and repudiated, but for the admiration of the sup- 
 posed infallibility of Sir Isaac Newton, so diligently and success- 
 fully inculcated, I have not hesitated to attempt a reformation, 
 by pointing out such defects and want of comeliness in the ob- 
 ject of adoration, as I supposed could not well be overlooked, 
 even though the charge of jealousy of the fame of Sir Isaac 
 Newton should be as industriously sounded abroad against my- 
 self as it has been against Liebnitz, Hook, Bernoulli, and others of 
 Newton's cotemporaries, who were to be properly disposed of, that 
 Newton might be regarded by the world as " the creator of nat- 
 ural philosophy." For such has been the constant practice of 
 the world ; nor have our ears been in anywise relieved from the 
 trump of fame, notwithstanding the modern disclosures in re- 
 spect to his treatment of Flamstead, and the indignation conse- 
 quent thereon, to allay which has even required the able pen of 
 Mr. Whewell. 
 
 Thus, in most English treatises upon science, we are con- 
 stantly reminded of the quarrel between Liebnitz and Newton, 
 in respect to priority of discovery of fluxions, or the calculus, 
 considered so important in the involution and evolution of curves, 
 &c., which, after having been introduced by the celebrated Mr. 
 Huygens in the legitimate application of the cycloid to pendu- 
 lums of clocks, has certainly been carried to great lengths in 
 speculative philosophy, even to those doubts and difficulties 
 which are acknowledged to be almost inexplicable, and which, 
 perhaps, of all the branches of science, best serve to give the his- 
 tory of the war between Newton and Liebnitz, which is often 
 set forth in such descriptions as might enable a Fuseli to depict 
 those giants, each having hold of a horn of the dilemma, and 
 contending manfully for the milk. But I protest, that it is both 
 the privilege and the duty of every free born son of earth, to be 
 jealous of the fame of any one, if such fame shall tend to check 
 or stop the upward progress of man, either moral or scientific, to 
 which he seems destined by a proper development of his capa- 
 cities. 
 
 It is said, however, that " the evolution and involution of 
 curved lines have great claims on our attention ; " and that " this 
 doctrine is peculiarly valuable to the speculator in the higher 
 mechanics." And in fact, we find that the thread of involution 
 and evolution has been industriously interwoven into the me- 
 chanics of the heavens, by ingenious processes, which are quite 
 12 
 
90 ON THE QUADRATURE 
 
 surpassing common comprehension, and therefore destined to re- 
 main, except to the initiated, as one of the mysteries of science. 
 
 Nevertheless, one great object of this branch of science is, 
 to ascertain the flexure or variation of curvature in any given 
 point of any curve other than that of the circle, (as that of the 
 parabola, for instance,) when compared with the uniform curva- 
 ture of the circle, whose magnitude is so adjusted to the occasion 
 as to give it the name of the equi- curve circle ; in which case the 
 curvature of such circle is conceived, or supposed, to be equal to 
 the curvature of the parabola in the given point. And it is said, 
 " had Appollonius but noticed a single fact, he would have dis- 
 covered the whole theory of evolution, and its importance in 
 speculative geometry. 
 
 Thus, in the case of involution and evolution, as in the case 
 of conic sections, the circle is considered to be the standard 
 measure, to which other curves must be referred in our investi- 
 gations ; and hence, if there be error in assigning the ratio of the 
 diameter of the circle to the circumference, such error is neces- 
 sarily visited upon every other curve of which the circle is neces- 
 sarily the standard. 
 
 The theory, then, of involution and evolution of curve lines, 
 as delivered by Mr. Huygins, (and which is said to have been 
 highly prized by Newton, who applied it to trochoids and epicy- 
 cles of all kinds, but which is said to have been finally eclipsed' 
 by Newton's fluxionary geometry,) may properly be said to have 
 been the progenitor of fluxions, and is said to be peculiarly 
 adapted to fluxionary equations, by which the fancy may con- 
 ceive infinities and infinitesimals to such an extent, that the great- 
 est of a never-ending series of one kind of magnitudes, shall be 
 infinitely less than the smallest of another successive series. This 
 extension of nothingness has, however, given occasion for such 
 eminent metaphysicians as Newton, Liebnitz, Bernoulli, &c., to 
 accuse each other of giving explanations of orders of curvature 
 that can have no existence. 
 
 Nevertheless, whoever shall carefully examine the subject will 
 perhaps perceive but little actual difference in the extension of 
 their respective imaginations : All agree that there is a differ- 
 ence between the simple contact of two curves, and the point in 
 which the circle shall osculate a curve, or the point of osculation ; 
 and hence, that there can be a closer union between the curve of 
 the circle and another curve, than that produced by simple contact. 
 
 Liebnitz speaks of different degrees or orders of osculation, 
 each of which is infinitely closer than the other; and it is con- 
 tended, in honor of Newton, that he had done this before, &c. ; 
 and that Liebnitz's allegation, that an osculation produced in the 
 evolution of a curve is equivalent to four intersections, is alto- 
 
OF THE CIRCLE. 91 
 
 gether erroneous, for it is contended " that the circle indicated by 
 the coalescence of three intersections is properly named the equi- 
 curve circle ; that since we measure all curvatures by that of the 
 circle, such circle is properly the circle of curvature, and its radius 
 is the radius of curvature ; for the flexure of the circle being the 
 same in every part, it becomes the proper index for all curves," 
 &c. Hence it is said that " where two intersections coalesce, 
 there is a common tangent ; and when three intersections coalesce, 
 there is an equal curvature, for then no other circle can pass 
 between this circle and the curve." 
 
 But it is again and again alleged " that there may be indefi- 
 nite degrees of this coalescence of closeness of contact between 
 a curve and the circle ; and that this gradation of more and more 
 intimate contact, or approximation to coalescence, may be contin- 
 ued without end : Neque novit natura limatem." And such has 
 been the extent to which this perplexed metaphysical reasoning 
 has been carried, that evolutrixes have even been conceived or 
 imagined, which coincide with straight lines, and others of in- 
 finitely greater rectitude which still are curves. 
 
 Now, however well such a course of reasoning may serve as 
 an exercise for the metaphysician or speculative geometer, it may 
 nevertheless appear to most people of plain common sense, illy 
 adapted to any beneficial purpose or practical use, and may re- 
 mind one of the eulogy pronounced upon Newton's defence of 
 his theory of optics, or analysis of light, against the attacks of 
 Hook, Huygins and others, in which it is said his minuteness of 
 detail was so inconceivably beyond the ken of mortals, as that 
 very few were able to comprehend his reasoning. 
 
 And it may be doubted whether those who contend for a 
 never-ending approximation to a closer and closer contact or co- 
 alescence, than that of a simple contact, have not left the sub- 
 stance to grasp at the shadow, and thereby extended their 
 conceptions beyond the reality of things ; for if a point be but 
 position without length or breadth, and if a line be but exten- 
 sion without breadth, it may be hard to conceive how two points 
 or lines can approximate nearer to each other than simple contact. 
 
 But such fantastic reasoning seems rather calculated to oper- 
 ate as conviction to the world, than as legitimate proof in the 
 premises ; for in view of the doubts and perplexities of those 
 who have essayed to give the evidence,- (and their mutual jeal- 
 ousies forbade their settling the matter by convention,) it is 
 doubtful whether a great portion of mankind will undertake a 
 reinvestigation of the subject; whether they will not rather pre- 
 fer to admit such reasoning to be beyond their comprehension. 
 And hence the world at large (with few exceptions) assent that 
 such may be an exposition of those laws, " according to which 
 
92 ON THE QUADRATURE 
 
 the Creator has caused worlds to revolve," in accordance with 
 the Newtonian theory and law of gravity, namely, by the sole 
 and constant operation of disturbing forces ; and hence it is said, 
 and generally believed, " that all these curves and angles of 
 contact, may be, and certainly are, every day described by bodies 
 moving in free space, and acted on by accelerating forces di- 
 rected to different bodies." 
 
 Those conclusions, arising from attempts to fathom infinity 
 by progressive steps, and in which we are as liable to pass the 
 goal as to fall short of it, have doubtless led to as much of error 
 as they have of curious and unprofitable speculation. They 
 have entangled the theory and law of gravity in those meshes 
 from which the world has been unable to extricate them ; and 
 although those mysteries seemed neither designed nor calculated 
 to develope or disclose the true theory or law of gravity, they 
 have nevertheless been supposed (some how or other) to estab- 
 lish them according to Newton. Perhaps the careless or unskill- 
 ed observer would never discover any reason why a piece of 
 complicated machinery might not possess the power of pro- 
 ducing perpetual motion, if it would but go ; but Newton has 
 wholly the advantage in the parable, for, thank heaven, the sys- 
 tem of worlds continues to move, however rickety it may have 
 become in consequence of having been transferred from the laws 
 of Nature to those of Newton. 
 
 Such, however, has been the struggle for the mastery in meta- 
 physical reasoning, with a view to establish the physical laws of 
 the universe to the satisfaction of the world, as to have enabled cer- 
 tain philosophers, high in estimation as such, to play off before 
 our blunted visions, their corpuscules or physical atoms, whereby 
 they would attempt to explain the centre of gravity between 
 them, the curves which they would describe while revolving 
 round the centre of gravity, and around each other, &c. ; and 
 thence, for our edification, transfer those imagined laws to the 
 solar system, or universe ; and so ready have we been to be 
 satisfied with the greatest, or most approved player, that opposi- 
 tion has given greater importance and impulse to the Newtonian 
 philosophy, than even passive assent or non-resistance. And it 
 may well be apprehended that it was more from some knowl- 
 edge of the fact, than from inspiration, that Milton was enabled 
 to furnish his Raphael with so appropriate an answer to Adam's 
 inquiry in respect to the movement of the heavenly bodies. 
 But a fig for the inspiration of a poet who must see events ful- 
 filled before he can sing them. 
 
 Notwithstanding, it is properly alleged, and generally assent- 
 ed to, that " curvature is a change of direction only ; " neverthe- 
 
OP THE CIRCLE. 93 
 
 less, the subject has been much perplexed through undue or 
 improper considerations ; and hence we find so many cases put, 
 the declared object of which is to assist our conceptions in 
 respect to the popular doctrines on what is termed a delicate 
 subject. 
 
 Now the same visionary mode of reasoning, (for it cannot be 
 philosophical,) which conceives a closer and closer contact in the 
 point of osculation as the magnitude of the circle is decreased, 
 leads to the idea or conception of curvilinear angles which are 
 incommensurable and incomparable ; thus an osculation is said 
 to denote an assignable angle ; and hence will arise that un- 
 meaning expression of an angular motion of a heavenly body ; 
 and although the circle is properly named the measure of all 
 angles, nevertheless, neither the periphery of the circle, nor of 
 any curve line, is supplied with angles, nor can they properly be 
 said to form angles with their tangents, or greater or less angles 
 with their tangents. 
 
 As is properly said, curvature is a change of direction alone. 
 This change of direction may be either greater or less ; and 
 hence may be said to denote, indicate, or determine, what may 
 be called the curvature or convergency ; and if referred to unity, 
 it will indicate what may properly be called the rate of curva- 
 ture ; or, as I have chosen to call it, while treating of the laws 
 of force and motion of the heavenly bodies, the rate of conver- 
 gency, which rate of convergency I proportion to that of the rate 
 of force or gravity by which it is produced, very differently from 
 Sir Isaac Newton. 
 
 Now it would seem if what may be called curvature, or con- 
 vergency, is an element for consideration, in an investigation of 
 the laws of planetary force and motion, that the same standard 
 should be assumed by which its rate may be measured and corn- 
 pared ; and such standard I have placed in unity; nevertheless, 
 we find the most approved Newtonian philosophers and com- 
 mentators, while treating of what is esteemed so very impor- 
 tant a subject as that of the Newtonian department of the invo- 
 lution and evolution of curves, treating the subject after this 
 manner : 
 
 " We speak of curvature that is greater or less ; and every 
 person has a general knowledge or conception of the difference, 
 and will say, that an ellipse is more curved at the extremities of 
 the transverse axis, than any where else. But before we can 
 institute a comparison between them with precision, &c., we 
 must agree about a measure of curvature, and say what it is 
 we mean by a double or a triple curvature. 
 
 " Now there' are two ways in which we may consider curva- 
 
94 ON THE QUADRATURE 
 
 ture as a want of rectitude ; we may call that a double curvature 
 which, in a given space, carries us twice as far from a straight 
 line ; or we may call that a double curvature by which we devi- 
 ate twice as much from the same direction. Both of these 
 measures have been adopted ; and if we would rigidly adhere to 
 them, there would be no room for complaint ; but mathemati- 
 cians have not been steady in this respect, and by mixing and 
 confounding these measures have frequently puzzled their read- 
 ers. All agree, however, in their first and simple measures of 
 curvature, and say, that the curvature of an arch of a circle is as 
 the arch directly, and as the radius inversely. This is plainly 
 measuring curvature by the deflection from the first direction. 
 In an arch an inch long, there is twice as much deflection from 
 the first direction as when the radius of the circle is of half the 
 length. If the radius be about 57 J inches, an arch one inch in 
 length will produce a final direction one degree different from 
 the first. If the radius be 114J inches, the deviation will be but 
 half a degree. The linear direction from the straight path will 
 also be one half. In the case of circles, therefore, both measures 
 will agree ; but in by far the greatest number of cases they may 
 differ exceedingly, and the change of direction may be greatest 
 when the linear deviation is least. 
 
 " Flexure, or change of direction, is, in general, the most sen- 
 sible and the most important character of curvature, and is 
 Understood to be its criterion in all cases. But our processes 
 for discovering its quantity are generally by first discovering the 
 linear deviation ; and in many cases, particularly in our philo- 
 sophical inquiries, this linear deviation is our principal object. 
 Hence it has happened that the mathematician has frequently 
 stopped short at this result, and has adapted his theorem chiefly 
 to this determination. 
 
 " These differences of object have caused great confusion in 
 the methods of considering curvature, and led to many disputes 
 about its nature, and about the angle of contact ; to which dis- 
 putes there will be no end, till mathematicians have agreed in 
 their manner of expressing the measures of curvature. At pres- 
 ent we abide by the measure already given, and we mean to 
 express by curvature or flexion the change of direction." 
 
 This, however, does not indicate any great, unity of purpose 
 in the consideration of curvature or convergency ; nor could the 
 mathematicians well settle the question by convention, so as to 
 avoid discrepancies in the application. 
 
 Now the same author above quoted says, that " the doctrine of 
 curves by involution and evolution is peculiarly valuable to the 
 speculator in the higher mechanics. The intensity of a deflecting 
 
OF THE CIRCLE. 95 
 
 force is estimated by the curvature which it induces on any rec- 
 tilineal motion ; and the variations of this intensity, which is 
 the characteristic of the force, or what we call its nature, is in- 
 ferred from the variations of this curvature." 
 
 And the inference which they have drawn is properly referred 
 to an examination of the Newtonian method of investigating 
 the laws of planetary force and motion, in which some proper 
 estimation is attempted to be given in respect to the curvature 
 or convergency produced in a given time at a given distance, &c. 
 
 That portion of the science, however, of the involution and 
 evolution of curves, which, from its supposed transcendentalism, 
 is so strenuously claimed by Newton's friends and admirers, to 
 be wholly of Newtonian origin, seems to have been left somewhat 
 perplexed, both as to enunciation, demonstration, and applica- 
 tion ; for there may yet be some doubt in respect to the magnitude 
 of the equi-curve circle, or rather in regard to the point in which 
 it osculates the curve ; for Newton himself perceived the closer 
 and closer contact as epicircles are drawn of less and less 
 magnitude, ad infinitum; and inasmuch as a mere point without 
 length or breadth is incapable of being curved, it may yet be 
 the case that the only method in which the equi-curve circle can 
 be settled and established, is by convention ; and more especially, 
 inasmuch as the osculating circle (as named by Liebnitz) forms 
 an actual or assignable angle with the curve supposed to be 
 measured. By such means, then, they obtain their fluent for 
 evolving the curve, when it is acknowledged, that after exhaust- 
 ing every device, no fluent could be found for evolving the circle. 
 This portion of the science, however, is not without its use to 
 the speculative geometer in the higher mechanics, as it enables 
 the expert symbolical analyst to equate the motions of the 
 heavenly bodies to their crinkled orbits, even though they assume 
 the force of- gravity to be inversely as the square of the distance. 
 
 Now in respect to the simple cycloid, if we conceive the prime 
 cylinder (namely, that the diameter of whose inscribed sphere is 
 unity) to be rolled upon a horizontal plane, so as to perform or 
 describe just one entire revolution, the area of the arch formed 
 by that line of the cylinder which rests upon the plane at the time 
 the revolution commences, as also when it ends, is 4, or ==. the 
 lateral surface of the circumscribed cube of such cylinder; 
 while the base of such arch formed on the plane by such revolu- 
 tion, is = the lateral surface of such cylinder ; or is = the entire 
 surface of the inscribed sphere of such cylinder. It is also = the 
 greatest ellipse or elongated circle that can be formed in the recti- 
 fied surface of such cycloidal arch ; and the bulk contained 
 between such cyloidal arch and its base, will be three times the 
 
96 ON THE QUADRATURE 
 
 bulk of the generating cylinder. The area of the greatest par- 
 abola that can be drawn within such arch, is = four times the 
 bulk of the inscribed sphere of the generating cylinder ; or is = 
 8-9ths of the area of one end of such cycloiclal arch, or is to the 
 area of one end of such cycloiclal arch, as two thirds of 1 is to 
 three fourths of 1. But the subject is easy of contemplation. 
 
 It will be seen from the gathered hints of Sir Richard Phillips, 
 that Sir Isaac and Sir Richard differ very widely in their estima- 
 tion of the fall of the moon from a tangent in one minute of 
 time, from which Sir Isaac deduced his proof of the law of 
 gravity ; Sir Isaac making the space about 16 feet, while Sir 
 Richard makes it about 130,000 feet. But admitting that Sir 
 Isaac has been mistaken, even to the extent alleged, such mis- 
 take was by no means the cause of the tremendous error com- 
 mitted by him in respect to the law of gravity. 
 
 Perhaps the world will yet come to the conclusion that what 
 is termed Newton's first law, or the axiom which he assumed for 
 the government of the world's philosophy, is wholly gratuitous 
 and uncalled for, being merely an assumption without having 
 any known foundation in the universe ; and that his second law 
 or axiom should at least have stated what proportion the motive 
 force impressed, beajs to the change of motion of a body. 
 Nevertheless, he ought to have been convinced, from the reason- 
 ing which naturally flows from his first two corollaries, that the 
 motion is the square root of the motive force applied ; and con- 
 sequently, that the force of gravity is inversely as the distance, 
 and not as the square of the distance. But, of ail the discov- 
 eries for the bewildering and misleading of philosophy, perhaps 
 nothing could have been contrived that could so perfectly mysti- 
 fy and darken counsel ; that could so effectually direct the mind 
 into an infinite and inexplicable mysteriousness, where it was 
 supposed ultimate truths alone existed, as do those lemmas upon 
 which his mathematical philosophy chiefly rests, and from which 
 those refined mathematics have emanated, which have enabled 
 La Place and others to subdue the phenomena of the heavens 
 to the Newtonian theory and law of gravity. 
 
 Thus the mystical terms ultimate ratios and ratios of equality, 
 have induced the belief that Sir Isaac Newton found that the 
 ultimate ratios of the sine, chord, and tangent of arcs infinitely 
 diminished, are ratios of equality ; and hence that in our reason- 
 ing, we may use the one for the other ; and having once com- 
 menced such a train of reasoning, we must come to the 
 conclusion that the ultimate form of evanescent triangles formed 
 by the arc, chord, and tangent, is that of similitude, and their ulti- 
 mate ratio is that of equality, and that each may be used for the 
 
OF THE CIRCLE. 97 
 
 other. And hence has been deduced that futile and impractica- 
 ble method of finding the length of the arc of a circle in terms 
 of the sine, versed sine, tangent or secant, by means of fluxions ; 
 that is, in geometrizing upon a purely geometric point, namely, 
 upon nothing, or mere position ; we may conceive it to be in 
 what shape we please ; that its ratios are ratios of equality, and 
 that they are all ultimate ratios ; and hence, that the evanescent tri- 
 angle may be in form of the circle, and vice versa. But such 
 has been the philosophy (if we may so speak) which has led the 
 world to strange conclusions. 
 
 Now it is said that Sir Isaac Newton showed that the space 
 which a body describes by any finite force, whether determinate, 
 or continually varied, are to each other, in the very beginning of 
 the motion, in the duplicate ratio of the forces. But this is di- 
 rectly deduced from the simple law of falling bodies, which had 
 long before been determined by Galileo. Nevertheless, the fact, 
 however disclosed, should have taught Newton, or any other 
 person, that the force of gravity is not inversely as the square of 
 the distance ; but that it is inversely as the distance. 
 
 13 
 
CHAPTER II. 
 
 On the Law of Gravity, and the Popular Deduction and 
 Promulgation of the Supposed Law. 
 
 SECTION FIRST. 
 
 GRAVITY may properly be defined to be (at least for the pur- 
 poses of this article) that physical power which pervades the solar 
 system, (and perhaps, judging by analogy, the universe,) dis- 
 played or manifested to our senses in and through the medium 
 of tangible matter, by which the phenomenon is constantly pre- 
 sented to our perception, of aggregate bodies of matter attracting 
 each other, at least under certain circumstances ; and hence we 
 observe bodies falling from a state of rest towards the centre of 
 the earth ; and hence the centripetal force which retains the 
 planets in their orbits around the sun, and the satellites in their 
 orbits around their primaries. To present and more readily 
 compare the popular notions of gravity with suggestions or hy- 
 potheses which I may attempt to intrude in place of them, I 
 will suggest the consideration of the subject under two heads, 
 namely : the theory of gravity, and the law of gravity. The the- 
 ory of gravity will mainly be referred to the momentous ques- 
 tion, whether tangible or aggregate matter, of itself, is endued 
 with or possessed of innate or inherent gravity, or an inherent 
 power to attract and be attracted, Justin proportion to the quantity 
 of aggregate matter, wherever situated, in the solar system, (for I 
 will not attempt to investigate beyond that,) which theory may be 
 termed the theory of inherent gravity ; or, whether the power 
 incident to a given quantity of matter to attract, or the susceptibil- 
 ity to be attracted, must depend upon some numerical law in 
 respect to its distance from the sun, or great centre of gravity of 
 the solar system, which may be termed incidental or incited 
 gravity. 
 
 The law of gravity, in this division of the subject, will simply 
 consist in the rate, intensity or efficacy which the force of gravity 
 
ON THE LAW OF GRAVITY. 99 
 
 exerts, between two given quantities of matter, in proportion to 
 the distance between them at which it shall operate ; and hence, 
 to determine the numerical law of gravity, is simply to determine 
 the rate or intensity in proportion to distance. And in the in- 
 vestigation of such ratio and proportion, unity or 1 occupies a 
 very conspicuous numerical point or position. If the theory of 
 inherent gravity shall prevail, it will be necessary to consider the 
 theory and law of gravity under distinct heads ; and consequent- 
 ly, that the theory may be brought to harmonize with the law in 
 some of the principal phenomena produced, some sufficient 
 cause must necessarily be assumed. Hence we find that Sir 
 Isaac Newton assumed that planets at a greater distance from 
 the sun were less dense than those which were nearer ; and con- 
 sequently that their powers of gravity were as their respective 
 bulks multiplied into their rates of density, or as their quantity 
 of matter in proportion to their bulk ; but as the supposed densi- 
 ties of those planets having no satellites, could not be as readily 
 determined and proportioned as those which had satellites, resort 
 was had to their supposed disturbing forces operating on other 
 planets ; and hence their supposed densities were variously as- 
 sumed, according to the supposed evidence obtained. 
 
 So nearly, however, have such men as Euler, LaGrange and 
 others found their supposed proportional densities to agree with 
 their proportional distances from the sun, (namely, in the inverse 
 ratio to that of distance,) that they, and especially LaGrange, 
 founded astronomical tables upon this hypothesis ; and long af- 
 ter, he remarked that he saw no occasion to vary from a strict 
 adherence to such hypothesis. To be sure, Herschel, the astron- 
 omer, assumed the density of the planet Herschel to be greater 
 than that of Saturn ; but modern discoverers have determined and 
 shown that Herschel was mistaken in respect to the motion of 
 Herschel's satellites ; and consequently in respect to the supposed 
 density of the planet ; and in regard to the supposed density be- 
 ing proportioned to the distance, as with other planets; and 
 hence upon the theory of inherent gravity, the proportional dens- 
 ities of the planets are assumed to be inversely as their respec- 
 tive distances from the sun, for the reason that the power of 
 gravity which they appear to exert is in proportion to their re- 
 spective distances from the sun, or the centre of gravity of the 
 solar system. But may it not well be doubted whether theon ti- 
 cal or hypothetical philosophy was well applied in attempting to 
 deduce the cause from the effect, from the facts and the phe- 
 nomena observed, when a Want of density in. proportion to the 
 bulk was assumed as the cause of the want of gravity in propor- 
 tion to the bulk ? This depends, however, upon the premises 
 
100 ON THE LAW OF GRAVITY. 
 
 from which we draw our inferences; as such a deduction of the 
 cause is at best but an inference, and should be drawn from the 
 best and greatest amount of evidence that may be brought into 
 the case, and should not be left to rest foe its truth or final sup- 
 port, upon some abstract hypothesis previously assumed, as 
 for instance, the innate or inherent gravity of matter. If the at- 
 tractive power of matter be inherent in if, and that too in pro- 
 portion to the quantity, wherever situated in the solar system, then, 
 to be sure, the popular inference, that the proportional densi- 
 ties of the respective planets are as their bulks multiplied into 
 the inverse of their respective distances from the sun, is well 
 founded, because their proportional attractive powers are such ; 
 but perhaps if the naked or abstract hypothesis of the inherent 
 gravity of matter had not been brought in as a make-weight, phi- 
 losophers would not have seriously thought of making the popular 
 deduction, or of drawing the popular inference in respect to the dif- 
 ferent densities of the planets, especially from any observed facts 
 or phenomena, but would have stricken from the declaration 
 such allegations as were not proved, and not founded upon the 
 evidence ; that is, they would have travelled upon facts, so far as 
 they were plainly to be discovered, and let the inferences or de- 
 ductions follow in their proper places ; and instead of alleging 
 that the proportional densities of the planets are respectively as 
 the product of the bulk X the inverse of the distance, they would 
 have said that the proportional power of gravity exercised in, by, 
 or through the respective planets, is as the product of the bulk X 
 by the inverse of the distance, (for here is a fact found,) and 
 would have left the cause (if not yet discovered through the me- 
 dium of facts or phenomena) to be assigned whenever warrant- 
 ed, instead of assuming a naked, unsustained allegation, as the 
 cause of the phenomena observed. 
 
 Let the inquiring mind, then, conceive, in accordance with 
 the popular notion or hypothesis, that the planets become less 
 and less dense in proportion to their distance from the sun, and 
 it will inquire for the cause of this very curious phenomenon, 
 this phenomenon apparently fraught with some unity of purpose, 
 some indicative of some necessary law, by which the solar 
 system (if not the universe) is governed ; and it would be slow 
 to believe that all this order and harmony is merely fortuitous, 
 or the result of chance ; nor would it be very likely, on serious 
 reflection, to base or assign the cause in a naked and wholly un- 
 supported assumption ; and especially if such assumption were 
 in direct derogation of all the physical laws and operations of 
 the universe, so far as we are able to enter into their examina- 
 tion and their operations, by means of such puny attempts and 
 
ON THE LAW OF GRAVITY. 101 
 
 trials as we are enabled to make. Neither would the inquiring 
 mind, with the view to cut short its inquiries, and dispose in- 
 stanter of a question or hypothesis in reference to which the uni- 
 verse is to be squared or quadrated, and upon which the whole 
 superstructure of astronomy is to be reared, be likely to adopt 
 as the cause of the most astonishing order and harmony in the 
 solar system, that naked hypothesis which makes the whole 
 physical power of the universe, a power to be constantly exer- 
 cised at immense distances, to be inherent in inert or dead 
 matter. If it be possible to have a clear conception of such a 
 matter, it is but conceiving a state of things existing per se, or, as 
 some have imagined the case to have been, when the earth was 
 without form and void. And will any one pretend that such 
 hypothesis is any improvement upon the heathen mythology, 
 which places fate beyond the power of the gods to control it ? I 
 say, the inquiring mind will hesitate before it adopts as the 
 basis of the whole fabric of astronomy and philosophy, a naked 
 allegation or hypothesis, which, if true, so fully dispenses with 
 any law of order by which we hope and believe the universe 
 is upheld, controlled and constantly governed, and converts the 
 whole into a system (if such it may be called) of disturbing 
 forces, that is, a system which constantly disturbs itself into an 
 eternal harmony. Unless, however, it had been believed, (as 
 many good and eminent men have believed implicitly, even to 
 the endorsement of the idea from the pulpit,) that the projectile 
 hypothesis of Sir Isaac Newton, namely, that the planets 
 were originally projected by the Supreme Being in tangents to 
 some given point of their respective orbits, was a sufficient 
 development of a Deity, and of his doings in upholding and 
 governing the solar system, and, by analogy, the universe, 
 such hypothesis could not have prevailed. But I dare not so 
 rashly presume or assume, but if I shall be able satisfactorily 
 to determine something in respect to a numerical development 
 of the great physical law by which the solar system is upheld, 
 sustained and governed, I shall be satisfied to dispense with 
 mere naked and presumptuous allegations, which, if issuing 
 from a man proudly sustained, as was Sir Isaac Newton, might 
 plunge the world into errors from which centuries could not 
 extricate it. 
 
 The true theory of gravity, then, is yet open to investiga- 
 tion ; and whether the force of gravity be a power inherent 
 in matter, or whether gravity or attraction be an omnipresent law, 
 operating on matter in proportion to distance from a centre of 
 gravity, is a question which ought to be determined upon the 
 best evidence the nature of the case will afford. If, however, it 
 
102 ON THE LAW OP GRAVITY. 
 
 should be discovered that the attractive power of a body of ag- 
 gregate matter in the solar system is only induced, and that, too, 
 in proportion to its inverse distance from the sun, or centre of 
 gravity of a system, the theory of gravity will become so far in- 
 volved in a consideration of the law of gravity, as to require no 
 distinct or separate consideration from lhat of the law. 
 
 In the consideration or investigation of this subject, it will be 
 my object to bestow most attention upon the law of gravity ; 
 namely, its rate or intensity as proportioned to the distance from 
 what may be properly termed the attracting body, or that which 
 induces, or from which appears to emanate the centripetal force 
 which retains a revolving body in its orbit ; treating, however, of 
 the Newtonian theory of gravity in an incidental manner, when- 
 ever it may seem necessary by way of explication. It may here 
 be remarked that the Newtonian method of investigating the law 
 of gravity by a consideration of the law of falling bodies towards 
 the earth, has been preserved with much care by those authors 
 who have quoted or given us the method, and who have com- 
 mented on it. See Maclaurin's account of the Newtonian philoso- 
 phy, in four books, Rees' Cyclopaedia, Vince's Universal System 
 of Physical Astronomy, &c. ; the investigation seems to have 
 been extremely simple in its character, and, from that cause alone, 
 would not seem calculated either to lead mankind into error, or 
 to keep them there, if peradventure they were thus led ; for the 
 investigation certainly contains nothing more intricate than 
 the consideration of a numerical formula, designed to explain 
 the law of falling bodies. The simplicity of the method, how- 
 ever, does not deny to it the susceptibility of affording ample 
 demonstration of the law of gravity ; namely, satisfactory and in- 
 dubitable proof to the mind, in respect to the rate or intensity of 
 gravity as proportioned to distance ; and whenever such indubi- 
 table proof is understandingly presented to the mind, whether by 
 a numerical, a mathematical, or a geometrical process, if so be 
 that ratio is the talisman that furnishes the proof, such proof may 
 properly be called a demonstration. I speak thus with a view to 
 the extreme simplicity in which demonstrations upon the most 
 important points relative to the science of astronomy may be 
 presented, and especially those relating to fundamental princi- 
 ples upon which the superstructure should be reared. Such is 
 the case in regard to the law of falling bodies towards the earth, 
 from which it has been supposed that the law of gravity has 
 been deduced ; which law of falling bodies, and the consequent 
 deduction of the law of gravity, depend upon a proper exposition 
 of one of the most simple numerical formulas that can be devised, 
 although the law of gravity may not be as simply or as beauti- 
 
ON THE LAW OF GRAVITY. 103 
 
 fully developed and presented to the mind, by a consideration of 
 the law of falling bodies, as by a consideration of Kepler's laws, 
 (leaving out his elliptical hypothesis.) Nevertheless, as the 
 method pursued for obtaining a knowledge of the law of gravity 
 has been, by a consideration of the law of falling bodies ; and as 
 a proper consideration of the same will deduce a proper knowl- 
 edge of the law of gravity ; the labor bestowed upon the subject 
 will be directed mostly to a deduction of the law of gravity from 
 a consideration of the law of falling bodies ; as thereby the pop- 
 ular methods may be quoted, and the popular modes or manner 
 of investigation compared with my own. 
 
 Nor is truth or demonstration the less valuable in consequence 
 of its simplicity, even though it may lie at the very foundation of 
 the science of astronomy ; for if it were so, the truths disclosed 
 by Kepler would be nearly valueless ; for a more simple disclo- 
 sure of the most important truths could scarcely be made, a 
 disclosure, too, developed and explained by the aid of our com- 
 mon numerical or Arabic characters, but from which, even yet, 
 all physical truths worthy of the science of astronomy, have em- 
 anated. 
 
 Let the world, then, understand, let mankind, who have 
 supposed demonstrated truths to be beyond their comprehension, 
 understand, that the value of truth consists in the benefit which 
 it bestows, not in its complexity ; and that demonstrable truths 
 are vastly nearer to their comprehension than many have been 
 led to believe. But notwithstanding the extreme simplicity and 
 ease of comprehension of the demonstrations of the most impor- 
 tant truths, (and especially in relation to the science of astron- 
 omy,) care should be taken that we do not extend the applica- 
 tion of a demonstration beyond its legitimate province, lest we 
 thereby deduce an inference which has no foundation in truth ; 
 which inference, when once engrafted into the science, is thence- 
 forward implicitly received as truth. And such I have supposed 
 to be the case in respect to the supposed demonstration, by Sir 
 Isaac Newton, of Kepler's elliptical hypothesis, which has con- 
 tributed so much to the fame of Sir Isaac Newton. 
 
 Kepler, in making calculations upon observations previously 
 made by Tycho, (in which observations Kepler had great confi- 
 dence,) came to the conclusion that the orbit of Mars was an el- 
 lipse, or an oval, as he called it, and he afterwards came to 
 the conclusion that the orbits of all the planets were elliptical ; 
 and he found from observation that the motions of planets, at 
 perihelion and aphelion, varied inversely as their distances from 
 the sun varied ; but he was unable to determine whether this was 
 a general law throughout the orbit, but he supposed it to be ; and 
 
104 ON THE LAW OF GRAVITY. 
 
 if so, the radius vector would always describe equal areas in equal 
 times. 
 
 Newton applied that proposition of Euclid which teaches that 
 " triangles upon equal bases, and between the same parallels or 
 equal altitudes, are equal to one another," which fully confirmed 
 the truth of what Kepler had suggested, that the variation of the 
 motion is in the inverse ratio to the variation of the distance ; 
 and that too in all parts of the orbit ; and consequently that the 
 radius vector would always pass over equal areas in equal 
 times. But that such demonstration goes to the extent of estab- 
 lishing Kepler's suggestions of elliptical orbits, perhaps none, on 
 reflection, will seriously contend. The demonstration itself is 
 only applicable to one of the ascertained laws of Kepler ; namely, 
 a law of motion depending for its phenomena upon the great 
 physical law of gravity, and not at all upon his elliptical hypothe- 
 sis ; for it will at once be seen that such demonstration applies 
 as well to a circular orbit, whether centric or eccentric, as to 
 an ellipse ; and that it will even apply to an orbit composed of a 
 polygon of rectilinear sides, so long as the planet shall be per- 
 forming one side of the polygon ; for such is the condition upon 
 which the proposition holds true, or upon which geometry is a 
 science to be relied upon. 
 
 Such demonstration, then, has nothing whatever to do with 
 the elliptical hypothesis, and should not have been used in its 
 support. 
 
 SECTION SECOND. 
 
 It is possible, that from the days of Aristotle to the present, 
 Physics and Metaphysics (especially so far as dialectics, or the 
 proper modes of reasoning, are concerned) have been treated of 
 with a distinction not wholly consonant with the best philosophy. 
 Those barriers, however, have been somewhat broken down by 
 some modern philosophers and profound thinkers, among whom 
 I will mention, by way of eminent example, Kepler and Kant, 
 the philosophy of whom, taken in connection, and carefully an- 
 alyzed and compared, would form the basis of a connected phi- 
 losophy, which, if the true intent and meaning were not frittered 
 away by false pretenders, or obscured by an occult philosophy 
 which no mortal can comprehend, would not only become ac- 
 cessible to a vast portion of mankind who now grope in the 
 darkness of philosophic error, in which it has been shrouded by 
 many who claim to dispense science to the world, but would be 
 worthy of that dignity of man to which science alone seems des- 
 tined to elevate him. If, however, either of those champions in 
 philosophy should, from too abstract a consideration of the sub- 
 
ON THE LAW OP GRAVITY. 105 
 
 ject on which they may be particularly treating, deviate in any 
 respect from those dialectic principles upon which all true philos- 
 ophy should be based, and which is not properly the result of 
 human logic, I nevertheless claim for mankind the right, the 
 privilege to pause for consideration, rather than implicitly follow 
 any file-leader to inexplicable conclusions. 
 
 Hence the monadology of Liebnitz with the incomprehensible 
 consequences to which he would extend the hypothesis, however 
 beneficial such hypothesis may be in disciplining the mind to a 
 remarkable degree of astuteness, may, however, savor somewhat 
 of that mysticism with which the Pythagorean theory of monads 
 and dyads has been accused ; there is not, perhaps, any philos- 
 ophical objection to either Kepler, Kant, or Liebnitz, so far 
 as they have been actuated in their researches by the Pytha- 
 gorean philosophy, or manner of investigation, notwith- 
 standing certain conclusions or consequences may be imag- 
 ined to result, which transcend the bounds of absolute investiga- 
 tion. It is said that the allegation of Pythagoras in respect 
 to his philosophy, was conclusive ; insomuch, that the phrase 
 " he said so " was sufficient authority without any proof. But it 
 is believed that the German philosophers, to whom I have re- 
 ferred, never aspired to be more than mortals while living, or to 
 have divine honors paid them after death ; this kind of ambition 
 being incident to another person and nation ; and hence, in what 
 are termed our highest and best works on astronomy, the whole 
 proof adduced in support of most material matters of which we 
 should like to know, or be informed, often consists in that ex- 
 pression so obnoxious to science, that " Sir Isaac Newton 
 found," &c. ; and it is somewhat surprising that, in this liberal 
 age of inquiry, even the learned should be more willing to rely 
 on such implicit faith, than to take the trouble to examine into 
 the truth or error couched under the naked allegation. But it is 
 yet to be hoped that, on serious consideration, infallibility will 
 be considered an attribute which no mere mortal should impute 
 to himself; and more especially, that the true dignity of man 
 can never condescend to award such a meed of praise to a mer 
 fellow mortal. 
 
 If Kepler, after an analogical course of reasoning upon the 
 eternal fitness of things, by which he deduced certain physical 
 laws, more beneficial to science than ever fell to the lot of any 
 other mortal, eventuatly fell into an absurd conclusion, from 
 placing too much confidence in the astronomical observations of 
 Tycho, it is certainly no more than justice to him and to the 
 world, to expose and correct such error. If Leibnitz became alto- 
 gether too sensitive at losing the credit of what he esteemed his 
 14 ' 
 
106 ON THE LAW OF GRAVITY. 
 
 own discoveries in mathematics, it should be recollected that 
 Mercator but just escaped a similar fate, and that poor Flamstead 
 was totally ruined by the same means. And it may be quite 
 doubtful, were Liebnitz to burst the cerement which shrouds 
 him, and to come forth at this period of time, and behold the ap- 
 plication of his discovery to what is, in the most emphatic sense, 
 termed " the Newtonian System," (viz. its application to the 
 Newtonian theory and law of gravity,) whether he would not 
 wish himself wholly freed from even the suspicion of having been 
 the discoverer. If, however, his supposed discovery of an infinite 
 series for the surface of the circle, is founded upon those errors 
 which were and still are extant, and consequently lead to erro- 
 neous results, his fame, like that of any other mortal, should, nev- 
 ertheless, stand upon its true merits, viz. that of the development 
 of beneficial truths. Kant, with all his profundity of knowledge, 
 doubtless applied his mind much more intensely, and reasoned 
 much clearer upon that compartment of philosophy which has in 
 general been termed metaphysics, than upon what is generally call- 
 ed physics ; hence, in the analyzation of the human mind, he was 
 doubtless far more astute and better versed, than in those merely 
 physical sciences, the apparent truths of which are but the results 
 of the mind ; and although he seemed well to understand the 
 machinery of the human mind, by which a knowledge of physi- 
 cal science is developed, nevertheless, inasmuch as physical re- 
 sults were but secondary objects in his Critique of Pure Reason, 
 and having written his Natural History of the Theory of the 
 Heavens according to the Newtonian System, previous to his 
 closer application to an analysis of the human mind, there, per- 
 haps, can be no good reason why he might not have fallen into 
 the same popular errors with others, in respect to the Newtonian 
 theory, inasmuch as the subject was popular, and generally re- 
 ceived as tantamount to demonstrated truth. 
 
 But after having so clearly and conclusively established his 
 categories in respect to what forms the pure intuitions of the hu- 
 man mind, viz. space and time, as also those which are indis- 
 pensable to our understanding, from which results and conclusions 
 are deduced, it would seem almost incredible that a mind like 
 his, with the elements of time and space so well considered and 
 understood, should not have discovered the most important errors 
 in the Newtonian theory and law of gravity ; and more especial- 
 ly, as the great law of Kepler, that the " squares of the periods 
 of the planets are as the cubes of their mean distances from the 
 sun," is the most lucid exposition or commentary .of his category 
 in respect to time and space, that can well be conceived. In 
 the Keplerean laws, we find mathematical results corresponding 
 
ON THE LAW OF GRAVITY. 107 
 
 with and proportioned to time and space, however varied such 
 time and space may be ; and for this very reason, it is perhaps 
 the best corroborative evidence extant, in favor of a true devel- 
 opment of the quadrature of the circle ; inasmuch as an abstract 
 mathematical consideration of the circle is destitute of time, as a 
 comparative element. And hence I will again suggest, that per- 
 adventure, physics and metaphysics, have, in general, been con- 
 sidered in too distinctive a manner ; for if, as taught by Kant, we 
 cannot perceive anything except by means of certain original, 
 necessary, unchangeable forms of thought ; if, to produce re- 
 sults, the categories must be applied to external objects, by which 
 application they only become subject to error ; if the demonstra- 
 tive certainty of mathematics, the objects of which are space, 
 time, form, &c., or whatever quantity may be numerically ex- 
 pressed and proportioned, lies in the necessity of the forms of 
 thought, and not in the range of error to which experience is 
 subject ; surely the most intimate connection exists between the 
 physical and metaphysical world, so far at least as the human 
 mind is concerned ; nor could we, perhaps, readily distinguish 
 this intimate connection from that of a unity of purpose in Him 
 who created all things. If it be then the object of the Critique 
 of Pure Reason, to mark out and define the categories, or tran- 
 scendental ideas or pure intuitions, or the necessary and un- 
 changeable forms of thought, and to direct their proper applica- 
 tion to external objects with the view to avoid error in our results, 
 the table of categories should not only be full and ample, but 
 most explicitly explained or defined; and although they may all 
 be comprehended in the four classes to which they are assigned 
 by Kant, viz., quantity, quality, relation and modality, it may per- 
 haps be questioned, whether Kant's division of the first class, or 
 rather whether those categories assigned to the first class, viz. 
 unity, multitude and totality, give a full and proper intimation 
 of all that would seem to be comprehended under that head. 
 To be sure, unity, multitude, and totality, are so far transcenden- 
 tal or intuitive, as not to depend on experience, or be subject to 
 error from a misdirected course of reasoning ; but when applied 
 to those passive qualities or elements, in which all operations 
 exist and take place, viz., space and time, which are as appli- 
 cable to quantity, numerically expressed, as are the elements of 
 force or motion, or even a^fy tangible form whatever, and in- 
 asmuch as quantity (however great or small) may always be 
 taken or assigned at unity, and consequently, quantity, however 
 great or small, may emphatically be the monad,, or unity of Py- 
 thagoras, and the source of all numbers, or more properly the 
 source of all ratio and proportion, (for I will not wholly discard the 
 
108 ON THE LAW OP GRAVITY. 
 
 Pythagorean philosophy of numbers, from which our happiest and 
 mosl beneficial results have been derived,) it would seem as 
 though what Kant has seen proper to call unity, might have been 
 denoted by the term quantity, in which case, its adjuncts would 
 have been denoted by those very expressions from which ratio and 
 proportion (those indispensable elements in human wisdom) ema- 
 nate or are deduced. Thus the adjuncts would have been a greater 
 or a less quantity, or an increase or decrease of quantity, &c., and 
 I am unable to comprehend, that when applied to space or lime, 
 the idea of more or less, or greater and less, is less transcen- 
 dental or intuitive, or that it is a less original or necessary 
 form of thought, than that of multitude or totality; and if so, the 
 former certainly would seem far better adapted to a proper applica- 
 tion of the understanding, a priori, to external objects or objects of 
 research. If, then, the mind cannot have any real conception of 
 quantity in the abstract, without having at the same lime, a con- 
 ception of greater or less quantity, I would certainly supply the 
 place of unity, multitude and totality, by quantity, a greater 
 quantity, and a less quantity. If quantity denoted by unity, or 
 assumed at unity, be an abstract idea, (and perhaps abstractly, 
 quantity cannot be considered otherwise;) and if quantity, con- 
 sidered in the abstract, be numerically accompanied with, or be 
 contained in the idea of more and less quantity, that is, that 
 unity, however great or small, may be either increased or dimin- 
 ished ; then, to be sure, that indispensable element to human 
 wisdom in the investigation of all phenomena, viz. that of ratio 
 or proportion, lies as much in the necessity of the forms of 
 thought, as that of time, space, quantity, &c., and perhaps on this 
 principle alone, rests the demonstrative certainty of mathematics. 
 
 I would suggest then, whether quantity may not properly take 
 the place of unity, and whether its adjuncts, or those inevitable 
 forms of the mind which accompany it, may not be wore and less, 
 or increase and decrease of any quantity which of itself is unity. 
 In such case, quantity, whether of force, motion, distance, 
 space, time, &c., taken at unity, becomes the standard by which 
 to proportion other quantities, whether they be greater or less ; 
 and this is to be done through the medium of ratio and pro- 
 portion as evinced in the economy of numbers. 
 
 Thus the laws offeree and motion, as displayed in time and 
 space, are so adapted to the econoifty of numbers, exhibited in 
 their powers and roots, or in that quality or principle in numbers 
 which emphatically denotes and determines by mathematical 
 precision, ihe ratio or proportion which exists between a quan- 
 tity taken at unity, and a greater or less quantity, (which ratio 
 or proportion, though existing in the mind as a necessary or 
 
ON THE LAW OF GRAVITY. 
 
 109 
 
 primitive idea or thought, is only definitely determined by means 
 of the powers and roots of numbers,) that unity of purpose will 
 always be discernible in those laws which operate in space and 
 time, if unity be made the standard by which to compare quan- 
 tities; as well those of space and time, as of the operations in 
 space and time. Thus in a system of planets revolving around 
 the same central force, it is immaterial which planet we adopt 
 as the standard by assuming its mean distance at unity, if we as- 
 sume its period, its rate of velocity, its rate of motion, &c, each 
 at unity ; for in such case, the elements of all the other planets of 
 the system may readily be proportioned to each other, and to 
 those of the standard planet, or that whose respective elements 
 are called unity. Suppose the planets A, B and C, to revolve in 
 centric orbits around the same centre of gravity ; A at the dis- 
 tance 1 ; B at the distance 2, and C at the distance 4; the peri- 
 od of A will be the square root of 1 ; that of B, the square root 
 of 8 ; and that of C, the square root of 64 ; and the rate of 
 motion of A will be the square root of 1 ; that of B the square root 
 of .5; and that of C the square root of .25; that is, in each case, 
 the rate of motion will be inversely as the square root of the 
 diameter, or the cube root of the inverse of the period. The dis- 
 tance of C, being the square of the distance B, the period of C 
 will be the square of the' period of B ; the rate of motion of G 
 will be the square of the rate of motion of B ; so also, the force 
 of gravity or attraction applied to C, will be the square of that 
 applied to B, &c. And such will be the case whenever the dis- 
 tance of one planet is the square of the distance of another; and 
 similar results will occur if the distance of one planet be the 
 cube or fourth power of the distance of another planet of the 
 same system. 
 
 In a centric orbit, what may be called the rate of deflection, 
 is always inversely as the square of the distance, and is con- 
 sequently the fourth power of the rate of motion ; hence the 
 square of the distance of the moon should have been referred to 
 the rate of deflection, and not to the rate of force which caused 
 the deflection ; for the rates of deflection of the planets A and 
 C, while passing over equal space, will be inversely as their dis- 
 tances ; but while passing equal time, they will be inversely as the 
 squares of their respective distances. Hence, if the force be in- 
 versely as the distance, their deflection, while passing over equal 
 space, will be as the force; but while passing over equal time, 
 will be as the square of the force. But if the force be inversely 
 as the square of the distance, their deflection while passing over 
 equal space, will be as the square of the force ; and while pass- 
 ing over equal time, will be as the force ; which in nowise cor- 
 
110 ON THE LAW OF GRAVITY. 
 
 roborates the law of falling bodies. For if a body fall from a 
 state of rest, the amount of fall in any given time, is the square 
 of the whole amount of force applied from the commencement 
 of the fall ; for the force applied is as the time. Hence, if, in 
 given time, the amount of fall of one body from a state of rest be 
 sixteen times that of another, the force that urged it must be four 
 times as intense as that which urged the other in its fall ; other- 
 wise the sum of all the spaces passed over at the end of any 
 given moment, could not be the square of the sum of all the 
 moments of time from the commencement of the fall. So if two 
 bodies, M and N, fall from a state of rest at the same time, and 
 at the end of any given moment, M shall have fallen four times 
 as far as N, the constant force that urged M will have been twice 
 as great as that which urged N ; but if M, at the end of any 
 given moment, shall have fallen sixteen times as far as N, in such 
 case, the force that urged M will have been four times as great 
 as that applied to N. Let us now endeavor to apply these prin- 
 ciples to the movement of planets in their orbits. 
 
 In a centric orbit the rate of motion is always inversely as the 
 square root of the distance : hence, if the rate of motion, is 
 the square root of the force, then the force is inversely as the 
 distance ; for the simple reason, that the inverse of the distance 
 is the square of the inverse of the square root of the distance. 
 Hence, in a centric orbit, because the rate of motion is inversely 
 as the square root of the distance, if the force is inversely as the 
 square of the distance, then the rate of motion must be the 
 fourth root of the force ; for the simple reason, that the inverse 
 of the square root of the distance is the fourth root of the in- 
 verse of the square of the distance. So in an eccentric orbit, in 
 the points of mean motion to which we must refer the rate and 
 progress of the motion, the motion is, of necessity, the inverse 
 of the square root of the mean distance, as much as it is in a 
 centric orbit ; else in a system of planets, the squares of their 
 periods cannot be as the cubes of their mean distances from the 
 sun. Hence in the points of mean motion, the motion will be 
 the square root of the force, if the force is inversely as the dis- 
 tance. But if the force be inversely as the square of the dis- 
 tance, then in the points of mean motion, the motion will be the 
 fourth root of the force. And here it may be proper to pause 
 for a moment, examine the original chart, and get our latitude 
 and longitude. 
 
 One of the famous discoveries of Kelper was, that the motion 
 of a planet revolving in its orbit, varies in an inverse ratio as the 
 distance varies. This phenomenon only appertains to eccentric 
 orbits, and is of the utmost importance in the explanation of 
 
ON THE LAW OF GRAVITY. Ill 
 
 the laws of force and motion, as applied to such orbits ; and 
 had not its force and effect, its spirit and efficacy, been fritterefd 
 away, and but the shadow been retained in lieu of the substance, 
 by those who succeeded Kepler, we might, long since, have 
 been reaping the benefits which the discovery of that principle 
 seemed so well calculated to bestow on science. Yes, frittered 
 away, until at length we find in place of it, in some modern 
 works on astonomy, the following, quoted as Kepler's law, 
 namely, " The areas described by the radius vector of a planet 
 are proportional to the times." I am aware that this rendering 
 of the law is in accordance with Sir Isaac Newton's notions of 
 its importance to the science of astronomy ; for Mr. Vince says 
 that Kepler, after the discovery of his said law, concluded, of 
 course, that the radius vector would describe equal areas in equal 
 times, but was unable to prove that it was so throughout the 
 orbit ; and that this supposition of Kepler's was what first sug- 
 gested to Newton the idea of his Principia ; and that Newton 
 proved or demonstrated Kepler's supposition to be true; and 
 the extended consequences, (for good or for evil,) flowing from 
 said demonstration, (though of no use whatever to the science 
 of astronomy,) I have endeavored in part to exhibit. Thus the 
 shadow was not only received for the substance, but the sub- 
 stance was perverted, made void, and useless ; insomuch, that 
 in lieu of its being a variation of the motion of a planet in the 
 inverse ratio as the distance from the sun varies, it has been more 
 commonly taught, or, at least, received, that the motion, or rate of 
 motion of the planet, is inversely as its distance ; from which the 
 grossest errors have arisen and been inculcated in respect to the 
 laws of force and motion. For there is not a simpler or plainer 
 principle in respect to the laws of motion of the heavenly bodies, 
 than that the rate of mean motion of a planet is inversely as the 
 square root of the mean distance, and that when it revolves 
 from those mean points, the motion varies inversely as the dis- 
 tance varies. If it were not so, the squares of the periods of a 
 system of planets could not be as the cubes of their mean dis- 
 tances from the sun ; for the very condition of this law is based 
 upon the principle, that the mean motion of a planet is the in- 
 verse of the square root of the mean distance ; as any one will 
 discover upon the slightest examination. 
 
 But if the motion of a planet were inversely as the mean dis- 
 tance, the squares of the periods of a system of planets would 
 be as the fourth powers of their mean distances. And in such 
 case, according to Newton's law of gravity, namely, " that force 
 or gravity is inversely as the square of the distance, " the force 
 would always be the square of the motion in any part of the 
 
112 ON THE LAW OF GRAVITY. . 
 
 orbit ; and upon this hypothesis has the base of astronomy been 
 laid. ' And hence T will allege that the only principle upon 
 which the hypothesis that the force is inversely as the square of 
 the distance, could be sustained, would be, that the motion of a 
 planet in all parts of its orbit, (whether centric or eccentric,) is 
 inversely as the distance from the sun ; in which case, if the 
 force were, inversely as the square of the distance, it would then, 
 of course, be the square of the motion. In such case, if the dis- 
 tance were 1, the period would be 1 ; if the distance were 2, 
 the period would be 4, in lieu of being 2.8284 ; if the distance 
 were 4, the period would be 16, or double what it really is. 
 That is, the period would always be the square of the distance, 
 in lieu of being the cube of the square root of the distance. 
 
 Let us exemplify the subject in the following manner : When- 
 ever it is found convenient or proper to compare the elements of 
 bodies revolving around different central forces, as of satellites 
 about two primary planets, if the attractive power of one of the 
 central forces be assumed at unity, and 4he other central force be 
 numerically proportioned to unity, we may then state the law 
 thus, namely, the square of the period is the cube of the distance 
 divided by the force of the central body ; which version will ap- 
 ply equally well when there is but one central body about which 
 several bodies revolve, but is quite convenient when there are 
 two central bodies. Thus, suppose two primary planets whose 
 attractive powers are as 1 to 4, each having a satellite revolving 
 around it, the satellites being at the same or at equal distances 
 from the centres of their respective primaries. Let the primary 
 whose force is 1 be denoted by A, and let the distance, and 
 consequently the period of its moon, be called unity or 1 ; and 
 let the primary whose force is 4, be denoted by B. Now be- 
 cause the mean force is the square of the mean motion, and con- 
 sequently is inversely as the distance ; as the period of A's moon 
 is 1, so the period of B's moon will be .5 ; and in either case, it 
 will be perceived that the square of the period is the cube of the 
 distance divided by the force of the central body ; which law 
 will be general, whatever be the distances of the respective satel- 
 lites. 
 
 If it be true, as found by Euler, LaGrange, and other emi- 
 nent astronomers and mathematicians, so far, at least, as could be 
 ascertained from observation and calculation, and even to that de- 
 gree of exactness which induced the latter to base certain as- 
 tronomical tables upon the hypothesis, and of which he after- 
 wards remarked that he saw no occasion for altering or changing 
 his opinion in respect to the hypothesis, that the power of 
 a planet to attract its satellite is conjointly as the bulk or mag- 
 
ON THE LAW OF GRAVITY. 113 
 
 nitude directly, and its distance from the sun inversely; then if 
 two primary planets of equal magnitudes revolve about the sun, 
 one of them at the distance 1, and the other at the distance 4, 
 each having a satellite revolving around it, each at an equal 
 distance from the centre of the primary, the period of the 
 satellite of that planet furthest from the sun would be double 
 the period of the satellite of the planet nearest the sun ; and in 
 this case, twice the motion would require quadruple the force ; 
 and in either case, the square of the period would be the cube 
 of the distance divided by the force of the central body. Kep- 
 ler's law may be properly described thus : The period is the 
 square root of the third power of the distance, or the dis- 
 tance is the cube root of the square of the period, divided 
 by the force. Let us assume the law of gravity to be inversely 
 as the distance, and let three satellites, A, B. and C, revolve 
 around one and the same primary, A at the distance 1, B at the 
 distance 2, and C at the distance 4. Then the period of A will 
 be 1, and its rate of motion 1 ; the period of B will be 2.8284, 
 and its rate of motion .707106 ; the period of C will be 8 and 
 its rate of motion .5. The rate of force exerted on A will be 1, 
 on B .5, and on C .25 ; thus fulfilling the law that twice the mo- 
 tion requires four times the force to retain the planet in its orbit. 
 Suppose now the Newtonian law of gravity to prevail in respect 
 to those bodies, namely, that the force exerted upon C is four 
 times that exerted upon B, and sixteen times that exerted upon 
 A. In such case, it would seem that inasmuch as B receives 
 quadruple the force that C receives, the motion of B should be 
 equal to that of A. And inasmuch as B receives but one fourth 
 the force that A receives, the motion of B should be only equal 
 to that of C. And whether in such case B is placed in a di- 
 lemma, I must leave the world to judge. 
 
 SECTION THIRD. 
 
 Perhaps no part of astronomy has caused the philosopher and 
 mathematician more perplexity than that arising from what is 
 usually termed the deflection of a planet from a tangent to its 
 orbit, accompanied as it has been with the supposition of its 
 identity, in its mathematical construction, with the simple and 
 well known law of falling bodies near the earth, and from the 
 attempts to reconcile and harmonize what is called deflection 
 from a tangent, with the legitimate elements for deducing the 
 laws of force and motion, namely, time, distance, motion and 
 gravity. Hence it is not passing strange that the world should 
 have received the doctrine that the law of gravity is inversely as 
 15 
 
114 ON THE LAW OF GRAVITY. 
 
 the square of the distance; having taken as their guide at the 
 setting out, a wholly factitious element in the laws of force and 
 motion, supposing it to possess a quality which it does not, 
 namely, identity in its mathematical or numerical construction 
 with the law of falling bodies near the earth. The error, however, 
 should not have been committed, as I will endeavor to show ; 
 even though deflection is identical with the law of falling bodies, 
 and for that purpose, in the present section, I will suppose de- 
 flection to be identical with the law of falling bodies, although 
 I think it demonstrable that it is not. 
 
 For the purpose of exhibiting Newton's method of assigning 
 the law of gravity, his method of investigation, and his re- 
 sult, (which has generally been supposed to be in conformity 
 with the law of falling bodies,) I will quote from Rees' Cyclo- 
 pedia, Vol. XXV, article Moon, wherein, after giving the magni- 
 tude, motion, density, &c., compared with the earth, together with 
 the irregularities of its motions, &c., the author proceeds as fol- 
 lows : " The irregularities above enumerated and some others 
 of a similar kind, have been urged as objections to the Newton- 
 ian theory of gravity, though they were anticipated by the illus- 
 trious author,, who not only evinced their consistency with it, but 
 suggested the explication of them which might be deduced from 
 that theory properly understood and applied." Mr. Rees, after 
 quoting from Mr. Vince's astronomy on the same subject, pro- 
 ceeds : " Mr* Euler also retracted his own erroneous opinion in 
 deference to the judgment of Mr. Clairaut, and concurs with 
 him in doing ample justice to the Newtonian theory," and ob- 
 serves that " others, and particularly Mr. Machin and Mr. Frisi, 
 have prosecuted a similar investigation of this theory and con- 
 tributed to establish it." Mr. Rees then proceeds in this very 
 remarkable manner : " It would not be consistent with the limits 
 nor the nature of this work, to investigate, by tedious and elabor- 
 ate processes, of an analytical and geometrical kind, the various 
 equations that have been explored for the illustration of these 
 laws. Much has been done in this way by several learned math- 
 ematicians, and of late by Professor Vince, who is eminently 
 qualified for the undertaking; and we shall therefore refer the 
 reader who may be desirous of further information, and who has 
 no access to a variety of other publications, to the second volume 
 of Vince's Complete System of Astronomy, Chapter XXXII." 
 Mr. Rees proceeds : " We shall, however, in this place, introduce 
 a general view of the Newtonian theory of gravity, as it is ap- 
 plied to the solution of the investigations of the moon's motions." 
 " We have already, under the article Gravitation, illustrated and 
 confirmed the Newtonian theory of gravity, as it regards the 
 
ON THE LAW OF GRAVITY. 115 
 
 moon and other planets ; but as the subject is of importance, and 
 as it is immediately connected with what follows, we shall here 
 give a concise statement of the leading facts by which the iden- 
 tity of the centripetal force as it respects the moon, and that of 
 gravity, was originally explained and established, referring for a 
 more detailed account, to the article just cited." " It is well 
 known and universally allowed that the planets are retained in 
 their orbits by some power which is constantly acting upon them ; 
 that this power is directed towards the centres of their orbits ; 
 that the efficacy of this power increases upon an approach to 
 the centre, and diminishes by its recess from the same, and that 
 it increases according to a certain law, namely, that of the 
 squares of the distances, as the distance diminishes ; and that 
 diminishes in the same manner as the distance increases. Now 
 by comparing this centripetal force of the planets with the force 
 of gravity on the earth, they will be found perfectly alike. This 
 we shall illustrate in case of the moon, the nearest to us of all 
 the planets." 
 
 u The rectilinear spaces described in any given time by a falling 
 body, urged by any powers, reckoning from the beginning of its 
 descent, are proportionable to those powers. Consequently the 
 centripetal force of the moon, revolving in its orbit, will be to 
 the force of gravity on the surface of the earth, as the space which 
 the moon would describe in falling any little time, by her cen- 
 tripetal force towards the earth, if she had no circular motion at 
 all, to the space which a body near the earth would describe in 
 falling by its gravity towards the same." " By a very easy and 
 obvious calculation of these two spaces it will appear that the 
 first of them is to the second, i. e., the centripetal force of the 
 moon revolving in her orbit is to the force of gravity on the sur- 
 face of the earth, as the square of the earth's semi-diameter to 
 the square of the semi-diameter of her orbit, which is the same 
 ratio as that of the moon's centripetal force in her orbit to the 
 same force near the surface of the earth. The moon's centripetal 
 force is therefore equal to the force of gravity. These forces 
 consequently are not different, but they are one and the same ; 
 for if they were different, bodies acted upon by the two pow- 
 ers conjointly, would fall towards the earth with a velocity 
 double to that arising from the sole power of gravity." " It is 
 evident, therefore, that the moon's centripetal force, by which she 
 is retained in her orbit, and prevented from running off in tan- 
 gents, is the power of gravity of the earth extended thither." 
 
 Mr. Rees, then, after the manner of others, supposes that the 
 moon is drawn from its tangent towards the centre of the earth 
 about 15f 3- Paris feet in a minute of time, which is the same 
 
116 ON THE LAW OF GRAVITY. 
 
 distance a body near the surface of the earth will fall in one sec- 
 ond of time, and proceeds thus : " But this force (as applied to 
 the moon) being known from the elliptical figure of her orbit to 
 be reciprocally proportional to the square of the distance, would 
 impel the moon, supposed to be at the surface of the earth, 
 through a space equal to 60 X 60 X 15^ feet in one minute." 
 " But bodies impelled by the force of gravity fall near the sur- 
 face of the earth through the space of 15^ Paris feet in one 
 second, and the spaces being as the squares of the time, through 
 60 X 60 X 15 T V in a minute. Consequently, as the force by 
 which the moon is retained in its orbit, and the force of gravity, 
 produce the same effects in the same circumstances, they are 
 the same forces." 
 
 This then is the law of gravity, as promulgated by Sir Isaac 
 Newton, by which the heavenly bodies move in their eternal 
 rounds. I will therefore give a short version of the foregoing 
 method to suit my own purposes and my own method of inves- 
 tigation, and will in due time, enter upon a confutation of his 
 deductions, and upon the support of my own. Thus a body 
 commencing its fall near the surface of the earth will, in the first 
 second of time, (which may be called the first moment,) fall 
 about 15 T ^ Paris feet, which may properly be called one space ; 
 in the second moment the body will fall over three like spaces ; 
 over five like spaces in the third moment, and so on; the 
 spaces increasing with the odd numbers in their order. Hence 
 if we set down all the common numbers in their order, as 1, 2, 3, 
 4, and so on, ad infinitum, and extract the square root of such 
 series, taking figure 1 at the left hand as the whole of the first 
 period in the process of the extraction, then the square root of 
 such series will be expressed by placing figure 1 over each odd 
 number in the series, in which case we shall have a very con- 
 venient formula whereby to express the law of falling bodies 
 near the earth ; in which, the square root of the series constantly 
 expressed by figure 1, will denote the individual and equal mo- 
 ments, (and also the equal forces in those moments,) in which the 
 body shall have been falling at the end of any given moment. 
 The sum of all the odd numbers of the series will be the whole 
 amount of spaces over which the body has fallen at the end of any 
 given moment, which will be the square of the sum of all the mo- 
 ments or of the forces in those moments in which the body has 
 been falling ; and the even numbers in the series will denote the 
 acquired motion, or the rate of motion at the end of any given 
 moment. The moon revolves about the earth in an orbit 60 
 times as far from the centre of the earth (or centre of gravity) as 
 the surface of the earth is ; and we will suppose it to be found 
 
ON THE LAW OF GRAVITY. 117 
 
 from observation and calculation, that the fall of the moon from 
 a tangent to its orbit, towards the centre of the earth, is one 
 space, or 15 T ^ feet, in 60 minutes, and hence that the fall of the 
 moon, at her distance from the earth, is but 3^0 part as rapid as 
 that of a body at the surface of the earth. 
 
 And from premises like these the deduction has been drawn 
 by Newton and endorsed by others, that the force is inversely as 
 the square of the distance ; notwithstanding the supposed iden- 
 tity shown between the centripetal force which retains the moon 
 in its orbit, and the law of gravitation at the surface of the earth 
 which causes bodies to fall and to accelerate their motion during 
 the fall. And hence it has been concluded that the force of the 
 earth's attraction at its surface is 3600 times as great as at the 
 distance of the moon, when in fact it is but 60 times as great ; 
 as I propose to show by a very simple process. And had 
 Clairaut and Euler but taken a common sense view of the sub- 
 ject, (having the elements all before them,) it would not have 
 been necessary for them to have retracted their rational doubts 
 as to the Newtonian law of gravity, or to have resorted to what 
 is called the problem of the three bodies, or any other mysterious 
 problem, for the purpose of accounting for the motion of the 
 moon's apogee, in corroboration of a law as simple as it is 
 universal. 
 
 In the present investigation of the subject, I will make use 
 of the element of time, in the abstract, as that of equal moments, 
 in lieu of our arbitrary division, as that of seconds, minutes, &c., 
 each of which moments being assumed at unity, or as a unit of 
 time, may be rationally and readily compared with the other re- 
 quisite elements in the investigation, each of which will be used 
 in the abstract, and each assumed at unity ; and such method 
 is as applicable to the law of falling bodies as to an investi- 
 gation of planetary motion. Now, " the squares of the periods 
 of the planets are as the cubes of their mean distances from the 
 sun" is emphatically the great astronomical chronometer from 
 which the most simple and rational deduction can be drawn in 
 respect to the law of gravity, and its application to planetary mo- 
 tion. Thus if unity be made, (as it emphatically is,) the base 
 or centre of all ratio, then, in respect to any one planet of a 
 system, if one of the elements used in any investigation be as- 
 sumed at unity, each and every other of the elements must be 
 assumed at unity, else the square of the period will not be the 
 cube of the distance. 
 
 Now, our great object of inquiry is, what is the rate or ratio of 
 the force of gravity to that of distance from the centre of gravity ? 
 This then naturally suggests, as the first or standard element, that 
 
118 ON THE LAW OF GRAVITY. 
 
 of passive distance in space, which has nothing whatever to do 
 in the performance of the operation. Nevertheless, a ponderous 
 body is not passed over, or made to describe a given quantity of 
 space, except in some given quantity of passive time ; hence 
 passive time may properly be regarded as the second element to 
 be considered. 
 
 The third element is that of the given space to be passed over 
 or described in a given time at a given distance, and is conse- 
 quently a passive element. 
 
 The fourth element is that of motion, and is of itself an effect, 
 an operation, being the phenomenon observed ; the rate of which 
 is to be proportioned to the rate of motion at a different dis- 
 tance. 
 
 The fifth element is that of the force of gravity which causes 
 the phenomenon, or passage of the body over the given space, 
 or which controls its passage over the given space, the rate of 
 which is to be proportioned to the rate of force at a different 
 distance, and is to be deduced from a proper consideration of 
 the other elements, and more especially from a proper con- 
 sideration of the different rates of motion, (or of the observed 
 phenomena,) at different distances from the centre of gravity. 
 And whether we attempt to deduce the rate of force or gravity 
 as proportioned to distance, from a consideration of two or more 
 planets revolving in their orbits at different distances from the 
 sun ; or from a consideration of two or more bodies commenc- 
 ing their fall towards the earth at the same time, but from differ- 
 ent distances from the centre of the earth ; that planet, or that 
 body whose distance is assumed at unity, must necessarily have 
 each and every one of its elements assumed at unity, by which 
 the elements of the other planet or body may be compared and 
 proportioned, as with ratio in unity. Notwithstanding, if two 
 planets revolving round the sun at different distances, or if two 
 bodies commencing their fall at the same time from differ- 
 ent distances from the centre of the earth, in which each and 
 every element must be assumed at unity for that planet or fall- 
 ing body whose distance is assumed at unity ; nevertheless, as 
 what is termed the amount of deflection of a planet from a tan- 
 gent to the orbit (even in any little time) while revolving in the 
 orbit, is not the same, or not equal in amount to what the recti- 
 linear fall of a body from a state of rest towards the centre of 
 gravity from the point of contact of the tangent and orbit, would 
 be in the same given time, however small, (although they were 
 assumed to be equal by Sir Isaac Newton,) such difference 
 will cause a much greater dissimilarity between the methods of 
 
ON THE LAW OF GRAVITY. 119 
 
 deducing the law of gravity from a consideration of two or 
 more planets, and the law of falling bodies, than would 
 otherwise exist. But notwithstanding the amount of deflection 
 of a planet from a tangent, even in any little time, is equal to 
 the versed sine of the arc described by the planet in such little 
 time, yet such versed sine, being greater in amount or not of 
 the same amount of space, as would be described by a body 
 falling from a state of rest towards the centre of gravity from 
 that distance in the same little time, would seem very strongly 
 to suggest that a like discrepancy must exist between any nu- 
 merical development of the law of falling bodies, and that which 
 is generally supposed to give a rational approximation of the 
 ratio of the circumference of the circle to the diameter, as would 
 exist between that of falling bodies and the deflection of a 
 planet from a tangent to its orbit. Nevertheless, either of these 
 methods should have led to the conclusion that the force of 
 gravity varies inversely as the distance varies ; and it may not 
 be amiss here to give some few hints as to the discrepancy 
 above named, that the mind may readily make the application 
 in respect to such discrepancy. 
 
 Conceive then the planets A and C revolving in centric orbits 
 about the sun ; A at the distance 1, and C at the distance 4 ; and 
 conceive a straight line to pass from the centre of the sun, inter- 
 secting both orbits, called the intersecting line ; let a tangent to 
 each orbit pass from the intersecting line in the direction in 
 which the planets are to revolve, and let the two planets be 
 conceived to start from the intersecting line at the same instant 
 of time. 
 
 Then if A shall have passed ever so small a distance in its or- 
 bit, from the intersecting line, C will have passed one half the dis- 
 tance that A has. Now some small distance from the intersect- 
 ing line on the orbit of A, may be called a unit of space ; and 
 the time in which A is describing it may be called a unit of 
 time ; hence in one unit or moment of time, C will describe half 
 a unit of space, or in two units or moments of time, will describe 
 one unit of space ; and when A has passed over one moment or 
 one unit of space from the intersecting line, the amount of its de- 
 flection from the tangent will be called unity or 1. And hence, 
 if such deflection be governed by the law of falling bodies, when 
 A has passed over two units of space, its amount of deflection 
 will then be 4, or four times that at the end of one space. But 
 as the orbit of C is four times as dilate as that of the orbit of A, 
 if the law of falling bodies were to prevail according to the 
 hypothesis of Sir Isaac Newton, then, when C has passed one 
 unit of space, its amount of deflection would be just one fourth 
 
120 ON THE LAW OF GRAVITY. 
 
 of that of A at the end of one space, and when C has passed 
 over one moment or unit of time, its amount of deflection would 
 be just one sixteenth that of A when A has passed over one 
 unit of time. If such be the case, (and it is so, excepting the 
 variation,) then at the end of equal time, the force applied to C 
 is (in the whole amount) one fourth of that applied to A in the 
 same time, for the reason that it would produce one sixteenth of 
 the deflection in C that the force applied to A gives in the same 
 time. And when C and A have passed over equal spaces, the 
 deflection of C will then be one fourth that of A, which, if 
 strictly true, would perfectly agree with the law of falling bodies. 
 It is manifest, then, that there is a resemblance between deflec- 
 tion and the law of falling bodies ; and were they identical, then 
 the law of deflection would be sufficient to prove conclusively 
 that the rate of force or gravity applied to C, is one fourth of 
 that applied to A, which rate of force properly refers to time, 
 viz., a continuation of the force long enough to make up a given 
 amount. Deflection, however, does so far resemble the law of 
 falling bodies, that those who undertook to prove them identical, 
 and to deduce the law of gravity therefrom, should have referred 
 the square of the distance to the amount of deflection in time, in 
 lieu of referring it to the rate or intensity of the force of gravity ; 
 for the rate of deflection in time would then have been assumed 
 as the inverse of the square of the distance, and as the square 
 of the rate of gravity directly, as in the case of falling bodies. 
 If then deflection were identical with that of falling bodies, then, 
 in either case, the amount of deflection or fall from the beginning 
 would always be the square of the whole amount of force ex- 
 pended during such deflection or fall. 
 
 In such case, the amount of deflection being as the square of 
 the force directly, and inversely as the square of the distance, 
 the rate of force must necessarily be inversely as the distance, as 
 the same is determined by the law of falling bodies. Here then 
 rested the Newtonian error, in referring the square of the dis- 
 tance to the force, instead of referring it to the fall or deflection 
 in time, to which it manifestly relates. Thus, to refer again to 
 the planets A and C, and suppose deflection and the law of fal- 
 ling bodies to be identical, A would receive twice the amount 
 of force that C would, while passing over equal space, because 
 the deflection of A would then be quadruple to that of C ; and 
 A would receive quadruple the amount of force that C would 
 while passing over equal time, because the deflection of A would 
 then be sixteen fold that of C. Hence A receives double the 
 force in half the time ; or C receives half the force in double the 
 time ; the consequence of which is, that in equal time A receives 
 
ON THE LAW OF GRAVITY. 121 
 
 four times the amount of force that C does ; or C receives 
 one fourth of the amount of force that A does in equal time ; 
 hence if the time of C be 4, and the time of A 1, they will then 
 each have received a like amount of force ; proving that the rate 
 of force is inversely as the distance ; that the rate of force ap- 
 plied to A is fourfold that applied to C ; and as the time of the 
 period of C is eight times that of A, hence C will receive, in one 
 of its entire revolutions or in the time of its period, double the 
 amount of force that A will receive in the time of A's period. 
 But if the rate of force were inversely as the square of the dis- 
 tance, then C, in the time of its period, would receive just half 
 the amount of force that A would receive in the time of A's 
 period. If the amount of deflection of C were just four times 
 as great in two units of time as it is in one, (in accordance with 
 the law of falling bodies,) then the amount of C's deflection in 
 one unit of time would be the square of the amount of C's de- 
 flection in one unit of space. The distance of C being 4, its pe- 
 riod is consequently 8, otherwise it would not fulfil the Keplerean 
 law ; hence if the distance of another planet be the square root 
 of the distance of C, the period of such planet will necessarily 
 be the square root of the rate of motion of C; and the same is 
 true of its rate of force or gravity ; and also of its deflection in 
 time. And similar will be the case whenever the distance of 
 one planet shall be the square root of the distance of another 
 planet of the same system ; for such result only will fulfil the 
 great law of Kepler. 
 
 Hence the distance of a planet divided by the rate of its mean 
 motion gives the time of its period ; or the distance divided by 
 the time of the period, gives the rate of mean motion. Hence 
 the mean distance is always the square of the reciprocal of the 
 mean motion ; or the mean motion is the reciprocal of the square 
 root of the mean distance, or of the cube root of the time of the 
 period. We find then these proportions to prevail between the 
 rate of motion and the time of the period, viz., when the dis- 
 tance is 1, the rate of motion is equal to the period ; when the 
 distance is 2, the rate of motion is equal to l-4th of the period ; 
 when the distance is 3, the rate of motion is equal to l-9th of the 
 period, &c. ; or in other words, the distance divided by the time 
 of the period gives the rate of motion ; and the rate of motion 
 is always, and by every consideration which may be given to 
 the subject, inversely as the square root of the distance ; for this 
 it is, which establishes the phenomena according to Kepler's 
 law. Hence if we take three planets, A, B and C, A at the 
 distance 1, B at the distance 2, C at the distance 4, the time 
 required for B to pass over a space equal to the circumference of 
 16 
 
122 ON THE LAW OP GRAVITY. 
 
 A's orbit, will be equal to the square root of B's distance, viz., 
 the square roo* of 2 ; and the time required for C to pass over a 
 like space will oe equal to the square root of C's distance, viz., 
 the square root of 4; the motion, or rate of motion of each 
 planet, being the reciprocal of the square root of its own distance. 
 And because the rate of motion is inversely as the square root 
 of the distance, hence if the rate of the force of gravity be in- 
 versely as the square of the distance, the rate of force or gravity 
 will be the fourth power of the rate of motion. But if the rate 
 of force or gravity be inversely as the distance, then the rate of 
 force or gravity will be the square of the rate of motion, in ac- 
 cordance with all we know in respect to the laws of force and 
 motion as resulting from a consideration of mechanics or dynam- 
 ics, as also with well established facts which have long been as- 
 sented to by mathematicians and astronomers, in matters wholly 
 analogous and corroborative : as that twice the motion requires 
 quadruple the force, whether it be the result of an impulsive force 
 or of that constant and uniform force which retains a planet in a 
 centric orbit ; or produced in any other manner in which mo- 
 tion may be conceived to be the result of force. In respect to 
 the difference in point of identity between deflection from a tan- 
 gent, and the law of falling bodies, inasmuch as I shall have oc- 
 casion hereafter to refer to some direct and conclusive evidence 
 on this point, I will defer the same for the present and proceed to 
 a short consideration of the law of falling bodies, from which 
 the law of gravity may also be deduced. 
 
 In a consideration of the law of falling bodies, the force ap- 
 plied is to be considered as constant and as uniform as the time 
 in which the force is applied ; and the whole amount of force 
 applied while the body is describing a given space, may be di- 
 vided into as many equal parts as the whole amount of time is 
 divided into in which the body is describing the given space. 
 
 The time required for a body to fall from a state of rest over a 
 given space (one unit of space) is as the distance, by all determi- 
 nations of the law of falling bodies. 
 
 Hence the rate or intensity of the force bears the same ratio or 
 proportion to the distance that it does to the time. It is as much 
 less or more than the distance as it is less or more than the time. 
 If the rate or intensity of the force which urges a body over a 
 given space, or unit of space, be inversely as the distance, it is 
 also inversely as the time. If it be inversely as the square of the 
 distance, it will also be inversely as the square of the whole 
 time, thus fulfilling the whole law in respect to the mechanical 
 powers. 
 
 If A and B are to fall from a state of rest at the same time, A 
 from the distance 1 from the centre of gravity, and B from the 
 
ON THE LAW OF GRAVITY. 123 
 
 distance 60, (each body to describe a like amount of space, 
 namely, a unit of space,) B, or the body at the distance 60, will 
 require sixty-fold as long time as A, or the body at the distance 
 1. Hence if the labor to be performed, (namely, the unit of space 
 to be described,) be at sixty-fold the distance from the centre of 
 gravity, (or centre of force,) the force at that distance is propor- 
 tionally so much less than at the distance 1, that sixty-fold the 
 time is required that would be required at the distance 1, for per- 
 forming the same amount of labor. 
 
 If, then, we proportion the rate or intensity of the force to the 
 whole amount of the time passed over, we thereby proportion the 
 rate or intensity of the force to the whole amount of the distance 
 from the centre of gravity. The space then to be described, or 
 the labor to be performed, being unity, the length of time deter- 
 mines the distance, or the distance determines the length of time. 
 And until we are able to obtain some kind of evidence that the 
 great, eternal, and universal law, that the intensity of the force is 
 diminished as'the whole time is increased, qr that the rate or in- 
 tensity of the force must be increased as the whole time shall be 
 diminished, in order that the task or labor shall be performed, 
 has been altered, we had better abide by the law as it was in the 
 beginning, is now, and ever shall be. The following numerical 
 formula is in accordance with those generally given for denoting 
 or explaining the law of falling bodies towards the earth, and 
 the same may also serve to indicate how extremely simple those 
 demonstrations are, and always have been, which lie at the foun- 
 dation of true science, and have rendered the greatest benefits ; 
 and hence, important demonstrations are not enveloped in a 
 cloud of darkness beyond the ken or sagacity of most mortals to 
 penetrate ; and whenever what may be alleged to have been de- 
 monstrated is so clothed, it is to be feared that it is but the at- 
 tempt to equate error to correspond with truth, which must ne- 
 cessarily require a mysterious process. It will be seen, also, that 
 the following formula is only intended to denote the numerical 
 law of a falling body in reference to the manner in which time is 
 apportioned to any given amount, of space described or passed 
 over by the body in its fall from a state of rest, without regard to 
 distance from a centre of gravity, and consequently without regard 
 to any rate, intensity or efficiency of force by which the body 
 should be urged, and without intention to ascertain the rate or 
 intensity of gravity as proportioned to distance from a centre of 
 gravity ; for, short of a comparison of certain elements of two 
 falling bodies, the rate of gravity could not be proportioned to 
 the distance. 
 
 In accordance with the popular formula above named, the 
 upper series, composed of number 1, repeated, may be called the 
 
124 ON THE LAW OF GRAVITY. 
 
 uniform series, and is used to divide the whole amount of time 
 in which a body is falling over a given space, into equal parts or 
 moments, as seconds, for instance. 
 
 The second column, composed of the odd numbers in their 
 order, may be called the accelerating series, the sum of which 
 odd numbers at the end of any given moment of the time, will 
 always be the square of the number of moments of time in which 
 the body has been falling ; or, in other words, the amount of 
 space described or passed over at any given instant of time from 
 the commencement of the fall, is the square of the whole time 
 that has been passed over from the commencement of ihe fall. 
 Thus the whole space passed over during the amount of time 
 allowed for the fall, is, by the series of odd numbers, divided into 
 as many equal parts (called spaces) as will make the square of 
 the number of equal moments into which the whole time is 
 divided ; the first space, or that denoted by unity, or figure 1, be- 
 ing the amount of space which the body will describe in the first 
 moment of time ; and the law, numerically expressed, is, that the 
 body will describe three times the space in the second moment 
 that it does in the first, five times in the third moment, and 
 so on. 
 
 Another part of the law, in respect to a single falling body, is 
 numerically expressed by the even numbers in their order, de- 
 noting the rate of velocity or motion acquired at the end of any 
 given moment of the fall. The acquired motion at the end of 
 any given moment of time, being sufficient, without further force 
 applied, to carry the body over one more space in the n xt suc- 
 ceeding moment than it passed over in such given moment. 
 But by the aid of the constant force applied, the body will pass 
 over two more additional spaces in each successive moment, 
 thus increasing the number of spaces as the odd numbers in- 
 crease ; and hence we find that the acquired motion is as ac- 
 tive as the force is efficient. 
 
 A formula, however, designed to express numerically the law 
 of a falling body, may always be most properly expressed in two 
 series or lines of figures, namely, the uniform series, or that ex- 
 pressed by the constant figure 1, and the series of natural num- 
 bers ; figure 1 occurring only once in each odd number of the 
 series of natural numbers. Then the uniform series to any extent 
 will be the square root of the series of natural numbers to the same 
 extent, in case we make figure 1, in the natural numbers, the first 
 period in the process of extraction ; and in such case the even num- 
 bers in the series of natural numbers, (namely, those figures which 
 denote the rate of acceleration,) will always come out as a remain- 
 der in the process of extracting the square root, or uniform series 
 
ON THE LAW OF GRAVITY. 125 
 
 The formula, with the uniform series carried or extended to 
 eight places, may be thus : 
 
 11111 1 1 1 
 
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 
 
 Now it is manifest that in calculating the law of the fall 
 simply of an individual body from a state of rest, in which 
 case distance from the centre of the earth or centre of gravity is 
 not material, for the reason that the law, numerically expressed, 
 will be the same, whether the force be more or less, (so that it 
 be uniform,) we may always consider the distance from the 
 centre of gravity to be unity, whether it be at the surface of the 
 earth or at the distance of the moon ; and that it is only when 
 the fall of two bodies, commencing their falls at different distan- 
 ces are compared, that it becomes necessary to assume one of 
 the bodies at unity, and the other at some other distance, either 
 greater or less than unity, as we may choose. 
 
 The comparison, however, may be made by means of two 
 simple formulas, as in case of the foregoing ; the first, or that 
 preceding the double line, being composed of two figures only, 
 each denoting unity, the upper figure of which may denote either 
 unity of distance from the centre of gravity, unity of time 
 in which the fall or operation is to take place, or unity of force 
 or gravity, namely, the whole amount of force or gravity expend- 
 ed on the body during the operation, namely, during its fall 
 over the given space ; and the lower figure in the first formula 
 denoting the given space to be described or passed over, namely, 
 a unit of space, or the same amount that is to be described or 
 passed over in the second formula, or that succeeding the double 
 line. 
 
 The uniform series in the second formula denotes as many 
 unities of distance from the centre of gravity, or as many unities 
 of time in which the operation is to be performed, as there are 
 figures in the series ; and the whole amount of force of gravity 
 (whatever such amount may be) is divided into as many equal 
 parts as there are figures in the uniform series ; namely, into as 
 many equal parts as there are units of distance, or units of time * r 
 and the odd numbers in the lower series of the second formula 
 are used to divide the given space to be described (namely, a 
 unit of space, or the same amount denoted by the lower figure in 
 the first formula) into as many equal parts as there are units con- 
 tained in all the odd numbers from 1 to 15 inclusive, namely, 
 64, or the square of the units of distance, or of the time of the 
 fall, as also of the whole amount of force expended during the 
 fall, or, if we please, of the sum of the equal parts into which the 
 
126 ON THE LAW OF GRAVITY. 
 
 whole amount of force is divided by the uniform series ; and 
 then l-64th of the space will be described while l-8th of the force 
 is being expended; 3-64ths more of the space will be described 
 while another l-8th of the force is being expended, and so on ; the 
 amount of space described increasing by the odd numbers, while 
 the force is uniform. Now the intensity or rate of force em- 
 ployed in the second formula is less by some given ratio than that 
 employed in the first formula, that employed in the first being 
 unity ; and the ratio of the intensity of the force employed in the 
 second formula, to that employed in the first, is what we are 
 seeking for, which, of course will give the ratio of the rate or in- 
 tensity of the force to that of the distance from the centre of 
 gravity. Now in either formula, the last 15-64ths of the unity 
 of space will be described, while an equal amount of force is 
 being expended, as while the first l-64th of the space is being 
 described ; for the law is the same in both formulas. But it is 
 too serious a question to be lightly determined, whether the 
 amount of force expended upon the body in the first formula, 
 while passing over the first l-64th of the given space, or the las-t 
 15-64ths of the given space, shall be eight-fold, or only equal to 
 the amount of force expended on the body in the second for- 
 mula while passing over the first l-64th, or the last 15-64ths of 
 the given space, or unit of space ; namely, whether effective force 
 shall perform the labor, or whether 7-8ths of the labor shall be de- 
 nied to the operation of force, and credited to passive time ; thereby 
 assigning or requiring the same amount of force to be expended 
 on the body in the first formula while describing the first l-64th 
 part of the unit of space to be described, as will be expended on 
 the body in the second formula, while describing the whole of 
 the unit of space ; which is readily perceived to be the case, if 
 the rate of force varies inversely as the square of the distance ; 
 while if the rate of force varies inversely as the distance, eight- 
 fold the amount of force will be expended upon the body in the 
 first formula, while describing the whole unit of space that is ex- 
 pended upon the body in the second formula, while describing 
 the first l-64th, or the last 15-64ths of the whole unity of space ; 
 that is, in either formula, the amount of space described at any 
 given instant of time during the fall over the given space, will be 
 the square of the amount of force then agd there expended from 
 the commencement of the fall ; and in such case passive time 
 will have effected nothing, or will have had no effective or phys- 
 ical agency in the matter* any more than space or distance has ; 
 it will only have awaited the operation of force according to the 
 rate or intensity of the force applied, precisely as in the case of 
 the mechanical powers, in which case the whole time recipro- 
 cates with the rate or intensity of the force employed, 
 
ON THE LAW OF GRAVITY. 127 
 
 But 1 will defer for the present a further investigation of the 
 law of gravity in and through a consideration of the law of fall- 
 ing bodies, as it is my intention to refer to and examine this 
 very important subject again ; and until I can induce a consider- 
 ation of the subject by others, I wish to present the subject in 
 various ways, (and they may be multiplied to any extent,) with 
 the view to ascertain whether that great, eternal and universal 
 law, upon which we all suppose we have a right to calculate, 
 (viz. that the whole amount of time employed or rather passed 
 over during a mechanical operation, is the simple inverse or 
 reciprocal of the rate of force employed) has anywhere, or in 
 any case, been abrogated, repealed or altered ; and whether time, 
 instead of passively awaiting an operation, has become the prin- 
 cipal operator. For if such be the case, it is certainly as im- 
 portant to philosophy that it be known, as Mr. Rees or any other 
 person can consider it. But surely such could not have been 
 suspected previous to its discovery, or its being found by New- 
 ton, after a search of some twenty years, (according to his biog- 
 rapher,) and even then, under such ecstatic agony as compelled 
 him to ask a friend to pick up and secure what had been found ; 
 and the importance of which discovery was such as to require 
 two years in penetrating its consequences. 
 
 In respect to the line of direction, which a body falling towards 
 the earth will form or describe, which was considered and moot- 
 ed between Newton and Mr. Hook, as though the same were to 
 be settled by convention, Hook suggested, that if the body fall 
 in a vacuum, the line would be an eccentric ellipse ; and also 
 suggested such eccentric ellipse as the consequence of a force 
 inversely proportional to the squares of the distances from the 
 earth's centre. In relation to this matter, it is said that Newton, 
 " having examined this suggestion by mathematical calculations, 
 found that an attractive force, emanating from a centre, and act- 
 ing inversely as the squares of the distances, would produce 
 motions exactly resembling planetary motions, both in regard to 
 the form of the orbit and the velocity of the body at each point;" 
 and this, he says, " was the secret of the system of the world ; 
 but it still remained to account for the discordance of t^ie moon's 
 motion, which had before (1665) embarrassed Newton." Now 
 what Sir Isaac Newton is said to have found by such mathe- 
 matical calculations, has been quite liberally endorsed to the 
 world as a demonstrated truth. But I predict that the world 
 (sooner or later) will esteem it as great an imposition as could 
 well have been palmed off upon it. Thus Mr. Rees, in order to 
 help Newton's moon story out of the fog in which it is envel- 
 oped, says, that the centripetal force of the moon is known from 
 
128 
 
 ON THE LAW OF GRAVITY. 
 
 the elliptical figure of the moon's orbit to be reciprocally propor- 
 tional to the square of the distance. 
 
 Nevertheless, even after this supposed mathematical discovery 
 in respect to Hook's suggestion, the motions of the moon were 
 still discordant until they were reconciled to the same mathe- 
 matical disclosure by a deduction of the Newtonian law of 
 gravity, from a consideration of the law of falling bodies, which 
 was almost impenetrable in its consequences, and laid the foun- 
 dation for the immortal " Principia." 
 
 We see, then, what supposed results are attempted to be 
 linked together, or placed in the same category, however discord- 
 ant, to induce a belief that the force of gravity varies inversely 
 as the square of the distance varies ; and that the same has been 
 so found or demonstrated by Sir Isaac Newton, the most con- 
 clusive method, and that of which the modus operandi is given, 
 (finished out by a friend of Newton,) being the one arising from 
 a consideration of the law of falling bodies. 
 
 The law of falling bodies, then, from which the law of gravity 
 may be easily deduced, and in which no error or mistake in the 
 course of the investigation need to occur, (unless it be in assign- 
 ing some active operation to time,) ought, again and again, to be 
 investigated. I am aware that time has been so far personified, 
 as to make it the principal actor in our world, and even in the 
 universe. We speak of the hand, the tooth, the foot, the track 
 of time ; of its ravages, of the work of time, &c. ; we speak of 
 beguiling time, and even of killing time. Such personification, 
 however, is not poetic enough to give us a clear idea of the 
 species of animated nature to which time belongs ; but when 
 we find time seated at the festive board, with ermine beard and 
 forelock gray, having laid down his scythe and glass, and be- 
 come a cheerful spectator of the proceedings, we can have but 
 little doubt of its human nature, even though it should not oth- 
 erwise partake of the festivities. 
 
 Nevertheless, after all this personification of time, in matters of 
 philosophy, we ought not to assign to time a task, a labor, or an 
 operation which it is wholly incompetent to perform. 
 
 \ 
 
 SECTION FOURTH. 
 
 In an investigation of the law of gravity by the use of the 
 means which have usually been resorted to by those who have 
 attempted the investigation, viz. a consideration of the law of 
 falling bodies, it may not be amiss, as preliminary thereto, to 
 quote from those authorities which may be supposed to contain 
 the best expositions of the Newtonian method of investigation. 
 
ON THE LAW OF GRAVITY. - 129 
 
 Colin Maclaurin, who was a celebrated mathematician, a con- 
 temporary of Sir Isaac Newton, and an admirer of his philoso- 
 phy and discoveries, and who wrote an account of them in four 
 books, which passed through several editions, thus gives us the 
 Newtonian exposition or deduction of the law of gravity from a 
 consideration of the law of falling bodies. " The computation 
 may be made in this manner, the mean distance of the moon 
 from the earth being sixty times the distance of heavy bodies at 
 the surface from its centre, (a body at the surface will fall 
 15xV Parisian feet in one second,) and her gravity increasing 
 in proportion as the square of the distance from the centre of the 
 earth decreases, her gravity would be 60 X 60 times greater near 
 the surface of the earth than at her mean distance, and therefore 
 would carry her through 60 X 60 X 15 T V Parisian feet in a 
 minute, near the surface ; but the same power would carry her 
 through 60 X 60 times less space in a second, than in a minute, 
 by what has been often observed of the descent of falling bodies ; 
 and therefore the moon, in a second of time, would fall by her 
 gravity near the surface of the earth 15 T V Parisian feet, which 
 therefore is the same with the gravity of terrestrial bodies." 
 I will also refer to Rees' Cyclopaedia, Vol. XXV, under the 
 article Moon, in which the subject under consideration is treated 
 of; when, after a comparison of the sun, moon, and earth, as to 
 density, proportional magnitude, &c., and after some remarks on 
 the motion of the moon's apogee, its irregularities, &c., we find 
 the following observation, viz., " The irregularities above enu- 
 merated, and some others of a similar kind, have been urged as 
 objections to the Newtonian theory of gravity, though they were 
 anticipated by the illustrious author, who not only evinced their 
 consistency with it, but suggested the explication of them which 
 might be deduced from that theory properly understood and ap- 
 plied." Mr. Rees then quotes from Mr. Vince, commencing 
 with the familiar expression, " Sir Isaac Newton having found," 
 &c., and finally observes, that " Mr. Euler also retracted his own 
 erroneous opinion in deference to the judgment of Mr. Clairaut, 
 and concurred with him in doing ample justice to the Newtonian 
 theory," &c. Mr. Rees proceeds to say, that " It would not be 
 consistent with the limits or nature of this work, [the Cyclopae- 
 dia,] to investigate, by tedious an,d elaborate processes of an 
 analytical and geometrical kind, the various equations that have 
 been explored for the illustration of these laws," and refers the 
 reader to Vince's Complete System of Astronomy, ' Vol. II. 
 Chap. XXXII. Mr. Rees then proceeds: "We shall, however, 
 in this place, introduce a general view of the Newtonian theory 
 of gravity, as it i? applied to the solution of the irregularities of 
 
130 ON THE LAW OF GRAVITY. 
 
 the moon's motions." "We have already, under the article 
 Gravitation, illustrated and confirmed the Newtonian theory of 
 gravity, as it regards the moon and other planets ; but as the 
 subject is of importance, and as it is immediately connected 
 with what follows, we shall here give a concise statement of the 
 leading facts by which the identity of the centripetal force as it 
 respects the moon and that of gravity, was originally explained 
 and established, referring, for a more detailed account, to the 
 article just cited." 
 
 " It is well known and universally allowed, that the planets 
 are retained in their orbits by some power which is constantly 
 acting upon them ; that this power is directed towards the cen- 
 tres of their orbits ; that the efficacy of this power increases upon 
 an approach to the centre, and diminishes by its recess from the 
 same ; and that it increases according to a certain law, namely, 
 that of the square of the distance as the distance diminishes ; 
 and it diminishes in the same ratio as the distance increases. 
 Now by comparing this centripetal force of the planets with the 
 force of gravity on the earth, they will be found perfectly alike. 
 This we shall illustrate in case of the moon, the nearest to us of 
 all the planets." " The rectilinear spaces described in any given 
 time by a falling body, urged by any powers, reckoning from 
 the beginning of its descent, are proportionable to those powers. 
 Consequently the centripetal force of the moon, revolving in its 
 orbit, is to the force of gravity on the surface of the earth, as the 
 space which the moon would describe in falling any little time y 
 by her centripetal force towards the earth, if she had no circular 
 motion at all, is to the space which a body near the earth would 
 describe in falling by its gravity towards the same." " By a very 
 easy and obvious calculation of these two spaces, it will appear 
 that the first of them is to the second, that is, the centripetal force 
 of the moon revolving in her orbit is to the force of gravity oil 
 the surface of the earth, as the square of the earth^s semi-diame- 
 ter is to the square of the semi-diameter of her orbit, which is 
 the same ratio as the moon's centripetal force in her orbit is to 
 the same force near the surface of the earth. The moon's cen- 
 tripetal force is therefore equal to the force of gravity. These 
 forces consequently are not different, but they are one and the 
 same ; for if they were different, bodies acted on by the two 
 powers conjointly would fall towards the earth with a velocity 
 double that arising from the sole power of gravity. It is evi- 
 dent, therefore, that the moon's centripetal force, by which she is 
 retained in her orbit and prevented from running off in a tangent, 
 is the very power of gravity of the earth extended thither." Mr. 
 Rees then proceeds to give the method of investigation after the 
 
ON THE LAW OP GRAVITY. 131 
 
 manner of Maclaurin, and also of Mr. Vince, namely, that Sir 
 Isaac Newton found that the moon is drawn by its centripetal 
 force from a tangent to its orbij; towards the centre of the earth, 
 about 15fV Parisian feet in one minute of time ; and that a 
 body will fall from a state of rest near the surface of the earth 
 15y^ Parisian feet in a second of time ; and that the distance 
 of the moon from the centre of the earth is about sixty times as 
 great as the surface of the earth is from its centre, &c. Mr. Rees 
 then says : " But this force (centripetal force of the moon) being 
 known from the elliptical figure of her orbit to be reciprocally 
 proportioned to the square of the distance, would impel the moon, 
 supposed to be at the surface of the earth, through a space equal 
 to 60 X 60 X 15^ feet in one minute." " But bodies impel- 
 led by the force of gravity fall, near the surface of the earth, 
 through the space of 15 T V Paris feet in one second, and the 
 spaces being as the squares of the times, through 60X60X15-^ 
 in a minute. Consequently, as the force by which the moon is 
 retained in its orbit, and the force of gravity, produce the same 
 effects in the same circumstances, and tend towards the same 
 point, they are the same force." Mr. Vince, who seems to have 
 been taught according to the perfect manner of the law of the 
 fathers, and who appears to be a very honest recorder of the phi- 
 losophy extant, from some cause seems to have avoided so de- 
 tailed an exposition of the Newtonian method of deducing the 
 law of gravity, as some others have presented ; and in place of it 
 find the following, namely: " Sir Isaac Newton found that if 
 we the force with which bodies fall upon the earth's surface were 
 extended to the moon, and were to vary inversely as the square 
 of the distance from the centre of the earth, it. would in one min- 
 ute draw the moon through a space which is equal to the versed 
 sine of the arc which the moon describes in one minute. He 
 concluded, therefore, that the moon was retained in its orbit by 
 the same force as that by which bodies are attracted upon the 
 earth." 
 
 I have thought proper to quote thus liberally from those text 
 books which have been received as authority, (upon a subject 
 which Mr. Rees considers important,) from the time of the dis- 
 pensation of the law by Sir Isaac Newton, until it became whol- 
 ly unnecessary to do more than simply to declare the law, name- 
 ly, " that the force of gravity varies inversely as the square of the 
 distance varies." 
 
 The subject is not less important than it is considered to be 
 by Mr. Rees, and consequently is entitled to as much consider- 
 ation as will serve to disclose a truth so important to be known. 
 The subject is not a new one, requiring only a simple disclosure 
 
132 ON THE LAW OF GRAVITY. 
 
 and development to the world, but as it has been husbanded and 
 fortified about by eminent tacticians and engineers, until at 
 length the world have assented that the premises are legitimately 
 and rightfully possessed, (especially after the adhesion to them 
 of such men as Clairaut and Euler,) something more must be 
 done than simply to disclose the truth to unprejudiced minds. 
 Error must be broken up and dispersed ; a breach and havoc 
 must be made in the citadel ; the old standard must be torn 
 down and a new one firmly placed in its stead, with legible in- 
 scriptions thereon which all may read ; and should such stand- 
 ard appear more republican or less regal than the old, it may 
 only indicate one step in the march of modern improvement. 
 Hence it may not be inappropriate, as preliminary to a more di- 
 rect investigation of the subject, that I should remark or com- 
 ment somewhat upon the text just quoted from Maclaurin, Rees 
 and Vince ; and also upon other extracts which may be taken or 
 considered in connection. Saying nothing about the discrepancies 
 which may appear in these quotations when taken in connec- 
 tion, I shall take the liberty to discard from the investigation or 
 comments, that barren, absurd and futile allegation which inex- 
 plicability required as a make-weight, and which in process of 
 time was so opportunely added ; and shall leave the same to be 
 inquired of by the world at their leisure, I refer to that allega- 
 tion of Mr. Rees, (which, to be sure, seems rather in the nature 
 of an exclusion of a conclusion,) that " this force [the centripetal 
 force of the moon] is known, from the elliptic figure of her orbit, 
 to be reciprocally proportional to the square of the distance," &c., 
 which naked allegation has been adopted as law, as in case of 
 Norton's Astronomy, " designed for use as a text-book in col- 
 leges and academies ; " in which, under the head of the " theory 
 of universal gravitation," he says, " It is also proved by the prin- 
 ciples of mechanics, that if a body, continually urged by a force 
 directed to some point, describe an ellipse of which that point is 
 a focus, the force by which it is urged must vary inversely as the 
 square of the distance." Now allegations like this may be ne- 
 cessary for the establishing of the Newtonian law of gravity, if 
 found to be inexplicable without them ; but they are not neces- 
 sary for the furtherance of philosophy, or the science of astrono- 
 my. If the orbits are elliptic, and if in an elliptic orbit the cen- 
 tripetal force is necessarily inversely as the square of the distance, 
 and if this were proved to be the case, then, to be sure, Rees' alle- 
 gation would be well enough, because true ; but as it is, the very 
 fact is assumed, a priori, for which the inquiry was being made. 
 An ellipse may vary from the circle by any given rate of ellipti- 
 city ; consequently the variation may be less than any assignable 
 
ON THE LAW OF GRAVITY. 133 
 
 quantity ; and consequently we all agree, that if the force of grav- 
 ity be inversely as the distance in the ellipse, it will also be so in 
 the circle; and, e converso, if the .force be inversely as the dis- 
 tance in the circle, it will be also in the ellipse, if a planet shall 
 be found revolving in such orbit. But the simple laws and prin- 
 ciples of dynamics, so readily evolved and explained, are driven 
 or warped from their legitimate sphere, for the purpose of estab- 
 lishing belief in a hypothesis which yet requires investigation to 
 determine its truth or error. It cannot be necessary that heaven 
 and earth should fail, rather than that Sir Isaac Newton should 
 be found to have been mistaken, either in his theory or law of 
 gravity. 
 
 As a prelude to the commentaries intended upon the quota- 
 tions before made, and also to such further matter as I may put 
 forth for consideration, perhaps no better can be offered than an ex- 
 tract from Colin Maclaurin, (the friend and admirer of Sir Isaac 
 Newton and his philosophy,) in which he sets forth the tribula- 
 tions of science in most appropriate remarks, as in the following : 
 
 " An entire liberty must be allowed to our inquiries, that natu- 
 ral philosophy may become subservient to the most valuable pur- 
 poses, and acquire all the certainty and perfection of which it is 
 capable ; but we ought not to abuse this liberty by supposing 
 instead of inquiring, and by imagining systems instead of learn- 
 ing from observation and experience the true construction of 
 things. Speculative men, by the force of genius, may invent 
 systems that perhaps will be greatly admired for a time. These, 
 however, are phantoms which the force of truth will sooner or 
 later dispel, and while we are pleased with the deceit, true phi- 
 losophy, with all the arts and improvements that depend upon it, 
 suffers." 
 
 " The real state of things escapes our observation, or if it pre- 
 sents itself to us, we are apt either to neglect it wholly as fiction, 
 or, by new efforts of a vain ingenuity, to interweave it with 
 our own conceits, and labor to make it tally with our favorite 
 schemes. Thus by blending together parts so ill suited, the 
 whole comes forth an absurd composition of truth and error." 
 " Of the many difficulties that have stood in the way of philoso- 
 phy, this vanity perhaps has had the worst effects," &c. " All oth- 
 er causes have not done so much harm as that pride and ambi- 
 tion which have led philosophers to think it beneath them to offer 
 anything less to the world than a complete and finished system 
 of nature ; and in order to obtain this at once, to take the liberty 
 of inventing certain principles and hypotheses, from which they 
 pretended to explain all her mysteries." Mr. Maclaurin, how- 
 ever, in speaking of Sir Isaac Newton's philosophy, evinces no 
 
134 ON THE LAW OF GRAVITY. 
 
 want of confidence in its perfection ; for he says that Sir Isaac 
 Newton " has secured his philosophy against any hazard of being 
 disproved or weakened by future discoveries." Mr, Maclaurin, 
 however, laments bitterly, approximating even to poetic agony, 
 that the Newtonian philosophy should have been so long resist- 
 ed and delayed. He says, " It was, however, no new thing that 
 this philosophy should meet with opposition, &c., and that it 
 should struggle with the prejudices of those who had accustom- 
 ed themselves to think only in a certain systematic way ; who 
 could not be prevailed upon to abandon their favorite schemes, 
 while they were able to imagine the least pretext for continuing 
 the dispute ; every art and talent were displayed to support their 
 falling cause ; no aid seemed foreign to them that could in any 
 manner annoy their adversary; and such often was their obsti- 
 nacy, that truth was able to make but little progress, till they 
 were succeeded by younger persons, who had not so strongly 
 imbibed their prejudices. " 
 
 Mr. Maclaurin having fully imbibed the Newtonian theory 
 and law of gravity, and also his projectile hypothesis, namely, 
 that the planets must of necessity have originally been projected 
 in tangents to their orbits, with forces so proportioned to their 
 powers of innate or inherent gravity, as to enable such innate 
 gravity to form the orbit, either more or less eccentric, according to 
 the original force of projection, (which he calls " the velocity 
 of projection,") then labors to show how a planet will proceed 
 in its orbit, what kind of curve it will form when projected in 
 a tangent to its orbit, with a greater or less force, at a greater or 
 less distance from the sun, what kind of an ellipse it would 
 form with a greater or less projection, which philosophy he has em- 
 bellished and elucidated by a multiplicity of diagrams, which 
 
 " Gird the sphere with centric and eccentric, scribbled o'er " 
 
 with all the fanciful notions in regard to the manner in which 
 bodies would proceed, what curves they would form, how 
 the force and the motion would alternately get the better of each 
 other, how a body might forever continue to recede from the 
 sun ; or upon the other hand, how it might directly or eventually 
 fall into the sun, &c., were the laws of nature which govern the 
 universe, altogether different from what they are; together with 
 a vast amount of similar matter, well calculated to bewilder the 
 mind into the belief that the force of gravity is inversely as the 
 square of the distance, that the planets were necessarily pro- 
 jected in tangents to their orbits; and hence, that a planet may 
 revolve in an eccentric orbit, and still obey the Newtonian law 
 of gravity, it is necessary to attach a kind of balance-wheel, 
 
ON THE LAW OF GRAVITY. 135 
 
 called the centrifugal force, to enable the force and motion alter- 
 nately to get the better of each other at the proper times and 
 places. And in respect to the centrifugal force, I will here give 
 a quotation from the author. " The gravity indeed at the lower 
 apside, is greater than the gravity at the higher apside, in pro- 
 portion as the square of the distance is less. But the centrifu- 
 gal force arising from the circular motion about the sun, increases 
 in a higher proportion, namely, as the cubes of the distances 
 decrease, for these centrifugal forces are in the direct proportion 
 to the squares of the velocities, and their inversed proportion of 
 the distances compounded together ; the first of these in the 
 inverse proportion of the squares of the distances, and the two 
 together compound the inverse proportion of the cubes of the 
 distances. The centrifugal force therefore increases in a more 
 rapid ratio than the force of gravity ; and though the force of 
 gravity prevails in the higher part of the orbit, the centrifugal 
 force gets the better in the lower part of it. The force of 
 gravity prevailing in the higher apsis, makes the planet approach 
 the sun, the centrifugal force prevailing in the lower apsis makes 
 the body recede from the sun ; and the continued action of these 
 two forces causes the planet forever to revolve from one to the 
 other." 
 
 Hence, if we are to believe Mr. Maclaurin, the burden of the 
 Newtonian philosophy in respect to the law of force and motion 
 by which a planet revolves in its orbit, is, to use the very language 
 or expression of Maclaurin, " that the force and motion alternately 
 get the better of each other." This then is the only possible 
 mode devised for compelling the planets to obey the New- 
 tonian law of gravity, as promulgated ; for even Mr. Maclaurin, 
 had there been "an entire liberty allowed to his inquiries," 
 (which he so well recommended in the outset,) could have de- 
 vised a much better law ; for after all his labors, wanderings and 
 perplexities, to which he was doomed in consequence of having 
 implicitly imbibed the Newtonian law of gravity, we find him 
 alleging and even repeating the allegation, that the proper limit 
 of gravitation, by which the planets could best revolve, would 
 be, " the inverse simple proportion of the distance," and he at- 
 tempts to show how beautifully a planet would revolve from one 
 apsis to the other, if operated upon by a force of gravity, de- 
 creasing in the reciprocal simple proportion of the distance. 
 
 I have thus far quoted from, and referred to the works of Mr. 
 Maclaurin, with a view to show what the obstinate world had 
 to contend with, inasmuch as he was contemporary with Sir Isaac 
 Newton, but long survived him ; and as he became the able 
 expounder of the Newtonian philosophy to the world, and 
 especially of the modus operandi by which the laws of planetary 
 
136 ON THE LAW OF GRAVITY. 
 
 force and motion were to operate in accordance with the New- 
 tonian theory and law of gravity, it might naturally be sup- 
 posed that through Maclaurin, we should drink from the highest 
 and purest streams of the Newtonian philosophy. But the 
 inquirer had better examine the work if it be accessible. 
 
 Gravity may be technically denned by saying that it is the 
 force which urges a planet towards the sun, and this force is 
 called the gravity of the planet, or its force of gravity towards 
 the sun ; and in case of a body falling towards the earth from 
 a state of rest, the motion of which is uniformly accelerated from 
 the beginning or commencement of the fall, there being no 
 space however small, without acceleration, the force which 
 constantly accelerates the motion may be called either the ac- 
 celerating force, the soliciting force, or the attractive force, or 
 force of gravity. 
 
 Now the specific manner in which the computation is made, 
 whereby the Newtonian law of gravity is supposed to have been 
 deduced from a consideration of the law of falling bodies, as 
 reported by Mr. Maclaurin and by Mr. Rees, is substantially the 
 same, and therefore requires the same comment; nevertheless 
 Mr. Rees has liberalized somewhat in his preparation, which 
 may also require a passing remark ; but from some peculiar am- 
 biguities in the version given by Mr. Vince the same may require 
 a somewhat different consideration. By Mr. Maclaurin and Mr. 
 Rees, the unit of distance from the centre of gravity is placed at 
 the surface of the earth, or the semi-diameter of the earth from 
 the centre of gravity ; while the moon is at the distance 60, or 60 
 semi-diameters of the earth from its centre. The unit space to 
 be passed over at the unit distance is 15 T V Parisian feet, and is 
 properly denoted by unity or 1, whether it be one rod, or more or 
 less than one rod. And the unit of time in which the unit of 
 space is to be described at the unit distance, is one second of 
 time. At the distance 60 the unit of space will be described in 
 60 units of time. 
 
 ' And from these premises it is required to ascertain and assign 
 the proportion or rate of gravity at the distance 60, to that at the 
 distance 1. Hence the rate of gravily or the force of gravity at 
 the unit distance, or distance 1, must also be assumed at unity; 
 for force is also quantity, and may as well have its quantity nu- 
 merically assumed as any other quantity, as that of distance, 
 space or time. Nor is there any other rational or proper way for 
 obtaining the proportion, rate or intensity of gravity at the dis- 
 tance of 60, to that at the distance 1, than that of assuming the 
 rate at the distance 1, at unity, by which that at the distance 60 
 is to be compared, as will be readily seen by any one on a little 
 
ON THE LAW OF GRAVITY. * 137 
 
 consideration of the economy of numbers, their use and perfect 
 adaptation in the expression of quantity of whatever denomin- 
 ation, and more especially in that transcendental property of 
 ratio and proportion, which is only developed by the powers and 
 roots of numbers, (the centre of all ratio and proportion being in 
 unity ;) and hence all quantities which may be numerically ex- 
 pressed, whether of the same or of different denominations, 
 may be proportioned to each other and to unity, that is, may 
 have their ratios numerically assigned. The proper elements, 
 then, for determining the law of gravity from the law of falling 
 bodies, are, distance from the centre of gravity, time, and space ; 
 namely, the time in which a given operation is to be performed, 
 and the space to be described or passed over in a given time ; 
 and force, or the force of gravity, which is the agent, the cause, 
 the physical power, which is to perform the labor or operation ; 
 which is to urge the body over a given space in a given time. 
 Force then is the active or efficient agent or element, in the oper- 
 ation ; while the time and space passed over, (I will not say the 
 time employed, or the time spent,) are as passive as is the dis- 
 tance. And if we have so far and so long been in the habit of 
 personifying time as being a physical agent or operator, that we 
 cannot divest ourselves of that form of thought, we had better 
 avoid the error by supposing or imagining that time waits for 
 the operation to be performed, or for the labor to be accomplish- 
 ed, that is the labor of urging a ponderous body over a given 
 space ; and by such consideration we shall find ourselves sus- 
 tained, corroborated and supported, by every principle of me- 
 chanics or dynamics with which we are acquainted, or which 
 has ever been disclosed. The rate, intensity or efficiency of 
 the force employed or required in raising a ponderous body 
 over a given space is inversely or reciprocally as the whole time 
 passed over during the operation. Every provident man pays 
 his hireling for the efficient labor performed, rather than for the 
 time passed over. Nevertheless, if the Newtonian law of gravi- 
 ty is true, no proposition can be plainer or more easily compre- 
 hended, than that fifty-nine sixtieths of the unit of space 
 (namely, of the 15 T V Parisian feet) passed over or described a 
 the distance of the moon, or distance 60, is wholly the operation 
 or the effect of time ; that is, but one sixtieth part of the physical 
 or effective force will be expended on the body at the distance 
 60, in urging it over the unit of space that will be expended on 
 the body at the distance of 1, in urging it over the same amount 
 of space, namely, the unit of space, or the 15yV Parisian feet. 
 
 I have said that distance, time, space, and the rate of gravity, 
 were the proper elements from which to deduce the law of grav- 
 18 
 
138 ON THE LAW OF GRAVITY. 
 
 ity, from the law of falling bodies ; and that the quantity of time, 
 and space, and the rate of gravity, must be assumed at unity, for 
 that body which is to commence its fall at the distance of unity ; 
 our object being to find the relative or proportional efficiency or 
 intensity of the physical agent or operator that is to perform the 
 operation in proportion to the distance. For, although the pro- 
 portions of the rates of velocity of the two bodies (conceiving 
 them to commence their fall at the same instant of time,) 
 are to be assigned, and consequently the rate of the velocity of 
 the body commencing its fall at the distance of unity, at any 
 given instant of time during the joint fall, or at any point of space 
 equi-distant from the point from which they respectively com- 
 mence their fall, is to be assumed at unity, that the rate of veloc- 
 ity of the other may be assigned or proportioned to unity ; 
 nevertheless, this proportion of velocity at any given instant of 
 time, and consequently at any point of equal space described by 
 both bodies from the commencement, is a deduction which 
 readily follows and discloses itself from a proper consideration 
 of the other elements, namely, distance, time, space, and the 
 rate of gravity. Nor has there been error in properly assigning 
 this rate of velocity, (calling that unity at the distance of unity,) 
 for all agree that the body at the distance 1, starts with a velocity 
 3600 times as great as does the body at the distance 60; that 
 when the body at the distance 1, has passed over half a second 
 of time, its velocity will be 3600 times as great, as that at the 
 distance 60, when the body has passed over half a second of 
 time ; that when the body at the distance 1 has passed over one 
 foot of space, its velocity will be 3600 times as great as that at the 
 distance 60, when the body has passed over one foot of space, &c. 
 We then speak properly, when we say the rate of the velocity 
 is inversely as the square of the distance. But we must never 
 forget, in our investigations, what all will allow to be an axiom 
 as sound as any which has been adopted as such, that the 
 velocity of a falling body is accelerated from the beginning of the 
 fall ; that is, there is no part of the space passed over, however 
 small, in which the motion is not accelerated ; and for this 
 reason, the determination will become absolute and conclu- 
 sive, that the rate or intensity of gravity, varies inversely as the 
 distance varies. But these principles will receive farther consid- 
 eration and elucidation in the proper place. 
 
 Now I will not hesitate to declare that a more bungling, erro- 
 neous and hap-hazard exposition of the law of gravity, could 
 scarcely have been devised by the ingenuity of man, than that 
 put forth by Maclaurin, which has so long stood the test of time, 
 for surely no mental or physical operation has affected it. And 
 
ON THE LAW OF GRAVITY. 139 
 
 on reading it, one might be well supposed to exclaim, Is it no 
 matter, of no consequence, upon \v%at premises the fabric of 
 astronomy is reared, or in what quagmire its foundations are laid ? 
 And is it possible that such an exposition could have thrown 
 mankind into those inexplicable difficulties under which they 
 have been laboring for the last two centuries ? 
 
 Mr. Maclaurin, after setting forth in his premises his unity of 
 distance, namely, the surface of the earth ; his unity of time, 
 viz., one second ; his unity of space, viz., 15 T V Parisian feet ; and 
 the distance of the moon, viz., 60 units of distance ; proceeds 
 thus : " And her (the moon's) gravity increasing in proportion as 
 the square of the distance from the centre of the earth decreases, 
 her gravity would be 60 X 60 (3600) times greater near the sur- 
 face of the earth than at her mean distance, and therefore would 
 carry her through 60 X 60 X 15 iV Parisian feet, (3600 units of 
 space,) in a minute, near the surface." 
 
 Now what the author actually meant by this passage must 
 perhaps rest somewhat upon conjecture. "Her gravity increas- 
 ing inversely as the square of the distance decreases, her gravity 
 would be 3600 times greater near the earth than at her mean 
 distance." If he intended to convey the idea that the ponderos- 
 ity of the moon would be 3600 times greater at the surface of 
 the earth than at the distance of the moon, he is entirely wel- 
 come to the conjecture, and to the consequences that would 
 follow, were the conjecture true in point of fact. The author 
 surely could not intend that a body commencing its fall at the 
 distance 60, would, when it arrived at the distance 1, have the 
 same velocity that a body would then have that commenced its 
 fall at the distance 1 ; for it is presumed that he defines gravity 
 after the manner of other authors ; viz., that it is the force which 
 urges the body towards the centre of gravity, and not the motion 
 or velocity produced by gravity. If by any possible construction 
 we could suppose that the author would only infer that if two 
 bodies commenced their fall at the same time, one at the surface 
 of the earth, and the other at the distance of the moon, that at 
 any given instant of the fall, the velocity of that at the surface 
 of the earth would be 3600 times greater than that at the distance 
 of the moon, it would be strictly true in point of fact, but his 
 premises assumed do not permit us to assign such as his mean- 
 ing or intention ; for he says, " her gravity, increasing in propor- 
 tion as the square of the distance from the centre of the earth 
 decreases, would be 3600 times greater near the surface of the 
 earth than at her mean distance." 
 
 And it is only upon the principle that the rate of gravity varies 
 inversely as the distance, (not inversely as the square of the dis- 
 
140 ON THE LAW OF GRAVITY. 
 
 lance,) that the velocity of a body at the distance 1, is 3600 
 times greater than that ar the distance 60, at any given instant of 
 their fall, both commencing at the same time. But why should 
 that be assumed as a fact for the truth of which we are seeking, 
 and the truth of which must necessarily be the result of the 
 evidence from which it is deduced ? Why commence by say- 
 ing that the gravity of the moon increases in proportion as the 
 square of her distance decreases, before the fact is ascertained, 
 inasmuch as the rate of gravity, as proportioned to distance, is 
 the very thing we wish to know ? But the premises and con- 
 clusion intended to exhibit cause and effect, preceding the fol- 
 lowing words, viz., " and therefore would carry her through 60 
 X 60 X 15 T '2- Parisian feet, (3600 unit spaces,) in a minute, near 
 the surface of the earth," would be equally vague, disjoined and 
 destitute of meaning, even though it were a fact, that the rate of 
 gravity increases from the moon to the earth in the same 
 proportion as the square of the distance decreases ; there being 
 no premises, conclusion or result whatever contained in the 
 sentence, nothing except absurdity. And it appears that this 
 part of the exposition, was in time, suspected of discrepancy ; 
 nevertheless, the law must not fail; and hence we find, in the 
 version given by Mr. Rees, that remarkable passage which I 
 have already quoted and given over to the world for investiga- 
 tion arid comment, viz., " but this force (meaning the centripetal 
 force that retains the moon in her orbit) being known, from the 
 elliptic figure of her orbit, to be reciprocally proportional to the 
 square of the distance, would impel the moon, supposed to be 
 at the surface of the earth, through a space equal to 60 X 60 X 
 15 T V Paris feet (3600 unit spaces) in one minute." This version 
 only assumes a previous knowledge of the fact that the rate of 
 gravity is inversely as the square of the distance; viz., "it being 
 known from the elliptic figure of the moon's orbit." 
 
 Now we know that by the law of falling bodies, a body fall- 
 ing from a state of rest at the surface of the earth, will fall 3600 
 times as far in one minute, as it will in the first second of time 
 of that minute; and hence, if a body were brought from the 
 moon and were to commence its fall near the surface of the 
 earth, it would doubtless be governed by the law of falling 
 bodies. But I am wholly unable to discover that the expositors 
 have in any way or manner connected this fact with that of the 
 Newtonian law of gravity, viz., that the rate of gravity varies 
 inversely as the square of the distance varies. 
 
 The result then of Mr. Maclaurin's reasoning seems to be this : 
 that a body falling from a state of rest near the surface of the 
 earth will fall 15^ Parisian feet in a second of time ; that if 
 
ON THE LAW OF GRAVITY. 141 
 
 the moon were to fall from a state of rest near the surface of the 
 earth, it would fall 3600 times as far in one minute, as it would 
 in one second, and that it will describe 3600 times less space in 
 one second than in one minute, and therefore, the moon, in a 
 second of time, would fall by her gravity, when near the surface 
 of the earth, 15 T ^ Parisian feet ; which therefore is the fact with 
 regard to the gravity of terrestrial bodies. 
 
 Thus far I have thought proper to comment on Mr. Maclau- 
 rin's exposition of the law of gravity ; wishing from the impor- 
 tance of the subject, (as Mr. Rees justly remarks,) to present the 
 same in as bold relief as possible, with a view to solicit an in- 
 vestigation of the subject. And whether he " was possessed of 
 many whims and conceits," as he alleged Kepler to have been, 
 when comparing him with the illustrious Newton, must, perhaps, 
 be inferred from his works. 
 
 Nevertheless, I think he has presented the world with a very 
 strange code of laws, and a strange body of philosophy ; yet no 
 one doubts its being strictly Newtonian. It is not consistent 
 with the limits of Mr. Rees ? Cyclopaedia to investigate the va- 
 rious equations that have been explored for the illustration of 
 these laws ; and in fact, to go into an elaborate process with a 
 view to present such as were adopted by Clairaut, and by mathe- 
 maticians in general, since Clairaut and Euler professed their 
 belief in the Newtonian laws, would be an onerous task indeed ; 
 for already has the rnoon been subject to between thirty and 
 forty equations for the illustration of these laws. Mr. Rees 
 alleges, in his preliminary to the final exposition of the Newton- 
 ian law of gravity according to Maclaurin, that i4 is well known 
 and universally allowed (among other things) that the efficacy 
 of the force of gravity increases or diminishes in the inverse 
 ratio of the square of the distance ; upon which allegation I am 
 bound to join issue. Mr. Rees also makes the following allega- 
 tion, which is similar in its consequences to the foregoing, and 
 consequently in accordance with the Newtonian law of gravity, 
 and with Mr. Vince's information of what Sir Isaac Newton 
 found, viz. : that the moon is drawn by its centripetal force from 
 a tangent to its orbit towards the centre of the earth about 15^ 
 Parisian feet, in one minute of time; Or, as expressed by Mr. 
 Vince in another place, "Sir Isaac Newton found that if the 
 force with which bodies fall upon the earth's surface were ex- 
 tended to the moon, and to vary inversely as the square of the 
 distance from the centre of the earth, it would in one minute 
 draw the moon through a space which is equal to the versed 
 sine of the arc which the moon describes in one minute. He 
 concluded, therefore, that the moon was retained in its orbit by 
 
142 ON THE LAW OP GRAVITY. 
 
 the same force as that by which bodies are attracted upon the 
 earth." Mr. Rees alleges that " the centripetal force of the moori 
 revolving in its orbit, is to the force of gravity on the surface of 
 the earth, as the space which the moon would describe in falling 
 any little time, by her centripetal force towards the earth, if she 
 had no circular motion at all, is to the space which a body near 
 the earth would describe in falling by its gravity towards the 
 same;" which sentence perhaps should be translated thus: 
 " The centripetal force of the moon is to the force of gravity on 
 the surface of the earth, as the space described by the moon in a 
 rectilinear fall towards the earth, in some little given time, caused 
 by the moon's centripetal force, is to the space described in an 
 equal time by a body falling from a state of rest near the surface 
 of the earth ; " and such translation appears to be in exact ac- 
 cordance with what Mr. Vince alleges that Sir Isaac Newton 
 found. And in corroboration of the version I have given, Mr. 
 Rees proceeds to say, that " the space at the moon is to the space 
 at the surface of the earth, i. e. the centripetal force of the moon 
 is to the force of gravity on the surface of the earth, as the 
 square of the earth's semi-diameter, (square of the distance 1,) 
 is to the square of the semi-diameter of the moon's 'orbit, 
 (square of distance 60,) which is the same ratio as that of the 
 moon's centripetal force to the same near the surface of the 
 earth." And hence it is concluded, and that too from such 
 premises, that the force of gravity on the surface of the earth, 
 and the centripetal force of the rnoon, are identical ; the intensity 
 varying in the inverse ratio of the square of the distance as we 
 recede from the centre of the earth. 
 
 If then it has been ascertained that the centripetal force acting 
 upon the moon would (if the moon had no circular motion in 
 its orbit) draw the moon from a state of rest by an accelerated 
 motion towards the centre of the earth as far in 60 seconds of 
 time, as a body near the surface of the earth will fall from a state 
 of rest in one second of time, we will consent that it furnishes 
 indubitable proof of the identity of the centrepetal force of the 
 moon with that of the force of gravity at the surface of the 
 earth ; the intensity or efficiency having its proportion assigned 
 in the powers and roots of numbers in some given inverse ratio 
 to that of the distance from the centre of the earth. But that 
 such fact has not been ascertained or disclosed by the methods, 
 or in the manner alleged by Mr. Maclaurin, Mr. Rees and Mr. 
 Vince, will, perhaps, yet become too manifest for contradiction. 
 For the very allegation which is calculated to sustain the reason- 
 ing upon the subject, namely : that gravity, or the intensity of 
 gravity varies inversely as the square of the distance varies, 
 throws the whole subject into inexplicable confusion, admitting' 
 
ON THE LAW OP GRAVITY. 143 
 
 of no rational result or conclusion. Nevertheless, " all nature 
 cries aloud, through all her works," that gravity at the surface of 
 the earth which causes bodies to fall, and the centripetal force of 
 the moon which retains it in its orbit, are identical, or one and 
 the same force, varying in its intensity in some given proportion 
 to the inverse of the distance, which ratio is assignable in the 
 powers and roots of numbers. Such was the opinion of Kepler 
 long before the days of Newton, when he declared that the moon 
 gravitated towards the earth, and that but for the moon's motion 
 in its orbit, the earth and moon would come together, and that 
 too, at a point as much nearer the earth than the moon, as the 
 earth is greater than the moon. Such, too, was the opinion of 
 Kepler when he disclosed and promulgated those eternal laws, 
 " that the squares of the periods of the planets are as the cubes 
 of their mean distances from the sun ; and that the motion of 
 any given or individual planet varies in its orbit inversely as its 
 distance from the sun varies." And this, too, may be indubita- 
 bly proved by a vast amount of evidence. 
 
 But the " a proof," which Mr. Vince says Sir Isaac Newton 
 gave Dr. Halley, to me is no proof at all ; it being a proof in 
 which the laws of Kepler were wholly thrown aside and disre- 
 garded, and in which the law of falling bodies was wholly 
 misunderstood and perverted. And as to what Mr. Vince says 
 that Sir Isaac Newton found, namely, u that if the force with 
 which bodies fall upon the earth's surface were extended to the 
 moon, and were to vary inversely as the square of the distance from 
 the centre of the earth, it would in one minute draw the moon 
 through a space which is equal to the versed sine of the arc 
 which the moon describes in one minute." I shall, in some fu- 
 ture section, in which the element of deflection, (namely, the 
 deflection of a planet from a tangent to its orbit,) which has been 
 adopted and so consequentially considered in reference to central 
 forces, will be somewhat liberally treated of, endeavor to show 
 that Sir Isaac Newton found no such thing ; for the reason that 
 no such thing existed in nature. 
 
 But I have already prolonged this section, perhaps quite too 
 far, in attempting to unravel the flimsy woof in which the law 
 of gravity has been enveloped, or in cutting the meshes in which 
 it has been entangled, inasmuch as the shreds and remnants, 
 however carefully saved, can be of no future use. 
 
 But what has Mr. Rees, or any other commentator on the law 
 of gravity, done with the point to be mooted ? Where are their 
 reasons, their evidence, their demonstrations, that show what the 
 law of gravity is ? Is it contained in a subsequent allegation of 
 Mr. Rees, " that this force is known from the elliptical figure of the 
 
144 ON THE LAW OF GRAVITY. 
 
 moon's orbit, to be reciprocally proportional to the square of the 
 distance ? " Or is it contained in a preceding allegation, which is 
 so boldly appended to certain principles which are well known 
 and universally allowed, that is, is it proved by his allegation, 
 " that gravity increases as the square of the distance diminishes, 
 and that it diminishes in the same ratio as the distance increases ?" 
 Or shall we look for the proof in the allegation of Mr. Vince and 
 others, that " Sir Isaac Newton found," &c. ? And if we do not 
 find the evidence so fully developed in these allegations as to 
 make it highly dangerous to doubt, I will say, let us reason together 
 upon the subject, and endeavor to comprehend and understand the 
 numerical developments of the great physical law of the universe. 
 For, that the intensity of gravity, is proportioned to the distance 
 from the centre of gravity, and that such proportion is inverse, or 
 reciprocal in some given power or root of numbers, as that of the 
 first or second, or some other, is not only universally allowed, 
 but is a necessary consequence synthetically deduced from the 
 great Keplerean laws, as well as from all phenomena arising from 
 the motion of the heavenly bodies. What this proportion be- 
 tween distance and intensity of gravity is, Mr. Rees considers of 
 importance ; at least, that it be established according to the New- 
 tonian hypothesis ; and hence he has thought proper to put forth 
 the leading facts by which such hypothesis was originally explain- 
 ed and established. But if others should think it of equal impor- 
 tance that the law of gravity should be determined according to 
 the truth, I have no fears as to the result. 
 
 Much stress, to be sure, has been placed upon the fact or cir- 
 cumstance, that M. Clairaut found out a very ingenious method 
 of extricating himself from the evils of a long rebellion against 
 the Newtonian law, by reconciling himself and the moon to its 
 control ; and M. Euler escaped by retracting his opinion in def- 
 erence to the judgment of M. Clairaut ; and if this were con- 
 clusive evidence in respect to the law, surely we need not search 
 farther, and may rest assured, that what Mr. Rees calls " the 
 various equations which have been explored for the illustration 
 of these laws," are in reality but the commencement of trouble. 
 
 I have already remarked upon Mr. Rees' premises and his 
 supposed consequent. Nevertheless, as I also consider the sub- 
 ject to be of importance, it may not be wholly amiss to repeat 
 the same and append a comment or two, by way of investigation. 
 The truth he alleges is thus : " The rectilinear spaces de- 
 scribed in any given time by a falling body urged by any powers, 
 reckoning from the beginning of its descent, are proportionable 
 to those powers." And his supposed consequent (and on this 
 point the investigation is particularly required) is this : " Con- 
 
ON THE LAW OF GRAVITY. 145 
 
 sequently the centripetal force of the moon, revolving in its orbit, 
 is to the force of gravity on the surface of the earth, as the space 
 which the moon would describe in falling, any little time, by her 
 centripetal force towards the earth, if she had no circular motion 
 at all, is to the space which a body near the earth would describe 
 in falling by its gravity towards the same." 
 
 Now although the author has confused the expressions in 
 which his supposed consequent is couched, beyond any of his 
 predecessors, (for what reason I know not,) nevertheless, it may 
 perhaps be rendered so as to obtain whatever of meaning it 
 may contain, and at the same time be in accordance with the 
 views of the illustrious author of the law of gravity, namely, that 
 the centripetal force exerted on the moon will be to the force 
 of gravity at the surface of the earth, as the fall of the moon 
 from a tangent, in some little time, (say one second,) will be to 
 the space described by a body falling from a state of rest at the 
 surface of the earth in the same time. Such would surely be 
 the case if the intensity of gravity were inversely as the square 
 of the distance. But the question in respect to what proportion 
 the intensity of gravity bears to the distance, is the one to be 
 solved, and that too, through the medium of the phenomena 
 which gravity produces. The law of gravity then is not to be 
 assumed, a priori, but must aw.ait its development in and through 
 its effects, or the phenomena which it causes, if so be it may 
 thus be obtained. If Mr. Rees' deduction were true, it would, 
 as he observes, be obvious that the centripetal force of the moon 
 is to the force of gravity on the surface of the earth as the square 
 of the earth's semi-diameter is to the square of the semi-diam- 
 eter of the moon's orbit ; which would be the same ratio as that 
 of the moon's centripetal force to the same force near the surface 
 of the earth ; and such centripetal force would in such case be 
 sufficiently identified with that of gravity near the surface of the 
 earth, as was fairly suggested by Kepler. But I think Mr. Rees' 
 conclusion of this part of his exposition, is quite too vague and 
 full of unknown consequents to merit very serious, attention; 
 and perhaps but little knowledge can be extracted from the same, 
 except it be of the confusion of the subject under investigation. 
 In treating of the identity of the centripetal force of the moon, 
 and that of gravity near the earth, he says, " these two forces 
 consequently are not different, but they are one and the same ; 
 for if they were different, bodies acted upon by the two forces 
 conjointly, would fall towards the earth with a velocity double to 
 that arising from the sole power of gravity. It is evident there- 
 fore, that the moon's centripetal force by which she is retained 
 in her orbit and prevented from running off in tangents, is the 
 19 
 
146 ON THE LAW OF GRAVITY. 
 
 very power of gravity of the earth extended thither." Now per- 
 haps there are those who can discover some explanation or elu- 
 cidation of the subject in the last quotation, and to such it must 
 be referred ; while it is my business to admire the unerring pre- 
 cision with which nothing but error has been produced. 
 
 Having in this work treated somewhat liberally of the law of 
 falling bodies, more with a view of disclosing those errors which 
 have resulted from attempts to elicit the law of gravity from a 
 consideration of the phenomena incident to falling bodies, than 
 to obtain therefrom the law of gravity, (believing that the law of 
 gravity may be much more easily, simply, and satisfactorily de- 
 duced from a direct consideration of the Keplerean laws,) I 
 shall not do more in this place than to offer a few brief and de- 
 sultory hints in respect to the law of falling bodies. I will use 
 figure 1 to denote unity ; what I will call one moment of time 
 may be considered as being one second ; the moon shall be con- 
 sidered at the distance 1 from the centre of the earth ; and con- 
 sequently, the surface of the earth will be at l-60th of 1 from 
 the centre of the earth, or centre of attraction ; the amount of 
 space described in one moment by a body falling from a state of 
 rest at the distance of the moon, or distance 1, will be called one 
 space ; hence the amount described in one moment by a body 
 falling from a state of rest at the surface of the earth, or at the 
 distance l-60th of 1, will be 3600 spaces, for the number of 
 spaces described in the given time is necessarily reckoned in 
 the sum of the odd numbers, commencing with the unit space 
 described in the unit of time by the body whose distance is 
 assumed at unity or 1 ; and hence, the number of spaces de- 
 scribed in the unit of time is always (properly considered) the 
 inverse of the square of the distance, (if so be that the greater dis- 
 tance is assumed at unity ) that is, making the body whose dis- 
 tance is assumed at unity the standard, the proportional amount 
 of space passed over in the given unit of time, will always be 
 inversely as the square of the distance, (but not of the time,) and 
 it is my purpose to show that it is always the square of the 
 whole amount of force applied during the fall. In this example, 
 then, the body at the distance 1, or distance of the moon, is the 
 standard ; and the time of the operation is one moment, or one 
 second ; and in that time the amount of force expended on the 
 body from the commencement of the fall is 1, so, also, the rate 
 or intensity of the force is 1, the body at the surface of the earth 
 is at the distance of l-60th part of 1, and the time of the opera- 
 tion is one moment or one second; and it is our business to as- 
 certain the ratio or intensity of the force at the surface of the 
 earth compared to that of unity, or that exerted at the distance 
 
ON THE LAW OF GRAVITY. 147 
 
 of the moon ; as also the proportional amount expended on the 
 two bodies in the second or unit of time ; that expended at the 
 distance of the moon being 1. 
 
 It having been well proved by Galileo and others, that the 
 numerical law of falling bodies may be simply presented thus, 
 
 11111 1 
 1 3 5 7 9 11 
 
 and so on, ad infinitum, from the commencement of the fall ; the 
 uniform figure 1 denoting an equal division of the time, and 
 also of the amount of force expended in the fall, whether the in- 
 tensity of the force be much or little ; which time or force thus di- 
 vided, may be called the equal times or the equal forces; and fig- 
 ure 1 in the series of odd numbers denoting the amount of space, or 
 the space described by the body in the first time, (called the unit of 
 time,) or while operated upon by the first force, or unit of space ; 
 each succeeding odd number denoting the number of unit 
 spaces described in another unit of time or unit of force ; and 
 the sum of all the odd numbers being the square of the sum of 
 all the units of time, or of all the units of force expended during 
 the fall ; I say, such having been determined or found to be the 
 numerical law of falling bodies, perhaps but few propositions 
 present themselves to the mind which can be more readily 
 solved than the following, viz. the amount of force applied from 
 the commencement of the fall during any given time, or while 
 passing over any given space, accumulates with, or in the same 
 ratio or proportion as the square root of the whole space then 
 described. Hence the whole amount of force that has been ap- 
 plied at any given instant of the fall, or at any given point of 
 space, is the square root of the whole amount of space described. 
 And this principle in respect to force and space as applied to 
 falling bodies, is invariable and universal ; whether the intensity 
 of the force be great or little ; or whether the body commenced 
 falling at the distance 1, or at the distance 60, or at the distance 
 l-60th from the centre of gravity. Thus, if the distance of the 
 moon be assumed at unity, then the space described in a given 
 moment of time, (however short,) called the unit of time, will be 
 the unit of space also ; and acceleration will have been the con- 
 sequence, even in this unit of space, however small ; and a body 
 commencing its fall at the surface of the earth or distance l-60lh, 
 will describe 3600 of those units of space in the same unit of 
 time. Hence, at the surface of the earth the amount of space 
 described in the unit of time is inversely or reciprocally as the 
 square of the distance ; and this principle is universal, and holds 
 good, whatever the distance may be which is assumed at unity. 
 
148 ON THE LAW OF GRAVITY. 
 
 This inverse or reciprocal attitude or position of time and space, 
 (the two passive elements observed in the phenomena and used 
 in the investigation of the law of falling bodies,) is necessarily 
 presented in this wise, viz. the unit of time being the same at 
 the distance of the moon as at the surface of the earth, if we 
 then assume the unit of space passed over in such unit of time 
 to be at the distance of the moon, 3600 of them will be passed 
 over in the unit of time at the surface of the earth. But if we 
 assume that as the unit of space which is described by the body 
 at the surface of the earth in the unit of time, then the amount 
 of space described at the distance of the moon in the unit of 
 time will be less than the unit of space at the surface of the 
 earth, or l-3600th part of the same. Thus the amount of space 
 described in a given time or unit of time from the commence- 
 ment of the fall is inversely or reciprocally as the square of the 
 distance at which the body commences its fall. Let, then, the 
 time be given (called one moment, or the unit of time, however 
 long or short) in which the operation is to be performed. Let 
 the space to be described in such moment (from the commence- 
 ment of the fall) be one space ; and when thus considered in the 
 abstract, the original form of the mind will not permit us to con- 
 sider either the rate or intensity of the force, or the whole 
 amount of the force expended while describing such unit of 
 space, to be other than unity, or more properly the square root 
 of unity (for we are endeavoring to ascertain ratio or proportion 
 as developed by the powers and roots of numbers.) Then sup- 
 pose the distance at which the fall is to commence to be less- 
 ened to l-60th part, and consequently that the number of units 
 of spaces described in the given moment will be increased to 
 3600, made up of the sum of the odd numbers in their order. 
 Now perhaps no proposition can become more manifest in its 
 solution than that the amount of force expended in the given 
 moment, is just the square root of the amount of the space de- 
 scribed ; for the force being uniform, will of course, in order to 
 correspond with the number of spaces, be divided into 60 parts ; 
 the same amount being expended in describing the three spa- 
 ces next succeeding the first, as is expended in the first ; and 
 consequently the amount of force expended in describing the 
 first space is equal to that expended in describing an equal 
 space or the unit space at 60 times the distance ; and conse- 
 quently, in either case, the amount of force expended is the 
 square root of the amount of the space described ; and hence 
 the conclusion is inevitable, that the rate or intensity of the force 
 is inversely as the distance. And now, aside from demonstra- 
 tion, whether such consideration of the subject be more or less 
 
ON THE LAW OF GRAVITY. 149 
 
 rational or satisfactory to the mind, than that which has so long 
 received the popular sanction, remains to be determined, viz. 
 whether physical force or whether passive time, which has no 
 physical agency in the matter, shall have the credit of perform- 
 ing the operation, viz. that of urging a body of ponderous mat- 
 ter over a given space ; for the fact will not be controverted for 
 a moment by any one, that by the popular determination in 
 respect to the law of falling bodies, time is a far more efficient 
 agent than force ; and that if one body commence falling at the 
 surface of the earth, and another at the distance of the moon, 
 when they shall have passed over equal spaces, 59-60th of the 
 fall of the body at the distance of the moon will be credited to 
 time alone, which, in fact, has only been passive, awaiting the 
 operation, without having any physical agency whatever in the 
 matter. Nevertheless, every provident person rather intends to 
 give his hireling credit for the efficient force expended in the 
 accomplishment of an operation, than for the time that may hap- 
 pen to be spent while the operation is being performed. 
 
 Thus have I, from a consideration of the importance of the 
 subject, again referred to the law of falling bodies, and its con- 
 sequent connection with the law of gravity, viz. the rate or 
 intensity of gravity as proportioned to the distance from the cen- 
 tre of gravity; and I can well hope that the importunity, the 
 urgency, and the anxiety which I have displayed on the subject, 
 will not be set to my account as an attempt to disparage the 
 judgment, the sagacity, or the perception of an enlightened 
 world. Whatever " Sir Isaac Newton found," the world has 
 been extremely loth to lose ; and perhaps it might be rationally 
 supposed, that, in a matter so simple, Sir Isaac could not have 
 found amiss ; and hence, that the subject could not require in- 
 vestigation. 
 
 SECTION FIFTH. 
 
 So far as philosophy depends upon numbers (whether nu- 
 merically, mathematically or geometrically considered) for its 
 development, it is in vain to attempt treating of subjects with 
 any degree of. unity of purpose, unless we assume unity, or 1, as 
 a standard in all our investigations on which to rely for a devel- 
 opment of the truth concerning the great law of ratio and pro- 
 portion between quantities, whether of the same, or of different 
 denominations ; for ratio properly flows from or to unity, as from 
 or to ratio in unity; and in unity the roots and powers are 
 equal. Hence if unity be always assumed as the standard in the 
 comparison of airquantities susceptible of a rational compari- 
 
150 
 
 ON THE LAW OF GRAVITY. 
 
 son by means of numbers, we thereby obtain our results with 
 ease and facility, and in that unity of purpose by which they 
 become manifest or conclusive to the mind. This I have en- 
 deavored to manifest in my investigations into the quadrature of 
 the circle ; as also when rising 
 
 " Higher in the line 
 And variation of determined shape, 
 Where truth's eternal measures mark the hound 
 Of circle, cube or sphere." 
 
 And it is my purpose to show that unity is the proper standard 
 in all our investigations in respect to the laws of force and motion 
 as compared with those passive elements, time and space ; 
 whether we consider the laws in reference to impulsive for- 
 ces and the rate of motion produced thereby, in which case it 
 has been fully determined and shown that the rate of motion 
 or velocity produced by an impulsive force is the root of the 
 amount offeree applied, as that four times the impulsive force 
 gives twice the rate of velocity ; sixteen times the impulsive 
 force gives four times the rate of velocity, &c. ; or whether 
 we consider the laws of force and motion in their most sim- 
 ple and common occurrence as produced by human or finite 
 operations; viz. in directly overcoming the ponderosity of a 
 body produced by the force of gravity in raising it through a 
 given space by a uniform motion, (neither accelerated nor retard- 
 ed,) by simple human force or power, unaided by any of the me- 
 chanical powers, properly so called, as that of the lever, or wheel 
 and axle, &c., in which case, each element proper to be consid- 
 ered is assumed at unity, or 1, as that of the ponderosity of the 
 body, the given space passed over, the time of the operation, 
 the rate or intensity of the force employed, the rate of ve- 
 locity with which the body moves, as also the whole amount 
 of force expended in the whole time of the operation ; or wheth- 
 er we consider the application of the mechanical powers in rais- 
 ing such body over a like space by a constant and uniform mo- 
 tion, but by the constant application of a less rate or intensity of 
 force; in which case the time required will be inversely or recip- 
 rocally as the rate or intensity of the constant force applied ; 
 thus, if the rate of force be half of unity, the time required will 
 be twice unity; if the time be 60, the rate of force will be l-60th 
 of unity, &c. and the rate of velocity will be as the rate of force. 
 Thus the elements of ponderosity, of space, and of the whole 
 amount of force employed in the operation, will be as those in 
 the foregoing example, viz. each will be assumed at unity, 
 while those of time and the rate or intensity of the force em- 
 ployed will be the inverse ratio or reciprocal of each other. 
 Thus the great lesson to be derived from the foregoing is, that 
 
ON THE LAW OF GRAVITY. 151 
 
 the same amount of force is required in raising the body over 
 the given space, whether the time be longer or shorter ; or 
 whether it be performed by the aid of the mechanical powers, 
 properly so called, or not. Time, in waiting for the accomplish- 
 ment of the object, necessarily waits until the amount of force or 
 labor required shall have been expended in the accomplishment. 
 Hence the whole amount of force expended in raising the body 
 over the given space, will be the same, whether the time be 
 longer or shorter, or whether the rate or intensity of the force be 
 greater or less. If we attempt to investigate the law of falling 
 bodies towards the earth, and to deduce therefrom the law by 
 which the force of gravity varies in proportion to distance from 
 the centre of gravity ; or if we attempt to investigate the laws of 
 force and motion as manifested in the revolutions of the heav- 
 enly bodies, and proportion the elements of force and motion to 
 those of time and the space passed over, our labors will scarcely 
 receive a liberal compensation unless we adopt unity as the 
 standard of comparison. In an investigation of the law of fall- 
 ing bodies towards the earth, with a view to ascertain by what 
 ratio the force of gravity varies with the varying distance from 
 the centre of the earth or centre of gravity, some of the premises 
 on which to proceed with the investigation, have been very 
 properly assumed ; as that of the comparison of two falling 
 bodies towards the earth, one commencing its fall from a state 
 of rest, at or near the surface of the earth, and the other at some 
 given number of semi-diameters of the earth from the centre of 
 the earth ; as that of four times the semi-diameter of the earth 
 from the centre, or sixty times the semi-diameter of the earth 
 from the centre, being the mean distance of the moon, &c. And 
 as the comparison of two bodies commencing their fall at dif- 
 ferent distances from the centre of gravity, is in most cases suf- 
 ficient for all the essential developments required, they may be 
 denoted by the letters A and D ; A commencing its fall at or 
 near the surface of the earth, which distance from the centre of 
 the earth or centre of gravity, is called unity or 1, while D 
 shall commence its fall from several times the distance from the 
 centre that A does, as at four times, eight times, &c., or at sixty 
 times, which is the distance of the moon, &c., both commencing 
 their fall at the same instant of time ; and the operation to be 
 performed and completed is for each to pass over an equal 
 space, called the given space, whatever difference there may be 
 in the time of the passage of the respective bodies over this given 
 space. 
 
 The given space is some abstract numerical quantity, denoted 
 by unity or 1, and may, if any prefer so to consider it, be de- 
 noted by some arbitrary measure, as one rod, that being very 
 
152 ON THE LAW OF GRAVITY. 
 
 nearly the space over which a body will fall from a state of rest 
 at or near the. surface of the earth, in the first second of time 
 after the commencement of its fall ; or in which a body at the 
 distance of the moon will fall in the first sixty seconds of time 
 after the commencement of its fall. Such given space, however, 
 should be some small quantity in which the intensity of the 
 force applied to either A or D, during the fall over the given 
 space, shall not be increased ; or so that there shall be no in- 
 crease to affect our calculations and the results arrived at, or 
 make them different from a perfect, uniform intensity, for so it 
 will be considered. 
 
 It will be perceived that the object or final result of our re- 
 searches is to ascertain the ratio between the intensity of the 
 force of gravity applied to A, and that which operates on D, dur- 
 ing their fall over the given space. The given space to be 
 passed over by each body, is an element in common between 
 the bodies ; and the operation or labor to be performed by the 
 force of gravity, is to pass each body over the given space, or an 
 equal space ; and as the time of the passage will not be the 
 same in respect to both bodies, we wish to ascertain what rela- 
 tion time has to the intensity of the force which urges the bodies 
 over the given space ; what relation the intensity of the force 
 has to the acceleration of the bodies while passing over the 
 given space, or to the instant velocity of the bodies at any given 
 instant of the fall, or at any given point of the space ; and 
 hence, what is the ratio of the intensity of the force of gravity 
 at the distance of D from the centre of gravity, to that at the dis- 
 tance A ; and hence, to ascertain the ratio or proportion which 
 the whole amount of force expended upon D while passing over 
 the given space, bears to the whole amount of force expended 
 on A while passing over the given space. The premises upon 
 which Sir Isaac Newton, Maclaurin, and other mathematicians, 
 proceeded in their attempts to investigate the law of gravity, viz. 
 to ascertain the intensity of the force of gravity as proportioned 
 to the distance from the centre of gravity, are based upon the 
 very proposition that a body commencing its fall at or near the 
 surface of the earth may be assumed at the distance of unity or 
 1 from the centre of gravity, viz. from the centre of the earth ; 
 and as the moon is sixty semi-diameters of the earth from the 
 centre of the earth, hence a body commencing its fall at the dis- 
 tance of the moon, commences at the distance 60 from the cen- 
 tre of gravity ; and a body at the surface of the earth will 
 fall from a state of rest as far in the first second or moment of 
 time, as the moon will fall from a tangent to its orbit in 60 sec- 
 onds. These proportional distances of A and D are not objec- 
 
ON THE LAW OF GRAVITY. 153 
 
 tionable otherwise than in the extension of a numerical formula 
 adapted to so high a number as 60, intended to aid the investi- 
 gation by an exhibition of the numerical law of falling bodies 
 in its most simple form, and in ringing those changes upon it 
 best calculated to place the phenomena in the most perspicu- 
 ous attitude. 
 
 I shall hence present the most simple numerical formula 
 which I can conceive of, in which the time of the fall of D over 
 the given space will be eight fold that of A ; or in which the 
 distance from the centre of gravity at which t> commences its 
 fall, is eight times that at which A commences its fall. This 
 will require that the given space over which D is to fall shall be 
 divided into 64 equal parts ; and that the time of the fall of D 
 shall be divided into 8 equal parts, called moments, each mo- 
 ment being of an equal length to that employed by A in falling 
 over the entire space. But it will be readily seen that the form- 
 ula given may be extended so as to denote D at 60 times the 
 distance from the centre of gravity that A is, viz. until the series 
 denoted by the constant figure 1 shall amount to 60 places ; 
 thereby not only denoting the proportional distance of D to that 
 of A from the centre of gravity, but also the proportional time 
 of the fall of D over the given space to that of A, (for it will 
 readily be seen that the proportional distance from the centre of 
 gravity will always be as the proportional times of the fall over 
 the given space.) And if the proportional distance or time of 
 D to that of A be as 60 to 1, the given space over which D 
 shall pass will properly be divided into 3600 equal parts, (or 
 into the square of the distance or of the time.) And hence in the 
 last moment of the fall of D, it will pass over 59 of the equal 
 parts of the given space. The formula, then, as applied to D, 
 may be as follows : 
 
 11111 1 1 1 
 
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
 
 Thus denoting the time or distance of D to be eight fold that 
 of A ; and that the given space passed over by D is divided into 
 64 equal parts, the first part being passed over in the first mo- 
 ment, three equal parts in the second moment, five in the third 
 moment, or in the time in which A would pass over the whole 
 64 equal parts. Now the number of equal spaces of the given 
 space passed over by D, is as well the square of the distance of 
 D as of the time; for the distance is as the time, each being eight 
 times that of A. 
 
 The formula also presents this phenomenon : viz. at the end 
 of the first moment figure 2 will denote the rate of velocity ; 
 20 
 
154 ON THE LAW OF GRAVITY. 
 
 figure 4 at the end of the second moment, figure 6 at the end 
 of the third moment, and so on by the even numbers. Hence, 
 at the end of the first moment the velocity would carry the body 
 over two of the equal spaces in one moment, by a uniform mo- 
 tion, without the aid of further force applied ; at the end of the 
 second moment the velocity alone, by a uniform motion, would 
 carry the body over four equal spaces in one moment, without 
 further force applied, and so on. If, then, the whole amount 
 of force applied to D, while passing over the given space, be 
 divided into as many equal parts as the time is into equal mo- 
 ments, so that the several equal parts shall be denoted by the con- 
 stant figure 1, then the velocity at the end of any given moment 
 of the fall may be said to be twice the amount of the whole 
 force expended from the commencement of the fall. If, then, 
 the whole amount of force be thus divided into equal parts, we 
 find that the whole number of those parts is the square root of 
 the whole number of spaces into which the given space is di- 
 vided ; and hence that an equal amount of force is expended in 
 such equal amount of time. Calling, then, the equal amounts 
 of force in the equal moments of time, the forces expended dur- 
 ing the fall over the given space, the sum of all the equal spaces 
 into which the given space is divided is the square of the sum 
 of the equal forces. Hence, the whole amount of space passed 
 over at any given instant of time or at any given point of space 
 during the fall, is the square of the whole amount of force that 
 has then been expended during the fall ; and hence the whole 
 amount of the given space is ihe square of the whole amount of 
 force expended while the body is passing over the given space ; 
 and this, too, whether the time be longer or shorter. Thus, pas- 
 sive time only waits for force to accomplish the labor, or object 
 to be accomplished ; and in such amount of time as shall be 
 required according to the intensity of the force employed ; and 
 that loo without the slightest interference on the part of time, 
 the whole amount of the time of the operation only varying 
 from unity in the reciprocal or inverse ratio to the intensity of 
 the force, precisely in accordance with an operation by the me- 
 chanical powers (as of the wheel and axle) in raising a ponder- 
 ous body. 
 
 Such, then, is the manifest operation of a falling body, which 
 seems to be much more in accordance with all we know of the 
 laws of force and motion, whether mathematically or experiment- 
 ally obtained, than that hypothesis which has been so carelessly 
 guessed off as the physical basis of astronomy, in which just 
 fifty-nine sixtieths of the space passed over by a body commenc- 
 ing its fall at the distance of the moon, or distance 60, is taken 
 
ON THE LAW OF GRAVITY. 155 
 
 or withheld from the labor of active or effective force, and is 
 credited to passive time, thus making time far more active 
 and effective than force, and far the greater operative. The 
 given space, whether applied to A or D, is unity, and is the 
 square of the whole amount of force expended during the fall 
 either of A or D over the given space; and in unity, the roots 
 and powers are equal ; hence the amount of force applied to D 
 is equal to that applied to A, while passing over the given space, 
 A being the standard, all its elements being assumed at unity ; 
 hence, the rate or intensity of the force applied to A is as the 
 whole time of the fall of A, (or figuratively, by the adoption of 
 ratio in unity, the intensity of the force applied to A may be 
 said to be in the inverse ratio or in the reciprocal ratio of the 
 whole time of A's fall.) And so in respect to D, for as 
 the intensity of the force varies from unity, it varies in the in- 
 verse ratio, or in the reciprocal ratio to that of the whole time ; 
 for such is the universal law or condition in respect to force and 
 motion in time, or as applicable to time. 
 
 Here, then, is a principle which always holds good in regard 
 to the effect of the force, (viz. the amount of labor performed,) 
 whatever be the intensity of the force employed, whether it 
 be greater or less than unity, for if the farther body be assumed 
 at unity, the intensity of the force applied to the nearer will be 
 more than unity ; and the time of its fall, less than unity. The 
 principle is this : in a falling body in which the velocity is con- 
 stantly accelerated from the commencement of the fall, the 
 amount of space passed over at any given inslant of time, or at 
 any given point of space, is the square of the whole amount of 
 force which has been expended on the body from the com- 
 mencement of the fall. And this principle holds good whether 
 the body commence its fall at the distance 1, or at the distance 
 60, or at any other given numeral distance from the centre of 
 gravity. Were it otherwise, the whole amount of force applied to 
 a body while passing over a given space from the commencement 
 of its fall, would always be in the inverse ratio to the distance 
 from the centre of gravity. If the distance were 8, the amount 
 of force required to pass the body over a given space from a 
 state of rest, would be but l-8th of what would be required 
 at the distance 1 ; so at the distance 60, the amount of force re- 
 quired would be but l-60th of what would be required at the 
 distance 1, &c. But such is the deduction of Sir Isaac Newton 
 and his followers in respect to the law of falling bodies, and 
 consequently in respect to the law of gravity, viz. the intensity 
 of gravity as proportioned to the distance from the centre of 
 gravity. But it is an operation, an effect produced by force, 
 
156 ON THE LAW OF GRAVITY. 
 
 for which we are seeking, with a view to ascertain or deduce 
 its proportional intensity at different distances from the centre of 
 gravity ; and although we may figuratively speak of the opera- 
 tion of time, of the effect of time, &c. (not of the intensity,) nev- 
 ertheless, such figures are not properly used or adopted in a 
 mathematical investigation of those operations and effects which 
 take place in time ; and it would seem far more natural in an 
 investigation of the intensity of gravity, that the space passed 
 over at any given instant of time, or at any given point of the 
 space from the commencement of the fall, should he referred to 
 the active or operative element of force, rather than to the pas- 
 sive element of time ; that the space passed over should be call- 
 ed the square of the whole force expended from the commence- 
 ment of the fall, which is true in each individual ease, (what- 
 ever the time may be,) rather than that the space passed over 
 should be called the square of the time which has elapsed dur- 
 ing the fall, which can never be true, except the time be assum- 
 ed at unity. For at the distance 1, if the given space be the 
 square of the whole time, at the distance 60, the given space 
 will be the square of only 1-60-th of the whole time. Hence, if 
 the time required at the distance l r for the body to fall over the 
 given space, be increased sixty fold or to 60 times, in conse- 
 quence of a body's commencing its fall at 60 times the dis- 
 tance, then to be sure, in order that the number of spaces shall 
 be the square of the number of times, the given space at the 
 distance 60 must be divided into 3600 equal spaces ; neverthe- 
 less, a formula so constructed, in which the space to be passed 
 over shall only be referred to the time of the passage, affords of 
 itself alone no direct clue to the proportional intensity of the 
 force of gravity at different distances from the centre of gravity. 
 
 Let us then always recollect, or keep in mind, the all-im- 
 portant principle, so full of meaning and so directly material 
 to the issue, and which will disclose itself in an infinity of the 
 most happy results and consequences, viz. that the space pass- 
 ed over at any given instant of time, or at any given point of 
 space from the commencement of the fall, is the square of the 
 whole amount of force which has then been expended during 
 the fall ; or in a falling body in which the velocity is continu- 
 ally accelerated by the force of gravity, the whole space passed 
 over is the square of the whole amount of force that has been 
 expended during the fall ; nor should we forget the important 
 principle that the rate or intensity of the force applied is in the 
 inverse ratio, or in the reciprocal ratio to the whole amount of 
 time in which the force is applied ; nor the no less important 
 principle, that the time is as the distance ; all which principles 
 
ON THE LAW OF GRAVITY. 157 
 
 are readily solved upon the simplest of the mechanical powers, 
 as that of the lever, or wheel and axle ; nor do we find that the 
 immutable law based upon those simple principles, has in any 
 case been abrogated or altered. And if these simple principles 
 were once applied to the mechanism of the heavens, in lieu of 
 that strange complication of human contrivance which now 
 assumes the control, new beauties would be displayed, not only 
 for the wonder and admiration of the learned, but for a vast mul- 
 titude who have been wholly unable to discover much beauty 
 in the works of creation through the medium presented. 
 
 Upon the foregoing simple principles we find this result ; that 
 if one body commence falling at the distance 1, at the same 
 time that a body commences falling at the distance 60, the 
 velocity of the body at the distance 1 will be 3600 times as 
 great as that at the distance 60, at any given instant of time 
 during their continued fall. For the force, viz. the rate or in- 
 tensity of gravity, at the distance 60, being only l-60th, or, 
 .01666, will be .0002777, or the l-3600th part of the given space, 
 which is the square of l-60th part of the whole force applied to 
 A while passing over the given space ; the final result of which 
 will be, that an equal amount of force will be expended on D 
 while passing over the given space, that will be expended on A 
 while passing over the given space. Nor can the operation be 
 carried on otherwise, being established in the immutable fitness 
 of things. Spite of all this, the Newtonian method of investiga- 
 tion makes the intensity of the force the inverse ratio of the 
 square of the reciprocal of the time, as well as of the square of 
 the reciprocal of the distance; all of which is equally absurd; 
 and hence, he allows but l-60th part of the force at the distance 
 60, that is allowed at the distance 1, to perform the same opera- 
 tion or same quantity of work. Thus, if one of the simplest 
 and best established facts in creation is to be regarded, viz. that 
 it is force which moves ponderous bodies, and that it requires 
 a like amount of effective force to move a given body over a 
 given space, whether it be in a longer or shorter time, if the 
 time be only passive, the whole time standing as the reciprocal 
 to the rate or intensity of the force applied, waiting its opera- 
 tion ; then at the distance of the moon or distance 60 from the 
 centre of gravity, the intensity of the force of gravity is J-60th 
 as great as at the surface of the earth, or distance 1 from the 
 centre of gravity. Thus the proportional velocities at any given 
 instant of time of two bodies commencing their fall at the same 
 time, but at different distances from the centre of gravity, are as 
 the spaces generated, or rather, as the space generated by each 
 respectively ; and the whole space generated by either at any 
 
158 ON THE LAW OF GRAVITY. 
 
 given instant of time, or point of space, is the square of the 
 whole amount of force that shall have been expended in urging 
 the fall thus far. If, then, the foregoing is a true exposition of 
 the law of falling bodies, it is also at the same time a very 
 easy and simple exposition of the law of gravity, determining 
 that the rate or intensity of gravity varies inversely as the dis- 
 tance varies ; and as it is demonstrable that such law of gravity 
 would afford to the motion of the moon's apogee just four times 
 the force afforded by the Newtonian law, (and consequently 
 twice the motion,) the adaptation of such law would have saved 
 Clairaut much perplexity in creating factitious force in ac- 
 counting for the observed motion ; and it would have deprived 
 others of the privilege of creating factitious force whenever they 
 found a revolving body not to quadrate with the Newtonian 
 law. The Newtonian method, then, is erroneous. But many 
 conspicuous names are engraved thereon, at an awful height ; 
 but those of Galileo and Kepler are not there ; and whoever 
 will search for those names must wander upon the rock far in 
 the back-ground. 
 
 But the subject is vastly important ; and it will be my en- 
 deavor to throw sufficient evidence before the public to induce a 
 thorough investigation of the subject, even though I should be 
 unjustly accused of resorting to the ingenious method of the 
 painter for distinguishing the man from the bear, which he had 
 painted on the same canvass. I would to Heaven that the world 
 would investigate the Newtonian method of deducing the law 
 of gravity ; for if the world shall still prefer that the deduction 
 shall be drawn from a consideration of the law of falling bodies 
 towards the earth, in lieu of being drawn from a far more nat- 
 ural and simple method, it is certainly proper that we should take 
 heed to our steps, even while travelling in the road which we 
 may prefer. 
 
 I need not here refer to the well known laws of force and motion 
 as applicable to the mechanical powers, as that of the wheel and 
 axle, the windlass, &c., in which the time is as the distance, or 
 length of lever, where the force is applied, and in which the in- 
 tensity of the force applied is inversely as the time; and conse- 
 quently inversely as the distance. And it would seem, also, a 
 matter of supererogation, to refer to the fact that the same law 
 prevails in respect to bodies falling towards the earth ; inasmuch 
 as it is so directly manifest in the phenomena of falling bodies, 
 as presented to view by the numerical formula by which the law 
 is so admirably explained; and more especially, as it is fully as- 
 sented to in that very process by which it has been supposed the 
 
ON THE LAW OF GRAVITY. 159 
 
 fact has been deduced, that the force of gravity is inversely as the 
 square of the distance ; that is, that a body at the distance 60, or 
 at the distance of the moon, requires sixty fold the time in falling 
 from a state of rest over a given space, that a body at the distance 
 1 or at the surface of the earth does. Hence the time is as the 
 distance, by any and every consideration of the subject. Hence, 
 if two bodies commence their fall at the same time at different 
 distances from the centre of gravity, each to perform, describe, 
 or pass over a given or like amount of space, each being acted 
 upon by a uniform intensity of the force of gravity equal to the 
 intensity by which it commenced its fall, the rate or intensity of 
 gravity of each will be to the distance from the centre of gravity 
 at which the fall commences, as it is to the time in which the 
 body shall be falling over the given space ; and this is univer- 
 sally assented to. Hence, it is as conclusively determined, in 
 respect to falling bodies, that the time is as the distance, as it is 
 in respect to the windlass or wheel and axle ; and in respect to 
 the windlass or wheel and axle, the whole time, as also the dis- 
 tance at which ihe force or power is applied, (or length of lever,) 
 is inversely as the rate or intensity of the force applied at the 
 given distance. 
 
 If, then, the rate or intensity of the force applied for accom- 
 plishing the object, namely, the urging of the body over the given 
 space, be inversely, or in the reciprocal ratio of the time, it will 
 also be in the reciprocal ratio of the distance. For as the dis- 
 tance is increased, the intensity of the force is decreased ; (and 
 for this reason alone is the intensity of the force decreased.) The 
 intensity of the force is decreased in some given inverse ratio to 
 that of time ; or is decreased in some given ratio to that of the 
 increase of time. This ratio is found to be, by every considera- 
 tion which the subject has yet received or can receive, to be sim- 
 ply the reciprocal ratio of the time. And hence, to repeat, the 
 time being as the distance, the intensity of the force of gravity 
 decreases in the same ratio that the distance increases, and not 
 in the same ratio as the square of the distance increases, as 
 taught by the Newtonian hypothesis. Let A be conceived to 
 commence its fall at the surface of the earth, or at the distance 
 1, and we will suppose A to fall over just 16 feet of space in one 
 second of time (called one moment) from the commencement of 
 its fall ; and let D commence its fall at four semi-diameters of 
 the earth from its centre, or at the distance 4 ; and the operation 
 to be performed by D is to fall from its commencement, or from 
 a state of rest, over 16 feet of space ; and with a view to certain 
 expositions, A and D shall commence their fall at the same in- 
 
160 ON THE LAW OF GRAVITY. 
 
 slant of time ; the 16 feet over which each body is to fall being 
 called the given space, which given space is denoted by unity, 
 or is that element in common between the two bodies which is 
 denoted by unity. This .is the task to be performed, and we 
 wish to know the proportional amount of labor required in each 
 case to perform the task. Now the 16 feet of space to be passed 
 over by either A or D, is unity or 1 ; the distance of A is 1 ; the 
 time of A is 1; the intensity or rate of force applied to A is 
 1 ; the instant velocity of A, namely, the velocity of A at any 
 given instant of its fall while passing over the given space, is 1 ; 
 and the whole amount of force applied to A while passing over 
 the given space, namely, the whole amount of labor bestowed on 
 A during its passage, is 1. Hence each and every element taken 
 into consideration when applied to A, is assumed at unity. 
 Hence, in respect to A, the given space, being unity, is the square 
 of the distance, of the time, of the rate or intensity of the force, of 
 the instant velocity, and of the whole amount of force expended 
 on A during the fall over the givjen space. 
 
 The given space, to be passed over by D, being also unity or 1, 
 is the square of l-4th of the distance, or of the time ; or is l-16th 
 of the square of the distance or of the time. If, then, the intensity 
 of the force be inversely as the square of the distance or time, 
 then the intensity of the force applied to D will be l-16th of the 
 given space ; and as the time of the application will be 4, hence 
 the whole amount of force applied to D during the fall over the 
 given space will be l-4th of that applied to A, and will be denoted 
 by .25, or l-4th of unity. But if the intensity of the force of gravity 
 be inversely as the distance or time, then the intensity of the 
 force applied to D will be l-4th as great as that applied to A, and as 
 the time of the application of the force to D is 4, hence the whole 
 amount of force applied to D during the fall over the given space 
 will be unity or 1, or just equal to that applied to Awhile passing 
 over the given space. Let not, then, the middle of a formula, ap- 
 plied to D with a view to ascertain the proportion which its respec- 
 tive elements bear to unity orl, or to like elements when applied 
 to A, perplex or thwart our calculations by considering the uni- 
 form series expressed by the constant figure 1, as only denoting 
 time or distance, or rather as simply denoting time. 
 
 The uniform series then expressed by the constant figure 1, as 
 well denotes equal quantities of force expended in equal times, 
 as it does the equal times into which the whole time is divided. 
 
 The great object of research, then, is directly this : to ascer- 
 tain the ratio or proportion which the whole amount of force 
 applied to D while passing over the given space, bears to the 
 
ON THE LAW OF GRAVITY. 161 
 
 whole time of the fall over the given space ; or what is the same 
 thing, to ascertain the ratio of the amount of force expended in 
 any given portion of the whole time, to that of the given portion 
 of the time in which such given amount of force shall have been 
 expended. But the given space is unity, as well when applied 
 to D as when applied to A, and it is an easy matter to divide 
 the given space into equal parts, so that one of them shall be per- 
 formed by D in the first moment, (or l-4th part of the whole time,) 
 three spaces in the second moment, five in the third, and seven 
 in the fourth. And such is the common presentation and extent 
 of the explanation of a numerical formula designed to exhibit the 
 law of a body falling from a state of rest towards the earth ; and 
 this is in fact all the exhibition required when but a single falling 
 body is to be considered, or when there is no ratio to be deter- 
 mined between its elements and those of another falling body 
 urged by a greater or less intensity of gravity ; or rather, when 
 the intensity of gravity is not brought into the calculation or in- 
 vestigation. But when the intensity or rate of gravity becomes 
 a matter of consideration and inquiry ; when the rate or intensity 
 of gravity at different distances from the centre of gravity, is to 
 be investigated by means of a consideration of the law of falling 
 bodies, and in which the time (being as the distance) must be 
 proportioned to the force, and which, consequently, requires a 
 comparison of two bodies, commencing their fall at different dis- 
 tances from the centre of gravity ; a consideration of the formula 
 of D, which only adapts the equal parts of unity, of which the 
 given space is compared, to the number of equal moments in 
 which I) shall be falling over the given space, will not be likely 
 to lead to any satisfactory result in respect to the ratio which the 
 intensity of gravity bears to the time of the fall over the given 
 space, and consequently, the ratio of the intensity of gravity, as 
 proportioned to the distance. 
 
 The uniform series of the formula for D, expressed by the con- 
 stant figure 1, always denotes the distance at which D commen- 
 ces its fall, (that of A being unity; ) it also denotes the units of 
 time (called the times) in which D is passing over the given 
 space, which is unity ; the time in which A is falling over the given 
 space, being unity or 1. So the uniform series expressed by the 
 constant figure 1, most emphatically denotes the number of equal 
 quantities into which the whole amount of force expended on D 
 during its fall over the given space, is divided ; in which case, 
 one of the equal quantities of the whole force expended during 
 the fall, will be expended in one of the equal times ; for the in- 
 tensity of the force is properly considered as uniform as the time. 
 
 Hence, what may properly be termed the accelerating series of 
 21 
 
162 ON THE LAW OF GRAVITY. 
 
 the formula, namely, the natural numbers in iheir order, or the 
 odd numbers in their order, applies equally well to the uniform 
 series when it denotes force as when it denotes time ; arid the 
 sum of the odd numbers of the accelerating series will be as well 
 the square of the forces into which the whole force is divided, as 
 of the times into which the whole time is divided, as also of 
 the distances into which the whole distance is divided, (the tims 
 and distances being but so many equal units of the time and dis- 
 tance of A.) Thus each of the times or of the distances denoted 
 by the uniform series, is unity, or is numerically equal to the en- 
 tire given space to be passed over. Hence, in respect to time 
 and distance only, an equal division of the given space into the 
 sum of the odd numbers in the accelerating series, would seem 
 to be of but little consequence, so far as the main question 
 is concerned ; such being the simple consequence of ratio and 
 proportion, (which always accompany unity of purpose,) the ben- 
 eficial results of which r.re manifested by an application of 
 such equal division of the given space, to an equal division of 
 the whole amount of force expended on the body while pass- 
 ing over the given space ; in which case the amount of force 
 denoted by the first figure 1, in the uniform series, is expended 
 on the first space in the accelerating series, the next amount of 
 force upon the next three spaces, and so on ; in which case 
 the whole amount of force, whatever be its ratio to that of 
 the time or distance, or to unity, is expended in the given space. 
 In such case the amount of space at any given instant of time, 
 or at any given point of the space, generated by the force, 
 (not by the time or the distance,) is the square of the whole 
 amount of force which has been expended from the commence- 
 ment of the fall ; and consequently it is so at the end of the given 
 space, or when the amount of space generated has become unity. 
 Let D, then, commence its fall at the distance 4, and let the 
 given space for A and for D be 16 feet, and let them commence 
 their fall at the same instant of time ; then, when A has fallen 
 over 16 feet, D will have fallen one foot. When D has fallen 
 over 16 feet, its velocity will be equal to that of A when A has 
 fallen over 1 foot ; and when D has fallen over 16 feet, its velo- 
 city will be one-fourth as great as that of A when A has fallen 
 over 16 feet. At any given instant of time while A is falling 
 over the given space, its velocity will be sixteen-fold that of D at 
 the same instant. At any given point of the given space, as at 
 the end of the first or second foot, &c., the velocity of A will be 
 four-fold that of D, at a like point of the given space; hence 
 D must fall over 16 times a given space, in order to acquire the 
 velocity of A at the end of such given space. As the time of A 
 
ON THE LAW OF GRAVITY. 163 
 
 is 1, and the time of D is 4, and as the whole time only stands 
 as the reciprocal of the intensity of the force, waiting its opera- 
 tion, the same amount of force will be expended on D as upon 
 A while passing over the given space. 
 
 It is easy to conceive upon what ground an error might occur, 
 in an attempt to deduce the law of gravity from a consideration 
 of the law of falling bodies, namely, by not properly considering 
 or regarding the simple principle or axiom, that in a falling body 
 there is no motion, not even the least, without acceleration ; 
 namely, that the whole motion is an accelerated motion. Con- 
 sequently in the first moment of the fall, however short we may 
 conceive such moment to be, if it be but time, the motion is ac- 
 celerated therein. If, however, in the consideration of a formula, 
 we conceive the first moment (and consequently the first space, 
 however short or small) to be performed without acceleration of 
 motion ; and of course upon a totally different principle from that 
 which occurs in the remainder of the time and space, in such 
 case it would at least become problematical whether the intensity 
 of the. force of gravity should be assumed as being inversely as 
 the square of the distance, or inversely as the distance. For 
 nothing would occur in the first moment or first space, to make the 
 amount of space then passed over, the square of the amount of 
 force then expended ; for this principle arises wholly from the 
 acceleration of the motion. But if there were no acceleration of 
 the motion in the first moment and space ; if in the first space the 
 motion is to be uniform, and only to be accelerated in future ; then 
 it might well be conceived that the force is as the motion in the first 
 -space or first moment ; and consequently that in the first space and 
 moment, the intensity of the force of gravity is inversely as the 
 square of the distance, or of the whole time ; for surely, in every 
 formula, the first space passed over is but one of the whole num- 
 ber of equal spaces into which the given space is divided, which 
 makes the square of the distance from the centre of gravity. But 
 if such error be once adopted, (and a more manifest error can 
 scarcely be conceived,) the consequences and results must ne- 
 cessarily partake of the same error. And as the force is to be 
 uniform throughout the fall over the given space, a like amount 
 of force as that expended in the first moment will be expended 
 in the next moment of equal length, and while the body is pass- 
 ing over three times the space, in consequence of an accelerated 
 motion acquired in the first moment, and while by supposition 
 the motion was uniform. The fact, however, is too manifest for 
 comment, that if we assign some given moment of time, as 
 one second for instance, to the first space in any given formu- 
 la, such moment being assumed at unity, the first space, how- 
 
164 ON THE LAW OF GRAVITY. 
 
 ever small, may also be assumed at unity ; in which case the dis- 
 tance, the rate of force, the rate of instant velocity, and the whole 
 amount of force expended while passing over such space, will 
 necessarily be assumed at unity. Hence the first term in what 
 is called the uniform series, and the first term in what is called 
 the accelerated series, always form a perfect formula for the 
 standard body, or the one commencing its fall at the distance 1, 
 by which to compare the elements of another; they being but 
 abstract numeral quantities assumed at unity. 
 
 Thus the distance of the moon may as well be assumed at 
 unity as the surface of the earth, and become the standard by 
 which to compare another body commencing its fall at sixty 
 times the distance of the moon from the centre of the earth. 
 Nevertheless, the motion of a body commencing its fall at the 
 distance of the moon, will be accelerated in its fall in the first 
 second of time, or in any shorter time ; and so will that of 
 the body commencing its fall at sixty times the distance of the 
 moon, however small the space may be which is passed over in 
 the first second of time. 
 
 If then the motion of the body be accelerated while passing 
 over the first space, (and no one will assume that it is not,) then 
 the force of gravity is inversely as the distance from the centre 
 of gravity. 
 
 Perhaps a still more simple exposition than has been given of 
 the formulas denoting the law of falling bodies towards the earth, 
 may more readily suggest that comparison of the elements to be 
 considered, from which to deduce the force of gravity as propor- 
 tioned to distance from the centre of gravity, than any which I 
 have heretofore presented. 
 
 When two bodies are conceived to commence their fall at 
 different distances from the centre of the earth, or centre of gravi- 
 ity, with a view of comparing the various elements of the two 
 bodies, they are conceived to commence their fall at the same 
 time, and one of them (called the standard) is conceived to com- 
 mence its fall at the distance of unity, or 1, from the centre of 
 gravity, whether it commence its fall at the surface of the earth 
 or elsewhere, and the other to commence its fall at some given 
 number of units from the centre of gravity, as at the distance 4, 
 8, 60, or any other given distance ; and the space to be perform- 
 ed or passed over by each, as the entire operation of each, is to 
 be equal one with the other, and is assumed at unity, or 1, in 
 respect to either body. Such space is called the given space ; 
 and may, if any prefer to consider it in our arbitrary measures in 
 preference to abstract numeral quantities, be assumed at 16 feet; 
 and that the body at the distance 1, will pass from a state of rest 
 
ON THE LAW OF GRAVITY. 165 
 
 over the given space, in consequence of the force of gravity, in 
 one second of time, permitting a second of time to be called a 
 moment, when used in the investigation, which moment is to be 
 understood as being one second of time. 
 
 As each and every element of the body which commences 
 falling from the distance 1, is properly or necessarily assumed at 
 unity, hence the formula for such body can only consist of two 
 figures, each denoting unity, thus : 
 
 1 
 
 the lower figure denoting the given space to be passed over, and 
 the upper figure denoting the second, or moment of time, in 
 which the space is performed ; also the rate or intensity of the 
 uniform force with which the body is urged, as also the whole 
 amount of force applied to the body while passing over the giv- 
 en space, or the given moment of time. If a body commence 
 its fall at the distance of 4 from the centre of gravity, its formula 
 will necessarily be thus : 
 
 1111 
 
 1234567 
 
 In this formula the given space is divided into sixteen equal 
 parts, denoted by the sum of the odd numbers contained in the 
 lower or accelerated series ; while the time of the passage of the 
 body over the given space is increased to four units, denoted by 
 the four figures in the upper series ; that is, the time being as the 
 distance, is in this formula increased to four seconds, or mo- 
 ments. So also the upper series may denote an equal division 
 of the whole amount offeree expended during the fall over the 
 given space, into four equal parts ; the first part to be expended 
 in the first moment,- the second part in the second moment, and 
 so on. Hence one fourth of the whole amount of force will be 
 expended in one fourth of the whole time ; and one foot of the 
 given space, or one sixteenth of the given space, will be passed 
 over in the first moment, or in one fourth of the whole time ; 
 three feet in the second moment, five feet in the third, and seven 
 in the fourth, in which case the number of equal spaces into 
 which the given space is thus divided, is the square of the num- 
 ber of seconds or moments, or units of time in which the body 
 is falling over the given space, or over unity, as also the square 
 of the number of equal parts into which the whole amount of 
 force expended during the fall is divided. It is then the whole 
 amount of the force expended while the body is falling over the 
 given space, denoted by unity, that we are seeking, and conse- 
 quently we wish to obtain its rate or intensity at the distance 4, 
 compared with that at the distance 1, or compared with unity. 
 
166 ON THE LAW OF GRAVITY. 
 
 Thus if at the distance 4 the rate be but l-16th of unity, then it 
 is inversely as the square of the distance. If at the distance 4, 
 its rate be l-4th of unity, then it is inversely as the distance. 
 Thus at the distance 4, the rate of force is in some given in- 
 verse ratio to the whole distance, or whole time ; and whether it 
 be in the inverse ratio to the first or second power of the whole 
 time, or whole distance, (for the time is as the distance,) is the 
 question. 
 
 It will be readily seen that the formula for a body that com- 
 mences its fall at the distance 4, is a perfect epitome of a formu- 
 la for a body commencing its fall at a greater distance than 1. 
 Nevertheless if the formula be ever so far extended, as for in- 
 stance, if the body be conceived to commence its fall at the dis- 
 tance 60. in which case the uniform series expressed by the con- 
 stant figure l,will be extended to sixty places, (denoting the num- 
 ber of unit distance s, and also the number of seconds or unit 
 times,) still the space to be passed over is unity, notwithstanding 
 it is divided into as many equal parts as unity is contained in 
 the sum of the odd numbers found in the second or accelerating 
 series, namely, 3600, if the distance or number of seconds or 
 moments be 60. The accelerating series then divides unity 
 into equal parts, but does not increase or decrease it, while the 
 uniform series denotes the increase of the units of time from that 
 of one only, which is required when the distance is 1, to that of 
 the number of figures in the uniform series. Each figure, then, 
 in the uniform series, so far as it refers to time or distance, is 
 unity. 
 
 The left hand figure, or figure 1, in the accelerating series, al- 
 ways denotes an amount of space equal to the reciprocal of the 
 square of the distance, or of the time in which the body is pass- 
 ing over the given space or the whole space to be passed 
 over, namely, unity ; nevertheless, the time of passing over the 
 amount of space denoted by figure 1 in the accelerating series, 
 is unity, however small such space may be; and it is manifest 
 that by extending the distance such space may become exceed- 
 ingly small. Nevertheless, if in the first moment of the fall there 
 be motion, (and the fall implies motion,) then space is passed 
 over in the first moment of time, or in one second of time, how- 
 ever small such space may be. 
 
 Now it may be set down as a lemma, or even an axiom, that 
 in a falling body there is no motion without a corresponding ac- 
 celeration of the motion; and that the same force which causes 
 the motion, at the same time causes the corresponding accelera- 
 tion of the motion. Hence if a body commence its fall from 
 ever so many units of distance, and consequently the unit of 
 
ON THE LAW OF GRAVITY. 167 
 
 space to be passed over be divided into ever so many equal 
 parts, (the number of which is always the square of ihe number 
 of units of distance or of time,) the motion of the body while 
 passing over the first equal part of the given or unit space, name- 
 ly, that denoted by the left hand figure in the accelerating series, 
 will be constantly accelerated. 
 
 Hence the uniform series, expressed by ihe constant figure 1, 
 may be used to denote the rate or intensity of the force, not as 
 units of force, as it does units of time or of distance, but in some 
 given reciprocal ratio to that of the whole lime or the whole dis- 
 tance, as that of the simple reciprocal ratio, or as the square of 
 the reciprocal ratio of the whole time or whole distance. 
 
 If the rate of force be in the simple reciprocal ratio of the 
 whole time or w T hole distance, namely, inversely as ihe whole 
 time or whole distance, then the whole amount of force expend- 
 ed while the body is falling over the given space, namely, unity 
 of space, will be unity of force ; that is, the whole amount of force 
 expended will be unity, and the amount of space passed over 
 will be the square of the whole amount of the force then expend- 
 ed. In such case the motion of the body will have been accele- 
 rated, throughout the whole given space, as well in that part de- 
 noted by the first figure in the accelerating series, (however small 
 or great, whether it be the l-16th part or the l-3600th part,) as in 
 any other part of the given space. That is, at any point of the 
 given space, or at ariy instant of the fall, the amount of space then 
 passed over, however small a portion of unity it may be, will be 
 the square of the amount of force which has been expended. But 
 if the rate or intensity of the force be the reciprocal ratio of the 
 square of the whole time or of the whole distance, namely, in- 
 versely as the square of the whole time or of the whole distance, 
 then the whole amount, of force expended on the body while 
 passing over the given space, namely, over unity of space, will 
 be the simple reciprocal ratio of the whole lime, or whole dis- 
 tance, (thus giving to the whole amount of force expended dur- 
 ing the fall, the same numerical quantity, or the same ratio com- 
 pared with unity, that should be given to the intensity,) in which 
 case, at the distance 60, the amount of force required to urge the 
 body over the given space, would be but l-60th of what would 
 be required at the distance 1. 
 
 Now, a more manifest proposition can scarcely be stated or 
 considered, than that such a principle wholly prohibits accelera- 
 tion to the motion of the body while passing over the first space, 
 or that denoted by the first figure in the acceleraling series, now- 
 ever great or small such space may be, and only permits accele- 
 ration to take place in future ; and hence the first space passed 
 
168 ON THE LAW OF GRAVITY. 
 
 over is, by such a course of reasoning, treated in the same man- 
 ner as the given space passed over by the body falling from 
 the distance 1, and in the time 1, and in which the intensity 
 of the force, as also the whole amount of force, are respective- 
 ly assumed at unity, and in which the element of acceleration is 
 not necessarily an element to be considered. If, then, the motion 
 over the first space be not accelerated, but uniform, then the 
 intensity of the force while the body is passing over the first 
 space, (however great or small,) is as the motion of the body 
 moving ; but if the motion be accelerated, then the intensity of 
 the force applied is not as the motion of the moving body. The 
 consequence then is most obvious ; for if we assume the motion 
 to be uniform in the first space, however small, we necessarily 
 make the intensity of the force inversely as the square of the 
 time, or of the distance ; and consequently the whole amount of 
 force expended while the body is passing over unity, or the giv- 
 en space, will be the inverse of the distance, or of the time. 
 
 For the amount of force expended in the first space, (however 
 small such space may be,) is the standard or criterion by which 
 to compare the future acceleration ; for the intensity of the force 
 being uniform, a like amount of force to that expended in the 
 first space will be expended in the three succeeding spaces, or in 
 the next five, and so on. 
 
 Hence the advocates of such hypothesis must say that the 
 same amount of force which produces a uniform motion over 
 the first space, (however great or small such space may be,) will 
 produce an accelerated motion over the three succeeding spaces, 
 and so on. The simple consequence then is, if the motion 
 in the first space be uniform, the rate or intensity of the force will 
 be inversely as the square of the distance. But if there be no part 
 of the given space in which the motion is uniform, or not ac- 
 celerated, or in which the rate or intensity of the force applied 
 is as the motion of the falling body, then the ratio or intensity 
 of the force applied is inversely as the whole distance, or the 
 whole time of the fall over the given space. 
 
 Mr. Vince says that " Sir Isaac Newton found that if the 
 force with which bodies fall upon the earth's surface were ex- 
 tended to the moon, and to vary inversely as the square of the 
 distance from the centre of the earth, it would in one minute 
 draw the moon through a space which is equal to the versed 
 sine of the arc which the moon describes in one minute. He 
 concluded, therefore, that the moon was retained in its orbit by 
 the same force as that by which bodies are attracted upon the 
 earth." That the power of attraction, which causes bodies to 
 fall to the earth, extends to the moon, and retains it in its orbit, 
 
ON THE LAW OF GRAVITY. 169 
 
 is perhaps too manifest for any one to doubt ; and this was the 
 conclusion of Kepler long before the lime of Newton. But how 
 Sir Isaac Newton should have come to such a conclusion from 
 his observaiions and calculations may appear somewhat mys- 
 terious. 
 
 Now that the versed sine of the arc which the moon would 
 d scribe in its orbit in 60 seconds of time, is equal to the 
 amount of deflection from a tangent in the same time, all will 
 agree. This part of the subject I have before partially ex- 
 amined; arid the reason for preferring the method of consider- 
 ing the amount of deflection from the tangent, in preference to 
 the versed sine of the arc, is because it gives a much more com- 
 prehensive range for investigation, 
 
 Now I will not undertake to say but that the versed sine of 
 the arc passed over by the moon in 60 seconds, is equal to the 
 fall of a body from a state of rest near the earth in one second ; 
 to be sure the space passed over in 60 seconds is, in respect to 
 the circumference of the moon's orbit, a very little distance ; 
 and so far as could be well ascertained from observation alone, 
 would not present an error that could readily be detected by ob- 
 servation. Nevertheless, such little distance would be sufficient to 
 contain a mathematical error, vast in its consequences, if it ex- 
 isted. But it must be presumed that Sir Isaac Newton drew his 
 conclusion, that the attraction of the earth extended to the moon, 
 from some calculation made upon said observations ; for other- 
 Wise, he might as well have supposed that Jupiter or Saturn 
 was retained in its orbit by the attraction of the earth. But 
 Mr. Vince says that Sir Isaac Newton found, &c., but the el- 
 ements upon which he > found are not given by Mr. Vince, and 
 hence it is fair to conclude that they are the same that are given 
 in other places, by which it is said the law of gravity was dis- 
 covered to be inversely as the square of the distance, viz : that 
 the moon being 60 times as far from the centre of the earth as 
 the surface is, would be deflected as far from a tangent to its 
 orbit, in 60 seconds of time, as a body near the surface of the 
 earth would fall from a state of rest in one second of time ; 
 and that this amount of deflection from a tangent would be 
 equal to the versed sine of the arc of the moon's orbit which it 
 would pass over in 60 seconds, and which induced the conclusion, 
 not only that the attraction of the earth extends to the moon, but 
 that it varies inversely as the square of the distance. Hence, the 
 statement of Mr. Vince may be resolved into this: that Sir Isaac 
 Newton, according to his calculation, found that if gravity de- 
 creases inversely as the square of the distance, that a body fall- 
 ing from a state of rest directly towards the earth, at the dis- 
 22 
 
170 ON THE LAW OF GRAVITY. 
 
 tance of the moon, would fall as far in 60 seconds, as a body 
 at the surface of the earth would in one second, and that the 
 amount, of such fall in 60 seconds would be equal to the versed 
 sine of the arc which the moon would pass over or describe in 
 the same time; or equal to the deflection of the moon from a 
 tangent in the same time ; and we must add also, that the moon 
 is 60 times as far from the centre of the earth as the surface of 
 the earth is. 
 
 Now that there is error in all these positions except that of 
 the distance of the moon from the centre of the earth, may per- 
 haps be safely alleged upon the truth of mathematics, geometry 
 and the law of falling bodies. 
 
 I am aware that Mr. Vince's statement of what he and others 
 suppose Sir Isaac Newton to have found, is rather vague and 
 intangible; and that if a proposition be drawn from it for the 
 purpose of demonstration, either by way of reductio ad absur- 
 dum, or otherwise, certain things not contained in the statement 
 of Mr. Vince, must be contained in the proposition ; as that the 
 moon is sixty times as far from the centre of the earth as the sur- 
 face of the earth is. Then the proposition may be put thus, viz: 
 If gravity decreases from the centre ol the earth to the distance of 
 the moon, inversely as the square of the distance, and a body at 
 the surface of the earth will fall from a state of rest 16 feet in one 
 second of time, how far will a body, at the distance of the moon, 
 fall from a state of rest, in 60 seconds ; or how far will the moon 
 be deflected from a tangent to its orbit in bO seconds ? And it ap- 
 pears to me that this proposition is so far based upon the state- 
 ment of Mr. Vince, as, when solved, to determine what Sir 
 Isaac Newton found, unless he found something that does not 
 exist. Inasmuch as the dimensions of what he found are not 
 given by Mr. Vince, either in length or breadth, we must en- 
 deavor to deduce them as well as we can from the premises ; 
 and first, a< to the distance a body would fall in 60 seconds, 
 from a state of rest, at the distance of the moon, or at distance 
 60, upon the assumption that gravity varies inversely as the 
 square of the distance from the centre of the earth to the dis- 
 tance of the moon ; and that a body falling from a state of rest 
 at the surface of the earth will fall 16 feet in one second of time ; 
 and also that the amount of fall at the distance of the moon is 
 to be determined by the law of falling bodies. Upon the New- 
 tonian law of gravity, then, a body falling from a state of rest, 
 at the distance of the moon, would fall in 60 seconds of time 
 the 1-. 3600th part of 16 feet, or a little more than l-20ih of an 
 inch; in which case it would fall in the first second of time the 
 l-12,600,000th part of 16 feet. It is a very little distance, to be 
 
ON THE LAW OF GRAVITY. 171 
 
 sure, compared with the time, but it is all that the law of falling 
 bodies will allow upon the Newtonian hypothesis of the law of 
 gravity. This quantity then, viz: the 1 -it3< h part of 16 feet, ac- 
 cording to said proposition, would be Sir Isaac Newton's versed 
 sine of the arc which the moon would describe in 60 seconds; 
 and consequently, according to his calculation, would be the 
 amount of deflection of the moon from a tangent in the same 
 time. Nevertheless, a body at the distance 60 will fall the 
 l-3600th part of 16 feet in the first second. Let us now sup- 
 pose that a body at the surface of the earth, viz: at the distance 
 1, will fall 16 feel from a state of rest in one second of time ; 
 and that a body at the distance of the moon, viz: at the distance 
 60, will fall 16 feet from a state of rest in 60 seconds of time, 
 which are the true proportions. This proposition will at once 
 shift the supposed law of gravity from the inverse of the square 
 of the distance, to the inverse of the distance, or to the law as 
 it is, if it be in anywise corroborated by the law of falling 
 bodies. In such case, the amount of fall at the distance 60 in 
 the first second of time, will be the 1 -3600th part of 16 feet; in 
 the next second, the l-1200th part; in the third second of time, 
 the l-720th part of 16 feet, and so on ; thus increasing in each 
 successive second of time in the ratio of the next succeeding 
 odd number ; and in which case, the whole 16 feet would be 
 accomplished in the 60 seconds, in accordance with the law of 
 falling bodies. But no one can rationally suppose that gravity 
 (which is as uniform at any given distance as time is) will, 
 at the distance 60, exert three times the force in the second sec- 
 ond of time that it does in the first ; five times as much in the 
 third, and so on, any more than it will at the distance 1, or at 
 the surface of the earth; or than time will; which it must do, 
 if gravity be inversely as the square of the distance, in order that 
 a body at the distance 60 may fall as far in time 60, as a body 
 at the distance 1, will fall in time 1. But this is palpably plain, 
 and it being within the reach of every person to examine and as- 
 certain the truth for himself, mankind should not have been lelt 
 in an error in respect to it for a moment. 
 
 I will acknowledge that in respect to the extent of what it is 
 supposed Sir Isaac Newton found, it is left somewhat vague 
 and intangible by Mr. Vince. Nevertheless, mankind have got 
 the idea, or notion, that he found that the moon will be deflected 
 as far from a tangent in 60 seconds of time, as a body would 
 fall from a state of rest in the same time, at the distance of the 
 moon. For it is said that the moon would be drawn towards 
 the earth by gravity, in 60 seconds, through the versed sine of 
 the arc that the moon would describe in that time in her orbit. 
 
172 ON THE LAW OF GRAVITY. 
 
 This is a serious allegation ; and when taken implicitly, and 
 without examination to see if these things are so, or can be so, is 
 well calculated to lead into a wilderness of error, in lieu of lead- 
 ing to the truth in matters depending upon such premises. And 
 I am confident the error contained in I he above statement of 
 what Sir Isaac Newton found, has been loo long continued for 
 the benefit of philosophy. 
 
 The rate of convergency of a planet is inversely as the period. 
 What may be called deflection in time, (so far as rate may be 
 said to apply to deflection.) is inversely as the square of the dis- 
 tance, or equal to Sir Isaac Newton's rate of force. And the 
 rate of deflection in space is inversely as the distance, or the 
 square root of the rate of deflection in time. Hence the rate of 
 convergency is a mean proportional between the rate of deflec- 
 tion in space and the rate of deflection in time. But as the 
 factitious element of deflection in time and space requires a 
 more labored explanation, we will not dwell upon it here. But 
 I will here make the allegation that no one will deny, namely, 
 that the greater the mean distance of a planet of the same sys- 
 tem is, the greater will be the motion in proportion to the rate of 
 gravity or force; and the less will be the rate of convergency in 
 proportion to the rate of gravity. For it is only upon this prin- 
 ciple that the squares of the periods of a system of planets can 
 be as the cubes of their mean distances from the sun. Thus, 
 upon the hypothesis that gravity is inversely as the distance, the 
 mean rate of gravity will always be a mean proportional be- 
 tween the mean rate of motion, and the mean rate of conver- 
 gency ; and in the same ratio that motion increases in proportion 
 to the rate of force as the distance increases, so the rate of con 
 vergency decreases in proportion to the rate of gravity. Thus 
 at the distance 4 the rate of motion is twice as s^reat in propor- 
 tion to the rate of gravity as at the distance 1 ; and the rate of 
 convergency is half as great in proportion to the rate of gravity 
 as at the distance 1. That is, by the hypothesis that gravity is 
 inversely as the distance, the rate of motion increases upon the 
 rate of gravity as the square root of the distance increases from 
 unity ; and the rate of convergency decreases upon gravity as 
 t e rate of motion decreases from unity. 
 
 But upon the hypothesis that gravity is inversely as the square 
 of the distance, the rate of motion will increase upon the rate of 
 gravity as the period increases from unity. Thus if the dis- 
 tance be 4, the rate of motion will be eight times the rate of 
 gravity. And upon such hypothesis of gravity, the rate of con- 
 vergency will increase upon the rate of gravity as the distance 
 increases, in the same ratio that the square root of the distance in- 
 
ON THE LAW OF GRAVITY. 173 
 
 creases from unity; so that when the rate of motion is eight times 
 the rate of gravity, the rate of convergeney will be twice the rate 
 of gravity. Hence the mean rate of motion is to the mean rate 
 of convergeney as the mean distance is to unity, by any law of 
 gravity ; nor do the mean rales of motion and convergeney, as 
 proportioned to each other and to the distance and period, de- 
 pend upon our knowledge of their proportions, or upon our know- 
 ing how gravity is proportioned to distance. But the reason for 
 introducing the foregoing principles in respect to the proportion 
 between motion and convergeney, is for the purpose of showing 
 that the motion of a planet in its orbit counteracts conver- 
 geney ; that is, if the motion be increased in proportion to the 
 force or gravity, the rate of convergeney will fall short of the 
 rate of motion in the ratio of the distance to unity by either hy- 
 pothesis of force. And hence, as the distance is increased, the 
 motion becomes greater, in proportion to the force, and also 
 greater in proportion to the convergeney by either hypothesis of 
 gravity. So then at the distance 4 the convergeney of a body will 
 not be as great in proportion to the motion as when the distance 
 is 1. The consequence then is apparent, that as the distance is 
 increased, not only the convergeney but the deflection is less in 
 proportion to the motion. This becomes apparent if we take a 
 planet \\hose distance is 1, and another whose distance is 4, and 
 conceive the planet whose distance is 1, to have passed over l-4th 
 of its orbit, and the planet whose distance is 4, to have passed over 
 l-8th of its orbit, or over twice the space of the other, and in quad- 
 ruple the time ; in which case, the deflection of the planet whose 
 distance is 1, is unity, while the deflection of that whose distance 
 is 4, is 1.171576, in which case, if deflection were to proceed 
 upon the principle of falling bodies, when the time became 
 doubled, namely, when the planet whose distance is 4 had pass- 
 ed over l-4th of its orbit, the amount of deflection would be 
 4.686304, when in fact it is but 4. Hence the deflection was not 
 three times as great in the last half of the time as it was in the 
 first; and hence deflection does not correspond with the law of 
 falling bodies. 
 
 I have made the example broad for the purpose of showing 
 the error in the expanse. Nor is the error mathematically avoid- 
 ed, by supposing a very little time to have been passed over; as 
 that passed over in 60 seconds of time by the moon. And 
 hence, Sir Isaac Newton did not find, by any law of gravity 
 whatever, that the versed sine of any small arc of the moon's 
 orbit would be the distance over which a body would fall from a 
 state of rest at the distance of the moon, while the moon was 
 describing or passing over such arc. We hence find that the 
 deflection of a planet, or fall from a tangent, is not likened in its 
 
174 ON THE LAW OF GRAVITY. 
 
 progression to that of falling bodies; and hence, as an element 
 for deducing the law of gravity, it is illy qualified, being too for- 
 eign and factitious. It may, however, be proper, inasmuch as so 
 much reliance has heretofore been placed upon it, to give it still 
 further consideration, in which not only its futility as such ele- 
 ment may be shown, but many examples given by which it will 
 be shown that most of the deductions which are apparently 
 drawn from its consideration are mere fictions, and not reali- 
 ties. This is the case in a previous section, notwithstanding it 
 presents much imporlant truth ; and such will consequently be 
 the case in the remainder of the present section so far as deflec- 
 tion is treated of upon the principle that it is one of the legiti- 
 mate elements of the planets and their orbits; and hence the 
 reader must not consider it as an element proper, but that most 
 of the apparent results drawn from its consideration are fictions; 
 however much it may serve as a discipline designed to exhibit 
 specious errors, for the purpose of avoiding them. 
 
 I have said that deflection may be properly divided into deflec- 
 tion in time and deflection in space ; and that the rate of deflec- 
 tion in time, is the square of the rate of deflection in space. 
 For example: Take the planets A and C, A at the dis- 
 tance 1, and C at the distance 4, and let them pass from their 
 tangents some little, but equal time; in which case C, will have 
 passed over one half the space that A has; and according to the 
 commonly received notions of deflections, (which, however, we 
 have shown to be erroneous,) A will have fallen sixteen times 
 as far from its tangent as C has. And such is what I have 
 thought proper to call deflection in time. 
 
 Suppose A should stop, and C pass on over equal space in its 
 orbit that A has, or twice the time that A was passing over equal 
 space, (which I have thought proper to call the deflection in space, 
 or the deflection of two planets while passing over equal space.) 
 Thus, if it were the case, that in equal time, the deflection of C 
 was l-16th that of A, and while passing over equal space was 
 l-4th that of A, the deflection of C in time, would be the square of 
 its deflection in space. Hence, I have made the allegation upon 
 such hypothesis ; and in such case, the deflection in time would 
 be the inverse of the square of the distance, while the deflection 
 in space would be the inverse of the distance. 
 
 If the deflection of C at the end of equal time, were l-16th of 
 that of A, and if its fall from a tangent were governed by the 
 law of falling bodies, then, at the end of equal space, its deflec- 
 tion must necessarily be l-4th that of A. But we have shown that 
 such are not the facts. It is then but simple hypothesis, based 
 upon the erroneous supposition that deflection is identical in its 
 construction with that of falling bodies. 
 
ON THE LAW OF GRAVITY. 175 
 
 We thus perceive that we are dealing in hypothesis, the re- 
 sults of which are fictions, and not those realities which are 
 material to the issue; or at least those fictions are but collateral 
 to the issue. But let us pursue a liitle the consequences arising 
 from the foregoing hypothesis; and to this end I will take A, 
 and another planet, at distance 60, denoted by D, or a body 
 revolving at the surface of the earth, and another at the distance 
 of the moon. 
 
 Now their relative motions will be as 1 to .12909, or inversely 
 as the square roots of their respective distances. And when the 
 spaces passed over by the two bodies in their orbits are as the 
 square roots of their respective distances; or when the times 
 passed over are as their respective distances; then, according 
 to the notions which the world have entertained in respect to the 
 correspondence between the deflection of a planet and the law 
 of falling bodies, the deflection or fall from a tangent will be 
 equal in the two bodies ; and in such case when the spaces 
 passed over in their respective orbits are as 1 to 7.74595, the 
 deflection of one will be equal to that of the other. 
 
 But however plausible such hypothesis may appear, it is nev- 
 ertheless untrue in iact, and should not be permitted to lead into 
 unbounded errors. 
 
 Recur again for explanation to the planets A and C; and 
 upon the hypothesis aforesaid, if C had passed over equal 
 space with A, (and in twice the time,) its deflection would be 
 l-4th that of A; the consequence of which would be, when C 
 had passed over twice the space of A, its deflection would be 
 equal to that of A, viz., when the spaces passed over were as the 
 square roots of their distances, or the times passed over were as 
 their distances. For if deflection be like the law of falling bodies, 
 then, if the deflection of C, in equal time, be l-16th that of A, 
 it will be l-4th in equal space, and in quadruple the time and 
 double the space, it would be equal to that of A. In which case 
 when C has passed over l-8th of its orbit, its deflection should be 
 just equal to that of A when it has passed over l-4th of its orbit, 
 which is not the case; and hence those proportions do not hold 
 in the deflection of two planets, neither in the expanse, nor in 
 any little distance. 
 
 Upon the foregoing erroneous hypothesis, the same amount of 
 gravity expended upon each of two planets of a system (if grav- 
 ity be inversely as the square of the distance) would give the 
 same amount of deflection in each ; and sixteen times the amount 
 of gravity or rate of gravity, would give twice the motion. But 
 if gravity be inversely as the distance, then the same amount 
 expended on two planets of a system will give the same 
 
176 ON THE LAW OF GRAVITY. 
 
 amount of deflection, if the times in which it is expended 
 are as the distances; and four times the amount of gravity 
 will give double the motion. But the results are neither of 
 them correct as to deflection. 
 
 Thus in respect to the factitious and troublesome element of 
 deflection, whether considered in reference to time or space, it 
 furnishes but little useful information in respect to the laws of 
 force and motion ; as it does not correspond with the law of 
 falling bodies, as has been too implicitly believed. To be sure, 
 from an attempt to get the rate of deflection in time, we obtain 
 an expression, denoting the inverse of the square of the dis- 
 tance; and by attempting to consider, or obtain, the rate of 
 deflection in space, we obtain an expression denoting the in- 
 verse of the distance ; both those expressions, we perceive, how- 
 ever, are wholly founded upon assumption. The fact is, there 
 has been a mistake in supposing that the progress of the deflec- 
 tion of a planet from a tangent, caused by the progress of a 
 planet revolving in its orbit, was identical with, or corresponded 
 with the fall of a body from a state of rest. And, but for this 
 mistake, the world might have been saved much trouble. For 
 it has introduced into the science of astronomy that kind of 
 scribe rule, by which things must be cut and tried until they fit; 
 so that already have they been compelled to equate some thirty 
 supposed irregularities of the moon, said to be the legitimate 
 offspring of physical astronomy, resulting from Clairaut's recon- 
 ciliation with the Newtonian law of gravity ; which reconcilia- 
 tion was brought about by more complicated ingenuity than most 
 astronomers and mathematicians are willing to reinvestigate. 
 For having adopted an hypothesis in respect to the law of 
 gravity, which would give but just half the motion to the apogee 
 of the moon which it actually has; and being compelled to sup- 
 ply the deficiency of gravity from extraneous materials ; the 
 moon has in consequence been thrown into more perturbations 
 than ever was a hunted hare ; while those who have essayed to 
 pursue, have been more sagacious than a pack, in tracing out 
 and exploring her devious paths caused by her perturbations. 
 
 I must confess, that I never have been reconciled to the idea 
 that the universe was all an error; and that truth was only to be 
 found at its furthest extremity, after groping through eternal 
 darkness; nor am I well satisfied with that class of truths 
 which are supposed only to be found in those ultimate ratios 
 which are supposed to end in nothing; but I prefer that ratio 
 should have, at least, a numerical point to stand on. But I know 
 not of a more notorious instance of the adoption of the principle, 
 
ON THE LAW OF GRAVITY. 177 
 
 that the path of eternal error is the road to truth, than that 
 evinced in the modern researches for the quadrature of the cir- 
 cle; in which not a single step that is taken in the progress (at 
 least, after taking up the line of march) develops or discloses a 
 single fact material to the issue. Neither the area, nor any 
 linear measure of a single polygon is measured, determined, or 
 known. 
 
 The summing up of the process at any stage of the proceed- 
 ing comes out error. Hence, by such method we -have no 
 criterion by which to determine whether we are or are not in 
 the right path. There is no progression or succession of results, 
 which serves constantly to confirm us in the correctness of our 
 pursuit, no landmarks by the way to cheer us onward, and 
 denote the progress we have made, no pleasant reminiscences, 
 like those of Homer's 
 
 " Wayfaring man, who wanders o'er 
 In thought, the length of road he trod before." 
 
 There is no series of results arising from the application of 
 the powers and roots of numbers, denoting demonstration, which 
 should as fully and conclusively determine the final result as any 
 other in the process. 
 
 Our whole hope and consolation in the process seems to arise 
 from the circumstance, that the farther we carry it, if we sum up 
 our accounts, so much nearer than at any previous step we find 
 we are to what we conceive to be the neighborhood of truth, and 
 we hence conclude that the path we are pursuing leads directly 
 there ; and that truth will be found at the terminus of our route. 
 But this is supposition ; it is not conscious truth ; and how near 
 the terminus of the process the truth lies, we know not. And 
 should it eventually be found that such process is wholly erro- 
 neous, it may serve to weaken the faith of mankind in the belief 
 that truth is only to be sought and obtained through an infinite 
 series of errors. 
 
 I will now conclude this section by a few desultory dictums 
 and remarks. 
 
 The mean distance of a planet divided by the period, gives 
 the rate of mean motion. The distance divided by the rate of 
 motion gives the period. Hence, when the distance is 1, the 
 rate of motion is equal to the period. When the distance is 2, 
 the rate of motion is l-4th of the period. When the distance is 3, 
 the rate of motion is l-9th of the period, and so on. 
 
 If gravity be inversely as the distance, the period divided by 
 the distance will give the amount of gravity expended during 
 the period. If gravity be inversely as the square of the dis- 
 tance, the distance divided by the period will give the amount 
 23 
 
178 ON THE LAW OP GRAVITY. 
 
 of gravity expended during the period. Thus, by the two hy- 
 potheses, the amount of gravity expended during the period of 
 a planet, will be reciprocals of each other ; thus, in one case, the 
 amount of gravity of C during its period, would be 2, and by 
 the other hypothesis, it would be .5. 
 
 If the spaces passed over by two planets of a system are as 
 the square roots of the distances of the planets, each planet will 
 have received an equal amount of gravity with the other. So 
 the gravity exerted on A in time 1, is equal to that exerted on D 
 in time 60, or on C in time 4. 
 
 If gravity be inversely as the distance, twice the motion will 
 balance four fold the gravity. 
 
 If gravity be inversely as the square of the distance, twice the 
 motion will balance sixteen fold the gravity. 
 
 Upon the hypothesis that gravity is inversely as the square of 
 the distance ; and also, that twice the motion balances quadru- 
 ple the force, it follows as a corollary, that if gravity were to be 
 increased so as to become inversely as the distance, the motions 
 of the planets throughout a system, would be equal one with 
 another, viz., the motion of C would, with but l-4th the rate of 
 gravity, be equal to that of A. 
 
 But the hypothesis that gravity is inversely as the square of 
 the distance, presents in its consideration strange dilemmas, 
 which I will not stop here to recount. Suffice it to say that the 
 advocates of such hypothesis claim for it, that twice the motion 
 of a planet balances quadruple the force ; when nothing is more 
 obvious than that the terms of the hypothesis require sixteen fold 
 the force to balance twice the motion. 
 
 But notwithstanding the hypothesis, that gravity is inversely 
 as the square of the distance, is so generally accepted and re- 
 ceived, astronomers have brought strong reasons to bear against 
 it ; as in case of Jupiter and his satellites, when compared with 
 the earth and moon ; for notwithstanding those unphilosophical 
 notions in respect to the different densities of the planets, they 
 have gone upon the principle that twice the motion requires 
 quadruple the force ; and consequently, that gravity is inversely 
 as the distance ; and hence, that the mean motion is always the 
 square root of the mean gravity. 
 
 From a consideration of the times of the periods of Jupiter's 
 satellites, and of the moon, astronomers have estimated the attract- 
 ing power of Jupiter to be something over three hundred times 
 as great as that of the earth. We will suppose then, for the sake 
 of round numbers, that the attracting power of the earth com- 
 pared with that of Jupiter, is as 1 to 324; that the distance 
 of our moon is 1, and consequently each of its elements 1 ; 
 and that Jupiter's first satellite is at the same distance from 
 
ON THE LAW OF GRAVITY. 179 
 
 Jupiter's centre, that the moon is from the centre of the earth. 
 Now in such case, all astronomical calculations have gone upon 
 the ground, that the period of our moon would be eighteen times 
 the length of the period of Jupiter, viz., that the rate of motion 
 in either case is the square root of the force ; and consequently, 
 that the force is inversely as the distance. 
 
 Thus the world, in relation to the law of gravity, seem to have 
 adopted an article of faith which they have not been able, in a 
 single instance, to practise upon. Nevertheless, the most crude 
 and palpable error of Sir Isaac Newton, is yet held more sacred 
 than that infinity of simple truths which are continually urging 
 us to reject it. 
 
 SECTION SIXTH. 
 
 The attempt of astronomers, to account for all the smaller mo- 
 tions of the planets, of the moon, &c., as the aphelia, the 
 nodes, the precession of the equinoxes, &c., upon the principle 
 of disturbing forces caused by the mutual attraction of those 
 bodies upon each other, upon the Newtonian theory of gravity, 
 and upon the principle upon which Clairaut is supposed to 
 have accounted for the motion of the moon's apogee, goes upon 
 the hypothesis that matter is alike endued with an innate princi- 
 ple of gravity or attraction, which operates on all other matter 
 with a force or power of attraction in an inverse ratio to the 
 square of the distance ; and hence the Newtonian theory of 
 gravity, has, through much curious contrivance, been converted 
 into a system of mutual disturbing forces. But if the foundation 
 upon which the superstructure is raised be unsound, the super- 
 structure, of course, must fall. 
 
 But it appears to me that much of the reasoning in support 
 of both the Newtonian theory and law of gravity, as applied to 
 what they are pleased to call physical astronomy, is somewhat 
 a priori ; as having discovered, for instance, the amount of 
 the motions of the aphelia of the planets, and having assigned 
 the cause to disturbing forces upon each other, it became neces- 
 sary then, in accordance with their theory and law of gravity, to 
 assign to each planet its proper share of the disturbing force, al- 
 though it has not been as easy to ascertain the disturbing powers 
 arising from what has been conceived to be the different densi- 
 ties of those planets which have no satellites, as of those which 
 have ; for La Lande and La Grange differ widely as to the den- 
 sity of Venus, La Grange assigning its density in proportion 
 to its distance from the sun, as he does also that of all the other 
 planets, and he avows that he saw no cause for assigning their 
 densities otherwise. 
 
180 ON THE LAW OF GRAVITY. 
 
 But I will quote a little from Mr. Vince, who has given 
 a general digest of this theory, with its reduction and equation to 
 a supposed coincidence with the facts, as found from observa- 
 tion. He says : " Sir Isaac Newton was the first who accounted 
 for this motion (the precession of the equinoxes.) Having prov- 
 ed lhat from the centrifugal force of the parts of the earth arising 
 from its rotation, the equatorial diameter must be greater than 
 the polar, he proceeded to show, that if we conceive a sphere to 
 be inscribed in the earth, the attraction of the sun and moon upon 
 the excess of the quantity of matter in the earth above that of the 
 sphere, will cause a motion in the plane of the equator, and make 
 the points where it intersects the ecliptic go backwards upon it. 
 But although he assigned the true cause of the precession, it is 
 acknowledged that he fell into an error in his investigations of 
 the effect," 
 
 This, then, is substantially the basis upon which they proceed : 
 and after an attempted demonstration, showing that from the ac- 
 tion of the sun alone, the annual precession would amount to 
 21" 6 /y/ , we find these observations, namely : 
 
 " This would be the precession of the equinoxes arising from 
 the attraction of the sun, if the earth were solid, of a uniform 
 density, and the ratio of the diameter as 229 to 230 ; but from 
 what follows, if the greatest nutation of the earth's axis be rightly 
 ascertained, the precession is only 14" 5'"; which difference be- 
 tween the theory and what is deduced from observation, must 
 arise either from the fluidity of the earth's surface, or increase of 
 density towards the centre, or the ratio of the diameters being 
 different from that which is here assumed ; or probably from all 
 the causes conjointly. This regression of the equinoxes (caused 
 by the plane of the equator moving backwards upon the ecliptic) 
 must necessarily cause the poles of the earth to describe circles 
 about the poles of the ecliptic, in a direction contrary to the order 
 of the signs, setting aside the effect of nutation." 
 
 Thus we perceive that heaven and earth may fail, but not one 
 jot or tittle of the Newtonian law or theory of gravity must fail ; 
 everything may be wrong but his theory and his promulgated 
 law of gravity ; when, in case of all his disturbing forces, as well 
 as in respect to the motion of the moon's apogee (even if those 
 motions depended upon disturbing forces) his law of grav- 
 ity would give but o^e-fourth part force enough to produce the 
 effect. 
 
 But in respect to the precession of the equinoxes, we find an- 
 other motion, namely, a nutation of the poles of the earth, which 
 is brought, in connection with the precession, to thwart and per- 
 plex the theory, and cause a division of the equation designed 
 for the precession. 
 
ON THE LAW OF GRAVITY. 181 
 
 And hence we find Mr. Vince, under the head of " Precession 
 and Nutation arising from the Action of the Moon," giving a dis- 
 sertation, mixed with commentaries from Dr. Bradley, upon his 
 observations upon the star Draconis and others, in which refer- 
 ence is made to his discovery of the aberration and progressive 
 motion of light, and in which it is said that Dr. Bradley was the 
 discoverer of the inequalities of the precession of the equinoxes, 
 and the nutation of the earth's axis, arising from the attraction of 
 the moon in different situations of its nodes ; and from the length 
 to which the dissertation is prolonged, I must refer the reader 
 mainly to the article itself. 
 
 Nevertheless, we find, toward the close of the article, this pas- 
 sage : " The surprising agreement, therefore, in so long a series 
 of observations, taken in all the various seasons of the year, as 
 well as in the different positions of the moon's nodes, seems to 
 be a sufficient proof of the truth both of this hypothesis and also 
 of that which I formerly advanced, relating to the aberration of 
 light." And then follows : " The conclusion derived from these 
 observations is, that the gradual diminution of the obliquity of 
 the ecliptic to the equator, does not arise from an alteration in the 
 position of the earth's axis, but from some alteration in the eclip- 
 tic itself; because the stars at the end of the period of the moon's 
 node, appeared in the same places with respect to the equator, as 
 they ought to have done, if the earth's axis had retained the same 
 inclination to an invariable plane," and finally the whole nutation 
 is fixed at 19". And again, Mr. Vince says : " If the annual 
 precession arising from the sun be taken at 21" 6 //; , as in Article 
 1022, and the whole precession be 50", then the part arising from 
 the action of the moon will be 28" 45'", hence (Article 1034) 
 the density of the moon to the density of the sun is as 28" 54'" to 
 21" 6"' multiplied by .988, equal to 20" 8'", which ratio does not 
 agree either with the proportion deduced from the tides, or with 
 the accurate observations of Dr. Bradley. The best method of 
 settling this point is from the greatest nutation." And finally we 
 find the point settled, and equated, so as to make that part of the 
 precession arising from the action of the moon, amount to 35" 
 39'", and that part arising from the action of the sun, 14" 36'". 
 
 And near the close of the chapter, we find, under the head of 
 objections, which appear to be raised for the purpose of answer- 
 ing and allaying them, the following : 
 
 " Another difficulty that may arise is in relation to our having 
 made the effect of the sun's force to be about one-third part less 
 than the quantity resulting from calculations founded on hydro- 
 statical principles and the hypothesis of an uniform density of all 
 parts of the earth. But, that the phenomenon cannot be truly 
 
182 ON THE LAW OF GRAVITY. 
 
 accounted for upon this hypothesis, appears from the concur- 
 rence of all experiments in general ; for, whether we regard the 
 mensuration of the degrees of the earth, the accurate observations 
 of Dr. Bradley on the proportions and times of the tides, the case 
 is the same, and requires a much less effect from the action of the 
 sun than results from, or can consist with the said hypothesis. 
 But if the density of the earth, instead of being uniform, is sup- 
 posed to increase from the surface to the centre (as there is the 
 greatest reason to imagine it does,) then the phenomenon may 
 be easily made to quadrate with the principles of gravitation ; 
 and that according to innumerable suppositions respecting the 
 law whereby the density may be conceived to increase." 
 
 Now, to me, this is a most extraordinary commentary upon 
 the Newtonian theory and law of gravity, as it must be to any 
 person who has not one particle of faith in either. 
 
 I have already remarked, that what are called the disturbing 
 forces of the planets upon each other, are by astronomers placed 
 on similar grounds to the supposed disturbing force of the sun 
 upon the moon ; and hence we find Mr. Vince saying, in treat- 
 ing of the motion of the moon's apogee, " The action of one 
 planet upon another to disturb its motion about the sun, is simi- 
 lar to the action of the sun upon the moon, to disturb its motion 
 about the earth. The general equation of the curve, therefore, 
 which the disturbed planet describes, may represent that which 
 the moon describes about the earth, and thence the irregularities 
 of the moon's motions may be investigated." 
 
 And in accordance with the foregoing, he further says, " These 
 fluxional equations are the same as those determined by Clair- 
 aut, Euler and Mayer, in their treatises upon the theory of the 
 moon, the integration of which is a problem of great difficulty," &c. 
 And Mr. Vince, after making what he calls a near approxima- 
 tion to the ratio between the mean motion of the moon's apogee 
 and the mean motion of the moon, concludes with these re- 
 marks : " Hence we may conclude that the theory of gravity is 
 sufficient to give the true motion of the moon's apogee." 
 
 Now, it is rather a sad commentary upon the intelligence 
 and independence of mind in enlightened communities, that it 
 should be the boast and admiration of the multitude, who are 
 ready to burst into pa3ans and songs of triumph at these suppos- 
 ed achievements in the science of astronomy, by those who are 
 conceived to have so far surpassed the ken or comprehension of 
 mortals, that very few persons upon the face of the globe can be 
 expected to understand the cause and effect, or comprehend the 
 modus operandi by which a knowledge of the operations have 
 been achieved ; and if there is too much arrogance on the one 
 
ON THE LAW OF GRAVITY. 183 
 
 part, and too much implicit faith on the part of the multitude, 
 such arrogance and implicit faith have doubtless, in a measure, 
 had their origin in a kind of national pride, and received their 
 impulse from those who were desirous to make the author of the 
 Principia illustrious ; and that, too, from the fact or circumstance 
 that not more than two or three of his contemporaries were able 
 to understand it, and that more than fifty years elapsed before its 
 great physical truth (the law of gravity according to Newton) 
 was finally understood, acknowledged and appreciated ; and only 
 then by the ingenuity of M. Clairaut, by means of a problem 
 which Mr. Vince often alleges to be one of extreme difficulty, 
 and which the most astute are scarcely able to investigate and 
 understand. 
 
 It is said that Sir Isaac Newton, in extending the consequen- 
 ces of the great physical truth contained in the Principia, gave 
 the true cause of the precession of the equinoxes, namely, 
 the attractive influence of the sun upon the surplus matter of 
 the earth over and above its greatest inscribed sphere ; al- 
 though Mr. Vince says, " it is acknowledged that Sir Isaac fell 
 into an error in his investigation of the effect ;" and that the cel- 
 ebrated English astronomer, Dr. Bradley, discovered a nutation 
 of the poles of the earth caused by the attraction of the moon up- 
 on the surplus matter of the earth over and above its greatest in- 
 scribed sphere ; which nutation, or motion of the poles of the 
 earth, he conceives and defines to be performed in small ellipti- 
 cal orbits, whose diameters are about 18" and 16" about the ac- 
 tual poles, or what he calls the mean poles of the earth, or per- 
 haps around what may properly be called the axis of the heavens. 
 
 This discovery of Dr. Bradley certainly caused much perplex- 
 ity, for a time, to the science of astronomy, until it was final- 
 ly settled (as it were by convention) what portion of the influ- 
 ence in producing the precession of the equinoxes should be 
 assigned to the moon, and what to the sun; for as no division 
 of influences would agree or correspond with the supposed pro- 
 portional densities of the sun and moon, as determined in case 
 of the tides; nor even with the critical observations of Dr. Brad- 
 ley ; together with many other difficulties that were found to 
 exist when tested by their assumed law of gravity ; it was at 
 length found necessary to cut the matter short, by considering 
 what might be produced by the nutation caused by the moon, and 
 assigning the remainder to the sun. And hence some 35" 39 //y 
 of the precession was assigned to the influence of the moon, and 
 14" 36' ' to the sun ; although some give only 30" to the moon, 
 and the remainder to the sun. 
 
 Now the division of the spoil in respect to the small motions 
 
184 
 
 ON THE LAW OF GRAVITY. 
 
 of the nutation of the poles, and the precession of the equinoxes, 
 which has been portioned out to the sun and moon, and to 
 certain great astronomers, seems to have arisen from Dr. Bradley's 
 critical and careful observations upon the apparent motions 
 of the fixed stars ; and in particular upon the star Draconis ; 
 from which phenomena observed, he made liberal deductions 
 as to causes and consequences ; as a nutation of the poles 
 of the earth, and a division of the influence of the sun and 
 moon in producing the precession of the equinoxes ; as also 
 a detection of the progressive motion of light in time, its ra- 
 pidity &c., all of which deductions still pervade our works on 
 philosophy in liberal treatises, embellished by diagrams, and what 
 is termed the higher mathematics. And inasmuch as we all 
 draw liberally upon the Dr.'s detailed observations in making 
 deductions as to the physical causes of the phenomena observed, 
 it is but just that the leading features in his account of them 
 should be liberally quoted, with the view that all may compare 
 the phenomena observed with the various deductions and hy- 
 potheses drawn from them as their physical cause. 
 
 In respect to the Dr.'s hypothesis of the progressive motion of 
 light in time, I certainly believe it to be a very vague conjecture, 
 which should never have been incorporated into philosophy as a 
 component part, clothed with all the dazzling brilliancy of the 
 higher branches of mathematics, and consequently, placed where 
 few would conceive themselves capable of investigating the truth 
 or error of the hypothesis. 
 
 If, however, the detailed observations of Dr. Bradley, on the 
 star Draconis, were alone to be considered, perhaps no phenom- 
 enon furnishes more conclusive evidence of any fact connected 
 with it, than that observed by Dr. Bradley in favor of the hy- 
 pothesis that light possesses no progressive motion. But in 
 respect to the nutation of the poles, it is believed that the Dr. 
 was not remiss in his attempt to assign the physical cause ; and 
 it is also believed, that he is the only one who ever made a suc- 
 cessful attempt to take from Sir Isaac Newton anything which 
 he had found ; nor is this robbery ; it is but a division of the 
 spoil, leaving to the lion his share, viz: his theory and law of 
 gravity ; he only took from the sun, and gave to the moon ; which 
 was certainly in accordance with the Newtonian theory. But 
 why, after the Dr. had so fully assigned what he and everybody 
 else then believed to be the physical cause of the phenomena, it 
 should afterwards have been alleged, that the Dr. was the first to 
 discover by observation, the apparent change in the declination of 
 the stars, (but that the physical causes were afterwards explained 
 by D'Alembert and others,) I know not, unless it was, that they 
 
ON THE LAW OF GRAVITY. 185 
 
 discovered a discrepancy in the Dr.'s deduction of the progressive 
 motion of light, from a supposition, that the apparent declination 
 of the stars was about twice the amount which would be 
 assigned to them, either in a consideration of the precession of the 
 equinoxes, or in a nutation of the poles; for surely the Dr. in 
 his speculations on the subject, transports us through a whole 
 period of the moon's nodes ; at the end of which he again finds 
 things as they were, except that the equinoxes have continued 
 their regressive motion. 
 
 There has been no disposition, however, in any of these as- 
 tronomers, to break loose from the Newtonian theory or law of 
 gravity ; and inasmuch as Sir Isaac Newton, in extending the 
 consequences of the great physical truth of the Principia, assign- 
 ed the phenomena of the precession of the equinoxes, to the 
 direct agency of gravity upon the surplus matter about the 
 equator, over and above the greatest inscribed sphere of the 
 earth, of course the same has never been questioned or doubted ; 
 and hence, as those motions of the earth were placed directly 
 under the Newtonian theory and law of gravity, and inasmuch 
 as a law would be useless, unless it were obeyed, it is said, that 
 D' Alembert and Laplace have succeeded by analysis in reducing 
 all these intricate phenomena to the law of gravity with the 
 most complete success ; and such is generally understood to be 
 the case. And inasmuch as the labor and ingenuity has been 
 so vast in reducing, adjusting and equating them to the Newto- 
 nian law of gravity, compared with what it would have been 
 to have simply accounted for them upon some true or rational hy- 
 pothesis, it might seem a pity to attempt to disturb the decis- 
 ion, even upon the discovery of ample and satisfactory evidence. 
 
 Nevertheless, I am bound to consider that those who have 
 reconciled these phenomena to the Newtonian law, have no 
 more abided by the two maxims laid down by Newton, than 
 Sir Isaac himself did ; namely, " No more causes are to be 
 admitted than are sufficient to explain the phenomena," and 
 that of "effects of the same kind, the same causes are to be 
 assigned, so far as it can be done." 
 
 Now aside from the question whether the rate or intensity 
 of gravity be inversely as the distance, or inversely as the 
 square of the distance, can it be rationally conceived, or even 
 pretended, that the position and motion of the earth requisite to 
 produce the precession of the equinoxes, or the nutation of the 
 poles, are any more dependent upon the direct operation, or force 
 of gravity, than is the position of the poles or equator of the 
 earth, in respect to the ecliptic, or the rotatory or diurnal motion 
 of the earth upon its axis ? 
 24 
 
186 ON THE LAW OF GRAVITY. 
 
 And if not, then there is no direct physical necessity, that the 
 force of gravity should be the direct or immediate cause, and 
 hence it should not be assigned as the immediate cause, unless 
 such shall be the necessity of the ease. That is, if the phenom- 
 ena may take place or exist without being directly or immediately 
 caused by gravity, we then have no business to assign gravity as 
 the direct cause. We then may draw something from analogy 
 in sustaining the hypothesis, that the earth may have the requi- 
 site motions for producing the phenomena under consideration, 
 without resorting to the immediate agency of gravitation; or 
 without resorting to the ridiculous idea of a projection of the 
 planets in tangents to their orbits in the outset of their career. 
 
 Now the nutation of the poles is called, and conceived to be, 
 Lunar Nutation ; and it is said that Physical Astronomy has 
 made known the existence of another nutation, called Solar 
 Nutation ; but that it is too small to be detected by observation. 
 Thus what is termed Physical Astronomy, based upon the 
 Newtonian theory and law of gravity, has already subjected the 
 moon to some thirty equations for the correction of supposed 
 perturbations ; and the earth seems destined to be hunted down 
 in much the same way, unless the world should happen to dis- 
 cover that the attractive influence of the sun at his distance, is 
 capable of producing more than two hundred times the motion 
 of the earth ; that the attractive influence of the moon is, at her 
 distance from the earth. 
 
 Nevertheless, Precession and Nutation, are actual motion, 
 and not merely position, as the obliquity of the ecliptic to 
 the equator, or to the poles of the earth would be, were it not 
 for the precession and nutation ; and as my theory is, that all 
 the motions of the heavenly bodies are caused and controlled 
 (either directly or indirectly) by force, or gravity ; that they are 
 the effects of the operation of force, and not of time; hence, 
 precession and nutation will fall within the same category as 
 that of any other motion of the heavenly bodies, and be subject 
 to the same laws. 
 
 SECTION SEVENTH. 
 
 It is not my design to give a theory of the moon's motions, 
 but to endeavor to relieve her from some of those perturbations 
 which have been caused by the attempts of astronomers and 
 mathematicians, to bring their theory to agree with observa- 
 tion ; and for authority on this subject, I shall mostly refer to 
 Vince's Astronomy, vol. 2, chap. 32nd, in which the theory of the 
 moon, as now understood, seems to have been pretty thoroughly 
 elaborated. 
 
ON THE LAW OF GRAVITY. 187 
 
 Mr. Vince, after some preliminary remarks on the subject, 
 commences thus : "As the attractive force varies inversely as the 
 square of the distance," &c. 
 
 Again, Mr. Vince says, " Sir Isaac Newton, having found 
 that the moon was retained in its orbit by a force, which at dif- 
 ferent distances from the earth, varied inversely as the square of 
 the distance ; and concluding from analogy, that the same law 
 of attraction might take place between all the bodies in the sys- 
 tem, applied this theory, (called the theory of gravity,) to com- 
 pute the effect of the sun's attraction upon the earth and moon, 
 so far as it might affect the relative situation of the latter as seen 
 from the former ; and hence he discovered, besides the irregular- 
 ities before observed, other small irregularities of the moon's 
 motion, whicli also were found to agree with observations. 
 From these and other applications of his theory, he was confirm- 
 ed in his conjectures concerning the principle of universal 
 gravitation ; and the same principle having since been further 
 applied, and found to produce conclusions conformable to obser- 
 vation, his theory of gravity is now firmly established." To 
 which is appended the following note, as farther explanation : 
 " Sir Isaac Newton found that if the force with which bodies 
 fall upon the earth's surface were extended to the moon, and to 
 vary inversely as the square of the distance from the centre of 
 the earth, it would in one minute draw the moon through a space 
 which is equal to the versed sine of the arc which the moon 
 describes in a minute. He concluded, therefore, that the moon 
 was retained in its orbit by the same force as that by which 
 bodies are attracted upon the earth." 
 
 And the truth in respect to the foregoing proposition is cer- 
 tainly a desideratum ; and should there be error, its detection 
 may serve to show how prone we are to receive those things for 
 truths, at the hands or suggestions of others, which it is our busi- 
 ness and our privilege, to examine for ourselves ; and that we 
 ought not too implicitly to rely upon the truth of those things, 
 which others profess to have found, (however renowned for 
 their wisdom,) lest we imbibe error in place of some great lead- 
 ing truth, and then call upon heaven and earth to yield their 
 laws to sustain and corroborate it. 
 
 I will now, in short, state the premises upon which I propose 
 to compare my result in respect to the law of gravity with that 
 of Sir Isaac Newton's, as deduced from the law of falling 
 bodies near the earth, and its identity with that of gravity which 
 retains the moon in her orbit. 
 
 1. For the purposes of calculation, Newton, Maclaurin and 
 others have gone upon the ground that the moon is sixty times as 
 
188 ON THE LAW OF GRAVITY. 
 
 far from the centre of the earth as it is from the surface of the 
 earth to its centre. 
 
 2. By the law of falling bodies, it is found that a body falling 
 from a state of rest near the surface of the earth will fall in one 
 second of time from its commencement, about sixteen feet, which 
 may, for the sake of ease in calculating, be called one rod, or 
 abstractly, one space ; and the second of time in which it is 
 falling, may, abstractly, be called one moment. 
 
 3. We will also consent, (for the present purpose,) that Sir 
 Isaac Newton found, from observation and calculation upon the 
 orbit of the moon, and her motion in such orbit, that the versed 
 sine of the arc of the moon's orbit described in 60 seconds of 
 time by the moon while revolving in her orbit, is just one rod, 
 or one space, or just as far as a body will fall from a state of 
 rest near the surface of the earth in one second of time, or 
 that the moon, at the distance 60 from the centre of the earth, 
 will fall just as far in 60 seconds or moments of time, as a body 
 at the surface of the earth, or at l-60th part of the distance of 
 the moon from the centre of the earth, will in one second or 
 moment of time. 
 
 Now these are the premises ; and the question to be deter- 
 mined is, whether, to produce these results, the force of gravity 
 towards the earth (or centre of the earth) be inversely as the 
 square of the distance, as taught by Sir Isaac Newton, or 
 simply inversely as the distance, as I allege it to be. 
 
 These are the premises from which Sir Isaac Newton and 
 others have drawn the conclusion that the force of gravity varies 
 inversely as the square of the distance varies. And these are 
 the premises from which I draw the conclusion that the force of 
 gravity varies inversely as the distance varies. 
 
 I contend that Sir Isaac Newton applied the inverse of the 
 square of the distance to the intensity or rate of the force ap- 
 plied ; when it only refers to the amount of the fall in given 
 time ; that is, in definite moments of time, to w r hich it does 
 apply. 
 
 For the purpose of making a few additional commentaries 
 on the subject, I will here again quote Mr. Maclaurin's exposi- 
 tion of Sir Isaac Newton's hypothesis, in respect to the law of 
 gravity, in view of the circumstance that he was a mathematician 
 of celebrity, and also a personal friend of Sir Isaac Newton. He 
 says, " The computation may be made after this manner : the mean 
 distance of the moon from the earth being 60 times the distance of 
 heavy bodies at the surface of the earth from its centre, (a body 
 at the surface of the earth will fall 15yV Parisian feet in a sec- 
 ond,) and her gravity increasing in proportion as the square of 
 
ON THE LAW OF GRAVITY. 189 
 
 the distance from the centre of the earth decreases, her gravity 
 would be 60 X 60 times greater near the surface of the earth 
 than at her mean distance, and therefore would carry her through 
 60 X 60 X 15 T V Parisian feet in a minute near the surface ; but 
 the same power \vould carry her through 60 x' 60 times less 
 space in a second than in a minute, by what has been often ob- 
 served of the descent of heavy bodies ; 'and therefore the moon, 
 in a second of time, would fall, by her gravity, near the sur- 
 face of the earth, 15-^ Parisian feet; which therefore is the 
 same with the gravity of terrestrial bodies." 
 
 Now this might be well enough for an article of faith, if a 
 person had no particular regard for its being true. But the 
 method of begging the question, or assuming without proof, in 
 matters that should be demonstrated, may not always make the 
 evidence as conclusive as it may be specious. 
 
 Mr. Maclaurin says, " a body near the surface of the earth 
 will fall ISyV Parisian feet in a second ; and her (the moon's) 
 gravity increasing in proportion as the square of the distance 
 from the centre of the earth decreases, her gravity would be 60 
 X60 greater near the surface of the earth than at her mean dis- 
 tance ; " (that is, in case that gravity increases as the square of the 
 distance decreases, the gravity of the moon would be 3600 times 
 as great at the surface of the earth as at, her mean distance ; 
 which would be a fair corollary or conclusion, if the premises 
 were true;) and therefore" (Mr. Maclaurin says) would carry 
 her through 60 X 60 X l&iV Parisian feet in a minute (or 60 
 seconds) at the surface;" thus making the force of gravity, or 
 urging force, increase with the acceleration, during the fall." 
 
 Mr. Maclaurin proceeds thus : " But the same power would 
 carry the moon 60 X 60 times less space in a second than in a 
 minute by what has been often observed of the descent of heavy 
 bodies ; " (which allegation is true, as found from observation and 
 experience, whatever be the law of gravity ;) " and therefore " (Mr. 
 Maclaurin says) the moon in a second of time would fall by her 
 gravity near the surface of the earth 15-& Parisian feet ; which 
 therefore is the same with the gravity of terrestrial bodies." 
 
 And this last allegation is true, for the simple reason, that a 
 body falling freely from a state of rest, near the surface of the 
 earth, will fall 15^ Parisian feet in one second of time. 
 
 But why the word therefore, should have been called in aid, I 
 know not, as the allegation, that the moon would fall 15 T V*Pa- 
 risian feet in one second of time near the surface of the earth, 
 certainly is not a corollary or consequent flowing from his first 
 allegation, viz : that gravity increases inversely as the square of 
 the distance decreases ; as his .last allegation would be true by 
 
190 ON THE LAW OF GRAVITY. 
 
 any law of gravity. Neither does it apply to his second allega- 
 tion, as that also would be true by any law of gravity ; for it is 
 an established fact, that necessarily applies to, and is controlled 
 by the law of gravity, whatever that law may be. 
 
 The whole of Mr. Maclaurin's or Sir Isaac Newton's sup- 
 posed demonstration, or explanation of the law of gravity, as 
 conceived to have been deduced from the law of falling bodies, 
 then vanishes in thin air. They have only seduced themselves 
 by a shadow of demonstration, where no substance exists. 
 
 They have laid no premises from which conclusions are 
 drawn, save the assumed or unproved and erroneous allegation, 
 that the force of gravity is inversely as the square of the dis- 
 tance ; and yet, upon those assumed premises, rests most of 
 the fabric of modern astronomy. 
 
 But what the world would be glad to know, is, whether the 
 force of gravity decreases as the distance increases, or as the 
 square of the distance increases ; and upon this point, Mr. Mac- 
 laurin has given us no proof or satisfaction whatever. 
 
 And I will here protest against this hap-hazard manner of 
 dispensing science to the world, either in respect to the quadra- 
 ture of the circle, or the law of gravitation, wherein no premises 
 are laid from which logical deductions can be drawn, although 
 conclusions are made and sent forth to the world, which too often 
 stifle further inquiry, from an implicit faith in those who have 
 assumed to make them. But I deny to the Pope himself, the 
 right thus to sell indulgences to the world. 
 
 Mr. Maclauriri has, however, well said, " that the moon would 
 fall through 3600 times less space in one second, than in one 
 minute, (or 60 seconds,) by what has often been observed of the 
 descent of falling bodies ; " thus consenting to the law of fall- 
 ing bodies as it is ; and because such is the law of falling bodies 
 the force of gravity varies inversely as the distance varies. 
 
 A falling body then, urged by a constant and uniform force 
 through 60 seconds of time, from a state of rest, will fall 3600 
 times as far as in the first second of its fall. Hence if a body 
 at the surface of the earth fall over one rod, or 198 inches, in the 
 first moment (or second) of its fall, the moon will fall over a lit- 
 tle more than l-20th of an inch in the first moment or second of 
 its fall. 
 
 Thus, if both bodies commence their fall at the same time, 
 that at the surface of the earth will fall 3600 times as fast, at any 
 given instant, as that at the distance of the moon will. So if 
 the time of the fall of the moon be 60 seconds, it will fall 119 
 times as far in the last second or moment, as in the first, and 
 with the same amount of force expended ; and the same ratio 
 will hold as to the fall of the body near the earth. 
 
ON THE LAW OF GRAVITY. 191 
 
 Thus we see that time only compensates for the lack of the 
 intensity or rate of force, just in accordance with the laws of 
 dynamics, from the days of Archimedes to the present ; but does 
 not supply the place of force, or compensate for it, in any other 
 way than that of a lack or want of intensity. 
 
 Thus Archimedes could move the world with his lever, if he 
 had a proper fulcrum and a place to stand on ; and the allega- 
 tion is true in theory. But we learn from his notions of the 
 operations of time, force, and motion, that he would expect no 
 assistance from time ; otherwise than that, if he would take time 
 enough for the accomplishment of his enterprize, it would com- 
 pensate for his weakness. 
 
 This principle in dynamics comes also under the most com- 
 mon observation ; as in the common use of the lever in raising 
 weights by a windlass, whether with a horizontal or upright 
 shaft ; as well as in the use of all the mechanical powers, in which 
 we make use of time to compensate for the lack or want of the 
 intensity of force. 
 
 Nevertheless, the same amount of force is necessary to accom- 
 plish the object, whether the time be longer or shorter ; for time 
 does not do the work, but only furnishes the time, and hence 
 does not compensate otherwise than for a lack of power to do it 
 sooner. 
 
 Hence there will be just as much force of gravity expended 
 in 60 seconds at the distance of the moon, as in one second at 
 the surface of the earth ; and in either of the 60 seconds at the 
 distance of the moon, there will bel-60th part as much force ex- 
 pended, as in one second at the earth's surface. 
 
 But if the force of gravity be inversely as the square of the 
 distance, I believe all will readily agree, that the whole amount 
 of force expended on the moon in sixty seconds of time, will be 
 but the l-60th part as much as that expended on the body near 
 the earth in one second of time ; in which case, at the distance 
 of the moon, time must actually supply the place of 59-60ths of 
 the force requisite to urge it over the given space, unless we 
 adopt another hypothesis, equally though no more absurd, name- 
 ly, by supposing the force to increase as the motion accelerates, 
 or as the odd numbers in their order ; in which case, 119 
 times the force that is expended in the first would be expended 
 in the last second of time. 
 
 For no one will contend, if the force of gravity be inversely 
 as the square of the distance, that more than l-60th part of the 
 force is expended on the moon that is expended on the body 
 near the earth while passing over equal space. 
 
 But if the force of gravity be inversely as the distance, the 
 
192 ON THE LAW OF GRAVITY. 
 
 same amount of gravity or force will be expended on the moon 
 while falling over a given space, as will be expended on a body 
 near the earth while falling over an equal space, but in l-60th 
 part of the time ; in which case the lack or want of intensity of 
 force at the distance of the moon, would be compensated for by 
 time, in accordance with all the laws of dynamics with which 
 we are acquainted. 
 
 Hence we arrive at this important fact in the law of falling 
 bodies, namely, if we conceive an individual body to fall over a 
 given space, and that the time of its fall be divided into any 
 number of equal moments, we must also conceive the amount 
 of force applied in each moment, to be equal one with another ; 
 then, in the first moment, if the amount of space fallen over be 
 called 1, the amount fallen over in the second moment will be 3, 
 only the like amount of time or force being expended as in 
 the first moment ; in the third moment, the spaces will be 5, and 
 so on, by the odd numbers ; in which case, at the end of any 
 given moment, the whole number of spaces fallen over will be 
 as well the square of the several amounts of force expended in 
 all the several moments, as of the sum of the moments them- 
 selves. 
 
 But if we compare two bodies commencing their fall at the 
 same time towards the gravitating or attracting power, at different 
 distances from such power, as in case of the moon and a body 
 near the surface of the earth, namely, one body at the distance 1, 
 and another body at the distance 60, for example, we shall 
 have the following results, namely : It is manifest that if 
 we conceive the two bodies, one near the surface of the earth, 
 at the distance 1, and the moon at the distance 60, to have fallen 
 over an equal space, namely, one rod for example ; that near the 
 earth in one second, and the moon in 60 seconds of time ; then, 
 taking the seconds as the time, the times of each, in falling over 
 equal space, will be as their respective distances from the centre 
 of gravity ; while the force employed on each will be as the 
 amount of space fallen over. Hence, in either case, the amount 
 of space fallen over or described, will be the square of the whole 
 amount of force employed in the fall ; while the amount of force 
 employed upon the moon during its fall, is but l-60th part of 
 the time of its fall. 
 
 Hence, the Newtonian law of gravity essays to convert time 
 into actual force, in lieu of permitting it to perform its legitimate 
 office of compensating for want of intensity of force, which has 
 been well known to be its legitimate office for thousands of years. 
 
 Thus have I endeavored fairly to present the evidence, the ma- 
 terials, the elements, from which Sir Isaac Newton deduced his 
 
ON THE LAW OF GRAVITY. 193 
 
 conclusion that the force of gravity is inversely as the square of 
 the distance, and from which materials or elements I deduce my 
 conclusion that the force of gravity is inversely as the distance. 
 
 And hence, as a corollary or consequent of my determination, 
 if the mean motion of a planet (whether revolving in a centric or 
 eccentric orbit) were to be doubled in order to retain it in its or- 
 bit, the mean force would only have to be quadrupled, or in- 
 creased four-fold, in lieu of being increased sixteen-fold, as would 
 be requisite by the Newtonian law of gravity ; and consequently, 
 the motion of the moon, whether in its simple revolution 
 about the earth, or in the motion of its apogee, (as well as 
 the motion of each heavenly body,) would be entitled to just 
 four times as much instant or constant force by my determina- 
 tion of the law of gravity, as by that of Sir Isaac Newton, as will 
 be readily perceived by recurring to a system of planets which 
 revolve about one and the same central power, as in case of the 
 planets about the sun. 
 
 And now, notwithstanding my belief is that I have, in the 
 most direct and conclusive manner, confuted the Newtonian deter- 
 mination in respect to the law of gravity, by showing that he as- 
 signed the square of the distance to the intensity of the force, 
 when it should have been assigned to the amount of the fall, (or 
 to what they have seen proper to call the amount of the deflec- 
 tion of the planet from a tangent to its orbit,) or the result of the 
 force applied, I will, byway of reductioad absurdam, present 
 some few of the difficulties and perplexities under which the phi- 
 losophers, the astronomers, and the mathematicians of Europe 
 for a long time labored, in their endeavors to reconcile the New- 
 tonian law of gravity with observed phenomena, and the facts 
 that were found from observation to exist ; and in which what 
 they supposed themselves required to adopt as a first principle, 
 namely, the Newtonian law of gravity, led to so much confusion 
 and mystery, as to cause them for a long time to conspire and 
 rebel against the law so promulgated; until at length, in despair 
 of extricating themselves from its unjust requirements, they as- 
 sented to and ratified the treaty so ingeniously devised by the 
 great French mathematician Clairaut. 
 
 And here I will render the only rational excuse which I can 
 devise in justification of their final acquiescence in, and acknowl- 
 edgment of the Newtonian law ; namely, that they could not con- 
 ceive an error to have been made in- respect to the well known 
 law of falling bodies; and hence, they even considered it use- 
 less, as well as derogatory to Newton, to investigate what he 
 had determined as being the result of so simple and easy a cal- 
 culation. 
 
 25 
 
194 ON THE LAW OF GRAVITY. 
 
 But this is a subject which I propose to press upon the world; 
 to ask the institutions of my own, and of foreign countries, 
 to reinvestigate ; and that I shall, eventually, have supporters, who 
 will dare to risk their reputation by venturing to think for them- 
 selves, I have no doubt. 
 
 But to refer to some of the difficulties under which the math- 
 ematicians and astronomers of Europe have labored, in respect 
 to the law of gravity, as promulgated by Newton, I quote from 
 Mr. Vince, who though a very honest recorder of scientific dis- 
 coveries, is nevertheless, quite partial to Sir Isaac Newton. 
 
 Mr. Vince, in the introductory remarks to his treatise upon the 
 Theory of the Moon, says, " To enter into a computation of all 
 the effects of the disturbing force of the sun upon the moon, 
 and their nature, would require the investigation of the nature 
 of a curve described by a body attracted to two points, called 
 the Problem of three Bodies ;" which problem, he suggests, has 
 been solved by but very few mathematicians, amongst whom 
 are Clairaut and some few others. And it appears, that for more 
 than half a century from and after the time that Sir Isaac New- 
 ton promulgated his theory and law of gravity, the learned 
 world essayed to apply them to the motions of the moon. 
 Mathematicians were exceedingly perplexed with Sir Isaac 
 Newton's law of gravity ; for Mr. Vince proceeds thus : " In 
 the year 1747, M. Clairaut, in a memoir read before the 
 Academy of Sciences, in Paris, made an objection to this law, 
 upon this ground; that it would not account for the motion of 
 the moon's apogee, it giving, according to his calculations, that 
 motion only one half of what it was found to be by observation ; 
 and he concluded, that it was necessary to change this law, 
 by adding something to correct it. He however soon after dis- 
 covered his mistake, and was the first who gave a complete the- 
 ory of the moon, in which he showed that Sir Isaac Newton's 
 law of gravity would not only account for the motion of the 
 moon's apogee, but also for all the irregularities of the moon," 
 and Mr. Vince proceeds to say, that, " M. Euler has done great 
 justice to M. Clairaut upon his solution of this important prob- 
 lem, in a letter to Rev. Casper Witstien ; in which he observes, 
 that, < This question is of the last importance ; and I must own, 
 that till now, I always believed that this theory did not agree 
 with the motion of the apogee of the moon.' M. Clairaut was 
 of the same opinion ; but he has publicly retracted it, by de- 
 claring that the motion of the apogee is not contrary to the New- 
 tonian theory. Upon this occasion, I have renewed my in- 
 quiries on this affair ; and after most tedious calculations, I have 
 
ON THE LAW OF GRAVITY. 195 
 
 at length found to my satisfaction, that M. Clairaut was in the 
 right, and that this theory is entirely sufficient to explain the 
 motion of the apogee of the moon. As this inquiry is of the 
 greatest difficulty, and those who hitherto pretended to have 
 proved this nice agreement of the theory with the truth, have 
 been much deceived, it is to M. Clairaut that we are obliged 
 for this important discovery, which gives quite a new lustre to 
 the theory of the great Newton ; and it is but now, that we can 
 expect good astronomical tables of the moon." 
 
 And in accordance with the foregoing suggestions of M. Eu- 
 ler, we find from Mr. Vince that various had been the attempts 
 to reconcile the Newtonian law of gravity with fact and obser- 
 vation in respect to the motion of the moon's apogee ; for Mr. 
 Vince says that " M. Walmsley, in his theory of the Motion of 
 the Apsides, has computed the mean motion of the moon's ap- 
 ogee, and his conclusion agrees very well with observation ; but 
 his principles are altogether wrong ; for he has entirely omitted 
 that part of the force which acts in a direction perpendicular to 
 the radius, which, as we have shown, produces just one half of 
 the motion ; he also assumes the mean disturbing force in the 
 direction of the radius as acting constantly, instead of the real 
 disturbing force ; and he also wrongly computed the periodic 
 time of the moon ; it was by accident therefore that he obtained 
 the mean motion ; in respect to the true motion his conclusions 
 are erroneous ; he says Mr. Machin has not given us his pro- 
 cess ; we cannot therefore say how far it was just. He says 
 also, in the Phil. Trans, of 1751, that Mr. P. Murdock has given a 
 method of computing the mean motion of the moon's apogee, 
 by first considering only that part of the disturbing force which 
 acts in the direction of the radius ; and then instead of supposing 
 the earth to be at rest, by conceiving the earth and moon to re- 
 volve about their common centre of gravity, he imputes about 
 one half of the motion to that cause, and thence deduced a con- 
 clusion agreeing with observation. 
 
 " What we have already observed (Art. 861) is sufficient to 
 show, that no part of the effect can arise from the latter circum- 
 stance ; and he has also (as we have already shown) omitted a 
 cause which produces about one half the motion ; by two mis- 
 takes he has therefore fallen upon a true conclusion." And Mr. 
 Vince, in conclusion, makes these remarks : "but in whatever 
 manner this subject is treated, some corrections are applied from 
 observations, in order to render the equations more perfect ; 
 not that the principle of attraction is insufficient to furnish 
 conclusions which shall agree with observation," &c. And 
 again : " Thus we have given the reader all the satisfaction we 
 
196 ON THE LAW OF GRAVITY, 
 
 are able upon this difficult subject, without entering into a direct 
 solution of the problem, which requires the integration of a 
 fluxional equation of the second order, and this can be done 
 only by an approximation of a very intricate nature, and of 
 great labor." 
 
 The reader or inquirer who will take the trouble to examine 
 the moon's motions, and especially the motion of its apogee, as 
 contained in Mr. Vince's Astronomy, will find that in all cases, 
 the little motions and inequalities of motion, are, in whole or 
 in part, assigned to the disturbing force of the sun upon the 
 moon ; and after much laboring of the author on the subject, we 
 find this allegation, namely : " If the force perpendicular to the 
 radius vector be neglected, the motion of the apsides comes out 
 one half of what is here determined, and this we know from 
 other principles. This, therefore, tends to confirm the legality of 
 the method here employed," which allegation, after all, I think, 
 must be taken for granted, rather than as being proved ; and in 
 respect to the whole matter, I have come to the following 
 doubts : 
 
 I think it doubtful, whether the student or inquirer, on a care- 
 ful examination of the whole subject, will readily agree with the 
 modus operandi by which the observed phenomena of nature have 
 been supposed to be reconciled with the Newtonian law. 
 
 Whether he will find the ratio between the force applied to 
 the moon, and the moon's motion in its orbit, clearly and dis- 
 tinctly assigned or stated. 
 
 Whether he will clearly discover how they have obtained the 
 force which acts perpendicular to the radius vector, which is said 
 to give just half the motion of the apogee. 
 
 Whether he will be able to discover by what law of gravity, 
 the distance may be assumed at unity, and its mean force of 
 gravity at less than unity ; or in other words, whether according to 
 Mr. Vince, the mean force of the moon towards the earth is 
 equal to the natural gravity of the moon towards the earth, 
 diminished by the disturbing force of the sun upon the moon in 
 the line of the radius vector, together with many other matters 
 equally dubious, which the inquirer is required to pass through, 
 in order that he may be brought to render assent to their results, 
 or rather to their conclusions. 
 
 And finally, I am doubtful whether the inquirer will readily 
 discover, that even Sir Isaac Newton understood the subject in 
 the same way and manner in which it is now supposed to be 
 understood, notwithstanding Mr. Vince's supposition of what 
 Newton must have meant by some very obcure passages in 
 his Principia ; and more especially, when it is acknowledged by 
 
ON THE LAW OF GRAVITY. 197 
 
 Mr. Vince, that Sir Isaac Newton left out the force which is now 
 said to act perpendicularly to the radius vector; and which is said 
 to give just one half of the motion to the apogee. 
 
 Nevertheless, Walmsley, Murdock, and others, have been 
 somewhat cavalierly treated, for their attempts to make the New- 
 tonian law of gravity, account for the phenomena of nature, 
 which, manifestly, are the result of some kind of law of gravity ; 
 notwithstanding, they made their conclusions agree very well 
 with observation. And I must say, that I think the treatment 
 they received from the inquisition from those who would settle 
 the controversy by establishing articles of implicit faith, however 
 my terious was somewhat harsh and arbitrary. 
 
 Walmsley and others, made use of all the force that was al- 
 lowed them by the Newtonian law of gravity ; and as to the 
 remainder that was found necessary for the purposes intended, 
 each and every one, (not even excepting Clairaut himself,) 
 picked it up from such sources as were in his view most avail- 
 able ; and I have yet to learn why others, (as well as Clairaut,) 
 had not the right to call to their aid, for the purpose of triumph- 
 ing over great difficulties, such auxiliary forces as they could 
 best command. 
 
 But wisdom would have been profitable to direct ; and had those 
 great mathematicians, in lieu of confiding in their high-priest, 
 just spent an hour or two in examining, to see from whence he 
 derived his authority, they would not have been thrown into all 
 these inexplicable dilemmas. They would have found just force 
 enough to account for all the observed phenomena, without 
 resorting to anything factitious or unnatural. 
 
 Nevertheless, however gloomy the stubborn fact may be, if Sir 
 Isaac Newton's law of gravity be erroneous, it wholly destroys 
 his whole system of philosophy ; and in such case, his iheory of 
 universal gravity, (as by him promulgated,) ceases to exist ; and 
 the universe will again be governed by immutable and controll- 
 ing laws, in lieu of depending on fortuity or chance. 
 
CHAPTER III. 
 
 Of the Proper Elements from which to determine the Laws of 
 Force and Motion, incident to the Heavenly Bodies in their 
 Eternal Rounds. 
 
 SECTION FIRST. 
 
 HAVING attempted to show that what astronomers have called 
 the deflection of a planet, or the deflection of a planet from a 
 tangent to its orbit, and which they have made use of as an ele- 
 ment, in their endeavors to account for the agreement between 
 the force and motion by which a planet revolves, is, at least, but 
 a factitious element, and not applicable to the laws of force and 
 motion, as operating in the curve of an orbit, nor identical in its 
 mathematical construction with that of the law of the rectilinear 
 fall of a body through space, I will now proceed to treat of the 
 proper elements of the planets from which to deduce the modus 
 operandi by which they are retained and controlled in their 
 orbits while performing their eternal rounds, which elements 
 I will call by the appellations of Time, Distance, Motion, Force, 
 and Convergence/ ; the qualities or principles of which I will 
 define as follows : Time always has direct reference to the Period, 
 namely the periodic time, or entire revolution of the planet 
 from any given point in its orbit, to the same point again ; which 
 period or periodic time, may be divided and subdivided, as occa- 
 sion may require, when comparing other elements with that of 
 time ; and hence the period or whole periodic time of a planet, 
 as also the mean distance of a planet from the centre of grav- 
 ity may, in contradistinction to the elements of Motion, Force, 
 and Convergency, properly be called positive quantities, by which, 
 what may be called the rates (ratios or proportions) of the ele- 
 ments of motion, force and convergency may be compared. 
 
 Distance, or the mean distance of a planet, may always be 
 abstractly expressed by some numerical quantity, by which 
 the mean distance of any given planet of a system is propor- 
 tioned to the mean distance of that planet of the system which 
 
ON PLANETARY FORCE AND MOTION. 199 
 
 is assumed at unity; thus, if the mean distance of our earth 
 from the sun be assumed at unity or 1, the mean distance of 
 Mercury will be expressed by the fraction .38710 ; that of Ve- 
 nus, by the fraction .72332 ; that of Mars, by unity plus the 
 fraction .52998, &c. So also if the orbit be eccentric, we speak 
 of the mean distance, the perihelion distance, &c. 
 
 The mean distance of a planet determines or denotes the cir- 
 cumference' of the orbit, and consequently, the amount of space 
 passed over in an entire period. It also determines the dilata- 
 tion of the orbit, which is consequently as the mean distance. 
 But the dilatation of the orbit which is denoted by the mean 
 distance, must not be confounded with the rate of convergency 
 of the planet, which is denoted by the time of the period. Mo- 
 tion, which I shall use as synonymous with Velocity, de- 
 notes a passing over space ; and as this takes place in time, it 
 also denotes the velocity or rapidity with which space is passed 
 over. 
 
 Motion manifestly depends upon the force of gravity, at 
 least, in retaining the planet in its orbit ; and if the force of 
 gravity be inversely as the distance, (as I contend,) the rate of 
 mean motion will always be inversely as the whole amount of the 
 force of gravity expended during the entire period of the planet. 
 
 Force, which I shall use as synonymous with gravity, or the 
 force of gravity, denotes the power of attraction by which one 
 body is drawn towards another, causing the tendency of one 
 body towards another ; as that, of a planet towards the sun, or a 
 satellite towards its primary ; the intensity, or rather, proportional 
 intensity of which manifestly depends upon the proportional dis- 
 tance ; and it increases or decreases in the inverse ratios of 
 some given root or power of the distance ; which intensity I 
 shall in general denominate the rate of gravity. 
 
 And as by my determination of the law of gravity, the rate of 
 mean motion will always be found to be inversely as the whole 
 amount of the force of gravity expended during the entire period 
 of the planet, so also, by the same law, the rate of mean force 
 will always be inversely as the whole amount of motion during 
 an entire period of the planet, thus reciprocating with each 
 other upon perfectly equal and mutual terms. 
 
 Convergency is measured by degrees, minutes, &c. of the 
 circle, and hence the amount of convergency in the time of the 
 period of any planet in 360 ; consequently the amount of con- 
 vergency in any one orbit is properly unity. 
 
 Convergency is no part of the motion of a planet ; but simply 
 the direction of a motion; nevertheless, there could be no con- 
 vergency without motion ; and although convergency is produc- 
 
200 ON PLANETARY FORCE AND MOTION. 
 
 ed by constant gravity, we perceive that motion is as essential 
 to this element as gravity is. Hence, motion and convergency 
 seem to be alike intimately connected with, and dependent upon 
 gravity, for the exhibition of their phenomena. Nor is it certain 
 that motion is an element that is any more independent of grav- 
 ity, than convergency is, or that gravity is any less attentive to 
 motion than it is to convergency, notwithstanding those notions 
 drawn from the Newtonian theory of the universal and equal 
 gravity of matter. 
 
 But this theory of universal gravity being established, and 
 inasmuch as gravity was to have little or nothing to do with 
 motion, otherwise than to accelerate or retard it, it was neces- 
 sary that a planet should be projected in a tangent to some point 
 of its orbit, thereby giving motion ; which, whether it were more 
 or less, would be taken charge of by the innate gravity of distant 
 matter. And as such force would not be likely to produce a 
 motion just sufficient to balance the force of gravity, those ele- 
 ments were left to keep the balance of power, by alternately 
 getting the better of each other ; which contrivance, however, 
 was found insufficient, until a centrifugal force was finally 
 attached to the machinery, endued with such forces as might be 
 found necessary to accomplish the object. 
 
 Next, the planets were found to be of different densities, in 
 proportion to their respective bulks or magnitudes ; because it 
 was found they did not possess the same power of attraction in 
 proportion to their magnitudes ; and notwithstanding, had anal- 
 ogy been applied to the train of facts which bear upon that ques- 
 tion, such hypothesis would long since have been overthrown. 
 But having adopted the Newtonian theory of innate and univer- 
 sal gravity, as the first great article of faith, neither fact nor 
 reason has made inroads upon it. 
 
 The hypothesis of universal gravity being thus far established 
 and put at rest beyond a wish or disposition to doubt, to ascer- 
 tain the ratio which the force of gravity or attraction bears to the 
 distance from the attracting body, became a disideratum ; and 
 M. Vince says, that Sir Isaac Newton gave Dr. Halley " a proof" 
 of the law of gravity as proportioned to distance, namely, that it 
 varies inversely as the square of the distance. And this hypo- 
 thesis is supposed to have been clearly deduced from a compar- 
 ison of the law of falling bodies near the earth, with the rate of 
 deflection of the moon from a tangent to its orbit. 
 
 Thus, in lieu of resorting to the natural elements of the 
 planets for the purpose of determining this very important ques- 
 tion, they have created for the purpose, the factitious element of 
 deflection, so wholly extraneous and foreign to the laws of force 
 
ON PLANETARY FORCE AND MOTION. 201 
 
 and motion, as to have been a very troublesome element to 
 astronomers and mathematicians ; for having suffered their imag- 
 inations to run off in a tangent to the orbit, where no force nor 
 motion is, (though but a very little distance,) in accordance with 
 the precaution from the highest authority, yet if a mathematical 
 error be contained in such very little distance, it may, nevertheless, 
 in its consequences, be exceeding broad. 
 
 Any amount of deflection is conceived to be measured or 
 denoted by direct linear quantity, namely, by a perpendicular 
 line from the given tangent to the revolving body in the orbit ; 
 and hence, if the rate of progression of the supposed fall or 
 deflection of the planet from the tangent, be ascertained or estab- 
 lished, it could not well be made to apply to the entire period or 
 revolution of the planet; nor could the whole amount of deflec- 
 tion in the time of the period be well ascertained, which should 
 be the case with any element which may properly be said to 
 have its rate, namely, its ratio to that of other elements of the 
 same planet. 
 
 Nevertheless, what may be termed the element of deflection, 
 is by no means as useless as that of a centrifugal force, but is 
 susceptible, when rightly considered, of furnishing or throwing 
 much light upon the law of gravity in respect to distance ; and 
 this too by comparing it with the law of falling bodies near the 
 earth ; in which comparison, we find many things in which 
 they corroborate each other; and hence, the identity of the 
 force which causes a body to descend or fall to the earth, and 
 that which retains the moon in its orbit, goes in no sense to 
 prove, that the deflection of a planet from a tangent, proceeds 
 by the same progression as that of a falling body from a state of 
 rest, namely, by a progression which shall be denoted by the suc- 
 cessive odd numbers, as 1, 3, 5, &c. 
 
 For if so, then the amount of deflection of a planet whose 
 distance is 4, when it has revolved over l-8th of its orbit, should be 
 equal to the amount of deflection of a planet of the same system 
 whose distance is 1, when it has revolved over l-4th of its 
 orbit ; for the motion of the planet whose distance is 1, must 
 be twice as great as that whose distance is 4, for otherwise the 
 squares of their periods could not be as the cubes of their mean 
 distances from the sun. 
 
 But in lieu of the rates of deflection being equal, when that 
 whose distance is 1, has passed over l-4th of its orbit, and that 
 whose distance is 4, has passed over l-8th of its orbit, their amounts 
 of deflection will then be proportioned to each other, as 1 1o 
 1.17156, and hence the amount of deflection of a planet while 
 revolving over l-4th of its orbit will be 2.41421 times as great in 
 26 
 
202 ON PLANETARY FORCE AND MOTION. 
 
 the last half of the time, as in the first half; when by the law of 
 falling bodies, it should be just three times as great in the last 
 half of the time as in the first half. 
 
 I am aware that Sir Isaac Newtonand others have said, that 
 if from the point of intersection, or touch of the given tangent 
 with the orbit, we take a very little distance, the deflection will be 
 likened to, or rather be identical in its mathematical construction 
 with the law of falling bodies near the earth ; which allegation 
 has certainly supplied the place of mathematical certainty for a 
 long time. But surely we ought to have been informed where 
 the little distance should stop ; for as the progress of deflection 
 in any assignable portion of its orbit, is different from that of a 
 falling body near the earth, it would seem necessary, in order 
 that such little distance might present a mathematical truth, that 
 it should be less than any assignable quantity, namely, in its 
 ultimate ratio, alias in 0, or no distance whatever ; in which 
 point or place, far too many important questions have been set- 
 tled ; as if ratios could as well exist in 0, as in actual numerical 
 quantities ; which, in point of demonstration, would seem 
 too much to resemble those important discoveries we have 
 somewhere read of, by which many of the laws of gravity and 
 attraction were deduced and explained, by geometrizing upon 
 two inconceivably small corpuscles, or physical atoms. 
 
 I will now return to a further consideration of convergency, 
 and the other proper elements of the planets. 
 
 Convergency is most directly connected with the phenomena 
 of motion in the heavenly bodies, and with the law of gravity ; 
 and being the description or delineation of the degrees, minutes, 
 seconds, &c. of a circle, produced in time, which becomes full 
 and complete in the time of the period, hence, the rate of conver- 
 gency is inversely as the time of the period ; and the amount of 
 convergency in the time of the period, is in all cases the same, 
 namely, 360. 
 
 Thus, while time, or rather the Period, and Distance, may be 
 called the positive elements of the planets, those of Motion, 
 Gravity, and Convergency, may properly be called the relative 
 elements ; performing relative operations in respect to each 
 other, and also in respect to Period and Distance. 
 
 The relative elements are therefore properly said to have their 
 rates of operation , not only in respect to each other, but in re- 
 spect to time or period, and distance ; which rates of operation 
 become complete and entire in the time of the period, which 
 makes what may be called the whole amount of the operation. 
 And thus, Motion, Gravity and Convergency, are said to have 
 both their rates and amounts of operation. 
 
 The period of a planet is the cube of the square root of the 
 
ON PLANETARY FORCE AND MOTION. 203 
 
 mean distance ; so then the mean distance is the square of the 
 cube root of the period ; for if it were otherwise, the square of 
 the period of a planet could not be the cube of the mean dis- 
 tance ; nor could the squares of the periods of a system of 
 planets be as the cubes of their mean distances from the sun. 
 And that the rates* and amounts of motion, gravity, and conver- 
 gency, are so proportioned to each other, and to the period and 
 distance, as to produce that grand proportional result, is what 
 no one will question or doubt. 
 
 Hence the rate of mean motion of a planet is inversely as the 
 cube root of the period, or the square root of the mean distance; 
 and the amount of motion in the time of the period, is as the 
 mean distance ; for the mean distance denotes or represents the 
 circumference of the orbit, and consequently the amount of 
 space passed over; which properly denotes or determines the 
 amount of motion. Hence, the mean rate of motion is to the 
 whole amount of motion, as unity is to the time of the period. 
 
 The rate of gravity or force, is inversely as some power or 
 root of the distance ; which I assume to be inversely as the first 
 power, or first root, of the distance, viz., inversely aa the dis- 
 tance. Hence, the mean rate of gravity of a planet is inversely 
 as the mean distance, and also inversely as the square of the 
 cube root of the period, and is consequently the square of the 
 mean rate of motion. The whole amount of gravity, then, in the 
 time of the period, will be the square root of the distance, which 
 is also the cube root of the period ; and hence the mean rate 
 of gravity is to the whole amount of gravity, as unity is to the 
 time of the period. 
 
 The mean rate of convergency is inversely as the time of the 
 period, or inversely as the cube of the square root of the dis- 
 tance ; and hence, is the cube of the mean rate of motion, and 
 the cube of the square root of the mean rate of gravity or force. 
 Consequently, the whole amount of convergency in the time of 
 the period is unity or 1 ; and hence, the mean rate of conver- 
 gency is to the whole amount, as unity is to the time of the 
 period ; or rather, as the inverse of the time of the period is to 
 unity. 
 
 Hence, the mean rate of motion, of gravity, or of conver- 
 gency, is to the whole amount in the time of the period, as unity 
 is to the time of the period. 
 
 Hence, the product of the mean rate of motion by the time 
 of the period, gives the whole amount of motion in the time of 
 the period. 
 
 The product of the mean rate of gravity or force, by the time 
 of the period, gives the whole amount of gravity or force in the 
 time of the period. 
 
204 ON PLANETARY FORCE AND MOTION. 
 
 And the product of the mean rate of convergency by the time 
 of the period, gives the whole amount of convergency in the time 
 of the period. 
 
 Thus the mean rate of gravity of a planet is always a mean 
 proportional between the mean rate of motion, and the mean 
 rate of convergency ; and is equally attentive to both. Nor does 
 it ever forsake its position, or neglect its trust ; for whether the 
 orbit be centric or eccentric, the element of gravity at all times, 
 and in all parts of the orbit, preserves the same position in re- 
 spect to the elements of motion and convergency, viz., a mean 
 proportional between them, and hence, the planets sing through 
 the heavens with as much harmony and unity of purpose, as 
 we can desire. 
 
 Kepler, in addition to the great law, " That the squares of the 
 periods of the planets, are as the cubes of their mean distances 
 from the sun," also discovered the important fact, " That the 
 motion of an individual planet revolving in an eccentric orbit, 
 varies in the inverse ratio to that of its varying distance from 
 the sun." But Kepler did not discover that the rates of motion 
 of the planets are inversely as their distances from the sun ; 
 although such has often been the version and the understand- 
 ing in respect to his discovery. And it requires but little con- 
 sideration to convince any one, that the rates of motion of the 
 planets are not inversely as their mean distances from the sun ; 
 for if it were so, the squares of the periods of the planets would 
 be as the fourth powers of their mean distances from the sun ; or 
 in other words, the time of their periods would be as the squares 
 of their mean distances from the sun ; as any one will discover 
 on a moment's reflection. For if the mean distance were 4, and 
 the mean motion were inversely as the mean distance, the period 
 would be 16, in lieu of 8. And hence, such hypothesis is un- 
 true. 
 
 And yet, this erroneous version of Kepler's law, has been a 
 great supporter of the hypothesis, that the rate of gravity is in- 
 versely as the square of the distance ; which error was com- 
 mitted by Sir Isaac Newton, in referring the square of the dis- 
 tance to the rate or intensity of gravity, in lieu of referring it to 
 the rate of deflection, or to the result or effect of the gravity. 
 
 Thus, once having adopted the well established fact in dy- 
 namics, that twice the motion requires quadruple the force to 
 retain a planet in its orbit, it necessarily follows as a corollary or 
 consequent, that if the mean motions of the planets of a system, 
 are inversely as their mean distances from the sun, the force of 
 gravity which retains them in their orbits, must, of necessity, be 
 inversely as the square of the distance from the sun. And the 
 
ON PLANETARY FORCE AND MOTION. 205 
 
 next corollary or consequent, which as necessarily follows, is, 
 that their periods will be as the squares of their mean distances 
 from the sun ; or that the squares of their periods will be as the 
 fourth powers of their mean distances from the sun ; and also that 
 the period of each planet of the system will be the square 
 of its mean distance from the sun, by which, error begets error. 
 
 And the same or like consequences would follow, in a sys- 
 tem of eccentric orbits, that would in a system of centric orbits, 
 viz., if the mean motions of the planets were inversely as their 
 mean distances from the sun; and also if from the points of 
 mean motion, the motion of the planet does vary in the inverse 
 ratio as the distance from the sun varies, as Kepler has shown ; 
 then their motions throughout their respective orbits would be 
 inversely as their distances from the sun ; presenting precisely 
 the same or similar results in this respect, as would be present- 
 ed in a system of centric orbits,. viz., the period of each would 
 be the square of the mean distance ; and the periods of the re- 
 spective planets of the system, would be as the" squares of their 
 mean distances from the sun ; or the squares of their periods 
 would be as the fourth powers of their mean distances from the 
 sun. And if such were the case, the consequence would con- 
 clusively follow, that the force of gravity would be inversely as 
 the square of the distance from the sun. 
 
 But Kepler taught no such doctrine ; he did not attempt so 
 readily to abrogate and make void his great law, so well founded 
 in observation and in the powers and roots of numbers, viz., that 
 the squares of the periods of a system of planets are as the 
 cubes of their mean distances from the sun ; nor to overthrow 
 the well established fact, that twice the motion requires quadru- 
 ple the force, when he thus declared the no less important dis- 
 covery : That the motion of a planet, viz., an individual planet, 
 revolving in an eccentric orbit, varies from the points of mean 
 motion, in the inverse ratio as the distance varies from the sun. 
 
 I say that the discovery by Kepler, that in an eccentric orbit, 
 the motion varies from the points of mean motion in the inverse 
 ratio as the distance from the sun varies, is no less important 
 than are his other discoveries. For if it were otherwise, the 
 revolution of a planet in an eccentric orbit would be wholly in- 
 explicable. . 
 
 It is this principle that chains the planet in its eccentric orbit, 
 and keeps it obedient to the power of gravity, as much so in 
 one part of the orbit as in another ; and hence prevents, or avoids 
 the fortuitous necessity, that gravity should get the better of the 
 motion at aphelion; and that the motion should get the better of 
 gravity at perihelion, in order to preserve the balance of power, 
 
206 ON PLANETARY FORCE AND MOTION. 
 
 by the help of a centrifugal force ; as the laws of gravity and 
 motion have been taught and explained by Maclaurin, Fergu- 
 son and others, to a world of too easy faith. 
 
 Hence, it will be found, (as I propose to show in the sequel,) 
 that both the gravity and convergency of a planet revolving in 
 an eccentric orbit, vary from their mean, in the same ratio that 
 motion does, viz., in the inverse ratio as the distance varies from 
 the mean, or from the sun ; which is one and the same thing. 
 And hence, the rate of gravity always remains a mean pro- 
 portional between the rate of motion and the rate of con- 
 vergency, controlling both of them, as well in one part of the 
 orbit as in another. 
 
 But in -respect to the Newtonian law of gravity, viz., that 
 gravity is inversely as the square of the distance, nothing can be 
 more manifest, than that, in order to sustain such law, the mean 
 motion of any planet of a system, whether revolving in a centric 
 or eccentric orbit, must, in reference to the mean motions of the 
 other planets of the system, be inversely as the mean distance, 
 which is not the case; but in such case, the squares of the 
 periods of the planets would be as the fourth powers of their 
 mean distances from the sun. 
 
 SECTION SECOND. 
 
 That the apparent motions of the planets are not equable in 
 their orbits, seems to have been familiarly known to ancient 
 astronomers ; and hence, that the orbits in respect to the earth, 
 as was then supposed, were eccentric. 
 
 This seems to have been the opinion of Ptolemy in ancient 
 times, and of Tycho in more modern times ; and upon this hy- 
 pothesis they endeavored to equate their motions; but when 
 it became more rationally understood, that the eccentricity of the 
 orbits referred to the sun directly, and not to the earth, the idea 
 that each planet revolved around some point in its orbit, about 
 which the motion was equable, was of consequence abandoned, 
 and gave place to new researches into the phenomena of the 
 motions of planets, preparatory to accounting for such eccen- 
 tricity. 
 
 In this respect, Kepler, who always seemed preeminent in 
 searching out physical cauess and phenomena, discovered the 
 fact, which is perhaps no less important in respect to eccentric 
 orbits, than is his great law that the squares of the periods of a 
 system of planets, are as the cubes of their mean distances from 
 the sun, in respect to orbits in general, - that the motion of 
 a planet revolving in an eccentric orbit, varies inversely as the 
 
ON PLANETARY FORCE AND MOTION. 207 
 
 distance from the sun varies ; which law is so essential and 
 important in respect to the motion of a planet in an eccentric 
 orbit, that any attempt to account, without the law, or to attempt 
 to supply its place by extraneous matter, would to me appear 
 like a forlorn hope. And why this law, in connection with the 
 other great law just quoted, should not long ere this have dis- 
 closed more important truths in respect to the laws of force and 
 motion than they have done, appears to me wholly unaccounta- 
 ble. 
 
 It will be readily seen by any one who will examine the sub- 
 ject but a little, that a misapprehension of said law, viz., that the 
 motion of a planet varies in its orbit inversely as the distance 
 from the sun varies, would lead to, and produce the same in- 
 explicable difficulties which now pervade the science of astron- 
 omy ; and that said law has been wholly misapprehended, there 
 perhaps is but little doubt. 
 
 If the mean motions of a system of planets (whether com- 
 posed of centric orbits, eccentric circles, or ellipses) were in- 
 versely as their mean distances from the sun, and if also in each 
 eccentric orbit, the motion should vary from the point of mean 
 motion inversely as the distance from the sun should vary while 
 revolving in its orbit, in such case nothing can be made more 
 manifest ; no more simple proposition exists than that force or 
 gravity must of necessity be inversely as the square of the dis- 
 tance. And it is, also, every way as simple a manifestation, 
 that if such were the case, Kepler's other great law would be 
 made wholly void ; for in such case, the squares of the periods 
 of a system of planets, would be as the fourth powers of their 
 mean distance from the sun. 
 
 In such case, the mean motion of a planet whose distance is 
 4, would have to be .25, when we know it must be .5 in order 
 to fulfil the great law. And the only possible condition, as I 
 have already shown, upon which the hypothesis that the force 
 is inversely as the square of the distance, could be even hypo- 
 thetically sustained, would be upon the ground, that the mean 
 motions of a system of planets are inversely as their mean 
 distances from the sun ; and also, that the motion of any planet 
 revolving in an eccentric orbit, varies from the point of mean 
 motion, inversely as the distance from the sun varies. 
 , But Kepler adopted but the last part of said hypothesis, viz., 
 that a planet revolving in an eccentric orbit does vary from the 
 points of mean motion inversely as the distance of the revolving 
 body varies from the sun ; and this permits his other law to 
 stand, and all the elements of the planets to operate in unison. 
 
 The mean motions of the planets then, are inversely as the 
 
208 ON PLANETARY FORCE AND MOTION. 
 
 square roots of their mean distances from the sun; that the 
 cubes of their mean distances may be as the squares of their 
 periods. 
 
 But what has, perhaps, served much to sustain mankind in 
 their misapprehension of Kepler's law, is that useless demon- 
 stration, (so far as astronomy is directly concerned or ever has 
 been,) that if a revolving body vary in its motion inversely 
 as its distance varies from the sun, it will describe or pass over 
 equal areas in equal times. This demonstration, although true, 
 has never been used in astronomy to any other effect than "to 
 perplex and dash maturest counsels." 
 
 In -any system of planets, whether all the orbits of such sys- 
 tem are centric, eccentric, or elliptical, if their mean motions 
 were inversely as their mean distances from the sun, the planets 
 of such system would in respect to each other describe equal 
 areas in equal times. 
 
 In such case, the mean motion of the planet whose mean 
 distance js 4, would be .25, or half what we know to be its ac- 
 tual motion ; and its period would be 16, or the square of its 
 distance, or twice its' actual period ; and the area passed over 
 in the time of its period, would be 16 times as great as that 
 passed over by the planet whose distance is 1, in the time of its 
 period. That is, in such case, the areas passed over by the 
 several planets of a system, would be inversely as their motions ; 
 or would be as the square roots of their respective distances ; 
 and hence, would be equal in equal times. Then, indeed, would 
 the force be inversely as the square of the distance ; and the 
 motion would be the fourth root of the force ; the squares of 
 the periods of a system of planets, would be as the fourth pow- 
 ers of their mean distances from the sun ; or their periods would 
 be as the squares of their mean distances from the sun. 
 
 If the distance of the earth from the sun were assumed at 1, 
 the distance of a planet whose period was 8, would be the 
 square root of 8, in lieu of the square root of 16, as now sup- 
 posed. Thus, the distances of the planets of our system would 
 be reduced from that of the third roots of the squares of their 
 respective periods, to that of the square roots of their respective 
 periods. The distance of Herschel from the sun would be 
 much less than half the distance it has heretofore been estimated 
 at, even though the earth were distant 95,000,000 miles. But as 
 Venus would have a greater proportional distance from the sun, 
 in proportion to that of the earth, than has been assumed, (un- 
 less from its greatest 'elongation or otherwise, its true propor- 
 tional distance has been determined,) hence, the actual distance 
 of the earth from the sun in miles, would have been overrated, 
 
ON PLANETARY FORCE AND MOTION. 209 
 
 in which case the actual distance of all the superior planets 
 would have been overrated in like proportion. 
 
 So it seems, in such case, the superior planets would, in a 
 measure, be relieved from the charge of a lack of density in 
 proportion to their respective bulks or magnitudes, which has 
 ben charged against them by some ; as their bulks or magni- 
 tudes would be vastly less than had been supposed ; with an 
 innumerable train of consequences which I will not here at- 
 tempt to follow out. 
 
 It then becomes conclusive, that if the mean motions of the 
 several planets of a system are inversely as the mean distances 
 from the sun, the periods of the planets will be as the squares 
 of their mean distances from the sun. 
 
 But Kepler only intended to declare, and only did declare, 
 that the motion of any individual planet of a system varies 
 from the point of mean motion (if the orbit be eccentric) in- 
 versely as the distance varies from the sun: In which case 
 the radius vector of such planet will pass over or describe equal 
 areas in equal times ; notwithstanding the mean motions of all 
 the planets of the system are inversely as the square roots of 
 their respective distances. And in which case also, the great 
 law that the squares of their periods are as the cubes of their 
 mean distances, will still apply to the system in all its force and 
 majesty. 
 
 I shall adopt, fully, Kepler's two said laws as he gave them ; 
 and consequently adopt the hypothesis, that the force or gravity, 
 of any individual planet, varies inversely as the distance varies 
 from the sun ; as that of necessity follows upon the adoption of 
 said two laws. And it is only upon the adoption of those prin- 
 ciples, that we are enabled in anywise to account for the mo- 
 tions of a system of planets revolving in eccentric orbits. And 
 that the primary planets of the solar system do all revolve in 
 eccentric orbits, is doubtless true ; and that such is the case with 
 all the heavenly bodies, is, at least, more than probable. 
 
 The adoption then of Kepler's law, that the squares of the 
 periods of a system of planets are as the cubes of their mean 
 distances from the sun ; and also, his law which teaches that the 
 motion of a planet revolving in an ecceniric orbit varies from 
 the points of mean motion inversely as the distance varies from 
 the sun in the course of its revolution, together with the neces- 
 sary and imperious consequent upon said law, viz. that the 
 force of gravity varies inversely as the distance varies from the 
 sun, furnishes a very easy and simple solution, accounting 
 for the movements of a system of planets in eccentric orbits ; 
 and we shall hence discover that those principles are entirely 
 
210 ON PLANETARY FORCE AND MOTION. 
 
 able to manage a planet throughout its entire period, without 
 the aid of any other machinery whatever. 
 
 I will suppose a system of planets revolving around the sun, 
 or around one and the same central power ; and that as occa- 
 sion may require, for the purposes of explanation, some of the 
 planets may be conceived to revolve in centric orbits, and others 
 in eccentric orbits, being nevertheless circles. 
 
 The line of the apsides is a right line extending through the 
 centre of the sun, from the perihelion to the aphelion ; hence the 
 time of the apsides will be the diameter of the orbit, and may 
 be thus denoted, even in a centric orbit ; although such orbit 
 cannot properly be said to have perihelion and aphelion. 
 
 In such system of planets, the squares of their periods would 
 be as the cubes of their mean diameters from the sun. For as 
 it is not necessary in order for a system of planets to fulfil this 
 law, that the orbits should be alike eccentric, hence if some of 
 them were centric and others eccentric, they would equally well 
 fulfil the law; hence, if the lines of apsides of an eccentric 
 orbit and of a centric orbit of the system were equal, the two 
 planets would pass over equal areas in the times of their re- 
 spective periods; the motion of one being uniform, and that 
 of the other constantly varying inversely as the distance from 
 the sun varies. 
 
 It will readily be perceived that as the force varies inversely 
 as the distance, and the motion also, that those two elements 
 preserve the same proportions to each other throughout the orbit, 
 namely, the same in any other part of the orbit that they do in 
 the mean points of force and motion. 
 
 And as the rate of convergency is inversely as the period, 
 hence, the rate of convergency depends upon the velocity in one 
 and the same orbit ; so that if the velocity be double in one 
 point of an orbit, in which the curvature is equal to what it is in 
 another point of the orbit, the rate of convergency will also be 
 double, &c. 
 
 Hence then, those elements, namely, motion, force, and con- 
 vergency, necessarily and consequently preserve the same ratio 
 to each other throughout the orbit, that they do in the mean 
 points ; and hence the rate of force, or gravity, is a mean pro- 
 portional between the rate of motion and the rate of convergency, 
 throughout the entire orbit ; nor are they under any necessity, 
 as will be clearly perceived, of alternately getting the better of 
 each other, in order to keep up the balance of power. 
 
 Thus the mean motion is inversely as the square of the mean 
 distance ; and the actual motion varies from the point of mean 
 motion inversely as the distance varies ; and hence, we deduce 
 
ON PLANETARY FORCE AND MOTION. 211 
 
 this general rule for finding the rate of motion at any given dis- 
 tance of the planet in its orbit from the sun, namely : 
 
 The product of the mean distance , by the rate of mean motion 
 divided by the actual distance at any given point of the orbit, 
 gives the rate of motion at such given point. 
 
 And upon no other principle can the motion vary inversely as 
 the distance varies ; nor the planet describe equal areas in equal 
 times, nor the square of the period be the cube of the distance. 
 And thus, gravity governs as perfectly throughout the orbit as 
 though it were a centric orbit. There is no more necessity of 
 calling to aid a centrifugal force, clothed with such powers as are 
 found necessary to keep the balance of power; no more necessi- 
 ty of resorting to the popular method of accounting, by alleging 
 that motion and force alternately get the better of each other, than 
 if the orbit were centric ; for the planet is as obedient to the 
 force of gravity, and sings through the heaven's with the same 
 harmony and concord in the eccentric, as in the centric orbit. 
 
 Thus, in any eccentric orbit, the motion of the planet at any 
 given point of the orbit varies inversely as the distance varies ; 
 hence, the motion at any given point of the orbit, will be the 
 square of what the motion would be in a corresponding point of 
 a like orbit to that of the given orbit, whose mean distance is 
 the square root of the mean distance of the given orbit; be- 
 cause the mean motion of a planet is always inversely as the 
 square root of the mean distance. 
 
 So if the mean distance of a planet be 4, and its perihelion 
 distance be 1, the actual motion at any given point of the orbit, 
 will be the square of what the actual motion of a planet revolv- 
 ing in a centric orbit would be, if revolving at just half the 
 distance of such given point. 
 
 Thus if the mean distance be 4, the motion at the mean dis- 
 tance will be .5, or the square of the motion of a planet revolv- 
 ing in a centric orbit, about the same central force, at the distance 
 of 2. And if the mean distance be 4, and the perihelion distance 
 be 1, the motion at the perihelion will be 2, or the square of the 
 motion of a planet revolving in a centric orbit about the same 
 central force at the distance .5. 
 
 Let us now suppose a case or two : and firstly, suppose a 
 planet revolving in an orbit whose mean distance is 4, and whose 
 perihelion distance is 1. In such case we know the rate of 
 mean motion is .5 ; the rate of mean force .25 ; and the rate of 
 mean convergency .125 ; and the rates of motion, force and con- 
 vergency, each, become quadrupled at perihelion, or when the 
 distance has become l-4th of the mean distance ; that is, when the 
 planet has arrived at perihelion, the rate of motion will be 2, the 
 rate of force 1, and the rate of convergency .5 ; thus retaining 
 
212 ON PLANETARY FORCE AND MOTION. 
 
 the same proportion to each other, and the same ability to per- 
 form their journey, or pursue the line of the orbit, as at any 
 other point of the orbit. 
 
 Thus we perceive that the force from aphelion to perihelion, 
 is constantly dividing its accumulating power between the mo- 
 tion and the convergency, according to their necessities ; and 
 from perihelion to aphelion, the force restrains and diminishes their 
 rates in like proportions to that of their increase from aphelion to 
 perihelion ; and in such case the motion becomes accelerated 
 and retarded, or rather increased and diminished, in the same 
 proportion that the convergency becomes increased and dimin- 
 ished ; or inversely as the distance is increased or diminished. 
 Hence, the motion is not accelerated by the same law of pro- 
 gression, in respect to time and force, that a falling body near 
 the earth is ; in whiqh the progression of acceleration is constant- 
 ly as the amount of the next succeeding odd number is to 
 unity. 
 
 Thus, when the foregoing planet arrives at perihelion, the 
 force is the same as would be required for it to revolve in a 
 centric orbit at that distance ; but the rate of motion having 
 become double what would be required for a planet to revolve 
 in a centric orbit at that distance, the rate of convergency is 
 consequently one half of what would be required ; and hence, 
 the planet pursues the curvature of the orbit dictated by the 
 mean distance. 
 
 In the foregoing we perceive that the general rule applies, 
 namely, the product of the mean distance, by the mean rate of 
 motion, divided by the actual distance of any point of the orbit, 
 gives the rate of motion at such point. So also this general 
 rule is equally applicable to the rate of convergency, namely, the 
 product of the mean distance by the mean rate of convergency, 
 divided by the actual distance at any given point of the orbit, 
 gives the rate of convergency at such given point ; and this gen- 
 eral rule also equally applies to the force. 
 
 This then is the simple and easy construction of the laws of 
 force, motion, and convergency, in an eccentric orbit, based upon 
 the hypothesis that force is inversely as the distance ; but if we go 
 upon the hypothesis, that gravity is inversely as the square of the 
 distance, and append to it that extraordinary though prevalent 
 version of Kepler's other law, namely, that the motion is inversely 
 as the distance, (and without which appendage, even the idea or 
 notion, that force is inversely as the square of the distance, could 
 not have been sustained for a moment,) and take the foregoing 
 planet whose mean distance is 4, and whose perihelion distance 
 is 1, we shall find among others, the following results, namely : 
 
 The period must be 16, and the mean rate of force .0625, or 
 
ON PLANETARY FORCE AND MOTION. 213 
 
 the inverse of the period, and equal to the mean rate of conver- 
 gency ; and the mean rate of motion must be .25, or inversely 
 as the mean distance. 
 
 When the planet has arrived at the perihelion, the force will 
 be 1, or 16 times its mean ; and the motion will be 1, or four 
 times its mean ; and what should hinder the planet from then 
 revolving in a centric orbit at that distance, may not readily be 
 perceived ; as the force and motion both conspire to that effect ; 
 there being no extra motion to cause a less rate of convergency 
 than that of the force or motion ; for it would be futile to say or 
 assume that the rate of convergency was any other than unity, 
 when there was no cause for it. 
 
 But it may be sard that the actual period is 8, the mean rate 
 of motion .5, and the mean rate of convergency 125, and that 
 those elements will always have the same proportions, whatever 
 be the law of force ; and hence, at perihelion, or distance 1, all 
 the elements would have the same ratio to the rate of force, 
 whatever be its laws ; that in such case, the rate of force at per- 
 ihelion would be a mean proportional between the rate of 
 motion and the rate of convergency. 
 
 This mean proportional, it will be perceived, could only hap- 
 pen at the point of perihelion, and that too when at the distance 
 1. But in all Bother parts of the orbit, while the rates of motion 
 and convergency preserved the same proportions to each other, 
 and varied inversely as the distance, the force would be wander- 
 ing, with little apparent regard to its change, over the motion and 
 convergency. 
 
 But I have already fully shown that the only condition upon 
 which the force can be inversely as the square of the distance, 
 would be that the mean motion of a planet should be inversely as 
 the mean distance, instead of being what it is, namely, inversely 
 as the square of the distance; than which, a more plain or 
 palpable fact could not well be desired. 
 
 I will now attempt to make some few simple applica- 
 tions of my determinations of the measures of the circle, 
 to the phenomena of planetary motion, as corroborated by 
 the great Keplerian law, and the eternal fitness of the econ- 
 omy of numbers for the development of the laws of nature, to 
 the human understanding; in which I will insist that the great 
 Architect of the Universe had Unity of Purpose in view, when 
 he created and established it by everlasting laws, which are as 
 necessarily the laws of order, regularity, and the eternal fitness 
 of things, as they are the laws of power and supremacy; 
 otherwise we should discover but little of the attribute of wis- 
 dom; and hence, truth, when once discovered, may at least, 
 be as beautiful as error. 
 
214 ON PLANETARY FORCE AND MOTION. 
 
 I will here remark, that the popular measures of the circle 
 apply to the great Keplerian law, that the squares of the periods 
 of a system of planets, are as the cubes of their mean distances 
 from the sun, in the same manner as (and no better than) they 
 apply to the measures of solids composed of plane surfaces. 
 Hence, in applying the popular measures of the circle to the law 
 by which a system of planets revolve in their orbits, as well as 
 in the application of the circle to the measure of plain solids, 
 it will always be found necessary in order to equate or finish out 
 the work, to have on hand a sufficient number of minim quanti- 
 ties, as a kind of calculi to chink up with ; and that too, in 
 proportion as the extension of the work may serve to place it 
 out of square. 
 
 I am riot going to assume that the infinity of coincidences and 
 reciprocities which continually flow from a comparison of my 
 measures of the circle with measures of plain solids, or with the 
 laws of planetary motion, as determined by Kepler's laws, is by 
 any means to supply the place of absolute proof of the correctness 
 of my determination ; but such must depend upon the re- 
 sults and absolute conclusions drawn from an inductive course 
 of geometric reasoning ; in which one portion of the task will 
 be to exhibit and show the errors of the popular methods, and 
 thereby to endeavor to unloose that implicit faith in those errors 
 which have become chained and riveted just in* proportion as 
 the inquirer has examined or attended to the subject; and for 
 the accomplishment of which, it will be necessary in the consid- 
 eration of the quadrature, to adopt certain appropriate (but 
 very simple) signs, characters or symbols, which cannot here be 
 used to advantage. 
 
 I will therefore simplify the present explanation as far as pos- 
 sible, and will call l-4th of the circumference of an orbit, the 
 quadrant of the orbit; l-8th of the circumference, the octant of the 
 orbit, &c. Radius of an orbit of course denotes one half of the 
 diameter of the orbit. 
 
 I will notice that what is properly the prime orbit, is that 
 whose radius is unity or 1, having its diameter 2, for the reason, 
 that the squares of the periods of a system of planets, are as 
 the cubes of their mean distances from the sun ; and consequent- 
 ly when the distances of the planets of a system are numerically 
 proportioned to any planet of the system whose distance is 
 assumed at unity ; the square of the period of each planet of 
 the system will be the cube of its own distance ; consequently, 
 when the distance of a planet is assumed at unity or 1, its period 
 is necessarily 1. Nevertheless, what is properly the prime circle, 
 is that whose diameter is unity, having its radius but half of 
 unity. Such circle will be found to afford vastly more facili- 
 
ON PLANETARY FORCE AND MOTION. 215 
 
 ties for deducing the true quadrature, than any other numerical 
 dimension. 
 
 But to proceed with a short application of my measures of 
 the circle to planetary motion. 
 
 1. By the Keplerian law, the square of the period is the cube 
 of the distance ; or in other words, the period of a planet is the 
 square root of the third power of the mean distance. And 
 according to my measures of the circle, when the quadrant of 
 an orbit is 4, (or the circumference 16,) the square of the quad- 
 rant of the orbit is the cube of radius. And this we know from 
 Kepler's law, that when the square of the quadrant is the third 
 power, or cube of radius, the quadrant of the orbit is conse- 
 quently numerically equal to the period. 
 
 But by the popular measures of the circle, in order that the 
 period may be equal to the quadrant of the orbit, the quadrant 
 must be very little less than 4, or the circumference of the orbit 
 a very little less than 16. 
 
 By Kepler's law the period of a planet is always the square 
 root of the third power, or cube of radius ; and by my determi- 
 nation of the circle, the quadrant of any orbit is numerically 
 equal to the third root of the square of twice the period. Hence, 
 when the quadrant of an orbit is 1, radius is the third root of 
 .25, and the period will be the second or square root of .25, or 
 equal to half of the quadrant, or to l-8th of the circumference of 
 the orbit ; and we know that when the diameter of an orbit is 8, 
 the period is equal to diameter, or to twice radius; for then the 
 third power of radius will be the square of the diameter as well 
 as of the period. So by my determination, when the quadrant of 
 an orbit is 2, the period of the planet will be the square root of 2, 
 and radius of the orbit will be the cube root of 2. And when the 
 diameter of an orbit is 2, (namely, when the orbit is prime,) the 
 period is necessarily equal to radius. 
 
 So when the diameter of the orbit is 4, the period will be the 
 square root, and radius the cube root of 8, for it will be recol- 
 lected that the period is always the square root of the same 
 numerical quantity of which radius is the cube root. 
 
 Again, by my determination, four times the square of the 
 period of a planet, is always the third power of the quadrant of 
 the orbit ; and radius of any orbit is always the farther of two 
 mean proportionals from unity to the time of the period ; for 
 radius is always the third root of the square of the period. 
 
 So by my determination, when the period of a planet is .5 or 
 half of unity, it is equal to half the quadrant of the orbit. 
 When the period is 4, the period is equal to the quadrant of the 
 orbit. When the period is 32, it is equal to twice the quadrant 
 of the orbit ; and when the period is 256, (or square of 16,) the 
 
216 ON PLANETARY FORCE AND MOTION. 
 
 period is equal to four times the quadrant, or is equal to the 
 circumference. 
 
 When the diameter of an orbit is 8, the period is equal to the 
 diameter ; and when the quadrant of an orbit is 4, the period is 
 equal to the quadrant. 
 
 By the first allegation, the square of the diameter or the third 
 power of the radius ; and by the second proposition, the square 
 of the quadrant, is equal to the third power of the radius. 
 
 To the first proposition, the world have assented, because it is 
 plain to see, that the square of 8 is the third power of 4. But 
 in respect to the last proposition, by the popular measures of the 
 circle, when the quadrant of an orbit is 4, the period is 
 4.063750, thus requiring a minim ratio of .063750 to equate the 
 work with. 
 
 By my measures of the circle, the square of the period when 
 the diameter of the orbit is the square root of 64, is the third 
 power of the period, when the quadrant of the orbit is the 
 third root of 64. Or thus, the square of the period, when radius 
 of the orbit is 4, is the third power of the period when the 
 quadrant of the orbit is 4. 
 
 But by the popular measures of the circle, the square of the 
 period when radius of the orbit is 4, would be the third power 
 of the period when the quadrant of the orbit is a little less than 
 4, namely, 3.978988. 
 
 I will now just refer to my prime table, upon which I propose to 
 explain many of the principles of the quadrature of the circle, for 
 the purpose of showing its applicability to the laws of planetary 
 motion also. Hence, if we take any term of the table for the 
 mean distance of a planet, we readily obtain in other terms of 
 the table, the period, the rate of mean motion, the mean rate of 
 force, and the mean rate of convergency of the planet, as also 
 the whole amount of motion, of force, and of convergency during 
 the period of the planet; the amount of force during an entire 
 period being always a mean proportional between the amount 
 of convergency, and the amount of motion during the period. 
 
 Thus, if the mean distance of a planet be the third root of 4, 
 for instance, the time of the period will be the third or cube root 
 of 8, the rate of mean motion will be the third root of .5, the 
 rate of mean force will be the third root of .25, and the rate of 
 mean convergency will be the third root of .125. The whole 
 amount of convergency during the period will be 1, the whole 
 amount of force during the period will be the third root of 2, 
 and the whole amount of motion during the period will be the 
 third root of 4. 
 
CHAPTER IV. 
 
 On the Hypothesis suggested by Kepler in respect to Elliptical 
 
 Orbits. 
 
 PERHAPS one reason that may be offered why it is not as im- 
 perious upon philosophy and astronomy that the elliptical hy- 
 pothesis of Kepler should be adopted, as that his other two 
 laws should, is, that any general law of gravity or motion, ap- 
 plies equally well, at least to an eccentric circular orbit, as to the 
 ellipse. 
 
 Thus Newton's famous demonstration of Kopler's hypothesis, 
 that the motion of a planet revolving in an eccentric orbit varies 
 inversely as the distance varies, and hence that the radius vec- 
 tor will pass over or describe equal areas in equal times, by an 
 application of the problem, " That triangles on the same base, 
 or on equal bases, having the same height are equal to each 
 other," is equally applicable to a circular orbit, whether centric 
 or eccentric, as to the ellipse ; nay further, it applies equally 
 well to the parabola, or hyperbola. 
 
 Hence there appears to be no necessity that a planet should 
 revolve in an ellipse with the sun in one of the foci, in conse- 
 quence of such demonstration ; as it can equally well revolve in 
 conformity with such demonstration, in a circle, having the sun 
 stationary in any given point within the circumference of the 
 orbit. 
 
 Nevertheless, it is not so much my intention to establish the 
 particular forms of orbits, as to promulgate the true laws of 
 gravity and motion ; to endeavor to reconcile their complicated 
 simplicity, and to remove that blur which has so long perplexed 
 the student and inquirer when endeavoring to get an insight 
 into the laws of force and motion through the medium of those 
 round and round explanations given by Maclaurin, Ferguson, 
 and others ; by which we are given to understand, that the ve- 
 locity sometimes gets the better of the force, and sometimes 
 the force gets the better of the velocity, and that it has been 
 found necessary, in order to regulate or equate the defect in the 
 28 
 
218 ON KEPLER'S HYPOTHESIS 
 
 eternal laws, to institute, as a kind of balance wheel, what they 
 term the centrifugal force, which, by the aid of such forces as 
 . they choose to bestow upon it, enables them to make the requi- 
 site equations. 
 
 Mr. Norton, in his Astronomy, gives the following version of 
 what are called Kepler's three laws, viz : 
 
 1. The areas described by the radius vector of a planet are 
 proportional to the times. 
 
 2. The orbit of a planet is an ellipse, of which the sun oc- 
 cupies one of the foci. 
 
 3. The squares of the times of revolution of the planets are 
 proportional to the cubes of their mean distances from the sun, 
 or of the semi-major axes of their orbits. 
 
 And Mr. Norton says that " The first two Kepler assumed as 
 hypotheses, (on partial examination,) after he had discovered 
 that the radius vector and angular motion of a planet were 
 equable, and afterwards verified them, or partially verified them, 
 and they have been completely verified by other astrono- 
 mers," and promises to show in the sequel, that they are ver- 
 ified by the results deducible from them ; and remarks, " that 
 these two laws being established, the third is obtained by simply 
 comparing the known major axes and times of revolution," and 
 then alleges, that " the apparent motion of the sun in space must 
 be subject to Kepler's laws." 
 
 That the apparent motion of the sun in space is subject to 
 Kepler's first and third laws, there can be*ho doubt. But that it 
 is subject to the second, must depend upon the proof of the el- 
 liptical hypothesis ; and that must, probably, be deduced from 
 observation alone, if proved at all ; as no physical cause appears 
 to come forth in support of it. 
 
 Mr. Norton says, " The orbits of the sun, moon and planets 
 being regarded as ellipses, the perigee and apogee, or the per- 
 ihelion and aphelion, are the extremities of the major axis of the 
 orbit." 
 
 This is true, if we regard the orbits as ellipses ; for the ellipse 
 is what presents a major axis, the extremities of which must be 
 the apogee and perigee. But it is the eccentricity of the orbit 
 which gives the apogee and perigee, and this whether the orbit 
 be an ellipse or a circle. 
 
 Mr. Norton again says, " The law of angular motion of a 
 planet about the sun, may be deduced from Kepler's first law." 
 
 This with proper qualifications is doubtless true ; but it has 
 nothing whatever to do with the elliptical hypothesis, as it 
 depends wholly upon the eccentricity of the orbit, and not upon 
 the ellipticity ; for when the orbit is eccentric, what astronomers 
 
RESPECTING ELLIPTICAL ORBITS. 219 
 
 call the angular motion, must of necessity exist. And if the 
 actual motion varies inversely as the distance varies, the radius 
 vector must of necessity describe equal areas in equal times, 
 whatever be the form of the orbit ; and this in accordance with 
 Newton's said demonstration, from which so much Astronomical 
 calculation has been drawn ; as that bodies may form their 
 perihelions in parabolic or hyperbolic curves ; and may revolve 
 about the sun in all kinds of ellipses however eccentric. 
 
 Nor do I think that Mr. Norton's explanations, given in Sec- 
 tions 184 and 185 of his Astronomy, have improved at all upon 
 Newton's demonstration, or that they in anywise prove that an 
 eccentric orbit must of necessity be an ellipse. Nor does he even 
 pretend to deduce the angular motion from any other cause than 
 that of the eccentricity of the orbit ; for he only assumes or 
 considers the orbit used by way of explanation, to be an ellipse. 
 
 Thus far it would seem as though nothing had been shown, 
 proved, or demonstrated, in favor of the ellipse, that does not 
 equally apply to the eccentric circular orbit ; and that the ellip- 
 tical hypothesis must still rest for its support upon evidence to 
 be wholly drawn from observation, (if supported at all,) in lieu 
 of being drawn from any physical cause that can be assigned. 
 
 But so general is the belief in the elliptical hypothesis, that 
 there remains perhaps as little apprehension that the orbits are 
 not ellipses at the present day, as that they were so previous to 
 the days of Kepler. And hence we find the constant attempt 
 in all astronomical works, by almost every mode of expression 
 that can be devised, to induce a belief in the elliptical hypoth- 
 esis ; and often, too, in a way and manner that would seem not 
 to be required for the purposes of explanation of the subject 
 directly under consideration, as though it were an imperious 
 point of faith, however reluctant the mind should be to adopt 
 it. 
 
 Thus Mr. Norton says, " The eccentricity of an elliptic orbit, 
 is the distance between the centre of the orbit and either focus ; " 
 and if a circular orbit be eccentric, the eccentricity is the distance 
 between the centre of the orbit and the sun ; the same as in the 
 ellipse. 
 
 Again he says, " The most accurate method of determining 
 the longitude and epoch of the perigee, rests upon the principle 
 that the apogee and perigee are the only two points of the orbit 
 whose longitudes differ by 180, in passing from one to the other 
 of which the sun employs just half of the year. This prin- 
 ciple may be inferred from Kepler's law of areas ; for it is a 
 well known property of the ellipse, that the major axis is the 
 <only line drawn through the focus, that divides the ellipse into 
 
220 ON KEPLER'S HYPOTHESIS 
 
 equal parts, and by the law in question equal areas correspond 
 to equal times." 
 
 Now the manner in which this sentence is constructed, has 
 somewhat the appearance of a design to sustain a favorite theory 
 which is lacking in the requisite proof, and hence must be sus- 
 tained by presumptive evidence however futile. 
 
 Now not a single important fact alleged in the sentence arises 
 from the properties of the ellipse ; for the truth is, that in either 
 an ellipse or a circular orbit having eccentricity, the apogee and 
 perigee are the only points of the orbit whose longitudes differ 
 by 180 in passing from one to the other, of which the sun em- 
 ploys just half the year; and that the line of the apsides is the 
 only line that can pass through the sun and divide the orbit into 
 equal parts at the same time, is as true of the eccentric circle as 
 of the ellipse. And the proof arising from the application of 
 Euclid's problem in support of Kepler's first law, viz, " That if 
 a body be made to revolve around a fixed point with a motion 
 varying inversely as the distance, a line drawn from such fixed 
 point to such revolving body, will describe, or pass over, areas 
 proportioned to the times," is as true in respect to the circle as 
 to the ellipse. 
 
 Having consulted Mr. Norton thus far, I am unable to dis- 
 cover that he has furnished the slightest evidence in favor of the 
 elliptical hypothesis; notwithstanding we find from his preface, 
 that, in his compilation, he has consulted the most eminent au- 
 thors upon astronomy and mathematics ; and hence, thus far 
 physical astronomy does not seem to furnish equal evidence in 
 support of the elliptical hypothesis that it does in favor of Kep- 
 ler's other laws. And hence the evidence in support of said hy- 
 pothesis seems rather to be the result of Kepler's observations, 
 to which an easy or ready assent has been given ; and especially 
 since the celebrated demonstration in respect to equal areas in 
 equal times, which seemed to operate as a charm, even upon 
 matters with which it had no concern. 
 
 But it is possible that the consequences resulting from the 
 adoption by Kepler, of the elliptical hypothesis, in connection 
 with Newton's demonstration of Kepler's first law, and his 
 especial application of it to Kepler's elliptical hypothesis, are not 
 such at the present day, in physical astronomy, as a better dis- 
 pensation would have produced; for the problem applying 
 equally well to an ellipse of extreme eccentricity as to any oth- 
 er, neither Newton, Halley, nor the world, have set any bounds 
 to the ellipticity of their orbits. For although both Newton and 
 Halley, previous to the supposed important demonstration, sup- 
 posed the comets to perform their perihelions in parabolic curves, 
 
RESPECTING ELLIPTICAL ORBITS. 221 
 
 they changed their views after such demonstration, even in re- 
 spect to cometary motion. 
 
 They hence allow 575 years for the period of the comet which 
 appeared in 1680, and give it an elliptical orbit so eccentric that 
 its curvature at its perihelion must be very rapid indeed, it being 
 166 times nearer the sun than the earth is. And notwithstanding 
 the astonishing velocity with which it is said lo move in its 
 perihelion, yet if the orbit were an ellipse with the sun in one 
 of the foci, the ellipticity would be so great or extreme, that if 
 another comet were to revolve in a circle with its line of apsides 
 of equal length and of equal eccentricity, (and Mr. Norton says 
 that the line of the apsides is what determines the length of the 
 period,) the motion of the comet revolving in the ellipse would be 
 but about 2-3ds as great as that of the comet revolving in the cir- 
 cle, in any corresponding part of the orbit, as will be self-evident 
 to any one who w T ill examine the subject but for a moment ; 
 while the amount of the force of gravity applied to the comet 
 revolving in the circle in the time of its period, would, upon any 
 hypothesis of the law of gravity, be vastly less than the amount 
 applied to the comet revolving in the ellipse in the time of its 
 period, or in the same time. 
 
 Bat this part of the subject belongs more particularly to the 
 laws of gravity and motion. 
 
 But notwithstanding the elliptical hypothesis is such a favorite 
 in the science of astronomy, it is somewhat surprising that it 
 should be so, seeing its application is so repugnant to, and irre- 
 concilable with, the laws of gravity and motion ; for though it is 
 alleged that the orbits of some of the comets are known from 
 observation to be very eccentric ellipses, yet there are but three, 
 namely, Encke's, Biela's, and Halley's, whose periods and orbits 
 have been determined. 
 
 But it is alleged by Mr. Vince and others, that astronomers do 
 not calculate the elements of a visible comet upon the princi- 
 ple that the orbit is an ellipse, but upon the principle that it is a 
 parabola, and afterwards deduce the elliptical form from those 
 calculations. 
 
 The elliptical hypothesis, then, seems to receive but little, if 
 any aid, by its application to cometary motion, it being sus- 
 tained neither by any known physical laws, nor by observation 
 upon which dependence can be placed. 
 
 Let us, then, recur to what Mr. Vince says in respect to Kep- 
 ler's investigations of the subject, and which induced him to 
 adopt and promulgate the hypothesis. 
 
 Ptolemy supposed that the orbits of the planets were circles, 
 and that the earth was not in the centre of those orbits, but situ- 
 
222 
 
 ated at some distance from the centre of those orbits, around 
 which the planets revolved with an equable motion ; and thus 
 he endeavored to account for the apparent inequality of a planet's 
 motion, requiring therefore an equation of its orbit. 
 
 Tycho adopted this theory in part, retaining the theory of the 
 equable motion of a planet around a centre, but placed that cen- 
 tre at a different distance from the earth from that at which Ptolemy 
 placed it, by which his observations and computations would 
 agree within a few minutes; but as the eccentricity, calculated 
 from the equation, would not agree with his supposition, he con- 
 cluded that the sun was not always at the same distance from the 
 central point, which made Kepler suspect that the centre was not 
 the point about which the motion was equal. 
 
 Hence Kepler instituted a series of calculations upon the orbit 
 of Mars, based mostly upon Tycho's observations, for the pur- 
 pose of ascertaining its eccentricity upon the assumption that it 
 was a circle ; and calling the distance of the earth 100000, he 
 found the aphelion distance of Mars to be 166780, and the peri- 
 helion distance 138500, and hence its mean distance 152640, 
 and its eccentricity 14140. 
 
 He next determined three other distances of Mars, and found 
 them to be 147750, 163100, and 166255 ; and on calculating 
 their distances upon the supposition that the orbit was a circle, 
 he found that they would corne out 148539, 163883, and 166605, 
 and that the errors, upon the hypothesis that the orbit was a cir- 
 cle, would be 789, 783, and 350. The distances determined be- 
 ing, to those calculated upon the ground that the orbit was a 
 circle, as 1 to 1.0053, as 1 to 1.0048, and as 1 to 1.0021. 
 
 But Mr. Vince says that " Kepler had too good an opinion of 
 Tycho's observations to suppose that this difference might arise 
 from their inaccuracy; and as the distance between the aphelion 
 and perihelion was too great, upon the supposition that the orbit 
 was a circle, he knew that the form of the orbit must be an oval 
 or ellipse, with the sun in one of the foci ; and upon calculating 
 the said observed or determined distances, he found they agreed 
 together. He did the same in regard to other parts of the orbit, and 
 found they all agreed ; and thus he pronounced the orbit of Mars 
 to be an ellipse, having the sun in .one of the foci. 
 
 Having determined this for the orbit of Mars, he conjectured 
 the same to be true of all the other planets, and upon trial he 
 found it to be so. Hence he concluded that the six primary 
 planets revolve about the sun in ellipses, having the sun in one 
 of the foci." 
 
 Mr. Vince says, " that Kepler, in calculating the eccentricity 
 
RESPECTING ELLIPTICAL ORBITS. 223 
 
 of the orbit of Mars, upon the hypothesis that it was a circle, and 
 on Tycho's principle that it had an equable motion around a 
 point, (supposing the sun and such point were bisected by the 
 centre of the circle,) and having calculated twelve oppositions of 
 Mars observed by Tycho, none of which differed more than 
 1' 47'' , he found that the hypothesis did not agree with the lati- 
 tude observed in opposition, nor with the longitude out of oppo- 
 sition, which differed sometimes 8' from observation; from 
 which want of agreement, and from the place of the sun, and 
 the point around which the motion was supposed to be equable, 
 not being bisected by the centre when thus computed, he was 
 persuaded that the orbit of Mars was not a circle." 
 
 But it is said that the circle, which Kepler says so w r ell repre- 
 sented the twelve Oppositions observed by Tycho, and on which 
 he based his calculations, had an eccentricity equal to 18544, 
 between the sun and point around which the motion was suppo- 
 sed to be equal, but he found this quantity to be divided by the 
 centre of the orbit as follows : between the sun and centre of the 
 orbit, 11332 ; and between the centre of the orbit and sup- 
 posed point of equable motion, 7232. 
 
 Now whether there was error in Tycho's observations, 
 whether there was error or discrepancy in Kepler's calculations, 
 or whether they may have been wrongly quoted, I know not. 
 But in one case upon Tycho's observations. Kepler finds the ec- 
 centricity of the orbit of Mars to be 14140, (calling the distance 
 from our earth to the sun, 100000 ;) and in another case upon Ty- 
 cho's observations, he finds the distance between the sun and cen- 
 tre of Mars' orbit, to be 11332, a difference of near 3000 in the 
 two calculations, the double of which subtracted from the line of 
 the apsides, as by him first determined, would serve doubtless to 
 fully compensate his oval or ellipse into a circle, according to his 
 determination of the three distances in other parts of the orbit. 
 
 But without more light being thrown upon this intricate sub- 
 ject, depending, as it must of necessity, upon observation 
 alone for its support, it is impossible to tell whether Kepler, in 
 his investigations from which emanated his elliptical hypothesis, 
 detected and rectified error to a greater extent than he estab- 
 lished it. 
 
 And notwithstanding we have all looked upon Kepler as the 
 father of physical astronomy, and that he has done more in lay- 
 ing its foundation than all who have attempted its superstruc- 
 ture, and that it is no idolatry to admire such wisdom in an 
 earthly shape, nevertheless, as the elliptical hypothesis seems 
 to be wholly abstract from any known physical law, or necessity 
 connected with unity of purpose or harmony of design, depend- 
 
224 
 
 ing for its support upon observation alone, let it again be cor- 
 rectly examined, and if found to be true, it is doubtless of more 
 extended consequence than we have heretofore conceived it to be. 
 
 But Kepler himself was somewhat perplexed with his ellipti- 
 cal hypothesis; for, although his first law, namely, that the 
 motion varies inversely as the distance varies, furnished the 
 means for equating an orbit, yet, upon the elliptical hypothesis, 
 the finding of the true anomaly from the mean, or the mean 
 anomaly from the true, was no easy task ; for Mr. Vince says, 
 "that although Kepler suggested a problem, which still goes by 
 the name of Kepler's problem, for finding the mean anomaly 
 from the true, and vice versa, of a planet revolving in an ellipse, 
 still the problem has remained unsolved to this day, otherwise 
 than by approximation, notwithstanding it has exercised the tal- 
 ents of eminent mathematicians." 
 
 Such, then, is the evidence upon which the elliptical hypoth- 
 esis rests ; and if it be of consequence to know whether it be true 
 or false, it is also of consequence that observations, depending, as 
 they necessarily must, upon much intricacy and exactness, should 
 be so critically, repeatedly and carefully made, as shall leave no 
 room to doubt. 
 
 Not being able to call to aid any evidence in support of the 
 elliptical hypothesis, I will endeavor to make some few compar- 
 isons between the ellipse and the eccentric circle, as orbits in 
 which planets may be conceived to revolve, for the purpose of 
 seeing if any light can be shed upon the subject by such a 
 course ; and here I will remark that in such examples or com- 
 parisons, one and the same central power is conceived to be used 
 for all the examples. 
 
 Conceive, then, an eccentric circular orbit, whose distance is 
 8 ; of course, the line of its apsides is 8 ; and let its perihelion 
 distance be .25, and its aphelion distance be 7.75, its motion at 
 perihelion would of course be 4, and its motion at aphelion, 
 .2666, or l-15th part as great as at the perihelion. Now sup- 
 pose an ellipse to be inscribed in the circular orbit, their re- 
 spective perihelions and aphelions touching each other, with the 
 sun in one of the foci of the ellipse. That is, according to the 
 idea of an elliptical orbit, it shall be as elliptical as it is eccen- 
 tric. Thus at the perihelion and aphelion, the rates of gravity 
 will be the same in both orbits, but will be greater in the ellipse 
 than in the circle in all parts of the orbits excepting those two 
 points. 
 
 For by Kepler's third law, the periods of the planets must be 
 the same, because the line of their apsides are equal ; at least, 
 according as Mr. Norton renders the law. 
 
RESPECTING ELLIPTICAL ORBITS. 225 
 
 Each planet in respect to itself, would describe equal areas in 
 equal times ; nevertheless, the planet revolving in the circle 
 would not only pass over more area than that revolving in the 
 ellipse, in the same time, but more in proportion to the velocity 
 or space passed over. 
 
 As the rate of gravity of the planet revolving in the ellipse 
 would in all parts of its orbit, or at all times, be greater than that 
 of the planet revolving in the circle ; hence, its amount of gravity 
 in the time of its period would be greater than that in the circle. 
 
 Hence, we have the strange anomaly, of two planets revolving 
 in equally eccentric orbits around the same central force, with 
 their mean distances equal ; their gravities equal at perihelion 
 and aphelion ; and yet one of them constantly passing over but 
 about two thirds of the space of the other, and a far less pro- 
 portion of area, in the same time, and yet constantly urged 
 towards the central power by a far greater force of gravity. 
 
 Now if planets were made to revolve in those two orbits, they 
 would have certain principles in common with each other ; in 
 each the product of the mean distance by the mean motion, 
 divided by the actual distance at any point of the orbit, would 
 give the rate of motion at such point; so each, in respect to 
 itself, would describe equal areas in equal times, if made to 
 revolve by a motion varying inversely as the distance; for the 
 principles are only reciprocal conditions of each other. 
 
 I will now suppose two orbits situated similar to the forego- 
 ing orbits, the lines of their respective apsides being 2, (hence 
 the mean distance of each, according to Newton, is 1,) and let 
 their eccentricities be such that the circumference of the ellipse 
 is just 2-3rds that of the circle ; for if planets can revolve in all 
 kinds of ellipses, however eccentric, they can revolve in such as 
 the foregoing. 
 
 In such case the rate of motion in the ellipse in corresponding 
 points of the orbit, should always be just 2-3rds as great as that 
 in the circle, and hence must be so in their point of mean mo- 
 tion. But if it be not so, we have no criterion whereby to pro- 
 portion the rates of motion between the two, any more than we 
 have whereby to obtain the true anomaly from the mean, or the 
 mean from the true, of a planet revolving in an ellipse ; in fact, 
 it would be all anomaly. 
 
 But what seems somewhat perplexing in respect, to the ellip- 
 tical hypothesis, is, that it seems always inclined to break the 
 law, and will not be subject to it ; but having fortified its posi- 
 tion behind the supposed correctness of Tycho's observations, 
 and Kepler's calculations made from the same, it has thus ac- 
 29 
 
226 
 
 quired a kind of invulnerableness to an attack, by any legitimate 
 kind of warfare. 
 
 Thus by a universal law of force and motion based upon 
 Kepler's great law, that the squares of the periods are as the 
 cubes of the mean distances, we find that the mean distance of 
 a planet divided by the mean motion gives the period ; or the 
 mean distance divided by the period gives the mean motion. 
 And although we find this law to prevail in a circular orbit, 
 whether centric or eccentric, it is always broken in the ellipse, 
 whether the eccentricity be much or little. 
 
 Thus, in the two last-named orbits, if the periods and mean 
 distances must be equal, they will each of necessity be 1 ; and 
 if the mean distance of each be divided by the period, the re- 
 sult will be the same in each, viz., unity or 1 ; but it will not 
 give the mean motion in the ellipse, for that is but 2-3rds as great 
 as in the circle. 
 
 So if the mean distance of each be divided by the mean mo- 
 tion, the period in the circle will come out but 2-3rds as great as 
 that in the ellipse. So that in either case, we find error, and a 
 want of conformity to the laws of force and motion. But 
 suppose in the two last-named orbits, that both planets move 
 from the perihelion at the same time, and that at perihelion, the 
 motion of the planet in the ellipse is just two thirds of that in 
 the circle. 
 
 Now while we are making planets revolve as we please, we 
 may imagine that the ratios of the motion of the two, will so 
 continue to aphelion, and throughout their orbits ; and such 
 would be the case if both planets were constantly attracted by 
 the same intensity of force ; for in such case, they would pre- 
 serve at the same height the same ratios or proportions of mo- 
 tion, whatever might be their respective routes in arriving at 
 such height ; at least, according to Sir Isaac Newton ; for Mr. 
 Maclaurin says, (Book 4th, Sect. 21,) " Our author (Sir Isaac 
 Newton) has shown that if one body move in a curve, and 
 another ascend or descend in a right line, acted on by the same 
 gravity, and their velocities be equal in any equal altitudes, they 
 will be equal in all other equal altitudes." Hence, if it were 
 possible to make the application of what " our author has shown," 
 (which by the way it is not; for I believe that no one can well 
 suppose that if the motions of the two planets were equal in 
 perihelion, that they could take their respective routes in their 
 orbits, having their velocities equal throughout their respective 
 orbits, and at the same time perform their respective periods in 
 the same time,) it would serve to establish my premises ; for the 
 amount of gravity applied to the planet in the ellipse in the time 
 
RESPECTING ELLIPTICAL ORBITS. 227 
 
 of its period, is vastly greater than what is applied to the other ; 
 and hence I will leave it to others, either to reconcile the incon- 
 gruities, or to complete the demonstration by way of reductio 
 ad absurdam. 
 
 The admiration which the wisdom of Kepler so justly merits 
 from the world, ought not to be bestowed indiscriminately with- 
 out regard to truth and error ; and we should as readily come 
 up to the help, against error which he may have promulgated, 
 as if promulgated by any other person. 
 
 I have already stated that his hypothesis in respect to ellip- 
 tical orbits is not founded upon those immutable laws, which 
 may be mathematically or numerically determined, as in respect 
 to his other two laws, but rests upon the supposition that Tycho's 
 observations, as well as his calculations made thereon, were 
 wholly correct; and that there could have been no error, either 
 direct or collateral in either. Nevertheless, while his other two 
 laws, which are immutable and eternal in the heavens, have as 
 yet been more or less misapprehended and misapplied, his 
 hypothesis in respect to an elliptical form of the orbits, has not 
 only taken deep root, but has grown to the overshadowing of 
 every other principle in respect to the planetary system. 
 
 Nor does it appear that Kepler, or others, who have so indus- 
 triously inculcated the elliptic hypothesis, have been aware that 
 it could not be true, in case the motion of a planet in its orbit 
 varies inversely as the distance from the sun varies. 
 
 The ellipse or elongated circle, may vary from that of the 
 circle, to one of extreme ellipticity, in which the two sides will 
 differ but little from straight lines ; and the distance from either 
 focus of the ellipse to the nearest or adjacent end or vertex, is 
 always less than half of the shortest axis or diameter of such 
 ellipse ; and it is gravely taught as a philosophic truth, that a 
 planet may revolve in an ellipse, however elliptical ; and also 
 that the motion of such planet, will vary inversely as the square 
 of the distance from the sun varies. 
 
 Now it would seem that the simple mention of these two 
 positions in connection, would be sufficient to show the import- 
 ant error contained therein, without any attempt whatever at 
 further explanation ; and that Kepler could not have committed 
 the error, in case the law of falling bodies had then been dis- 
 closed by Galileo, his contemporary. But that Newton, who 
 seems not to have been culpable for any error or imposition 
 which he might entail upon the learned world, should have 
 committed the error, with the law of falling bodies before him, 
 (not to say with the full knowledge of the law of falling bodies,) 
 is to me not at all surprising. 
 
228 ON KEPLER'S HYPOTHESIS 
 
 For notwithstanding his attempts at a disclosure of the law 
 of gravity from a consideration of the law of falling bodies, 
 (the very process which should have detected the fallacy of the 
 elliptic orbit,) the bungling and palpable errors therein com- 
 mitted might well induce the belief that no error was too pal- 
 pable or gross for him either to commit or overlook. 
 
 The ellipse, or elongated circle, may vary from the circle by 
 any assignable quantity ; for as New T ton would say, Neque novit 
 natura limitum. 
 
 If a planet revolve in a centric orbit, (viz., with the sun in the 
 centre,) the motion will be uniform ; the whole amount of cen- 
 tripetal force during the time of the period, being expended in 
 producing the convergency of the planet ; and consequently, no 
 portion of the force is expended in accelerating or retarding the 
 motion of the planet. 
 
 If such orbit shall become eccentric, (still retaining its circu- 
 lar form,) inasmuch as the planet in the course of its revolu- 
 tion approaches and recedes from the sun, then upon the prin- 
 ciple of falling bodies, its motion must alternately be accelerated 
 and retarded, and consequently a portion of the constant force 
 applied is expended in such acceleration and retardation, and 
 the remainder in the convergency. 
 
 Now the convergency of a planet is no part of the motion, 
 and may simply be denned to be the direction of the motion; 
 and so we find the definition in treatises upon the involution and 
 evolution of curves, where it is said that "curvature is a change 
 of direction only," although it is frankly acknowledged in those 
 treatises, that curvature has been illy defined, and the subject 
 very much perplexed in its consideration. Nor is it passing 
 strange that the subject should have been perplexed, considering 
 the manner in which it has been treated. 
 
 I have heretofore attempted to show that convergency is not 
 only a proper, but an important element in the consideration of 
 planetary motion ; that it is a tendency to the same point again, 
 from which the planet has revolved ; that in the time of the 
 period, the convergency is always 360, and consequently the 
 amount of convergency in the time of 'the period is always 
 unity, and hence might be assumed as a standard element in 
 any orbit, whatever may be the distance ; that it is directly con- 
 nected with planetary motion and the law of gravity ; and that 
 the rates of motions, of gravity, and of convergency, are so pro- 
 portioned to each other, and to the period and distance, as to 
 perform the grand results required by the Keplerian law. That 
 the mean rate of convergency is always inversely as the time of 
 the period, consequently, inversely as the cube of the square 
 
RESPECTING ELLIPTICAL ORBITS. 229 
 
 root of the distance, it is hence, the cube of the mean rate of 
 the motion, and the cube of the square root of the mean rate 
 of the gravity or force, &c. And hence, the mean rate of 
 gravity or force, is always a mean proportional between the 
 mean rate of motion, and the mean rate of convergence ; and 
 gravity throughout the period is equally attentive to conver- 
 gency as it is to motion; and whether the orbit be centric 
 or eccentric, the element of gravity or force, at all times and in 
 all parts of the orbit, preserves the same position in respect to 
 the elements of motion and convergency, viz., a mean propor- 
 tional between them ; for upon no other condition can the 
 motion of the planet vary inversely as the distance varies from 
 the sun ; nor should we be much alarmed at the immutable pre- 
 cision by which the force is thus divided or proportioned. 
 
 Suppose, then, the force not to be thus proportioned between 
 the motion and convergency ; that is, suppose the orbit to be 
 an ellipse, in which case it is manifest that the force cannot be 
 thus proportioned to the motion and convergency. And to lay 
 our premises broad, we will suppose such ellipse to possess very 
 great ellipticity ; in which case the planet, in descending from 
 its aphelion to its perihelion, shall describe a line of but little 
 curvature, or varying but little from a rectilinear fall ; in which 
 case, the motion of the planet at aphelion will be as near to as 
 the ellipticity of the orbit is to extreme ellipticity. 
 
 Now the rates of commencement of motion, or of continued 
 motion of two bodies falling from a state of rest at the same 
 time from different distances, towards the same centre of gravity, 
 are inversely as the squares of the distances at which they com- 
 mence their respective fall, for the very reason that the force of 
 gravity is inversely as the distance ; so also, the motion of each 
 body from the commencement of its fall, will vary inversely as 
 the square of the distance varies ; and such would be the condi- 
 tion of a body revolving in the extreme ellipse, (if, however, 
 such impossibility could be performed; but I will not, like Mac- 
 laurin, be wiser in respect to laws that have no existence in 
 nature, than those which have.) 
 
 The consequence must then necessarily be, that as we vary 
 from the circular orbit, whether centric or eccentric, in which the 
 motion varies inversely as the distance varies, and approximate 
 from the ellipse of least ellipticity to that of the greatest, we 
 must necessarily approximate from the inverse variation of the 
 motion to the distance, to the inverse variation of the motion 
 to the square of the distance. 
 
 I have, therefore, in my investigations, become rather satis- 
 fied that no other explication co^Ul well be given of the opera- 
 
tion of force and motion as applicable to a planet revolving in 
 an ellipse, than that given by Maclaurin and others, viz., that 
 the force and motion alternately get the better of each other. 
 And whether it shall continue to be said, as heretofore, that, 
 " As it has been proved that a planet can revolve in an ellipse, 
 having the sun in one of the foci, it consequently follows as a 
 general law, that bodies may revolve in all kinds of ellipses, 
 however eccentric," or, whether it shall hereafter be said that, 
 as it has been shown that bodies cannot by the laws of force 
 and motion, revolve in all kinds of ellipses, however eccentric, 
 it consequently follows that a body cannot, by the laws of force 
 and motion, revolve in any kind of ellipse, are questions 
 which are submitted for consideration. 
 
CHAPTER V. 
 
 Outline of a Theory of the Solar System. 
 
 I have already intimated that I might offer an outline of my 
 theoretic notions in respect to the great physical law of the uni- 
 verse, under its various modifications, in its constant operations 
 in the adjustment and control of the Solar System; referring for 
 evidence in support of such theory, so far as I am able to avail 
 myself of them, to such facts, phenomena and analogies in na- 
 ture, already before the world, as will, in my opinion, best serve 
 to sustain or support such theory or hypothesis. 
 
 In assigning causes to the phenomena of nature, perhaps we 
 cannot well adopt a better criterion by which to be governed, 
 than that contained in the first two rules laid down by Sir 
 Isaac Newton, namely : " No more causes are to be admitted 
 than are sufficient to explain the phenomena;" and 2d, "Of 
 effects of the same kind, the same causes are to be assigned, as 
 far as it can be done." And it might be well for the advance- 
 ment of science and philosophy, were these rules more strictly 
 adhered to. But we are all quite too apt to deviate from those 
 rules which we may have prescribed for our future conduct. 
 
 But in attempting to theorize where the immutability of num- 
 bers does not come directly to our aid, there are many things as 
 yet too high for our understanding, too vast for our compre- 
 hension, the causes too remote to be assigned, and the opera- 
 tions too abstruse and evasive in their detection, to be well de- 
 fined or explained, without more facts and data than the accu- 
 mulated knowledge of the world is yet in possession of, to enable 
 us to form a theory (even of the Solar System,) that is not in 
 many respects liable to constant demolition, arising from the dis- 
 covery of new facts and analogies ; all of which, doubtless, tends 
 to show the simplicity, harmony, reciprocity and analogy which 
 pervade all the laws of nature, and their operations. 
 
232 OUTLINE OF A THEORY 
 
 I will not here make the sacrilegious attempt to theorize upon 
 the primitive formation of the universe, as many have done, 
 for the purpose of accounting for all things; or, at least, of 
 being understood by others to have done so, as how adventitious 
 nature, with her gravity on hand, accumulated her atomic matter 
 into its present aggregate masses, from which gravity is con- 
 stantly emanating in search of prey, and that nothing but the 
 speed of the smaller bodies prevents their being dragged to the 
 larger, or the no less vague, unwarranted and uncalled for as- 
 sumption that all the revolving bodies of the solar system were 
 originally projected in tangents to their respective orbits, with 
 such forces as would produce the present momenta of the revolv- 
 ing body in that point of the orbit where it was first taken charge 
 of by gravitation, whatever point in the orbit that might have been 
 so that all the people might know by what means the planets 
 and other revolving bodies first got under way. But as none of 
 these projections have happened in our day, our faith in the hy- 
 pothesis (if we have any) must rest upon a supposition or ap- 
 prehension which is of no use whatever to science, as it is not a 
 phenomenon of nature or of nature's laws; we see it not, we 
 know it not, and at best it is but an attempt to show a necessity, 
 consonant with human contrivance, which Deity was under, of 
 making use of certain means for the accomplishment of his pur- 
 poses in creating and upholding the universe. 
 
 Not so with the laws of nature, which are continually operat- 
 ing, and the phenomena which they produce, and which it is 
 our highest privilege to investigate, thereby exalting the human 
 mind by a more comprehensive view or understanding of cause 
 and effect, which, more than anything, distinguishes man from 
 the brutes. 
 
 Nor will I undertake to fancy that this earth, at some former 
 period, was detached from the sun in one molten mass, the out- 
 side of which as yet is but just cooled to a crust, and that we 
 are yet in danger of having it broken up in some terrible erup- 
 tion, and we plunged into the eternal fires, &c. 
 
 But I would rather confine my speculations to the solar sys- 
 tem, as it now exists, and as it now appears to be governed by 
 those immutable laws founded in the fitness of things, and from 
 which all the phenomena of nature are doubtless results, effects 
 and consequences. 
 
 Of gravitation, or the attraction of gravitation, we at present 
 know but little, otherwise than through its effects and operations 
 upon palpable matter, or matter in aggregate form. Nor is our 
 philosophy sufficiently astute to reason intelligibly upon the ope- 
 rations of gravity, or attraction, upon two inconceivably small 
 
 ' : 
 
OF THE SOLAR SYSTEM. 233 
 
 corpuscles or physical atoms, and thence to transfer the analogies 
 of those operations to that of tangible masses of matter, as has 
 been attempted and pretended by high philosophers. 
 
 Nevertheless, we have sufficient facts, data and phenomena, 
 resulting from the laws of gravitation or attraction, from which 
 to deduce many important results, if analogically considered, 
 amongst which is that of the ratio in respect to distance, by which 
 the attraction of gravitation operates upon matter. But in what 
 way or manner the influence of gravitation is exerted on matter, 
 has perhaps never been well defined or determined. Neverthe- 
 less, it is not absolutely necessary that we should conceive it to 
 operate through absolute space, without a material medium in 
 and through which to operate ; and which medium, however at- 
 tenuated, has a density varying in some given ratio to the dis- 
 tance from bodies of aggregate matter, namely, in the same ra- 
 tio as that of the force of gravitation. 
 
 Perhaps there are but few astronomical facts better ascertained 
 from observation and calculation, than that the attractive powers 
 exerted by the several primary planets of the solar system, are 
 as their bulks or magnitudes directly, and their inverse distances 
 from the sun conjointly. And if such be the case, it furnishes 
 strong presumption, at least, that their attractive powers are not 
 innate, and that attraction or gravitation is not an essential prop- 
 erty of matter ; but that the planets of the system obtain their 
 gravitating and attracting powers from the circumstance of their 
 being within the attracting power of the sun, in the same man- 
 ner as soft iron obeys the magnetic influences when brought 
 near a permanent magnet or an electro-magnetic battery. 
 
 I know that Sir Isaac Newton, having adopted the theory of 
 the equal and innate gravity of all matter, and observing that the 
 attracting power of Jupiter and Saturn upon their respective sat- 
 ellites was not as great in proportion to their apparent magni- 
 tudes as that of the earth upon the moon, adopted the concomi- 
 tant hypothesis, that those primary planets are less dense than 
 the earth is. And such has since been the progress in ascertain- 
 ing the different densities of the primary planets, that Euler cal- 
 culated their densities to be inversely as their distances from the 
 sun ; and the astronomer and mathematician, LaGrange, long 
 since adopted the same ratio, as a standard from which to de- 
 termine their disturbing forces upon other matter ; and after 
 much observation on the subject, remarked that he saw no cause 
 for altering that ratio, believing it to be true. 
 
 To be sure, the present Herschel supposed the planet Hers- 
 chel to have a greater density than would be given it by su^h 
 oO 
 
234 OUTLINE OF A THEORY 
 
 ratio, but modern observations have determined that he overrated 
 the density of Herschel. 
 
 But I know of no analogies in nature to sustain the hypothe- 
 sis of the different densities of the planets, and shall therefore en- 
 deavor to seek for the cause of the phenomena upon which such 
 hypothesis is based, in the analogies arising from observed phe- 
 nomena, as well as such direct proof as may have been adduced 
 by those who have been most forward in establishing and sus- 
 taining the hypothesis of the different densities of the planets. 
 But surely there can be no analogy between the hypothesis that 
 the force of gravity varies inversely as the square of the distance, 
 and that which assumes the densities of the planets to be in- 
 versely as their distances from the sun. But that there are anal- 
 ogies between the hypothesis that the force of gravity varies in- 
 versely as the distance, and that the planets vary in their re- 
 spective gravities in proportion to their respective magnitudes, 
 in the inverse ratio to their respective distances from the sun, I 
 will endeavor to show. 
 
 In respect to the Newtonian law of gravity, namely, that grav- 
 ity varies inversely as the square of the distance, we find in 
 Vince's History of Astronomy, Vol. 2, p. 279, the following, 
 namely : 
 
 " That in 1684, Dr. Halley turned his thoughts to the relation 
 between the periodic times and distances of the planets, and 
 concluded from it that the centripetal force must vary inversely 
 as the squares of the distances ; but not being able to prove it, 
 he applied to Mr. Hook and Sir Christopher Wren ; they, how- 
 ever, not being able to give him satisfaction, he went to Cam- 
 bridge, to Mr. (afterwards Sir Isaac) Newton, who soon gave 
 him a proof of his position." 
 
 And this a proof ] as also that of Newton's theory of universal 
 gravity, seems not only to have satisfied the mind of Dr. Halley, 
 who was searching for truth, and longing to be satisfied, but 
 also that of the world in general ; and it appears that no farther 
 inquiry was made upon the subject. Thus we find in the Amer- 
 ican Encyclopaedia, under the head of Gravity, that the author 
 asserts, that gravity appears to be an essential property of mat- 
 ter ; that it is proportional to the masses of bodies; that it 
 varies inversely as the square of the distance in proceeding 
 from the surface of the body outwards, or from its centre ; that 
 it varies directly as the distance, in descending from the surface 
 to the centre in uniform spherical bodies ; and that it is trans- 
 mitted instantaneously from one body to another, &c. Some of 
 which allegations are certainly worthy of consideration and 
 remark in due time. 
 
OF THE SOLAR SYSTEM. 235 
 
 But a vast amount of what is generally conceived by the 
 world to be evidence of the law of gravity, as declared or pro- 
 mulgated by Sir Isaac Newton, may be found in Vince's Physi- 
 cal Astronomy, Vol. 2, Chap. 31, under the head of " The gen- 
 eral principles of centripetal forces," in which frequent refer- 
 ence is made to the Principia, as authority ; and in which, to 
 me, the reasoning appears to be vastly more deep or dark, than 
 profound ; as by attempting to geometrize upon a geometric 
 point, or upon the law of gravity which two geometric points 
 or physical atoms will regard in consequence of their innate 
 gravity upon each other, aided by the use of expressions like 
 the following: the limiting state, the limiting ratio, a ratio of 
 equality, ultimately proportional, &c., which being applied to 
 the circular orbit, the transition is then made to the ellipse, the par- 
 abola, and the hyperbola. (And, perhaps, no one will doubt but 
 that the same law of gravity which operates in the circle, will 
 operate also in the ellipse, the parabola, and the hyperbola, so 
 far, at least, as distance is concerned.) And in the result, we find 
 Mr. Vince coming to the following conclusion, viz. : " Hence, 
 we conclude that each planet, is attracted to the sun by a force 
 which varies inversely as the square of the distance of its 
 centre, and that the constituent particles also attract each other 
 by a force varying according to the same law," and that the 
 measure of force will be obtained by dividing the square of the 
 motion by the distance ; which, if true, would consequently make 
 the force inversely as the square of the distance ; which explan- 
 ation has too well satisfied the world. 
 
 I will now proceed to a consideration of the attractive power 
 of the planets as proportioned to their magnitudes and distances 
 from the sun. 
 
 Mr. Vince says, " That to understand the principle upon 
 which this determination rests, we may observe that the effect 
 of attraction at equal distances will be in proportion to the 
 quantity of matter in the attracting body ; and at different dis- 
 tances, as the quantity of matter and the inverse square of the 
 distance conjointly. The quantity of matter is also in propor- 
 tion to the magnitude of the body and its density conjointly. If, 
 therefore, we know the effects of the attraction of different 
 bodies, together with their magnitudes, we can find their densi- 
 ties, and thence their quantities of matter." 
 
 And this is true, in case the theory of equal innate gravity of 
 matter be true ; in which case, the density of a central body, as 
 of Jupiter for instance, will be numerically expressed in the 
 reciprocal of the product of the magnitude of the central body 
 by the square of the period of the revolving body ; which meth- 
 
236 OUTLINE OF A THEORY 
 
 od, if the principle were but correct, would furnish a very easy 
 method of obtaining the different densities, (so called,) or the 
 proportional quantities of matter of those planets having satel- 
 lites revolving around them. 
 
 But the world are wishing for the best evidence by which to 
 account for the phenomenon, that the power of a planet to 
 attract its satellite varies in proportion to the magnitude of the 
 planet, in the inverse ratio to its distance from the sun. And 
 for the purpose of presenting some natural and consequential 
 evidence upon the subject, I will present, in short, a method 
 which I would pursue in the deduction of consequences from ob- 
 served phenomena, with a view to some unity of purpose ; and 
 to this end, I would assume the distance of the moon from the 
 earth to be unity or 1. In which case, if the distance of the 
 earth from the sun be 95,000,000 miles, and the distance of the 
 moon from the earth be 237,500 miles from the earth, (as esti- 
 mated by some,) the proportional distances between the earth 
 and moon, and between the earth and sun, will be as 1 to 400, 
 which I will assume for the sake of round numbers. And as 
 the relative motion of the moon in its orbit about the earth, is 
 to that of the motion of the earth in its orbit about the sun, as 
 1 to 29.9207, the square of which is 895.2428, this multiplied 
 by 400, (the relative distance of the sun,) gives 358099, and a 
 fraction, as the relative power of the sun to that of the earth; 
 that of the earth being called unity. But the exact proportion 
 between the power of the earth and sun must depend upon 
 getting the true relative distances between the earth and moon, 
 and between the earth and sun, which, perhaps, cannot be ob- 
 tained to a great degree of exactness. 
 
 So, if we take the mean between the estimates of Cassini and 
 Sir Isaac Newton, as the mean distance of Jupiter's fourth satel- 
 lite from Jupiter, its relative distance from Jupiter to that of 
 Jupiter from the sun, will be as 1 to 427, and the relative mo- 
 tion of such satellite in its orbit to that of Jupiter in its orbit, 
 will be as 1 to 1.6463, in which case the relative power of at- 
 traction of Jupiter to that of the sun would be as 1 to 1157.8501. 
 Hence, the relative power of attraction of the earth to that of 
 Jupiter (calling the distance of the moon from the earth 237,500 
 miles ; or rather, calling the relative distance of the moon from 
 the earth to that of the earth from the sun as 1 to 400) would 
 come out as 1 to a little more than 309. Or, if we call the 
 relative distance of the moon from the earth to that of the earth 
 to the sun, as 1 to 391, then upon the foregoing calculation, the 
 relative force to that of Jupiter would be about as 1 to 312, 
 which is in general about the proportion assumed. But it must 
 
OF THE SOLAR SYSTEM. 237 
 
 be manifest that the estimates of relative power must depend 
 upon the relative distances we assume ; and those cannot be 
 ascertained to a very great degree of exactness. 
 
 The foregoing method goes upon the principle that the force 
 exerted by the sun upon the planets is in the inverse ratio to 
 their respective distances from the sun; and hence that the mean 
 force exerted by the sun upon a planet is the square of the mean 
 motion of the planet; and that such is the case in respect to the 
 earth and moon, as in respect to Jupiter and its satellites. And 
 hence, that the force exerted by the sun upon the earth at the 
 distance 400 is a little over 895 times as great as that exerted by 
 the earth upon the moon at the distance 1. 
 
 Thus the force exerted is always inversely as the distance ; 
 and hence the proportional power of gravity or attraction of the 
 respective planets of the solar system, are as their respective 
 magnitudes and their inverse distances conjointly. That is, if 
 one planet be twice the distance of another planet from the sun, 
 its power of attraction will be but just one half as great in pro- 
 portion to its magnitude as that of the other. 
 
 And when inquiry into the phenomena of nature presents 
 and submits itself directly to mathematical investigation and 
 solution, corroborated by well known laws, taking for data 
 such elements only as would most naturally assign the true 
 cause from the result produced, without seeking further for an 
 hypothesis by which to account for the cause of the phenomena; 
 we may feel much confidence in a conclusion which so naturally 
 follows from such investigation. 
 
 Thus, the square of the mean motion of a revolving body 
 multiplied into its mean distance from the central body, gives the 
 ratio of force between the two bodies. Or thus : if the relative 
 motion of a planet compared with the relative motion of its 
 satellite, be multiplied into the relative distance of the planet 
 from the sun, compared with the distance of the satellite from 
 the planet, we thereby obtain the relative power of the sun com- 
 pared with that of the planet. 
 
 But as the mean motion of a planet is inversely as the square 
 root of the mean distance from the sun, how the fact should 
 have escaped observation, that the mean force is the square of 
 the mean motion, and consequently inversely as the distance, 
 I know not; for the world has certainly acknowledged those 
 principles in the methods they have pursued for finding what 
 they have conceived to be the different densities of the planets. 
 
 Thus, they make the mean motion of a satellite, the square 
 root of the mean force of the planet, exerted upon the satellite ; 
 and so do I. 
 
 So if there were three planets, A, B, and D, of equal magni- 
 
238 OUTLINE OP A THEORY 
 
 tudes, revolving about the sun, A at the distance 1, B 2, and 
 D 4 ; each having a satellite revolving around it at like distances, 
 one with the other ; in such case, by my calculation, as well as 
 by that made by the world in their investigations, the attractive 
 powers of the planets compared with each other, will be inverse- 
 ly as the squares of the periods of their respective satellites ; and 
 upon this ground the world have placed the comparative, or 
 rather relative attracting powers of the earth and Jupiter ; and 
 so have I. 
 
 The consequence then is, that the reciprocal of the square of 
 the distance of a satellite, divided by the square of the period of 
 the satellite, gives the attractive power of the primary. 
 
 Then these being facts and results from all calculation, it only 
 remains that the true cause should be assigned to the phenom- 
 ena thus observed. 
 
 But whatever may be the cause why the planets exert a force, 
 thus proportioned to their respective distances from the sun, 
 upon their respective satellites, we do find that the force thus 
 exerted, is the square of the motion produced in the satellite ; 
 thus emphatically determining line law of force to be inversely as 
 the distance ; and this goes far towards overturning that fortuitous 
 notion, that matter, gross matter, contains, or is possessed within 
 and of itself, all the requisite powers and principles for its gov- 
 ernment and control throughout eternity. 
 
 Thus do we find that not only is the force of gravity inversely 
 as the distance ; but that a primary planet only exerts a force 
 upon its satellite in the inverse ratio to the distance of such 
 primary from the sun ; which fact I shall make use of as an 
 unequivocal inference hereafter. 
 
 I have already intimated, that from the many analogies which 
 magnetism bears to that of attraction in the solar system, it de- 
 serves consideration ; especially in the analogy presented in the 
 powers of attraction of the planets as proportioned to their 
 respective bulks and distances from the sun ; and from which 
 analogies, much evidence is certainly derived to justify the 
 hypothesis, that the sun is the great magnetizer of the bodies 
 which compose the solar system ; and that the primary planets 
 possess no other attracting power than that communicated to 
 them by the sun; and also, that the attracting power of the pri- 
 mary planets is in the inverse ratio to their respective distances 
 from the sun, and their respective bulks or magnitudes conjoint- 
 ly ; and that the density and intensity of what I will here call 
 the magnetic atmosphere of the sun, varies inversely as the 
 distance varies from the sun. 
 
 It could scarcely be expected from our present limited 
 knowledge of magnetism, (even since the discoveries of Galvani, 
 
OF THE SOLAR SYSTEM. 239 
 
 and Oerstead,) that we should be able to trace out all the analo- 
 gies existing in its various forms and modifications; or of 
 solar magnetism, in all its various forms and modifications ; but 
 nevertheless, there would seem to be enough under all circum- 
 stances, to make it more than probable, that the laws of magnet- 
 ism are precisely those of solar attraction. 
 
 Although we may well suppose the sun to be a permanent 
 magnet, yet perhaps there are no direct means for determining 
 whether the magnetism of the planets is permanent, or whether 
 they are only dependent upon the sun for their magnetism, 
 while lying within his magnetic influence, in the same manner 
 as are particles of iron filings, which become magnetic and attract 
 each other when brought within the influence of a permanent 
 magnet; or like soft iron when under the influence of the electric 
 or galvanic fluid ; in which cases the magnetism ceases when the 
 inducing fluid is withdrawn. 
 
 If then the attractive power of a planet is not an innate 
 principle of matter, but is only induced by the magnetic influ- 
 ence of the sun or centre of attraction, in the manner in which 
 iron filings are operated upon when under the magnetic influ- 
 ence, or soft iron, when operated upon by the Galvanic, or 
 Electro-magnetic fluid ; then a planet revolving in an eccentric 
 orbit, would possess a greater power of attraction when in 
 perihelion, than when in aphelion ; and its increasing force and 
 motion while passing from its aphelion to its perihelion, would 
 tend to dilate the orbit of its satellite ; and the reverse while 
 proceeding from perihelion to aphelion ; and were the orbit of 
 such planet sufficiently eccentric for determining so nice a test, 
 doubtless, the oscillations of a pendulum of a given length, or of 
 a compass-needle, would be more rapid when the planet was in 
 perihelion, than when in aphelion. 
 
 So likewise, if a soft piece of iron be made to revolve about 
 a strong magnet, and within its magnetic influence, its power of 
 attraction will vary in some inverse ratio to its distance ; but 
 I do not know but that it would be as dense in one part of 
 its orbit as in another. 
 
 I think then there are analogies and unequivocal inferences in 
 support of the hypothesis that attraction is not necessarily an 
 innate or inherent principle of matter; but I know of no anal- 
 ogies to support the Newtonian hypothesis of universal gravity 
 or attraction. 
 
 But perhaps those analogies between magnetism and solar 
 attraction, which may serve to explain, (at least, theoretically,) 
 the more hidden or occult recesses of nature, depend upon 
 the polarity of magnetism in all its various forms and modi- 
 fications ; and here we bring directly to our aid, the phenomena 
 
240 OUTLINE OF A THEORY 
 
 exhibited by electricity and Galvanism, which appear to be so 
 intimately connected with magnetism as to induce the belief 
 that they are but modifications of one and the same great law, 
 but presenting somewhat different phenomena under different 
 circumstances and modifications ; and that the magnetic fluid of 
 the solar system is the medium in and through .which all their 
 various operations take place, either in the government of inert 
 matter, or in the production of organic matter, and in the con- 
 trol of final causes. 
 
 Each of these modifications, (if such they be,) has its poles, 
 or inverse influences, tending, nevertheless, to a junction with 
 an ardency or intensity corresponding with the degree to which 
 they appear to have been separated ; which poles, or adverse 
 influences, are so inverse in their operations, as often to perplex 
 the mind in respect to them. Thus it is said, " what we call the 
 north pole, (of the magnet,) because attracted by the north pole 
 of the earth, is, for this very reason, the south pole of the mag- 
 net ; " going upon the ground or well known fact, that poles of 
 the same kind repel each other, while those of opposite kinds 
 attract each other ; and why this circumstance has not given rise 
 to a different method of investigating the operations of terres- 
 trial magnetism, I know not. 
 
 I know it has generally been supposed that the earth was a 
 great mass of inert matter; but at the same time possessed 
 of power over other inert matter, by means of an innate or 
 inherent principle of attraction ; and that from some unaccount- 
 able circumstance attached to this mass of dead matter, it has 
 two poles, a north and a south, with a magnetic equator, &c. 
 Some who have believed in the volcanic theory of internal fires, 
 as connected with the Huttonian theory of geology, have been 
 apprehensive lest, those fires were not yet sufficiently exhausted, to 
 prevent danger from a breaking through of the -crust in some 
 volcanic eruption, or violent earthquake. And many a curious 
 calculation has been gone into, to show how long it would take 
 a mass of matter as large as the earth, to finally cool through 
 and become one solid mass. 
 
 Some have imagined that there was latent heat or caloric 
 enough combined with the solid matter of the earth, to reduce 
 the whole mass to what they conceive to be its primitive gasses, 
 (should that heat once become active,) and again fill a wondrous 
 space with its rare and primitive elements ; while others have 
 been perplexed with the geology of the earth ; the inequalities 
 of its surface ; the vast apparent changes that have been going 
 on in time, at and near the surface ; the vast regions that have 
 manifestly been raised from under the waters of the ocean, even 
 into elevated mountains, &c, ; many of which phenomena are 
 
OF THE SOLAR SYSTEM. 241 
 
 truly wonderful when unaccounted for ; and that they are satis- 
 factorily accounted for by the causes which have generally been 
 assigned, will not be contended for by a large portion of the 
 community. 
 
 But had geologists, while endeavoring to assign causes for 
 those phenomena, in lieu of placing the adverse poles of the 
 magnetic fluid, (in its various modifications,) upon the surface of 
 the earth, where they appear to have but little efficiency in the 
 great operations of nature, or but little use. except in directing 
 the compass needle, only placed them in contact, at the centre 
 of the earth, they would at once have assigned a cause sufficient 
 to account for all the phenomena that have so much perplexed 
 them, by establishing a laboratory sufficient for all the internal 
 operations of the earth ; as well as to account for most of the 
 geological phenomena upon and near its surface. The intensity 
 of heat would then pervade the whole internal region of the earth, 
 and fuse the whole internal mass. There would be the process 
 of combining, separating, and mineralizing the various metals, 
 minerals, and other substances, to the utmost capacity. 
 
 As certain portions of the solid parts of the earth became thin- 
 ner, by the fusion of its inner parts, they would yield to internal 
 pressure and gradually recede from the centre ; while the thicker, 
 and less fused portions, would gradually become central por- 
 tions; until their inner portions in turn, should become molten, 
 while other portions were cooling and becoming thicker; and 
 by thus alternating, (but without much apparent uniformity,) 
 would, in connection with the volcanic flues or vents to the in- 
 ternal fires, which occasionally and alternately pervade the sur- 
 face of the earth for periods of time, cause all the phenomena 
 upon the surface; as that of the inequalities upon its surface, 
 geological strata, the veins of metals, &c. This theory would 
 most admirably account for the independent coal formations 
 which are found to exist at and near the surface of the earth, 
 more commonly in latitudes remote from the equator, and 
 which are doubtless caused by deposits of vegetable matter, 
 conveyed from towards the equator by ocean currents, as that of 
 the Gulf-stream, for instance, which in process of time, under 
 pressure and heat, is converted into coal, and thence raised 
 above the ocean for the future use of man. 
 
 This gradual rising and falling of different portions of the 
 crust, (and with such uniformity only as all the circumstances 
 would permit, some perhaps being raised or sunk several times, 
 while others were stationary, or nearly so, together with the 
 alteration of the internal portions of the solid parts, by the action 
 of heat, as well as by new volcanic vents produced, or old ones 
 31 
 
242 OUTLINE OF A THEORY 
 
 closed, or opened anew,) would cause the magnetic or gal- 
 vanic fluid to become somewhat varied in its direction; and 
 thereby cause a corresponding variation in the magnetic needle. 
 And should the magnetic meridians be found to correspond 
 with the thicker or thinner parts of the crust, or with lines having 
 reference to the great chains of volcanic vents, or to their greater 
 or less activity; in such case, the declination of the needle 
 would only serve to indicate where those sections were. 
 
 There are certainly strong indications that the magnetic (or 
 galvanic) current, is in the direction from the equator (or rather 
 the magnetic equator) of the earth, towards the external poles of 
 the earth ; and that in all this distance, at least until we arrive 
 at a high latitude, it is received and conducted by the surface 
 of the earth, in a curve corresponding with the surface or nearly 
 so ; increasing in quantity or intensity as we proceed towards the 
 higher latitudes ; until the whole shall at length be conveyed, by 
 this conducting curve, to the internal poles ; and hence, from 
 this constant accumulation in quantity, as we advance towards 
 the higher latitudes, the accepted pole of the needle becomes 
 more strongly attracted or more strongly inclined to correspond 
 with the magnetic current ; and hence, the oscillations of the 
 needle in selling to its meridian will be more rapid as we 
 approach the higher latitudes, than they are at the magnetic 
 equator. 
 
 But it is by no means certain that the magnetic fluid becomes 
 more abundant or intense up to 90 degrees of latitude ; and if 
 not, (and so the inference would seem to be,) then, whenever the 
 point of greatest intensity was arrived at, the dip or inclination 
 of the needle, would be at 90 degrees, having lost all horizontal 
 traverse or direction, and thence would remain in that situation 
 or nearly so, up to 90 degrees of latitude; and hence, the com- 
 pass-needle might become quite useless as a guide to the 
 mariner in the highest latitudes. 
 
 But as the poles of magnetism, electricity, and galvanism, 
 (which in their effects and operations appear to be but different 
 modifications of the same agent or power,) are so inverse in 
 their apparent operations, it behoves us to have a care, lest we 
 assign to them effects that do not belong to them. Thus it is 
 said, "what we call the north pole of the magnet, because 
 attracted by the north pole of the earth, is, for this very reason, 
 the south pole of the magnet." 
 
 Now the ground of this dictum, is the assumption that 
 the external poles of the earth are the only actual and effec- 
 tive poles of the magnetic or galvanic fluid ; and if such were 
 the fact, the allegation in respect to the compass-needle or magnet, 
 
OP THE SOLAR SYSTEM. 243 
 
 would be correct, could we but reconcile the idea with the dip 
 or inclination of the needle. But aside of the little consequence 
 that we are able to deduce from the external poles of the earth, 
 (except that of giving direction to the compass-needle, and which, 
 by the way, becomes less and less a direction as we advance 
 into the high latitudes, antipodes as those external poles are to 
 each other, and so contrary to all experience, in producing 
 effects by the adverse poles of the magnetic, electric, or galvanic 
 fluids;) the inferences, would seem almost unequivocal, that 
 the fluid of what we commonly term the north pole of the 
 magnet, is identical with that at the external north pole of 
 the earth ; and that the error has arisen from supposing the 
 external poles of the earth to be the actual poles of the fluid ; 
 and not that the earth formed a conducting circle from its sur- 
 face to its centre ; and that the pole which has been conceived 
 to have its direction towards the north, has in fact, its direction 
 towards the south, at the centre of the earth, where it meets the 
 opposite pole, and where the battery performs its operations. 
 
 And hence, the fluid of either pole of the magnet, or needle, 
 is always impelled in the line of direction of its own identical 
 fluid; seeking connection or contact with the opposite pole at 
 the centre of the earth ; but when fixed, chained and iron bound 
 to a steel bar, and not at liberty to pursue its wonted course, it 
 is always ready to turn aside, and enjoy connection with the 
 adverse fluid, whenever opportunity or inducement may offer ; 
 but such does not offer when fastened at opposite ends of the 
 same straight bar; and those subtle fluids never mistake in 
 their object. 
 
 I will then assume for the fluid of the respective poles of the 
 magnet, its wonted inclination in fact ; viz. that of flowing in 
 the current of its own identity when left free to do so ; and 
 hence, in the direction assumed by the compass needle. 
 
 I have said that the adverse currents flow all the way from the 
 magnetic equator towards the external poles of the earth ; and 
 thence to their respective poles at the centre. This was only 
 by way of exemplification ; for there is much evidence to show 
 (by way of inference at least) that the whole surface of the earth 
 is both a receiver and conductor of the fluids for both poles ; but 
 that each end of the earth only conducts its respective fluid to the 
 centre, or central poles. Thus, on all parts of the earth, so far 
 as we know, artificial magnets can be formed possessing the 
 adverse fluids, each being inclined to pass with the fluid of 
 which it is identical. 
 
 If this theory in respect to magnetism and the earth, be true ; 
 if it be sustained by all the analogies arising from experiments 
 
244 OUTLINE OF A THEORY 
 
 made with magnetism, electricity and galvanism, either singly 
 or combined ; if it be impossible otherwise to rationally account 
 for many of the phenomena observed in geology, and especially 
 that hypothesis so strongly inferred as to compel most people to 
 implicitly believe the hypothesis a fact, and to endeavor to ac- 
 count for it by the most vague, absurd, and unknown cause, 
 viz. the central fires of the earth ; then let us adopt the theory, 
 and transfer all the analogies to the other primary planets, and 
 to the sun ; all of which, being governed by the same great laws 
 of nature as the earth is, especially by that of attraction, or the 
 great physical law of the universe, apparently require internal 
 fires as much as the earth does ; and also that they should be 
 made in the same way ; and if beneficial, that they should be 
 continued, and not liable to be extinguished, nor to frighten those 
 mortals who inhabit the surface, nor put them in danger of fall- 
 ing into them by breaking through the crust. 
 
 Conceive then that the sun is the centre of our system : and 
 that it is surrounded, to an unknown extent, by what we must 
 probably conceive to be a highly attenuated and subtle fluid, 
 which is the medium of attraction ; and that this fluid, or 
 medium, becomes less dense or less intense in proportion to the 
 distance, or as we recede from the sun ; 
 
 That by and through this medium of attraction, the sun be- 
 comes the magnetizer of all gross matter in the solar system ; in 
 which case, the primary planets are magnetized in the propor- 
 tion to their inverse distance from the sun ; thereby giving them 
 attractive powers in proportion to their bulks directly and their 
 inverse distance conjointly ; and hence their inverse distances 
 from the sun denote their respective rates of gravity towards 
 the sun ; 
 
 That each planet, from being thus magnetized by the sun, be- 
 comes itself imbued by a portion of the magnetic fluid, the 
 density of which is proportioned to the bulk of the planet direct- 
 ly and its inverse distance from the sun conjointly; and hence, 
 its power to attract other matter ; and in such case, it would 
 be hard to conceive that the magnetic atmosphere of the sun 
 would be a resisting medium to the planet, from the circum- 
 stance that the fluid as readily attaches to the planet as to the 
 sun ; in which case, all resistance to the motion of the planet 
 would be annihilated far in advance ; 
 
 That the sun and planets are supplied with internal fires in 
 the same manner that the earth is ; and that too in proportion to 
 their respective surfaces and their powers of attraction ; and also 
 with volcanic flues proportional, as outlets to such internal fires ; 
 
 That neither light nor attraction, or gravitation, are emana- 
 
OF THE SOLAR SYSTEM. 245 
 
 tions from matter, in the sense in which they have generally 
 been considered ; that neither of them are progressive in time; 
 but are instantaneous in their effects at all distance subject to 
 their effects ; arid hence, that the effect, or rate of intensity of 
 each, is inversely as the distance; nor has light anything to do 
 with filling space with particles, or rays of light, as has in gene- 
 ral been taught and understood; 
 
 That the atmospheres of the sun and planets, properly so 
 called, (viz. that which consists of the gasses, gross vapors, and 
 impalpable matter that surrounds those bodies,) may well be 
 conceived to have some near proportion to the size, and to the 
 power of attraction of the bodies which they surround; the den- 
 sity of such atmosphere depending upon its elasticity and the 
 power by which it is attracted, must necessarily depend on the 
 amount of matter which it contains, and the power of attraction 
 with which it is drawn towards the body which it surrounds ; 
 and hence its capability to affect light by refraction, reflection or 
 otherwise ; and hence its power for raising gross vapors from the 
 body which it surrounds ; which in case of the sun, must be 
 with great rapidity, and to a great height; 
 
 That when we view the sun through a telescope, we see its 
 solid parts ; but through an extremely dense and brilliant atmos- 
 phere of vast extent ; 
 
 That the spots often observed upon the sun are caused by 
 volcanic eruptions, or discharges of fire, smoke, vapors and oth- 
 er substances from the vents to the internal fires ; which, rising 
 in vast quantities, with great rapidity, and to a great height, often 
 spread the umbra to a vast extent, so as to hide large portions of 
 the sun's surface from our view ; and while the eruption con- 
 tinues, the boundary between the nucleus, or matter immediately 
 ejected, and the umbra, will be well defined; and also, that the 
 nucleus will naturally vanish before the umbra ; and on their 
 both vanishing, a red hot lava may for a time present to view 
 what is termed a facula, until it shall at length take the appear- 
 ance of other parts of the surface of the sun. 
 
 If, however, the sun, or centre of the solar system, be sur- 
 rounded by a vast magnetic atmosphere of ever so great tenuity, 
 if the same vary in density in the inverse ratio of the distance, 
 and is sufficiently dense to refract light, (the central battery of 
 which fluid is at the centre of the sun,) the same would most ad- 
 mirably account for the phenomena observed by Dr. Bradley, from 
 which he deduced his hypothesis of the progressive motion of 
 light in time, as also for the zodiacal light. Notwithstanding, 
 La Place would reject the last suggestion, for the reason that it 
 would interfere with the Newtonian theory of gravity, which he 
 has so liberally endorsed to the world. 
 
CHAPTER VI. 
 
 On the Tides. 
 
 IN accounting for the tides upon the principle of the attractive 
 powers of the sun and moon upon the earth, and the waters of 
 the earth ; those who have attempted it, have assumed as 
 a lemma, or at least an hypothesis, that the force of gravity 
 is inversely as the square of the distance ; and as the appa- 
 rent diameter of the moon is to that of the sun when viewed 
 from the earth, as 1.091 to 1, and as according to their lav/ 
 of gravity, the respective influences of the sun and moon upon 
 the earth, should be as their apparent diameters, in case their 
 densities were equal, hence, according to their apparent diam- 
 eters, the moon would have but very little more influence upon 
 the tides than the sun has. 
 
 But finding that the greater part of the tides are caused by the 
 influence of the moon, it has been concluded that the moon is 
 much more dense than the sun ; which densities as proportioned 
 to each other, they have weighed and determined, so far at least 
 as to satisfy themselves and many others, by resorting to a 
 comparison of their supposed respective influences upon the 
 tides ; which, to be sure, has cost them no small trouble ; nor 
 have their conclusions been very uniform or satisfactory, even 
 to themselves. 
 
 Bernoulli estimates the influence of the moon to that of the 
 sun upon the tides, as 2.5 to 1 ; Newton, as 3.5 to 1 ; and others 
 have even estimated it as 1 6-7th to 1 ; La Lande, as 2 7-lOths 
 to 1 ; and La Place 3 to 1. But Bernoulli's ratio is in general 
 settled upon, as being thought the most probable. Hence the 
 conclusion, that if the joint influence of the sun and moon will 
 raise the tides 7.445 feet, the moon's portion will be 5.412 feet, 
 while that of the sun will be 2.033 feet. 
 
 But after all, they seem to labor under difficulties in adapting 
 their theory to observation ; for Mr. Vince, after enumerating 
 many difficulties of this kind, observes, " Thus the theory alone 
 will afford no practical conclusions." And in justification of 
 
ON THE TIDES. ' . 247 
 
 certain tables based or founded on theory, after consenting that 
 they will not agree with observation, says : " To determine, 
 therefore, the true time at any port, we must find from observa- 
 tion what is the difference between the true time and that shown 
 by the table, and then 'that difference added to the time shown 
 by the table will give the time of the high tide." 
 
 In the American Encyclopaedia or Conversations Lexicon, the 
 reader, under the article Tides, will find a somewhat racy expo- 
 sition of the commonly received theory of the tides, and as the 
 articles in that work, though often giving but an epitome of the 
 subjects of which they treat, are nevertheless written with ability, 
 giving the most approved philosophy of the day in which they 
 were written, as is the case with the article alluded to, which 
 is an article also well calculated to satisfy the minds of most 
 people upon a subject of which they had thought and cared but 
 little, and who were not very curious to know if these things 
 were so, and as it is somewhat convenient to have it here 
 inserted, with a view fo commentary and comparison, I will 
 make a liberal extract from the same, notwithstanding the work 
 itself is readily accessible to most readers. 
 
 " The ebb and flow of the sea are evidently connected with 
 the moon's motions. The level of the ocean is slightly disturbed 
 by the attraction which is alternately exerted and withdrawn. 
 The waters for a large space under the moon, being more at- 
 tracted than the great body of the earth, are thus rendered light- 
 er than those parts of the ocean which are at the same distance 
 as the earth's centre, and being lighter, they are forced upwards 
 a little by the surrounding mass, which is heavier, just as water 
 and oil will stand at different heights in the two branches of a 
 syphon tube ; or just as ice, which is lighter than water, is made 
 to rise a little higher on that account, when placed in water. 
 
 " If the earth rested immovable upon a fixed support, there 
 would be a tide or rising of the waters only on the side towards 
 the moon. But the great body of the earth is just as free to 
 move as a single particle of the ocean, and if suffered to yield 
 to the moon's attraction, would be carried just as fast. Hence 
 for the same reason that a particle of water, on the side of the 
 earth towards the moon, is drawn away from the centre, or has 
 its downward tendency diminished, so the solid earth itself is 
 drawn away from the mass of waters on the side of the earth 
 farthest from the moon. It is the difference of attraction in both 
 cases between the surface and the centre, which causes the light- 
 ness of the waters and the consequent elevation. It will be 
 seen, therefore, that, taking the whole earth into view, there are 
 always two high tides diametrically opposite to each other, and 
 
248 ON THE TIDES. 
 
 two low tides also midway between the high ones. The high 
 tides are two great waves, or swells, of small height, but extend- 
 ing each way through half a right angle. These waves follow 
 the moon in its monthly motion round the earth, while the earth, 
 turning on its axis, causes any given place to pass through each 
 of these swells and the intervening depressions in a lunar day, 
 or twenty-four hours and fifty minutes. What we have said with 
 respect to the moon's influence in disturbing the level of the 
 ocean, may be also applied to that of the sun ; only in the case 
 of the sun, although its absolute action is about double that of 
 the moon, yet on account of its very great distance, its relative 
 action upon the surface of the earth compared with that at the 
 centre, is but about one third as great as that of the moon. At 
 the new and full moon, when the sun's and moon's action con- 
 spires, the tides are highest, and are called spring tides. But at 
 the first and last quarters of the moon, the action of one body 
 tends to counteract that of the other; and the tides, both at ebb 
 and flow, are called neap tides." " That the high tide when the 
 moon is above the horizon, exceeds the high tide when the moon 
 is below the horizon. And that 50 towards the nearest pole 
 from where the moon is vertical, there will be only one tide in 
 twenty-four hours," &c. 
 
 Now the theory contained in this running account of the 
 phenomena of the tides, (though it were the result of La Place's 
 greatest intellectual effort,) is not calculated to give full satisfac- 
 tion upon the subject. 
 
 1st. The waters on the side of the earth towards the moon, 
 are more attracted than the great body of the earth, made 
 lighter than the waters at the distance of the centre of the earth, 
 and hence forced up by the surrounding mass of waters. 
 
 2d. If the earth were made immovable, (viz., so that its centre 
 and solid parts could not be attracted towards the moon,) there 
 would be a tide only on the side of the earth towards the moon. 
 But as it is not fixed, the solid earth is drawn from the mass of 
 waters on the side of the earth opposite the moon, and hence, 
 creating a tide on the side of the earth opposite the moon. 
 
 3d. That the tides on both sides of the earth are caused by 
 the difference of attraction between the surface and centre of the 
 earth. 
 
 4th. That the high tides are two great waves or swells, ex- 
 tending each way through half a right angle, following the moon 
 in its monthly motion round the earth. 
 
 5th. That the sun has a like influence upon the waters that 
 the moon has, only, though the sun has about twice the abso- 
 lute action upon the earth that the moon has, yet from its great 
 
ON THE TIDES. 249 
 
 distance, its relative action upon the surface of the earth com- 
 pared with that at the centre, is but about one third as great as 
 that of the moon. 
 
 In the foregoing theory there is certainly an ample number of 
 causes assigned to produce the effect or phenomena, in case they 
 are such as are sufficient. Thus, if the attraction of the moon 
 makes the waters on the side of the earth towards the moon 
 lighter than they are at the distance of the centre of the earth from 
 the moon, so that they are forced upward by the heavier sur- 
 rounding waters, this cause would be sufficient for causing the 
 tide on the side of the earth towards the moon ; and would 
 seem to dispense with that of their being attracted away from 
 the central parts of the earth. But as the surrounding waters 
 could only press those upward which were made lighter on that 
 side of the earth towards the moon, this cause alone could not, 
 therefore, account for the phenomena of the opposite tide ; and 
 hence, inasmuch as the opposite tide must be accounted for 
 from some other cause, and by one which would as well account 
 for both as for either, the whole cause seems rather at length to 
 be assigned to that of the waters on the side of the earth towards 
 the moon being attracted from the central parts of the earth ; 
 and the central parts of the earth being attracted and drawn 
 from the waters on the opposite side of the earth. 
 
 This doubtless goes upon the ground, that if the earth were 
 an entire mass of waters, and were not disturbed by any foreign 
 power, the mass \vould be in shape of a perfect sphere ; but if 
 attracted by the moon, (for instance,) it would become elongated; 
 its greatest diameter being directed towards the moon, in like 
 manner as Sir Isaac Newton conceives the moon to be in 
 respect to the earth ; with this difference, however, that while 
 Newton supposed the end of the moon which hung next to the 
 earth was heavier than the other end, the contrary would be the 
 case in respect to the ends of the earth. 
 
 But we will suppose the internal parts of the earth to be solid, 
 or composed of solid matter, having its surface covered with 
 deep waters ; and then suppose the tides to be two broad waves 
 directly opposite each other ; that the one nearest the moon shall 
 be caused by being attracted from the solid parts of the earth, 
 and the opposite one shall be caused by the solid parts of the 
 earth being attracted from the opposite waters ; the high water of 
 that wave nearest the moon being, however, some three hours 
 or 45 behind the moon, (or the position of the moon, caused 
 by the diurnal motion of the earth ;) and then let us endeavor to 
 calculate the motion of the solid parts of the earth, (as well as of 
 32 
 
250 ON THE TIDES. 
 
 those two great waves) caused by the disturbing force of the 
 moon. 
 
 Will it be said in such case, that the solid parts of the earth, 
 in consequence of the attraction of the moon, make or perform 
 a revolution in an orbit, in about twenty-four hours and fifty 
 minutes; the direction of which orbit will be subject also to all 
 the wanderings, or northings and southings of the moon; and 
 that the direction of the solid parts of the earth in such orbit is 
 always towards the point of high tide, in that wave or swell 
 nearest the moon ? 
 
 If so, to calculate and adjust the equations of the earth's an- 
 nual orbit, required in consequence of this disturbing force, would 
 puzzle even a La Place. 
 
 But it is said the sun has about twice the influence upon the 
 earth that the moon has, but from its very great distance, its rela- 
 tive influence upon the surface of the earth, is but about one 
 third as great as that of the moon compared with that at the 
 centre. Hence, according to this hypothesis, the influence of 
 the sun upon the centre of the earth, must be vastly greater 
 than that of the moon ; and I will here merely query, how 
 those joint, opposing, or varied influences of the sun and moon, 
 are to affect the centre and surface of the earth, in all the wan- 
 derings of the moon; how in conjunction, in opposition, in 
 quadratures, &c. 
 
 To me, the whole theory is but a tissue of absurdities and 
 dilemmas ; as though cause and effect had been guessed off, 
 without much consideration with regard to their connection, 
 mutuality, or reciprocity; and as if the connecting links had 
 been buf little regarded. 
 
 Nor does the foregoing theory differ essentially from that of 
 Mr. Vince. Mr. Vince, after observing that Kepler was the 
 first who assigned the true physical cause of the tides, namely, 
 " that the waters of the sea gravitate towards the moon, and 
 cause the tides;" and that Sir Isaac Newton had shown that 
 from the principles of gravity, the phenomena of the tides may 
 be solved, proceeds with an attempted exposition of the pheno- 
 mena, upon the Newtonian principles, by referring to the sup- 
 posed operation of corpuscles, of infinitely small magnitudes, up- 
 on each other, (by which the world has been so much darkened 
 in its researches after truth ;) and after alleging difficulties that 
 constantly prevent fact from agreeing with theory, as for instance, 
 " that the free motion of the waters in the open seas is hindered 
 by shallow places, rocks, and islands, in consequence of which, 
 the tides in some of the open seas, at the time of the conjunction 
 of the luminaries, are found to rise only to the height of about 
 
ON THE TIDES. 251 
 
 three feet, 5 ' proceeds to remark thus, in which the case seems 
 rather to be summed up : 
 
 " The general phenomena of the tides from observation, 
 agree very well with the conclusions deduced from the theory of 
 gravity, indeed much more accurately than could have been 
 expected, when we consider how many circumstances there are 
 which take place, and which cannot be reduced to computation. 
 The theory supposes the w r hole surface of the earth to be covered 
 with deep waters, that there is no inertia of the waters, that 
 the major axis of the spheroid is constantly directed to the moon, 
 and that there is an equilibrium of all the parts. But the inertia 
 of the waters will make them continue to rise after they have 
 passed the moon, although the action of the moon begins to 
 decrease, and they come to their greatest altitude in the open 
 seas about three hours after, at which time there is not a general 
 equilibrium, but the waters rise and fall by a reciprocation ; 
 hence, the longest axis is not directed to the moon, nor is the figure 
 a perfect spheroid. The waters have not a free motion on account 
 of the shallow places, rocks, islands, and continents, the force of 
 currents and winds; also as the waters approach the equator 
 where the earth has a greater velocity about its axis, they must 
 necessarily be left behind, and obstruct the regular motion of the 
 water when it moves from west to east, but conspire with that 
 from east to west. 
 
 " All these circumstances must affect the measures of the phe- 
 nomena as deduced from theory ; it may, however, in many cases, 
 give the relative measures without any great error, so that by 
 accurate observations once made, on their absolute quantity, in 
 some one particular case, the measures, in all other cases, may 
 be ascertained to a considerable degree of accuracy." 
 
 Now, from the phenomena of the tides, as found from obser- 
 vation, perhaps no one would doubt Kepler's hypothesis, that 
 they were caused by the influence of the moon ; and had not New- 
 ton and Bernoulli subterfuged and perplexed the theory, in 
 Newton's mysterious law of gravity, in connection with the ease 
 and facility with which La Place has been able to equate the 
 effects of all disturbing forces that might be suggested, it is 
 highly probable that the world, from observation, would long 
 ago, have come to more rational conclusions as to the manner in 
 which the influence of the moon is exerted in causing the tides. 
 
 Mr. Vince, who always treats general subjects with the utmost 
 candor, seems rather cautiously to have avoided descanting 
 much upon the cause of the tide on the side of the earth oppo- 
 site the moon ; and hence, has left this part of the subject rather 
 destitute of explanation. 
 
 I have thus presented some of the palpable and insurmount- 
 
252 ON THE TIDES. 
 
 able difficulties under which the popular theories labor in at- 
 tempting to account for the phenomena of the tides upon the 
 Newtonian law of gravity, and more especially upon the New- 
 tonian hypothesis of universal gravity, which, in my humble 
 opinion, lays the whole foundation of the fabric of astronomy 
 in fortuity and atheism. And if. as is so whimsically alleged, for 
 the purpose of giving it a marvellous effect, this hypothesis was 
 the offspring of the fall of an apple, the consequences of that fall, 
 (comparing small things with great,) were not less deleterious to 
 the scientific world, than was the forbidden fruit to the moral 
 world. 
 
 I will now, after a few preliminary remarks upon the law 
 of gravitation according to my determination, (viz., that the 
 force of gravity varies inversely as the distance varies,) give 
 an outline of my theory of the tides, based upon my theory of 
 universal gravity as connected with my theory of the solar system. 
 
 By rny determination of the law of gravity, if the moon were 
 to revolve about the earth in an orbit but l-4th the distance that 
 it now does, its influence upon the earth would be four times what 
 it now is, and by the Newtonian law of gravity, its influence 
 would be sixteen times what it now is. But according to my the- 
 ory of the tides, I should, according to my law of gravity, make its 
 influence upon the tides a little more than four times what it now 
 is ; and by the Newtonian law of gravity, I should make its in- 
 fluence upon the tides a little more than sixteen times what it now 
 is ; for the motion of the moon in its orbit must of necessity, be 
 just double w^hat it now is, if it revolve four times as near the 
 earth ; and hence, the motion of the moon in its orbit being 
 doubled, its apparent motion over the surface of the earth, caused 
 by the diurnal motion of the earth upon its axis, would not be 
 quite as great as it now is. But whether upon the popular 
 theories of the tides, what are conceived to be the great tidal 
 waves of the ocean would swell to four times the height in the 
 one case, or to sixteen times the height, in the other case, that they 
 are now conceived to swell, I will not undertake to determine. 
 
 The estimated bulk or magnitude of the moon to that of the 
 earth (which is doubtless correct) is as 1 to 49, and if their 
 influences or attractive powers upon each other are in propor- 
 tion to their respective bulks or magnitudes, (and from all anal- 
 ogy as yet deduced from observation, there is little or no doubt 
 of the fact,) their respective influences upon each other will be 
 as 1 to 49. 
 
 If we now take round numbers, and consider the distance of 
 the moon from the earth to that of the earth from the sun as in the 
 proportion of 1 to 400, which is in accordance with the estima- 
 tion of some, and is sufficiently near for our present purposes, 
 
ON THE TIDES. 253 
 
 although it makes the distance of the moon somewhat less 
 than the common estimate, if we call the distance of the sun 
 95,000,000 miles from the earth, in such case, according to 
 my determination of the law of gravity, the attractive power of 
 the sun upon the earth or moon would be some eight hundred 
 and ninety-live times as great as that of the attractive power of 
 the earth upon the moon ; as denoted and determined by the mo- 
 tion of the earth in its orbit about the sun, and the motion of the 
 moon in its orbit about the earth ; and consequently, that the at- 
 tractive power of the sun upon the earth would be more than 
 forty thousand times as great as that of the moon upon the 
 earth. 
 
 And, yet, according to my theory of the tides, no portion of 
 them are caused by the direct influence of the sun upon the 
 earth, or upon its waters ; although its indirect influence exerted 
 through the moon, causes the variation from spring" to neap tides. 
 And in fact, if my determination of the law of gravity be 
 correct, it is quite certain that the influence of the sun upon the 
 tides compared with that of the moon (whether direct or indi- 
 rect) must be exceedingly small, when compared with their 
 respective attractive powers upon the earth. 
 
 But I will conclude my investigations of the theories extant, 
 in the words of Mr. Vince, in which he again sets forth, by way 
 of repetition or recapitulation, the difficulties to be encountered 
 under present theories ; and will thence proceed to a theory of my 
 own. After many attempted explanations of the causes and 
 phenomena of the tides upon the Newtonian theory of gravity, 
 he says, " It has been here supposed, that the high tide was 
 under the luminary, and that there was a general equilibrium of 
 the waters ; but the high tide is at some distance from the lumi- 
 nary, and the waters rise and fall by a reciprocation ; also the 
 free motion of the waters in the open seas is hindered by shal- 
 low places, rocks and islands ; in consequence of which, the 
 tides in some of the open seas, at the time of the conjunction of 
 the luminaries, are found to rise only to the height of about 
 three feet," and concludes by saying, " thus the theory alone will 
 afford no practical conclusions." 
 
 So much for Mr. Vince, and other authorities upon the theory 
 of the tides. 
 
 I will now present a short outline of my own theory of the 
 tides, based upon my theory of universal gravitation, and upon 
 my theory of the solar system, submitting it to the consideration 
 of an inquiring world, who have ever been anxious, from the 
 days of Aristotle and Virgil, to know 
 
 Why flowing tides prevail upon the main, 
 And in what dark recess they shrink again. 
 
254 ON THE TIDES. 
 
 Malte Brun, who was somewhat of a whimsical and credu- 
 lous philosopher, (however eminent he was as a geographer,) 
 treats of the tides mostly after the popular methods ; he has 
 some remarks, however, not unworthy of notice, as "that they 
 have tides in the islands of the South Sea of only one or two 
 feet elevation, whilst upon the western coasts of Europe and 
 the eastern coasts of Asia the tides are extremely strong and 
 subject to many variations." Now Europe and Asia make a 
 very broad continent ; and the western shores of Europe and 
 the eastern shores of Asia are washed by broad oceans. 
 
 The same author says, " In the torrid zone, the flood-tides run 
 from east to west with the motion of the stars," but he says 
 nothing as to the direction of the ebb of such tides, and might 
 leave us to infer that the tides in the torrid zone may have some 
 connection with those ocean currents with which he has made 
 such a display, that it has been thought prudent to suggest, that 
 he may have accounted for more than existed. 
 
 Such ocean currents, however, as are known to exist, and 
 which have been traced, or the courses of which have been ascer- 
 tained, seem very well calculated to aid and corroborate a true 
 theory of the tides; nor do I conceive it a matter so inexplicable as 
 the author above named would intimate, that a current should 
 proceed from the Ethiopian ocean to the coasts of Brazil, that 
 a current should proceed from the coast of Brazil and extend 
 towards the eastern coasts of America, and prevail between 
 30 north, and 10 south latitude ; and that within the above 
 limits (of the degrees of latitude no doubt) there should be a cur- 
 rent running the contrary way, or toward the coast of Africa. 
 These currents, however, all seem inclined to run towards conti- 
 nents, or large tracts of land, and thence to float along the shores 
 or near them, and to pass strongly through straits, or narrow 
 passes, as between New Holland and Van Diemen's Land, 
 &c. But few of the ocean currents, however, have been very 
 distinctly traced ; nor will I do more than merely to suggest or 
 hint at a subject so little known or understood, in conroboration 
 of my theory of the tides. 
 
 SECTION SECOND. THEORY OF THE TIDES. 
 
 My theory is, that the tides are caused by the direct influence of 
 the moon, the sun exerting no other influence in raising the 
 tides than an indirect one exercised through the moon ; al- 
 though the attractive influence of the sun upon the earth is many 
 thousand times greater than that of the moon upon the earth ; 
 
THEORY OF THE TIDES. 255 
 
 That the earth possesses \wofixed poles in respect to the sun, 
 which are susceptive of, and obedient to solar magnetism, (or the 
 magnetic fluid of the solar system,) in conducting the same to the 
 centre of the earth, for the purposes designed; which solar mag- 
 netism so wholly and powerfully pervades the whole earth, that 
 solar magnetism is not perceptible in one part of the earth more 
 than in another, if we except the stronger influence that may 
 exist at the poles; but that the earth has no movable poles in 
 respect to the sun ; 
 
 That the moon has but one fixed pole in respect to the earth, 
 and none in respect to the sun ; but that its poles in respect to 
 the sun, are movable and revolving ; one always being towards 
 the sun, and the other opposite to the sun. And that similar to 
 this, is the earth in respect to the moon, so far as the mag- 
 netic influence of the moon is capable of creating polarity 
 in the earth ; the poles of the earth created by the moon being 
 movable, or revolving poles, one always being directed towards 
 the moon, and the other in the opposite direction. And although 
 those poles may be said to have some influence over the whole 
 of both hemispheres of the earth, yet the greatest attractive 
 power of those poles would always be directly underhand op- 
 posite to the moon, at which points matter would always be 
 inclined to accumulate. 
 
 That the moon has but one fixed pole in respect to the earth, 
 may fairly be inferred from the circumstance that one and the 
 same side of the moon is always retained towards the earth. A 
 circumstance which Newton attempted to account for upon 
 the hypothesis that the moon might be elongated, and that, 
 consequently, one end would hang towards the earth, while 
 others have as gravely supposed, that one side of the moon 
 might be heavier than the other. If, however, our present con- 
 ceptions of polarity and magnetism will not permit the idea of 
 a fixed or permanent pole without its opposite, there surely could 
 be no objection to conceiving the moon to have two fixed poles 
 in respect to the earth, both of them situated on the side of the 
 moon next to the earth (exercising their influence in and through- 
 out the moon,) any more than to a magnet having both its poles 
 in one direction. 
 
 If the foregoing hypothesis in respect to the magnetic or at- 
 tractive influences of the earth, sun, and moon, upon each other, 
 be adopted, the cause and phenomena of the tides are readily 
 accounted for. 
 
 In such case, the movable poles of the earth created by the 
 moon, would be constantly tending to attract matter toward 
 those poles ; and should we conceive the earth composed of a 
 fluid, as of water, or to be covered with deep waters, and that 
 
259 THEORY OF THE TIDES. 
 
 one side of it is stationary towards the moon, for any con- 
 siderable length of time, we might well suppose the earth 
 would become elongated in the direction towards, and from the 
 moon, to the extent of the moon's polarizing influence. 
 
 But if the whole globe were formed of water, or if it were cov- 
 ered with deep waters, and were to revolve upon its axis, as it now 
 does, it may be hard to conceive how a tide could well be formed ; 
 for notwithstanding the earth would be polarized in a direction to 
 and from the moon, yet, as those poles must travel on the sur- 
 face of the earth, at the rate of about one thousand miles an 
 hour, in consequence of the diurnal motion of the earth, there 
 would be no time for the accumulation of the waters at those 
 moving poles ; and hence the waters must remain at rest, without 
 rise, or without reciprocation of rise and fall. Nor can I conceive 
 how the most imaginative or fanciful theorist, could, under those 
 circumstances, assign the place where the waters should rise, or 
 the reciprocation occur. 
 
 But the surface of the earth being divided into oceans and 
 continents, and the matter upon the surface of the earth having 
 a constant tendency (caused by the moon) to accumulate at the 
 poles ; hence, when the moon is passing over a broad continent 
 and vast solid parts of the surface are under the influence of this 
 polarization for a long time, or for several hours, the continent 
 being itself fixed and immovable upon the surface of the earth, 
 serves to produce an effect similar to that which would be pro- 
 duced, if the whole surface were covered with deep waters, and if 
 one side of the earth should be made to remain stationary un- 
 der the moon for hours together. 
 
 That is, the waters of the surface of the oceans adjoining the 
 continent over which the moon should be passing, (and like phe- 
 nomena would occur on the side of the earth opposite the moon,) 
 would flow or tend towards such polarized continent, and pile 
 upon its shores, ascend its rivers, &c. 
 
 And hence, the flowing of the tides is always towards con- 
 tinents or large bodies of land, caused by the fixedness of 
 the land, and its polarization by the moon ; and the height 
 of the tide, or the piling of the waters upon the shores, will 
 be mostly governed by the magnitude or breadth of the continent, 
 and the situation of the moon in respect to the continent. 
 
 So also the flowing of the tide may be east, west, south 0*' 
 north, or in any direction, as the body of land shall dictate ; and 
 as the flowing' of the tide is towards the land, its ebb is necessa- 
 rily from the land ; and hence the rising and falling of what is 
 termed the great tidal ivave, by reciprocation, seems to savor 
 more of fancy than of possibility or reality. 
 
THEORY OF THE TIDES. 257 
 
 Upon this hypothesis, the waters would rise upon the shores of 
 islands near continents, the same as upon the snores of the con- 
 tinents ; but in the middle of broad oceans, or far from large 
 bodies of land, the tides would be small; so also on the shores 
 of continents, where the continent was narrow, &c. 
 
 While this hypothesis accounts for the tides being greater on 
 the shores of continents than in the broad oceans, it also accounts 
 for the reason of the tides not being so long behind the meridian 
 of the moon, in the midst of broad oceans, as upon the shores 
 of continents ; as well as why it is behind the moon's meridian 
 at all ; without resorting for a reason, to the tardiness of the 
 water in obeying the power of attraction, arising from its inertia, 
 for the water will continue to pile upon the shores of a conti- 
 nent, so long as the continent, from polarization, possesses an 
 attracting influence sufficient to draw the waters towards it ; and 
 it will not be denied, that time is requisite for the flowing of 
 water, whether it be in a river or upon the surface of the ocean. 
 
 Should it be objected, that the time of high water on the west 
 side of a continent, should not be so long behind the meridian 
 of the moon as on the east side, I will only remark, that the 
 whole cause of the tide arises from the influence which polar- 
 ized terra firma has upon that substance, (namely water.) which 
 is always at liberty to flow towards the fixed earth, as the moon, 
 from its apparent rapid motion, cannot of itself move the waters 
 at all. And each side of the continent is under the influence 
 of the moon for an equal length of time ; and most clearly, such 
 objection would apply with equal, if not greater force to any 
 other theory extant 
 
 When the moon is in conjunction or in opposition, namely, in 
 syzygies, the tides are highest ; and when in quadratures, the 
 tides are lowest, or very nearly at those times, the altitude of 
 the tides at quadratures to those at syzygies being proportioned 
 to each about as 1 to 1.3756, although it is said the ratio will 
 vary, from many circumstances. The tides also increase in alti- 
 tude from quadratures to syzygies, and diminish from syzygies to 
 quadratures ; which increase has been attributed to the attraction 
 of the sun conspiring with the moon at those times, and that 
 the decrease is occasioned by the attraction of the sun, conspir- 
 ing against the moon, or not in the line of direction with the 
 moon ; and hence, that the sun is gradually abstracting, or 
 separating its own tides from those of the moon, as the moon 
 approaches its quadratures ; at which times the respective tides 
 of the sun and moon are in quadratures, which must necessarily 
 throw the earth into the shape of a double oblate spheroid, the 
 33 
 
258 THEORY OF THE TIDES. 
 
 altitude of the moon's tides being to those of the sun's as 2.662 
 to 1, or as 5.412 feet to 2.033 feet. 
 
 That those four tides may exist when the moon is in quadra- 
 tures, (according to the Newtonian theory,) both the sun and 
 moon must draw away the solid parts of the earth from the mass 
 of waters on the side of the earth opposite the given luminary; 
 and hence, .the motion of the solid parts of the earth could not 
 be in a direct line toward either, (arid whether an equation 
 has ever been resorted to in consequence of those directing 
 forces, I know not.) 
 
 So when the sun and moon are in conjunction, their united 
 influence must draw the solid parts of the earth directly towards 
 the sun and moon, leaving the mass of waters on the opposite 
 side. 
 
 But when the sun and moon are in opposition, it may be 
 hard to determine which way the solid parts of the earth will 
 be drawn, as the tides would be equal under each luminary. 
 
 This then, must be a very perplexed question ; nor do I think 
 it in anywise solved, or the difficulty obviated, by the more 
 modern and supposed ingenious theory of La Place. 
 
 For according to some authors, the sun exerts about twice the 
 influence upon the earth, taking both its external and internal 
 parts, that the moon does ; " but from its very great distance" 
 it does not exert so much upon the surface. But in this case, it 
 would seem that it might exert its whole influence fairly, as it is 
 the internal and solid parts of the earth that are to be drawn 
 away from the opposite waters. 
 
 It is alleged by some, that as the moon subtends a greater 
 angle than the sun does, when viewed from the earth, its in- 
 fluence upon the earth is greater than that of the sun ; and 
 having concluded, from the amount of tide raised by each, 
 (having no other method whereby to ascertain the fact,) that the 
 density of the moon to that of the sun is about as 2.5 to 1, and 
 hence, that the difference between their attractive influences upon 
 the earth where their respective subtended angles and densities 
 are compounded, is about as 2.662 to 1, by which it would 
 seem that the moon has quite the advantage over the sun in its 
 attractive influence upon the earth. 
 
 But to me all the theories extant in respect to the tides seem 
 to be fraught with inexplicable dilemmas. I will therefore leave 
 them, and will return and finish my own. 
 
 If then the sun exerts an influence on the moon wholly re- 
 gardless of. the fixed pole or poles of the moon, and similar to 
 that exerted by the moon upon the earth, causing two moving 
 or revolving poles of the moon, always directed to and from the 
 
THEORY OF THE TIDES. 259 
 
 sun, performing their revolution in the same time that the moon 
 is revolving about the earth, or in revolving upon its axis in 
 respect to the sun ; then, when the sun and moon are either in 
 conjunction, or in opposition, one of the movable poles of the 
 moon will be extended directly towards the earth, in which case 
 its full influence will be exerted in connection with the fixed 
 pole of the moon, upon the earth, in increasing the influence of 
 its movable poles ; and hence, an increase of the tides. But 
 as the moon proceeds towards quadratures, the influence of the 
 movable pole of the moon upon the earth becomes gradually 
 abstracted, as the line of its direction varies more and more from 
 the earth, until quadratures ; after which, the opposite movable 
 pole of the moon commences and increases in influence until 
 syzygies ; and so on, alternating from syzygies to quadratures, 
 and from quadratures to syzygies. And hence, no portion 
 of the tides is caused directly by the sun, nor in fact, by the 
 moon ; their influence being only exerted in polarizing the earth ; 
 which polarization can only have an effect in raising the tides 
 when situated in a continent, or terra firma, against which the 
 flowing waters may pile. 
 
 This theory then, relieves from all disturbing forces incident 
 to other theories of the tides; from the hypothesis that the earth 
 is constantly drawn from a true curve, and hence performing its 
 revolution in wholly an inexplicable orbit ; performing " cycles 
 and epicycles " in consequence of the attraction of the moon ; 
 and even from being disturbed in its regular course by any 
 disturbing 1 force of the sun. 
 
CHAPTER VII. 
 
 An Investigation of the Theory of the Progressive Motion of 
 
 Light. 
 
 PERHAPS neither the conjecture of Cassini the elder, (that acute 
 observer in astronomy,) that light was progressive in time, from 
 observations by him made on Jupiter's first satellite, (which con- 
 jecture, however, he shortly abandoned,) nor the supposition 
 of Roemer from a like cause, that light is progressive in time, 
 would ever have been adopted into philosophy as an infallible 
 truth, and been the cause of so much time and trouble in en- 
 deavoring to equate the time in which light is conceived to be 
 passing, had not Dr. Bradley, (astronomer royal to George II.,) 
 some few years after the supposed discovery by Roemer, come 
 to the conclusion of a progressive motion of light, from being 
 unable otherwise to account for an apparent motion in the fixed 
 stars, some of which, he had been very critically and carefully 
 examining with a view to ascertain whether they had an appa- 
 rent parallax or motion. From these observations he came 
 to the conclusion, (which the world have ever since assented to,) 
 that light had a progressive motion in time ; and that its velocity 
 could be determined by his method, with far greater accuracy 
 and certainty than could be hoped for or expected from obser- 
 vations made upon Jupiter's satellites. Since this time, in 
 adapting equations, (even in respect to Jupiter's satellites,) re- 
 course has been had to Dr. Bradley's hypothesis in making 
 those equations, so far as the motion of light is to be considered. 
 
 But there are many reasons or causes why Jupiter's satellites 
 furnish no conclusive evidence of the progressive motion of 
 light, which naturally fall out from those astronomers who have 
 most observed the satellites with a view to furnish correct tables 
 of their eclipses, for the purpose of aiding the science of as- 
 tronomy, and for the better ascertaining or finding of longitude. 
 And it is even possible that from the equations of light which 
 they have been led to adopt, they have also been led, for the 
 purpose of forming correct tables, to adopt other and even em- 
 
PROGRESSIVE MOTION OF LIGHT. 261 
 
 pyric equations, which otherwise would have been dispensed 
 with. Some of the causes why Jupiter's satellites would not 
 be a safe criterion whereby to calculate the progressive motion 
 of light, I will here recount. 
 
 1. The various modes or methods of ascertaining a synodic 
 revolution of the satellite, as by an eclipse ; the passage of the 
 satellite over Jupiter's disk, &c., which may lead to more or less 
 error. 
 
 2. In an eclipse, the satellite does not always pass through 
 the centre of Jupiter's shadow; hence from this cause some 
 eclipses are longer than others. 
 
 3. Jupiter is a spheroid, and its shadow an ellipse ; and the 
 position of its shadow is not always the same in respect to the 
 satellite. 
 
 4. The immersions are observed before Jupiter's opposition, 
 and the emersions after ; and those must be compared with each 
 other at distant times in order to get their mean, or the true con- 
 junction of the satellite. 
 
 5. Some of the satellites are said to be longer at some times 
 than at others in passing through the centre of Jupiter's shadow, 
 on account of the eccentricity of their orbits ; and that the dura- 
 tion of eclipses are unequal. 
 
 6. It is said that from the unequal motion of Jupiter, the true 
 conjunctions of the satellites compared with the mean may vary 
 by twice the greatest equation of Jupiter's orbit, which amounts 
 to near forty minutes in the first satellite, and more than an hour 
 in the second. And on this equation a mean synodic revolution 
 as well as the eclipses must depend. And even this equa- 
 tion must be variable, for the reason that the eccentricity and 
 greatest equation of Jupiter's orbit is subject to change. 
 
 7. After making the equations for the supposed motion of 
 light, a number of empyric equations are necessary in order to 
 make the tables and the theory agree with observation. 
 
 8. Maraldi, whose tables have been esteemed, observed that 
 even Cassini's tables erred, as to the time of the fourth satellite, 
 near two hours, for which error several astronomers have at- 
 tempted to assign a cause, as the eccentricity of its orbit, the 
 situation of Jupiter in its orbit, &c. 
 
 9. Error of time in calculating the entry of the satellite into 
 the shadow, or upon his disk; and that the calculations deduced 
 therefrom are different by different observers, and said not to be 
 depended upon. 
 
 10. The first satellite is said by astronomers to be the best for 
 finding the longitude ; the tables of that being more correct, 
 on account of its vanishing and appearing more instantly, 
 in an eclipse, than the others, and therefore being more certain. 
 
262 ON THE THEORY OF THE 
 
 11. Astronomers say it is better to compare eclipses observed 
 under different meridians, than to depend on the tables, on ac- 
 count of their incorrectness ; and then, that the observer should 
 have the same kind of telescope. 
 
 12. Astronomers say, the time when a satellite becomes vis- 
 ible or invisible, depends upon many circumstances ; as the 
 telescope the proximity of the satellite to Jupiter the altitude 
 above the horizon the state of the weather, &c. 
 
 13. Dr. Bradley found from observations made in Cassini's 
 time, that the eclipses of Jupiter's first satellite lasted six minutes 
 longer at some times than at others, which indicates that its 
 motion was not uniform ; but Bailie and some others, concluded 
 it arose from the disturbing force of some of the other satellites. 
 
 14. Authorities say, that the satellite will disappear later and 
 appear earlier, the better the telescope is. And some have 
 observed, that the disappearance of the satellite depended on 
 the distance of Jupiter from the earth and sun ; and others 
 have observed, that different states of the atmosphere, different 
 altitudes, and their distance from Jupiter, have a like influence. 
 
 15. The authorities also say, that the time when the satellite 
 becomes invisible at an immersion, or visible at an emersion, 
 depends upon the quantity of light, which the telescope receives, 
 and its magnifying power, as well as on other causes, as upon 
 the altitude of Jupiter above the horizon, the twilight, the 
 clearness of the air, the proximity of Jupiter to the moon, and 
 the eye of the observer ; all combining to affect the time at 
 which the satellite becomes visible or invisible. 
 
 It is highly probable that Cassini doubted the correctness of 
 his own observations, so far at least as to cast away the idea of 
 the progressive motion of light, which Roemer picked up and 
 adopted. 
 
 Cassini only suspected the motion of light, from observing 
 that the immersions of Jupiter's first satellite took place sooner 
 and sooner in respect to the computed time, as they are observed 
 from the conjunction of Jupiter to his opposition ; and that the 
 emersions took place later and later, as they are observed from 
 Jupiter's opposition to his conjunctipn. 
 
 Now good philosophy would dictate, as it did to Cassini, 
 that it would not be proper to ingraft a hypothesis or theory into 
 the code of philosophy, upon so slight or doubtful evidence as 
 the small apparent difference between the observed and computed 
 time of a revolution of Jupiter's first satellite, lest we adopt false 
 premises, and deduce wrong conclusions, which will only serve 
 to bewilder and perplex us, in accounting for phenomena, and 
 in making those necessary and rational deductions and conclu- 
 
PROGRESSIVE MOTION OF LIGHT. 263 
 
 sions, so necessary in support of a true theory ; while hasty 
 and absurd conclusions or deductions often serve only to lead 
 us into a thousand empyric adaptations and adjustments, in 
 making out and establishing our theories, from having in the 
 outset overlooked some important point in the logical analogies 
 of nature, which eventually lead on to false deductions. 
 
 From the foregoing, and from many other causes and diffi- 
 culties that have existed and still exist, in the forming of correct 
 tables of the eclipses of Jupiter's satellites, it may well be 
 doubted, (as I have already remarked,) whether the theory of a 
 progressive motion of light, would have been adopted as an 
 established truth in philosophy, from a renewal by Roemer of 
 the momentary and cast off hypothesis of Cassini, had not Dr. 
 Bradley soon after seized upon the hypothesis, to aid him in 
 accounting for an apparent motion of the fixed stars; which 
 was supposed not only to confirm the theory of Roemer, but to 
 a degree of exactness and certainty that could never be hoped 
 for or expected from Jupiter's satellites. 
 
 But fully consenting to the very careful and critical observa- 
 tions of Dr. Bradley, upon the fixed stars, and their apparent 
 motions, as by him observed, should his theoretic conclusions 
 deduced therefrom prove unphilosophical and absurd, on further 
 investigation, it may yet be hoped that we shall again observe 
 the sun in its true place ; that we shall see it where it is, and 
 not where it is not ; and that astronomers will be relieved from 
 a thousand empyric equations, adopted for the purpose of making 
 error as beautiful as truth. 
 
 I now propose to examine the evidence furnished by Dr. 
 Bradley, in support of the theory of the progressive motion of 
 light in time, as deduced from his observations. And in doing 
 so, I shall avail myself of the common terms and phrases 
 usually applied to light, considered both as a substance, and 
 progressive in time, as the rays of light, the particles of light, 
 the motion of light, the emission of light, &c. ; believing, however, 
 that a better vocabulary of terms, as applied to light, might be 
 adopted, in which specific terms would not convey to the mind 
 a whole body of false philosophy. 
 
 I will not say that I am intending to prove a negative, to 
 prove that light is not progressive in time, or that time is not 
 necessary to the perception of a distant luminous body ; but 
 rather, that sufficient facts or data have not as yet been ascer- 
 tained, to prove that light is progressive in time. And should 
 those views prove correct, they will further serve to show the 
 propriety of giving to a hypothesis a logical investigation previ- 
 ous to applying to it mathematical and geometric adaptation. 
 
264 ON THE THEORY OF THE 
 
 and supposed demonstration, lest paradoxes, dilemmas and 
 absurdities, intrude themselves, to the overturning of the theory, 
 if it be not true in fact. 
 
 Dr. Bradley's observations upon the fixed stars, from whence 
 he derived his proof as to the progressive motion of light, were 
 made as early as the year 1728, an account of which he com- 
 municated to Dr. Halley, in a letter containing the results of his 
 observations, and the final conclusions he had drawn from the 
 same. And that Dr. Bradley may be heard in respect to his 
 observations upon the fixed stars, together with his deductions 
 drawn therefrom, I will here give such parts of his letter as directly 
 refer to or serve to show his manner of observation, together 
 with his conjectures and conclusions ; not quoting those portions 
 of his letter which only serve to show the method of fixing up 
 his instrument, &c. He says : 
 
 " Mr. Molyneux's apparatus (a zenith sutor) was completed and 
 fitted for observing, about the end of November, 1725, and on 
 the third day of December following, the bright star in the head 
 of Draco, (marked y by Bayer, called Draconis,) was for the first 
 time observed, as it passed near the zenith, and its situation 
 carefully taken with the instrument. The like observations 
 were made on the 5th, llth, and 12th of the same month ; and 
 there appearing no material difference in the place of the star, 
 a farther repetition of them at this season of the year, seemed 
 needless, it being a part of the year wherein no sensible altera- 
 tion of parallax in this star could soon be expected. It was 
 chiefly, therefore, curiosity that tempted me, (being then at Kew, 
 where the instrument was fixed,) to prepare for observing the 
 star on December 17th, when, having adjusted the instrument as 
 usual, I perceived that it passed a little more southwardly this 
 day, than when it was observed before. Not suspecting any 
 other cause of this appearance, we at first concluded that it was 
 owing to the uncertainty of the observations, and that either this 
 or the foregoing was not so exact as we had before supposed ; for 
 which reason, we purposed to repeat the observation again, in 
 order to determine from whence this difference proceeded ; and 
 upon doing it on December 20th, I found that the star passed 
 still more southwardly than in the former observations. This 
 sensible alteration the more surprised us, in that it was the con- 
 trary way from what it would have been, had it proceeded from 
 an annual parallax of the star ; but being now pretty well satis- 
 fied that it could not be entirely owing to the want of exactness 
 in the observations, and having no notion of anything else that 
 could cause such an apparent motion as this in the star, we 
 began to think that some change in the materials, &c., of the 
 
PROGRESSIVE MOTION OF LIGHT. 265 
 
 instrument itself might have occasioned it. Under these appre- 
 hensions we remained some time, but being at length fully 
 convinced, by several trials, of the great exactness of the instru- 
 ment, and finding by the gradual increase of the star's distance 
 from the pole, that there must be some regular cause that 
 produced it, we took care to examine nicely, at the time of each 
 observation, how much it was ; and about the beginning of 
 March, 1726, the star was found to be 20" more southwardly 
 than at the time of the first observation. It now indeed seemed 
 to have arrived at its utmost limit southward, because in sev- 
 eral trials made about this time, no sensible difference was 
 observed in its situation." 
 
 By the middle of April, it appeared to be returning back 
 again towards the north ; and about the beginning of June it 
 passed at the same distance from the zenith as it had done in 
 December, when it was first observed. " From the quick altera- 
 tion of this star's decimation about this time, (it increasing a 
 second in three days,) it was concluded that it would now pro- 
 ceed northward, as it before had gone southward of its present 
 situation ; and it happened as was conjectured, for the star con- 
 tinued to move northward till September following, when it 
 again became stationary, being then near 20" more northwardly 
 than in June, and no less than 39" more northwardly than it was 
 in March. From September the star returned towards the south, 
 till it arrived, in December, to the same situation it was in at that 
 time twelve months, allowing for the difference of declination 
 on account of the precession of the equinox. 
 
 " This was a sufficient proof that the instrument had not been 
 the cause of this apparent motion of the star, and to find one 
 adequate to such an effect seemed a difficulty. A nutation of 
 the earth's axis was one of the first things that offered itself upon 
 this occasion ; but it was soon found to be insufficient ; for 
 though it might have accounted for the change of declination in 
 7 Draco nis, yet it would not at the same time agree with the 
 phenomena in other stars, particularly in a small one almost 
 opposite in right ascension to 7 Draconis, at about the same dis- 
 tance from the north pole of the equator; for though this star 
 seemed to move the same way as a nutation of the earth's axis 
 would have made it, yet it changed its declination but about 
 half as much as 7 Draconis in the same time, (as appeared by 
 comparing the observations of both made upon the same days 
 at different seasons of the year ;) this plainly proved that the 
 apparent motion of the stars was not occasioned by a real nu- 
 tation, since if that had been the case, the alteration in both stars 
 would have been nearly equal. 
 34 
 
266 
 
 ON THE THEORY OF THE 
 
 " The great regularity of the observations left no room to 
 doubt but that there was some regular cause that produced this 
 unexpected motion, which did not depend on the uncertainty 
 or variety of the seasons of the year. Upon comparing the ob- 
 servations with each other, it was discovered that in both the 
 fore-mentioned stars, the apparent difference of declination from 
 the maxima was always nearly proportional to the versed sine 
 of the sun's distance from the equinoctial points. This was an 
 inducement to think that the cause, whatever it was, had some 
 relation to the sun's situation with respect ta those points. But 
 not being able to frame any hypothesis at that time sufficient to 
 solve all the phenomena, and being very desirous to search a 
 little farther into this matter, I began to think of erecting an 
 instrument for myself at Wanstead," &c. 
 
 After giving the preparation of his instrument, he says, " my 
 instrument being fixed, I immediately began to observe such 
 stars as I judged most proper to give me light into the cause of 
 the motion already mentioned. There was variety enough of 
 small ones, and not less than twelve that I could observe through 
 all the seasons of the year, they being bright enough to be seen 
 in the daytime when nearest the sun. I had not been long 
 observing before I perceived that the notion we had before en- 
 tertained of the stars' being farthest north and south when the 
 sun was about the equinoxes, was only true of those that were 
 near the solstitial coloure ; and after I had continued my obser- 
 vations a few months, I discovered what 1 then apprehended to 
 be a general law, observed by all the stars, viz., that each of 
 them became stationary or was farthest north or south when 
 they passed over my zenith at six o'clock either in the morning 
 or evening. I perceived likewise that whatever situation the stars 
 were in with respect to the cardinal points of the ecliptic, the 
 apparent motion of every one- tended the same way when they 
 passed my instrument about the same hour of the day or night ; 
 for they all moved southward while they passed in the day, and 
 northward in the night; so that each was farthest north when it 
 came about six o'clock in the evening, and farthest south when 
 it came about six in the morning. 
 
 " Though I have since discovered that the maxima in most of 
 these stars do not happen exactly when they come to my instru- 
 ment at those hours, yet not being able at that time to prove the 
 contrary, and supposing that they did, I endeavored to find out 
 what proportion the greatest alterations of declination in differ- 
 ent stars bore to each other; it being very evident that they 
 did not all change their declinations equally. 
 
 " I have before taken notice that it appeared from Mr. Moly- 
 
PROGRESSIVE MOTION OP LIGHT. 267 
 
 neux's observations that y Draconis altered its declination about 
 twice as much as the fore-mentioned small star almost opposite 
 to it ; but examining the matter more particularly, I found that 
 the greatest alteration of declination in these stars was as the 
 sine of the latitude of each respectively. This made me sus- 
 pect that there might be the like proportion between the maxima 
 of other stars; but finding the observations of some of them 
 would not perfectly correspond with such an hypothesis, and 
 not knowing whether the small difference I met with might not 
 be owing to the uncertainty and error of the observations, I 
 deferred the farther examination into the truth of this hypothesis 
 till I should be furnished with a series of observations made in 
 all parts of the year; which might enable me not only to deter- 
 mine what errors the observations are liable to, or how far they 
 may safely be depended upon ; but to judge whether there had 
 been any sensible change in the parts of the instrument itself. 
 
 " Upon these considerations I laid aside all thoughts at that 
 time about the cause of the fore-mentioned phenomena, hoping 
 that I should the easier discover it when I was better provided 
 with proper means to determine more precisely what they were. 
 
 " When the year was completed I began to examine and 
 compare my observations, and having pretty well satisfied my- 
 self as to the general laws of the phenomena, I then endeavored 
 to find out the cause of them. I was already convinced that 
 the apparent motion of the stars was not owing to a nutation 
 of the earth's axis. The next thing that offered itself, was an 
 alteration in the direction of the plumb line with which the in- 
 strument was constantly rectified; but this upon trial proved 
 insufficient. Then I considered what refraction might do, 
 but here also nothing satisfactory occurred. At last I conject- 
 ured that all the phenomena hitherto mentioned, proceeded from 
 the progressive motion of light and the earth's annual motion in 
 its orbit. For I perceived if light was propagated in time, the 
 apparent place of a fixed object would not be the same when 
 the eye is at rest, as when it is moving in any other direction 
 than that of the line passing through the eye and object ; and 
 that when the eye is moving in different directions, the apparent 
 place of the object would be different." 
 
 This is " Dr. Bradley's account of this very important discov- 
 ery," and many pages are filled with supposed geometrical and 
 mathematical explanations or demonstrations of the truth of his 
 hypothesis, and its adaptation to the equation of light, &c., to 
 which explanations and adaptations I must refer all who may 
 be curious to know the vast amount of labor and investigation 
 that has been bestowed upon this hypothesis. 
 
268 ON THE THEORY OF THE 
 
 I have thought proper to bring forward Dr. Bradley's own ac- 
 count of his observations, and his theory or hypothesis deduced 
 therefrom, that they may be readily compared with my attempt- 
 ed refutation of his theory, and with that which I may suggest 
 in its stead. 
 
 We find also in the works on astronomy this further reference 
 to Dr. Bradley's observations and deductions upon the same 
 subject, viz. 
 
 Dr. Bradley further observes that in seventy observations 
 made in a year on 7 Draconis, there was but one (and that is 
 noted very dubious on account of clouds) which differed more 
 than 2" from the theory, and that did not differ 3". And in about 
 fifty observations made in a year on n Ursce majoris, he did not 
 find a difference of 2", except in one marked doubtful on ac- 
 count of the undulation of the air, &c. and that did not differ 
 3". And this agreement between the theory and observation 
 leaves no room to doubt but that the cause is rightly assigned. 
 And, if this be the case, the annual parallax of the fixed stars 
 must be extremely small. " I believe," says the Dr. " that I may 
 venture to say that in either of the above mentioned stars, it does 
 not amount to 2". I believe if it were 1", I should have per- 
 ceived it in the great number of observations that I made, 
 especially on / Draconis ; which agreeing with the theory (with- 
 out allowing anything for parallax) nearly as well in conjunc- 
 tion with, as in opposition to this star, it seems very probable 
 that the parallax is not so great as one single second ; and conse- 
 quently that it is above four hundred thousand times further from 
 us than the sun." The observations of Mr. Flamstead of the 
 different distances of the pole star from the pole at different times 
 of the year, and which was looked upon as a proof of its an- 
 nual parallax, was undoubtedly owing to this cause. For he 
 concluded that the star was 35", 40" or 45" nearer the pole in 
 December than in May or July; and according to this hypothe- 
 sis, it ought to appear 40" nearer in December than in June. 
 This agreement is greater than could have been expected from 
 observations made with his instrument." 
 
 Hence Dr. Bradley deduced the following conclusions : 1. 
 " That the light of all the fixed stars arrives at the earth with 
 equal velocities ; for the major axis of the ellipse is the same in all 
 the stars, that is 40", according to his last determination. 2. That 
 unless their distances from us are all equal, which is very im- 
 probable, their lights are propagated uniformly to all distances 
 from them. 3. That light moves from the sun to the earth in 
 8' 7" .5, and its velocity is to the velocity of the earth in its orbit 
 as 10314 : 1. 4. That the time thus determined can scarcely err 
 
PROGRESSIVE MOTION OF LIGIJT. 269 
 
 from the truth by above 5" or 10" at most, which is such a de- 
 gree of exactness as can never be expected from the eclipses of 
 Jupiter* s satellites. 5. That as this velocity of star light comes 
 out about a mean of the several velocities found from the 
 eclipses of Jupiter's satellites, we may reasonably conclude that 
 the velocity of these reflected lights are equal to the velocity of 
 direct light. 6. And as it is highly probable that the velocity of 
 the sun's emitted light is equal to that of star light, it follows 
 that its velocity is not altered by reflection into the same medium." 
 I have thought proper to insert the foregoing reference to Dr. 
 Bradley's observations and theory, as furnishing (after most am- 
 ple applications to practical utility as is conceived) a string of 
 corollaries extending to such things as were conceived to fall 
 within analogies. 
 
 Dr. Bradley in rendering a hypothesis to the phenomena 
 which were the result of his observations, has made this assump- 
 tion. " He perceived, that if light was propagated in time, the 
 apparent place of a fixed object would not be the same when the 
 eye is at rest, as when it is moving in any other direction than 
 that of the line passing through the eye and object ; and that 
 when the eye is moving in different directions, the apparent 
 place of the object would be different." 
 
 Now it appears to me this is begging the question ; for not- 
 withstanding the Dr. has carefully inserted the word if as a salvo, 
 yet the world have assumed the premises as being true, and have 
 reasoned more critically on the subject than I am able to com- 
 prehend as being either sound or true, notwithstanding their ex- 
 planation of the laws of motion upon the principles of a par- 
 allelogram or resolution of forces, and their finally assumed 
 proportions between the velocity of light, and that of the earth 
 in its orbit, as sometimes attempted to be explained by con- 
 ceiving a very long telescope or tube of sufficient calibre only 
 to permit a single ray or particle of light to flow down in the 
 tube ; and so adjusted in respect to the motion of light and the 
 motion of the earth in its orbit, that the ray or particle of light 
 would flow freely down the tube or telescope ; and that the eye 
 placed at the lower end of the tube would perceive the ray or 
 particle of light giving the appearance of the star or object from 
 which the light is emitted, in the line of the tube; notwithstand- 
 ing the tube or telescope would not lie in the line from the eye 
 to the object ; but so situated as to catch the rays or particles of 
 light in advance of such direct line from the eye to the object. 
 But this mode of reasoning appears to me to overlook some of 
 the plainest] and best established principles and facts in the 
 science of optics; for although I have long considered the 
 
270 ON THE THEORY OF THE 
 
 adopted theories on light to be vague and unphilosophical, nev- 
 ertheless, the theory of optics, so far as regards the effects or op- 
 erations of light, is in many respects as beautifully and satisfac- 
 torily defined and explained as that of any other branch of 
 science or philosophy ; among its laws are those which teach 
 us that the eye discerns an object in the precise direction in 
 which the eye receives or obtains the light, whether the light 
 which presents the object to the eye be direct from the object to 
 the eye, or whether it may have been reflected or refracted ; but 
 when the light that presents the object to the eye has been refract- 
 ed, for the very reason that the eye discerns the object in the 
 exact direction in which it receives the light which presents 
 the object, the apparent place of the object will not be its real. 
 
 And as well might we expect to see the sun at noon day by 
 looking in any other direction than towards it, (inasmuch as its 
 rays are falling thick all around,) as to perceive the star by look- 
 ing at those rays or particles of light (as they are called) which 
 are yet in advance of a direct line from the eye to the star; and 
 the idea is preposterous that we can see a ray or particle of light 
 flowing down the telescope or tube, not directed towards the 
 eye, but to a place in advance of the eye, merely because the 
 eye and ray or particle will presently meet in the angle of the 
 requisite parallelogram. And it is not pretended even by way 
 of explanation of Dr. Bradley's theory, that the line of direction 
 of the telescope or tube is in that from the eye to the star, but 
 as in the case of a composition of forces, an angle is formed by 
 the line of the eye and object with the line of the motion of the 
 eye, equal to what is conceived to denote the proportional ve- 
 locities of light, and of the eye. 
 
 And to show how the progressive motion of light may affect the 
 place of the star seen by the naked eye, they have in reality, only 
 made the eye a shorter tube, or telescope ; and hence it is said 
 that, " if a ray of light fall upon the eye in motion, its relative 
 motions in respect to the eye, will be the same as if you were 
 to impress equal motions in the same direction, upon each at 
 the instant of contact; for equal motions in the same direction, 
 impressed upon two bodies, will not affect their relative motions, 
 and therefore, the effect of one upon the other, will not be 
 altered." And another parallelogram of forces, or of motion, is 
 presented by way of diagram, to exemplify the truth of the 
 position. 
 
 But, if philosophers had abided by all experience, and by 
 what all sound philosophy has taught, as to the eternal fitness of 
 things, in adapting the mechanism of the eye to our best conve- 
 nience, so that when it discerns an object, the crystalline lens, 
 and other parts of the eye, are so constructed, as to present the 
 
PROGRESSIVE MOTION OF LIGHT. 271 
 
 object in a direct line from the eye, unless the light has been 
 reflected, or refracted ; and which, of all the laws of optics, is 
 most dear to us, as presenting the object so that we may see it 
 where it is, rather than where it is not ; they would, perhaps, not 
 have attempted to filch enough from the well known and best 
 established laws of optics, to make a philosophic truth of Dr. 
 Bradley's conjecture and final perception, that if light was propa- 
 gated in time, it would solve all the phenomena arising from 
 his observations, by assuming, for the purpose of explanation, 
 that such is the fact; and by those subtleties which may some- 
 times " make the worse appear the better reason," impose the 
 hypothesis upon a credulous world, as being demonstrated or 
 proved ; and therefore requiring no farther investigation. But I 
 think, however specially, or critically, they may have pleaded in 
 the case, that the world, on a re-examination of the case, will 
 adjudge that they take nothing by their plea. 
 
 One result of the Doctor's theory would be, that the nearer a 
 planet is to the sun, the greater will be the difference between 
 the true and apparent place of the sun, as observed from such 
 planet. This, however, would be no direct or conclusive argu- 
 ment against his theory or hypothesis. 
 
 I will now attempt to show some results, that must necessarily 
 follow or flow from Dr. Bradley's hypothesis, when tested by an 
 attempt at demonstration. But, before suggesting a diagram, I 
 will quote from explanations of Dr. Bradley's theory, in order to 
 justify myself in the form or construction of the diagram which 
 I have chosen to adopt. 
 
 1. It is assented to by the authorities, and will be readily 
 assented to by all, that for all purposes of explanation or of demon- 
 stration, so far as the principle of the aberration of light may 
 be concerned, a circular orbit, with the sun in the centre, is as 
 proper as an eccentric or elliptical orbit ; the only difference 
 consisting in this : that in a circular orbit, the motion of the earth, 
 and of course, of the eye of the observer, would be uniform ; while 
 in an eccentric orbit, it would be something greater at the peri- 
 helion than at the aphelion ; but even our earth's orbit is so 
 nearly a centric orbit, as to permit astronomers to consider it as 
 such, so far as the aberration of light is concerned, as well as 
 in many other calculations. 
 
 2. It is said that, " Whilst the earth makes one revolution in 
 its orbit, the curve, parallel to the ecliptic, described by the 
 apparent place of a fixed star, is a circle," to which all will 
 assent. 
 
 3. It is said also, that " in the plane of the ecliptic, the sine of 
 the star's latitude being 0, the orbit becomes a straight line," &c. 
 
272 ON THE THEORY OF THE 
 
 That is, to an observer, placed in the plane of the ecliptic, at a 
 great distance from the line of the earth's orbit, if such line 
 were a visible line, it would also appear to be a straight line. 
 
 4. It is also said " to find the aberration in latitude or 
 longitude, let A B C D, be the earth's orbit, supposed to be a 
 circle, with the sun in the centre," &c. 
 
 Hence, in the diagram suggested, I shall conceive the star to 
 be observed, for the purpose of getting its apparent motion, or 
 what is termed its aberration, to be in the plane of the ecliptic, 
 but at such an infinite or indefinite distance from the earth, in 
 any part of its orbit, as to have no sensible parallax, or actual 
 motion or change of place, even by the nicest observations ; in 
 which case, I think, the diagram will be assented to, as wholly 
 proper for investigating the apparent motion of such star, caused 
 by the revolution of the earth in its orbit. 
 
 And to avoid any inconvenience or perplexity in considering 
 the different hours of the day, when the star observed comes to 
 the meridian where the observer is situated, occasioned by the 
 diurnal revolution of the earth on its axis, I have conceived only 
 the eye of the observer, to revolve around the sun in the orbit of 
 the earth, with the same motion with which the earth revolves ; 
 in which case, the eye of the observer would at all times see the 
 star ; and that in such case, the only apparent motion of such 
 star, would be that of a backward and forward motion, as in a 
 straight line, in the plane of the ecliptic, to an amount of 40" in 
 the course of an annual revolution of the eye of the observer. 
 
 I will here, again recapitulate some of the phenomena arising 
 from Dr. Bradley's observations, which appear most essential to 
 be considered, with a view to a proper solution. 
 
 1. The apparent motion of the star, is always in the same 
 direction in which the earth would appear to be moving, to an 
 observer situated at such star. 
 
 2. In March, the star 7 Draconis, appeared stationary for many 
 days, at its utmost southern limit. 
 
 3. In June, the star had passed 20" north of its place in 
 March, and was continuing northward, having then its most 
 rapid motion. 
 
 4. It continued to pass northward until September, when it 
 was 40" north of its place in March, and where it again became 
 stationary for some days. 
 
 5. In September, it commenced returning to the south, and in 
 December, it was 20" farther south than it was in September, 
 having again in December, its most rapid motion. 
 
 6. In March following, it had again arrived at its most south- 
 ern limit, and after remaining stationary for a time, again 
 commenced its return to the north. 
 
PROGRESSIVE MOTION OF LIGHT. 273 
 
 Let us now conceive a diagram of the orbit of the earth, 
 with its requisite appendages, with the star to be observed, situ- 
 ated at an indefinitely great distance, but in the plane of the orbit. 
 
 Conceive a straight line (called the zenith line) to be drawn 
 from the fixed star to be observed, through the centre of the orbit, 
 intersecting the line of the orbit in two points ; the point of inter- 
 section next the star being denoted by A, and signifying June, 
 or the summer solstice ; and the opposite point of intersection 
 by C, and signifying December, or the winter solstice. 
 
 Conceive another line drawn through the centre of the orbit, 
 at right angles to the central line, intersecting the line of the 
 orbit in two points ; one being denoted by B, and signifying 
 September, or the autumnal equinox ; and the other denoted by 
 D, signifying the vernal equinox ; which said line so intersecting 
 the orbit, may be called the equinoctial line. 
 
 The points of intersection of the orbit by the central line, may 
 be called the zenith points, and the others the equinoctial points. 
 
 Conceive tangent lines to the orbit to be drawn from the equi- 
 noctial points, extending (parallel to each other) to the fixed star 
 to be observed ; for it must be understood, that notwithstanding 
 the great diameter of the earth's orbit, it will not form the base to 
 a sensible triangle, the vertex of which is in a fixed star ; and 
 hence those tangent lines will be parallel to each other to an ex- 
 actness beyond detection. That tangent proceeding from B, 
 being called the B tangent, and that proceeding from C, the C 
 tangent. Then a body revolving in the orbit would, to an ob- 
 server at such fixed star, appear to pass backward and forward 
 upon the equinoctial line. 
 
 Now conceive the eye of the observer revolving in the orbit in 
 the direction D A B C, and to have arrived at D, or the vernal 
 equinox, when the star will be stationary, and 20" further south 
 than it will be when the eye arrives at A, and 40" further south 
 than when the eye arrives at B. And, according to Dr. Brad- 
 ley's theory, the reason why the star appears stationary at D or 
 at B, is because the eye is then passing in the line of the eye 
 and object, or star. Hence at D the star must appear 20" south 
 of the tangent line at D, or 20" out of its true place for days, 
 and at the time and place when the eye is passing in a line 
 towards the star, and not transverse to the line of the eye and 
 object. When the eye arrives at A its motion is directly transverse 
 perpendicular to the line of the eye and star ; and the star then 
 appearing 20" further north than when the eye was at D, and 
 20" further south than it will appear when the eye arrives at B ; 
 and hence a mean between the two extremes, it will then at A 
 appear directly in the zenith, or zenith line ; and this at the time 
 35 
 
274 ON THE THEORY OF THE 
 
 when its apparent motion northward is most rapid ; and when, 
 according to Dr. Bradley's theory, the aberration of the star 
 should be greatest. 
 
 When the eye arrives at B it views the star 20" to the north 
 of the tangent line drawn from B ; and there the star again be- 
 comes stationary for days, because the motion of the eye is in 
 the line of the eye and star, or object viewed. As the eye pro- 
 ceeds from B to C, the star returns southward ; and when the 
 eye arrives at C, the star will again appear in its true place, in 
 the zenith line, although its apparent motion will then be most 
 rapid towards the south, and the motion of the eye will then be 
 transverse to the line of the eye and star ; and the motion of the 
 star will continue southward till the eye again arrives at D, 
 when it will again become stationary at 20" out of its true place ; 
 when it will commence going north again. 
 
 Now it would seem as though no further explanation need 
 be made to convince the most casual observer of the entire ab- 
 surdity of Dr. Bradley's theory. For I will ask whether any one 
 can reconcile the Doctor's theory with the fact, that when the 
 motion of the eye transverse to the line of the eye and star is 
 greatest, the star should be in its true place ; and when the requi- 
 site motion of the eye for producing an aberralion is nothing, 
 because it is in a line of the eye and object, the aberration of the 
 star should not only be the greatest, but continue so for days, 
 namely, while the star should appear stationary ? 
 
 But again, the amount of aberration at D or at B is only 
 8 I -4th minutes in time. Hence v if the eye were to remain still, 
 or were to be passing in the line of the eye and star for that 
 length of time, the star must necessarily come to its true place, 
 even upon Dr. B.'s theory. But at D or B the eye has no requisite 
 motion for producing any aberration of the star for days, and 
 yet the star stands out south atD and north at B, for all this time. 
 
 But when the eye comes to A or to C, the star appears in its 
 true place, being a mean between its north and south declination, 
 notwithstanding the motion of the eye is then transverse to the 
 line of the eye and star ; and as the motion of the star is then the 
 greatest, then according to the Doctor's theory, that if light be 
 progressive, and this be the cause of the apparent motion of the 
 star, its greatest aberration should be when the eye is at A or C ; 
 but it is directly the reverse, and goes as far as a theory can to 
 prove that light is not progressive in time. 
 
 This would seem to indicate that were the motion of the eye 
 to cease in any part of the orbit, the star would continue to re- 
 main at the place where it appeared when the motion of the eye 
 ceased ; and so doubtless would be the fact ; for it is too obvious 
 that the aberration of light, upon Dr. Bradley's theory, is not the 
 
PROGRESSIVE MOTION OF LIGHT. 275 
 
 cause of the phenomena, which doubtless depends on a much 
 more natural, simple and beneficial cause, which I may hereaf- 
 ter venture to suggest. 
 
 Perhaps the only rational way of accounting for the error 
 which those have fallen into who have undertaken to ex- 
 plain the Doctor's theory, is, by supposing that they erred as to 
 what places or points of the orbit the star would be seen in its 
 true place ; and we must conclude that they made the true place 
 where the star would appear when stationary; for three days 
 were allowed for it to come to its true place, when at most but 
 a few minutes would be required ; not considering that at those 
 points where the star appears stationary, lines drawn to the two 
 apparent places of the star from those points would form an 
 angle of 40" ; and that at A and C the star from both points 
 would appear in the same point in the heavens, and that, too, in 
 the mean and the true place. 
 
 But it will be perceived that the star appears in its true place 
 only when the motion of the eye is in a direct transverse direction 
 to the line of the eye and star ; and that in no other parts of the 
 orbit will the eye perceive the star in its true place, going upon 
 the true theory, whatever that may be. 
 
 And hence I think it clear that if the phenomena observed by 
 Dr. Bradley in respect to the apparent motion of the fixed stars, 
 prove anything, they prove directly the opposite of what has gen- 
 erally been supposed; and I think it conclusively shown that 
 the motion of the eye, whether it be much or little, has nothing 
 to do in accounting for the phenomena observed by Dr. Bradley; 
 and if any have supposed that the star appeared in its true place 
 at B and at D, let them attempt to extricate themselves from 
 such a dilemma. 
 
 A nutation of the poles of the earth is sometimes defined to 
 be " a kind of tremulous motion of the axis of the earth, by 
 which in its annual revolution, it is twice inclined to the ecliptic, 
 and as often returns to its former position " and so far as the 
 actual poles of the earth partake of a circular motion, or revolu- 
 tion about the mean poles of the earth, such revolution is found 
 to be in the order of the signs. 
 
 Dr. Bradley found in his critical observations that the stars 
 which he observed, apparently had their most rapid motion near 
 the winter and summer solstices, and their least apparent motion 
 near the vernal and autumnal equinoxes ; and that a star sit- 
 uated near the solstitial colures, would have a greater apparent 
 motion either north or south, than those situated near the equi- 
 noctial colures. The Dr. also found, that it was not a general 
 law of the apparent motion of the stars that they would be 
 
276 PROGRESSIVE MOTION OP LIGHT. 
 
 farthest north or south when the sun was about the equinoxes ; 
 but that such only held good in respect to those stars that were 
 near the solstitial colure ; nevertheless, any star would become 
 apparently stationary (being farthest either north or south) when 
 it passed over his zenith about six o'clock in the morning, or 
 six o'clock in the evening ; being farthest south when it passed 
 at six o'clock in the morning ; and farthest north when it 
 passed at six o'clock in the evening ; the apparent motion of 
 any star tending the same way when it passed his instrument 
 about the same hour of the day or night, with the exception of 
 some small variation. 
 
 Now notwithstanding all this evidence, the Dr. says that on 
 examining and comparing his evidence, and having pretty well 
 satisfied himself as to the general laws of the phenomena, he 
 then endeavored to find out the cause of them ; that he was 
 already convinced that the apparent motion of the stars was not 
 owing to a nutation of the earth's axis. But I am certainly at 
 a loss to know from what cause his conviction arose inasmuch 
 as all the phenomena detailed or set forth would seem to refer 
 the nutation of the earth's axis and the apparent motion of the 
 stars to the same cause. 
 
 But the Dr. says, at last he conjectured the cause &c., which 
 conjecture has ever since been received and acted upon as the 
 true hypothesis. But it would seem as though most philos- 
 ophers with all this evidence before them, would have con- 
 jectured that one and the same cause produced the nutation of 
 the earth's axis, the precession of the equinoxes, and the appa- 
 rent motion of the fixed stars ; not however, unless they con- 
 ceived the solar system to be governed by some immutable law 
 of order, in lieu of being governed by fortuity, or the Newtonian 
 theory of universal gravity, which would indeed very illy ac- 
 count for those phenomena. 
 
 But if we conceive the solar system to be governed by laws 
 of order, those phenomena, viz. the nutation of the poles, the 
 precession of the equinoxes, and the apparent motion of the 
 stars observed by Dr. Bradley, would seem to be the direct and 
 natural results of such law, and some of the most happy eluci- 
 dations of the law. In such case there would be no lunar nu- 
 tation, but the whole would be solar nutation ; nor would it be 
 said as is now said, that physical astronomy has made known a 
 solar nutation, but that it is not sufficient to be detected by ob- 
 servation ; but if a law of order for the solar system were 
 known and acknowledged, it would, dictate to us that such phe- 
 nomena must necessarily occur ; hence on observing their occur- 
 rence there would be no difficulty in assigning the true cause. 
 
APPENDIX. 
 
 I have thought proper, in a short appendix, to add something 
 further in reference to the quadrature and its general application ; 
 as also some few additional remarks in reference to the law of 
 gravity. 
 
 As heretofore, the p whose M a and C are conceived to be 
 adverse, is denoted by figure 1 at the right of its respective deno- 
 tations ; and the p whose B and C are conceived to be adverse, 
 and whose M a and D coincide in %, is denoted by figure 2 at 
 the right of its respective denotations. 
 
 M a and B of the p of twice the sides of the p 2 occupy the 
 identical places that C and D of the p do whose A and B coin- 
 cide in ,. 
 
 C and B of the p 2 occupy the identical places that B and D 
 do of the p whose C is in ^. 
 
 M a and C of the p 1 occupy the identical places that A and 
 M a do of the p whose C is in J^. 
 
 The adverse of A of any given p is a mean between 2, and E 
 of a like /?, and is consequently = m 2 when the area is = E. 
 
 The adverse of M a of any given p is a mean between ^ an( i 
 G of a like p, and is consequently = m 2 when the area is = G. 
 
 The adverse of a mean between A and M o, namely, the ad- 
 verse of C, is a mean between ^ and E of twice the sides, and 
 is consequently = m 2 when the area is = E of twice the sides. 
 
 The mean place of M a is a mean between M a and C of a p 
 of half the sides. 
 
 The mean place of D is a mean between B of the same p, and 
 D of a p of half the sides. 
 
 The square of the mean place of M a is a mean between the 
 square of the true place, and the square of E of a p of twice the 
 sides. 
 
278 APPENDIX. 
 
 The square of the mean place of D or of C, is a mean be- 
 tween the square of the true place of D, and the square of the 
 true place of C ; and the square of the mean place of A is a mean 
 between the square of the true place of D of half the sides, and 
 the square of the true place of C of half the sides. 
 
 The product of Ma 4 by C, is jl above the square of "Ma 8, 
 the product of M a 8 by C 8, is L above the square of M a 16, 
 the product of M a 16 by C 16, is ^ above the square of M a 32, 
 and so on, until the product of Ma 1 by C 1, is the square of 
 M a 2. And if we attempt to continue a like progression, we 
 shall find a uniformity which has not previously existed; for 
 the product of Ma 2 by C 2, is = the square of M a of twice 
 the sides, and such will continue to be the case, however far we 
 attempt to continue the progression. Hence, the product of M a 
 by C, can never be below the square of Ma of a p of twice 
 the sides ; and hence, from the p 1 onward, our progression is 
 futile and void. 
 
 The product of M a by D, is always = the product of B by 
 C , or of B of the given p by A of &p of twice the sides ; and as in 
 the circle, M a actually coincides with C, hence, our conception 
 necessarily is, that in the circle, M a coincides with A of twice 
 the sides. 
 
 When D is conceived to be JL below Ma, such D is the 
 adverse of the popular D, or B of the circle ; and such M a is 
 the adverse of the popular Ma or C of the circle; and such Ma 
 is then a mean between D and B of the same J9, and is conse- 
 quently = B of a p of twice the sides. 
 
 B 2 is conceived to be the farther of two means from B 1, to 
 ^ and the popular B of the circle, is the farther of two means 
 from B 2 to ^. 
 
 , is the farther of two means from A to B, hence, when A 
 and B coincide in ^, such point is as well the farther of two means 
 from A to B of twice the sides, as from A to B of half the sides. 
 
 If we conceive a p of twice the sides of that whose A and B 
 coincide in ^ , we necessarily conceive M a of such p to be 2_ 
 below^, or in the place of C 1, or of A 2 ; and also, that A of 
 such p is j?. above ^, or in the place of M a 1 ; and that C of 
 such p, is in ^, and B of such p, will be conceived to be J_ 
 below ^ and D JL above ^. If in the p 1 we conceive 1 to 
 po^s ss the same value as in the p 8 and still conceive B to con- 
 tinue _L above M a and D to continue to be J_ above C, and attempt 
 to continue the progression beyond the p 1 and until B and D co- 
 incide J_ above Ma and C, we necessarily place B and D final 
 | above ^, and M a and C final f below ^,. But, as before 
 remarked, from the p 1 onward, we do not displace Ma from its 
 constant mean position between M a and C of p of half the sides. 
 
. APPENDIX. 279 
 
 So also, if we conceive the progression to be continued after 
 A and B coincide in ^, we shall arrive at the same conceived 
 result, namely, shall place B and D final | above J^, and M a 
 and C, final f below J^ ; but it is manifest that such continued 
 progression will be by an apparent retrograde movement of the 
 denotations from that p which is conceived to have twice the 
 sides of that whose A and B are in J^,. That is, when C is in 
 ^, we conceive M a to be 1. below M a when A is in , ; and 
 if B continues to be _L above M a, and if B of the circle is 1_ 
 above M a of the circle (if such _L then has any value, whatever 
 value it may then have,) the final B and D must be ^ (of such 
 value) above J^, and M a and C final must be f (of such value) 
 below J^ ; but if in the circle there is no value to _L, then M a 
 and D of the circle are in J^. 
 
 In such case, A in J^, must be conceived to be A of twice the 
 sides of Ma of the circle; and consequently, we must conceive 
 A of half the sides of A in J^, to be in reality the utmost A, 
 and it is manifest that we can only conceive such A of half the 
 sides to be = Ma 1. And the product of Ma 1 by C 1 is the 
 square of the true place of Ma 2, and of the mean place of. Ma 
 of four times the sides of the p 1, and the like may be conceived 
 in respect to the circle. Hence, in ^, the true and mean places 
 of M a coincide, as they do in the p 4. 
 
 Hence M a in J^,, nas risen to a mean between M a and C of a p 
 of half the sides ; and the popular methods do not attempt anything 
 farther. M a S is 1_ below a mean between M a and C of a /; of 
 half the sides; hence, _L, which in the p 8 denotes a mean be- 
 tween Ma, and a mean between Ma and C of a p of half the 
 sides, is equated to JL in the p 2 ; and no one expects that Ma 
 will either rise above a mean between M a and C of a p of half 
 the sides, or that the law of progression will so' far change, that 
 Ma will continue (while the progression continues) to be a 
 mean between M a and C of a p of half the sides. 
 
 M a then coincides with M a and C of half the sides, at the 
 time and place when it becomes a mean between them, and then 
 it necessarily coincides with C of the same jt?, thus making the 
 four-fold chord of the periphery of the circle. 
 
 Thus, by the law of polygonal progression, M a cannot be 
 below J^ unless it be above a mean between M a and C of half 
 the sides ; and by any method, it is supposed that when M a 
 becomes a mean between M a and C of half the sides, it coin- 
 cides with them. 
 
 Suppose then, that the difference between Mai and B 1, or 
 between M a 2 and B 2, to have even been increased in value 
 from the p S, according to the popular increase, and then attempt 
 to obtain M a of twice the sides of the p 1 in the popular man- 
 
280 APPENDIX. 
 
 ner, that is, by dividing the product of Ma 1 by C 1, by half 
 the sum of M a 1 and C 1. 
 
 In such case, M a of a p of twice the sides of the p 1 will be 
 proportionally as far below ^ as an equal mean between Mai 
 and C 1 is above J^ ; that is, if M a 1 and C 1 are not in ^, by 
 such process, M a of twice the sides, will be below J^, and con- 
 sequently, must be above a mean between Mai and C 1 
 which cannot be the case neither can such Ma be below ^ 
 for when M a and C are adverse, M a of twice the sides is neces- 
 sarily in J^. Thus, showing direct error, in the popular mode of 
 conducting the progression. 
 
 As C is a mean between M a of the same p and C of half 
 the sides, if after obtaining the popular M a 2, we then attempt 
 to obtain C 2, we shall obtain it proportionally, half as far too 
 low as the popular M. a 2 will be ; and it will be manifest to 
 any one, that the error will continue to increase during the pro- 
 gression. But as M a cannot become a mean between M a and 
 C of a p of half the sides till it coincides with them, and as 
 M a in ^ is a mean between M a and C of a p of half the sides, 
 hence, Ma in ^ coincides with Ma and C of half the sides. 
 
 And as in the case of M a in J^,, in which M a has then risen 
 to a mean between M a and C of a p of half the sides, so D in 
 , has fallen to a mean between B of the same JP, and D of a p 
 of half the sides. Hence, in J^, the relative situation of D will 
 have altered in respect to a mean between B of the same />, and 
 D of a p of half the sides, from what it is in the p 8, proportion- 
 ally half as much as the relative situation of Ma in J^, in 
 respect to a mean between M a and C of half the sides will 
 have shifted from what it is in the p 8. 
 
 Hence, in ^ the mean places of both M a and D will be their 
 true places ; nor can D by rising above J^, fall below a mean 
 between B of the same jo, and D of a p of half the sides, any 
 more than M a by falling below J^, can rise above a mean be- 
 tween M a and C of a p of half the sides. Consequently, in J^ 
 the progression is ended in respect to M a and D, and hence, inj^ 
 M a and D become M a and I) of the circle, which are the final 
 quantities sought. And as the mean place of D is a mean 
 between D and C, hence, when the true and mean place of D 
 coincide, D is then = C, hence, when D is in J^, C is in ^, as 
 also Ma. 
 
 Such then, gives the true or actual ratio between the diameter 
 and circumference of the circle, the proportions of which may be 
 expressed in the powers and roots of definite numerical quanti- 
 ties ; making the circle in its infinite use and application a 
 pleasant, rather than irksome task. 
 
 I will here notice a few additional facts, by way of application. 
 
APPENDIX. 281 
 
 The fifth power of M a of the circle, is d of the circle, whose 
 circumference is 1, and if such be an orbit, the period of the 
 planet will be = one sixteenth of the circumference. 
 
 If the circumference of an orbit be 4, the period is one eighth 
 of the circumference ; thus, if the circumference be quadrupled, 
 the ratio of the period to the circumference is reduced one half. 
 So when the circumference is 16, the square of the period is = 
 to the circumference ; and this occurs when the diameter of the 
 orbit is half the square of the circumference of the prime circle. 
 
 If the diameter of the inscribed circle of a given /?, be = to 
 the square of the circumference of a like major/?, the circumfer- 
 ence of such polygon, will be the third power of the circumfer- 
 ence of such major p ; hence, if the diameter of the circle be 
 10.07936 (square of the circumference of the prime circle) the 
 circumference of such circle will be 32. 
 
 The area of a circle is always twice the square of IT below m ; 
 so if m of the circle be IT above two thirds of 1, (namely, when 
 m of the circle, is .839946) the area is .888, ad infinitum ; which 
 area is = the area of the p 6, when m of such p 6 is = B 6 ; as 
 .888 is the area of the circle whose m is two thirds of b a of the 
 circle, so also, it is half the area of the p 4 whose m is two thirds 
 of b a of the p 4, and is twice the area of the p 4 whose m is 
 two thirds of da of the p 4. 
 
 When the area of the circle is .888, ad infinitum, m of such 
 circle is the cube root of two thirds of the area, and is the recip- 
 rocal of the cube root of the square of M a 3. So four thirds of 
 M a of the circle is b of the circle whose area is .888, ad infinitum, 
 and four thirds of M a of the p 4 is b of the square whose area 
 is .888, ad infinitum. 
 
 Three times the square of M a 4 is the reciprocal of one-third 
 of M a 4, and three times the square of M a of the circle is the 
 reciprocal of two-thirds of M a of the circle, or of the bulk of the 
 prime sphere. So the square of M a 4 is half of b a 4, and the 
 square of M a of the circle is half of b a of the circle ; but such 
 is not the case with any p between the p 4 and circle. 
 
 The diameter of any regular solid is the diameter of its in- 
 scribed sphere ; and four times the square of M a 3 is the third 
 power of the reciprocal of the bulk of the sphere whose diameter 
 is 1, namely, 6.75 is the third power of 1.889881 ; and when the 
 diameter of the sphere is 1.889881 its surface will be 12. 
 
 The third power of the entire surface of any prime prism -=- by 
 the third power of the lateral surface of such prism gives 
 3.375, or twice the square of M a 3. And of course, the third pow- 
 er of the surface of the prime cube -r- by the third power of the sur- 
 face of the prime sphere, gives four times the square of M a 3. 
 36 
 
282 APPENDIX. 
 
 When the diameter of any given prism is =- d a of a like pj 
 the bulk of such prism will be the square of the diameter, for 
 the reason that bulk flows by a triplicate ratio to that of diame- 
 ter, or any other linear measure of the given solid ; and in such 
 case, the surface will be six times the diameter, for the reason 
 that surface flows by a duplicate ratio to that of diameter. 
 
 We know but little or nothing of the mode of operation of that 
 power or force (better expressed by the word attraction than by 
 any other) which, like a messenger of Omnipresence, operates 
 throughout the solar system, (and by analogy, throughout the 
 universe,) in causing the gravity or ponderosity of the bodies of 
 aggregate matter which compose the solar system, and also in 
 controlling their motions during their eternal rounds. 
 
 The phenomena, however, caused by the force of attraction, 
 enable us to contemplate many of its physical effects, and to in- 
 vestigate and determine their ratios and proportions by means of 
 the powers and roots of numbers, whatever the number of ele- 
 ments may be which enter into the calculation, as that of time, 
 space, distance, force, convergency, motion, &c. 
 
 From the phenomena observed in the solar system, which are 
 manifestly caused by the power or force of attraction, we proper- 
 ly conceive the sun to be situated in, or to be the centre of at- 
 traction for the solar system ; and hence, that all bodies of matter 
 in the solar system, which revolve around the sun, are attracted 
 and retained in their orbits by the sun. Nevertheless, those 
 bodies which revolve around the sun are observed to possess the 
 power of attraction ; hence each planet having satellites, attracts 
 and retains them in their orbits. 
 
 But a question which may yet require a further and a better 
 consideration of philosophers and astronomers, is, whether a like 
 quantity of matter, wherever situated in the solar system, possess- 
 es a like power of attraction, or a like power of attraction for an- 
 other body of matter ; or whether the power of attraction inci- 
 dent to a body of matter situated in the solar system, may not 
 depend upon its distance from the sun, or centre of attraction of 
 the system ? If the power of attraction of a given quantity of 
 matter be the same, wherever situated, then Jupiter must be far 
 less dense than the earth ; but if the power of attraction of a 
 given quantity of matter in the solar system be inversely as the 
 distance of such matter from the sun, or centre of attraction of 
 the system, then the density of the matter of which Jupiter and 
 the earth are composed may be considered as being equal, or 
 substantially so. And in such case we should say that the pow- 
 er of attraction of a given quantity of matter in the solar system 
 is inversely as its distance from the sun, or centre of attraction of 
 
APPENDIX. 283 
 
 the solar system. And that this hypothesis is the true, would 
 seem to be well established from the observations and calcula- 
 tions of some of the most astute mathematicians and astrono- 
 mers, who have found the law to be so general, uniform and 
 exact, as to construct nice astronomical tables upon its cor- 
 rectness. 
 
 But it must be manifest to any person, that the theory of the 
 different densities of the planets can have no other support in 
 favor of it, than that of the Newtonian hypothesis, that all matter, 
 wherever situated, is equally endued with an innate or inherent 
 principle of attraction ; which theory, on examination, will per- 
 haps be found not to be supported or sustained by a single fact 
 or phenomenon found to exist in the universe. Nor is it at all 
 certain that the atheistical tendency of such theory was relieved 
 by the vague conjecture that the Almighty originally pro- 
 jected the heavenly bodies in tangents to their orbits, and 
 thereby placed them under the eternal control of this original, 
 innate power of attraction ; for such hypothesis only presupposes 
 that attraction had previously aggregated the masses of matter, 
 each of which was waiting for the impulse to be given. Never- 
 theless, this naked conjecture or hypothesis, wholly unsustained 
 or supported by any physical law or phenomenon with which 
 we are acquainted, is too generally conceived to be a full and 
 ample belief in, or acknowledgement of the wisdom and power 
 of Deity. But I must dissent from those philosophers who at- 
 tempt to inculcate such notions by mingling them with the do- 
 ings of a deity of their own making. Such is certainly not an 
 exposition or explanation of natural philosophy, but is emphati- 
 cally what Newton's admiring biographer alleges, a creation 
 of natural philosophy. 
 
 It is not, perhaps, the better way, in the development of nat- 
 ural philosophy, to reject everything as being vulgar which falls 
 under common observation, and strike at some bold and original 
 hypothesis, as the foundation to which everything must be 
 squared, (or rather equated,) in order that one may have the name 
 of creating an entire system ; nor to cast aside as useless all the 
 actual evidences we possess, which can aid us in our investiga- 
 tions, even though such evidence may (by certain philosophers) 
 have been used in a whimsical or futile manner, by attempting 
 to prove more by it in respect to the phenomena of nature than 
 is warranted by true philosophy. Nevertheless, the most com- 
 mon observer must perceive some analogy between the law of 
 attraction, as operating in the solar system, and the operation of 
 the laws of magnetism, electricity, galvanism and electro-mag- 
 netism, which we are able to elicit for our use and edification 
 
284 APPENDIX. 
 
 from the great store-house or laboratory of nature ; which are 
 perhaps but modifications of one great physical law of the uni- 
 verse ; all tending, however, to induce the belief that it is an 
 omnipresent law, and is not under the necessity either to wait 
 for its prey, or to reach for it. 
 
 The equal innate gravity of matter, wherever situated, necessa- 
 rily (if adopted as an axiom) draws after it the concomitant the- 
 ory of the different densities of the planets ; and if either hypoth- 
 esis is true, they both doubtless are. So also if either of them be 
 untrue, they doubtless both are. But to me both hypotheses 
 appear to be wholly uncalled for, and unsupported by any legiti- 
 mate evidence of which we can avail ourselves ; and that in lieu 
 of the round-about process in attempting to ascertain the differ- 
 ent densities of the planets having satellites, supposed to exist 
 in consequence of the different attractive powers which they ex- 
 ert upon their respective satellites, in proportion to the bulks or 
 magnitudes of the planets, we have only to consider the quantity 
 of matter in the respective planets to be as their bulks or magni- 
 tudes, and that their powers of attraction, in proportion to their 
 respective magnitudes, are inversely as their respective distances 
 from the sun or centre of gravity of the system ; and especially 
 when we are sustained by the evidence, the fact, in accordance 
 with the general and well known law, that their respective rela- 
 tive powers of attraction are given by multiplying their respec- 
 tive bulks or magnitudes by the inverse of their respective dis- 
 tances from the centre of gravity of the solar system ; for here is 
 a fact found, according with a law found to be uniform and 
 universal, that the squares of the periods of the planets are as 
 the cubes of their mean distances from the sun. 
 
 I am aware that the adoption of my premises would abstract 
 from mankind a vast amount of admiration ; for " still they gazed, 
 and still the wonder grew," that human intellect could become 
 so sublimed as to be able to weigh off the planets, and determine 
 their different densities, when in reality the exploit supposed to 
 have been accomplished, was but a corollary flowing from one of 
 the greatest absurdities that was ever interwoven into a code of 
 philosophy. 
 
 Let us now examine a little into the attractive powers of Ju- 
 piter and the earth. The time of the period of the earth to that 
 of Jupiter is as 1 to 11.8568 ; and the mean motion of Jupiter to 
 that of the earth is as 1 to 2.2802. The mean distance of the 
 earth from the sun to that of Jupiter is as 1 to 5.1993, and the 
 cube or third power of 5.1993 is the square of 11.8568 ; hence 
 the square of the reciprocal of the rate of mean motion is the 
 mean distance. 
 
APPENDIX. 285 
 
 Hence, the mean attractive power of a planet applied to its sat- 
 ellite to retain it in its orbit, is the square of the mean motion of 
 the satellite ; hence the whole amount of motion of a satellite 
 during its entire period is the square root of the whole amount 
 of the force of attraction expended upon the satellite during the 
 period. Thus the rate of mean motion, or the square root of the 
 rate of mean force, multiplied by the time of the period, will 
 give the time of the period of the satellite when operated upon 
 by a different rate of force, but at the same distance from the at- 
 tracting body. 
 
 According to La Grange, the power of attraction of the earth 
 to that of Jupiter, is as 1 to 313.0640, and the square root of 
 313.064 is 17.6936, which of course would be the rate of mean 
 motion of a satellite revolving around Jupiter, if the rate of mean 
 motion of a satellite revolving around the earth at the same dis- 
 tance were 1. The distance of Jupiter's first satellite is found to 
 be a little greater from the centre of Jupiter than the moon is 
 from the centre of the earth. The period of Jupiter's first satel- 
 lite is one day and .7639 of a day, namely, 1.7639 days. 
 
 If, then, the time of the period of Jupiter's first satellite be 
 multiplied by the square root of the attractive power of Jupiter, 
 compared with that of the earth, which is assumed at 1, namely, 
 by 17.6936, it gives 31.20974 days, which would be the time of 
 the period if such satellite were revolving around the earth at the 
 same distance from the centre that it revolves from the centre of 
 Jupiter. And the square of the period of the moon is to the 
 cube of its mean distance from the earth as the square of 
 31.2097 is to the cube of the mean distance of Jupiter's first sat- 
 ellite from the centre of Jupiter. Hence the rate of mean motion 
 of Jupiter's first satellite is as much greater than it would be if 
 revolving about the earth at the same distance, as 17.6936 is 
 greater than 1 ; and this calculation will be found to rest, or to 
 be based upon the principle, that if the bulk or magnitude of the 
 earth be 1, and its distance from the sun be 1, and its attractive 
 power be 1, the relative or proportional power of attraction of 
 Jupiter, compared with the earth, will be obtained by multiply- 
 ing the bulk or magnitude of Jupiter, when compared with the 
 earth, by its inverse distance from the sun when compared with 
 the earth. 
 
 If we assume the mean distance of the earth from the sun at 
 95,000,000 of miles, and the distance of the moon from the earth 
 at 237,500 miles, the relative distance of the moon from the earth 
 to that of the earth from the sun will be as 1 to 400. These dis- 
 tances are however assumed for the sake of round numbers, the 
 actual mean distance of the moon from the earth being general- 
 ly stated at 240,000 miles. 
 
286 APPENDIX. 
 
 The periodic time of the moon around the earth is to that of 
 the earth round the sun as 1 to 13.3683. If we then adopt the 
 foregoing assumption as to distance, the mean motion of the 
 moon in its revolution about the earth will be to that of the earth 
 about the sun as 1 to 29.9207, and in such case, while the square 
 of the mean motion of the moon is 1, the square of the mean 
 motion of the earth will be 895.2482 ; and in such case the rela- 
 tive power of attraction of the earth upon the moon to that of the 
 sun upon the earth would be as 1 to 895.2482. But if we as- 
 sume the distance of the moon from the earth at 240,000 miles, 
 the relative attraction of the earth upon the moon to that of the 
 sun upon the earth would be as 1 to 876.7402. Hence the force 
 of attraction exerted by the sun upon the earth would be some 
 40,000 times as great as that exerted by the moon upon the 
 earth ; nor would this power of attraction exerted by the sun upon 
 the earth have a direct operation in any wise upon the tides. 
 
 But I will close this article by a few doctrines relative to the 
 laws of force and motion, with an additional reference to the law 
 of falling bodies. 
 
 Twice the motion of a revolving body requires quadruple the 
 force of attraction to retain the planet in its orbit. 
 
 The period of any planet is the square root of the cube of its 
 mean distance from the centre of gravity. 
 
 If A denote a planet at the distance 1, and B a planet at the 
 distance 2, the time in which B will revolve over a space equal 
 to the circumference of A's orbit, will be = to the square root of 
 B's distance, or of the diameter of A's orbit. Hence the time in 
 which B will revolve over a space = to the circumference of A's 
 orbit, will be inversely as the rate of the mean motion of B. 
 
 Hence the rate of mean motion of a planet as proportioned to 
 the mean distance, is inversely as the square root of the mean 
 distance ; or the square root of the mean distance is the inverse 
 of the rate of mean motion. Hence the mean distance is a mean 
 proportional between the time of the period and the square root 
 of the mean distance. 
 
 If D denote another planet whose mean distance is 4, the time 
 required for D to pass over a space equal to the circumference of 
 A's orbit will be equal to the square root of D's distance. Hence 
 if ever so many planets revolve round the same central force, 
 and the mean distance of either of them be assumed at unity, 
 the time required for any other planet to revolve over a space 
 equal to the circumference of the orbit whose distance is 1, will 
 be equal to the square root of the mean distance of the planet, 
 compared with that whose distance is 1. This, however, can 
 only be true in respect to centric orbits. 
 
APPENDIX. 287 
 
 So when the mean distance is 1, the rate of mean motion is 
 = to the time of the period. When the distance is 2 the rate of 
 mean motion is equal to one fourth of the period. When the 
 distance is 3 the rate of mean molion is equal to one ninth of 
 the period ; and so on. Hence, if the force of attraction be in- 
 versely as the distance, it is the square of the mean motion ; 
 but if the force of attraction be inversely as the square of the dis- 
 tance, it will be the fourth power of the mean motion. 
 
 If we take two planets of a system, as B and D, if the distance 
 of B is the square root of the distance of D, the period of B, the 
 rate of motion of B, and the rate of force of B, will respectively 
 be the square root of like elements of D. 
 
 If the mean distance of A be 1, and that of D be 4, if the force 
 of gravity varies inversely as the distance, D will receive twice 
 the force in the time of its period that A will in the time of its 
 period. But if the force of gravity varies inversely as the square 
 of the distance, D will receive but half the force in the time of its 
 period that A will in the time of A's period. 
 
 The mean distance of a planet multiplied by the square root 
 of the mean distance, gives the period ; and the reciprocal of the 
 period of a planet is the product of the rate of motion by the rate 
 of force. Hence when the circumference of the orbit is 16, and 
 the time of the period is 4, the radius of the orbit or the distance is 
 then the third root of 16, and the square root of such radius is 
 the third root of four ; and the product of such radius by its 
 square root gives the time of the period, or one fourth of the cir- 
 cumference of the orbit. But when the period is 8, the product 
 of radius by the square root of radius, gives as well the diame- 
 ter as the period. Or the version maybe thus, the cube root 
 of the period multiplied by the cube root of the square of the 
 period gives the period. 
 
 By the popular determination of the quadrature, when the 
 period of the planet is = one fourth of the circumference of the 
 orbit, such period is 2_ below 4 ; for when the period is 4 by the 
 popular mode, one fourth of the circumference of the orbit is f 
 below 4. Hence by any system of the quadrature, when the peri- 
 od of the planet is = one fourth of the circumference of the orbit, 
 the time of the period will be the third power of twice M a of 
 the circle. 
 
 So by any system, when the period is = one fourth of the cir- 
 cumference of the orbit, the circumference of the orbit is half of 
 the third power of the circumference of the circle whose diam- 
 eter is 1. . 
 
 Nevertheless, by the popular determination of the quadrature, 
 if the period of a planet be made commensurable with the di- 
 ameter of the orbit in the powers and roots of definite quan. 
 
288 APPENDIX. 
 
 titles, it will not be commensurable with the circumference of 
 the orbit, or with any part of it ; and if the period be made 
 commensurable with the circumference of the orbit, or with any 
 part of it, (namely, with the path in which the planet travels,) it 
 will not be commensurable with the diameter. But by the true 
 determination of the quadrature of the circle, the period of a 
 planet is as well commensurable with the circumference of its 
 orbit as with its diameter. 
 
 I have heretofore alleged, that by the true law of gravity or 
 attraction, the earth would exert quadruple the amount of the 
 force of attraction upon the moon, that would be exerted by the 
 Newtonian law ; and hence the true law would be just what 
 seemed to be required by Clairaut, Euler and others, for the 
 accounting for the motion of the moon's apogee, upon the princi- 
 ple of attraction. So by the true law of gravity, any planet in 
 the solar system will be attracted towards the sun by quadruple 
 the force that it would be by the Newtonian law. 
 
 And lest any one should not readily apprehend the truth of 
 the allegation, I will shortly explain, nevertheless, some part of 
 the explanation may appear somewhat paradoxical, or similar 
 to Maclaurin's explanation of what would be the various move- 
 ments of the planets were they operated upon by various laws of 
 gravity which have no existence in nature. Nevertheless, by all 
 philosophy, it requires quadruple the force of attraction to bal- 
 ance duplicate the motion ; and the motion of a planet is doubled 
 at one fourth the distance, for the reason that the force is quad- 
 rupled. If, then, the Newtonian law of gravity gave but just 
 one half the force requisite for the motion of the moon, or of the 
 moon's apogee, as found by Clairaut, Euler and others, then in 
 order that the Newtonian law might give the actual or observed 
 motion, it would be necessary for the moon to revolve at one 
 fourth the distance from the earth that it now does. And the 
 like in respect to any planet in the solar system ; the motion be- 
 ing just double that which would be sustained by the Newtonian 
 law of gravity at the distance at which the planet revolves. And 
 in order that it might revolve about the sun retaining the mo- 
 tion it now has, it must revolve at one fourth the distance it now 
 does, in order to be sustained in its orbit by the Newtonian law 
 of gravity. 
 
 In respect to a body commencing its fall towards the earth, 
 as there is no portion of the fall, from its commencement, how- 
 ever small, in which the fall or motion is not accelerated, it fol- 
 lows as a consequence to such axiom or proposition, that at any 
 given point of space of the fall, or at any given instant of time 
 during the fall, the amount of space then fallen over is the square 
 
APPENDIX. 289 
 
 of the amount of force then expended on the body during the 
 fall, and consequently that the intensity of the force varies inverse- 
 ly as the distance varies. 
 
 Nothing could be devised to show more clearly the truth which 
 I allege, than the simplest formula that can be constructed for 
 the purpose of exhibiting the numerical law of falling bodies. 
 If we conceive a body to fall from a state of rest over a given 
 space in a given time, (as for instance, over one rod in one sec- 
 ond of time,) we may denote the operation by 
 
 1 
 
 the space fallen over, (denoted by the lower figure 1,) being a 
 unit of space, and its square root denoted by the upper figure 1, 
 will as well denote the unit of force employed during the fall 
 over the unit of space, as it will the unit of time in which the 
 body is falling over the unit of space. 
 
 If we now conceive the unit of space to be divided into four 
 equal parts, the formula will be thus, 
 
 1 1 
 
 123 
 
 in which case the odd numbers in the lower series will denote 
 the four equal parts into which the unit of space is divided. If 
 then we extract the square root of the lower series, taking figure 
 1 for the first period in the process of extraction, the square root 
 of the lower series will be expressed by figure 1 over each odd 
 number of the lower series ; which square root expressed by the 
 upper series, will denote as well a division of the unit of force, 
 as a division of the unit of time, into equal parts. Hence, while 
 the body is falling over that portion of the unit of space denoted 
 by figure 1 in the lower series, one half of the unit of force, as 
 also one half of the unit of time, will be expended ; and the re- 
 maining half of the unit of force, and also of the unit of time 
 will be expended while the body is falling over three fourths of 
 the unit of space, namely, that denoted by figure 3. 
 
 So the unit of space may be divided into as many equal parts 
 as we please, even to an infinite or indefinite number ; in which 
 case the lower series of the" formula will be expressed by the nat- 
 ural numbers in their order, while the equal parts of such unity 
 of space will be denoted by the odd numbers of the series ; and 
 a like extraction of the square root of the series, (taking figure 1 
 at the left as the first period in the process of extraction,) will 
 give figure 1 over each odd number, as the square root of the 
 series ; in which case the unit of force, as also the unit of time, 
 will be divided into as many equal parts as there are figures in 
 such root. And it is manifest that however far we extend such 
 37 
 
290 APPENDIX. 
 
 division, at any instant or point of the fall, no matter how little 
 space has been described or fallen over, the sum of the odd 
 numbers of the lower series described by the fall thus far, will be 
 the square of the number of equal moments, or of the equal 
 amounts of force which have been expended thus far. Hence 
 the space passed over or described in the fall is as well the 
 square of the amount of force, as of the amount of time expend- 
 ed ; and consequently there is no space described or passed over, 
 without acceleration. 
 
 Such, then, enables us, in the most simple and easy manner, 
 to determine the law of gravity, namely, by comparing the de- 
 scent of two bodies falling from a state of rest, but from different 
 distances from the same centre of gravity. 
 
 SUGGESTIONS IN REGARD TO CYLINDRICAL SECTIONS. 
 
 I will here only suggest in respect to the three principal 
 planes that may be cut from the prime cylinder, or cylinder, the 
 diameter of whose inscribed sphere is unity or 1. 
 
 If we conceive the prime cylinder standing upon one end, 
 upon a horizontal plane, the base will denote the prime circle, 
 and if such cylinder be cut. horizontally through, the plane will 
 be the prime circle ; and if a plane be cut perpendicularly 
 through the centre of the cylinder, the plane will be the prime 
 square. 
 
 If the cutting plane commence on the side of the cylinder, at 
 any point between the base and altitude, and terminate on 
 the opposite side, at the point touching the periphery of the base, 
 the plane will be an ellipse (or elongated circle,) the area of 
 which will be the product of the longest by the shortest diameter, 
 multiplied by the area, or one fourth of the circumference of 
 the prime circle. The longest diameter of such ellipse may be 
 said to have its upper and lower ends ; the upper end being the 
 one above the base of the cylinder. 
 
 As the shortest diameter of any ellipse cut from the prime 
 cylinder is always unity, it is therefore manifest that the area of 
 any such ellipse is to the longest diameter, as Ma of the circle 
 is to 1, or as 1 is to d a or b a of the circle ; and hence, that the 
 area will increase or decrease by the same ratio that the longest 
 diameter will ; the consequence is, that the longest diameter (or 
 major axis) of any ellipse cut from the prime cylinder, multiplied 
 by the area of the prime circle, gives the area of such ellipse ; 
 consequently, if the major axis of such ellipse be = da or ba 
 of the circle, the area of such ellipse will be 1, or is = m of the 
 circle whose area is = d or b of the circle. Thus, the area of 
 
 
APPENDIX. 291 
 
 any ellipse cut from the prime cylinder, is = m of a circle whose 
 b or d is = the major axis of such ellipse ; and hence, if the 
 major axis be at its maximum, namely, the square root of 2, the 
 area of such ellipse is necessarily a mean between Ma and 
 twice Ma of the circle ; and such ellipse may properly be termed 
 the prime ellipse, the area of its circumscribed circle, or of any 
 circumscribed parallelogram, being double that of a like inscribed 
 figure. The plane of such ellipse as situated in the cylinder, 
 forms an angle of 45 with the base ; and as the focal distance 
 of any ellipse cut from the prime cylinder is = the altitude of 
 the upper end of the longest diameter, or major axis, hence, 
 the focal distance of the prime ellipse is unity. And as the 
 length of the describing thread of any ellipse is = the major 
 axis, plus the focal distance, hence the length of the describing 
 thread of the prime ellipse is = the sum of the longest and 
 shortest diameters, or of the major and minor axes. Thus, while 
 the major axis of the prime ellipse is = ba 8, the length of the 
 describing thread, is twice d a 8, consequently, from one end of 
 the prime ellipse, to the nearest focus, is = v a 8 for in respect 
 to any p, da plus v a is = b a. 
 
 So the area of the prime ellipse is = b or d of the circle, whose 
 area is 1 or is = b of the p 8 whose area is the reciprocal of 
 b or it is a mean between the area of the square whose cir- 
 cumference is 4, and the area of the circle whose circumference 
 is 4. 
 
 But it is not my intention here, to treat of the qualities of 
 ellipses ; nor to do more than to suggest some unity of purpose 
 in their consideration, which is certainly to be desired ; if, (as is 
 generally supposed,) planets can revolve in ellipses, however ec- 
 centric or in ellipses of extreme eccentricity, and in which, 
 (according to the modern version of Kepler's law) "the squares 
 of the periods are as the cubes of their semi-major axes." Now, 
 according to this version, if we conceive two planets to revolve 
 around the same central force in orbits, the respective circumfer- 
 ences of which are 4, one orbit being a circle (either centric or 
 eccentric,) and the other an ellipse of extreme ellipticity. In 
 such case, the time of the period of the planet revolving in the 
 ellipse must be 1, while that in the circle will be .5. Hence, in 
 the ellipse, the time of the period will have no regard to the 
 space, or length of road passed over, or to the area described by 
 the radius vector, or even to the amount of force of attraction 
 expended on the planet during the period. And if we conceive 
 a planet to revolve in an ellipse, we must conceive it to revolve 
 without obeying any of the requirements of the great physical 
 law of the universe. 
 
., 
 
 * 
 
 292 APPENDIX. 
 
 But in respect to the circular orbit, (whether centric or eccen- 
 tric,) by any system of the quadrature, if the diameter of the 
 orbit be = b a of any given j, the time of the period will be the 
 cube or third power of B of a like p. So also, if the diameter 
 is = da of any given jt?, the time of the period is the cube or 
 third power of D of a like p. So when the diameter is equal 
 da or b a of the circle, the period is the cube, or third power of 
 B or of D of the circle. 
 
 Hence, the reciprocal of A of any given p, is the diameter of 
 an orbit in which the period is the third power of B of a like p. 
 And the reciprocal of M a of any />, is the diameter of an orbit 
 in which the time of the period is the third power of D of a 
 like j9, by any system of the quadrature. 
 
 Hence, the time of the period of a planet, is the third power 
 of the square root of radius of the orbit. 
 
 So if the diameter of an orbit be = m of the circle, whose 
 area is 2, namely, = b of the equal square, or square whose 
 area is = to the area of the prime circle, or whose diameter is 
 one eighth of the square of the circumference of the prime cir- 
 cle, the time of the period in such orbit will be the sixth power 
 of m, or the third power of the area of such equal square. 
 

 
 
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