UC-NRLF GIFT OF Berkeley Public Librai fs GENERAL HISTORY MATHEMATICS FROM THE EARLIEST TIMX5, TO THE MIDDLE OF THE EIGHTEENTH CENTU8Y. FROM THE FRENCH J O, H N i B O $ S U T, OF THE FRENCH NATION Al INiTJTUT* OF A&T AND AND OF JHE ACADEMIIS TO WHICH 1$ ATMXS&, A CHRONOLOGICAL TAE1T, OF THE MOST IM2KEKT MATHEMAT3C2AKS FOK J. JOHNSON, ST. PAVX*S ClXUXCHT V SYE AND 1 AW, ST. JOHN'S SQVAR S 1803. A EDITOR'S PREFACE. I HE mathematical or exact sciences, as they are frequently called, having been reckoned, in all ages, among the most useful and sub- lime productions of the human mind, it must appear a matter of surprise, that, till within a few years past, no regular or well connected history had ever been gken of their origin and progress ; or to shew by what steps they had ad- vanced from their first rude beginnings to their present state of improvement. While many other branches of knowledge, both civil and literary, have been traced through all their stages, and elucidated in every possible variety of method and language, this interesting and important subject, which is so well calculated to display the reasoning and inventive powers of man, has been almost wholly neglected. This defect appears to be owing to various causes, and has long been perceived and la- mented by those, who were able to estimate the advantages of such inquiries. Lord Chancellor Bacon accordingly remarked, near A Z two IV two centuries ago, that most of the histories, which had then appeared, might be compared to the trunk of a tree, deprived of it's most noble branches. And in more ancient times, the elder Pliny, speaking on the same subject, has observed, with regret, that no writer of his time had undertaken to transmit to poste- rity the names and labours of those eminent men, who by their meditations and re- searches had enlarged the boundaries of sci- ence, and dignified and solaced life by their inventions and discoveries. It is, indeed, obvious, that a good history of Mathematics might be considered as a re- gister of experiments, to ascertain the strength of the human understanding, in some of it's highest attainments; which would also serve, as far as they have been successful, to guide and encourage our future researches. And, even in those cases where they have failed, they might prove of nearly equal importance, in preventing the repetition of useless trials, and unprofitable labour. How many, for instance, have wasted a great part of their lives in at- tempts to square thq circle, to discover the perpetual motion, &c., who, if they had only read an account of what had been done by others in that way, would probably have been deterred deterred from entering upon these hopeless and illfated speculations. But among the advantages -which may be expected from a comparison of the early state of these sciences^ with their present extension and improvement, one of the most consider- able is, that it will lead to the full discovery of the origin of many theories and practices, which though, at first* defective and insuffi- cient, have given birth to others more correct and commodious in their application. Such, for example, has been the case of the Flux- ional and Differential Calculi^ which may be traced through the methods of Slusius, Bar- row, and the Indivisibles of Cavallerius, to their present improved form. The young mathematician, in reading a work of this kind, will, also, be highly grati- fied, in following the accounts which have been given of the contention of rival talents, in the attempts that have been made to excel in the solution of some new and difficult pro- blems ; many of which are now become com- paratively easy ; but which, at the time they were first published, required the utmost ex- ertions of the most vigorous understanding, to complete their investigation. This he will find to be particularly the case with many of the A 3 questions vi questions proposed by Leibnitz, the Ber- noullis and several British and french ma- thematicians ; which being but little known to the generality of english students, cannot fail of proving extremely entertaining and curious. Another advantage which such a work may produce, is that of showing the difficulties which the learner has to encounter, before he can hope to arrive at any degree of eminence in the pursuits which he has embraced. He will here soon perceive, that it is not an ac- quaintance with the mere elements of the various branches composing the great body of these sciences, which can entitle him to rank among mathematicians ; but that he must look to the example of Newton, Euler, Lagrange, and other eminent men, w r ho have risen to a height so far above that of their predecessors. M. de Monmort, a celebrated french ma- thematician, being impressed with the utility of such an undertaking, observes, in one of his letters to Bernoulli, that " it is much to be wished some person would take the pains to inform us, how and in what order mathe- matical discoveries succeeded each other, and to whom we owe the obligation. We have histories Vii histories of painting^ music and medicine; but a good history of geometry .would be a Work much more curious and useful. What pleasure would it afford us to perceive the connection of methods* and the chain of new theories, proceeding from the earliest periods to our own time ! Such a work, well executed, might be considered as a history of the human mind ; because it is in this science, njore than, in any other, that man shows the excellence and intelligence which God has given him, for the purpose of elevatuig him above all other creatures/'' It may, also, be added, that M. de Mon-* mort did not content himself with mere wishes* but undertook to perform the task* which he had recommended to others ; and no one could have been better qualified to execute a work of this kind, than this learned geo- meter ; who, to a profound knowledge of the subject, joined the advantage of an extensive correspondence and connection with most of the ablest mathematicians in Europe. But after having made considerable advances in the undertaking, as we learn from a fragment of one of his letters, which has been pre- served in the Leipsic Acts, he was prevented, by death, from finishing the work. And from A 4 the via the inquiries which were made among his executors by M. de la Condamine,, who had been one of his most intimate friends, he had the misfortune to find, that all his manuscripts had been totally destroyed or dispersed. There were even among the ancients some writers, Who were led to cultivate this species of erudition ; the first of whom, in order of time, was Theophrastus, w r ho wrote the his- tory of Arithmetic, Geometry, and Astronomy; and the two last of these sciences also found an historian, about the same time, in Eude- mus, another philosopher of the school of Aristotle. Geininus, likewise, a little before the commencement of thq Christian era, wrote anew the history of geometry ; but of all these works, which would have proved so interesting to the moderns, nothing has been transmitted to us, except the little that Proclus has extracted and employed in his prolix commentary on the first Book of Euclid, The other writers of antiquity, who have left us some slight accounts of the history of these sciences, are Diogenes Laertius, in his Lives of the Philosophers, Plutarch in his Placita Pkifamphorum, Stobeus in his Eclogce Physicce, and Tatius in his Isagoge ad Arati Ph&nomena ; but these are only a small number of isolated particulars. IX particulars, disfigured by credulity and art ignorance of the subject. With respect to more modern writers, the principal work, of an early date, which treats .expressly on this subject, is that of Vossius, entitled Universe Mathtseos &c. ; but this consists of little more than divisions and sub- divisions of the mathematics, and an arid enumeration of authors and their works, with a few pompous eulogiums on some of the most remarkable, distributed without taste or judgment. Beside the writers here mentioned, the his- tories of some particular branches of these sciences have been given by various authors; the most celebrated work of this kind being that of the unfortunate Bailly, who has writ- ten the history of ancient and modern Astro- nomy, in 5 Vols. 4to, which, allowing for some singularities of opinion, is one of the most agreeable and instructive performances, that has hitherto appeared on any scientific subject. To this may also be added the his- tory of Optics by Dr. Priestley, published in 1772 in one Vol. 4to, which has been trans- lated into german, with considerable notes and explications by Mr. Kliigel; but this work. is, in fact, more physical than mathe- matical. Many other slight and imperfect accounts bf particular parts of these sciences have been given, both in the prefaces of some ma- thematical works, and in separate tracts* But the only general and extensive history of this kind, which has yet appeared in any lan- guage, is that of M. Montucla in 4 Vols. 4to; which, in it's present state, embraces almost all the parts of this great and important branch of human knowledge. The first part was published in 1758, in 2 Vols. 4to. ; and a new edition of it was given in 1798, with very considerable improvements and augmenta- tions, by the author ; who was afterward in- duced to continue his work, till his death, which happened in 1799* put a stop to the undertaking. Having left behind him, how- ever, a number of materials for the further prosecution of his plan, they were revised and methodized by his friend M. de la Lande, who published the two additional volumes in the latter end of the year 1802. This great work, like most other extensive undertakings, which require a variety of ta- lents and acquirements, must be acknow- 9 ledged xi ledged to have many defects ; but when we consider the difficulties which the author had to encounter, in so new and hazardous an enter- prise, every candid and liberal mind will readily make allowance for imperfections of this kind ; more especially when it is well known, that few men possess the requisites necessary to the more complete execution of this plan, which, beside a very considerable acquaintance with the various branches of science, requires a knowledge of most of the ancient and modern languages, and an unwearied spirit of re- search, which nothing but a strong and ar- dent mind could support. His chief faults are, that his style is often inelegant, and too much embarrassed with repetitions ; and in his account of some mo- dern discoveries, he displays a spirit of na- tionality, which ought never to be found in a strict and impartial historian. But these blemishes are of trifling importance compared with the general excellence of the work, which every where abounds with interesting details, and the most perspicuous explanations of the various inventions and improvements/ which, at different times, have contributed to the progress of these sciences. If he be not so profound as some other writers, he is fre- quently Xll quently less obscure, and may often be con* suited with advantage, upon points where the original authors would be nearly unintel- ligible to common readers. In short, there is, perhaps, no work which is capable of affording more pleasure and instruction to those, who propose to devote themselves to these studies, or which is likely to create a more earnest desire to prosecute them : so that it is probable he will long continue to deserve the title, which he has hitherto exclusively possessed^ of The Historian of Mathematics. In the interval, however, between the pub- lication of the first and second parts of M. Montucla's performance, M. Bossut, a mem- ber of the National Institute of France, and a mathematician of considerable eminence, prepared and executed a new work on this subject, of which that now offered to the public is a translation. Though given, in the original, under the modest title of an Essay, the author, like his predecessor, describes the origin and progress of most of the mathe- matical sciences, with a discrimination and judgment which do him great credit. Upon most subjects of a curious and difficult na- ture, he appears to have thought for himself; and on many occasions will be found to dis- play X1U play a shrewdness and perspicacity of obser- vation, accompanied with certain anecdotes, which afford a pleasant relief to the dryer parts of the narrative, and tend to impress the subject more strongly on the mind. As it appeared, therefore, that this per* formance contained a clear and well digested detail of the subject, within a moderate com- pass, it was judged, that in it's present dress, it could not but prove acceptable to the bri- tish student, who proposes to devote himself to mathematical pursuits ; since, if he should be desirous, in the prosecution of his in- quiries, to take a general survey of the rise and progress of these sciences, and by these means to make comparisons and draw conclu- sions for himself, he would seek in vain for the necessary information in any english work. But, as the author has given an account in his Preface of the plan and design of the undertaking, it remains only to observe, that he alone must be considered as responsible for the opinions he maintains, with respect to certain discoveries, and other points, which, involving some of the highest claims of genius and invention, have given occasion to many violent disputes, both of a personal and even of a national cast. The only addition that has been XIV been made, is the annexed biographical table of the most eminent mathematicians of ancient and modern times, exclusive of those now liv- ing; which it was conceived would make a proper appendage to a work of this kind ; especially as it will be found to contain a far more complete list of the most respectable names than any that has hitherto been pub- lished. For the rest, nothing more has been here undertaken, than to give to the public a correct and faithful translation of the original work, which it is hoped will be found to pos- sess sufficient merit, to entitle it to their ap- probation. Royal Military Academy, RONNYCASTLE Woolwich, Feb. 21, 1803. J ' BUJN JN Y CA& 1 1-U THE AUTHOR'S PREFACE. OEVEK.AL authors have written detached parts of the history of mathematics, without any regard to proportion, either in prefaces to their own works, or in separate tracts ; but Montucla is the only person, who has at- tempted the whole of the subject, in a method adapted to the nature and extent of each par- ticular branch. His History of Mathematics first made it's appearance in 1758, and gave an account of the progress of the sciences comprised under this term, from their origin to the commencement of the last century. In 1798 it Avas reprinted with considerable addi- tions, but still confined to the same period. The author had collected materials for bring- ing it down to the present day ; but they were not prepared for the press at the time of his death, which happened in 1799- His ma- scripts, revised, corrected, and enlarged with the necessary additions, have lately been printed; but I know nothing more of them, than what is said in the periodical publica- tions. The XVI The work of Montucla has received from the learned the applause it justly merits. It contains an immense number f interesting re- searches, chiefly respecting ancient mathema- tics ; yet it must be owned, it has not passed without various criticisms. Generally speak- ing, it is somewhat defective in method and style ; it is also disfigured by mixing together matters that do not coalesce; and would be improved by the suppression of certain plea- santries, which do not accord with the gravity of the subject. It is besides, adapted only to professed mathematicians; for though it is true, that treatises on almost every part of the ma- thematics are to be found in it, yet, as these treatises do not succeed each other in a clas- sical and elementary order, they are capable of being understood only by readers who are already acquainted with the fundamental principles of them. And it is to be wished, that Montucla had entered a little more into the spirit of the authors, whose discoveries he records : for instance, we have to regret, that, in speaking of the conic sections, he has nei- ther given us a tolerable abstract of the Conies of Appollonius, nor sufficiently made known the method of this ancient geometrician ; a subject XVli subject highly interesting to the admirers of elegant synthesis. Whether these criticisms be just, or not, Montucla will ever enjoy the honour of having produced a very learned and useful work, of a kind so much the more rare, because those who are enamoured of the mathematics, have usually more inclination to enrich them with their own discoveries, than to relate those of others. We ought therefore to allow him all the merit due to such a sacrifice* My object is not to give a minute history of the mathematical sciences ; but to consider, in each of them, only the leading branches, and the chief consequences that flowed from them. Having always felt a curiosity, during the course of my studies, to trace all the know* ledge I acquired to it's origin, and filled with a profound veneration for those to whom we are indebted for it, I began about thirty years ago, to write down from time to time such re- flectioiis, as this disposition produced. The first fruit of this was a sketch, which I pre- fixed in 1784 to the mathematical part of the Encyclopedic mitliodique* This sketch received some applause; though it was very imperfect, both from the narrow limits to which I was confined, and from some irregu- a larities xtiii larities in my plan, on which at that time I had not sufficiently meditated : and to aggra- vate these defects, japme things of essential import were omitted, or scarcely mentioned. Some intelligent friends therefore urged me to correct what I had done, and to produce a work, that might gratify curiosity, and instruct the mind. These views I have endeavoured to accomplish, as far as my feeble means would permit ; and I shall deem myself happy, if I can inspire youth with a taste for the study of those sublime sciences, which are truly worthy to employ a thinking being. Perhaps I may be suspected of partiality in their favour ; but I trust I shall find little dif- ficulty in exculpating myself. I am firmly persuaded, as on several occasions I have de- clared, that men of superiour endowments are almost equally rare in every branch of know- ledge ; and that nature has established a kind of equilibrium between all her productions : as a consequence, therefore, of the same principle, I must contest the point with those who confine genius to the faculties of imagination ; as I believe, that with a common understand- ing, and assiduous labour, any man may at- tain the first rank in the sciences. The ex- amples, ort which they rest their opinion, are far XIX far from conclusive. It is true we have seen men of assiduity, possessed of good memories, and with moderate natural parts, acquire in the World the reputation of great geome- tricians. But is it in the least surprising, that the ignorant or superficial many should con- found the fruits of that knowledge, which is ac- quired by study, with those new and original truths to which genius alone can give birth ? To be just, we must weigh the great mathe- maticians of well established reputation against the great poets, and the great orators. Thus on the one side let us place Homer, Virgil, Racine, Pope, Demosthenes, Cicero, Bossuet; on the other Archimedes, Hipparchus, Galileo, des Cartes, Huygens, Newton, and Leibnitz; and it will not then be so easily determined, to which side the balance ought to incline. I shall likewise combat, or at least en- deavour to weaken, a reproach made to ma- thematicians, to which it must be confessed some of the most illustrious have been liable, though perhaps it has been still more justly merited by their adversaries ; that of being vain. Such, for example, was John Bernoulli, as appears in this work. But why should the World so rigidly require of superiour men, to appear entirely ignorant of their own worth? a 2 I have XX I have sought for the reason, and I believe I have found it. Modesty is a desertion of our- selves, a kind of avowal of inferiority, at which mediocrity catches greedily as a source of consolation, which it endeavours to interpret in the literal sense, and which it frequently employs as a weapon to keep at a distance the timid man of genius, destitute of support, and the victim of his own candour. Experience shows, that there is more danger in humbling ourselves too much, than ridiculousness in proclaiming our own merits. Let me add, that what is sometimes taken for self love is but the estimable ingenuousness of a man of letters, almost always a solitary in the midst of society, ignorant of the maxims and customs of a corrupt World, where men seek only to deceive each other, and to feign sentiments which they do not possess. This history concludes with the fatal years of 1782 and 1783, in which the sciences w^ere robbed of Daniel Bernoulli, Euler, and d'Alem- bert. I refrain at present from speaking of the labours of living mathematicians : but of these likewise I have made a sketch, which I shall publish under the title of Considerations on the present State of Mathematics. The reader is aware how much circumspection such a 9 work xxi work must require, as it is my intention to be perfectly just, and to pay to every real inventor that tribute of praise and acknowledgement which is his due. THE EDITOR takes the liberty of adding, by way of a note, that he can have no doubt but the projected work, here mentioned, by M. Bossut,, would prove highly acceptable to Mathematicians ; and that many of them will wish it may be speedily executed. cotf- ERRATA. Page line 3, ji, dele some 1 8, 12 f. ^ for was read were 44, 12 situated situate 78, 4 this that 88, 3, <&/* that he 99, 6 f. b. for and was m? 176 CHAP. VII. State of the Sciences among the Chinese and Hindoos 177 CHAP. VIII. State of the Sciences among the modern Greeks - 179 CHAP. IX. State of the Sciences among the Christians in the West, to the end of the thirteenth Century - - 185 CHAP. x. State of the Sciences among the Christians in the West* continued through the fourteenth and ffteenth Cen- turies - - 191 PERIOD THE THIRD. PROGRESS OF THE MATHEMATICS, FROM THE END OF THE FIFTEENTH CENTURY TO THE INVENTION OF THE METHOD OF FLUXIONS. - - 204 CHAP. I. The Progress of Analysis - - '_ .":, - 206 CHAP. II, Progress of Geometry - - -223 CHAP. in. .Progress of Mechanics - 23 CHAP, iv, Progress of Hydrodynamics - - - 251 CHAP. Page CHAP. V. Progress of Astronomy - - - -257 CHAP. VI. -Progress of Optics . - - 288 PERIOD THE FOURTH. PROGRESS OF MATHEMATICS, FROM THE DISCOVERY OF FLUXIONS TO THE PRESENT DAY - 308 CHAP. I. The Discovery of the Analysis of Infinites : Leibnitz first published it's Elements ; Newton employed a similar Method in his PRINCIPIA MATHEMATICA 310 CHAP. II. Leibnitz continues to extend his new Analysis, se- conded by the two Bernoullis. Various problems proposed and resolved. The Marquis de VHopitaVs Analysis of Infinite* - - - 31G CHAP. in. Extraordinary progress in the theory of maxima and minima. Dispute between the two Bernoullis on the problem of isoperimetrical figures - - 331 CHAP. IV. Solutions of various problems. Leibnitz invents the method of differencing de curva in curvam. Justi- fication of the marquis de VHopital. Newton's works. Account of some other geometricians - 345 CHAP. V. An Examination of the Claims of Leibnitz and Newton to the Invention of the Analysis of Infinites - 359 CHAP. VI; 'Continuation of the dispute'. War of problems between John Bernoulli and the english. Miscellaneous . .. - - 380 b CHAP. CHAP. VII. Continuation of the Progress of geometry. Solutions of various problems - - - - 391 CHAP. VIII. Problem of isochronous curves in resisting mediums, General reflections on problems of pure theory. Al- gebra of sines and cosines. Utility of methods of approximation, and in particular of infinite series 400 CHAP. IX. Continuation. Progress of the methods for resolving differential equations. New step in the problem of isoperimetrical figures. The integral calculus with partial differences - 406 CHAP. X. Of some works on analysis - - - 415 CHAP. xi. Progress of Mechanics -. * * 423 CHAP. XII. Progress of Hydrodynamics * 440 CHAP. XIII. Progress of Astronomy - - - -45-$ SECTION i. Practical Astronomy - SECTION ii. Physical Astronomy CHAP. XIV. Progress of Optics ? - T HISTORY Of MATHEMATICS, INTRODUCTION. A general mew of the mathematical sciences. Nations by whom they have been cultivated* THE term mathematics, implying from it's etymo- logy discipline, science, represents with justice and precision the high idea that we ought to form of what is signified by it. In fact mathematics are a methodical concatenation of principles, reasonings, and conclusions, always accompanied by certainty, as their truth is always evident : an advantage that particularly characterises accurate knowledge, and the true sciences, with which we must be careful not to associate metaphysical notions, conjectures, or even the strongest probabilities. The subjects of mathematics are the mensura- tion and comparison of magnitudes; for instance, numbers, distances, velocities, &c. They are di- vided into pure and mixed', what is -understood by mixed mathematics being sometimes called the phy~ weo-mathcniatical sciences. 3 Pur* Pure mathematics consider magnitude generally, simply, and abstractedly; whence they have exclu- sively the prerogative of being founded on the ele- mentary notions of quantity. This class compre- hends 1, Arithmetic, or the art of computation ; 2, Geometry, which teaches us to measure extension ; 3, Analysis, or the calculation of magnitudes in general ; and 4, mixed Geometry, or the combination ;, of common geometry with analysis. Mixed "mathematics borrow from physics one or >"' ;-riii)Fe' incontestable experiments, or suppose in bodies some principal and necessary quality ; and then, by 2. methodical and demonstrative chain of reasoning, they deduce, from the principle established, conclusion* as evident and certain, as those which pure mathe- matics draw immediately from axioms and defini- tions. To this class belong 1, Mechanics, or the science of the equilibrium and motion of solid bodies; 2, Hydrodynamics, in which the equilibrium and motion of fluids are considered ; 3, Astronomy, or the science of the motions of the celestial bodies ; 4, Optics, or the theory of the effects of light; 5, and lastly, Acoustics, or the theory of sound. I have here arranged the different parts of mathe- matics in that order which appears to me best cal-< ciliated for exhibiting at one view their reciprocal concatenation, in the state in which they are at pre* sent; but this order is not altogether analogous to their actual and historical developement. It is not possible to fix the origin of mathematics with precision ; though we are able to affirm, that it goes back to the remotest ages. When mankind, re- linquishing linquishing the wandering and savage life, united in societies ; and general laws or conventions had esta- blished it as a rule, that each should provide for his own subsistence, without seizing what was in the possession of another; want and convenience, the two great springs of industry, soon invented the most necessary arts. Huts were built; iron was forged; the land was divided; the course of the stars was observed. It was seen, that the earth yielded spontaneously, and in every season, various productions for the food of fome animals ; but that for others, of still greater utility, and in greater abund- ance, it required the assistance of a cultivation regu- lated by the seasons : thus the ground was sown, and the harvest reaped. All these observations, all these performances, at first extremely rude and unskilful, were con- nected with mathematics by a secret and unknown tie: though for a long time they had no rule or guide but experience and blind custom. The assiduous labour required in hunting, fishing, and the business of the field, did not allow men to ascend to general and abstruse ideas : the circle of their physical wants bounded that of their thoughts. By imperceptible degrees, several of them acquiring a sort of superfluity, either by superiour industry or abundant harvests, gave themselves up to that idle- ness, to which all animals have a natural propensity. Happiness they imagined was to be found in this state of indolence and repose ; a seductive illusion, which soon undeceives us, but to which we are at least in- debted for the first flights of the human understand- ing in those days, The languor of inaction, the 3 torment torrtient of wcarisomencss annexed to it, and the ac- tivity of the thinking principle which we carry within, us, snatched man from this disgraceful lethargy, and gave an impulse to that spirit of curiosity and re- search, which incessantly agitates us, and which de- mands sustenance no less imperiously than the body itself. Man then beheld with new eyes the magnifi- cent spectacle, which Nature exhibited on all sides to his senses and imagination : he learned to ex- amine things, and compare them together. Ideas acquired from physical objects were detached from them as it were, and transported to an intellectual world : orators, poets, painters arose : the phenomena of nature were studied with discriminating attention, and the mind was impressed with a desire to know the causes, by which they were produced. Geo- metry, confined at first to the art of measuring the fields, was extended to new purposes, and proposed to itself loftier and more difficult problems. Astro- nomy was enriched by regular observations, and by several instruments adapted to increase their number, arid to give them the requisite accuracy and con- nexion. Machines were invented, in which a skilful combination of wheels and levers was employed to raise or transport the heaviest loads : in a word, all parts of the mathematics successively advanced. Their progress would have been more rapid, if fana^ ticism and the insatiate love of power, while they ravaged the Earth, had not too frequently obscured the flame of genius for a long series of ages : but, as a fire concealed beneath the embers, it resumed it's lustre in happier times, and by degrees the edifice of science science arose. Let us hope, that posterity will feel the honourable ambition of pursuing the work, with- out being discouraged by the apprehension of neve? being able to complete the roof. The most general and best established opinion is, that mathematics began to acquire a certain solidity among the primitive chaldeans, and the primi- tive egyptians, that is to say the two most ancient people known, almost at the same period. Accord- ing to a permanent tradition, transmitted from age to age, the shepherds of Chaldea, dwelling under a clear sky, laid the foundations of astronomy, during the leisure of their quiet occupation. If their ob- servations were too imperfect, to serve as the basis of any theory, they at least afforded some general hints to the first astronomers, and saved them the trouble of some mistaken attempts. The magi, or priests of Egypt,, directed by the laws of their institution, to study and collect the secrets of nature, were become the depositaries and dispensers of all human knowledge. From all parts men came to consult them, and to acquire from them instruction. They would have deserved the un- bounded respect and gratitude of the world, if, COJL- tent to enlighten it, they had not sometimes endea- voured to impose upon it likewise, and to conceal the proud desire of sway under the sacred veil of reli- gion. Nations, like individuals, seek to swell their origin, and carry it backward to remote ages. The Chinese and hindoos are particularly accused of this national jnania, If we believe their own accounts, they were 3 tie the first inventors of all the sciences, and of all the arts. As they rest their pretensions more particu- larly on the antiquity of astronomy among them, I shall defer my examination of them, till I come to speak more at large of the progress of this science. With the mathematics of the ancients we are ac- quainted only through the writings of the greeks : and to estimate the instructions, which these derived from their intercourse with the magi, we possess not the necessary documents. Some authors have said, that Thales, in one of his visits to Memphis, taught the egyptians how to measure the height of the pyra- mids by the extent of their shadow, a proposition that ranks very low in the elements of geometry. If this were true, we must infer, that the egyptians trere but little versed in the science : but the fact is not probable, and it is the safest way to assert no- thing on the subject, since all the records of egyp- tian science perished with the alexandrian library. We ought simply to admit, that, if the egyptians were the first masters of the greeks, they were soon surpassed by their scholars. As soon as the mathe- matics began to take root in Greece, we see them shoot up with a strong and rapid growth, and suc- cessively enrich themselves with a number of impor- tant discoveries, in which the mutual connexion of principles and consequences marks the unity and continuance of the plan. The greeks became in some measure the preceptors of all other nations : they alone have had the glory of excelling in every branch, in the military art, poetry, oratory, , paint- ing, the accurate sciences, &c. Most of the illus- trious 7 tnous men assembled in the museum of Alexandria*, the centre of aits and sciences, were greeks by birth. All this grandeur had the fate of every thing human, it gradually decayed. Already the jealousy that reigned among the dif- ferent states of Greece had excited in it's bosom several bloody wars, fatal to it's political constitution. As long as the nation at large preserved it's morals, as long as it continued invariably attached to the principles of justice and moderation, it triumphed over it's foreign enemies. People came from distant countries to study it's laws and institutions. Weak- ened by intestine discord, it fell at length under the yoke, which the romans imposed on all the Earth; but while it yielded to the power of arms, it retained in a great measure it's superiority in the domains of genius. If Virgil and Cicero have equalled Homer and Demosthenes; if Herodotus, Thucydides, and Xeriophon, have been surpassed by Livy, Sallust, and Tacitus ; in two extensive regions, those of the fine arts and the accurate sciences, the greeks remain absolute masters. The ambition of the romans, ever active, ever young, was to enlarge their dominion abroad : at home, the eternal rivalry, which divided the senate and the tribunes of the people, from the expulsion of the kings to the fall of the republic, stimulated the minds of the romans, and produced a crowd of great orators, who were followed by a num- ber of great poets. Painting, sculpture, and archi- * The museum of Alexandria was founded by Ptolemy PhiladeU phus, king of Egypt, about 320 years before the christian era. The mathematics flourished in that city near ten, centuries. B 4 tecture tecture were far from having equal success at Rome. Yet we must acknowledge, that the work of Vitru- vius, on the subject of architecture, written in the time of Augustus, is a valuable record of curious in- formation respecting that art. In the accurate sci- ences, which require cool attention, silence, and profound meditation, the romans never went beyond mediocrity. Useless as means of attaining the chief offices in the state, they were the occupation of a few obscure individuals, remote from the turmoil of public affairs. The roman mathematicians were little more than translators or commentators of Archi- medes, Apollonius, &c. We can remark among them only a few learned astronomers under Augustus and his earliest successors : after this period every thing fell into decay. On the death of Theodosius, the division of the empire between his two sons, Arcadius and Honorius, having weakened that huge body, the western por- tion, long ravaged, dismembered, and at length sub- jugated by the barbarians, sunk into the profoundest ignorance ; while the schools of the East were wholly employed in wretched theological disputes. The ac- curate sciences had taken refuge in the museum of Alexandria, almost exclusively ; and there, destitute of support and encouragement, they could not fail to degenerate. Nevertheless, they still preserved, at least by tradition or imitation, that ancient and strict character, which had been stamped on them by the greeks. Of this asylum they were soon deprived. About the middle of the seventh century of the Christian era, era, the arabs, conducted by the immediate successors of Mohammed, spread carnage and devastation throughout the East: the museum of Alexandria was destroyed : artists and men of science perished or were dispersed. However, though the chain of mathematical dis- covery was broken by this fatal catastrophe, a few links remained, which this very nation of destroyers, softened by the charms of peace and idleness, strove to collect and unite afresh. In less than a century we find the arabs cultivating astronomy, of which they had before some general notions. This taste for a particular science gradually extended to all the branches of human knowledge. For the space of seven hundred years the mathematics flourished in. ail the countries that were subject to the dominion of the arabs, and afterward of the persians, when these two peopie became united. By the moors they were carried into Spain ; and some rays of them pe- netrated into Germany. The conquests of the turks brought back igno- rance and barbarism into the delightful countries, which the arabs inhabited. At the taking of Con- stantinople by Mohammed II, a persecution arose against artists and men of learning, by which many were destroyed : but some escaped by flight, and carried with them the remains of the mathematical sciences into Italy, France, Germany, and England ; countries in which, in Italy particularly, a taste for literature and the arts had already begun to take root, From id From this period every thing was changed: the human mind was regenerated in every part. Algebra, Geometry, Astronomy, proceeded with rapid steps : and at length, in the last thirty years of the seven- teenth century, the grand discovery of the method of fluxions was made. Here a new order of things, for which men could not have ventured to hope, took place in the accurate sciences. By the method of fluxions we have been put into possession of an infinite number of pro- blems, inacceffible to all the mathematicians of antiquity. Let us not forget, however, that these great men were our first masters : let us not ima- gine, that the moderns of Europe have excelled the greeks in genius : but let us be satisfied with saying, that, in consequence of the natural pro- gress of knowledge, they have surpassed them in science. In the arts of imagination, as poetry, elo- quence, painting, &c., perfection is the work of genius, not of time : and in this point of view, the only glory to which the moderns can pretend, is that of having equalled the ancients. But in the sciences the discoveries of ages are added to each other; they are disseminated by writing or print- ing; and at length a general mass of information is accumulated among a studious people, as it would be by an individual who should live for many centuries. Were Archimedes to return to this World, he must pursue a long course of study, ere he could place himself on a level with Newton*, though it is perhaps very difficult to decide, which of the two excelled the other in genius. The 11 The Chinese and hindoos partook not in this great movement made by the sciences, and in this respect they cannot enter into competition with the people of Europe. It appears, that the americans have never had any distinct notions of the mathematics. Before their communication with europeans, they were acquainted only with those mechanical arts, which are most ne- cessary to the wants of life : their minds never had any tendency to reflection. My design in this work is to give an historical ab- stract of the mathematical sciences, from their origin to the present day, and at the same time to honour the memories of those great men, by whom their limits have been extended. I shall not enter into systematic discussions, frequently founded on very dubious grounds ; and I shall avoid the formality of geometrical demonstrations, as I write chiefly for those readers, who add to a general taste for erudi- tion a true and steady desire of being acquainted with the progress of the human mind in the noblest exercise of it's faculties. Sometimes however I shall explain different methods sufficiently at large, to enable the professed mathematican to discover the demonstration of those conclusions, to which I must necessarily confine myself. If I cannot satisfy him entirely, I shall at least point out to him the sources, whence he may derive more ample instruc- tion. In the history of mathematics I remark four ages. The first exhibits in the commencement faint gleams of their origin, then their rapid progress among the 2 greeks, la greeks, and at length their languishing state till the destruction of the school of Alexandria. In the se- cond period they are revived and cultivated by the arabs, who carry these sciences with them into some of the countries of Europe. This reaches nearly to the end of the fifteenth century. Some time after this they are diffused, and make a rapid progress among all the nations in Europe of any consequence; and this third period brings us to the discovery of fiuxions, where the fourth and last period begins. These four periods will constitute the general divi- sions of this work. At first view it might seem, that for the sake of perspicuity I should go through each branch of the mathematics successively without interruption : but this method, applied indiscriminately to every part and every age, has some inconveniences. The dif- ferent branches of the mathematics have been formed and developed by degrees, and frequently one has promoted another. A proposition in mechanics has given birth to a complete theory of geometry ; and then it would be impoffible to give an account of the one without explaining the other, and thus being led into details, often prolix, and even foreign to the true and principal object. Besides, a disagreeable void in the general picture, or too striking a dispn> portion in the parts, would sometimes occur ; for all the sciences have not advanced with equal pace, some appearing at times stationary, while others have been making a rapid progress. These observa- tions are more particularly just with regard to the se- cond and fourth ages of mathematics: and frequent instances instances of them will be seen, when we come to the application of fluxions to mechanics and astronomy. The first age is that, in which the thread of each science is most uniform and distinct," so that every part of the mathematics may be kept separate. Of this advantage I have availed myself as much as possible ; but in the following periods I have not been able completely to preserve the same order. I must request the reader's assent, therefore, to a plan, which the nature of the subject appeare^ to me to exact It is superfluous to make another observation, which will naturally suggest itself. Frequently it will be seen, that the historical documents, necessary to form a complete and uninterrupted narrative, are very rude, or very defective : on the other hand, or- nament and fiction are incongruous to the strictness of my subject. In such sterile parts, therefore, I can hope for attention only from those readers, who find treasures even in the ruins of the edifice of science, CHAP. PERIOD THE FIRST. STATE OF THE MATHEMATICS, FROM THEIR ORIGIN TO THE DESTRUCTION OF THE ALEXANDRIA^ SCHOOL. CHAP. I. Origin and Progress of Arithmetic. THERE is no idea more simple, or more easy to con- ceive, than that of number or multitude. As soon as the understanding of a child begins to unfold itself, he can count his fingers, the trees around him, and the other objects that are before his eyes. These operations are performed at first without order, with- out method, and by the help of memory alone : but means of extending them, and of subjecting them to a kind of regular form, are soon found. Different as the objects to be counted might be, as the same method was always pursued, it was easily perceived, that their nature might be left out of con- sideration ; and to represent them, general symbols were invented, which afterward assumed particular values, adapted to each question, that was to be re- solved. Thus for instance little balls were employed, strung 15 strung together like the beads of a rosary, or knots in a cord : each ball denoted a sheep, or a tree ; and the whole assemblage of balls, the flock, or the grove. The invention of writing advanced the art of nu- meration a step farther. On a table covered with dust characters chosen arbitrarily to express numbers were traced, and thus calculations, to a certain extent, were capable of being performed. All nations, if we except the ancient Chinese, and an obscure tribe mentioned by Aristotle, distributed numbers into periods, composed each of ten units. This custom can scarcely be attributed to any thing but our habit of counting in childhood by the fingers, which, with some very rare exceptions, are uniformly ten. The ancients equally agreed in representing numbers by the letters of the alphabet ; and the dif- ferent periods of tens were distinguished by accents put over the numeral letters, as was the practice of the greeks, or by different combinations of the nu- merals, as was done by the romans. All these me* thods of notation, particularly that of the romans, were very complex and embarrassing when calcula- tions of any extent were to be performed. Strabo, who lived in the reign of Augustus, says in his Geography, that in his time the invention of arithmetic, as well as that of writing, was ascribed to the phenicians. The establishment of this opinion would find the less difficulty, because the phenicians, being the most ancient mercantile nation, must natu- rally have improved a science, of which they were making constant use : but the principles of arithmetic were known to the egyptians and chaldeans long be- fore 16 fore we hear any thing of the phenicians, who prtf- bably acquired them from their egyptian neighbours. The mathematics had already taken root in Greece, when Thales appeared : but the impulse he gave them constitutes the era, from which we begin to reckon their real advancement. [A. c. 640,] We know not whether this philosopher made any particular discoveries in arithmetic : his inclination led him principally to the study of geometry, physics, and astronomy. He travelled a long while in Egypt and India. Enriched with the knowledge he had acquired in foreign countries, and which he improved by his own reflections, he returned to Miletus, the place of his birth, and there founded the celebrated ionian school, which divided itself into several branches or sects, embracing every part of philosophy, and spreading themselves through several of the grecian cities. Some time after, Pythagoras of Samos rendered himself illustrious by his vast erudition, and the singularity of his philosophical opinions. Never man more ardently sought glory; never man more deserved it, or raised himself to a higher pitch of reputation. He had all the ambition of a conqueror : full of zeal for extending the empire of the sciences, and not contented with having instructed his coun- trymen, he went to Italy, and founded a school, which in a short time acquired such celebrity, that he reckoned princes and legislators among his disciples. That almost every part of the mathematics has im- portant obligations to him, will be seen as each come* under consideration. The 17 The combinations of numbers constituted one of the principal objects of his researches : and all anti- quity testifies, that he had carried them to the highest degree* He clothed his philosophy in emblems, which, necessarily differing from the ideas they were intended to represent, became still more obscure in. process of time, and occasioned whimsical systems to be attributed to him, which we can hardly suppose to have been the productions of so great a genius. According to some authors, Pythagoras is at the head of the inventors of the ancient kabbala : he attached several mysterious virtues to numbers, and swore by nothing but the number four, which was to him the number of numbers. In the number three likewise he discovered various marvellous properties, and said, that a man perfectly skilled in arithmetic possessed the sovereign good, &c. But if he did advance such propositions, were they to be taken strictly according to the letter? Is it not more probable, either that his words were erroneously reported, or that they in- cluded allegories, with the meaning of which we are unacquainted ? This conjecture appears to be the better founded, as, according to other authors, Pythagoras never hav- ing written any thing on the different subjects of phi- losophy, his doctrines were preserved for a long time solely in his own family and among his scholars ; and afterward Plato, with other philosophers, committed them to writing, and corrupted them from vague and confused tradition. On this obscure question, how- ever, which is but little interesting in the present day, I shall say no more. Of all the real or supposed dis- c coveiies 16 fore we hear any thing of the phenicians, who bably acquired them from their egyptian neighbours. The mathematics had already taken root in Greece, when Thales appeared : but the impulse he gave them constitutes the era, from which we begin to reckon their real advancement. [A. c. 640,] We know not whether this philosopher made any particular discoveries in arithmetic : his inclination led him principally to the study of geometry, physics, and astronomy. He travelled a long while in Egypt and India. Enriched with the knowledge he had acquired in foreign countries, and which he improved by his own reflections, he returned to Miletus, the place of his birth, and there founded the celebrated ionian school, which divided itself into several branches or sects, embracing every part of philosophy, and spreading themselves through several of the grecian cities. Some time after, Pythagoras of Samos rendered himself illustrious by his vast erudition, and the singularity of his philosophical opinions. Never man more ardently sought glory; never man more deserved it, or raised himself to a higher pitch of reputation. He had all the ambition of a conqueror: full of zeal for extending the empire of the sciences, and not contented with having instructed his coun- trymen, he went to Italy, and founded a school, which in a short time acquired such celebrity, that he reckoned princes and legislators among his disciples. That almost every part of the mathematics has im- portant obligations to him, will be seen as each come*, under consideration. The "v 17 The combinations of numbers constituted one of the principal objects of his researches : and all anti- quity testifies, that he had carried them to the highest degree. He clothed his philosophy in emblems, which, necessarily differing from the ideas they were intended to represent, became still more obscure hi process of time, and occasioned whimsical systems to be attributed to him, which we can hardly suppose to have been the productions of so great a genius. According to some authors, Pythagoras is at the head of the inventors of the ancient kabbala : he attached several mysterious virtues to numbers, and swore by nothing but the number four, which was to him the number of numbers. In the number three likewise he discovered various marvellous properties, and said, that a man perfectly skilled in arithmetic possessed the sovereign good, &c. But if he did advance such propositions, were they to be taken strictly according to the letter? Is it not more probable, either that his words were erroneously reported, or that they in- cluded allegories, with the meaning of which we are unacquainted ? This conjecture appears to be the better founded, as, according to other authors, Pythagoras never hav- ing written any thing on the different subjects of phi- losophy, his doctrines were preserved for a long time solely in his own family and among his scholars ; and afterward Plato, with other philosophers, committed them to writing, and corrupted them from vague and confused tradition. On this obscure question, how- ever, which is but little interesting in the present day, I shall say no more. Of all the real or supposed dis- c coveries 20 as would lead to absurd results : but in the second, the choice of some unknown values constitutes of it- self an indeterminate problem, which is to be resolved only by a particular art. It was in this art that Dio- phantus displayed a sagacity truly original. If, for example, the following questions were proposed : to divide a square number into two other square num- bers ; to find two numbers, the sum of which should be in a given ratio to the sum of their squares ; to find two square numbers, the difference of which should be a square : nothing could be more easy than to resolve them, if we were allowed to employ any kind of numbers. But if it were made a condition, that the numbers sought should be rational, and frac- tions be excluded, the solution would require some address. Diophantus found the method of subjecting all questions of this nature to certain rules, exempt from every kind of conjectural proceeding. His me- thods bear an evident analogy to those we now employ for the resolution of equations of the first and second order, and hence some authors have taken occasion, to ascribe to him the invention of algebra. He wrote thirteen books of arithmetic, the first six of which have reached us : the rest are lost, if a seventh, which is found in some editions of Diophantus, be not his work. This seventh book contains some learned investigations of the properties of figurate numbers. This writer had a number of interpreters among the ancients, but most of their works are loft. Of these we regret the commentary of the learned Hy- patia. A. D. 410. The talents, virtues, and misfor- tunes 4 tunes of this illustrious victim of fanaticism have a claim to the homage of posterity, and we cannot dispense with paying her this tribute. The philosopher Theon, her father, had taken such pains to instruct her, and she made so considerable a progress in a short space of time, that she was chosen, when very young, to teach mathematics in the school of Alexandria. All historians agree in saying, that in Hypatia^ personal beauty was united with uncommon modesty, purity of morals, and consummate pru- dence. These advantages procured her great respect at Alexandria, particularly from Orestes, the governor of that city. Some wretched theological disputes having excited a bitter dissension between Orestes and St. Cyril, the monks of St. Cyril's faction stirred up the people to massacre Hypatia, as the author of the troubles, in consequence of the advice she gave the governor. ( This action,' says the historian So- crates, ' brought great reproach upon Cyril, and the church of Alexandria ; for such acts of violence are totally inconsistent with Christianity.' Fleury, a man of justice and moderation, but perhaps too much at- tached to the dogmas of an intolerant religion, does not express with sufficient energy the horrour, with which such an abominable crime should have inspired him. c 3 CHAP. CHAP. II. Origin and Progress of Geometry. DIFFERENT origins, more or less ancient, are ascribed to geometry. Most authors assign Egypt for it's birthplace; as Herodotus for example, the first who began to write history in prose, A. c. 450. ; for in the remotest antiquity, the memory of past events was preserved only in a feeble and muti- lated state in some rude songs ; and afterward it was confounded with fiction in the poems of Hesiod and Homer, where every thing else was sacrificed to the embellishment of the subject. We will recite the ac- count, which Herodotus gives of what he himself learned respecting it at Thebes and at Memphis. * I was told,' says he, ' that Sesostris had divided Egypt among all his subjects, and that he had given each an equal quantity of land, on condition of pay- ing annually a proportionate tribute. If the allot- ment of any one were diminished by the river, he repaired to the king, and related what had befallen his land. The king then sent to the spot, and caused his land to be measured, that he might know what diminution it had undergone, and require a tri- bute only in proportion to what remained. I believe,' adds Herodotus, ' that here geometry took it's birth, and that hence it was transmitted to the greeks.' In this passage we perceive two distinct things : the account of a verification depending on geometry, and 23 and the private opinion of Herodotus respecting the origin of this science. If, as many chronologists suppose, Sesostris be the same with Shishak, who made war on Rehoboam, the sou of Solomon, it would follow from the opinion of Herodotus, that the origin of geometry preceded the Christian era about a thousand years only : but it may be carried much higher; for the measuring of the fields, di- rected by Sesostris, is not only far from fixing pre- cisely the origin of geometry, but even appears to indicate, that the science must have made some pro- gress at that time. If we were inclined to indulge in frivolous conjec- tures, we should carry back the origin of geometry to the invention of the rule, square, and com- passes, since it makes the greatest use of these in- struments in practice : but the same argument of their use should rather lead to the supposition, that they were invented at the commencement of society, and that the invention was prompted by simple ne- cessity, without the aid of any theory, for the pur- pose of constructing huts or houses. If we confine ourselves, in beginning this abstract of the history of geometry, to the time when it assumes, at least to us, the character of a real science, we shall at once transport ourselves to Greece, and the age of Thales. Whether this philosopher taught the egyptians, or learned from them, the method of measuring the height of the pyramids of Memphis by the extent of their shadows, we discover, that he was versed in the theory and practice of geometry. A. c. 40. All c 4 the the ancient writers indeed speak of him as a very learned geometrician. To him is ascribed the first employment of the circumference of a circle for the measure of angles. No doubt he made other disco- veries in geometry, now lost, or confounded among those, which have been collected and transmitted to posterity by the authors of elementary works. He possessed an ample share of knowledge in every branch of mathematics and physics, as we have already observed. In astronomy he will appear again before us with distinction. The name of Pythagoras is rendered immortal in the annals of geometry, by the discovery which he made, that the square of the hypothenuse of a rec- tangled triangle is equal to the sum of the squares of the other two sides. A. c. 590. Some authors re- late, that he was so transported with joy and grati- tude to the gods for having inspired him with it, that he sacrificed to them a hundred oxen : but we can hardly reconcile this hecatomb with the moderate for- tune the philosopher possessed, still less with his reli- gious opinions concerning the transmigration of souls. But be this as it may, never had enthusiasm a better foundation. This problem of Pythagoras ranks in the first class of geometrical truths, both from the singularity of it's result, and the number and importance of the cases to which it is applica- ble in every branch of mathematics. The author himself derived from it this consequence, that the diagonal of a square is incommensurate to it's side : it led also to the discovery of several ge- neral 25 neral properties of other incommensurate lines or numbers. In the long series of grecian philosophers, which extends from Thales and Pythagoras to the destruc- tion of the Alexandrian school, there is scarcely one by whom the mathematics were neglected. Astro- nomy is in general the science on which they were chiefly occupied ;* but the most celebrated of them applied themselves to geometry as the principal, without which all the rest must remain lifeless and inactive. The propositions, which constitute the bulk of what we now call elementary geometry, were almost all invented by the philosophers of Greece. One of the most ancient of these geometricians, mentioned after Thales and Pythagoras, is (Enqpidus of Chios, the author of some very simple problems, as of letting fall a perpendicular upon a right line from a given point, making an angle equal to ano- ther angle, dividing an angle into two equal parts, &c. A. c. 4 80. Zen odor us, his contemporary, and the first of the ancients of whose geometrical writings any have readied us, one being preserved by Theon in his Commentary on Ptolemy, went farther. He showed the falsity of the opinions then entertained, that figures with equal circumferences must have equal areas. The demonstration of this was not easy to discover, and proves that geometry had then niade considerable progress. The ingenious theory of the regular bodies originated about the same time in. the pythagorean school, Hippocrates Hippocrates of Chios distinguished himself by the quadrature of the famous lunulce of the circle, which bear his name. A. c. 450. Having described three semicircles on the three sides of a rightangled isos- celes triangle as diameters, that on the hypothenuse being in the same direction as the others, he found, that the sum of the areas of the two equal Junes, comprised between the two quadrants answering to the liypothenuse and the semicircles answering to the other two sides, was equal to the area of the triangle. This is the first instance of a curvilinear space being found equal to a rectilinear, and it has been extended to other more abstruse and curious quadratures, in proportion as geometry has improved. The attainments of Hippocrates of Chios in geo- metry were very extensive. He wrote Elements of Geometry, much esteemed in his time, though other works of the same kind, particularly Euclid's, have occasioned his to be lost and forgotten. He appeared witli honour in the Jist of geometricians, who at- tempted to solve the celebrated problem of the dupli- cation of the cube, which at that period began to be pursued with ardour. The object of this problem was to construct a cube, that should be double a given cube ; not in respect to it's sides, about which there could be no question ; or even it's superficies, for this could have been easily accomplished by the geometry of that time ; but in it's solidity, so that it's weight should be double that of the other, supposing them both to be made of one liomogeneal substance. And it was to be re- solved 27 solved without employing any instruments beside the rule and compasses : for, by the ancients, no opera- tions were considered as geometrical, unless performed by means of these two instruments alone ; those that required others being called mechanical. According to an old tradition generally spread through Greece, a public calamity, in which reli- gion was concerned, gave rise to this research. It was said, that, Apollo having afflicted the athenians with a dreadful pestilence, to revenge an affront he had received from them, the oracle of the temple of Delos, being consulted on the means of appeasing his wrath, answered : Double the altar. The altar of Apollo at Athens, to which the oracle alluded, was a perfect cube ; and the problem was immediately pro- posed to all the geometricians of Greece. The priests, who never forget their own interests, added a condition, which they represented as a religious duty, but which happily did not increase the diffi- culty of the problem : they required the material of the new altar to be gold. The question at first sight appeared easy ; but this mistake was soon corrected, and all the sagacity of the geometricians of Greece was baffled by it. On turning the problem every way, it was per- ceived, and the discovery is attributed to Hippocrates of Chios, that if two geometrical mean proportional lines could be inserted between the side of the cube given and the double of this side, the first of these two lines would be the side of the cube sought. This new point of view revived for a moment the hope of accomplishing the solution by means of the rule and compasses ; 28 compasses : but the difficulty was only disguised ; it had merely changed it's form : thus it was still insur- mountable, and the geometricians, tired with the labour they had already exerted on the problem, let it sleep for a time. Still, however, geometry advanced. A. c. 390, Plato cultivated it with care, and acquired profound knowledge in the science. It is true we have no work of his written expressly on the subject : but we see by various passages in his other works, that he was master of it ; and the ancient historians have trans- mitted to us the results of several discoveries, with which he enriched it. He placed it in the first rank of mental acquirements, and made it the principal object of the instructions he gave his scholars. He had written over the door of his school : * Let no one enter here, who is ignorant of geometry.' The problem of doubling the cube could not fail of en- gaging his attention. Having attempted in vain to solve it with the rule and compasses^ he invented, for the purpose of rinding the two mean propor- tionals, an instrument composed of two rules, one of which moved in the grooves of two arms at right angles with the other, so as always to continue pa- rallel with it But this solution was of the me- chanical kind, and did not satisfy the wish of geo- metricians. He was more fortunate in another speculation of a kind entirely new. Before his time the circle was the sole curve considered in geometry: he introduced into it the theory of the conic sections, or those ce- lebrated curves, which are formed on the surface of a cone. cone, when cut by a plane in different directions. On attentively examining the generation of these curves, he discovered several properties of them. These first notions, being spread through his school, germinated there rapidly. His principal scholars or friends, Aristeus, Eudoxus, M^nechmus, Dinostra- tus, &c. penetrated deeply into this branch of geome- try. In a short time it was so extended as to form a distinct part of the science, of a more exalted order than the common geometry, whence it derived the name of the higher or sublime geometry. Some other ancient curves, which I shall have occasion to no- tice, were afterward comprised under the same de- nomination. Aristeus composed five books on conic sections, of which the ancients have spoken with the greatest eulogies, but unfortunately they have not reached us. A. c. 380. Of Menechmus we have two learned applications of the same theory to the pro- blem of doubling the cube. The properties of the conic sections, and those of geometrical progressions, led him to remark, that, on constructing, conformably to the conditions of the problem, two conic sections, which should intersect each other, the two ordinates corresponding to the point of intersection might re- present the two mean proportionals. Hence he framed two solutions : in the first he constructed two parabolas, having one common summit, with their axes perpendicular to each other, and for their respective parameters the side of the given cube and the double of that side : then the two ordinates, drawn to the point of intersection of the two curves, are the two mean 50 mean proportionals sought. The second solution proceeds by the intersection of a parabola and an equilateral hyperbola : the parabola has for it's para- meter the side of the given cube, or the double of this side ; it's summit is the centre, and it's axis one of the assymptotes of the equilateral hyperbola ; and the power of the hyperbola is the product of the side of the given cube by the double of that side. Lastly the ordinates of the two curves, drawn to the point of intersection, are the two mean proportionals re- quired. The reader who is tolerably versed in geo- metry will make out the demonstrations of these theorems without difficulty. Thus it appears, that, if we possessed the means of describing conic sections with one continued mo- tion, and in as simple a manner as we trace a circle with the compasses, the solutions of Menechmus would have all the .advantages of geometrical con- structions, in the sense which the ancients applied to the term. But there exists no instrument for de- scribing the conic sections in this manner. These so- lutions, therefore, do not fulfil the desired purpose in practice, though they are perfect in theory, and must be considered as an effort of inventive genius. It was afterward found, that the same end might be attained by the intersection of a circle and a para- bola; an easy simplification of the problem, which detracts nothing from the honour due to Me- nechmus. This discovery is so much the more remarkable, as it has been the source of the celebrated theory of loci geometrici t of which ancient and modern geo- metricians have made so many important applica- tions. 31 tions. Let us add, that the method of Menechmus includes likewise the germe of geometrical analysis, or of that art, by which, comsidering a problem as solved, and treating the unknown quantities as known, we proceed from reasoning to reasoning, from consequence to consequence, till we obtain an expreffion, which we may call the geometrical trans- lation of all the conditions of the problem. This art is not algebra ; but algebra lends it important assist- ance : and in this respect the moderns have a great advantage over the ancients, though these became versed in geometrical analysis after the solutions given by Menechmus. The problem of the trisection of an angle, which is of the same nature with that of doubling the cube, was equally agitated in the school of Plato. Without attaining it's solution by means of the rule and compasses, it was reduced at least to a very simple and very curious proposition. This consists in drawing a right line from a given point to the semi- periphery of a circle, which line shall cut this peri- phery, and the prolongation of the diameter that forms it's base, so that the part of the line comprised between the two points of intersection shall be equal to the radius : a result which gives rise to several easy con- structions. The intersection of the conic sections is applied to this problem likewise, as it was by Menech- mus to the duplication of the cube. According to the modern methods, each of these two problems leads to an equation of the third order, with this difference, that the equation relative to the duplication m duplication of the cube has but one real root, whits that relating to the trisection of the angle has three. Most of the ancient geometricians were so possessed with the hope of resolving these problems by means of the rule and compasses, that they could not bring themselves to give it up. They made many fruitless attempts ; and this eagerness became a kind of epi- demic disease, which has been transmitted from age to age down to the present day. But it must find an end ; and in fact it was relinquished by those who kept pace with the progress of mathematics, when, in modern days, algebra began to be applied to geo- metry. At present, the disease is incurable in those, who attack these questions with the weapons of the ancients ; for, as they are ignorant of the present state of the sciences, there are no means of curing them. Though the ancient geometricians, of whom I have just spoken, did not attain their principal ob- ject, their researches were useful in other respects : ge- ometry is indebted to them for several new theories, and some ingenious instruments for solving the two pro- blems in question, so as to approximate the truth, and more than sufficient for practical purposes. Most of these methods are lost : but we have those of four illustrious geometricians, Dinostratus, Nicomedes, Pappus, and Diocles, who deserve to be mentioned with honour. The first was of the school of Plato, and contemporary with Menechmus, of whom he is even supposed to have been the brother : the other three flourished in the school of Alexandria. Dinostratus 33 Dinostratus invented a curve, which would have possessed the double advantage of giving the trisec- tion or multiplication of an angle, and the quadra- ture of the circle, whence it derived the appellation of quadrat rLc 9 if it could have been described with one continued motion by means of the rule and com- passes. It is formed by the intersection of the radii. of a quadrant with a rule, which is made to move uniformly and parallel to one of the extreme radii of the quadrant : but it is of the number of mechanical curves, and does not rigorously fulfil either of the objects for which it was designed. The conchoid of Nicomedes is a geometrical curve, which applies equally to both problems. A. c. 280. It is generally constructed by fixing a rule on a table, and revolving round one of it's extremities another rule furnished with two points, which are kept constantly equidistant from each other: one of these points traverses the fixed rule, and the other de- scribes the curve. This mechanism is susceptible of several variations. The position of the polar axis, and the distance of the two movable points, are deter- mined by the conditions of whichever of the two problems is to be solved. Newton, in an appendix to his Arithmetic, passes the highest encomium on the invention of Nicomedes; he prefers it for the geometrical construction of determinate equations of the third or fourth order to the methods derived from the intersections of the conic sections. Pappus, in his Mathematical Collections^ proposes an ingenious method for finding the two mean propor- tionals in the problem of doubling the cube, or rnul- D tiplying 34 tiplying it in general. A. c. 450. Of the two ex- treme lines he forms the two sides of a rectangular tri- angle; and from the summit of the right angle, taking the longest side as the radius, he describes a semicircle, the diameter of which is consequently double that side: then from the two extremities of the diameter he draws two indefinite right lines, one of which has the same direction as the hypothenuse, the other cuts this pro- duced, as also the shortest side of the triangle pro- duced, and the semiperiphery : and he orders it so, that the middlemost of these three points of intersec- tion is equidistant from the other two. The distance from this middle point to the centre will thence the greater of the two mean proportionals required. This method, it is obvious, requires a proceeding by supposition, which is liable to some uncertainty. A. c. 460, Diocles improved it by means of the curve, called the cissoid, which bears his name. This curve *is constructed by describing a semicircle on the double of the greatest extreme line as a diameter ; raising on one of the extremities of the diameter an indefi- nite perpendicular which serves as a directrix ; draw- ing from the other extremity an infinite number of "transverse lines cutting the semiperiphery and the directrix ; and taking on each transverse line a point, the distance of which from the commencement of the line rs equal to the portion comprised between *tlic directrix and the semiperiphery. This series of points forms the cissoid. The rectangled triangle of "Pappus is then constructed, and the cissoid cuts the line produced from thd hypothenuse in a point, through which is to pass the transverse line, that de- termines, 35 termines, on the line produced from the shortest side pf the triangle, the middle point of Pappus. I must now turn back, and resume the history of geometry from a little after the time of Plato. In proportion as this science was extended, partir cular treatises appeared from time to time, in which all the known propositions were collected and ar- ranged in systematic order. Such ' was the object proposed by Euclid, a geometrician of the alexan- drian school, in his celebrated Elements. A. c. 300. This work is divided into fifteen books, eleven of which belong to pure geometry : the other four treat of proportions in general, and of the principal pro- perties of commensurate and incommensurate num- bers. Though the theory of the conic sections was considerably advanced at the time when Euclid wrote, he has not mentioned them, as his object then was simply elementary geometry : but it appears by his data, and by some fragments of other works, that he was well versed in their theory. No book of science ever met with success com- parable with that of Euclid's Elements. They have been taught exclusively for several centuries in every mathematical school, and translated and com- mented upon in all languages : a certain proof of their excellence. The ancient geometricians sought the utmost strictness in their demonstrations. From a small number of axioms, or selfevident propositions, they deduced in an incontestable manner the truth of the secondary propositions, which they aimed to esta- blish, without indulging themselves in any of those . D 2 suppo- 36 suppositions, sometifnes savouring of boldness, which the moderns occasionally employ to simplify their reasoning and the consequences deduced from them. One of their grand principles was the reductio ad ab- surdum : they concluded that two ratios must be equal, when they had proved, that, on the supposi- tion of their being unequal, one must be at the same time both greater and less than the other, which im- plies a contradiction. For instance, were it required to demonstrate, that the circumferences of two circles are as their diameters ; they would have imagined, that they were offending against the strictness of geo- metry, if, after having proved, that the perimeters of two regular and similar polygons, inscribed in two circles, were always as the diameters, whatever might be the number of the sides of the polygons, they had finished with confounding the peripheries of the circles and the perimeters of the polygons, and consequently the two ratios, by multiplying the number of sides of the polygons to infinity. Their mode was less diffuse. They began by establiming, that if we continue to subdivide into two equal parts each of the arcs subtended by the sides of the poly- gons, the perimeters of the new polygons, still pro- portional to the diameters, would continually ap- proach the peripheries of the circles, so as ultimately not to difter from them by any assignable quantity. They then showed, that we could not suppose the ratio of the two peripheries to be greater or less than that of the perimeters of the two last rectilinear poly- or of the diameters, without an absurdity : whence 37 whence they concluded, that the two ratios \vere the same. Euclid has confined himself, in his Elements, to this strict method, sanctioned by the unanimous, assent of the ancient geometrickins. But for this very reason his demonstrations are sometimes long, ', indirect, complicated, and difficult to be followed by a beginner* This has induced several moderns, in the editions they have given us of Euclid's Elements, to employ more simple and easy demonstrations than those of the author. To this inconvenience attached to the ancient methods perhaps we must ascribe the difficulties, which Ptolemy Philadelphia, king of Egypt, in other respects a man of understanding, experienced in studying the mathematics. Weary of the extreme attention it required, he one day afked Euclid, whether he could not make the way smoother for him : to which the philosophical geometrician in- genuously answered : ' no sire ; there is no royal road to geometry/ In Euclid's Elements we find all the principles neces- sary for determining the perimeters and areas of right lined polygons, and the superficies and solidities of po- lyedra terminated by plane rectilinear faces : it wants, however, the method of measuring the circumference of the circle, though the author has entered into se- veral particulars respecting the properties of this curve, and it's different uses in determining and com- paring angles. It is true, he demonstrates, that the peripheries of two circles are as their diameters ; that their areas are as the squares of their diameters; that a cylinder is equal to the product of it's base multi- D 3 plied plied by it's altitude ; and that a cone is equal to one third of a cylinder having the same altitude and hase: but all these propositions are incomplete or steril, while we remain ignorant of the length of the peri- phery of the circle relative to it's radius or diameter. The knowledge of this, if we possessed it, would enable us to find the area of the circle, or in other words it's quadrature. In fact we see from Euclid himself, that by inscribing in a circle regular poly- gons, the number of the sides of which go on con- tinually increasing ad infinitum, the area of the circle is equal to that of a triangle, the base of which would be the periphery drawn out into a straight line, and it's altitude the radius ; whence it follows, that we should have a square equal to the area of thfc circle, by taking a mean geometrical proportional between the periphery and half the radius: but Euclid has not given this necessary supplement. Archimedes, the greatest geometrician of antiquity, was the first who discovered the ratio which the peri- phery of the circle bears to it's diameter ; not indeed with geometrical strictness, but by a method of approxi-* ihation, admirable in it's kind, and the source and model of all the approximate quadratures of curvili- near spaces, when we are destitute of the means of determining them exactly, and without any omis- ion. A. c. 250. Having found, that, if we inscribe a regular po- lygon in a circle, and circumscribe another of the same number of sides about it, the periphery of the circle, being between their perimeters, will be greater than the one, and less than the other \ and by going on 39 on continually increasing the number of their sides, the difference will ultimately become less than any assignable quantity. H? supposed the first two poly- gons had six sides each, the second twelve, and thus continuing the geometrical progression to the number of ninety-six, he perceived, that at this term, at which he stopped, the perimeters of the two polygons approached pretty near to equality. He took, in con- sequence, the arithmetical mean between them as the, approximate value of the periphery of the circle : and the conclusion of his calculation was, that, if the diameter were represented by seven, the circum- ference would be comprised between the numbers twenty-one and twenty-two, but much nearer the latr ter than the former. The same method, if carried farther, gives the ratio of the circumference to the diameter more accurately, but that of seven to twen- ty-two is sufficient in practical problems, which dp not require very great precision. Since the time of Archimedes a number of useless attempts have been made to assign the precise ratio of the circumference to the diameter: but adepts in geometry consider this problem, if not absolutely ir- resolvable in itself, at least as incapable of a perfect solution by any of the means the present state of ge- ometry affords, If the hope of resolving it could be conceived for a moment, it was at the discovery of * fluxions; for this method has rectified and squared curves, by which the ancient geometry was, baffled : but the circle has resisted it, and there are now ncftie but beginners., or persons altogether ignorant of p 4 geometxy, 40 geometry, who seek for the absolute and rigorous quadrature of the circle. The numerous discoveries, with which Archimedes enriched the mathematics, have placed him among the small number of those rare and inventive geniuses, who from time to time have given a great impulse to the whole body of science. Beside his work of the Dimension of the Circle, of which I have just given an abstract, we have his treatises of the Sphere and Cylinder, of Conoids and Spheroids, of Spiral Lines [De Spiralibus S$ Helicibus}, of the Quadrature of the Parabola, of Equiponderants, of Bodies floating on a Fluid, and his Arenarius, or numbering of the Sand, with his Lemmata. In all these we aclmire the force of his genius.' The titles of these works sufficiently indicate their subjects, and I shall not give an analysis of them here, but content myself with relating some of their principal results. In the treatise on the Sphere and Cylinder Archi- medes determines the ratio both of the superficies and solidities of these bodies to each other. He shows, that the superficies of the sphere is equal to the con- vex superficies of the circumscribed cylinder, or, which is the same thing, to the quadruple of one of it's great circles : that the superficies of a spherical segment is equal to the corresponding cylindrical su- perficies, or to that of- the circle which has for it's radius the chord drawn from the summit to a point in the circumference of the base : that the solidity of the sphere is two thirds that of* the cylinder : &c. The treatise on Conoids contains several properties of solids produced by the revolution of the conic sections round 41 round their axes. Archimedes compares these solids with one another ; and determines their ratios to the cylinder and the cone of the same base and altitude: he also demonstrates, for example, that the solidity of the paraboloid is only half that of the circumscribed cylinder : &c. In his work on the Quadrature of the Parabola he proves in two equally ingenious modes, that the area of the parabola is two thirds of that of the circumscribed rectangle : which is the first instance of an absolute and rigorous quadrature of a space comprised between right lines and a curve. The treatise on Spirals is built on a very profound geometry. Archimedes compares the length of these curves with arcs of corresponding circles, and the spaces they include with circular spaces; he also draws tangents and perpendiculars to them, &c. All these researches, so easy, since the invention of fluxion?, were extremely difficult to the geo- metry of those times. We must not be surprised therefore, if the demonstrations of Archimedes be somewhat complex ; on the contrary, we ought to admire that force of intellect, which was requisite to retain such a great number of propositions, and pre- serve the chain unbroken. This abstract is sufficient to give a general idea of the geometrical discoveries of Archimedes : to which I shall add, that he extended and clearly demon- strated the use of geometrical analysis, the principles of which were given by the school of Plato. We shall also see other proofs of the genius of this great man, when I come to speak of mechanics, hydro- statics, and optics, Archimedes 42 Archimedes was a lover of glory; not of that vain phantom, which mediocrity pursues, yet cannot reach ; but of solid glory, of that reputation, of that respect, which are due to the man of genius, who enlarges the limits of science. He desired, when he was dying, that a sphere inscribed in a cylinder might be engraved on his tomb, to perpetuate the memory of his most brilliant discovery. His desire was obeyed, but the Sicilians, his countrymen, having their minds turned on objects very different from geometry, soon forgot the man, who was their chief honour in the eyes of posterity. Two hundred years after his death, Cicero, being then quaestor in Sicily, gave Archimedes to the light, as he himself expresses it, a second time. Unable to learn from the Sicilians the place of his tomb, he sought for it by the symbol, which I have just mentioned, and six verses in greek inscribed on it's base. After much labour it was at length discovered overgrown with thorns in a field near Syracuse; and the Sicilians blushed for their ignorance and ingratitude. Fifty years had scarcely elapsed after the death of Archimedes, when another geometrician appeared, by whom he was almost equalled, and who is at least beyond dispute the second of the ancient geome- tricians. A. c. 200. This was Apollonius, born at Perga in Pamphilia, whence he is called Apollonius Pergseus. His contemporaries styled him the great geometrician : and posterity has confirmed this ho- nourable title, without detracting from the merit of Archimedes, to whom it assigns the first rank. Apollonius 43 Apollonius had composed a great number of works on the higher geometry of his times, most of which are lost, or exist only in fragments ; but w$ have almost the whole of his treatise on Conic Sections, which alone is sufficient to justify the high repu-, tation of it's author. This treatise was divided into eight books. The first four have reached us in their original language, the greek : the following three have been preserved only in an arabic translation, made about the year 1250, and rendered into latin in the middle of the seventeenth century: the eighth was entirely lost. The celebrated Halley very accu- rately revised and corrected both the text of Apol- lonius, and the translation from the arabic ; and he has himself restored the eighth book, conformably to the plan of the author ; the whole forming a mag- nificent edition, which was published at Oxford in 1710. In the- first four books Apollonius treats of the generation of the conic sections, and of their principal properties with respect to their axes, foci, and dia- meters. Most of these things were already known : but when Apollonius borrows some propositions from his predecessors, he does it like a man of genius, who improves and augments the science. Before him the conic sections were considered only in a right cone : but he takes them in any cone whatever, having a circle for it's base, and he demonstrates several theorems, which are either new, or given in a more general form than they had been before. The following books contain a number of theorems remarkable problems, altogether unknown before his 44 his time ; and hence Apollonius has merited princi- pally the title of the great geometrician. From these I shall quote a few particulars. In the fifth book Apollonius determines the greatest and the least lines, that can be drawn from a given point to the circumference of a conic section. He at first supposes, that the given point is placed in the axis of the conic section, and he solves a great number of curious problems on this subject, with a simplicity and elegance that cannot be too much ad- mired : he then extends the investigation to cases where the point is situated out of the axis, which affords a new field for problems still more difficult. For in- stance, in proposition LXII he determines the shortest line that can be drawn from, a given point placed within a parabola, and out of the axis, by a very in- genious construction, in which he employs an equi- lateral hyperbola, which cuts the parabola at the poin^t sought. In the same book we also find the germe of the sublime theory of evolutes, which modern geo- metry has carried so far. The subject of the sixth book is the comparison of similar or dissimilar conic sections or portions of co- nic sections. Apollonius here teaches us to cut a given cone in such a manner, that the section shall have given dimensions ; also on a cone similar to a given cone he determines a conic section of given dimensions ; and every where we find a simplicity, an elegance, and a perspicuity, which afford infinite satisfaction to the admirers of the ancient geometry, In the seventh book, of which the eighth was a part or a continuation, Apollonius demonstrates, and it * was 45 was here these important theorems appeared for the first time, that, in the ellipsis or hyperbola, the sum or difference of the squares of the axes is equal to the sum or difference of the squares of any two conjugate diameters ; and that in either of these curves the rec- tangle of the two axes is equal to the parallelogram formed about any two conjugate diameters. I pass over other propositions, very curious, and not less profound. The age of Archimedes and Apollonius was the most brilliant era of ancient geometry. After these two great men we meet with no other geometrician of the first order in the period we are now considering: yet there are several others, who enriched geometry with interesting theories or discoveries, and have thus merited the esteem and gratitude of posterity. It appears, that men of great inventive powers, too much addicted perhaps to abstract and theoretical speculations in geometry, attached too little import- ance to the applications that might be made of them in practice. This no doubt was the cause, why the first origin of trigonometry, or that branch of ge- ometry by which we find the relations between the sides and angles of a triangle, has fallen into oblivion. Yet trigonometry affords curious problems, which must naturally have excited the researches of the early geometricians. For instance, a person might have wished, or even found it necessary, to ascertain the breadth of a large river, without being obliged, or without having it in his power, to measure it di- rectly ; or he might have been desirous of knowing the distance between the summits of two mountains separated 46 separated by precipices. Now these, aadmany other problems of a similar kind, are to be resolved by the construction of a triangle, one of the elements of which shall be the quantity sought, and in which three of the six things that constitute it, namely three sides and three angles, are known ; with this con- dition alone, that among the three things known there is one side of the triangle, which is capable of being measured directly, or having it's length de- termined by some other known distance. Hence we perceive, that the principles of rectilinear trigono- metry are very simple. There are circumstances which indicate, that they were not unknown to the egyptians : and we are certain, that they were familiar to the greeks. Beside their use in the mensuration of terrestrial distances, they are applicable to several astronomical problems. From the consideration of plane triangles the ge- ometrician rose to spherical triangles, or those formed by three arcs of great circles of a sphere intersecting each other; a theory particularly useful in astronomy, to which it is in some degree indispensable. It is also somewhat. complicated, because, in a space extended according to the three dimensions, we must discover the ratios of the sides and angles of a triangle, the three sides of which are arcs of a circle. Accord- ingly the rise of spherical trigonometry was slow. We have no reason to imagine, that it had made any progress, or such at least that deserves notice, before the time of Menelaus, who lived v about the year 55, and who was both a skilful geometrician and a great astronomer. He wrote a treatise on Chords, which is 47 Js lost : but we have his work on Spherical Triangles, a learned performance, in which we find the con- struction of these triangles, and the trigonometrical method of resolving them in most cases necessary in the practice of ancient astronomy. There is also another geometrical theory, perspective, with which it is doubted whether the ancients were acquainted. For my own part, I do not see how it can be questioned with regard to linear perspective : for this science, if we can give it the separate name, is nothing more than a very simple and easy applica- tion of the theory v bf similar triangles. In fact it con- sists only in representing on a plane, or given super- ficies, an object as it would appear when seen from a given point : or, in geometrical language, in pro- jecting ojPa given surface the parts of an object by means of lines drawn from a fixed and given point to every point of the object. Now is not such a problem more tlian virtually contained in the Ele- ments of Euclid ? not to mention, that perhaps it has been explicitly solved in some of the works, which have not reached us. If, however, any one should not be satisfiea^with this indirect proof, I will pro- duce him\ direct one, taken from Vitruvius. The passage that includes it has not been translated quite agreeably to the true sense by Claude Perrault, and I cannot avoid adopting in preference the following translation given by Mr. Jalabert- in the Memoirs of the Academy of Belles Lettres, vol. xxni, p. 341. * Agatharchus was the first who painted decora- tions for the theatre, which was at the time when JEschylus exhibited tragedies at Athens From his 6 Ms example Dernocritus and Anaxagoras wrote or*. the subject, how, a point being fixed in a certain spot with regard to the eye and visual rays, certain lines proportionate to the natural distances were made to answer to it, in such a manner, that from a thing con- cealed, or which it would be (difficult to guess at, images resembling objects arise; so that, for example, they represent buildings on the stage, which, though painted on a flat surface, appear to project in certain parts.' Vitr. book vn, pre Here, it seems to me, linear perspective is plainly described. With regard to aerial perspective, which depends on the opposition and gradation of colours, the question is not so easily solved. that the ancients had only imp .'bunded on a kind of customar. tice : but I con reasons adduced by count Cayin>. in support of the opposite opinion, have ; the reader weigh with attentio, extracted from a disser which that learned critic discusses this subject. * Ancient painting, a: finished, no longer exist ancients had carried t 1 It is certain, that even in the time of Augustus the works of Zeuxis, Protogenes, and the otncr gruu ^amia's of the most flourishing era of Greece, were scarcely distinguishable, so much had the colours flown off and become effaced, and so wormeaten was the wood : for portable pictures were painted on no othe: material, at lea^t as far as we can learn from any historian. 49 historian. What now remains therefore, on which we can found our judgment, either for or against it ? A few paintings on walls, which we may think our- selves happy in possessing, but which our taste for the antique should not lead us equally to admire. Beautiful as they are in certain respects, it is un- questionable, that they are not to be compared with those superb paintings, on which ancient writers have passed such high encomiums, while speaking to men who admired them as well as themselves, to men who felt all the merit of those masterpieces of sculp- ture, with regard to which we cannot suspect these authors of partiality, since we are able to judge of them ourselves, admire them daily, and know that they were both equally designed for the decoration of their temples and their public edifices. These arts follow each other : on this I must continue to insist, and I shall add, it is naturally impossible that the one, sculpture, should be elegant and sublime; while the other, painting, was reduced to a degree of insipidity and imperfection, such as must be the case with a picture destitute of relief, without any gra- dation of colour, in short void of what we call skill in harmony.' M6ra. de TAcad. cles Belles Let. vol. xxiii, p. 323. tVere I writing a minute history of mathematics', might give an ample list of the geometricians, flourished from the time of Archimedes to the destruction of the Alexandrian school. I should qucre Conon and Dositheus, both very learned, and both friends of Archimedes; Geminus, a ma- thematician of Rhodes, who wrote a work entitled E Enarra-* 50 Enarrationes Geometric? ; c. : but I shall confine myself here to giving the reader a succinct view of those, some of whose works have come clown to us, and of whom therefore we can speak with some know- ledge, without being wholly led by the mere narratives of historians. Theodosius is the first who presents himself with his treatise on Spherics, in which he examines the properties, which circles formed by cutting a sphere in all directions have with respect to each other. A. c. 60. This work, excellent in itself, may be con- sidered as an introduction to spherical trigonometry. Most of the author's propositions now appear evident at the first view : but, faithful to the maxims of the ancients, he demonstrates every thing with the greatest strictness, and with much elegance. We have like- wise two other treatises by Theodosius, entitled, Of Habitations, and Of Days and Nights, which contain an explanation of the celestial phenomena, to bej perceived in different parts of the Earth. After Theodosius we proceed for three or four hundred years, without meeting with any geometri- cian of eminence, except Menelaus, who has already been mentioned. At length we come to Pappus and Diodes, of both of whom I have likewise spoken with culogium on occasion of the two particular pro- blems of the duplication of the cube, and the tri- ion of an angle, and who reappear in this place i other accounts. A. D. 385. We shall meet with spine other geometricians also of distinguished mejit. The mathematical collections of Pappus exhibit one of the most valuable monuments of ancient ge- 1 ometry. 51 ometry. In them the author has assembled together the substance of a great number of excellent works, almost all of which are now lost ; and to these he has added several iie^w, curious, and learned propositions of his own invention. This collection therefore is not to be considered as an ordinary compilation; though even in this view it would deserve very high esteem, as it gives us almost a complete view of the state of ancient mathematics. It was divided into eight books : the first two are lost ; the subjects of the others are, in general, questions in geometry, with a few in astronomy and mechanics. Among other researches Pappus proposed to him- self the problem of geometrical loci in all it's extent, and advanced a great way in it's solution. As it's completion required the assistance of algebra, I shall defer entering into it, till I come to the geometrical discoveries of des Cartes, in the third period. Pappus gave the solution of another very curious problem, of a kind which at that time was absolutely novel : it was that of finding on the superficies of a sphere spaces capable of being squared. By means of the theorems of Archimedes he demonstrates, that, ' if a movable point, setting out from the summit of a hemisphere, traverse a quarter of the circum- ference, while this quarter of the circumference makes a complete reatfJution round the axis, the space, com- prised between the circumference of the base and the spiral of double cuIWTure described on the sphe- rical surface by the movable point, is equal to the square of the diameter/ The proposition may easily be generalised ; and we find, that, all the other cir^ E % cumstances 52 cumstances continuing the same, if the quarter of the circumference, instead of making a complete re- volution, make but a given part of one, the spherical space, comprised between the quarter of the circum- ference in it's initial positiosi, the arc corresponding to the base, and the spherical spiral, is to the square of the radius, as the arc of the base is to a quarter of the circumference. Many great geometricians have treated generally the question of determining spaces that may be squared on a given surface, as will be seen in the fourth period. To the praise of Pappus must likewise be added, that at the end of the preface to his seventh book we find a tolerably distinct idea of the celebrated the- orem commonly ascribed to father Guldin, a Jesuit: ' that the superficies or solid, generated by the mo- tion of a line or a plane, is equal to the product of the generating line or plane multiplied by the path de- scribed by it's centre of gravity.' Though few of the works of Diocles have come down to us, we have enough to inform us, that he was endowed with great sagacity. Beside his cissoid, he discovered the solution of a problem, which Ar- chimedes had proposed in his treatise on the sphere and cylinder, and which consisted in cutting a sphere by a plane in a given ratio. We know not whether Archimedes himself had resolved this c^stion, at that time very difficult, and which leads to an equation of the third order in the fflJRrn methods. The so- lution of Diocles, which is learned and profound, terminates in a geometrical construction by means of two conic sections cutting each other. It has been trans- 53 transmitted to us by Eutocius, who was himself a good geometrician, and whose commentaries on part of the works of Archimedes and Apollonius in par- ticular are much esteemed. A. D. 520. Serenus, another learned geometrician, is placed about the time of Diocles. We have of him two books on the section of the cylinder and the cone, which Halley has published in greek and latin at the end of the edition of Apollonius. In his first book Serenus considers the ellipsis as an oblique section of the cylinder, and shows that the. curve formed in this manner is the same as the ellipsis of the cone. He likewise teaclies us to cut a cylinder and a cone, so that the two sections shall be equal and similar. The second book treats on sections of the right and oblique cone by planes passing through the apex, which produce right lined triangles, and by their comparison give rise to a great. number of cu- rious problems and theorems, from the different ra- tios that may subsist between the axis, the radius of the base, and the angle which the axis makes with the base. TJie whole work of Serenus is a chain of interesting propositions very perspicuously demon- strated. We know no particulars respecting the author. M must not forget to men t fen Proclus, the head of the platonic school established at Athens. A. D. 500. He has rendered imj^^nt service to the sciences ; lie encouraged those, ^ffembraced their pursuit, by his example, instruction, and acts of kindness ; and he has left a commentary on the first book of Euclid, E 3 which 54 which contains many curious observations respecting the history and metaphysics of geometry. His successor was Marinus, the author of a pre^ face or introduction to the Data of Euclid, which is commonly printed at the head of that work. We have none of the writings of Isidorus of MU letus, a disciple of Proclus : hut his name must not be omitted here, as he is said to have been very learned in geometry, and mechanics, and was em- ployed in erecting the temple of St. Sophia at Con-. stantinople, under the emperor Justinian, jointly with Anthemius, of whom we have a valuable fragment, on. which I shall treat more at large, when I come to speak of the burning mirrors of Archimedes. A. p. 530. Hero the younger, so called to distinguish him from Hero of Alexandria, who will be noticed under the article of hydrostatics, is also mentioned among the ancient geometricians. His Geodemt. a work in other respects of little importance, contains the me- thod of finding the area of a triangle by means of it's three sides, but without a demonstration This pro- position is supposed to belong to some preceding and more profound mathematician. It is useless to swell this historical abstract with the names of a few other geometricians, from whom their contemporaries perhaps derived instruction ; who, not having perceptibly contributed to van cement of science, scarcely merit^the notice of posterity. arcey CHAP; CHAP. III. Origin and Progress of Mechanics. As the ancients were unacquainted with the theoret- ical principles of mechanics till a very late period, it is not a little surprising, that the construction of machines, or the instruments of mechanics, should have been pursued with such industry, and carried to such perfection by them. Vitruvius^ in his tenth book, enumerates several ingenious machines, which had then been in use from time immemorial. We find, that for raising or transporting heavy burdens they employed most of the means, which we at pre- sent apply to the same purposes ; such as the capstan, the crane, the inclined plane, pulley, &c. Diffi- culties give birth to resources. For instance, when, Ctesiphou the architect, employed in building the temple of Ephesus*, had procured the pillars, which were to support or adorn that vast edifice, to be falhioned in the quarries, and they were now to be conveyed to Ephesus, he was aware, that, if they were placed on a common waggon, their enormous weight would sink the wheels into the ground, and render them incapable of moving. Accordingly he had recourse to anotbagtery simple mode. He fixed v&^p * We know not the date of the building of this temple ; but it was buvnt by Herostratus, on the night when Alexander was born,, in the year before Christ 356, E 4 into 56 into the centres of the opposite ends of a pillar two stout iron pins, which turned in holes cut in two long beams of wood joined together hy a cross piece. Oxen being then harnessed to this sort of frame easily rolled along the pillar. It is by a similar mechanism, that we smooth our terraces, the gravel walks of our gardens, &c. In like manner Metagenes, the son of Ctesiphon, who succeeded his father in continuing the building, having to transport to Ephesus the stones, which were to form the architraves of the temple, fastened these stones between two wheels of twelve feet diameter, which, from their proximity, formed as it were but one cylinder. I might quote a multitude of other instances of the genius of the ancients in practical mechanics, of which the art of war alone would afford me several ; for- we know, that with their catapulta^ scorpions, x balistae, &c. they produced a part of those terrible effects, which, to man's misfortune, the invention of powder has too much facilitated. In the theory of mechanics the ancients were not so successful. We see by some of the writings of Aristotle, that even he had only confused or errone-* ous notions concerning the nature of equilibrium and motion, and of course all his predecessors must have been still more deficient. A. c. 320, The true theory of the equilibrium of machines, dates no higher than the time of Archimedes, to whom we are indebted for it's elem^Jp In his book of Equi* ponderants he considers a balance supported on a ful- cvum, and having a weight in each basin. Taking as a fundamental principle, that, when the two arms of 57 r all the mass wiil be in equilibrio, when each particular molecule is equally pressed upon in every direction. This equality of pressure, on which he makes the state of equilibrium essentially depend, is demonstrated by experiment. The author afterward examines the conditions, which are requisite to pro- duce and preserve the equilibrium of a solid floating on a fluid. He shows, that the centre of gravity of the whole body, and that of the immersed part of it, must be in the same vertical line ; and that the whole weight of the body is to the weight of the quantity of fluid displaced, as the specific gravity of the fluid is to that of the floating body. This general theory he illustrates by various examples taken from the tri- angle, the cone, the paraboloid, &c. We readily perceive by the 7th proposition of the first book, that two bodies equal in bulk, and im- mersed in a fluid lighter than either of them, lose equal quantities of their weight; or inversely, that two bodies, when they lose equal quantities of their weight in a fluid, are of equal volume. I cite this the- orem, because it is the general opinion of mathema- ticians, that Archimedes employed it to solve a well known problem, proposed to him by king Hiero on the following occasion. For this monarch a goldsmith of Syracuse had made a crown, which, by the terms of the agree- ment, was to have been of pure gold. But the king suspecting, that silver had been mixed with it, had recourse to Archimedes, to discover the truth without injuring the crown. It is very probable, that Archi- medes tnedes accomplished it in this manner. He begati by forming two ingots, one of pure gold, the other of pure silver, each of them equal in bulk to the crown ; weighing the three bodies, that is the crown and the two ingots, one after the other in water, and diminishing or enlarging the ingot of gold, and the ingot of silver, till each lost in water the same weight as the crown. This preliminary operation being performed, Archimedes weighed the same three bodies separately out of the water, or in the air : and having found, that the crown weighed less than the ingot of gold, but more than the ingot of silver, he concluded, that the crown was neither pure gold, nor pure silver, but a mixture of the two* All that was wanting to complete the solution of the problem was, to discover the proportion of the metals* This he effected by a very simple arithme* tical calculation, which consists in estimating the proportion of the gold to the silver in the ratio of the excess of the weight of the crown over the ingot of silver to that of the weight of the ingot of gold over the crown. Some authors say, that Archimedes being in the bath when these ideas presented themselves to his mind, he immediately leaped out in a transport of joy, and, without thinking of the condition in which he was, ran through the streets of Syracuse, exclaim- ing ' I have found it ! I have found it !' I have no intention to detract from this ingenious discovery, which would be as unjust as misplaced : but for the sake of some of my readers I shall ob- serve, that, if the crown, instead of containing merely merely silver and gold, as was supposed, had con- tained more than two metals, as gold, silver, and copper, for instance, the crown might have been, made of the same weight, though these three metals had been combined in several different proportions, which could not have been detected in this manner*. The screw of Archimedes, as it is called, is a very simple hydraulic engine, and very convenient for raising water to a small height. According to Dio- dorus Siculus, Archimedes invented this machine when on his travels in Egypt, and it was employed for draining marshes, rivers, &c. : but Vitruvius, a contemporary of Diodorus, does not include it among the discoveries of Archimedes, of whom however he was a great admirer. Claude Perrault, the translator and commentator of Vitruvius, observes, that the remarkable use assigned to this machine by Diodorus, that of having contributed to render Egypt habitable, by draining off the water with which it was formerly inundated, may lead us to presume, that it was much more ancient than the time of Archimedes. Vit. book x, chap, ir. If this conjecture have any foundation, let us not mix with the legitimate property of Archi- medes an invention, his title to which may be dis- puted : he is too rich in other respects, for us to he- sitate about sacrificing an equivocal claim. About a century after Archimedes, two mathema- ticians of the alexandrian school, Ctesibius and his To this may be added, that the specific gravity of a compound of two metals in many cases differs from the mean of the two simple metals composing it, and sometimes considerably, T, F disciple disciple Hero, invented pumps, the siphon, and fountain which plays by the compression of the air and still retains the name of Hero* A. c. 150. We are indebted more especially to Ctesibius for a ma- chine of the same kind, composed of a sucking and a forcing pump, which are so disposed, that by their alternate action the water is continually drawn up and forced through a tube ascending between them. At present we know, that the medium through which the moving principle is applied to all these machines is the pressure of the atmosphere, which forces the water up into the vacuum made by the piston in ascending or descending: but the effects produced by them are very curious, and must have appeared at first not a little extraordinary. Accord- ingly the ancients, not knowing to what they were ascribable, had recourse to their grand scheme of occult qualities, so convenient for explaining the phenomena of nature. The water, said they, ascends in the pump, because Nature abhors a vacuum, so that the moment the piston is raised, the place it quits must be occupied by the water. The natural philosophy of the ancients was full of these secret powers, which were infinitely diversified according to the occasion. The ideas of hatred and affection were transferred from the moral to the physical world ; celestial or terrrestrial bodies had a sympathy or antipathy with respect to each other ; and a phe- nomenon was supposed to be explained, if it could be any way brought under the dominion of these chi- merical agents, 67 The measuring of time by means of clepsydra?, or water clocks, is ascribed to the egyyptians. These blocks indicated the hour by the gradual rise of wa- ter, running into a vessel in quantities regulated ac- cording to the divisions of time; or by the move- ment of a hand, which this water turned by means of a wheel, on which it was made to impinge. Cte- sibius and several others of the ancients proposed dif- ferent machines of this kind, as may be seen in Vi- truvius, book x. Hourglasses with sand were after- ward substituted instead of those with water. The different kinds of persian wheels likewise come to us from the ancients, but we know not at what periods they began to be in use. Before mills moved by water, or by wind, were in- vented, pestles were used for pounding corn, and re- ducing it to meal. After this two millstones were employed, the under one being fixed, and the upper one turned round upon it by the hand simply, or with the assistance of a rope wound round a capstan ; whence these implements were termed handmills, or horsemills. They were much used by the romans at the origin of their republic, and no doubt came to them from more ancient nations. By the french likewise, under their first race of kings, they were equally employed with success. They have since been too much neglected ; for they might not only Supply the place of wind or watermiils, when these are prevented from working by frosts or calm wea- ther, but be of use in a besieged town ; and they would at all times employ for the benefit of the state F 2 a nurn- a number of useless hands, which now remain idle in our prisons. An epigram in the greek anthology has given room to suppose, that watermills were invented in the time of Augustus : but Vitf uvius, who flourished under that potentate, does not say, in the description he gives of them, that they were of recent invention. Probably, therefore, they were known long before. Windmills came into use much later. Some au- thors assert, that they were invented in France in the sixth century. Others say they were brought thither iduring the croisades from the East, where they were then of very ancient use, and where they were pre- ferred to watermills, because rivers and springs are there less common than in Europe. Whether, how- ever, they were invented or borrowed by the french, it is certain that their use was established among them slowly and with some difficulty ; they, on the contrary, preferring watermills, as more convenient, and more regular in their working. I cannot avoid remarking by the by, that the me- chanism of mills, particularly that of windmills, is one of the masterpieces of human industry. The man of feeling and gratitude, when he reflects on so many labours, so many monuments of genius, is naturally prompted to inquire: To whom are we indebted for all these sublime and useful discoveries? What honours, what rewards, have these benefactors of mankind received from their countrymen, and from the world at large ? To these questions History commonly is able to give no answer : but she takes 5 great 69 great care to transmit to us the names and exploits of those conquerors, by wh.om the Earth has been laid waste. For a considerable time the action of fluids was ap- plied as the moving principle to various machines, without the knowledge of any theory, by which it's effects could be previously determined. The defects of one machine were lessons for constructing another less faulty, and thus by guessing and experiment a certain degree of perfection was gradually attained Sextus Julius Frontinus is said to be the first, wbp had any thing like clear ideas of the motion of fluids. He was inspector of the public aqueducts at Rome under the emperors Nerva and Trajan, and left a work on the subject entitled, A Treatise on the Aque- ducts tf the City of Rome, A. D. 300. In this he .considers the motion of water running in canals, or flowing out from apertures in the vessels in which they ate contained. He first describes the aqueducts of Rome, mentions the names of the persons by whom they were constructed, and notes the periods when they were built : in the next place he deter- mines and compares together the measures or modules, which were then used at Rome for ascertaining the quantity of water discharged by the ajutages ; and lastly lie proceeds t the modes qf distributing, the waters of an aqueduct or fountain,. On these differ- for in the time of Strabo the science of the magi had so much declined, that they concerned themselves only about sacrifices, and explaining the different ceremonies employed in them to strangers. The reader no doubt will be surprised, to see the jews appear on the stage as astronomers. It is not the fault Of their historian Josephus, if we do not consider the hebrew patriarchs as the inventors of astronomy and geometry. In the 2d and 3d chapters of his first book On the Antiquities of the Jews, he thus expresses himself. ' To their genius and la- bours we are indebted for the-science of astrology * ; and as they had learned from Adam, that the world would be destroyed by water and by fire, their appre- hensions lest this science should perish, before man- kind were acquainted with it, led them to erect- two pillars, the one of brick, the other of stone, on which they engraved the knowledge they had acquired ; in order that, if a deluge should destroy the pillar of * Astrology here is synonymous with astronomy. brick, fcrick, the pillar of stone should remain, to preserve to posterity what they had written. Their precaution succeeded ; and it is affirmed, that the pillar of stone is still to be seen in Syria Beside this out ancient forefathers were particularly beloved by God; as the work which he had made with his own hands ; and the diet on which they fed was well adapted to the preservation of life ; God prolonged their existence, both on account of their virtue, and to afford them an opportunity of bringing to per- fection the sciences of astronomy and geometry, vhich they had invented. This they could not have done, had the period of their lives been less than six hundred years, because the completion of the grand year requires the revolution of six centuries.' A few very simple reflections will enable us, justly to appreciate this fine tale; I will not inquire, whether it be sufficiently proved, that the Jewish patriarchs lived as long; as Josephus relates ; still less shall I endeavour to fathom die rea- sons, for which God might have granted them such length of life; I shall content myself merely with putting the few following questions to Josephus. If your patriarchs were in reality such great astro- nomers, how is it that all their knowledge has va- nished? and how happens it not to have been trans- mitted to posterity by Noah, who was himself one of the most eminent, and undoubtedly one of the best informed of the patriarchs ? How is it that the jews have never shown the least acquaintance with astro- nomy on occasions, on which it would have been of great use to them ? why, for instance, when the ce- lebration 79 lebration of the Passover was to be fixed by the ne\f moon, did they wait till some one had seen it; and had made his report to the assembly of the people, since a moderate degree of skill in astronomy would have made it known in a much more simple and ac- curate manner? Lastly, what proof is there of the absurd fable of the two pillars ? As to the period of six hundred years, though per- haps it does not deserve all the praise which modern writers have lavished on it, and though it wants one of the principal advantages of a period, that of being included within no very distant limits, I confess it indicates a great number of accurate observations, and a scientific employment of astronomical calcu- lation : but for these very reasons I conceive, that it's discovery, if a fact, must not be ascribed to the Jewish patriarchs. Who, indeed, can believe, that a nation, whose forefathers were capable of such an effort of science and attention, had degenerated and sunk to such a degree, that from the time of the de- luge, and as long as it has been separated from other people, it has displayed only the vilest superstition, and the most stupid ignorance? For what other opi- nions can we entertain, when those historians, whom it considers as sacred, coolly tell us, that Joshua stopped the course of the Sun, that the shadow on. the sundial of Hezekiah went back ten degrees, and that plants are formed by putrefaction, with a thou- sand other absurdities of a similar stamp ? Is it not very probable, that Josephus, from a blind zeal for the honour of his nation, or from other reasons with which we are unacquainted, has endeavoured to give it II It the reputation of a discovery, whether true or the notion of which he himself had derived from the writings of chaidean, egyptian, or grecian astro- nomers ? When the jews were led captive to Bahylon under Nebuchadnezzar, their intercourse with a learned people naturally gave them some taste for the sciences; and many of their rabbis began to study geometry, astronomy, optics, Sec. A. c. 588. These first ru- diments of learning, slight as they were, spread and were propagated among them. In the sequel, the total dispersion of the jews, after the taking of Je- rusalem by the romans, made of them in some sort a new people ; and they adopted the customs, occu- pations, arts, &c. of the nations among whom they were scattered. We find Jewish mathematicians in Greece, and among the arabs. They translated the elements of Euclid, the works of Archimedes, the writings of Apollonius, and the Almagest of Ptolemy. Several rabbis are even mentioned as very learned in these subjects : but we do not find, that they ever made in them any discovery of importance, or one truly conducive to the progress of the human intellect. The Chinese come before us with more advantage. The wisdom of their political institutions, the ex- cellence of their morals, and acquaintance from time immemorial with the liberal and mechanical arts be- neficial to society, combine to announce a people of application and industry, and versed in the sciences for a number of ages. Astronomy in particular at- tracted their early attention, the climate in which they dwell being very favourable to observations. But 81 But the Chinese, dissatisfied with an honourable an- tiquity which history avows, have exaggerated it to such a degree, that we could scarcely give credit to it, were the foundations on which it rests as sure and solid, as they are in reality false and fragile. I am ohliged, therefore, to controvert pretensions, that cannot be admitted, unless we shut our eyes against the incontestible truths, with which they are incon- sistent. , In the first place, the ancient annals of the Chinese contain nothing but a heap of absurd fables, which they themselves have been forced to abandon : , but they persist in maintaining, on the authority of some of their writers, whom they suppose to have been well-informed, that the Chinese nation, already in a flourishing state, began to acquire a knowledge of the motions of the heavenly bodies under the emperor~Yao, about two thousand three hundred years before the commencement of the Christian era. They assign nearly the same date to the foundation of their celebrated mathematical tribunal, which strll subsists, notwithstanding the vicissitudes it has experienced 'in such a long series of ages. The missionaries sent to preach the Christian religion in China about the end of the seventeenth century, led by some appear- ances of truth, or from condescension to the weak- ness of a vain people, whom they wished to convert, and. whom they conceived it impolitic to affront, adopted it's marvellous history, and spread it over Europe. For a long time no one troubled himself to inquire into it's authenticity. At length, however, the eyes of criticism were opened to this strange - G system. 82 system, and two formidable adversaries, Chronology and Astronomy, united their forces to subvert it *. In the first place, to speak of Chronology. It was observed, that the succession of emperors, up to the period at which we presume the Chinese history to be authentic, has several considerable breaks ; that most of these potentates are known only by their names, whether true or fictitious ; that there is a great barren- ness of historical facts, and of those which occur some are manifestly absurd^ that there are numerous contradictions in the order of the dates ; and lastly, that the Chinese history does not acquire connexion, or any character of certainty, till the time of Confu- cius, that is to say, about the year 460 before Christ. With regard to astronomy, the defenders of the great antiquity of the Chinese in the sciences have imagined, that in the Shoo-king, a fragment of the ancient an- nals of China collected by Confucius, they find the mention of an observation of the solstices made in the time of the emperor Yao, and of an eclipse of the Sun almost as ancient. But this account is so obscure, and so brief, that europcan astronomers, having attempted to calculate the appearance of these phenomena, have not agreed in their conclusions. The observation of the solstices has no precise date, or mark of truth : and the eclipse is placed by some in the year 2154 before Christ, by others in 2007. A very uncertain observation of the solstices between the years 1098 arid 1104 before Christ is likewise * See the Memoirs of the Academy of Belles Lettres, vol. xxxvi, p. 164. quoted. 83 quoted. But the most ancient Chinese observation, to which any authority can be allowed,, would be that of a solar eclipse, supposed to have been made in the year 776* before the Christian era, could we be certain, that it was not given from a subsequent calculation. The annals collected by Se-Ma -Quang, a Chinese his- torian of the eleventh century, mention, in the reign of the emperor Tshwen-Yo, which began a hundred and fifty years before that of Yao, a conjunction of five of the planets, Saturn, Jupiter, Mars, Venus, and Mercury, in the constellation called by the Chinese Ska; and to characterise this conjunction, the year of the cycle in which it occurred, the day of the s y z ygj> and tne position of this syzygy with respect ta the constellation Ska are added *. From these indications, Mr. Kirch of Berlin, and after him father Mailla, the Jesuit, having calculated by astronomical tables the conjunctions of the planets, which might have taken place in ancient times, found a con- junction of the four planets Saturn, Jupiter, Mars, and Mercury, in a space of some degrees in the neighbourhood of the constellation Ska, in the year 2449 before Christ. But, beside that this pretended Conjunction is incomplete, since Venus is wanting, it does not accord with the year of the cycle, the syzygy^ or the position of the syzygy. Cassini places the same conjunction in the year 2012; and his Calcu- lation gives with more accuracy than the others the position of the four planets in the constellation Sha, * Memoirs of the Academy of Belles Lettres, vol, x, p. 392, and vol. xviii, p 4 284* G 21 but 84 but it equally fails of fulfilling the other conditions of the problem. Some other fruitless attempts have been made to reconcile the whole ; but all these un- certainties give rise to a strong probability, that the Chinese never observed a conjunction of these five planets. It is very possible, that it was the invention of flattery; for the Chinese, considering conjunctions of the planets as very fortunate presages for the reigns of their emperors, make no scruple to forge them sometimes, or to be very accommodating on the sub- ject. For this we may appeal to what occurred in the year 1725, the second of the reign of Yong- Tshing, when the approximation of Mercury, Venus, Mars, and Jupiter, was given out as a conjunction, and inscribed as one in the public registers. It is the opinion of father Gaubil, a Jesuit missionary and learned astronomer, that the pretended conjunction in the time of the emperor Tshwen-Yo has no other foundation, than a calendar published under the dy- nasty of Han, who ascended the throne in the year 207 before Christ, and considered by the most learned of the Chinese as a forgery, which did not even con- tain the conjunction in question in the text, but in a comment that has slipped into it. Finally it has been demonstrated by Freret, that this calendar was the work of some blundering forger, who was even ig- norant of the art of making calculations *. It appears unquestionable, that we cannot affix to the astronomy of the Chinese any real and positive date earlier than the, year 722 before Christ, or twenty * Memoirs of the Academy of Belles Lettres, Vol, xvm, p. 289. five 85 five years after the era of Nabonassar. In the work entitled Tshu-Tseu, Confucius notes a succession of thirty six eclipses, from that period to the year before Christ 480, of which thirty one have been verified by modern astronomers. From that time the Chinese astronomy was continually enriching itself with new observations, the fruits of patient industry, not of genius ; for we have every reason to believe, that the- chinese were never very expert in astronomical cal- culations, and that they have frequently had re- course to foreign astronomers, to enlarge or correct their theoretical knowledge. Thus, for instance, in the time of the khalifs, several mohammedan astrono- mers went into China, and were placed at the head of the mathematical tribunal ; an honour, which our astronomical missionaries likewise have frequently received. I must not conceal, that a powerful objection to the antiquity of the claims, which the Chinese have to the sciences, is drawn from the period when their astronomical observations begin to acquire certainty. This era being postcriour to that of Nabonassar, which serves as a base to the computations of the chaldean and grecian astronomy, it has been concluded with some probability, that the astronomers of Babylon, or those of Greece, carried their knowledge into China; particularly as we are certain, that some in- tercourse between those nations subsisted about that period. To conclude, w r e have before our eyes a striking proof of the mediocrity of the Chinese in the science of astronomy. Notwithstanding the concurrence of G 3 so 86 so many favourable circumstances, a serene sky, and the encouragement of the emperors ; which should naturally have accelerated it's progress in China, it has always remained there nearly in the same state : it abounds in observations, but to any new theory it is a stranger. Superstitiously attached to it's ancient customs, to the barren imitation of it's forefathers, and to the opinion that they knew every thing necessary to be known, the Chinese nation appears destitute of that restless activity, which endeavours to extend it's acquirements, and gives birth to discoveries. Some of the learned consider Hindostan as the cradle of all the sciences, and particularly of astro- nomy, which ttcy date from the remotest antiquity. As a proof of this they quote the celebrated hindoo pe- riods, which would leave no doubt, were they perfectly clear and exact, that the hindoos were formerly well versed in the knowledge of the celestial motions. But all that relates to this origin is enveloped in thick darkness : every thing in it is the .work of system : it proceeds only by the help of conjectures and bold suppo- sitions, is frequently contradictory, and always ques- tionable. Others, perhaps erring in the opposite extreme, assert that the astronomy of the Hindoos,, far from having had such an ancient origin, is the work of the arubs, who carried it into Hindostan about the middle of the ninth century. A third and more probable opinion refers the origin of astronomy in Hindostan to the time, when Pytha- goras travelled 'into that country, and there diffused the philosophical attainments of every kind, with which his mind vyas enriched. A. c. 540. It 87 It is not my intention to bewilder myself in these long and obscure discussions, which the reader no doubt would find tedious, and from which he would derive little instruction. Accordingly I shall confine myself here to a very brief sketch of the knowledge we have, that has any pretension to certainty, respect- ing the astronomy of Siam and the coast of Coro- mandel, from the works of Dominic Cassini and le Gentil. Mr. de la Loubere, ambassador from France to the Siamese in 1687, brought home with him an indian manuscript, which contained a method of calculating the motions of the Sun and Moon. This method was founded on a multitude of additions, subtractions^ multiplications, and divisions, to discover the inten- tion and use of which, required profound astrono- mical knowledge. The celebrated Dominic Cassini unravelled this chaos *. He discovered in it two different eras, the one civil, answering to the year 544 before Christ; the other astronomical, answering to our year 633. According to his explanations, about the time of the first epoch the Siamese were acquainted with the tropical solar year and the ano- malistic year, the equation of the centre of the solar orbit, the two principal equations of the Moon, and the cycle of nineteen solar years, which comprises, two hundred and thirty five lunations. All these theories may have been simply the result of a long series of accurate observations : but it may be susr peeted, that Cassini, through the illusion of his own * Ancient Memoirs of the Academy of Sciences, vol. viu, G 4 pipfound 83 profound knowledge, rather conjectured or intro- duced these theories into the indian manuscript, than that he really found them there. Those however, who would avail themselves of the authority of Cassini in this case, to date back the origin of indian astronomy, can only carry it up to the time of Pythagoras : and then it is possible, that this philosopher taught astro- nomy in India, as I have already observed. The si- amese of our times have greatly degenerated from the real or pretended learning of their forefathers, for they are scarcely able to make a rude calculation of an eclipse. About thirty years ago ; during a residence of three and twenty months at Pondichery, le Gentil, one of the astronomers of the academy of sciences, had an opportunity of inquiring into the astronomy of the brahmens ; which must not be confounded with that of the Siamese, and of which I shall proceed to give a general idea, from the account he laid before the academy and the public. The peninsula of India on this side the Ganges is well known to be inhabited by two very different na- tions : the western coast by the malabars, who have given it their name ; and the eastern, called the coast of Coromandel, OB which is Pondichery, by the gen- toos. The brahmens, originally from Tanjore and Madura, form the first cast, or the, privileged cast of these, of whom the lowest cast may be considered as slaves. All transition from one cast to another is severely prohibited by the laws. To the brahmens all knowledge is confined, the inferiour casts being, de- voted to ignorance, We 89 We may judge of the astronomy of the early brah- mens from the science of the present. For a long time they have ceased to take observations of the feeavenly bodies : astronomy is to them merely a tra- ditional science, to which they have added no new views, no discovery that has in the least tended to advance it: their principal object is to know the motions of the Sun and of the Moon, which they calculate according to the methods of their fore- fathers. The ancient astronomy of the brahmens was a chaos of rude observations, when one of their kings, named Salivagena, or Salivaganam, whose death is placed about the year 78 of our era, made a consider- able reform in it, and carried it to that degree of advancement, at which it has stopped. The reign of this prince is an era as famous in India, as that of Nabonassar was among the chaldeans. The brahmens are very vain, little communicative, and consider themselves as infinitely superiour to the europeans in knowledge of every kind. Le Gentil found it very difficult to dive into their mysteries, which at first were concealed from him with con- temptuous reserve. At length however, by dint of money and caresses, he was enabled to acquire a sufficient idea of their astronomy. He discovered, that it was reducible to five leading points : the use of the sundial, the length of the year, the precession of the equinoxes, the division of the zodiac into twenty seven constellations, and the calculation of eclipses of the Sun and Moon. The knowledge of -all these is extremely defective among the brahmens, 4 while 90 while the europeans have carried them, as well as all the other branches of astronomy, to a very high de- gree of precision. We certainly cannot be allowed to place the phe- nicians, the first merchants in the World, among the number of astronomers. A. ex 900. Yet it is undeniable, that they had sufficient practical know- ledge at least of the motions of the stars, to guide them in the distant voyages which they undertook. When they had the courage to venture out to sea, they began with directing their course by stars in the north, of which they never lost sight. By degrees, they made long voyages on the Mediterranean ; planted colonies on it's shores ; passed the strait of Gibraltar; built Cadiz on the coast of Spain ; spread themselves along the shores of Africa ; and it is even said, that they doubled it's southern extremity and formed settlements on it's eastern coasts, &c. The learned Huet has entered into very curious details on this subject, in his history of the Trade and Na- vigation of the Ancients, which the reader may con- sult. Many other people, following the example of the phenicians, or led by their own industry, addicted themselves to navigation and commerce. We are not unacquainted with the colonies of Marseilles, Tarentum, and Sicily, founded by the ancient greeks, previous to those great astronomical discoveries, by which they acquired scarcely less glory, and perhaps greater fame, in the history of the sciences, than by the works of their geometricians. Thalesi . 91 Thales of Miletus is considered as the first, who propagated any truly scientific knowledge of astro- nomy in Greece. He acquired no doubt it's elements in Egypt; but he extended them by his own me- ditations, and to him we must ascribe the remarkable movement, which then took place in the science, and which went on increasing for several centuries. He taught his countrymen the cause of the inequality of the clays and nights ; he explained to them the theory of eclipses, and the manner of predicting them; and he gave them a practical example of his art in an eclipse of the Sun, which happened a short time after conformably to his prediction. All these things ap- peared so novel, and so extraordinary, that they acquired Thales the highest reputation, and brought him a multitude of illustrious scholars. In the num- ber of these is mentioned particularly the philosopher Anaximander, who succeeded him as head of the school of Miletus. A. c. 600. Anaximander had some idea of the sphericity of the Earth : the invention of celestial globes and ge- ographical maps is also ascribed to him : and he con- structed a gnomon at Sparta, by means of which he ascertained the obliquity of the ecliptic, with the solstices, and the equinoxes. The advantage, or even in certain cases the ne- cessity, of readily distinguishing the stars, had long before given rise to the contrivance of arranging them in groupes, or constellations, as we divide the surface of the habitable world into continents, king- doms, provinces, districts, &c. This division could not be otherwise . than very imperfect at first, on account account of the unavoidable want of accuracy in the enumeration of the stars, or in the mode of classing them : it was brought to perfection by the greeks about the time of Thales and Anaximander. The first names given to the stars were derived from implements of husbandry, animals, useful oc- cupations, &c. The greeks altered, augmented, or improved this nomenclature, which was in some in- stances rude, and in others whimsical. The rich and lively imagination, that guided all the conceptions of this ingenious people, concealed the natural dry- ness of the subject under pleasing and graceful images. Thus there is a constellation consisting of several stars very near together, and followed by a single star remarkable for it's brilliancy and magni- tude: this groupe they called by the name of Pleiades, derived from a word signifying a multitude, and to the great star they gave that of the man Orion. The Pleiades they feigned to be the daughters of Atlas and the nymph Pleione, and Orion a giant enamoured of them, and pursuing them incessantly. Thus all the sky of the greeks was full of fabulous or historical emblems, which amused and eased the me- mory, without distracting the attention. Among these constellations, those, to which the Sun, Moon, and planets answer by their true or ap- parent motions from west to east, occupy the space called the zodiac. This is a spherical band about six- teen degrees in breadth. Different nations have their peculiar zodiacs, that is to say, a zodiac com- posed of a greater or smaller number of constellations, or of more or fewer stars in each constellation. The most 93 most ancient and probable opinion is, that the zodiac of the greeks was borrowed from the egyptians ; and an inscription lately found in Egypt tends to confirm this conjecture. In the time of Thales it assumed a regular form; it afterward spread throughout Europe; and we have now no other. This is divided into twelve constellations, the names and order of which from west to east are expressed in the two following verses. Sunt Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libraque, Scorpius*, Arcitenenst, Caper ||, Amphora J, Pisces, It has been disputed among the learned, whether the five planets Saturn, Jupiter, Mars, Venus, and Mercury, were known before the time of the greeks. We cannot easily suppose, that they escaped obser- vation in the remotest ages of astronomy, or that astronomers had not acquired general ideas of their revolutions from west to east, and even of the varia- tions which occasion them to appear sometimes sta- tionary, sometimesdirect in their motions, and at other times retrograde. But it is very doubtful whether the grecian astronomers, at the time of the first for- mation of their zodiac, had sufficiently accurate ideas of the inclinations of the orbits of the planets to the plane of the ecliptic, to comprise these orbits in the extent now assigned them. In fact, according to the opinion of the most learned astronomers, the first accurate observations made of the motions and * Scorpio, t Sagittarius. -|| Capricornus. J Aquarius. appearances 54 appearances of Saturn, Jupiter, Mars, Venus, and Mercury, do not ascend above three centuries beyond the Christian era. All the intricacies of these mo- tions require much time and observation, to unravel and explain them in a plausible manner. Mercury, being frequently immersed in the rays of the Sun, occasioned most difficulty in this respect. It is pro- bable, that the first zodiac of the greeks comprised only the paths of the Sun and Moon, the orbits of which cut each other at an angle of about five de- grees. We know at present that the comets are solid bo- dies, like the Moon and the Earth, wandering through space in all kinds of directions. The ideas of the ancients respecting the nature of these bodies were altogether erroneous : they considered them as simple meteors, which the Supreme Being made to appear from time to time, for the purpose of manifesting his displeasure, or announcing some extVaordinary event. The sudden and unfrequent appearance of these comets, their irregular movements, those long tails, or trains of light, which accompany them, and display themselves in various singular forms, were terrible at first to the eye and to the imagination : every thing led a credulous and superstitious people, to consider comets as a particular class of temporary phenomena, intended by the Creator as indications of his will, which was interpreted as man thought fit. Whatever were the opinions entertained of comets by astronomers, they scarcely took the trou- ble to observe bodies, which, after they had appeared above the horizon for very short periods, suddenly vanished 95 Vanished without leaving any expectation of their return. The astronomy of comets is a modern sci- ence, of which I shall speak hereafter. Here, how- ever, justice demands me to render Seneca due homage : by an effort of philosophy superiour to the notions of his age, he did not adopt the received opi- nions respecting these bodies. * I am not of the opi- nion,' says he, ' of our philosophers : I do not consider comets as transient fires, but as one of the eternal works of nature Is it any wonder, that comets, so rarely as they are seen, should not yet be subjected to determinate laws; and that we do. not know the beginning and end of the revolutions of these bodies, which reappear only after a long interval ? Time and research will at length afford a solution of these problems The day will come, when our posterity will be astonifhed, that we were ignorant of truths so clear/ Nat. Qutest. lib. 7, cap. 22, 24, and 25. The school founded by Pythagoras in Italy made astronomy a particular study. Seconded by his earliest scholars, he clearly demonstrated the sphe- ricity of the Earth, which Anaximander had only conjectured. Having observed, that any given star appears to occupy a higher or lower station in the sky to a traveller, who goes to a place at any distance, they concluded, contrary to the testimony of the senses, that the surface of the Earth must not be a simple plane extended in a right line, but a curved and spherical superficies. Pythagoras had another idea, not less true, but much more extraordinary for the time in which he lived : he judged the Sun to be fixed, in the centre of the planetary world, and the Earth 96 Earth to move round it with the other planets in the celestial space : a system, which has been developed and demonstrated in modern times. But as this opi- nion was directly repugnant to appearance, and to the vulgar prejudices of the times,. Pythagoras con- tented himself with communicating it in secret to his scholars ; whether, unable to confirm it by a suffi- cient number of observations, he considered it onlv as a very probable hypothesis ; or feared, if he openly promulgated it, that he should expose himself to public derision, or even, which would be of more dangerous consequence, that he should render him- self obnoxious to the persecutions of ignorance and fanaticism. In fact, these two enemies of human reason have exercised their persecuting power and despotic sway in all ages ; and we need not recur to modern days, to find striking instances of it. We know, that about a hundred years after his time the philosopher Anaxagoras was accused of impiety, and condemned to banishment, for having said, that the Sun w r as a mass of fiery matter ; and some au- thors add, that nothing but the influence of Pericles, his scholar and his friend, saved him from being put to death. The measuring of time being the principal object, or rather the foundation of all astronomy, both an- cients and moderns have made the greatest efforts, to determine accurately, and compare with each other, the motions of the Sun and Moon, on which this jneasure universally rests. Some imperfect observations had at first made it supposed, that the solar year consisted of 365 days : by 97 -by degrees it was found to be sensibly longer ; and the egyptians and the first grecian astronomers ex- tended it to .365 days 6 hours, which exceeds it's real duration about 1 1 minutes. This important point of astronomy has been gradually improving to the present day ; when at length, by the combination of a great number of ancient and modern observa- tions, it is determined at 365 days, 5 hours, 48 mi- nutes, and 48 or 49 seconds. Though the Moon is much nearer to us, and more rapid in it's motion than the Sun, more difficulties occur in measuring it's revolution. An immense number of observations and calculations are requisite, to determine it's length with respect to the first point of the ecliptic, the Sun, the fixed stars, and the apogee and nodes of the lunar orbit. It was for a long time supposed, that the synodical month consisted only of 29 days and a half; and to avoid the fraction, the 12 synodical months, com- prised in the solar year, were considered as of 29 and 30 days alternately ; the former being called menses tavce, the latter menses plence. This determination was very defective, since it gave only 354 days for the duration of the lunar year, or of 12 synodical months, while the true duration of this year must be the same as that of the solar year, or very nearly 365 days, 5 hours, 48 minutes, 48 seconds. When the inaccuracy of this comparison was dis- covered, different means of correcting it were sought, either by the intercalation of some days, or of some lunar months in a certain number of solar revolutions. These however were only palliatives, and errours H were 98 were continually returning in the course of time. The egyptians, having very soon perceived the difficulty of establishing an accurate correspondence between the solar and lunar motions, took the course of the the Sun alone as the basis of the fundamental mea- sure of time; and contented themselves with refer- ring to it by approximation that of the Moon, the knowledge of which was necessary for the calculation of eclipses. For similar reasons other astronomers, the arabs in particular, regulated the measure of time by the motion of the Moon. The greek astronomers persisted in their endeavours to reconcile the motions of these two celestial bodies. An indefatigable perseverance in this research led them to undertake a great number of fresh observa- tions, in which they employed such accuracy, and such critical nicety, that we may consider this effect as the chief cause of the progress of astronomy among the greeks. A little after Thales [A. c. 550], Cleostratm, an astronomer of the isle of Tenedos, proposed a luni- solar period of eight solar years, composed of four subordinate periods of two years each, in which one lunar month of thirty days was intercalated only 'three times; and the three intercalary months were added at the ends of the third, fifth, and eighth years. This period was called octaeteris. It is very simple, as is obvious ; and it would be perfectly ac- curate, if the solar year were of 365 days 6 hours, and the lunar year of 354 days : for the eight solar years would then contain 2-922 days, and the eight lunar years, with the addition of the 90 days form-. ing 99 ing the three incercalary months, would contain 2922 days likewise. But the two bases of the perioa being inaccurate, it leads to errour, and it was soon perceived to differ widely from the truth. Several other attempts of a similar kind were not more successful. The object however was gradually approached more and more near ; and two athenian astronomers, Mcton and Euctemon, enjoyed for a time at least the glory of having attained it. By sagaciously combining all the observations then known, they formed a lunisolar period, or cycle, of nineteen solar years, twelve of which were composed of twelve lunations each, and the other seven, of thirteen, making in all 235 lunations. The unequal numbers of lunations they distributed by intervals among the whole number of years that composed the cycle, intercalating them in the 3d, 6th, 8th, llth, 14th, 17th, and 19th. Beside this, instead of supposing, according to the common cus- tom, that the lunar year consisted of six months of thirty days each, and as many of twenty-nine, they made up their 9,3d lunations of 125 months of thirty clays each, and only 110 of twenty-nine days, which gave 6940 days in the whole number of lunations. This is likewise very near the amount of the days in nineteen solar years. The use of this cycle was adopted on the 16th of July, in the year 433 before Christ ; and was called the metonic cych, no doubt because Meton was the principal person concerned in it's invention. This discovery, in which we may observe extensive astronomical knowledge, and every appearance of great accuracy, had such success and reputation in H 2 Greece, 100 Greece, that the order of the period was engraved in letters of gold, whence it acquired the name of the golden number. For a long time it served as a basis in calculating the calendar among all the nations of Europe ; and it is even still retained in use, by the help of certain modifications and changes, which it is found occasionally to require; for in astronomical strictness it is defective with regard to the motion both of the Moon and of the Sun. The 6940 days exceed the real duration of 235 lunations about 7 hours 28 minutes, and that of 19 solar years about t) hours 28 minutes : besides, the new and full moons, and other phases of this satellite, do not re- cur precisely at the same periods from one cycle to another. These defects having become perceptible in the course of four or five cycles, Calippus, another athe- nian astronomer, in the year 338 before Christ, pro- posed a new cycle of 76 solar years, or four metonic cycles, retrenching one day from them at the con- clusion of this period ; so that his cycle comprised three subdivisions of 6940 days each, and a fourth of 6939. Thus, while he deviated from the simpli- city of the metonic cycle, he obtained greater accu- racy; but the motions of the Sun and Moon were not yet either of them represented with sufficient precision, and the grand problem of their coincidence remained still to be solved. This difficulty the sub- sequent greek astronomers endeavoured in vain to surmount. All nations have had their cycles, and their pecu- liar calendars : .but none have succeeded, in bring- ing 101 ing the motions of the Sun and Moon perfectly to coincide with each other. Every reader versed in the theory of the universal gravitation of the celestial bodies will readily con- ceive the reason. A perfect cycle, as it is continu- ally renewed, must bring back the Sun and the Moon to the same point of the heavens at, the, end' of 'each' revolution ; and the new moons, the full moons, &c. to the same periods, from one cycle to another. But the union of all these conditions may be consi- dered as impossible. In fact, 1st, the motion of the Moon round the Earth, being incessantly altered by the attraction of the Sun, and of the other bodies of our planetary system ; and in like manner the appa- rent motion of the Sun round the Earth, or the real motion of the Earth round the Sun, being disturbed by the attraction of the Moon, and of the other planets ; must it not be the mere effect of chance, if in two consecutive cycles, particularly were they not very long, the Moon and the Earth should J32 both precisely in the same situation with re- spect to the forces by which they are acted upon, .and that the times of the cyclical revolutions should be exactly equal? dly, should the times of the , cyclical revolutions be equal, the intervals of the time comprised between the phases of the same kind, jn the succession of the cycles, would not be equal : as, for instance, the time from one new Moon to another varies continually, and is subject to numer- ous inequalities, produced by the attraction of the surrounding orbs. Thus we have another source of imperfection in the cycles. Hence we may conclude, that they can serve only to indicate nearly the corre- H 3 spend- 102 spondence between the motions of the Sun and the Moon. Astronomical calculation is incomparably more certain and exact ; and accordingly societies of the learned have been accustomed, for more than a century;. ;to publish Ephemerides, for previously making kridw'n U> mariners and astronomers the state ;of:tte'6eay:Qns^And these publications are in fact of great utility to persons of either description. From the time when the school of Plato was esta- blished, several astronomers were formed, most of whose useful labours have perished, and of the rest only the substance or fragments have been preserved in the works of other ancient writers. Among these Eudoxus, who has been already mentioned as a geo- metrician, is particularly distinguished. He was a great observer, had written several books on astro- nomy, and the observatory he built at Cnidus, his birthplace, was shown long after his death. For se- veral years lie published celestial ephemerides, which were highly famed, and put up in public places, such as the Prytaneum at Athens. Some authors vaguely mention a sphere of Eu- doxus, which they date twelve or thirteen hundred years before Christ, but we have no other account of this ancient astronomer. Hence it has been supposed by others, with greater appearance of probability, that the explanation of the celestial motions, known by the name of the sphere of FAidoxus, was the work of our platonic philosopher, and consequently not older than the fourth century before the Christian era: It's intention was to show, for the climate of Greece, the rising and setting of the Sun and Moon, those of 103 of -the constellations, the new moons, c. On these subjects the Eudoxus of whom we are speaking hud composed two works, known to the ancient astro- nomers, and quoted by them : one was a description of the constellations, the other treated on the times of their rising and setting. Eudoxus is blamed for having endeavoured to ac- count for the appearances of the planets by a very complicated .mechanism, little consistent with pro- bability; in which he employed a number of circles, included one with another, and moving in contrary directions scarcely compatible. But unacquainted with the motion of the Earth, which explains every thing in such a simple manner, or not .daring to avow it, could he .do better at the time in which he lived? and do we not owe him some gratitude at least for having suggested the idea of calling in mechanics to the aid of astronomy ? Under Antiochus Gonatus king of Macedon, Aratus, by order of that monarch, reduced all that was known of astronomy at that time into greek verse. A. c. %lQ. His poem, which has reached us entire, is divided into two books. The first of them, under the title of Phenomena, contains an expla- nation of the sphere of Eudoxus : the second, en- titled Prognostics, though not in the sense of judicial astrology, with which astronomy was not at that time infected, exhibits the physical signs, which are the harbingers of fair weather, of rain, or of storms. It ^enjoyed considerable reputation among the ancients. Cicero translated the Phenomena into latin ; and we have likewise a great part of the poem translated into the H 4 same 104 same language by that Germanicus, who was so dear to the romans, and a victim to the jealousy of Tibe- rius ; and even a third translation of it exists, the performance of Avienus, who lived in the reign of Theodosius. ; While astronomy made such great progress in Greece, it was cultivated with success by some of the western nations of Europe, among whom we may reckon the ancient gauls. A. c. 380. Cxsar relates in the first book of his Commentaries, that the druids, in their instruction of youth, particularly taught what concerned the motion of the stars, and the ex- tent of the Earth and the heavens, in other words, astronomy and geography. If the gauls left no ob^ servations behind them, or if they have been lost in the lapse of time, we know at least, that they were well skilled in navigation, which is essentially connected with" astronomy. Dominic Cassini, in his- Essay or* the Origin and progress of ancient Astronomy *, says, that they had founded colonies on the coasts of Spain, on the Euxine sea, and in several other places. Pytheas, a celebrated astronomer of Marseilles, ob- served in that city the meridian altitude of the Sun, at the time of the solstices, by means of a gnomon, 380 years before Christ. The purpose of these ob- servations was merely to determine the latitude of Marseilles. On comparing their result with that of modern observations, some astronomers have con- cluded, that the obliquity of the ecliptic has di- * Ancient Memoirs of the Academy, Vol. vin. minished 105 minished since that period about a minute in a cen- tury : but this fact is not sufficiently established. The philosopher of whom we are speaking did not content himself with observing the phenomena of nature in his own country. He visited remote re- gions, and proceeded through the Atlantic very far toward the north. As he advanced, he observed ai* evident diminution of the length of the night at the summer solstice; and when he reached an island, which he calls Tliule, he perceived that the Sun rose presently after it hacl set. As this is the case in Iceland, and the northern part of Norway, it has been inferred, that he reached these countries. The ancients, by whom these parts were deemed uninhabit- able, treated the narrations of Pytheas as fables ; but modern navigators have confirmed the truth of the facts he related, and have confirmed to him the glory of having been the first person, who taught us ta distinguish climates by the difference in the length of days and nights. Several other discoveries are ascribed to Pytheas; as having made known to the greeks, that the polar star is not precisely at the pole itself, but forms with three other stars in it's vicinity a trapezium, of which the pole is nearly the centre ; he, also, pointed out the connexion of the phenomena of the tides with the piotion of the moon ; &c. The taste of Alexander for the sciences, and par- ticularly his desire of making known to posterity the countries to which he had extended his conquests, were of great advantage to astronomy, as well as to every part of natural philosophy in general. A. c. 330. 106 330. On these subjects Aristotle wrote by his order a great number of works. In that entitled De Calo he proves the spherical shape of the Earth from the circular shadow it casts on the moon in the eclipses of this satellite ; and also from the variation, that ap- pears in the altitudes of the stars, as we approach or recede from the poles. The book De Mundo, ascribed to the same philosopher, contains a description of the ancient World, which the author divides into three great continents, Europe, Asia, and Africa, But the most important service, rendered the .sciences by Alexander was that of causing an accurate and particular account of the countries under his do- minion to be taken, not merely from estimation, and the accounts of travellers which are always uncertain, but by direct measurements, and observations of the correspondence of terrestrial objects with the po- sitions of the stars. From this time geography gra- dually became, through it's alliance with astronomy, a real science, which was extended and improved, and has conferred on commerce the greatest ad- vantages, by the intercourse it has established be- tween distant nations. Callisthenes, who has al- ready been mentioned, was appointed to superintend this task. The hypothesis of the sphericity of the Earth was very ancient: it had sprung up, as before noticed, in the time of Anaximander and Pythagoras. It was also discovered, that the Earth was separate from the heavens, that it remained suspended in space, and that it was not of excessive magnitude. All these ideas were founded on the observation of the daily motion of 107 of the stars from east to west, and on the chaiigei observed in the position of these stars, when V?Q travel nearly under the same meridian toward the north, or toward the south. And soon the com- parison of the apparent change in the stars with the corresponding lengths of the distance passed over on the Earth suggested the idea of measuring the circum- ference of the globe by means of observations of the stars. Aristotle, the most ancient author of whom we have any writings on this subject, expresses him- self as follows in his second book De ctzlo, chap, xiv. * In eclipses of the Moon the line that bounds the eclipsed part is always a curve: and as the Moon is eclipsed by the shadow of the Earth, it is certain, that this appearance is caused by the circumference of the Earth, which is spherical. Indeed it is evident from the appearances of the stars, that the Earth is round. "It's 'extent too cannot be very considerable: for, if we travel ever so little either toward the North or toward the South, the horizon manifestly varies in such a manner, that the stars over our heads arc altered, not being the same to those who travel north, as to those who travel south.' Aristotle adds : * those mathematicians, who have attempted to de- termine the magnitude of the Earth's circumference, say, that it is 400000 stadia.' By those mathematicians, we have every reason to presume, Aristotle means the pythagoreans, who con- sidered the Earth as a star, and made it revolve around the centre of the World, so as to produce the alterations of day and night; an opinion, which Aris- totle himself attempts to refute in the preceding chap- ters. 108 ters. He clearly speaks only as an historian, when he mentions the measure of the Earth. Horace fur- nishes us with a proof, that this measurement is to be ascribed to the pythagoreans ; for in the 28th ode of the 2d book he calls the pythagorean philo- sopher Archytas, who had been Plato's master, the measurer of the Earth. Eratosthenes, librarian of the alexandrian museum, is the first of the ancients, from whom we have a measurement of the Earth by a method consistent with the principles of geometry arid astronomy. A. c. 280. This measurement, admired in it's time as a prodigy of human sagacity, has been transmitted to vis by Cleomedes : Ci/cl. Tkeor. book I, chap. 10. Eratosthenes was informed, that, at the time of the summer solstice, the Sun at noon was vertical to the city of Syene, situate on the borders of Ethiopia, imder the tropic of Cancer. A well is particularly mentioned to have been constructed in this city, which was illuminated throughout it's whole depth by the Sun at noon on the day of the solstice. He knew likewise, or at least he imagined, and he was not far from the truth, that Alexandria and Syene were both under the same meridian. On these data he constructed a concave hemisphere at Alex- andria, from the bottom of which arose a vertical style, terminating at the centre of curvature of the hemisphere. Then, supposing the city of Syene to be in the vertical direction of the style, he ob- served, that the arc, included between the foot of the style and the extremity of it's shadow projected on the concavity of the hemisphere by the meridian Suu 109 Sun at the solstice, was equal to a fiftieth part of the whole circumference. Hence he inferred, that the arc of the Heavens comprised between Alexandria and Syene must he the same; and that the distance between the two cities must likewise be a similar arc, or a fiftieth part of the whole circumference of a great circle of the Earth. Now on measuring this distance, or the length of this arc, it was found to be 5000 stadia, which gave 250000 stadia for the length of the entire circumference of the Earth, and 6944 stadia to a degree. Some later astronomers, desirous of avoiding the fraction, and supposing, that it was impossible to answer for the accuracy of the measure to five or six stadia in a degree, extended this length to 700 stadia, which give 252000 stadia for the mea- sure of the whole circumference. There is another ancient measure of the Earth re- ported likewise by Cleomedes, which is that of the phi- losopher Posidonius, who was contemporary with Pom- pey. A. c. 6(). This philosopher, having observed, or been informed, that the star Canopus did but just ap- pear on the horizon at Rhodes; while at Alexandria, which he placed under the same meridian, it rose a forty eighth part of the circumference of the heaveus above it, which answers to a forty eighth part of the circumference of the Earth : and supposing, that the distance between Rhodes and Alexandria was 5000 stadia ; he reckoned 240000 stadia for the entire cir- cumference of the globe, or 666- to a degree. But it was soon after found, that these two determi- nations exceeded the truth, because Posidonius had made the distance of Rhodes from Alexandria much greater rid greater than it really was. Strabo, who wrote h& geography in the time of Augustus, asserts, that Era- tosthenes had measured this distance, and found it to* be only 3750 stadia : whence we should have only 180000 stadia for the length of the entire circum- ference of the Earth, and 500 stadia for that of a degree. It remains for us to ascertain the proportion of the stadium to some of our present measures, that we may be enabled to compare the length assigned to a degree by the ancients with what it has been deter- mined to be by the moderns. Some authors affirm, that both Eratosthenes and Posidonius employed the greek stadium, which is 607 feet and a half english measure ; others, the egyptian stadium, which is 731 feet and a half. Supposing it V> have been the greek stadium, the first measure of a degree by Eratosthenes would be 421875 feet; the second, 425250: the first by Posidonius, 404999; and the second, 303750. Of these four different measures three are more or less erroneous in excess^ while the fourth errs by deficiency ; the degree being 365640 feet, or thereabout, according to the measures of the moderns. Supposing the egyptian stadium to have been used, the first three err greatly by ex^ cess; but the fourth, giving 365672 feet, differs little from the modern measures. But this agreement can be no more than the effect of chance, or a false esti^ ination of the stadium ; for the methods of Era- tosthenes and Posidonius were not susceptible of great precision, and cannot be compared with the modern in this respect. The discussion of this sub- ject, ill ject, on which the reader may consult several excellent memoirs among those published by the Academy of Beiles-Lettres, I shall pursue no farther ; but return to the general history of astronomy, at the time of Alexander, The impulse, which this prince had given to the astronomy of the greeks, was powerfully seconded by the encouragements and liberalities of the new kings of Egypt, who sought out the men most illustrious for their learning in all parts of the World, and in- vited them to the museum of Alexandria. A. c. 300. It was here, during the space of twenty six years, reckoning from the year 295 before Christ, that Aristillus and Timocharis made an immense series of observations, as well on the position and number of the fixed stars, as on the motions of the planets : and these observations afterward served, as the basis, on which Ptolemy founded his theory. About the same time flourished Aristarchus of Samos, who rendered himself illustrious in astronomy by several interesting opinions or discoveries. He observed a solstice in the year 281 before Christ, according to Ptolemy's calculations, which fixes with precision the age of this astronomer, respecting whom historians, not well informed, express themselves doubt- fully. We have a very simple method of his, if not very accurate, for determining the ratio of the distances of the Moon and the Sun from the Earth. It con- sists in observing the moment, when the plane of the circle, which, in the different phases of the Moon, separates the dark from the enlightened part, is di- rected towards the eye of an observer on the Earth? and lit projected on the lunar disk in a right line ,* and then measuring the arc of the heavens comprised be- tween the Moon and the Sun ; and lastly, in con- ceiving a rectangled triangle, the right angle of which terminates at the Moon, while it's three sides are formed by the three lines which join the Earth, the Moon, and the Sun. In this triangle, it is obvious, the three angles are known, from which the ratio of it's three sides to each other may consequently be deduced. In this manner Aristarchus found, that the Sun is eighteen or twenty times as far from the Earth as the Moon ; which is not very accurate, for it's distance is three or four hundred times greater : but to have attempted the solution of a problem at that time so difficult, and so complicated, was no trifling matter. By the strong probabilities, derived from observations, with which he supported the pytha- gorean system of the Earth's revolution round the Sun, Aristarchus acquired more real and permanent glory as a geometrician and astronomer. Accord- ingly this grand truth was matured by degrees in minds capable of conceiving it, till at length it ac- quired strength enough to appear in open day, like Minerva issuing armed from the brain of Jupiter. The emulation of the philosophers who addicted themselves to astronomy was not the sole cause of it's progress : but for this it was partly indebted to the invention of some new instruments, with which it was gradually enriched, and by means of which ob- servations were rendered more easy, more accurate and more numerous. Among others of these are mentioned the rings, which Eratosthenes caused to 5 be 113 be constructed in the museum of Alexandria. These, according to the description given by Ptolemy, were an assemblage of different circles, not much unlike our ancillary sphere, which probably derived it's origin from them. First there was a great circle performino* the office of a meridian : the equator, ecliptic, and two colures, formed an interiour assemblage, turning on the poles of the equator. There was likewise a circle turning on the poles of the ecliptic, and fur- nished with vanes diametrically opposite to each other, it's concave side nearly touching the ecliptic, or car- rying an index, to point out the division at which it stopped. Such was the general form of the instru- ment. It was applied to several uses : the following, for instance, is the manner, in which it was em- ployed to determine the equinoxes. The equator of the instrument being placed with great care, as it always ought to be, in the plane of the celestial equator, the observer watched the mo- ment, when the upper and lower surfaces were no longer illumined by the Sun ; or rather, which was less liable to errour, when the shadow of the an- teriour convex portion of the circle completely co- vered the concave part, on which it was projected. It is evident, that this point of time was that of the equinox. When this did not take place, which indi- cated that the equinox occurred during the night, twt> observations were selected, in which the shadow was projected on the concave part of the circle in opposite directions, and the mean of the interval between these observations was considered as the instant of the equinox. i Not 114 Not contented with having rendered it easier for others to take observations, Eratosthenes made a great number himself. He likewise wrote several books on astronomy, which are quoted by the ancients, but of which one alone, a description of the constellations, has escaped the ravages of time. His genius led him to things out of the common track, one proof of which is his measure of the Earth. Of all the ancient astronomers, no one has so much enriched the science, or acquired so great a name, as Hipparchus, a native of Nice in Bithynia. A, c. 142. Among these he holds nearly the same rank, as Archimedes among the geometricians. He began by making observations at Rhodes, and after- ward settled at Alexandria, where he executed all those labours, which established ancient astronomy on certain foundations, and furnished the moderns with points of comparison for a multitude of astro- nomical theories. One of his first cares was to rectify the duration of the year, wliich before his time was made to consist of 365 days, 6 hours, and which he found to be a little too long. By comparing an observation of his own at the summer solstice with a similar one made a hundred and forty five years before by Aristarchus of Samos, he shortened it about seven minutes ; which however Mas insufficient. But that Hipparchus did not come nearer the truth must unquestionably be ascribed to some inaccuracy in the observation of the samian astronomer: for those of Hipparchus himself, being compared with modern observations, give 365 clays, 5 hours, 49^ seconds, for the duration of the year ; a result 115 a result differing scarcely a second from what is found on comparing the best observations of our own day with those of Tycho Brahe. In general, modern ob- servations, in which the assistance of glasses is em- ployed, are much more accurate than those of the ancient astronomers, who observed the stars merely through vanes by means of the naked eye. But in, questions, where the inevitable errours of observations are diffused over a long interval of time, as on the present occasion, the comparison of ancient with mo- dern observations may afford a result nearly as ac- curate, as that which is derived from a comparison of the latter alone. The ancient astronomers supposed, that the Sun, in it's annual motion, proceeded uniformly in a cir- cular orbit : but this uniformity, believed to be real, was altered, in appearance at least, with respect to the Earth. The general effect was known : but Hippar- chus investigated and assigned it's cause. He ob- served, that the Sun was about 94 days 12 hours in proceeding from the vernal equinox to the summer solstice, and only 92 days 12 hours in it's progress from the summer solstice to the autumnal equinox ; which gave 187 days, nearly, for the time spent by the Sun in traversing the northern part of the ecliptic, and 178 days only for it's progress in the southern portion. Of course the Sun must either move, or ap- pear to move, with greater velocity in the southern part of the ecliptic, than in the northern. Without relinquishing the hypothesis of the real uniformity of the Sun's motion, Hipparchus explained the ine- quality of the motion with respect to the Earth by i 2 placing ne placing the Earth at a certain distance from the centre of the ecliptic. This distance, which he termed the eccentricity of the solar orbit, gave rise to an equation between the real and apparent motions, sometimes additive, at others subductive, by means of which the two motions were made to correspond in every instant. He determined the quantity of the eccentricity with respect to the radius of the ecliptic, as well as the position of the line of the apses, or the line which joins the diametrically opposite points, in whidh the Sun is at it's greatest and least distance from the Earth. He made similar remarks and cal- culations for the lunar orbit. From these data he constructed tables of the motions of the Sun and Moon, svhich are the first of the kind that are men- tioned. All these determinations were offered as at- tempts, which time and farther observations were to bring to perfection. Hipparchushad formed the pro- ject of constructing similar tables for the motions of the five planets Mercury, Venus, Mars, Jupiter, and Saturn: but judging, that the observations then known were insufficient, to furnish elements suffi- ciently accurate, he relinquished this task. Though the eccentricities of the orbits of the Sun and Moon, as determined by Hipparchus, differed not widely from the truth, it must be remarked, that they were affected by a radical errour : they supposed these orbits to be perfect circles. The ancients never suspected, that the planets in reality described ellipses : much less did they imagine, that these ellipses them- selves are continually varied and altered in their figure 117 figure by the universal and reciprocal gravitation of the celestial bodies. Hipparchus, however, made another discovery, which, having been confirmed and brought to perfec- tion by time, is become one of the principal founda- tions of astronomy. On comparing his observations with those of Aristillus and Timocharis, made a hun- dred and fifty years before, he found that the stars constantly preserved the same positions with respect to each other, but that they all had, or appeared to have, a trifling motion in the order of the signs of the Zodiac, or from west to east, the quantity of which was two de- grees in a hundred and fifty years, or 48 seconds in a year. This motion having been since studied and observed with persevering attention, has been found to be a little more than 50 seconds a year. Hence it follows, that the Sun and a given star, setting out together from the same point of the ecliptic, and proceeding from west to east with velocities* which should be in the ratio of 360 degrees to 50 seconds of a degree, the Sun would return to the point of de- parture in a shorter time than it would to the star by a quantity corresponding to 50 seconds of a degree. By calculation we find, that, the former period of time, which constitutes the tropical year, being 365 days, 5 hours, 48 minutes, 48 seconds, the latter, or sidereal year, is 365 days, 6 hours, 9 minutes, 10 se- conds. The tropical revolution, we see, brings back the solstices and equinoxes before the sidereal revo- lution is finished, or the equinoctial points appear to recede with respect to the stars. Hence is derived the term of precession of the eguinoxes, given to this i 3 motion 118 motion by which the equinoxes are found to anticipate the sidereal revolution. The physical cause of this precession of the equinoxes, with that of the vari- ations to which it is subject, will appear hereafter. The quantity and cause of the third kind of year, the anomalistic, will likewise be pointed out. The method given by Aristarchus of Samos for de- termining the ratio of the distances of the Sun and Moon from the Earth was very imperfect, as has been already observed : and besides it was incapable of making known the absolute quantities of these distances. To this method Hipparchus substituted others more complete, in which he made use chiefly of the parallaxes. The reader is not to be informed, that the parallax of a star is the angular quantity comprised between the point in the heavens to which a star is referred, when seen from a given point on the surface of the Earth, and the point to which it would be referred, were it observed from the centre of the Earth. This is equal to nothing, when the star is in the zenith of the observer, and greatest when it is in his horizon. The parallaxes of the common planets, as the Moon, Mars, Jupiter, &c. are easily ascertained, and thence the distance of these planets from the Earth are af- terward deduced. The distance of the Sun from the Earth is a subject of more delicate investigation, and more liable to errour. To discover it, Hipparchus began with calculating the distance of the Moon from the Earth, taking the Earth's radius for a basis, or by means of the horizontal parallax of the Moon. Jn this there was no difficulty, since the sine of the horizontal horizontal parallax of a star is as the sine of the angle under which it's horizontal semidiameter is seen, and in the case before us, we have a right angled triangle, of which the three angles are known, and also one of the sides, namely the radius of the Earth, by the measure of Eratosthenes : whence the hypothenuse, or the distance of the Moon from the centre of the Earth, may be obtained. Having then measured the appa- rent diameter of the Sun, as he had measured that of the Moon ; and having calculated, by the dura- tion of a lunar eclipse, the breadth of the cone of the shadow traversed by the Moon ; from all these data he formed triangles and analogies, which led him to conclude, that the distance of the Sun from the Earth was equal to about twelve or thirteen hundred times the length of the Earth's radius, or that the horizontal parallax of the Sun was about three mi- nutes. This conclusion is very remote from .the truth : but it will not be a matter of wonder, when we consider, that Hipparchus has employed in his calcu- lations a multitude of elements, which could not be determined with sufficient precision in his time. In fact the moderns, enriched by all the knowledge of their predecessors, and furnished with the best instru- ments, were long before they had determined with ac- curacy the horizontal parallax of the Sun. It is but little more than a century ago, that la Hire and the Cassinis made it fifteen seconds ; while in reality, ac- cording to the best observations of our days, it is about eight seconds only : which carries the Sun to a prodigious distance in celestial space. i 4 An 120 An extraordinary phenomenon, the almost sudden disappearance of a large star, in the time of Hip- parchus, excited that indefatigable astronomer to number the stars, and to note down their configura- tions, respective situations, &c. ; in order to enable posterity to know, whether they were permanent bo- dies, invariably fixed in the vault of Heaven, and preserving always the same position with regard to each other ; or whether, exclusive of the motion that produces the precession of the equinoxes, they be not subject to other irregular and unknown motions, in which case the motions of the planets could no longer be referred to them as points of comparison. This immense labour laid the founda- tion, on which the whole superstructure of astronomy was to be raised. He was admired and celebrated in all nations, where learning was pursued. Pliny speaks of him with enthusiasm. ' Hipparchus, ' says he, l has never been sufficiently praised : no one has given proofs like him, that man is allied to Heaven, and that his mind is a portion of the Deity He lias braved the anger of the gods, in making known to man the number of the stars. . , thus bestowing the heavens as a portion on those who are capable of making use of it.' Hist. Nat. lib. n, chap. 26. To so many important researches, immediately connected with the progress of astronomy, Hippar- chus added the merit of applying this science to fa- miliar purposes of the greatest utility to geography and the promotion of commerce. He reduced to certain and invariable principles the method of de- termining termining the situation of terrestrial objects by their latitude and longitude, of which men had some no- tion as early as the time of Alexander. The prin- cipal points being once directly fixed by astronomical observations, the topographical details, by which they are connected together, are nothing but easy operations, which may be executed and abridged by means of various instruments, as the graphometer, the plane table, c. The limits of this sketch oblige me to pass over in silence the rest of the works of Hipparchus, such as his researches respecting the calendar, astronomical calculation, &c. He likewise undertook to correct the measure, which Eratosthenes had given of the circumference of the Earth, but we know not what he made it. In these discoveries he was followed by several astronomers, who, without equalling him in genius and learning, contributed notwithstanding to the progress of the science, by the new observations with which they enriched it, or the works in which they displayed it's theory. In the number of these benefactors of astronomy pos- terity reckons the philosopher Posidonius, whom I have already mentioned on the subject of measuring the Earth. He lived in the island of Rhodes, A. c. 60, where he made many observations. To represent the state of the heavens he constructed a movable sphere, of which Cicero speaks with admiration. .Tusc. i, de Nat. Deo. i. If Posidonius were not an astronomer of the first order, he deserves our attention for a moment by his 4 moral moral character and private life. He was a celebrated stoic, enjoying the highest reputation in his own country, and in no less degree the esteem of the ro- mans. Upon a certain occasion, Pompey, passing through Rhodes, went to pay him a visit, and forbad his lictors to knock at the door, as was the custom : ' thus, ' says Pliny, c he, before whom the east and the west bowed their heads, lowered his own fasces before the door of a philosopher.' The strictness of the stoical principles of Posidonius is known by a re- markable circumstance. In a speech which he deli- vered before the same Pompey, he was suddenly seized with such a violent fit of the gout, that the sweat flowed down his face. This acute anguim he bore at first stoutly, without uttering a complaint, without altering the tone of his voice, without interrupting his discourse. At length nature proving the stronger, this exclamation escaped him, but wa,s immediately stifled by philosophic pride : ' Pain, thou shalt not get the better of me: I will never confess that thou art an evil. * Cleomedes, who lived a little later, has left us a work entitled, The cyclical theory of Meteors, or of the Celestial Motions ; in which he treats of the sphere* of the periods of the planets, of their dis- tances, of their magnitudes, of eclipses, &c. He acknowledges that he derived all this knowledge from Pythagoras, Eratosthenes, Hipparchus, and Posidonius, either by their writings or by tradition : yet his work is valuable, as it is the most ancient that has reached us on these subjects. Nearly 123 Nearly the same may be said of the Elements of Astronomy by Geminus, whom some circumstances indicate to have been a contemporary of Cleomedes. Geminus speaks very much at length of the observa- tions of the chaldeans, and of the lunisolar periods they invented. The system of the arrangement and motions of the planets, which he proposed, was de- veloped and explained a hundred and fifty years after by Ptolemy. Perhaps the reader will little expect to find Julius Caesar in the list of astronomers : yet it is incumbent on us, not to deny him this honour, since he was really well skilled in this science, and in particular did an important service to the roman calendar. A. c. 46. This calendar was established by Numa Pompi- lius, the second king of Rome. Some inaccuracies in it's fundamental principles, and the accumulation of fresh errours, had gradually brought it into such confusion, that at the time of Ceesar the autumn oc- cupied the place of the winter months of the calen- dar, and the winter occurred in the months of spring. When Caesar became dictator, he invited the astro- nomer Sosigenes from Athens to Rome, to assist him in correcting this disorder. They began by giving the year of Rome 708 fourteen months, to reesta- blish the order of the seasons. They then assumed as a basis, that the duration of the common year was 365 days 6 hours : and this has been called the Julian year, from the name of Julius Caesar. But as this term exceeded that of the old egyptian year by six hours ; and it would have been inconvenient for civil and political purposes, to have made the year begin 124 begin sometimes at one hour of the day, and sometimes at another; it was determined, that the beginning of each year should fall constantly on the same hour of the day, that the ordinary year should consist of 365 days, and that the six hours should be left to accumulate for three successive years, on the expira- tion of which a day should be added, so that every fourth year should have 366. The additional or in- tercalary day was assigned to the month of february : so that in the common year the 24th of february was called the 6th of the calends of march, or the 6th day before the calends of march ; and this day Cresar ordered to be reckoned twice every fourth year. Thus there were two days in this year, each of which was called the sixth of the calends of march ; and in consequence these years derived the name of bissex- tile, from bis sexto ante calendas Martii. The form of this calendar was very simple ; but it rested on the hypothesis, that the duration of the year is of 365 days 6 hours ; which is not accurate, it being in reality about eleven minutes shorter. These differences accumulating rendered at length a reform of this calendar necessary, to which I shall return hereafter. Some other illustrious romans are mentioned as very skilful astronomers^ as Cicero, Varro, &c. ; but no record of their, observations, or of their ac- quirements in this science, has reached us. In the reign of Augustus, A. c. 6, appeared the latin poem of Manilius, entitled Astronomicon *. It * Pingre published a tranflation of Manilius into French, in 1783, and added notes to it, of more value than the poem itfelf. is 125 is divided into six books; and, like the poem of Aratus, contains an explanation of the celestial mo- tions according to the sphere of Eudoxus. The poe- try is excellent; and the exordiums of the several books, with the moral digressions, are particularly admired. Unhappily it is stained with all the reveries of judicial astrology. It is the first work of the ancients, in which this deceptive art shows itself, and is exhibited in a body of systematic doctrines. We find no trace of it in the poem of Aratus, or in the accounts of the labours of Thales, Pythagoras, Hip- parch us, &c. It has derived it's birth from the na- tural propensity of mankind, particularly of princes and the great, to believe in the marvellous, and to receive without examination every thing that tends to flatter their vanity. Greedy quacks, acquainted with some of the secrets of nature, availed them- selves of these, to acquire credit with the great, and to persuade them, that their destiny, and the fate of empires, were written in the skies : they ventured to utter equivocal and mysterious predictions, which it was always easy to make conformable with events :, the errour spread abroad, and soon became general : it has continued more than sixteen hundred years, and at last has yielded only to the repeated attacks of philo- sophy. But by a lamentable fatality, which seems to condemn mankind to be everlastingly deceived, quackery is perpetually reviving under new forms, more or less gross ; and we see it in all ages usurping without a blush the places and rewards due to real talents, to virtue, and to genius, Menelaus, 126 Menelaus, who has been mentioned already as a geometrician, A. D. 55, distinguished himself like- wise in astronomy by several excellent observations, and by the discovery of the principal theorems of spherical trigonometry, which are highly necessary and useful in subjecting observations to calculation. Astronomy had begun to languish in the school of Alexandria, when the celebrated Ptolemy began to revive it, to augment it's stores, to reduce all it's parts to more order and consistency, and to collect it's scattered fragments from ail the writings and traditions that existed in his time. A. D. 140. Some make him a native of Pelusium, others of Ptolemais in Egypt. But this is of little importance : it is enough that we know he came very early to Alexandria, and that his immense labours were executed there. His principal work, entitled Almagest *, contains all the ancient observations, and theories, to which his own researches being added, he may be considered as having formed of the whole the completest collection of ancient astronomy, that ever appeared ; a work that may supply the place of those by which it was pre- ceded, and which the hand of time has destroyed. Ancient observations, and particularly the cata- logue of the stars composed by Hipparchus, having taught Ptolemy, that these stars constantly retained the same position with respect to each other ; he had fixed bases, to which he could refer the motions of * A word derived from the arable, and signifying The grtat Col- lection, the 127 the planets ; and he applied himself to the determi- nation of their courses through the heavens their re- spective arrangement, and their distances from the Earth, with more accuracy than had hitherto been obtained. If we consult appearances, the Earth seems to oc- cupy the centre of the universe, and all the motions in the heavens to be made around it. Pythagoras, however, opposed this idea : he placed the Earth among the planets, and made it revolve round the Sun with the rest of them. Aristarchus of Samos afterward embraced the opinion of Pythagoras, and supported it by powerful arguments. But the prejudice in favour of the immobility of the Earth was too deeply rooted, and too agreeable to the testimony of the senses, easily to give place to a truth, which the force of genius might rather be said to have di- vined, than to have proved, or rendered comprehen- sible to the vulgar. Ptolemy embraced the common opinion. He supposed that the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn, re- volved at different distances round the immovable Earth, in the order in which they are here given. All his explanations of the motions of the planets rested on this hypothesis, to which his authority as an astronomer ensured a universal reception, and which has descended to posterity under the name of the Ptolemaic system. As soon as he began to make the application of his system, the apparent motion of the planets with regard to the Earth presented difficulties, which he could neither elude nor surmount, but by new and very 128 very embarrassing hypotheses. It has already been observed, that the planets Mercury, Venus, Mars, Jupiter, and Saturn, sometimes appear to move di- rectly forward with respect to us, sometimes back- ward, and sometimes to be stationary. To account for all these appearances, Ptolemy supposed, that each particular planet describes in space a small cir- cle, which he called a deferent ; and that all these cir- cles, carrying each it's planet with it, described other concentric or eccentric circles round the Earth. The motion of the planet round it's deferent circle being combined with the motion of this deferent round the Earth, a compound motion is produced, which explains the successive aspects of the planets with regard to the Earth. But it may be imagined, that such a complication of motion, and of real or optical appearances, must form a chaos difficult to unravel. Every person is acquainted with the witti- cism of Alphonsus x, king of Castile, surnamed the astronomer. Though he believed in this mechanism of the heavenly bodies, the perplexity he observed in it led him to say : ' if God had summoned me to his council, when he created the World, I would have advised him better.' This speech was consi- dered at that time as impious, because it was sup- posed, no doubt, that Ptolemy had assisted at the council of the Creator. The progress of the stars in longitude, which was discovered by Hipparchus, was adopted and con- firmed by Ptolemy, who only thought it necessary to diminish it a little. According to Hipparchus, this progress, or the consequent retrogression of the equi- noxiai 129 ttoctial points, amounted to two degrees in a hundred and fifty years, or forty eight seconds of a degree in a year; which is somewhat below the truth. Pto- lemy reduced it to one degree in a hundred years, or thirty-six seconds annually, which is still farther from the true quantity. From this errour arose a perceptible increase in the duration of the year, which Ptolemy assigned from a comparison of the observations of his time with those of Hipparchus: he made it 365 days, 5 hours, 55 minutes, which was upward of six minutes too much. In his other inveftigations of the theory of the Sun and Moon he was more fortunate. Hipparchus had noticed the eccentricities of the orbits of these two bodies : Ptolemy demonstrated the same truths by different means. He made likewise a very im- portant discovery, which was wholly his own : he re- marked that noted inequality in the motion of the Moon, which is familiar to us at present by the name of ejection. It was known in general, that the ve- locity of the Moon in it's orbit is not always the same; and that it increases or diminishes, in propor- tion as the diameter of this satellite appears to in- crease or diminish : it was known too, that the greatest and least velocities take place at the extre- mities of the line of the apses of the lunar orbit : but astronomers had gone no farther than this. Pto- lemy observed, that the absolute quantities of these two extreme velocities varied from one revolution to another; and that the more remote the Sun was from the line of the apses of the Moon, the greater in proportion was the difference between these two velocities. 130 'velocities. Hence he concluded, that the first in- equality of the Moon, that which depends on the eccentricity of it's orbit, is itself subject to an annual inequality, dependant on the position of the line of the apses of the lunar orbit with respect to the Sun. Modern observations have fully demonstrated the truth of this theory ; and they have likewise made known a great many other inequalities in the motion of the Moon, which will be mentioned when I speak of the progress of astronomy in modern times. Beside the Almagest, of which a succinct account has been given, there exists another great work of Ptolemy's, his Geography , in which he determines the situation of places by their latitude and longitude according to the method of Hipparclms. If Ptolemy Lave made many mistakes in the situation of the towns and countries of which he has spoken, it must be remembered, that geography is the work of time ; that, at the period when Ptolemy lived, only a small portion of the old World was clearly known ; and that even at present, when the knowledge of astro- nomy is without comparison more widely diffused, .the situation of a multitude of places in both hemis- pheres is not accurately ascertained. It must not be omitted too, that this work contains the first princi- ples of the ingenious theory of projections now em- ployed in the construction of maps. Some books, in which judiciary astrology is treated of and explained, are published under Ptolemy's name : but learned critic* have shown, that he was not their author ; no doubt, therefore, some im- endeavoured to shelter their pernicious reveries 131 reveries under so great an authority. Thus much is certain, neither the Geography nor Almagest, the two greatest works of Ptolemy, contains the least vestige of it. Ptolemy had the ambition, like Archimedes, to transmit the remembrance of his labours to posterity by a public monument. In a fragment, which Bou- illaud published in 1668, Olympiodorus and Theo- dorus, two astronomers of Mytilene, relate that Ptolemy had put up in the temple of Serapis at Ca- nopus an inscription engraved on marble, in which the hypotheses of his astronomical system wqre ex- plained ; such as the duration of the year, the eccen- tricities of the solar and lunar orbits, the dimensions of the epicycles of the planets, &c. If men of greater genius than Ptolemy have ex- isted, at least no man ever collected a greater body of profound knowledge, or more truly conducive to the progress of astronomy, considering the age in which he lived. From Ptolemy to the time of the arabs no astro- nomer of the first order can be reckoned among the greeks, except perhaps Theon of Alexandria [A. o. 3^5], of whom we have a learned commentary on the Almagest. Among the different applications that have been made of astronomy to common use, gnomonics, or the science of dialling, much occupied the ancient astronomers : and indeed it merited all their atten- tion, by it's universal utility at that time in pointing out the hours of the day for civil purposes. At pre- sent it is not necessary, either in the country or in K 2 cities, 132 cities, where sundials serve only to regulate clocks and watches. Solar, lunar, and aftral dials were - formerly con- structed: but of these the first were without compa- rison mot in use. A dial is commonly a simple plane, on which the hours, and parts of hours, are marked by the projection of a shadow, or by a lumi- nous point passing through a perforated plate. Some- times however dials are drawn on curved surfaces, as on those of a cone, a cylinder, a sphere, &c. The principles of the construction are the same in every case : the only difference is in the length and multi- plicity of the operations requisite. I shall confine myself, therefore, to giving a general idea of sun- dials, traced on planes, and denoting the hour by the projection of shadows. The solution of this pro- blem is easily reducible to a simple question in geo- metry, as will clearly appear. Let us suppose, that the Sun moves daily round the concavity of an immense sphere, the centre of which is that of the. terrestrial globe considered as immovable; and let us conceive an axis passing through this centre perpendicularly to the equator, as well as to all the parallels which the Sun successively describes : it is evident, that, if we attribute a cer- tain magnitude to this axis, the Sun will continue to project it's shadow on the dial, that is in this instance on a plane in a given position, and passing through the centre of the celestial sphere. Hence it follows, that, to mark the hours of the day on the dial, we have only to determine the intersections of the plane of the dial by the series of' planes, which pass through tbe 133 the Sun at each instant of it's motion and through the axis of the globe ; a problem, which to thfe geo- metrician has no difficulty. The principle of this construction supposes, as we see, that the radius of the terrestrial globe is infinitely small in proportion to the radius of the circle de- scribed by the Sun's diurnal motion ; and this may be considered as true in practice. On the dial no more lines are traced, than are in- dispensably necessary. The style fixed in the dial,, and making part of the axis of the World, may be longer or shorter, at pleasure. Sometimes the artist js content with marking the hours by the arrival of the shadow of the point of the style on the lines drawn to denote them. There are dials, however, on which not only the hours and parts of hours are marked, but in addition some re- markable points of the path traced by the shadow of the point of the style, and the entrance of the Sun into the different signs of the zodiac. For instance, let us suppose a horizontal sundial made for the city of Paris : the solar ray, which passes by the point of the style, being indefinitely prolonged, and consi- dered as a physical and inflexible line, we see, that, during the revolution of the Sun, this line will de- scribe the surfaces of two cones joined together at their summit, which is the point of the style ; and that the shadow, projected by this point, will form on the dial a portion of an hyperbola, for each day, or each parallel, sipce by prolonging the plane of the dial it will cut the two opposite cones. Another pa- rallel gives another portion of an hyperbola. Now K 3 as 134 as all these portions of an hyperbola, being different in magnitude and position, would produce confusion on the dial, if they were traced out completely, the diallist contents himself with marking the points of the shadow for the entrance of the Sun into each sign of the zodiac ; and these points being joined together, a series of arcs is formed, which are called the arcs of the signs. The inventions of dials is very ancient Diogenes Laertius ascribes the first idea of them to Anaxi- menes. In the ninth book of Vitruvius we find a concise description of several ancient dials, the names given to them, and those of the persons by whom they were invented. To this work therefore I refer my reader, and likewise to the excellent notes, with which Claude Perrault has accompanied his translation, of this author CHAP* 135 CHAP. vi. Origin and Progress of Optics. IT would be waste of time to enter into the physical explanations, which the ancients, and Aristotle in particular, have given of the phenomena of vision. In these the abuse of occult qualities is carried to excess : but sometimes they were satisfied with ex- ploring nature by the torch of experiment, from which they failed not to reap advantage. Thus the school of Plato, for example, was clearly acquainted with the first principles of optics, namely the propa- gation of light in a straight line, and the property it has of being reflected at an angle equal to that of it's incidence. Long before this mirrors were constructed of metal; and the use of glass was likewise known, for which, according to Pliny, mankind were indebted to ac- cident. ' Some merchants, who traded in mineral alkali [iiitrunt], having occasion to dress their victuals on the borders of the riyer Belus, in Phenicia, where they had landed, and not rinding any stones at hand to support their kettles, fetched some pieces of the alkali from their ship for the purpose. These the fire melted with the sand of the shore, and caused to flow in. valuable translucent streams, whence originated the jnaking of glass.' Nat. Hist. lib. xxxvi, chap. 26. K 4 In 136 In the time of Socrates [A. c. 433] the jnanufao ture of glass had made considerable progress, and the use of burning glasses was already become very com- mon. This the following passage from the second act of the Clouds of Aristophanes shows. The author introduces Socrates giving lessons in, philosophy to Strepsiades, a citizen of Athens, and a man of low cunning. The subjects of these lessons are silly trifles, intended to make Socrates appear ri- diculous. Strepsiades, after having asked him how he should avoid paying his debts, proposes the fol- lowing expedient himself, ' Strepsiades, YOU have seen at the druggists that fine transparent stone, with which they kindle fires. Socrates. You mean glass, do not you? Strepsiades. The very thing. Socrates. Well, what will you do with that ? Strep" siades. When a summons is sent to me, I will take this stone, and, placing myself in the sun, I will melt all the writing of the summons at a distance.' The, writing, as we know, was traced on wax spread upon, a more solid substance. Such a proof of the antiquity of burning glasses is irrefragable. However, the effect mentioned by Strepsiades may easily be explained, in three ways ; either a concave mirror, reflecting the rays of the Sun, might be employed for the purpose ; or a con- vex glass, transmitting the rays ; or an assemblage of several plane mirrors, operating by reflection. But in the first case the summons must have been held up high betwe.en the mirror and the Sun, at the spot where the solar rays, after having struck en the con-r cavity 137 fcavity of the mirror, would unite by being reflected at an angle equal to that of their incidence : a very inconvenient situation for the summons, and it may be presumed, not what was meant by Strepsiades. In the second case, the summons would be held lower than the glass, at the focus in which the solar rays would unite, after having traversed the thickness of the spherical lens, which would occasion no difficulty, ijo embarrassment, in putting it into practice. The third mode is equally easy to be employed, for it re- quires only to dispose plane mirrors in such a manner, that the solar rays falling on them should be reflected in lines intersecting each other in a point, where they would form a burning focus. Several other ancient observations on the same phenomenon exist Pliny mentions globes of glass, or of crystal, which, being exposed to the Sun, would burn clothes, or the flesh of a patient when cauteri- zation was requisite. Hist. Nat. lib. xxxvi and xxxvij. JLactantius, who lived about the year 303, sa}'S, * a globe of glass filled with water, and ex- posed to the Sun, will kindle a fire, even in the coldest weather.' De Ira Dei, The most memorable effect of burning glasses in all antiquity would be that of the mirrors of Archi- niedes, if the fact were sufficiently established. This is a disputable question, which I think it incumbent on me to examine with all the brevity possible, with- out omitting any of the arguments, that may be al- leged on either side. It is related by several ancient authors, that Archi- medes sev fire to the rorjian fleet, at the siege of Sy- racuse, 138 racuse, with burning glasses. This fact was con* sidered by some of the moderns as fabulous and im- possible : by others it is admitted as certain, and even as easy to be carried into execution. I will begin with the arguments of the sceptics, at the head of whom we find the celebrated, des Cartes, Diopt* Disc. vui. In the first place they have observed, and this all the world concedes to them, that Archimedes could not employ a dioptric lens, or a burning glass by re- fraction, if local circumstances would have permitted it ; because such a glass would not have collected into one focus a quantity of solar rays any thing like sufficient, to produce a fire of such magnitude ; and because the radius of the sphere, of which it made a part, must be immense. This defect could not be remedied by employing several glasses of the same kind : for all these glasses, exposed at the same time to the Sun, to produce a simultaneous conflagration, must have had the same curvature, the same focus, and the same position with respect to the Sun and the object to be burned ; whence it is obvious, that they would mutually exclude each other. For similar reasons des Cartes and his followers reject the catoptric mirror, saying, as is very true, that to unite the rays at the distance to which a dart might be thrown, that is about a hundred and fifty feet, the radius of sphericity must have been three hundred feet, which would render such a mirror im- possible to be made with any degree of accuracy. Besides, it would not have furnished a sufficient i quantity 139 quantity of rays; and if, in order to augment thi$ quantity, the extent of the mirror had been enlarged, the solar rays, ceasing to be physically parallel, would have been diffused over a wider space, whence their density and power would have been porportionably diminished. Lastly, in this case, as in the preceding, only one mirror could have been employed. In this view of the question, it is certain, that des Cartes would completely gain his cause. JBut why should the mirrors be restricted to curves, that admit but one focus, and exclude the combination of se- veral mirrors ? Is it not possible, to collect a great number of little plane mirrors, and arrange them in such a manner, that they should reflect to one point, or to one small space, a quantity of solar rays suf- ficient to set fire to wood, cordage, or other parts of a ship's furniture? Certainly in this there is no the- oretical impossibility. As to it's execution, can we suppose such a man as Archimedes, who possessed the genius of invention in mechanics in the highest degree, could be puzzled to find means of combining together several pieces of glass, and making them movable on joints or hinges, so as to assume at pleasure different inclinations, according to the ex- igencies of the case ? It seems to me therefore, that the whole question is reducible to the point of fact, whether Archimedes really set fire to the roman fleet with burning glasses, or not. On the one hand, not a word of this is mentioned by Polybius, Livy, or Plutarch : on the other hand, it has* been positively affirmed by Hero, Diodorus Siculus, 140 Siculus, and Pappus^. The works in which the former speak of the siege of Syracuse exist; those of the latter are lost : but these existed in the twelfth century, and the passages, in which the burning glass of Archimedes is particularly mentioned, are quoted by Zonaras and Tzetzes, writers of that period. The silence of Polybius, Livy, and Plutarch, is of the class 6f negative proofs, which should yield to a positive assertion, when the fact announced includes no im- possibility. Besides, as Plutarch speaks generally with admiration of the efTcct of the engines of Archi- medes, without specifying any particulars, the burn- Ing glasses might be included among these. Be this as it may, Zonaras and Tzetzes, being writers of very moderate abilities, are entitled from this very circum- stance to our credit : as they were incapable of in- venting such a story, their testimony must be con- sidered as that of the authors they quote. Now Zonaras affirms, from the ancients, that Archimedes set fire to the roman fleet by means of the solar rays collected and reflected by a polished mirror : he then adds, that Proclus, taught by this example, burnt with mirrors of brass the fleet of Vitalian, who besieged Constantinople under the emperor Anastasius, in the year 514 Tzetzes, referring to the same authorities, gives a particular explanation of the mechanism of the burning glasses of Archimedes. ' When Marcellus/ says he, ' had removed his fleet out of reach of the * The authorities, taken together, are nearly equal in point of antiquity. Hero lived before Polybius ; Diodorus and Livy wer% contemporaries ; Pappus was subsequent to Plutarch. darts. 141 darts, Archimedes brought into play a hexagon mirror, composed of several other smaller mirrors, each of \vhich had twenty four angles, and which could be moved by means of their hinges and of certain plates of metal. He placed this mirror in such a position, that it was intersected through the centre by the meridian both in winter and summer; so that the rays of the Sun, being received on this mirror, and reflected by it, kindled a great fire, by which the romaii vessels were reduced to ashes, though as far off as a dart can be thrown.' Whether this passage contain an exact or defective description of the burning glasses of Archimedes, or whether it be sup- posed, if you please, to exaggerate their effects ; at least it points out nearly the manner, in which the parts of the mirror turned, to assume the position suitable to their purpose, it's situation with respect to the Sun, and the distance at which it exerted it's burning properties ; all of them circumstances neither impossible, nor improbable. Some persons, struck with these arguments, but still a little incredulous on the point in question, have made an objection, which has been deemed more forcible than it really is. Admitting, say they, that Archimedes might have set fire to the roman ships, if they had remained fixed in the same place ; it will be very different, when we come to consider, as w6 ought to do, that a ship is coming nearer or moving farther off: for, they add, at every movement it makes, a considerable time will be required, to give the faces of the mirrors the positions, that will be rendered necessary by the change in the distance of the the mirror from the object to be set on fire. To thb I answer, 1st, Archimedes having once seized the fa- vourable moment for his purpose, without the romans having any suspicion of his design, his scheme might have been executed very quickly, and before any thing was done to prevent it : 2dly, with all the re- sources his mind possessed, he could find means of changing the inclination of his mirrors, without dif- ficulty, so as to follow, at least for a time, any ship endeavouring to escape : 3dly, and lastly, he might have had in reserve several mirrors of different foci, which is certainly possible in this instance, for all the cases that could occur, and which it was easy to fore- see. The mobility of the ships, therefore, is not an Insurmountable obstacle to the action of the mirrors : and modern writers, paying no regard to this objec- tion, have imagined they might rest the reality of the effect proposed on experiments, in which the objects to be fired were immovable. In a work entitled The great Art of Light and Shade, father Kircher, the Jesuit, says, that from the description of Tzetzes he had constructed a mirror, composed of several plane looking glasses, which, all reflecting the light of the Sun to one point, produced there -considerable heat. In 1747 Buffon executed the same experiment on a large scale, and thus irrevocably stamped with the seal of truth the effects of the mirrors of Archimedes. He caused a reflecting mirror, composed of a hundred and sixty eight plane glasses, movable on hinges, and capable of being made to change their position all at once, or only part of them at a time, to be con- structed 143 striicted by Passement, an excellent optician and ma thematical instrument maker. By means of this assemblage of glasses, in the month of April, and with no very strong Sun, Buffon set wood on fire at the distance of a hundred and fifty feet, and melted lead at a hundred and forty. These etfects are more than sufficient to overturn every argument that has been adduced against a fact evidently possible. The question was in this state, when, in 1777, the learned Dupuy, a member of the Academy of Belles Lett res, published a translation of a fragment of Anthemius on the same subject. Anthemius, we know, flourished under the emperor Justinian. He was a man of uncommonly profound skill in the mathematical sciences, particularly in mechanics. In conjunction with Isidore he began the building of the celebrated cathedral of St. Sophia at Constanti- nople, which he finished by himself after the death of his colleague ; and to him is ascribed the first in- vention of domes. The fragment translated by Dupuy contains some problems in optics, among which the author treats particularly of the mirrors of Archimedes, on the effects of which he neither entertains nor leaves any doubt. He begins by observing, that Archimedes could not have employed a concave catoptric mirror, 1st, because such a mirror must have been of immense size ; 2dly, because in mirrors of this sort the object to be burned must be placed between the mirror and the Sun; a position, which the situation of the roman ships with respect to Syracuse would not admit He afterward explains the mechanism of the mirrors 5 which which Archimedes employed, nearly as it has becti transmitted to us by Tzetzes, and executed by Buffon. Perhaps I have been rather too diffuse on this par- ticular subject ; but I thought it incumbent on me to elucidate, as far as I possibly could, an interesting" problem, on which there still rests a little obscurity. I shall conclude with a few general observations. In the succession of human knowledge there is ail unhappy fatality : what is most useful, and most ne- cessary for our wants, is almost always the last ac- quired. The ancients, who knew how to employ with so much art and success the property of burning, which glasses possess, were ignorant of the much more important and advantageous use now made of them, that of magnifying objects, and assisting the feebleness of sight. I know this opinion is not agree- able to some antiquarian fanatics, who are resolved to maintain, right or wrong, that the ancients invented every thing, and left us only the poor glory of dis- covering their meaning, and commenting on thck works. The historian of the Academy of Belles Lettres expresses himself as follows, on the single testimony of the learned Valois, without deigning to quote any ancient authority. * We read that one Ptolemy, king of Egypt, built a tower, or an obser- vatory, in the island in which the Pharos of Alex- andria was erected ; and that upon this tower he caused to be placed telescopes of such prodigious power, as to discover an enemy's ships, when coming to make a descent on the coast, at the distance of sixty miles/ Acad. des Belles- Lettres, Vol. I, p. 1 1 1. But 145 But if the ancients in reality possessed an invention so pleasing, so useful, and so simple, is it probable, that it would have been lost even in the most bar- barous ages ? Should we not see very distinct traces of it in ancient authors ? Would it not have furnished a number of metaphorical expressions in their lan- guages, as it has done in modern tongues? How happens it that Seneca, who lived subsequently to this pretended Ptolemy with his telescopes, for Egypt became a roman province after the death of Cleopatra, should have had no knowledge of this circumstance ; for he simply says, in his Quest. Nat., that ' small let- ters, seen through a glass bottle filled with water ap- pear larger/ without adding the least allusion to per- spective glasses ? The ancients, misled by their erro- neous opinions with regard to the nature of vision, never imagined, that, by a mechanism similar to that which collects the solar rays into a burning focus, a mild and gentle light could likewise be collected, forming a bundle of luminous rays, that would assist the eyes in performing their functions, without in- juring them. If we stick to certain proofs, and not give into mere conjectures, which may always be formed by twisting the expressions of ancient au- thors, we shall be convinced, that the invention of spectacles, dates no older than the thirteenth century ; and that of telescopes is about three hundred years later. The glasses proper for constructing these instruments must either be very large spheres, the use of which would be very inconvenient, and almost impossible, or very small portions, of large spheres, which is easy L iu 146 in practice, by the method usually adopted. But this method requires a knowledge of the art of cutting glass, which appears to have been altogether unknown to the ancients, who simply knew how to blow glass, and form it into hollow vessels, CHAP. 147 CHAP. VII. Origin and Progress of Acoustics, THE term acoustics, unknown to the ancients, has been invented by the moderns, to denote in a succinct manner that part of mathematics, which considers the motion of sound, the laws of it's propagation, and the relation which different sounds bear to each other. There is a striking analogy between acoustics and optics, both in their theory, and in the instru- ments by which the sight and hearing are assisted. The air is the vehicle of sound. When a sonorous body is struck, it trembles, and makes vibratory mo- tions, which it communicates to the circumambient air ; and this fluid transmits them by successive un- dulations, arising from it's elasticity, to the tympa- num of the ear ; a kind of drum, at which the audi- tory nerve terminates. The more dense and elastic the sonorous body is, and the more violently it is agitated, the more fulness and strength has the sound which is produced. A series of sounds, succeeding each other un- equally and without order, forms merely a simple noise, which is often very disagreeable. But when the sounds have measured intervals, and their relations to each other are subjected to constant and regular laws, the result is a harmony, a modulation, pleasing L 2 to 148 to the ear. This is the source of that pleasure, which all nations receive from music. On a mutual comparison of two sounds, one is more grave, one more acute than the other. This difference arises from the greater or smaller number of vibrations, which the sonorous body makes in a given time. Take, for instance, two strings of a violin, equal in thickness, and equally tense, but of which one is double the length of the other ; and draw them both out of the rectilinear position, so as to make them vibrate. In this case, while the shorter string makes two vibrations, that of double it's length will make but one ; and the sound of the former will be acute, that of the latter grave. We say likewise, that one is an octave to the other, be- cause they constitute the two extremes of eight notes in the musical gamut. If the tension of the two strings be greater or less, but still equal in both, the sounds produced will be proportionately more or less loud, but they will bear the same relation to each other. If you would obtain the ratios of the eight musical notes, you have only to take eight strings, equally tense, of equal thickness, and the lengths of which shall be to each other as 1, 4-, , , , f, |, -J. The number of vibrations, which these eight strings will make in a given time, will be reciprocally pro- portionate to the preceding numbers ; and you will hear the fundamental or gravest note, the minor third, the major third, the fourth, the fifth, the minor sixth, the major sixth, and the octave. The 149 The same ratios may be obtained by means of a single string, by giving it different degrees of ten- sion, so that the forces of tension shall be as the All these proportions, and several others, spring from the following theorem : The number of vibra- tions, made by a string in a given time, is generally as the square root of the weight by which it is stretched, divided by the product of the weight of the string multiplied by it's length. Though this theorem was invented by the modern mechanists, I thought it proper to introduce it here, as it will en- able us to appreciate the experiments ascribed to Py- thagoras, the author of the first discoveries made on the subject Nicomachus, an ancient writer on arithmetic [A. c. 400], relates, that Pythagoras, passing one day by a blacksmith's shop, where the workmen were hammering apiece of iron on the anvil, was surprised to hear sounds, which accorded with the intervals of the fourth, fifth, and octave: that, reflecting on the cause of this phenomenon, he conceived it to depend on the weight of the hammers : and accordingly, having caused them to be weighed, he found the weight of the heaviest hammer, answering to the fundamental note, being represented by unity, fche weights of the other three, answering to the fourth, fifth, and octave above, were as -J-, ~, and -J-. Ni- comachus adds, that Pythagoras, on his return home, was desirous of verifying this first experiment by the following. Fastening a string to a fixed point, and L 3 passing 150 passing it over a peg in a horizontal line with this point, he stretched the string more or less by dif- ferent weights ; and on causing it to vibrate, he found the weights corresponding to the fourth, fifth, and octave above, to be to each other as the weights of the smith's hammers. On applying the theorem above quoted to these ex- periments, we find, either that they were inaccurate, or that they are erroneously related. The length of three strings, of the same uniform thickness, which, being stretched by the same weight, would give the fourth, the fifth, and the octave above, are as the three fractions, .J, , J-: but to make the same string give the fourth, fifth, and octave above, by stretching it with different weights, these weights must be to each other as V 6 > f> 4. Thus there is a mistake in the proportions between the weights of the hammers found by Pythagoras, or in the manner in which his experiments are related. Undoubtedly it was natural to suppose, that the three different weights, which should produce the fourth, fifth, and octave, by stretching one and the same string, would be to each other as the lengths of three different strings, which, being equally stretched, should pro- duce these three notes : but this is not the fact. Be this as it may, it is unquestionable, that these first ideas of Pythagoras were the true source of the theory of music. As what is properly called the art of music, however, derives very little assistance from mathe- matics, I shall enlarge no farther on the music of the ancients; particularly as it's history may be 1 found 151 found in several works, more especially in the Me- moirs of the Academy of Belles-Lettres. But I shall hereafter resume the geometrical theory of vibrating cords, and of the motion of the air in a tube ; a theory which has originated in modern days. END OF THE TIRST PERIOD. L 4 PERIOD 152 PERIOD THE SECOND. STATE OF THE MATHEMATICS, F'ROM THEIR RE- VIVAL AMONG THE ARABS TO THE END OF THE FIFTEENTH CENTURY. THE mathematics still flourished in Greece, and more particularly in the school of Alexandria, when a little before the middle of the seventh century a tremendous storm arose, which threatened their total destruction in those climes. A. D. 638. Filled with all the enthusiasm a militant religion inspired, the successors of Mohammed ravaged that vast extent of country, which stretches from the east to the south- ern confines of Europe. All the cultivators of the arts and sciences, who from every nation had assembled in Alexandria, were driven away with ignominy. Some fell beneath the swords of the conqueror : others fled into remote countries, to drag out the re- mainder of their lives in want. The places and the instruments, which had served for making an im- mense number of astronomical observations, were in- volved in one common destruction. In fine, that precious depository of human knowledge, the library of the Ptolemies, which had already suffered ty fire under Julius Caesar, was entirely delivered to the flames 153 flames by the arabs. The khalif Omar directed all it's books to be burnt ; because, observed he, if they agree with the Koran, they are useless ; if they differ from it, they ought to be held in detestation, and destroyed : an argument worthy of a fanatic robber. The fate of the sciences, thus attacked and anni- hilated in the heart of their empire, must seem abso- lutely desperate. But that same vicissitude, which produces so many misfortunes and so many crimes, sometimes brings about revolutions, by which man- kind is benefitted. Such was the change, that speedily took place in the manners of the arabs. This people, like all those of the east, had formerly possessed some notions of the sciences, and in par- ticular of astronomy. If the fanaticism of a san- guinary religion at first overwhelmed these seeds, it did not totally dry up their germes. When the dif- ferent nations were weary of mutually exterminating each other, their ferocity was softened, and the leisure of peace recalled the active minds of the arabs to less empty and more agreeable occupations, than disputing on the dogmas of the Koran. Scarcely had a hundred and twenty years elapsed after the death of Mohammed, when they began themselves to cultivate those arts and sciences, which they had endeavoured to proscribe. Soon they had their poets, orators, mathematicians, &c. -In this num- ber are reckoned several of the khalifs of the arabs ; and afterward several of the emperours of the per- sians, when these people became a separate empire. The arabs derived the principles of all parts of the mathematics from an assiduous study of the greek. writers. 154 tfriters. Furnished with these acquisitions they be- came emulous of their masters, and rendered them- selves capable of translating, commenting, and some- times adding to their discoveries. Hence the substance of several works of the greeks has reached us only through the medium of arabic versions. By the arabs other nations were taught, and the sciences were revived with such success, as posterity ought never to forget. Let us proceed to particulars. CHAP. 155 CHAP. I. The Arithmetic and Algebra of the Arabs. FOR the ingenious system of arithmetical numera- tion, which all modern nations employ, they are in- debted to the arabs. It has the advantages of clear- ness and simplicity over all those of the ancients. With ten characters, made to occupy different places, we are able to express in the most commodious man- ner a number immense in multitude. Some writers assert, that the arabs derived this idea from the hin- doos : but the arguments they adduce do not appear to me very convincing. Without entering into this idle disquisition, I shall content myself with observ- ing, that we derive our arithmetic, as we practise it at present, immediately from the arabs. The cele- brated Gerbert, afterward pope Silvester n, went to Spain, then under the dominion of the arabs, to study this science, and thence diffused it over the rest of Europe, about the year 9^0. The first notions of algebra, which are found in Diophantus, were unfolded by the arabs. Cardan even considers these people as the real inventors of algebra. The celebrated analyst Wallis, adopting this opinion, assigns as a reason for it, that the arabs in the denominations of the powers employed a sys- tem different from that of Diophantus ; whence he concludes, that their principles were likewise different In 156 In the greek author the second, third, fourth, fifth, and sixth power, and so on, are called the square, cuhe, quadrato-quadratum, quadrato-cubus, cubo-cubus, &c. ; so that each power took it's deno- mination from the two hiferiour powers, by which it was produced. With the arabs they are the square, cube, quadrato-quadratum, sursolid, quadrato-cubus, second sursolid, &c. ; where we find, that those powers, which are not the product of two powers of the same kind, are called sursolids. For instance, in Diophantus the quadrato-cubus, or cube multiplied by the square, forms the fifth power : the arabs mean by the same expression the square of the cube, or the cube of the square, which with them forms the sixth power. The reader will weigh the force of this conjecture of Wallis. We do not accurately know the extent of the pro- gress, which the arabs made in algebra ; but we have some indications, that they had advanced so far as to resolve equations of the third order, and even some particular cases in the fourth ; in which they went farther than Diophantus, who did not get be- yond the second. As a proof of this it is asserted, that in the Leyden library there is an arabic manu- script, entitled, The Algebra of cubic Equations, or the Solution of solid Problems. CHAP, 157 CHAP. II. Geometry of the Arabs, AMONG the arabs are reckoned many skilful geome- tricians. Their first care was to translate the ele- mentary works of the greeks, such as the Elements of Euclid, the Treatise on the Sphere and Cylinder by Archimedes, the Spherics of Theodosius, the Treatise on Spherical Triangles by Menelaus, &c. Soon after they rose to the higher geometry, or that of the ancient curves ? Apollonius's doctrine of conic sections became familiar to them ; and the fifth, sixth, and seventh books of his work on this subject have reached us only in an arabic version. By degrees their knowledge extended to statics and hydrostatics, and through their means we have received the work pf Archimedes on Bodies floating on a Fluid. Practical geometry and astronomy owe the arabs eternal gratitude, for having given to trigonometrical calculation the simple and commodious form, which it has at present. They reduced the theory of the resolution of triangles, both rectilinear and spherical, to a small number of easy propositions ; and by the substitution of sines, which they introduced instead of the chords of double arcs employed before, they made abridgments in calculations, of inestimable value to those who had a great number of triangles to resolve. These discoveries are ascribed chiefly to a geome- 158 a geometrician and astronomer of the name of Mo- Tiammed Ben Musa, the author of a work still ex- tant, entitled Of Plane and Spherical Figures ; and to another geometrician and astronomer much better known, Geber Ben Aphla, who lived in the eleventh century, and of whom we have a commentary on Ptolemy. On geodesia, or the art of surveying, we have an elegant work by Mohammed of Bagdat, which som authors have ascribed to Euclid, without assigning any reason for it. CHAT. 159 CHAP. III. Astronomy of the Arabs. OF all the mathematical sciences astronomy is that, which the arabs have most cultivated, and in which they have made the most remarkable discoveries. A great number of their khalifs themselves were excel- lent astronomers. Nothing could equal the magnifi- cence of the observatories and instruments, which they caused to be constructed, to promote the pro- gress of this science, which has more need than any other of the patronage of monarchs. I shall here mention only the principal arabian astronomers; and among them I shall particularly notice the khalifs, who have merited this distinction ; because those exemplary princes, who add to the glory of governing well that of enlightening the men they govern, have a peculiar claim to the re- spect, admiration, and gratitude of posterity. The arabs regulated time by the motion of the Moon. Their months were of twenty nine and thirty days alternately, which gave 354 days for the dura- tion of the lunar year. But as the synodical month, or the duration of each lunar revolution with respect to the Sun, consists of 29 days, 1 2 hours, 44 minutes, 3 seconds ; the term of the arabian lunar year was 8 hours, 48 minutes, 36 seconds, less than the true length of twelve lunar revolutions with respect to the Sun. 16*6 Sun. Accordingly, to remedy this difference, which left the Moon behind the Sun, in their course from West to East, and to make the positions of these two heavenly bodies coincide, a day was occasionally added to this period of 354. Among the different branches of astronomy, the theory of the Sun first drew the attention of the arabs, and occupied them for a long time. They quickly remarked, that Ptolemy had found or supposed the obliquity of the ecliptic a little too great. Flamsteed, in his Plistoria celestis, records the progress of their labours on this subject. We see them continually approaching the truth; and at length, at the end of about seven hundred years, they determined the obliquity of the ecliptic with nearly the same precision, as it is now given by the best modern observations ; a result so much the more singular, as they had not, like us, the assist- ance of telescopes. The khalif Abou Giafar, surnamed Almansor, or the victorious, who ascended the throne in 754, and died in 775, ranks among the first of the arabian astronomers. This studious and philosophical prince had a taste for all the sciences, but particularly for astronomy, on which he bestowed every moment, that his indispensable duties left at his disposal. His reign is the epoch, at which the whole circle of the sciences received an impulse, that continued to in- crease among the arabs for several centuries. Almost all the successors of Almansor thirsted like him after knowledge. "His grandson Haroun, sur- named Al Rashid, who reigned from 786 to 809, cul- tivated 161 tivatecl astronomy and mechanics. On occasion of a solemn embassy, which the great fame of Charle- magne, king of France, drew from him in 799, he sent this monarch as a present a clepsydra, or water- clock, of a very ingenious construction. In the dial were twelve small doors, forming the divisions of the hours ; and each of these doors opened in succes- sion at the hour it marked, and let out little balls, which, falling on a brazen bell, struck the hour. The doors continued open till twelve o'clock, when twelve little knights, mounted on horseback, came out together, paraded round the dial, and shut all the doors. This machine astonished all Europe, when men's minds were employed chiefly on futile questions of theology or grammar. Haroun had two sons, who reigned in succession after him. The second, named Al Maimon, who held the sceptre from 8J3 to 833, was taught the sciences by Musva, a Christian physician and neg- lected neither gifts, exhortations, nor example, to induce his subjects to pursue them with ardour. He caused all the writings of the greeks he could procure to be translated, and in particular Ptolemy's Almagest. Some authors even assert, that in a treaty of peace with Michael nr, when he made his own terms with the emperor, he demanded several greek manuscripts, which were in the imperial library at Constantinople. He also made observations himself, and directed the making of others, for which affairs of state did not allow him time; as for instance those at Bagdat and Damascus, undertaken by his command, to ascertain the obliquity of the ecliptic, which was M found found to be twenty three degrees thirty five minutes ; a result much nearer the truth, than any of those of the ancient astronomers. In the plain of Singiar, on the borders of the Red Sea, he caused a degree of the Earth to be measured : but unfortunately we have only a vague and doubtful knowledge of the propor- tion, which the arabian measure employed bears to our measures, and cannot tell how far this mensura- tion agrees with those of modern date. Lastly, still more to facilitate the study and advancement of astronomy, Al Maimon directed a work to be com- posed by men who possessed most skill in the science; and it yet exists in manuscript in several libraries, under the title of Astronomia elaborata a complurihus DD. Jussu Regis Maimon. The city of Bagdat, oc- cupying nearly the same site as the ancient Babylon, was enlarged and embellished by his care, and be- came the usual residence of the khalifs. In this city- there were schools for all the sciences, and one in particular for astronomy. Al Maimon carried with him to the grave a glorious character, that of having been the wisest, most humane, and most learned prince, that had ever occupied the throne of the khalifs. In the time of Al Maimon several celebrated astro- nomers flourished ; among whom Alfragan, Thebit Ibn Chora, and Albategni, are particularly distin- guished. Alfragan, A. D. 8,50, composed Elements of Astro- nomy : a book almost classical formerly, even in the west, and of which several editions have been pub- lished siiisc the invention of printing. He likewise wrote 163 wrote treatises on Solar Clocks, and on the Astro- labe, which are preserved in manuscript in some libra- ries. He is said to have possessed extreme facility in. making the most complicated calculations, whence he was surnamed the Calculator. Thebit was an algebraist, geometrician, and astro- nomer. A. D 86'0. An observation of his on the obliquity of the ecliptic is quoted, in which he found it to be twenty three degrees, thirty three minutes, thirty seconds. He suggested the idea of referring, the Sun's motion to the fixed stars, instead of the equinoctial points, which are movable : and he deter- mined the length of the sidereal year to be nearly what . it has been found to be in modern days ; but this was a happy conclusion, which can scarcely be ascribed to any thing but chance ; for Ptolemy, whose system was generally followed by the arabs, had a little per- plexed the elements of the problem. The force of this reflection is the stronger, when we consider, that Thebit had not a very just idea of the position of the stars with respect to the firmament. He supposed with Hipparchus and Ptolemy, that they had a slight motion from west to east ; but he added, and his opinion obtained credit, that at the end of a certain time they returned back to their places again, and then resumed their former direction, to recede anew: thus they proceeded each way alternately, whence arose a kind of trepidation, the partial motions of which were still subject to farther irregularities ; a; system overturned by observation. Thebit admitted a similar motion of trepidation in the obliquity of the ecliptic. M 2 Albategni 164 Albategni was one of the greatest promoters of astronomy. A. P. 879. His numerous observations, and the important knowledge he derived from them, acquired him the surname of the arabian Ptolemy ; a comparison perhaps to the honour of the grecian in point of genius. He was governor of Syria under the khalifs, and made his observations partly at Antioch, the capital of his province, partly at Aracte, a large city in Mesopotamia. The following is a succinct view of his labours. An accurate investigation of the observations of the ancients, and a comparison of them with his own, taught him, that Ptolemy had made the longi- tudinal motion of the stars too slow, when he sup- posed it only one degree in a century ; and he found nearly the same result as Hipparchus, namely, that this motion was one degree in seventy years. Ac- cording to the observations of the moderns it is one degree in seventy two years. Albategni came still nearer the truth in his exami- nation of the eccentricity of the solar orbit. lie was very little indeed from finding it such as it is given by the observations of the moderns. There are even astronomers of the present time, who consider the measure of Albategni as very exact, exclusive of the trifling errours, which are inevitable in the re- sults of the best observations. His calculation of the length of the year, which he makes to be 365 days, 5 hours, 46 minutes, 24 seconds, differs about two minutes only from the truth : and the celebrated Halley has shown, that the crrour of Albategni arose from his too great confi- dence 165 dence in the observations of Ptolemy, , for -he would have come much nearer the true length, if he had compared his own observations directly with those of Hipparclms. Before Aibategni the Sun's apogee was reckoned immovable: but he showed, that this point has a small motion in the order of the signs, which a little exceeds that of the stars ; a nice observation, the necessity and importance of which have been demon- strated by modern astronomers, and the theory of universal gravitation. Lastly, having discovered the insufficiency and the defects of Ptolemy's theories of the motions of the planets, Aibategni exerted his utmost endeavours to correct them, and bring them to perfection. The discovery which he made of the motion of the Sun's apogee led him to suspect similar inequalities in the motions of the other planets ; and this conjecture likewise has been converted into a certainty by mo- dern theories. By means of all the .knowledge he had thus attained, Aibategni substituted new tables instead of Ptolemy's; and thereby rendered an essen- tial service to astronomers, that of facilitating or abridging their calculations for a. time, I use the words for a time^ as we know, that even at present the best tables require to be corrected and adjusted, in proportion as observations increase and are im- proved. The works of Aibategni have been collected in one quarto volume, entitled DeScientiaStellarum; of which there are two editions, one dated 1537, the other 1646. M 3 A num- 166 A number of learned arabs are quoted, who for several centuries continued to observe the heavens, and improve all the branches of astronomy. These people not only cultivated the mathematical sci- ences themselves, but endeavoured to spread them through all the nations, that were subject to their sway. Montucla, in his History, gives an ample list of mathematicians, who were either of arabian birth, or scholars of the arabs, with some account of their works. Most of these particulars, however, being associated with barbarous names, which I could not recapitulate without wearying the reader, I shall confine myself to the leading points, which may serve to make known the obligations, that the sci- ences are under to the arabs. In Egypt, under the patronage of the khalif Azir Ben Akim, Ibn lonis made several observations, which have come down to us with those of many other astronomers, in a kind of history of the hea- vens which he wrote, and which exists in manuscript in the library at Leyden. In this work are twenty eight observations of eclipses of the Sun or Moon, jpade by arabian astronomers from the year 829 to the year 1004 ; seven observations of equinoxes from 830 to 851 ; and one observation of the summer sol- stice in 832. Three eclipses observed near Cairo in the years 977, 978, and 979, afforded a remarkable result, proving that the mean motion of the Moon is subject to a small acceleration, which, accumulating in 'a succession of ages, must be admitted into the elements of astronomical calculation. All these richet 167 riches having led the national institute of France, to wish for an inspection of the Ley den manuscript, the Batavian Republic has entrusted it to it's care, and transmitted it through the hands of it's ambassador at Paris. It has been carefully examined ; but no' observations, except those I have mentioned, have been found in it. It gives us none of the informa- tion, that was hoped for, respecting the instruments of the arabs, and their manner of observing : but it has afforded some interesting corrections of the frag- ment, of which Delisle had obtained a copy, at pre- sent in the possession of citizen Messire, member of the national institute ; and of which citizen Caussin, professor of arabic in the national college of France, has made a translation, now printing with the ori- ginal on the opposite page. Ibn lonis likewise con- structed some astronomical tables, which were long celebrated and useful in the East The arabs established -in Spain, of which they had conquered the greater part in the eighth century, cultivated the sciences there with the same ardour, and the same success, as in the East Astronomy was the principal object of their labours; and they built observatories in several cities in Spain. Arsa- cjhel, who was one of the most distinguished of their astronomers, improved fhe theory of the Sun. A. D. 1 020. By a method also, more simple, and more capable of accuracy, than those which Hipparchus and Pto- lemy employed, he made some happy alterations in the dimensions they had assigned to the solar orbit. He is likewise believed to have discovered certain inequa- lities in the Sun's motion, the existence of which has M 4 since 168 since been confirmed both by observations and the newtonian theory ; whence he has been considered as a very accurate and attentive astronomer. He composed a collection of Tables, entitled Tabulae Toledanav from the name of the city of Toledo, where he resided. Alhazen, another celebrated arab settled in Spain, has left us a treatise on optics, which contains the first attempt we have at a theory of refraction and the twilight. A. D. 1100. He makes them depend, not on vapours accumulated in the vicinity of the horizon, but on the difference of transparency, that exists between the air of the atmosphere and an ethe- real matter placed beyond it. He even teaches, how we may ascertain by observation the difference be- tween the true and apparent place of a star produced by refraction. According to him the cause of the extraordinary size of the Sun and Moon near the horizon is not to be sought in refraction, but rather a contrary effect. Malebranche has since employed and enlarged upon the same doctrine ; and as he does not quote Alhazen, we may presume he was unac- quainted with his works. Some authors assert, that Alhazen only translated and commented on a work, which Ptolemy had composed on the same subject; and which is quoted by other arabian writers, though now lost : but this we may venture to contradict, since the ancient astronomers, and Ptolemy himself, paid no regard to the effect of refraction in their astronomical observations. To Alhazen at least must be ascribed the honour of having 'clearly pointed out this 169 this effect, and made astronomers sensible of the ne- cessity of taking it into account. Several other arabian mathematicians are recorded in Spain about the same period : as Geber, mistakenly considered, on account of his name, as the inventor of algebra ; but in reality the author of a translation of the Almagest, and of two theorems in spherical trigonometry very commodious for the resolution of rightangled triangles : Almansor, or Almaon, who made a very good observation of the obliquity of the ecliptic : Averroes, a celebrated physician of Cordova, who abridged and commented on Ptolemy, and who was very learned for his time both in natural philo- sophy and the mathematics : c. Some of these learned arabs were prompted by in- clination to remove into more northern parts of Eu- rope, where the knowledge they brought with them is so confounded with that of their scholars, that it is impossible at present, to assign to each their just share. 170 CHAP. IV. State of the Sciences among the Persians, 1 HE persians, who till the middle of the eleventh century made but one people with the arabs, having then shaken off the yoke of the khalifs, did not re- linquish the study of the sciences amid the troubles' of war. They had algebraists, geometricians, and par- ticularly astronomers, who were highly distinguished. Loggia Nassir, or doctor Nassir, composed several works of great reputation in their time. We- have a commentary of his on Euclid, printed in 1590, in his native tongue, the arabic. Nassir Eddin, another geometrician better known, has given several very ingenious demonstrations of the forty-seventh pro- position of the first book of Euclid, which are recited by Clavius. They proceed by a simple transposition of parts, with which Nassir Eddin composes either the square of the hypothenuse, or the squares of the other two sides of the right angled triangle. He also made an accurate translation of the Conies of Apol- lonius, and added a commentary, of which Halley availed himself with advantage in translating the fifth, o D ' sixth, and seventh books of this important work. We find at the same period another very celebrated persian geometrician, Maimon Rash id. He com- mented upon Euclid, and his passion for geometry was so 171 so great, that he always wore some favourite geome- trical figures on the sleeves of his garment. All these ancient persian geometricians had care- fully collected the works of the greeks, and were thoroughly acquainted with their doctrines. It is even said, that several grecian works, which we do not possess, are still preserved in Persia. CHAP. 17S CHAP. V. Of the Persian Astronomy in particular, 1 HE ancient persians, from the time of Darius Ochus a had made a great number of observations. They particularly applied themselves to determine the length of the solar year, to which they referred all the mea- sures of time. Having fixed it's duration to 365 days fix hours, they sunk these six hours, or frac- tional part of a day, by intercalating a month of thirty days once in a hundred and twenty years ; which amounts to the same as a day every four years in the Julian account They likewise placed this intercalary month succeflively the first in the year, the second, and so on ; by which means it made a complete re- volution through the year, and gave occasion to va- rious religious ceremonies. When the persians were subjected to the yoke of the arabs, the practice of computing by lunar revolutions, which was followed by the conquerors, became that of the conquered likewise. But on the recovery of their freedom, the} T resumed their ancient method about the year 1079. At that time the persian astronomer Omar Cheyam, in order to rectify the ancient calendar of his nation, founded on the hypothesis of a year about eleven minutes too long, conceived the idea of adding a day every fourth year, seven times following, and then a day on the fifth year; which is the same thing as if he 173 he had added eight days every thirty-third year. This system, which comes very near to the truth, was adopted, and has been retained by the persians. Several of the persian emperors also zealously pa- tronised astronomy. It was a kind of religion of the state. Chioniades, a greek author, who lived in the thirteenth century, relates that the persians were so jealous of their acquirements in this science, that the communication of them to foreigners was prohibited by a law, except on very rare occasions, which were left to the decision of the emperor. This prohibition was founded on a prophecy, which foretold, that the Christians would one day overturn the persian em- pire by means derived from the science of astronomy. Chioniades himself found great difficulty in being admitted to receive lessons from the persian astro- nomers, though he had been strongly recommended by the emperor of Constantinople, then connected with the persian emperor by the ties both of friend- ship and interest. From this intercourse he brought back into Greece some astronomical tables, which, according to Bouillaud, were very accurate for the time when they were calculated. A descendent of Genghis khan, called by some Holagu Ilecou khan, by others Houlagou khan, who conquered Persia about the year 1264, honoured the sciences, which he himself cultivated ; and during the remainder of his life seemed to bend his mind solely to render them flourishing throughout the vast ex- tent of his dominions. In the city of Maragha, near Tauris, the capital of Media, he built an observatory; at which he assembled a great number of astronomers; and 174 and appointed Nassir Eddin, who has been mentioned above, their president. This society was a kind of academy, which was the more flourishing, as it re- ceived every sort of encouragement from a generous prince, who was himself very learned. Nassir Eddin. eornposed several astronomical works, among which were a Theory of the Motions of the heavenly Bodies, a Treatise on the Astrolabe, and some astronomical tables, which he entitled Itecalic Tables, as a record of his gratitude to his benefactor. It is said that Houlagou, feeling himself near his end, caused him- self to be removed to the residence of these learned men, in whose arms he wished to resign his last breath, considering them as the children of his glory, and it's best heralds. His example was surpassed by a tatar prince, the famous Ulugh Beg, grandson of Tamerlane. Ulugh Beg not only encouraged the sciences as a sovereign, but is himself reckoned among the most learned men of his age. In the city of Samarcand, his capital, he established a numerous assembly or academy of astro- nomers; and caused larger and more perfect instru- ments, than had ever been seen, to be constructed for their use. He acquainted himself with all their labours, and observed the heavens with affiduity. Some historians relate, that, to determine the latitude of Samarcand, he employed a quadrant, the radius of which was equal to the height of the church of St. Sophia at Constantinople, or 180 feet: but the con- struction of so large a quadrant is physically im- possible ; and we have every reason to presume, that these historians, little acquainted with astronomy, 8 mistook 175 mistook a simple gnomon for a quadrant. The la- titude of Samarcand was found to be 29 degrees 37 minutes. By means of the fame instrument, the ob- liquity of the ecliptic was fixed at 23 degrees, 30 minutes, 20 seconds : which, surpassing that of mo- dern observations about two minutes, has led to the belief, that the obliquity of the ecliptic suffers a constant diminution ; but this is a point, on which we have not sufficient information. Ulugh Beg com- posed several works, some of which are printed, and others are preserved in manuscript in a few libraries. The chief of them are a Catalogue of the Stars, and Astronomical Tables, the most perfect then known in the East This prince deserved by his virtues and talents the homage of all the Earth ; yet he was as- sassinated at the age of fifty-eight by his own son. The troubles that followed this atrocious deed : plunged the kingdom of Persia into a state of bar- barism. .The men of learning soon disappeared ; and astronomy continued declining in this country to such a degree, that it is now a mere heap of astrolo- gical visions ; and the persians- scarcely know how to make a rude calculation of an eclipse by means of practical rules, which they follow by rote, without having the least knowledge of the theories, en which they, were founded, CHAP, 176 CHAP. VI. State of Science among the Turks. rays from the sciences of the arabs penetrated to the turks. On the foundation of their empire, about the year 1220, medresses, or colleges, were in- stituted, in which geometry and astronomy were taught, as they still are. The first impulse given to the turks carried them to some length in the mathe- matics; but it gradually grew feebler, as it did with their masters. However, even in the present day the turks are not altogether so ignorant, as they are com- monly supposed to be. Mr. Toderini, an italian author, who has written a work on the Literature of the Turks, asserts, that they are well skilled in arith- metic; that they make numerical calculations with ex- traordinary quickness; that some of them have carried algebra as far as we have done ; that geometry is taught with success in their colleges ; and lastly, that they cultivate astronomy for two cogent reasons, one of which is the necessity of regulating their time, and the other, their fondness for judiciary astrology, which cannot dispense with the assistance of astronomy it- self. On this subject I shall say no more, and I shall not return again to a people, who, after all, never made any one discovery in the sciences. CHAP. 177 CHAP. VII. State of the Sciences among the Chinese and Hindoos* ' IF we were to discuss the high opinion, which has been hitherto entertained of the acquirements of the Chinese in all branches of science, we should find no very solid foundation for it in the period under our consideration. The arithmetic and geometry of that nation still remain very imperfect ; and we find among them no new theory, no interesting application, of the principles of mechanics. It is true the Chinese have been assiduous observers of the stars : but all their observations are confined to the most common objects in astronomy, such as eclipses, the positions of the planets, the solstitial altitudes of the Sun, and occul- tations of stars by the Moon : while no deduction of importance to the progress of the science stands for- ward to meet the eye. I shall only remark, that the emperor Hupilay, the fifth successor of Genghis khan in China, and the founder of the dynasty of Ywen in 1271, was a great patron of astronomy. He was brother to Houlagou, already mentioned, and re- sembled him in disposition. He appointed Co-Cheou- King chief of the tribunal of mathematics ; an in- dustrious observer, who carried the Chinese astronomy to a degree of precision, which it had never reached before. But this lustre was transient: the Chinese astronomy fell baek into it's former languor, and did N not 178 not again raise it's head, till about a century after, under the emperors of a new dynasty, who gave the direction of the mathematical tribunal into the hands of mohammedan astronomers. On the history of the sciences among the hindoos at the same time we may be still more Brief. Their knowledge never extended beyond the elements of mathematics ; and their astronomy had nearly the same fate as t that of the persians after the death of Ulugh Beg. CHA!>< 179 CHAP. VIII. State of the Sciences among the modern Greeks. + ON the destruction of the alexandrian school, the men of science, who were dispersed over all parts of Greece, contributed at first to keep up a taste for the mathematics in that country ; but in the state of neglect, to which they were reduced, they could not fail to decline. Indeed many ages elapsed, before any modern greek exhibited the least spark of that genius, which had animated Euclid, Archimedes, Apollonius, &c. Zonaras and Tzetzes, who have been quoted on occasion of the burning glasses of Archi- medes, were mere compilers, and in many cases but little acquainted with the subjects on which they treated. At length, in the beginning of the fifteenth century [A. B. 1420], Emanuel Moschopulos, a greek monk, made the very ingenious discovery of magic squares. It is true this was of no practical utility; but it ranks among those theoretical and subtle speculations, Which exercise the mind while they amuse it : and as I cannot dispense with mentioning it here, 1 shall give at once a general sketch of the labours of modern geometricians on this subject, that I may not have to return more than once to a matter of mere curiosity. Let a geometrical square, each side of which is represented by a given number, as for example the v 2 number 180 number 5, be traced on a plane; and let every side, both vertical and horizontal, be divided into five equal parts, and the points that mark the divisions be joined by vertical and horizontal lines. Thus the square will be divided into twenty-five equal cells; and if, beginning from one of the angular cells, and proceeding successively through all the ho- rizontal or vertical rows, the series of numbers, 1, 2, 3, 4, 5, 6, &c. be written, the last cell will contain the number 25, which is the square of 5. This dis- .position of the figures in their natural order forms in :consequence a natural square; the numbers in each row compose an arithmetical progression ; and the -sums of all these progressions are different But if the order of the numbers be changed, and they be arranged in such a manner, that all the rows, in- cluding even the two diagonal rows, produce the same sum; the square obtains the name of magical. This epithet may have been derived from the singular property of these squares, at a time when the mathe- matics were considered as a sort of magic : but it is not improbable, that it was taken from the super- stitious application of jthese squares to the. con- struction of talismans in an, age of ignorance. For instance, Cornelius Agrippa, who lived in the fifteenth century, has given in his treatise on .Occult Phi- losophy the magic squares of all the numbers from three to nine : and these squares, according to Agrippa and the followers of the same doctrine, are planetary: the square of three belongs to Saturn ; of four, to Jupiter; of five to Mars; of fix, .to the Sun.; of seven, to 181 to Venus ; of eight, to Mercury ; and lastly, that of nine to the Moon. Hist, de 1'Acad. 1705, p. 71. .The methods of Moschopulos for the formation of* magic squares extend only to certain particular cases, and required to be generalized. Bachet de Meziriac, a very learned analyst about the beginning of the seventeenth century, found a method for all squares with uneven roots; such as 25, 49, 81, &c., the roots of which are 5, 7, 9, &c. In these cases there is a central cell, which facilitates the solution, of the problem ; but Bachet could not solve it com- pletely for squares, the root of which is an even number. Frenicle de Bessi, one of the oldest members of. the academy of sciences, a profound arithmetician, considerably increased the numbers of cases and combinations that produce magical squares, both for even and uneven numbers. For instance, a skilful algebraist had supposed, that the sixteen numbers, which fill the cells of the natural square of four, could produce no more than sixteen magical squares ; but Frenicle showed, that they were capable of form- ing 880. To this research he added a new difficulty. Having formed, for example, one of the magic squares of the number 7 ; if the two extreme hori- zontal ranks, and the two extreme vertical ones, be taken from the 49 cells which composed it, a square will remain, which iia general will not be a magic square, but may be so, if the primitive magic square be properly chosen for the purpose. Frenicle taught N 3 how how to rtakfTHrii choice. By his method, on taking away the outer circumference from a magic square, or any circumference when the square has enough to admit this, or even several circumferences together, the remaining square is still magic. He likewise in- verted this condition, requiring that any circumfer- ence taken at pleasure, or several circumferences, -should be inseparable from the square ; that is, that it should cease to be a magic square* if these were taken away, and not if others were. In 1703 Mr. Poignard, a canon of Brussels, pub- lished a book on magic squares, in which he made two innovations, that extend and embellish the pro- blem. 1st: Instead of taking all the numbers that fill a square, for instance the thirty six consecutive numbers that would fill all the cells of the natural square qf six, he takes only so many of these numbers as there are units in the side of the square, that is in this case six ; and these six numbers alone he dis- poses in such a manner, in the thirty six cells, that no one is repeated twice in the same rank, either horizontal, vertical, or diagonal : whence it neces- sarily follows, that all the ranks, taken in any direc- tion whatever, uniformly produce the same sum. 2dly : Instead of taking these numbers only in the natural series, or in arithmetical progression, he takes them likewise in geometrical and harmonical progres- sion ; but with the last two progressions the magical contrivance of the square necessarily changes. In squares filled by numbers in geometrical progression, the products arising from the multiplication of the 2 numbers 183 numbers in each rank must be equal; and in the harmonic progression the numbers of each rank in like manner follow this progression. Poignard like* wise makes squares of these three progressions re- peated. La Hire, a geometrician of the academy of sci- ences, having his attention called to the subject by these researches, in which mere conjecture had fre- quently been employed, investigated and demon- strated it's principles in two very curious papers in the Memoirs of the Academy for 1705. In these he adds several new problems, which, extending and ge- neralizing the theory still farther, render it the more interesting to those, who are fond of the combinations of numbers. The demonstrations of all these learned men ap- pearing to Sauveur, another geometrician of the academy of sciences, too complex, and too little con- nected, he undertook to subject this theory to ana- lytical calculation, and uniform methods ; whence he might afterward deduce as corollaries simple and easy means of constructing magic squares in all cases. Pajot Osembrai considered the question in the same point of view ; and to him we are indebted for a new analytical method for magical squares of even num- bers, those for uneven numbers having been suffi- ciently investigated. Lastly Rallier des Ourmes still farther improved and extended all these methods in an excellent memoir presented to the academy of sciences. We have every reason to suppose, that the subject is now exhausted. N 4 This 184 ; This discovery of magic squares by Moschopulos may be called the last breath of the greek mathema- ticians. After the taking of Constantinople by Mo- liammed ir, we hear of them in these climes no longer. CHAP. 185 CHAP. IX. State of the Sciences among the Christians in the West, to the End of the thirteenth Century. 1 HE Christians in general for a long' time "displayed a great aversion to the sciences. Subjected from the origin of Christianity to a multitude of superstitious opinions, which tended to convert man into a con- templative automaton, they looked with indifference or disdain on all occupations foreign to religious wor- sliipj or to the labours absolutely necessary to procure them subsistence. However, when they had begun to drive the arabs out of some parts of Spain, ill the beginning of the tenth century, the voluntary or compulsive intercourse which they had with these people excited the electric fire of genius among the Christians, and many of them were eager to acquire knowledge from those moors, whose religion they held in abhorrence. We have already said, that pope Silvester n had learned arithmetic from the arabs of Spain. Alphonsus IT, king of Castile, founded in his capital a sort of college or lyceum for the advancement of astronomy, and entrusted it's principal direction to some arabs. He himself made observations and cal- culations with them. These mutual labours produced the celebrated Alphonsine Tables, more accurate and complete than any that had preceded them. And the study of astronomy was pursued in Castile long after the death of Alphonsus* But the interests of ambU ' tion, 186 tion, which nothing can withstand, perpetually che- rished the seeds of hatred and discord between thearabs and the Christians. The latter, never losing sight of their project of retaking all Spain, gained ground every day : in proportion as their victories multiplied, the sciences were neglected : and these at length received their mortal blow, when the moors were completely expelled from Spain by the loss of Granada, A. D. 1492; an event to be regretted in the annals of the human mind, and advantageous to nothing but the catholic religion, the empire of which it extended on the ruins of islamism. In the other Christian countries of Europe we find many men distinguished for the extent of their know- ledge, considering the time in which they lived ; or by the proofs of genius, which they exhibited, and from which society might have derived the most striking benefits, had not ecclesiastical authority, ever intole- rant, and ever clothed in thunder, too frequently checked or totally stopped their career. The Italians are the first, that here present them- selves to our view ; and at the outset their attention was drawn to algebra by an accidental circumstance. One Leonard, a rich merchant of Pisa, made several voyages to the East in pursuing his commerce ; and his intercourse with the arabs affording him an op- portunity of attaining a knowledge of algebra, which was then considered as the sublime part of arithmetic ; and he imparted his acquirements to his countrymen about the beginning of the thirteenth century* It was supposed till very lately, on the authority of Vos- sius and some modern Italian authors, that Leonard of Pisa 187 Pisa flourished toward the end of the fourteenth cen- tury . but Mr. Cossali, a canon of Parma f has dis- covered and quoted* a manuscript by that algebraist dated 1202, which was enlarged and rewritten ia 1228. Leonard of Pisa was very skilful in algebra, particularly in the analysis of problems of the Dio- phantine kind. The extract from his manuscript, which Mr. Cossali has given, shows, that the author had proceeded as far as the resolution of cubic equa- tions, and those of higher powers capable of being re- duced to the second or third order. This impulse given to algebra was propagated through Europe, and extended to all parts of the mathematics. The thirteenth century produced a great number of learned men in every branch, in Italy, France, Germany, and England. The chief of those, who rendered any service to the mathe- matics, I shall proceed to mention. Jordanus Nemorarius was eminent for his time [A. D. 1230] in arithmetic and geometry, as may be judged by his treatise On the Planisphere, and his ten books of Arithmetic. He had a contemporary much better known, John of Halifax, vulgarly called Sacrobosco, which signi- fies the same thing in the barbarous latin of those days. Sacrobosco, who was born in England, came to reside as professor of the mathematics at Paris. We have a treatise of his on the sphere, which has been commented on by Clavius the Jesuit, and re- * Origine, Transporto in Italia, e primi Progress! in Essa del Algebra, &c 1.797, printed 188 printed a great number of times. He likewise left us treatises on the Astrolabe, on the Calendar, and on the Arithmetic of the Arabs. He died at Paris in 1256, and his tomb was still to be seen in the cloister of the Matlmrins before the french revolu- tion, A. r>. 1250, Carnpanus of Novara translated and commented on Euclid's Elements, and wrote a trea- tise on the Sphere ; another on the Theories of the Planets, the object of which was to make known the ancient astronomy, and the corrections introduced in it by the arabs; &c. A. D. 1260, Vitellio, a native of Poland, but set- tled in Italy, wrote a treatise on Optics in ten books. This work is, in fact, nothing more than that of Al* hazen, only rendered more perspicuous and methodical. We have another work of the same period on Optics by Thomas Pecham, who from a simple obser- vantine monk became archbishop of Canterbury. This work has been repeatedly printed, and was for a long time a classical book. The sciences found a zealous patron in the great emperor Frederic u, amid the continual wars he had to sustain against the popes. This prince ascended the throne in 1219, and died in 1250. He founded the university of Naples, composed several works, and caused those of Aristotle, with the Almagest of Pto- lemy, to be translated into latin. For these transla- tions he employed Gerard of Sabionetta, .commonly called Gerard of Cremona, by whom we have also a translation of Geber's commentary on the Almagest, and of Alhazcn's treatise on Twilight. To him like- wise 189 wise is ascribed a treatise on the Theories of the Planets. I should say nothing of Albertus Magnus, as. he was named by his little contemporaries [A. D. 1260]: if he had not written works on arithmetic, geometry, astronomy, and mechanics, useful in their day, but now lost. He distinguished himself chiefly in the construction of machines. It is reported, that he made an automaton figure of a man, which went and opened the door when any person knocked at it, and uttered some sounds, as if speaking to the person who entered, Roger Bacon, an english cordelier, who was born in 1214, and died in 1294, has still more claim to the. notice of posterity. He enjoyed a very high reputa- tion among his contemporaries, and stili retains it in the eyes of the learned. His numerous works, in which great genius and invention are displayed, have been successively printed. His treatise on Optics is particularly remarkable for the ingenious, just, and at that time new ideas it offers on the subjects of astro- nomical refraction, the apparent magnitudes of ob- jects, the extraordinary size of the Sun and Moon near the horizon, the place of spherical foci, c. Some english writers, a little too much prejudiced in favour of their countryman, have fancied they discovered in this treatise, that the author knew the use of specta- cles, and even of the telescope : but Mr, Smith, an englishman of more impartiality, and an irrefragable judge, has controverted this opinion by an accurate and critical discussion of the passages, that gave rise to it. Others have been desirous of attributing to Bacon 190 Bacon likewise the discovery of gunpowder : in fact he was on the verge of it, being a great chemist for his time, and was acquainted with the effects of salt- petre ; but it was not explained, and thoroughly krrowfy till some years afterward. By the monks he was persecuted, accused of magic, and confined in a dungeon ; from which he was not liberated, till he had fully convinced his superiours and the pope, Nicholas iv, that lie had never held any correspond- ence with the Devil. The invention of spectacles is due to the close of the thirteenth century, and we are indebted for it to the italians. Incontestable proofs exist, that the first glasses of this kind were constructed by Alexander dq Spina, a jacobin friar, who died at Pisa in 1313, CHAP, 191 CHAP. X. "State of the Sciences among the Christians m the continued through the fourteenth and fifteenth Cen- turies. THE fourteenth century, fertile in theologians, alchy- mists, and even men of reputation in the department of letters, was barren of mathematicians among all the western nations of Europe. It is true a few geo- metricians, and astronomical observers or theorists appear, who, tl^nigh they did not advance the sci- ences, at least maintained their honour, till they could receive more effectual assistance. In Italy, Peter of Albano, a celebrated physician, wrote a treatise on the Astrolabe: Cecchi Ascoli, professor of mathematics at Bologna, composed a commentary on the Sphere of Sacrobosco, which was several times reprinted. Both these acquired the re- putation of sorcerers and heretics, and Albano was burnt in effigy, as the more unfortunate Ascoli was in person at Bologna, in the year 13S8, at the age of seventy. In England there were many geometricians and astronomers ; but of their works or observations we have only a few fragments, most of which are dis- persed in manuscript in different libraries. In Germany, John of Saxony, an augustin friar, wrote on the Alphon^ine Tables, and on eclipses; and and Henry of Hesse, professor at the newly founded university of Vienna, treated on the theory of the planets : but these wovks have never been printed. France too quotes some mathematicians, as John cle . Muris, author of the system of our modem music, and likewise versed in astronomy, for we have a manuscript treatise of his on this science: John de Lignieres, an astronomer, born at Amiens, and pro- fessor of mathematics at Paris, some of whose obser- vations, collected by Gassendi, are still in being : Nicholas Oresme, who translated Aristotle de Mundo^ and composed a treatise on Proportions, which has been preserved in manuscript. To the last of these mathematicians we have another obligation : he was .preceptor to Charles v, surnamed the Wise, and was the chief instrument in creating the library of the kings of France, which was founded by that men jiarch. Notwithstanding the inactive state, in which the theory of mathematics remained, practical mechanics produced some ingenious machines, which ought not to be unnoticed. Paper had been fabricated for a long time; but in the fourteenth century, Ulman Strame, a senator of Nuremberg, contrived a pecu- liar machine for grinding rags, and passes for the in- ventor of the paper-mill. Wheelwork clocks, both fixed and portable, are of the same period. Richard Wallingford, -an.english benedictine, made for thp convent of St. Albans, of which he was abbot, a clock of this sort, that pointed out the hours, the courses of the Sun and Moon, the time of high water, c. ; .and he wrote a book on this subject, which is still 193 still preserved in manuscript in the Bodleian library. After his example James cle Donclis, a citizen of Padua, very learned in Physic, Astronomy, and Me- chanics, constructed for his own country a clock, which was then considered as a wonder. This clock marked the hours, the courses of the Sun, Moon, and planets, and the days, months, and festivals throughout the year. Whether these machines "be- long entirely to the age in question, or were more or less accurate imitations of the clock, that was sent to Charlemagne by the khalif Haroun al Rashid, are points, however, on which we cannot decide, for want of the necessary documents. We nov/ advance toward happier times: the fif- teenth century produced a great number of learned mathematicians, particularly of very skilful astrono- mers. We will begin with geometry and algebra. Among those who cultivated these two sciences at that time is chiefly to be distinguished Lucas Pac- cioli, commonly called Lucas de Borgo, because he was born at Borgo San Sapocha in Tuscany. He was a Franciscan monk, and flourished toward the end of the fifteenth century. After having travelled a con- siderable time in the East, either in quest of know- ledge, or to execute the commissions with which he was privately entrusted by his superiours, he taught mathematics at Naples and Venice, and afterward at Milan, where he was the first who occupied the mathematical chair founded by Lewis Sforsa, called the Black. He composed several works for his pu- pils ; and translated Euclid into latin, or rather re- vised the translation of Campanus, to which he o added 194 added learned annotations. In 1494 he published a treatise on algebra in Italian, entitled : Summa de Arithmetica^ Geometria, Proportion^ et Proportion- alita, c. : in which we find the common rules of arithmetic ; some inventions of the arabs, such as tjie rules of false position, and the resolution of equations of the first and second order ; and lastly elements of geometry. We are likewise indebted to Lucas de Borgo for two other works : one de Divina Proportione^ which includes various subjects, as per- spective, music, architecture, c. ; the other is a treatise on the regular bodies, under a long latin title, which it would be useless to copy. Astronomy made great progress in this age. It's first benefactors were John Gmunden, a professor of this science in the university of Vienna about 1416; and the celebrated Peter Dailli, who in 1414 proposed to the council of Constance some means of reforming the calendar, which was then become very incorrect, and reconciling the motions of the Sun and Moon. The cardinal Nicholas de Cusa, who was born in 1391, and died in 1454, is also celebrated among men of learning for his attempt to revive the pythagorean system of the Earth's motion. This idea, though true, had not yet attained all the maturity, which observations could give it ; and it cannot but appear a little extraordinary, that a cardinal should maintain at that period an opinion, without giving offence to any one, for which two hundred years afterward, when it was supported by more substantial argu- ments, Galileo was confined in the dungeons of the Inquisition. Purbach, 195 Purbach, who was born in 1421, and died in 1461, and his scholar Regiomontanus, are considered as the restorers, or the two greatest promoters of astronomy in the fifteenth century. The former of these, after having travelled a long time, to gain from intercourse with men of learning a full know- ledge of astronomy, the principles of which he had studied under John Gmunden, settled at Vienna ; to which place he was attracted by the liberalities of the emperor Frederic in, and where he succeeded Gmunden in his professorship at the university. On this occasion he undertook a useful and necessary work, that of producing a good translation of Pto- lemy's Almagest ; for all that had hitherto been pub- lished in latin abounded with mistakes, from the ignorance of the translators in the science of astro- nomy. He was unacquainted both with the greek and with the arabic, but his perfect knowledge of the subject enabled him to correct the erroneous versions, and supply, at least as to the sense, the real work of Ptolemy. Soon after this he wrote for the use of his pupils several treatises on arithmetic, geometry, the solstitial altitudes of the Sun, the description and use of portable timepieces, the calculation of the length of the degree in each parallel of latitude com- pared with it's length at the equator, &c. As he united manual dexterity with theoretical knowledge, he constructed himself instruments of use in 2:110 mo- o nics, and celestial globes, on which was marked the progress of the stars in longitude from Ptolemy's time to the year 1450. He determined the obliquity of the ecliptic by his own observations : he also made o 2 various various corrections in the theory of the motion of the planets, which the ancient tables exhibited in a defective manner ; and lastly he introduced some ab- breviations into trigonometrical calculation. His greatest glory is, that he was the instructor o Hegiomontanus. They observed the heavens toge- ther at Vienna for ten years. After the death of Puibach, the genius and avidity of the latter, for all the sciences, induced him to make a journey to Rome, that he might acquire the greek language with more facility, and enable himself to read the other greek mathematicians, as well as Ptolemy, in their native tongue. His progress was so rapid, that in a very short time he translated from the greek into latin the Conies of Apollonius, the Cylindrics of Serenus, the Mechanical Questions of Aristotle, the Pneumatics of Hero, all the works of Ptolemy, &c. The old version of Archimedes by James of Cremona he corrected from the original greek ; and he did not confine himself to translation, for he was the author of several excellent works. His treatise on Trigono- metry is remarkable for several novelties, particularly for an elegant method, which was likewise the first ever given, for resolving any spherical triangle in general, when the three angles, or three sides, are known. The fame of Regiomontanus induced the senate of Nuremberg, to invite him to that city. There he formed an observatory ; furnished it with excellent instruments, invented or improved by him- self, and with these made observations, that enabled him to correct and enlarge the ancient theories. From certain observations erroneously interpreted, of which 197 which he gives the particulars, many astronomers had attributed to the stars an irregular motion, some- times directed to the east, at others to the west. This opinion is successfully combated by Rcgiomon- tanus. In 1472 he had an opportunity of observing a comet, the motion of which, at first very slow, soon acquired such a degree of velocity, that toward it's perigee it traversed a space of more than thirty degrees in twenty four hours. This comet drew after it a tail of more than thirty degrees in length. Pope Sixtus iv, being desirous of reforming the calendar, invited Regiomontanus to Rome, to superintend and execute this important operation ; made him splendid promises ; and even nominated him to the bishopric of Ratisbon. Regiomontanus obeyed the summons ; but after having resided a few months at Rome, he died there in the year 1476, at the age of forty. A rumour was propagated, that the children of George of Trebisond, one of the translators of Ptolemy and Theon, procured him to be poisoned, because he had publicly pointed out se- veral of their father's errours. When Regiomontanus quitted Nuremberg, he left there a pupil well qualified to pursue his plans, and add new ones to them. This was Walt her. a rich citizen, who procured the construction of all the in- struments invented by Regiomontanus; and who, after the death of his master, continued to observe the heavens till 1504, when he died, being seventy four years of age. ; All these observations, which present a multi- tude of various phenomena, form a valuable treasure o 3 t 198 to the astronomer. Unfortunately astronomical in- struments did not then possess all the perfection they have since acquired ; and besides, the aid of the telescope was then wanting, Walther was as jealous of his astronomical acquirements, as a lover is of his mistress : he shared them with no one ; and he is even charged with having reserved to himself ex- clusively the use of the manuscripts of Regiomontanus, which were deposited in his hands. In the fifteenth century we find likewise several learned mathematicians. In France, James Lefevre cultivated the mathematics with success, and ren- dered them some service by his translations and other performances. In Italy, John Bianchini, of Bologna, constructed astronomical tables much esteemed in their time; James Angelo, a florentine, translated Ptolemy's geography ; and Dominic Maria Novera^ a bolognese, initiated Copernicus into astronomy. In Germany, John Engel, a bavarian, publifhed Ephemerides of the celestial motions, and proposed a scheme for reforming the calendar. In Spain, Fer- dinand of Cordova commented upon Ptolemy's Al- magest; Bernard of Granolachi also published Ephe- merides in the vernacular language, beginning with the year 1488, and calculated as far as 1550; c. All these labours contributed to feed the sacred flame of science. Navigation is too essentially connected with astro- nomy, independent of it's particular utility, to allow the vast progress it made in the fifteenth century, particularly towards it's conclusion, to be passed over in silence. For this it was chiefly indebted to the iise 199 use of the compass, of which consequently the origin requires first to be made known, as well as the me^ns it affords of guiding a veflel on the ocean. The property which the loadstone possesses of attract- ing iron was known by the greeks as early as Thales, and by the Chinese more than live centuries before our era. But it was not known, at least in Europe, before the commencement of the twelfth century, that a loadstone freely suspended, or floating on water by means of a cork, would always point in one direction toward the poles of the World ; still less, that it could communicate this property to a bar or needle of iron. It appears by the works of Guy de Provins, a french poet of the twelfth century, that the mariners of France were the first who employed the compass for directing the course of vessels, whence it had the name of marinette. The practice of sus- pending the needle on a pivot is veiy ancient in France. The Italians, germans, and english, how- ever, dispute with us the invention of the compass. These reciprocal pretensions are capable of being supported ; either because it is possible, that the same thing may be discovered in different places at the same time; or because, the compass having been brought to perfection by degrees, those nations that contributed to it, each for it's own use, have imagined they might arrogate to themselves the whole of the invention. With regard to the Chinese, if it be true, as some historians assert, that they employed the compass in navigation long before any european, they have always confined themselves at least to a rude method ; for their constant practice of floating o 4 the 200 the magnet on water is by no means comparable to it's suspension on a pivot. The ancients, who had nothing to guide them by sea but the observation of the stars, seldom ventured to any considerable distance from the land. Modern navigators, furnished with the compass, gradually abandoned this slow and timid practice of coasting along the shore ; and, trusting to their new guide, not less safe than commodious, they launched out into the open sea, voyaging by night as well as by day, and in the cloudiest weather, with full confi- dence in it's power, which was justified by success. Thus the compass may be said to have put man in possession of the dominion of the sea, and to have opened a communication between all the people, that inhabit the different parts of the terraqueous globe. Toward the middle of the fourteenth century, the Spaniards had begun to navigate the Atlantic ocean, and discovered the Canaries, or Fortunate Islands, with which the ancients were acquainted, but which had long been neglected and forgotten. In the fifteenth century navigation took a still greater and bolder flight, and owed it's first success, of a new kind, to the genius and courage of the portuguese. The sciences cultivated by the arabs had been intro- duced into Portugal, as into Spain, by the moors and jews, who were very numerous in these countries, Under John I, one of the greatest princes that ever swayed the sceptre of Portugal, a small fleet sailed in 1412, to attack the moors settled on the coast of Barbary ; while other vessels were navigating by his direction along the western shore of Africa, to dis- cover 201 cover the countries that lay in that stuation. These first attempts proved fortunate, and were the prelude to the great discoveries about to be made. Henry duke of Visco, the fourth son of John, had accompanied his father on his expedition to Barbary, and distinguished himself there by different acts of bravery. Instructed in all the science of his time, particularly in geography, by the lessons of the ablest masters, and the narratives of travellers, he had ac- quired a thorough knowledge of the configuration of the globe ; and conceived the project, as well as the possibility, of pushing these acquisitions still far- ther. He assembled a great many naval officers, who had already acquired much experience; and communicated to them his designs, which they em- braced with enthusiasm. Fleets were fitted out, and, proceeding toward the south, they not only discovered rich and extensive countries along the western coast of Africa, but, by standing to sea, felljn with several islands, as Madeira, the Cape Verdes, the Azores, &c. So that at the death of prince Henry, in 1463, the portuguese navigators had advanced within five degrees of the equator. The discovery of prince Henry, which belongs most particularly to our subject, is that of sea charts, known by the name of plain charts, to, represent the course a vessel ought to follow, and to direct it ac- cording to this course. Tbe use of terrestrial globes was very ancient : but tl^it of maps, which were more recent, had the preference, since Ptolemy and the arabs had taught geometrical methods of projecting the circles of the sphere on a simple plane surface : but 202 but prince Henry, who wanted to mark out by straight lines the several points of the compass for the steering of a ship, could not employ these, and was obliged to invent a different construction. He supposed the meridians to be represented by parallel right lines, and the circles parallel to the equator by other pa- rallel right lines perpendicular to the former : he then describes a compass on the chart ; and to mark the course of a vessel, which he supposes to hold on in one direct line, he draws a straight line from the place of departure to the place for which it is bound, and sup- poses the rhumb which is parallel with this is to be the course desired. But these charts can be of real utility only for small portions of the globe. When the spaces are considerable, the degrees of the circles parallel to the equator cannot be represented by equal lines in each, as the author supposes ; for we know, that the cir- cumferences of these circles are continually diminish- ing from the equator to the pole. Besides, the course by one and the same rhumb is not a simple straight line even in this construction, except in the two cases, which must be of very limited occurrence, when the ship sails uniformly on the same meridian, or on the same parallel. These inconveniences were soon perceived, and in the course of the two suc- ceeding centuries they were remedied. The impulse which prince Henry had thus given to na- vigation was soon carried to the utmost point. Nothing- was thought of throughout Europe but distant voyages, with projects of acquiring new countries, and forming new settlements, in seeking which, beyond the seas, men exposed themselves to the most fearful dangers. At 203 At the death of prince Henry the throne of Portugal was filled by Alphonsus, who, having to maintain his pretensions to the crown of Castile, and a war against the moors of Barbary at the same time, could but feebly pursue the chain of discovery along the african shore. It was followed up with ardour however by his son, John II, who was endowed with all the spirit and learning of his great uncle, prince Henry. In 1484 the portuguese equipped a powerful fleet, which, after making itself master of the kingdom of Benin, advanced far beyond the equator, and for the first time exhibited to european eyes a new firmament, and new stars. Two years after, Bartholomew Diaz reached the Cape of Good Hope ; which was doubled in 1 192 by Vasco de Gama, who proceeded to found several portuguese settlements in the East Indies. Toward the west the celebrated Christopher Co- lumbus, formed in the school of the portuguese na- vigators, undertook in the same year, 1492, to cir- cumnavigate the globe, with a small fleet equipped at the expense of Isabella, queen of Castile, and Fer- dinand, king of Arragon, her husband. If he did not completely accomplish this vast design, at least he immortalized himself by the discovery of America, the greatest and most important, that was ever made in the art of navigation. The. particulars of these celebrated expeditions however are foreign to the purpose of this Work. END OF THE SECOND PERIOD. PERIOD PERIOD THE THIRD. PROGRESS OF THE MATHEMATICS, FROM THE END OF THE FIFTEENTH CENTURY TO THE INVENTION OF THE METHOD OF FLUXIONS. THE progress, which was made in the sciences by the* western nations of Europe, from the commence- ment of the sixteenth century up to the present day, so completely effaces that of all other nations, that I shall confine myself to it alone in the remainder of this work. What indeed are the astronomical observa- tions of the Chinese and hindoos, compared with all the valuable discoveries, with which the europeans have enriched algebra, geometry, mechanics, astro- nomy, &c. ? In the history of the sciences it is not as in the common history of nations. The narrative of political affairs must be given at length, and ar- ranged conformably to the succession of \vars and negotiations, changes in manners, revolutions in go- vernment, &c., so as to give a body to chronology, and make known the ranks, which different nations occupy in the World. In science, where new truths are $e events to be recorded, if a discovery come to be 205 be connected with a more extensive and more im- portant theory, it loses it's individual existence, and it may be expunged from the general picture of hu-r man knowledge without detriment. CHAF, ftM CHAP, r. V The Progress of Analysis. UNDER the generic name of analysis I comprise here arithmetic and algebra, to both of which it agrees, since in fact they are fundamentally one and the same science. Arithmetic operates immediately on numbers, and algebra operates in a similar manner on magnitudes in general. Frequently algebra lends very important and even necessary assistance to arith- metic, to guide it through the labyrinth of certain abstract combinations ; because, as numerical calcu- lations leave no trace of the road through which we have passed, we have need, on several occasions, of recurring to general principles, and being able to follow their regular chain. The analytical works of Leonard of Pisa having remained in manuscript, and as it were absolutely unknown even in Italy, Lucas de Borgo's treatise, Summa de Arithmetica e Geometria, which has been already mentioned, exhibited the state in which al- gebra then was ; that is to say, confined to the com- plete resolution of equations of the first and second order. The passage to the higher orders was difficult. In this respect Italy had the honour of enabling algebra to take a new flight, by the general resolution of equations of the third and fourth order. Cardan, 207 Cardan, who was born in 1501, and died in 15/6, relates in his book De Arte magna, pub- lished in 1545, that Scipio Ferrei, professor of ma- thematics at Bologna, was the first who gave the formula for resolving equations of the third order: that about thirty years afterward, Florid o, a Venetian, informed of this discovery by his master Ferrei, pro* posed to Nicholas Tartaglia, a celebrated mathema- tician of Brescia, several problems, the solution of which depended on this formula : and that Tartaglia discovered it, by meditating on these problems. In another place Cardan confesses, that, at his urgent intreaty, Tartaglia communicated to him this for- mula, but without adding the demonstration : and that, having discovered this demonstration, with the assistance of his scholar Lewis Ferrari, a young man of great penetration, he thought it his duty, to give the whole to the public. But Tartaglia was much displeased with the conduct of Cardan. He claimed the sole invention of the formula ; maintained, that Florido himself was unacquainted with it; and as- serted, that Cardan was guilty both of treachery and plagiarism, in having made public a rule entrusted to him under a promise of secrecy, and to which he had not the least claim. The resolution of equations of the fourth order closely followed that of equations of the third. And here too, we learn from Cardan, that Lewis Ferrari made this new discovery. His method, known at present to all analysts by the name of the Italian method, consisted in disposing the terms of the equation of the fourth order in such a manner, that, by adding to each side one 208 one and the same quantity, the two sides might be resolved by the method of the second order. By ac- complishing this condition we are brought to an equation of the third order, so that the complete re- solution of the fourth order is connected with that of the third, the difficulties of which equally aifect the other. I say the difficulties ; for in fact there is one case in the third order, which has embarrassed all algebraists, and for this reason is called the irreducible case. This case embraces equations of which the three roots are real, unequal, and incommensurable to each other: also the formula*, which represent them, include imaginary terms; and we should at first be induced to believe, that these expressions were imaginary, if an attentive examination of their nature did not prevent us from being too hasty in our judgment. Tartaglia and Cardan did not venture to decide upon the subject. The latter only applied himself to the resolution of some particular equations, which ap- peared referable to it, and in which the difficulty fortuitously disappeared. Raphael Bom belli of Bologna, who was a little posteriour to Cardan, first showed in his algebra, printed in 1579, that the terms of the formula which represents each root in the irreducible case, form when taken together a real result in all eases. This proposition was at that time an actual paradox : but the paradox disappeared, when Bombelli had demon- strated by geometrical constructions, nearly of the same nature as those which Plato had employed to find the two mean proportionals in the problem of the duplication of the cube, that the imaginary quan- S tities 209 tities included in the two sides of the formula must necessarily destroy each other by the opposite nature of their signs. In support of this general demon- stration the author produced several particular ex- amples, in which, by extracting the cube roots of the two binomials, which compose the value of the un- known quantity, according to the usual methods for known quantities, and then adding the two roots, real results were obtained. Mathematicians have since arrived at tl*e same conclusion by other means more direct and simple; but this first effort of 13om- belli was a great step in the analysis of equations for the time. It was natural to suppose, that the methods for the third and fourth orders should be extended farther, or at least give rise to new views respecting the forms of roots in still higher orders. But if we except those equations, which by transformations in their calcu- lation are reducible, in their ultimate analysis, to the first four orders, the art of resolving equations in general, and with the utmost rigour, has made no progress since the labours of the Italians just men- tioned. Maurolicus, abbot of Santa Maria del Porto in Sicily, who was born in 1494, and died in 1575, was profoundly skilled in all parts of the mathematics. He applied himself to another branch of analytical calculation, at that time almost unknown. This was the summation of several series of numbers, as the series of natural numbers, that of their squares, that of triangular numbers, &c. On this subject he gave p theorems 210 theorems remarkable for subtilty of invention and simplicity of result. The reader will perceive, that I feel pleasure in doing justice to learned foreigners : but the same equity requires me, to ascribe to one of my illustrious countrymen the glory of having generalized the al- gorithm of algebra, and made several important discoveries in the science. This was Vieta, born in 1540, and who died in 1603. Before him equations of the kind called numerical were alone resolved : the unknown quantity was represented by a particular character, or by a letter of the alphabet ; and the other quantities were absolute numbers. It is true, that the method applied to one equation could afterward be applied equally to another similar equation. But it was desirable, that all the quantities indifferently, should be represented by general characters, and that all the particular equations of the same order should be only simple translations of one general formula. This advantage Vieta conferred on algebra, by intro- ducing into it the letters of the alphabet, to represent all sorts of quantities, known or unknown : an easy and commodious method of notation, both because the use of letters is very familiar to Us, and because a letter may express indifferently a weight, a distance, a velocity, &c. He himself applied this new al- gorithm very happily in several cases. He taught us to make various transformations in equations of every order, without knowing their roots ; to deprive them of the second term ; to free them from fractional coefficients; to increase or diminish the roots by a given a given quantity; and to multiply or divide the roots by any number whatever. He also gave a new and ingenious method for resolving equations of the third and fourth order. Lastly, in default of a rigorous resolution of equations of all orders, he came to an approximate resolution ; which is founded on this principle, that any given equation is but an imperfect power of the unknown quantity : and he employs nearly the same modes of proceeding, as for finding by ap^ proximation the roots of numbers which are not perfect powers. If we at present possess more simple and commodious means of attaining the same end, , we ought not the less to admire these first efforts of genius. About the same time several algebraists published very useful treatises for diffusing a knowledge of the science, though they contained no new views, that have any thing in them remarkable* The commencement of the seventeenth century is marked by the noble discovery of logarithms, which has rendered, and will never cease to render the most important services to all the practical parts of science, and particularly to astronomy, by it's abridgment of numerical calculations, without which the most un- conquerable patience must have been obliged to re- linquish a number of useful researches. For this in- vention we are indebted to baron Napier of Mer- chiston in Scotland, who was born in 1550, and died in 1617. Every one knows, that of the four fundamental rules of arithmetic, addition, subtraction, multiplication, and division, the former two are easily performed with p % accu- accuracy, if a little attention be bestowed on them ; but both the others, particularly division, require operations frequently very long, and tiresome, and capable of wearying the calculator, or exposing him to commit important mistakes. An observation that had long been made respecting the correspondence of geometrical progression or proportion with arith- metical, but from which nothing had been deduced, suggested to baron Napier the idea of constructing tables, by means of which multiplication and division are avoided, and all numerical calculations are re- duced to simple addition and subtraction. This observation is, that every thing effected by means of multiplication and division in geometrical progression or proportion is effected in arithmetical progression or proportion by means of addition and subtraction, For instance, in geometrical proportion the fourth term is equal to the product of the two middle terms divided by the first; in arithmetical proportion the fourth term is equal to the sum of the two middle terms, subtracting from them the first : in geometrical progression, one term is equal to another multiplied by the ratio of the progression, as- many times, plus one, as there are terms between them and in arithmetical progression one term is- equal to another, plus the difference of the pro- gression, added as many times, plus one, as there are terms between them. Hence baron Napier made two progressions, corresponding term to term, one geometrical, the other arithmetical ; the terms of the iirst he considered as the principal numbers, and those of the second as their logarithms, or the measures of their ratios ; and he taught how to con- struct 213 struct tables containing these two kinds of numbers. Then, when a multiplication or division is required, nothing more is necessary, than to work with the lo- garithms by addition or subtraction ; and the new logarithms, thus obtained, will answer in the tables to the numbers, which, without this assistance, must have been sought directly by niultiplication or cji-r vision. The Choice of the two progressions Is equally arbi- trary with regard to the theory. Napier took for arithmetical progression of the logarithms, that of natural numbers, 0, 1, 2, 3, 4, 5, 6, &c. ; making the logarithm 0, or nought, answer to the unit of the geometrical progression ; and this he regulated $p that it's terms, being represented by the abscisses of an equilateral hyperbola, in which the first absciss an4 the first ordinate are each equal to 1, the logarithms are represented by the series of hyperbolic spaces. Then the fundamental number of the geometrical progression, that is to say, the number, which forms by it's successive powers the terms of the geometrical progression, and by it's exponents those of the arith- metical, equals 2.71828 nearly. If this number once found, be raised successively tq the square, the cube, the fourth power, the fifth, and so on, the numbers arising, 7-382, 20.08(>,' 54.59.9, 148.425, &c., are the following terms of the geometrical pro- gression, to which the logarithms, 2, 3, 4, 5, &c. correspond. But this is not sufficient : it is ne- cessary, to determine likewise the logarithms of the numbers intermediate to the terms of the geometrical progression; in order to construct tables, which, from p 3 the 214 the proximity and extent of the numbers to be operated upon, may be applicable to all the cases required in practical calculation. Arithmetic alone furnishes sufficient assistance for this, but the end is more readily obtained by the help of algebra. Such was the system of logarithms, which Napier exhibited in his book entitled Logarithmorum Canords Descriptio, sen Arithmetica Supputationum mirabilis 'Abbreinatio^ published for the first time at Edin- burgh, in 161 4. The system has this inconvenience, that the terms of the fundamental geometrical pro- gression, the first excepted, are numbers accompanied with fractions, while those in the arithmetical pro- gression of the corresponding logarithms are whole numbers, which would have occasioned inconveni- ences in the use of tables constructed according to this hypothesis. The author himself acknowledged this defect, and conferred on the subject with his friend Henry Briggs, professor of mathematics in Gresham college. They both agreed in substituting for the geometrical progression first proposed the decuple progression 1, 10, 100, 1000, &c., which serves as a base to the numeration, and retaining all the rest. By this change the construction of the tables was rendered more easy, and their use more commodious. We may add, that, when the logarithms are once calcu- lated for either of the two systems, they may be found for the other, by multiplying them by a con- stant given number. This reciprocal correspondence of the two systems has occasioned the use of the former to be retained in the logarithmic formulas of . the 215 the inverse method of fluxions, in which it's applica- tion is very simple and convenient. Napier dying before he was able to calculate tables according to the new system, the whole labour de- volved upon Briggs, who applied to it with indefa- tigable ardour. In 1618 he published a table of the common logarithms for the first thousand natural num- bers : in 1624 he published a second, containing the logarithms of the natural numbers from 1 to 20000, and from. 90000 to 100000. Gelibrand, Gunther, and Hadrian Vlacq, men distinguished for their learning, and pupils or friends of Briggs, tilled up the defi- ciencies he had left : they also published new tables,' which contained the logarithms of the sines, tan- gents, Sec. for 90 degrees. All these tables have since been carried still farther ; and I should never end, were I to examine all the forms that have been given them, though constantly according to the system adopted by Briggs. Nothing more remains to be desired with regard to them ; and most of the enlargements, that some have occasionally endea- voured to give to the tables, are but illusory super- fluities. It must not be omitted, that Justus Byrge, a ger- man geometrician, printed in 1620 a table con- structed according to the inverse order of our ordi- nary tables of logarithms. Instead of considering the numbers relative to the geometrical progression as the principal numbers, to which the logarithms ought to be subordinate, he on the contrary regarded the logarithms as the principal numbers, to which he made those depending on the geometrical progression p 4 correspond* 216 correspond. But this system met with no success, and was little calculated to succeed, from the im- mense tables it would have required. While arithmetic was enriched by the discovery of logarithms, algebra made a distinguished progress in the hands of Harriot, an english analyst, who pub- lished in l6Q a work entitled, Artis Analytics fraxis. This work contains every thing of much importance that had been written on the subject of algebra, and many novelties, which belong to the author. In the first place Harriot simplified the no- tation of Vieta, by substituting small letters for capitals, and new signs to abridge the statements. Some persons, perhaps, will ascribe very little merit to these alterations ; but a different judgment will be given by those who know, that the simplicity of an algorithm has frequently produced remarkable discove- ries. Harriot was also the first who thought of placing all the terms of an equation on one side, and thus distinctly saw, what Vieta had only pointed out in a confused manner, that, in every equation, the coef- ficient of the second term is the sum of the roots taken with contrary signs; that the coefficient of the third is the sum of the products of the roots taken two by two : that the coefficient of the fourth is the sum of the products of the roots taken three by three with contrary signs ; and so on, to the last term, which is the product of all the roots taken with con- trary signs. We are likewise indebted to him for hav- ing observed, that all equations beyond the first order may be considered as produced by the multiplication uf equations of the first order ; so that by substi- tuting 217 luting, in the place of the unknown quantity, one of the values given by these component equations, the totality of the terms of the equation proposed becomes equal to nought. And these theorems have greatly facilitated the complete resolution of some particular equations, as well as other researches. Harriot was born in 1560, and died in 1621. To the general advancement of the science of ana- lysis no one has contributed more than our illustrious des Cartes, who was born in 1596, and died in 1650. Nature had bestowed on him the genius and energy necessary to extend all the boundaries of human knowledge. In his Method, he taught mankind the art of seeking truth ; and in his mathematical works he illustrated his precepts by his example. The glory he has acquired by his writings will never perish, be- cause the truths he discovered are truths for all ages ; though it must be avowed, that most of his philoso- phical systems, begotten by Imagination, and con- tradicted by Nature, have already disappeared, and produced no advantage, except that of having abo- lished the tyranny of the peripatetic philosophy. Algebra stands indebted to him for several important discoveries. He introduced, instead of the repeated multiplications of one and the same letter, the notation of the powers by exponents, which simplifies the cal- culation, and has been the germe of the method for developing the radical quantities in series. The ana- lysts who preceded him were unacquainted with the use of negative roots in equations, and rejected them as useless : he showed, that they are all as real, and as proper for resolving a quest ion ; as positive roots ; the 218 the distinction that ought to be made between them having no other foundation, than the different man- ner of considering the quantities, of which they are the symbols. Pie taught how to distinguish, in an equation containing real roots only, the number of positive roots, and that of negative roots, by the combination of the signs which precede the terms of the equation. The method of indeterminate*, of which Vieta had a glimpse, was developed by des Cartes, who made a clear and distinct application of it to equations of the fourth order : he assumes, that the general equation of this order is the product of two equations of the second, which he affects with indeterminate coefficients; and by the comparison of the terms of this product with- those of the equa- tion proposed, he arrives at an equation reducible to the third order, which gives the unknov/n coeffi- cients. This method is applicable to an infinite number of problems in all parts of the mathematics. I shall not here mention several learned algebraists, who, soon after the death of des Cartes, studied and even improved his methods. One, however, deserves particular notice, the celebrated Hudde, a burgo- master of 'Amsterdam, who died at a very advanced age in 1704. He published in 1658, in Schooten's Commentary on the Geometry of des Cartes, a very ingenious method of discovering whether an equation of any order contain several equal roots, and of de- termining these roots. Pascal, who died in 1662 at the age of thirty-nine, opened to himself a new path in analysis by his well- known Arithmetical Triangle. This is a kind of ge- nealogical nealogicaltree, in which, by means of an arbitrary num- ber written at the point of the triangle, he forms suc- cessively, and in the most general manner, all the figurate numbers ; and determines the ratios, which the numbers of any two cases have to each other ; with the different sums, which must result from the addi- tion of all the numbers in one rank, taken in what- ever direction you please. He afterward makes many interesting applications of these principles. That in which he determines the odds, to be laid between two persons playing at various games, de- serves particularly to be noticed, since it gave rise to the calculation of probabilities in the theory of games of chance. Some authors have ascribed the elements of this calculation to Huygens, who pub- lished in 1657 an excellent treatise, entitled, De Rat lo- ci-nils in Ludo Akce : but Huygens himself informs us, with a modesty worthy of so great a man, that this subject had already been handled by the greatest geometricians in France, and that he laid no claim to the honour of the invention. In fact we see by the letters betweeen Pascal and Fermat, printed in Fer- mat's Works, that the principles of the arithmetical triangle were spread through France as early as 1654, though the tracts in which Pascal explains them at large were not published till after his death. * While Pascal was thoroughly investigating the na- ture of figurate numbers at Paris, Fermat discovered several elegant properties of them at Toulouse, by pursuing a different method. These two great men frequently hit on the same things in the course of their researches :< and this was so far from altering 4he friendship, to which the similarity of their stu- dies dies had given birth without their having ever seen each other, that each rendered the other that disin- terested justice, to which mediocrity is a stranger. The predilection of Fermat for numerical researches led him particularly to the theory of prime numbers, which had not yet been examined, and in which he made profound discoveries. It is known, that every number is no more than a ratio of the unit : but it is often difficult to perceive, whether this ratio be sim- ple, or produced by the multiplication of several others. Fennat established general and distinguish- ing characters, calculated to discriminate, on a variety of occasions, those numbers which have divisors, from those which have none. The Analysis of Diophantus equally exercised his genius. Bachet de Mezhiac, editor and commentator of the greek algebraist, had already resolved several new problems dependent on the doctrine of his author ; but Fermat carried the same subject still farther* Since which time, all these researches have been extended and improved by other great geometricians. In 1&55, Wallis, an english mathematician, whom I have already quoted, published his Arithmetic of In- finites : a work full of genius, and the object of which, like that of the arithmetical triangle, was to determine the sum of different series of numbers. By this method curves are squared, when the ordinates are expressed by a single term : as may likewise curves with com- plex ordinates, by resolving these ordinates into series of which each term is a monomial. The author's dispute with Pascal on the subject of the cycloid shall be mentioned hereafter. Wallis was a profound analyst ; 221 analyst : to him we are indebted for the notation of radicals by fractional exponents ; as we are for that of negative exponents ; des Cartes having employed exponents for whole and positive powers only. As the path of truth is beset with difficulties, at which the feebleness of the human understanding is perpetually liable to stumble, the means of avoiding them, or of approaching our object when it is im- possible to reach it completely, cannot be rendered too numerous. Such is the advantage procured by the theory of continued fractions, when an irreducible fraction is expressed by numbers too great to be em- ployed in practice, in it's direct form. Instead of a complicated expression it substitutes one that is simple, and nearly equivalent. William lord Brounker, viscount of Castle Lyons in Ireland, who was born about 1620, and died in 1684, gave the elements of this theory ; which was afterward enlarged, improved, and applied to various important uses, by Huygens and other celebrated geometricians. All these particular branches of analysis did not induce mathematicians to lose sight of the problem of the general resolution of equations. Newton, who was then young, was a long while engaged in this re- search : he did not find it ; but in other respects he considerably enlarged the bounds of algebra. He gave a method for decomposing an equation into com- mensurable factors, whenever it is possible; a method extending to all the orders, and as simple as can be desired : he summed up the powers of the roots of an equation, be they what they might: he taught the 4 art 222 art of extracting, when there is occasion for it, the roots of quantities partly commensurable, and partly incommensurable: he instructed us how to form in- finite series, to find by approximation the roots of numerical and literal equations of all orders : &c. most of which researches have been illustrated and commented upon in modern works* CHAP. CHAP. II. Progress of Geometry. FROM the commencement of the sixteenth century the ancient geometry was cultivated in Europe with rapid success. The greek geometricians, most of whom were translated into latin or Italian, were taken as guides: and the study of the ancient languages, being then much in vogue, multiplied both the ob- jects and means of instruction. Werner, who died in 1528 at the age of sixty, is mentioned as a learned geometrician. In 1522 he published some tracts at Nuremberg, almost all of which related to the theory of the conic sections. Tartaglia and Maurolicus, who have already been mentioned, also rendered themselves useful to geo- metry, not only as translators of several ancient works; but as authors. The former composed a treatise in Italian on Numbers and Measures; which is the first modern work, where the determination of the area of a triangle by means of it's three sides, without letting fall a perpendicular on one of the sides from the angle opposite, is to be found. The latter wrote on various subjects ; and his treatise on conic sections as re- markable for perspicuity and elegance. La Hire af- terward did no more than amplify the method of the Sicilian geometer, and apply it to new purposes. We 224 We ought not to forget Nonius, who was born in Portugal in 1492, and the author of several very esti- mable works. To him we are particularly indebted for the subdivision of the small parts of a mathe- matical instrument by transverse lines, called Nonius's division. He died in 1577. Commandin, who was born in 1509, and died in 1575, was very learned in the dead languages, as well as in mathematics. He translated into latin Euclid's Elements, great part of the works of Ar- chimedes, Ptolemy's treatises on the Planisphere and the Analemma, the book of Aristarchus of Samos on the Magnitudes and Distances of the Sun and Moon, Hero's Pneumatics, the Geodesia of the arabian geo- meter Mohammed of Bagdad, the Mathematical Col- lections of Pappus, &c. Every where Commandin displays a thorough knowledge of the subject ; and he clears up the difficult passages of his authors by instructive notes, of great perspicuity and precision ; a rare merit, which places Commandin far above the generality of translators and commentators. The celebrated Ram us made no discovery in ma- thematics ; and his Elements' of Geometry and of Arithmetic are not above mediocrity: but he de- serves well of the sciences for the zeal with which he defended them, and for the sacrifice he made to them of his peace, his fortune, and even his life. He was a professor in the College of France, where he founded a chair, which still subsists ; but being of the protes- tant religion, he fell in the horrible massacre of St. Bartholomew's day by the hand of one of his fel- low 225 low professors, a zealous catholic, of the narfie of Charpentier. Fernel, physician to Henry II, king of France^ acquired a great name by different medical works, and some treatises and observations on the mathe* matics. It is said, that the great favour he enjoyed at court arose from his having taught the precious secret of removing the barrenness of Catharine de Medicis. We have a book on pure mathematics by him, entitled De Proportio?iibus; and two astrono- mical works, his Monalospherioii) a kind of analemma, and his Cosmothcoria. His greatest celebrity as a mathematician is founded on his being the first of the moderns, who gave the measure of the Earth. From the number of turns made by a coach wheel on the road from Amiens to Paris, till the altitude of the pole-star was increased one degree, he estimated the length of a degree of the meridian at 56746 french toises : a conclusion pretty near the truth ; but it is obvious, that it's exactness can be ascribed only to ' chance. It would be as useless as tiresome, to quote here a number of geometricians, who wrote, at this period, works that deserve much praise, though they display little depth of science, and are now nearly forgotten. I shall mention, however, two german mathema- ticians, Peter Metius, and Hadrian Romanus, and one of Holland, Ludolph van Ceulen ; all three authors of different methods for approaching much nearer to the ratio, which the circumference of the circle bears to the diameter, than had hitherto been done. Peter Metius made the remark, highly worthy our attention. & and 226* and gratitude, that, if the diameter be represented by 113, the circumference will be 355; a result singu- larly near the truth, considering the small number of figures by which it is expressed. Neither must I forget the celebrated Snell, another celebrated dutch mathematician, who began at the age of seventeen to writer works in geometry ; in which we find, among other curious things, a new determination of the ratio of the circumference of the circle to the diameter; and who afterward acquired great reputation by his inquiries concerning refraction. The works of Regiomontanus, Tartaglia, ami Bom- belli, contain some geometrical problems resolved by means of algebra. But these isolated solutions, where in every particular case simple numbers were em- ployed to express the known lines, were not founded on a regular and general method of applying algebra to geometry. Such a method was first given by Vieta. The mu- tual assistance, which these two sciences lend each other, proved to him a source of many important discoveries. For instance, he observed, that every equation of the third order, containing in general either one real root only and two imaginary ones, or three real roots, the real root in the first case was to be found by the duplication of the cube, and the three real roots in the second by the trisection of an angle, It must not be forgotten, however, that he had only a confused idea of negative roots, and that they began to be made known distinctly by des Cartes. The elements of the doctrine of angular sections are likewise the invention of Vieta. It is well known, that that the object of this theory is to find the general expressions of the chords or sines, for a series of arcs that are the multiples of each other ; and inversely the expressions of the arcs, when the chords or sines are known. It has since received great additions from the hands of Hermann, James Bernoulli, and Euler. Some authors have published, what others have re- peated after them; and we hear every day in conversa- tion, that des Cartes invented the application of algebra to geometry. But this is not accurate. It is ascribing more to des Cartes, than is justly his due ; and forget- ing the claims of his predecessors, particularly those of Vieta. The mistake is certainly pardonable, when we consider the happy, original, and extensive use, which des Cartes made of the discovery ; but strict justice ought to be done, and the truth established. By this des Cartes will lose very little : he will have the glory of being the first, who in this way com- pletely solved the following general problem, which the ancient geometricians, Euclid, Apollonius, and Pappus, proposed to themselves, and pf the solution of which they merely gave a sketch : ' the positions of any number of right lines, on a plane surface, being given, to find a point, from which as many other right lines may be drawn, one to each of the given lines, which shall make with them given angles; with this condition, that the product of two of the lines thus drawn shall be in a given ratio to the square of the third, if there be only three; or to the product of the other two, if there be four ; or if there be five, that the product of three shall be in Q 2 a 228 a given ratio to the product of the other two and a third given line ; or if there be six, &c.' Des Cartes began by observing, that the question thus proposed is indeterminate; and that there are an infinite number of points, from which the lines required may be drawn : he conceived, that all these points might be regarded as placed in the curve de- scribed by a style, made to move on a plane according to the conditions of the problem : and he expressed this condition by an equation between the given quantities and two variable lines ; so that by assuming at pleasure one of these lines, the other would be de- duced from the equation ; which made known at every instant the position of the describing point. Soon after, by a fresh effort of genius, the honour of which no one shares with him, he arrived at a general method of representing the nature of curve lines by equations, and of distributing them into dif- ferent classes according to the different orders of these equations : a vast and fertile field, which des Cartes laid open to the sagacity of every mathematician. By this, the law according to which a curve is to be described being given, it's course is readily traced ; and it's tangents, perpendiculars, finite or infinite branches, points of inflexion or contrary flexure, arid in general all the affections which characterize it, are O ' determined. This method combines simplicity and generality in one point of view. Thus, for instance, one and the same equation of the second order be- tween the absciss and ordinate, combined with con- stant quantities, may represent in general the nature of the three conic sections; then the values and ratios 229 ratios of the constant quantities restrict the equation, to express, in particular cases, a parabola, an ellipsis, or an hyperbola. We are likewise indebted to des Cartes for the manner of considering and constructing curves of double curvature by projecting them on two planes perpendicular to each other, on which they form or- dinary curves, which have a common absciss and ordinate. Of all the problems he solved in geometry no one, he said, gave him so much pleasure, as his method of drawing tangents to curve lines; by which, how- ever, geometrical curves only are to be understood- This method gives the tangents by means of per- pendiculars to the points of contact. The author assumes, that, from any point taken in the axis of a curve, a circle may be described, cutting the curve at least in two points : after which he seeks the equation that expresses the places of intersection : he then sup- poses, that the radius of the circle diminishes, till two adjacent intersections coincide: then the two corre- sponding radii form but one, which is perpendicular to the curve : and the, question reduces itself to form- ing, from these elements, an equation, which contains two equal roots. Des Cartes afterward proposed another method for tangents. In this he takes a point out of the curve, and on it's axis produced, on which he makes a curved line revolve, cutting the curve at least in two points : and the two points of intersection he makes to coincide, by subjecting, as in the former instance, the equation of the intersections to contain two equal Q, 3 roots, 230 roots. Both these methods are evidently founded on the same principle, and are very ingenious, though they are much less simple, and less direct, than that of fluxions. The geometry of des Cartes appeared in 1637. Before this period Fermat had invented his me* thod of determining the maxima and minima, in quan- tities which increase at first, and then diminish ; or which begin with diminishing, and afterward in- crease. It turns on this remark, tnat on each side of the point of the maximum^ and also of the mi- nimum, there are two equal magnitudes. Fermat seeks the expressions of two magnitudes at an arbi- trary distance, and makes them equal to each other; then supposing the distance proposed to become infinitely small, or less than any assignable finite quantity, he obtains an equation, which gives the maximum, or the minimum. The same method serves to determine the tangents of geometrical curves, by first considering the tangent as a secant, and then making the portion of the absciss, comprised between the two ord mates answering to the two points of intersection, to vanish. The method of fluxions rests on the same basis ; yet Fermat cannot be called the inventor of fluxions. His method is not reduced to an algorithm : it is simply a general indication of the calculations to be made in each particular case ; it applies only to geo- metrical curv.es ; and even in this case it requires, that the radical quantities, which the equations may contain, be made to disappear; which frequently leads to calculations that are intractable, either on 4- account account of their length, or from the difficulty of discovering the root that satisfies the conditions of the problem. We must refer to the head of mixed geometry se- veral works, that appeared in the seventeenth century previous to the rise of the direct and inverse methods of fluxions : not -that the methods employed in them are founded on algebraical calculation, but because they are all more or less guided by it's spirit. One of the most original is Cavaleri's Geometry of Indivisibles, which appeared in 1635. The method of the ancients for determining the superficies and solidities of bodies was very strict, but it had the. in- convenience of requiring many collateral processes: it was necessary, that polygons should be inscribed in a figure, and circumscribed about it; or solids inscribed in a solid, and circumscribed about it f and then the limits of the ratio between the last inscribed and the last circumscribed polygon, or that of the last inscribed and the last circumscribed solid, were to be sought. Cavaleri proceeds more directly to the object : he considers plane superficies as com- posed of an infinite number of lines, and solids of an infinite number of planes; and he assumes as a principle, that the ratios of these infinite sums of lines, or planes, compared with the unit in each case, are the same as those of the superficies or solids, that were to be measured. The work of Cavaleri is divided into seven books. In six of these the author applies his new theory to the quadrature of the conic sections, the cubature of the solids formed by their revolution, and other questions of a like nature on spirals : the Q 4 seventh 232 seventh is employed in demonstrating the same things by principles independent of indivisibles, and to establish the perfect accuracy of the new method by the coincidence of their results. The french geometricians, on their part, resolved similar but more difficult problems. For instance, Fermat, Roberval, and des Cartes squared parabolas of the higher orders, and determined the solids which all these curves form by revolving round the absciss or ordinate, as likewise the centres of gravity of these solids ; thus completing the theory, which Archimedes had given for the common pa- rabola. The method of Roberval, like that of Cavaleri, was founded on the principle of indivisibles ; but it was exhibited in a point of view more conformable to geometrical strictness, because Roberval considered the planes, or solids, as having for their elements rectangles of infinitely small altitudes, or sections of infinitely little thickness, and not simple lines or planes. There are proofs, that he employed this method as early as 1634, and consequently he bor- rowed nothing from Cavaleri, About the same time Roberval applied his me^ thods to the cycloid, a curve that had become cele- brated by it's numerous and singular properties. He determined the area of this curve, and the solids it generated by turning round it's base or it's axis : he likewise found the centre of gravity of the area of the same curve, and those of it's parts on each side of the axis. These 233 These new problems having been proposed to Fer- mat and des Cartes, they both resolved them. They likewise taught how to draw tangents to the cycloid, which, being a mechanical curve, required methods different from those already employed for drawing tangents to geometrical curves. Roberval had framed a general method for tan- gents, which was applicable to geometrical and me- chanical curves indifferently, and by this he determined the tangents of the cycloid. This method deserves notice for it's analogy, in respect to the metaphysical principle, with that dfjferions, which Newton pro- duced long after. A curve being supposed to be described by the motion of a point, Roberval con- siders this point as acted upon at every instant by two velocities given by the nature of the curve : he constructs a parallelogram, the sides of which are proportional to these velocities : and he assumes as a principle, that the direction of the element, or of the tangent, must fall on the diagonal ; so that the po- sition of this diagonal being known, we have that of the tangent. Thus, for example, in the ellipsis, where the sum of the two lines drawn from the foci to one and the same point in the curve is always the same, if one of the lines be diminished in any degree, the other will be increased by an equal quan- tity : then the parallelogram becomes a rhomboid; and consequently the tangent must divide into two equal parts the angle formed by the prolongations of the two lines proposed. But the method does not apply to all cases with the same facility; and fre- quently it even becomes impracticable by the diffi- culty 234 culty of determining the two velocities of the de- scribing point : while in the method of fluxions, the metaphysical principle being reduced to an algorithm of calculation, disencumbered of all superfluous ope- rations, one and the same general formula gives, without the least difficulty, the tangents for all curves, of which we have the equation. Unhappily, Roberval united with great geometrical talents a vain and peevish disposition. He was con- tinually at war with des Cartes and other french geo- metricians, and very often he was in the wrong. On the subject of the problems of the cycloid he mortally offended Torricelli. This illustrious Italian geometer having published solutions of these pro- blems, as of his own invention, in 1644, Roberval claimed them ; maintaining that they were funda- mentally the same with his own, which one Beau- grand had communicated to Galileo, on whose death they fell into the hands of Torricelli, his pupil, and the inheritor of his papers. Torricelli was so much affected by this charge of plagiarism, that it brought him to his grave in the flower of his age. If any one follow Torricelli attentively in his demonstrations, he will be fully convinced, that they are his own; and that probably he had never read either the pre- tended copies of RobervaFs solutions, sent to Galileo, or the Universal Harmony of father Mersenne, pub- lished in 1637, where these same solutions are given. The Jesuit Gregory St. Vincent, who was born in the Netherlands in 1584, and died in 1667, acquired considerable reputation in the mathematics by a work, in which he attempted to square the circle, and failed, but 235 but which notwithstanding abounded in accurate and profound theories on the measure of the ungulss of different bodies formed by the revolution of the conic sections. Herigon, also, deserves to be mentioned here, not as a mathematician of the first order, but for having col- lected into one course all the branches of the mathe- matical sciences, in the state in which they were at that period. His book, published in 1644 in french and latin, was widely circulated, and was of great utility. Beside the general knowledge of arithmetic, algebra, geometry, mechanics, astronomy, geography, &c., Herigon has introduced into his collection several works of the ancient geometricians ; such as Euclid's Elements, his Data, his Optics and Catoptrics, the Geometry of Tactions of Apollonius, the Spherics of Theodosius, c. In all of which his neat, dear, and strict method of demonstration deserves praise. The celebrated inverse method of Tangents origi- nated from a problem, which Beaune proposed to his friend des Cartes in 1647. This was, 4 to find a curve, such that the ordinate shall be to the subtan- gent, as a given line is to the part of the ordinate comprised between the curve and a line inclined at a given angle.' Des Cartes pointed out the construction of the curve, and several properties of it ; but he could not completely accomplish the solution, which was reserved for the method of fluxions. While Roberval, and some others of the french geometricians, endeavoured to depreciate the geometry of des Cartes, it found a crowd of admirers of the greatest merit in foreign countries. Of these the chief 236 chief was Schooten, professor of mathematics at Leyden, who displayed and extended it in an ex- cellent commentary, published for the first time in 1649, and afterward reprinted with considerable ad* ditions. He had already distinguished himself by his Geometrical Exercitations, published in 1646. In England geometry acquired new treasures of a different kind. By his Arithmetic of Infinites Wallis resolved a great number of elegant problems relating to the quadrature of curves, the cubature of solids, the determination of centres of gravity, &c. When parabolas of every order had been squared, it was natural to think of determining their curva- tures, or generally to find a right line, which should be equal in length to the perimeter of a given curve. This new problem was then attended with the greatest difficulty. As early as the year 1657 Huygens gave, by letters, some openings for solving it. His country- man Van Heuraet reduced the question to geome- trical constructions, which were a little embarrassing, but which at length led him to a very beautiful dis- covery. He found, that the second cubic parabola, where the squares of the ordinates are as the cubes of the abscisses, is equal to a right line, which he as- sig^s. This discovery was published in 1659, at the end of the second edition of Schooten's Commentary on the Geometry of des Cartes. The other parabolas are not susceptible of recti- fication algebraically : but they may be measured at least by methods of approximation, by employing series, or quadratures, of certain curvilinear spaces easy to be calculated : for example, the rectification of 237 of the common parabola depends on the quadrature of the hyperbola, or on logarithms. Huygens, in the geometrical demonstrations of his Horologium os- cillatorium, which appeared for the first time in 1673, rectifies curves, squares superficies, or reduces their expressions to others more simple, with an elegance and address, which the lovers of the true or linear geometry will never cease to admire* It is commonly supposed, from the assertion of Wallis in his treatise on the Cissoid, that William Neil, his pupil, was the first who rectified the second cubic parabola. Huygens on the contrary maintains, that the theorem of Van Heuraet was circulated among geometricians, before the english had turned their thoughts to the question. As the methods are different, it may very easily have happened, that both Neil and Van Heuraet arrived at the same re- sult, without either of them having borrowed any thing from the other. However, all these problems are but trifles, since the invention of the method of fluxions. The cycloid began to be a little forgotten by ge- ometricians, when Pascal brought it forward again in 165 8, by proposing new problems relating to this curve, and offering prizes to those who should solve them. The total area of the cycloid, the centre of gravity of this area, the solids which the curve de- scribes by revolving round it's basis, or the diameter of the generating circle, and the centres of gravity of these solids, had been determined: Pascal de- manded what was then much more difficult, inde- finite measures ; that is to say, the area of any seg- ment 238 ment of the cycloid whatever, the 'centre of gravity of this segment, the solids, and centres of gravity of the solids, which this segment describes in revolving round the ordinate, or round the absciss, whether it make a complete revolution, half a revolution, or a quarter of a revolution. Huygens squared the segment included between the vertex and as far as a fourth of the diameter of the generating circle. Sluze measured the area of the curve by a very elegant method. The celebrated english architect sir Christopher Wren determined the length and centre of gravity of the cycloidal arc comprised between the vertex and the ordinate, and the superficies of the solids which the revolution of this arc produces. Fermat and Roberval, also from the bare enunciation of the theorems of the english geometrician, found the demonstrations of them. But all % these investigations, though very profound and beautiful, did not answer the questions of the challenge, at least completely ; neither were they sent in, to compete for the prize. Wallis, and father Lallouere, a Jesuit, were the only persons, who, having treated all the problems proposed, thought themselves qualified to claim the prizes. Yet Pascal demonstrated to them both, that they were mistaken in several points; and that they had advanced false conclusions, founded on errours, not in their calculations, but in their methods. He alone gave, in 1659, a true and complete solution of the problems, as well as of several others still more difficult 239 In each of these investigations the common cycloid alone was the subject in question. Pascal determined in addition the dimensions of all the cycloids, curtate or prolate. He showed, that the length of these curves depends on the rectification of the ellipsis, and assigned the axes of the ellipsis in each case: when one of these axes becomes nought, the ellipsis is changed into a right line, the curve becomes the common cycloid, and Pascal concludes from his method, that the cycloidal arc is then double the corresponding choYd of the generating circle; which comprises the theorem of sir C. Wren as a particular case. He likewise deduced, from his method, another very remarkable theorem, which is, that, if two cy- cloids, one prolate the other curtate, be such, that the base of one is equal to the circumference of the generating circle of the other, the length of these two cycloids will be equal. In all Pascal's inventions in mathematics we discern a genius, of the most powerful kind, for the advance- ment of science, that the earth ever bore. Geome- tricians regret, that he did not dedicate to them the whole course of his short life : but then we should have lost those celebrated Provincial Letters, and those profound Thoughts, which are perhaps the masterpieces of french eloquence. Barrow had a happy idea, which may be considered as a fresh step toward fluxions, in forming his diffe- rential triangle, for drawing tangents to curves. This triangle, as is well kiiown, has for it's sides the ele- ment of the curve, and those of the absciss and or- dinate. Barrow's method is in fact nothing more than 240 than Fermat's abridged and simplified, as he treats the three sides of the triangle immediately as in- finitely small quantities, and thus saves some length of calculation : but it still wants the essential cha- racters of the method of fluxions, that is to say, a uniform algorithm for all cases, and the advantage of giving by one general formula the tangents of all sorts of curves, whether geometrical or mechanical Accordingly Barrow stopped at the problem of tan- gents, and this even limited to the single case, where the equations are algebraical and rational, while the method of fluxions applies to an infinite number of uses besides. The ancients set great value on simplicity and ele- gance in the construction of geometrical problems. And Sluze, their imitator in this respect, carried the use of geometric loci for the resolution of equations to the highest degree of perfection. One of the greatest discoveries made in modern geometry was the theory of evolutes, invented by Huygens, which may be found in his Horologium oscil- latorium already mentioned. A curve being given, Huygens constructs another curve, by drawing a se- ries of right lines perpendicular to the former, which touch the latter : or inversely, the second curve being given, he constructs the first. From this general idea he deduces a number of remarkable proposi- tions ; such as divers theorems on the rectifications of curves ; the singular property, which the cycloid bas, of producing by evolution an equal and similar cycloid, placed in a reverse position ; &c. The uses of this, theory, in all parts of the mathematics, are are without number. Apollonius had given a ge- neral idea of it ; but it remained barren , and Huy- gcns, who, not content with clearing the ground, produced from it himself an abundant harvest, will always retain the glory of having transmitted it as a possession to future geometricians. The english continued to enrich geometry with no- velties, at that time very striking. Brouncker gave an infinite series to represent the area of the hyper- bola; and Nicholas Mercator separately made the same discovery. Wallis had long taught how to square curves with monomial ordinates ; and his me- thod equally applied to curves, the ordinates of which were complex quantities raised to entire and positive powers, by developing these powers by the common principles of multiplication. He sought to extend this theory likewise to curves with complex and radical ordinates, by endeavouring in this case to interpolate the series of the former kind with new series; but he could not succeed. Newton surmounted this difficulty : he did more ; he solved the problem in a direct and much more simple manner, by means of the formula he discovered for expanding into an infinite series any power of a bino- nomial, whatever the exponent of the power might be, integral or fractional, positive or negative. The infinite series thence resulting for the quadra- ture of the circle was found in another manner by James Gregory, who formed several other very curi- ous series. In a work never published, but of which a summary has been preserved, he gave the tangent and secant by the arc, and inversely the arc by the R tangent 242 tangent or secant : he also constructed series to find directly the logarithm of the tangent or of the secant, when the arc is given ; and reciprocally the logarithm of the arc by that of the tangent or secant : lastly, he applied this theory of series to the rectification of the ellipsis and hyperbola. The use of series in geometry made some progress in Germany likewise. Leibnitz gave a method far transforming one curvilinear surface into another, the parts of which, supposed equal to those of the for- mer, should have such a figure and position, that the methods of Mercator and Wallis would be applicable to the quadrature of this latter curve. CHAF. 243 CHAP. III. Progress of Mechanics* IN this period, as in the two preceding, a great number of very ingenious machines were invented ; but the theory of mechanics remained in a state of stagnation till the sixteenth century. Stevinus, a flemish mathematician, who died in 1635, appears to be the first person, that made known directly, and without the assistance of the lever, the laws of equi- librium of a body placed on an inclined plane. He also examined with the same success many other ques- tions in statics. The manner, in which he determined the conditions of equilibrium between several forces concurring to one point, comes in fact ,to the fa* mous principle of the parallelogram of forces ; but he was not aware of all it's consequences and advan- tages. In 1592 Galileo composed a little treatise on statics, which he reduced to this single principle : it requires an equal power, to raise two different bodies to heights in the inverse ratio of their weights ; that is to say, the same power will raise two pounds to the height of one foot, as will raise one pound to the height of two feet. Hence it was easy to infer, that, in all machines in equilibrium, the powers, which coun- teract each other, are inversely proportional to the spaces, which they would pass through in a given a 2 time. time. The only question then is accurately to de- termine the spaces, from the arrangement and action of the parts of the machine. Thus, for instance, in the common screw, where the weight rises the height of one thread of the screw, while the power describes the periphery of a circle in a horizontal di- rection, the weight is to the power as this periphery is to the height of the thread of the screw. A long while after this des Cartes employed the same princi- ple, to determine the equilibrium of all machines, in a small work entitled, an Explanation of Machines and Engines. He ought however to have quoted Galileo. It does not enter into my plan, to relate the prac- tical applications made of the principles of me- chanics : yet I cannot avoid remarking here inciden- tally, that Claude Perrault, so much decried by Boileau, who was incapable of appreciating his talents, displayed no less mathematical and physical knowledge, than he did genius, in the machines he invented to raise the enormous stones, that compose the pediment of the colonnade of the Louvre. A de- scription of them may be seen in his commentary on the eighteenth chapter of the tenth book of Vitru- vius. The general theory of motion, of which the an- cients were acquainted only with the paiticular case where it is uniform, originated with Galileo. He discovered the law of the acceleration of bodies fall- ing freely by the power of gravity, or sliding down inclined planes ; and on this subject he established the general properties of motion uniformly accele- rated. The conformity of his theory with the phe- nomena 245 nomena of nature is one of the largest strides, that the modern science of physics ever took; -it formed the first step in the system of universal gravitation. Every person, who sees a stone fall, may suppose it's motion to be accelerated, and become so much the more rapid in proportion to the height from which it falls ; since the stone, the weight of which remains the same, strikes a blow so much the harder, as the height of it's fall is greater. But in what proportion does this acceleration take place ? This is the new problem, which Galileo solved ; and he was led to it by one of those simple reflections, which may enter into any mind, but turns to account only in the mind of genius. Since ajl bodies are heavy, said Galileo, and into whatever number of parts we divide any mass, whe- ther an ingot of gold, or a block of marble, all these parts are themselves heavy bodies, it follows, that the total weight of the mass is proportional to the number of material atoms, of which it is composed. Now the weight being thus a power always uniform in quantity, and it's action never undergoing any in- terruption, it must in consequence be continually giving equal impulses to a body, in every equal and successive instant of time. , If the body be stopped by any obstacle ; as, for example, if it be placed on a horizontal table: the impulses' of the weight, as they are incessantly renewed, are incessantly destroyed by the resistance of the table. But if the body fall freely, these impulses are incessantly accumulating, and remain in the body without alteration, the re- sistance of the air alone being deducted: whence it R 3 follows, 245 follows, that the motion must be accelerated by equal degrees. Experience has fully confirmed this sound reasoning. Fortunately Galileo brought to this question a mind free from all prejudice, and from any systematic opinion concerning the cause of gravity : for if he had supposed, for example, as some philosophers did after him, that the impulses of gravity are produced by the impulse of some subtile ambient fluid, he would have missed the truth; the impulses in ques- tion not being proportionate to the masses of the falling bodies, and decreasing continually as the ve- locity increases. Among the philosophers who were the first to em- brace and comment on Galileo's theory of the fall of heavy bodies, we must distinguish his pupil Torricelli, who published a very elegant work on this subject in 1644, entitled: De Motu Gravium naturallter acce* lerato. He added several very curious propositions to those, which Galileo had given on the motion of projectiles. Huygens considered the motion of heavy bodies on given curves. He demonstrated generally, that the velocity of a heavy body, which descends along any curve, is the same at every instant in the direc- tion of the tangent, as it would have acquired by falling freely from a height equal to the correspondent vertical absciss. Then applying this principle to a reversed cycloid, the axis of which is vertical, he found, that a heavy body, from whatever part of the cycloidal arc it falls, always arrives at the lowest point, or the inferiour extremity of the arc, in the same 247 same space of time. This very remarkable proposi- tion includes what is commonly called the isochronism of the cycloid, and would alone have been sufficient, to establish the fame of a geometrician. From the motion of an isolated body a transition was made to the motions, which several bodies com- municate to each other, whether they act by impact, or by the interposition of levers, cords, &c. The most simple of these problems was that of a body proceeding to strike against another at rest, or moving before it with less velocity, or approaching towards It. Des Cartes, misled by his metaphysical principles, which had induced him to suppose, that the same quantity of absolute motion always exists in the World, concluded, that the sum of the motions after the impact was equal to the sum of the motions before it. But the proposition is true only in the first and second of these cases : it is false when the two bodies meet each other, for in this case the sum of the motions after the impact is equal to the difference of the motions before it, not to their sum. Thus des Cartes discovered only part of the truth. In 1661, Huygens, Wallis, and sir Christopher Wren, all discovered the true laws of percussion se- parately, and without any communication with each other, as has been completely proved. The base of their solutions is, that, in the mutual percussion of several bodies, the absolute quantity of motion of the centre of gravity is the same after as before the shock. Farther, when the bodies are elastic, the respective velocity is the same after as before per- cussion. B 4 Two 248 Two other celebrated and more difficult problems concerning the communication of motion, proposed by father Mersenne in 1635, exercised the skill of geometricians for a long time. One was, to deter- mine the centre of oscillation of a compound pen- dulum : the other, to find the centre of percussion of a body, or a system of bodies, turning round a fixed axis. In the first, several heavy bodies connected to- gether, at invariable distances, by rods considered as destitute of weight, are supposed to oscillate round a fixed horizontal axis. Then, as all these bodies restrict the motion of each other, and do not acquire the same velocities as if each oscillated separately, the bodies nearest the axis lose a part of their natural motion, and transmit it to the most remote. Thus there is an equilibrium between the motions lost and the motions gained. In whatever way this equili- brium is established, there exists in the system some point, where if a small isolated body were placed, it would oscillate in the same time as the compound pendulum; whence tjiis point has been called the centre of oscillation. The property of the centre of percussion is of another nature. What characterises this point is, that it must be found in the direction of the resul- tante of all the motions of the bodies of a system turning round a fixed axis, and occupy in this system a place analogous to that, which the centre of gra- vity occupies in a heavy body. I have said of another nature : for, though it is demonstrated, that the cen- tre of oscillation and the centre of percussion are situated 349 | situated in one and the same point of the system, and that the two problems are resolvable by the same principles of mechanics, the application of these prin- ciples is more simple and easy in the second case, than in the first, and the two questions are different. Des Cartes and Roberval, persuaded that they were the same, and finding it more easy, to consider them under the second point of view, than under the first, determined the point sought with accuracy in some particular cases : but in several others they deceived themselves. Their methods, being likewise founded on vague and uncertain suppositions, were very pre- carious and insufficient. Huygens was the first, who resolved in a complete and general manner the most important of these problems, that of the centres of oscillation. He as- sumed as a principle, that, after the centre of gravity of a compound pendulum has descended to the lowest point, if all the bodies should separate from one another, and each ascend singly with the velocity it had acquired, the centre of gravity of the system in this state would ascend to the same height, as that from which the centre of gravity of the pendulum had descended. At first this solution was not very well understood: some philosophers attacked it's principle, perfectly incontestable in itself, yet it must be confessed a little strained, and in consequence not presenting, at least to every mind, a very evident connexion with the elementary laws of mechanics. It was afterward demonstrated in the most luminous and indisputable manner ; and is now every where known by the name of &50 of the principle of the conservation of active forces. The problem of centres of oscillation is the firstborn of that numerous family of problems of dynamics, so long agitated among geometricians. Though the investigation of the centre of per- cussion offered but few difficulties to geometricians versed in mechanics, many of them resolved the problem badly, or gave incomplete solutions of it. Wallis himself was mistaken in it, in his treatise De Motu. Long after him James Bernoulli, of whom I shall have occasion to take much notice in the sub- sequent pages, gave an accurate and general solution of it by means of the principle of the lever. CHAP, CHAP. IVc Progress of Hydrodynamics, IT has been seen, that Stevinus contributed a little to the advancement of statics ; and the same must be said of him with regard to hydrostatics. He showed, that the pressure of a fluid on the bottom of a vessel is always as the product of the area of that bottom multiplied by the height of the fluid, whatever the shape of the vessel may be : but he does not ap- pear to have had a thorough perception of the reci- procal connexion of all parts of hydrostatics. The first methodical and truly original treatise on hydrostatics, published by a modern writer, is that of Pascal on the Equilibrium of Fluids. The author demonstrates the properties of the equilibrium of fluids by this simple and extensive principle : when two pistons, applied to two apertures made in a vessel full of any fluid whatever, and close in every other part, are acted upon by forces inversely proportional to the apertures, they are in equilibrio. He solves all the difficulties, which certain propositions might still offer ; such as, for example, the once celebrated paradox, which is no longer so, that a slender tube of water, and a column of the same fluid of any dia- meter, being of the same height, and pressing on the same bottom, exert equal pressures. The 252 The weight of the air, of which the ancients were ignorant, was likewise unknown to Galileo, even long after he had discovered the theory of the acceleration of falling bodies. Apparently from the invention of pumps to the time of this philosopher, no one had found occasion, or had thought of placing the piston in a sucking pump at a height exceeding thirty-two feet from the reservoir; otherwise the difficulty would have occurred, which was proposed to Galileo by the engineers of Cosmo de Medicis, great duke of Flo- rence. Be this as it may, we are indebted to an ex- periment attempted by these men for the discovery, or more properly the irrefragable proof, of the weight of the air. They had constructed a sucking pump, in which it. was necessary, that the water should be raised under the piston to a greater height than thirty- two feet ; and finding they could not draw it above this height, they applied to Galileo for the reason. The honour of philosophy would not allow him to confess himself ignorant, or to defer his answer. The ancients ascribed the ascent of water in pumps to nature's horrour of a vacuum : this cause Galileo assigned to the engineers, adding with regard to the present case, that nature's horrour of a vacuum ceased, when the water had reached the height of thirty-two feet. This explanation was considered as an oracle, and no one thought of contradicting it. But on reflecting a little more closely, Galileo sus- pected, that this horrour of nature for a vacuum, and the limit he had assigned it, were equally chimerical He went no farther however ; anc [ though he begau to 253 to understand the weight of the air by experiments of another kind, he did not think of employing this agent here. His pupil Torricelli imagined, that the weight of the water might be an obstacle to it's rising in pumps : a simple and happy idea, incompatible with the system of horrour for a vacuum ; for why should the weight of the water limit the force of this horrour? Guided by this gleam of light, he made an experi- ment, analogous to that of the pump, with an in- strument; whence the common barometer derived it's figure and origin : and he found that quicksilver, the weight of which is fourteen times as great as that of water, kept itself at a height fourteen times less. Hence Torricelli concluded, that the two phenomena were produced by the same cause ; and proceeding a step farther he affirmed, that this cause was the weight of the air. The inveterate partizans of the system of the hor- rour of a vacuum started some doubts against Tor- ricelli's explanation ; but these doubts were com- pletely removed by the celebrated experiment of Puy- de-D6me, near Clermont in Auvergne. In this ex- periment, projected by Pascal, and executed by his brother-in-law Perrier, the quicksilver was seen for the first time to descend in the barometer, in pro- portion as it was carried higher up the mountain, or as the column of air diminished in altitude and weight. The course of water flowing on the surface of the Earth drew the attention of Castelli, another disciple of Galileo. In a little treatise which he published on 54 on this subject in 1628, Castelli explains some phe- nomena of the motion of water in a natural or ar- tificial channel of whatever figure. He showed, that, when the water has once taken a regular and per- manent course, the velocities at different sections, made perpendicular to the line of motion, are in the inverse ratio of the surfaces of these sections : a prin- ciple true in itself, and from which Castelli deduced many just conclusions ; but he deceived himself af- terward in the absolute measure of the velocity, which he makes proportional to the fall of the channel, or the height of the water. Torricelli was the first, who proposed an accurate theory in a particular case of the motion of water. On reflecting that water, at it's issuing from a small perpendicular ajutage, rises, at least nearly, to the height of the reservoir, he thought, that it's initial ascen- sional velocity must be the same as that which a heavy body would have acquired in falling from the height of the reservoir : whence he inferred, conformably to the theory of his master, that, the loss by friction and the resistance of the air being deducted, the ve- locities of the jets will follow the subduplicate ratio of the pressures. This idea \vas confirmed by experi- ments, which Raphael Magiotti made at this time on the products of different ajutages under different charges of water. Torricelli published his discovery in 1 6*44, in his book De Motu Gravium naturaliter accelerate, already mentioned. Hydraulics then became truly a science, in .that branch which relates to the flow of water through small orifices, from which the greatest ad- vantages 255 vantages were derived in practice. But where the water issues through orifices of some size in pro- portion to the horizontal sections of the reservoir, the velocity follows a much more complicated law, to the discovery of which geometry in Torricelli's time was unequal. Among those, who were the first to introduce Tor- ricelli's theorem into practice, Mariotte deserves to be mentioned with distinction. Born with a singular talent for inventing and executing experiments, and having opportunities of making a great number on the motion of water at Versailles, Chantilly, and se- veral other places, he composed a treatise on this subject, which was not printed till after his death, In this he has committed some mistakes, as well as but slightly touched on several questions, and was unacquainted with the effect of the contraction of the jet at issuing from the ajutage : yet, notwithstanding it's imperfections, the work has been of great utility, and contributed much to the progress of practical hydraulics. At the time when the discoveries of Galileo on motion began to turn the studies of the learned that way, des Cartes conceived the idea pf explaining the general motion, which carries the planets from west to east, by the laws of hydrodynamics. The ancients considered the planetary heavens as composed of solid movable orbs^ each of which carried along the planet attached to it The terrible confusion, or rather the absolute incompatibility of all these mo- tions, particularly in the ptolemaic system, was felt. Des Cartes transported to the firmament the infinitely 4 more 256 more simple mechanism of a boat floating on a river, and carried along by the stream. He imagined, that the planets swam in a similar mode in a vast whirl- pool, which turned from west to east; yet in such a manner, that, amid the general vortex, there were particular currents, or vortices, for each planet, which cut the ecliptic with different degrees of obliquity. This idea, which is very imposing at first view, seduced many illustrious philosophers, who publicly avowed themselves it's partisans. At that time the theory of the motion of solid and fluid bodies was too much in it's infancy, to be capable of undertaking a critical examination of this system, which even main- tained itself a long time against the most powerful objections : at length however it was obliged to be given up, as equally contrary to the laws of astronomy and mechanics. CHAP. 257 CHAf. V. Progress of Astronomy. ASTRONOMY made great progress at this period, ia which we find many astronomers of the first order. At their head is the celebrated Copernicus, whose labours began with the sixteenth century; for though he was born in the year 147^, he did not give him- self up entirely to his inclination for astronomy till about the year 1507. His understanding at once revolted against the explanations, which Ptolemy had given of the mo- tions of our planetary system. He found in thems a perplexity and obscurity, which he could not reconcile with the general simplicity of the laws of nature. Knowing that the pythagoreans had transferred the revolving motion of the ecliptic from the Sun to the Earth, and that other ancient philosophers had at- tributed to the Earth a rotatory motion round it's axis every twenty-four hours, to explain the suc- cession of night and day, he adopted both these ideas. He made Mercury, Venus, the Earth, Mars, Jupiter, and Saturn, revolve round the Sun in the order here given ; while he let the Moon still continue to turn round the Earth. Thus the celestial phenomena, the direct, stationary, and retrograde appearances of the planets, were explained with a degree o? facility, at which he himself was astonished. The principal s objec- 253 objections made to him he victoriously refuted : and those which still left some doubts were afterward re- moved by observations, as he had predicted. The whole of his doctrine is explained in his celebrated book De Rccolutionikus codestibus, which was written about the year 1530, but did not appear till 1543; the au- thor dying on the very day, on which he received the first complete printed copy of it. The system of Copernicus was so simple, satis- factory, and conformable to all the laws of mechanics and natural philosophy, that it would have been adopted immediately by all astronomers, if a mistaken religious zeal had not imagined, that it was confuted by certain passages in the Bible : as if, in a book in- tended to teach religion, and not astronomy, the lan- guage should be conformable to astronomical truth, which the learned alone could understand, instead of that in common use, which is intelligible to all men. We have to regret that Tycho Brahe sacrificed his ta- lents, and perhaps his inward conviction, to supersti- tious considerations : but let us forgive this errour, or this weakness, in return for the numerous observations -and discoveries, with which he enriched astronomy. As Tycho could not entirely adopt the ptolemaic system, which every thing united to condemn, he at least restored to the Earth it's fancied immobility, and then made first the Moon revolve round it, and afterward the Sun, carrying with it in the sphere of it's revolution the planets Mercury, Venus, Mars, Jupiter, and Saturn. Thus he explained, in what he conceived to be a satisfactory manner, the appear- of the celestial motions then known ; but he was 259 was too enlightened not to perceive, that his system was in fact almost as repugnant to the laws of me- chanics as that of Ptolemy. His true glory is the hav- ing been an excellent observer, and having laid or strengthened the foundations of the new astronomical theories, either by his own labours, or by those of his disciples and coadjutors, whom he had assembled in his little town of Uraniburg. We know, that the motion of the Moon is subject to a great number of inequalities. The four principal of these are the equation of the centre, the evection, the variation, and the annual equation. The first, we have seen, was discovered by Hipparchus, and the second by Ptolemy ; and what they consist in has been explaineck Tycho discovered both the others. The variation is an alternate increase and dimi- nution of the motions of the Moon, which depend on it's position with regard to the syzygies, or the line which joins the centres of the Sun, Moon, and Earth, when these three bodies are in conjunction or oppo- sition. Tycho observed, that in setting off from the point of conjunction, for instance, the velocity of the Moon diminished till the first quarter; that it increased from the first quarter to the point of op- position ; that it diminished in the third quarter pf it's revolution, and again increased in the fourth : and so on alternately in it's succeeding revolutions. The annual equation arises from an inequality, which is found in the duration of the lunar months, according to the different seasons of the year. It is to be observed, that the periodical revolutions of the Moon arc of the same duration only at the same s 2 seasQns; 260 seasons ; while from erne season to another they in- crease or diminish. The longest take place in the months of december and January; the shortest hi those of June and July. Hence result in the theory of the Moon three small equations, proportional to the equation of the Sun's Centre : one for the motions of the Moon in it's orbit, the other for the motion of it's apogee, and the third for the motion of the node* of the lunar orbit. Beside these four principal inequalities, which have deen detected by the immediate assistance of obser- vation, the motion of the Moon is subject to several other little inequalities, which the theory of universal gravitation has occasioned to be remarked; and which we are at present obliged to introduce into astro-, nomical calculation, when we would have it represent the state of the heavens with all the accuracy, which it is possible to attain. Tycho likewise improved the theory of the Moon in another essential part : he determined the greatest and least inclination of the lunar orbit to the plane of the ecliptic with more care, and with greater pre- cision, than had ever before been done. The same research he also extended to the different planets. The ancients had a general knowledge of the ef- fects of refraction. Every person might perceive, that the brightness of the Sun, when seen at the horizon, was much less, than when it has reached the meridian. The reason of this is, the Earth being surrounded with a dense atmosphere, which extends fifty miles from it's surface according to the common supposition, the, .rays of the Sun coming from the 2 horizon 2(5 1 horizon traverse a greater space in the atmosphere* and consequently experience more resistance, and are more enfeebled, than the rays that come from the meridian. This difference should have led the ancients to suspect, that refraction might produce some, change in the apparent position of the heavenly bodies above the horizon ; a change which in fact is known to take place. But we do not find, that the ancients paid any attention to it. Tycho was the first who felt the necessity of introducing this impor- tant element into astronomical calculation, and who began to employ it. But as the laws of refraction were not yet known in his time, he could only give general and somewhat vague results. To the same astronomer we are indebted for the elements of the theory of comets. The opinion, that comets are only meteors, was not yet subverted; notwithstanding' the judicious reflections of Seneca, quoted above. Tycho completed the demonstration, that they are solid bodies like the planets, and subject to the same revolutions round the Sun. He observed a great number of comets in which he recognized this character of resemblance, which ought naturally to have dissipated the prerogatives ascribed to them. But his authority and his reasonings did not prevent comets from being still foj a long time considered as the harbingers of great events : so powerfully is the unhappy race of mankind enchained by errours, with which religious superstitions are connected. The great star which suddenly appeared in the constellation of Cassiopeia, in 1572, attracted the attention of every astronomer, and Tycho has re- s 3 corded 262 corded the history of this extraordinary astronomical event. It was first seen on the 7th of november, both at Wittemberg and Augsburg at the same time. Bad weather prevented Tycho from observing it be- fore the llth, when he found it almost as bright as Venus when stationary. It continued thus for some weeks, when it's magnitude gradually diminished. It was seen for seventeen months, at the end of which time, in march 1574, it disappeared entirely. Ac- cording to all probability, had astronomers been as- sisted by the telescope, it would have been longer visible. Tycho very accurately observed the periods of magnitude through which it passed during the time of it's appearance : and he noted with similar attention the singular changes of colour it underwent. At first it was of a bright whiteness : then it became of a reddish yellow, like Mars, Aldebaran, and the right shoulder of Orion ; after which it changed to a leaden white, like that of Saturn ; and thus it re* mained, till it disappeared : it twinkled like a com- mon star : &c. Similar phenomena have been seen on many other occasions. The ancient poets, particularly Ovid in his Fasti, book iv, relate that one of the stars, of the Pleiades was extinguished. Pliny says, that Hipparchus undertook to number the stars, in conse- quence of the appearance of a new one in his time. Nearer our own times, in the years 945 and 1264, it is said, that a new star was seen in the same place of the heavens. In 1600 a star was. observed for the first time in the breast of the Swan, which appeared and disappeared alternately: in 1616 it was of the third 26$ fhird magnitude ; after which it diminished for some years, and then disappeared. It was seen again in 1655, and disappeared once more, to reappear in 1665; &c. In the neck of the Whale there is a star, which changes it's magnitude periodically, and appears and disappears at regular intervals. It would be superfluous here to adduce a greater number of these extraordinary facts, and I shall hereafter relate the reasons, which modern astronomers have given to explain them. Contemporary with Tycho several eminent astro- nomers flourished, among whom William iv, land- grave of Hesse Cassel, and Kepler, deserve particu- larly to be distinguished. Of both of them I shall speak, after I have given a brief account of the reform of the calendar in 1582, under the pontifi- cate of Gregory xin. Great confusion had long taken place in the per- plexing and defective method, which the church had adopted for fixing the celebration of Easter every year, which, as is well known, regulates all the other movable feasts. The jews celebrate their Passover on the fourteenth day of thefrst month ; that is to say, of that lunar month in which the fourteenth day fell on the vernal equinox, or nearest after it. The pri- mitive church made no change in this system, except directing, that the Christian festival of Easter should be kept on the Sunday next following the fourteenth day. When this fourteenth day happened to be on Sunday, some churches made no scruple of cele- brating their Easter on that day, notwithstanding it's coincidence with the Jewish Passover : but the coun- cil held at Nice in 325 prohibited this practice, and s 4 ordered^ 264 ordered, that on such occasions Easter should not be kept till the Sunday following. After this general arrangement, nothing more was necessary, but to fix the day of the equinox, and the age of the Moon with respect to the Sun. The vernal equinox happening on the 21st of march in the year 325, the council of Nice believed, or supposed, that the same phenomenon would always occur on the same day, and at the same hour, throughout the course of time. On the other hand it decreed, that the age of the Moon should be regu- lated by the metonic cycle; so that all the years which had the same golden number, or which should be equally distant from the beginning of each period of nineteen solar years, must have their new Moons. on the same days, The fathers assembled at this council, however, though in other respects very ignorant, having some confused notions of the im* perfection of the metonic cycle, directed the patriarch of Alexandria, in which city the celebrated school of mathematics flourished, to verify the paschal moons by astronomical calculation, and communicate the results to the pontiff at Rome, who should announce the precise day of Easter to all the Christian World ; but this judicious regulation was neglected. In the system of the calendar adopted by the .council of Nice there were two little astronomical errours, the effects of which, accumulating through a series of ages, were become very considerable, One of these was, that the duration of the solar year ex- tended to 365 days, six hours ; the Other, that 235 lunations composed exactly nineteen solar years. The first supposition errs in excess about eleven mi-r nutcs ; 265 nutes ; and hence it followed, that the vernal equi- nox, which fell on the 21st of inarch in the year 325, happened on the llth in 1582. The second errs by defect ; and about the middle of the sixteenth cen- tury, the new moons indicated by the calendar pre- ceded Hie true new moons, as shown by observation, four days. The defects of the calendar had long been known; and several attempts had been made to correct them, but always in vain. The great progress of astronomy in the sixteenth century gave hopes of better success to Gregory xiii, who was desirous of rendering his pon- tificate illustrious by a signal and necessary reform, in which his predecessors had failed. -Accordingly he issued a solemn invitation to all the astronomers of Christendom, to deliver their thoughts on the best means of correcting the calendar, and giving it an accurate and permanent form. This invitation produced a multitude of schemes, among which that of Aloysius Lilius, an astronomer of Verona, obtained the preference, and was esta- blished by a bull, issued in the month of march 1582. It is a little complicated; and, to acquire a perfect knowledge of it, recourse must be had to the works, which treat of it expressly *. I shall confine myself here to a few general remarks. In the first place it was decreed, that, in the year 1582, the clay after the 4th of October should be called the 15th, or that this month should be reduced to twenty-one days only, to make the equinox of * For a fuller account of the reformation of the Calendar, see Bonny castle's Introduction to Astronomy. the the following year, 1583, fall on the 21st of march, 2dly : To prevent the return of the anticipation of the equinoxes in future, as well on account of the eleven minutes surplus of the Julian year, as of the precession of the equinoxes, the quantity of which then began to be pretty accurately known, it was re- gulated, that of four secular years, all which would be bissextile according to the Julian calendar, in future one only should be so, and the other three re- main common years : thus for example, of the four secular years 1600, 1700, 1800, and 1900, the first alone would be bissextile. 3dly : With respect to the Moon, the motion of which makes here the most perplexing part of the problem, Lilius substituted in- stead of the golden numbers, or metonic cycle, the epacts, which are the numbers that express the Moon's age at the commencement of every year, or the ex- cess of the solar year above the lunar. This ar- rangement, which easily admitted the addition or subtraction of certain clays at determinate epochs, had the advantage of reconciling the motions of the Sun and Moon better than the simple metonic c) T cle. The days of the year were preceded by letters inch eating little calculations, which it was necessary to make, in order to find the age of the Moon at every moment, and to regulate the festival of Easter with the other movable feasts. This new calendar was received and adoptee} with universal applause in all catholic countries. But it had not the same success among the protestants, who retained the Julian calendar as far as regarded the Sun's motion, and had recourse to astronomical cal- culation 267 culation for fixing the celebration of Easter. How- ever, as the practical form of the gregorian calendar is level to every capacity, the protestants of Germany at ler 4th adopted it in 1700, as the english did in 175., It is equally in use among the other nations gf the north, Prussia alone excepted. On this subject I shall add only one word more. The convenience of any calendar whatever is not a sufficient reason for adopting or retaining it : the es- sential condition is, that it should be perfectly accu- rate. But, whatever plan be pursued, this is unat- tainable. Happily the common calendars are of very little use, since the most celebrated academies in Europe have begun to publish ephemerides, the utility of which I have already had occasion to make known, when speaking of the ancient cycles, The landgrave of Hesse Cassel, William iv, who died in 1592, at the age of sixty, was very early instructed in astronomy ; and not only patronized the science, but gave himself up to the practice of observations with a zeal and success, which would have done ho- nour to a private individual He built an observatory in his capital, and furnished it with the best instru- ments then to be obtained. Among his excellent observations are quoted those he made of the positions of several stars, and of the solstitial altitudes of the Sun in the years 1585 and 1587. Tycho has been called the great observer ; and for a similar reason Kepler should be styled the creator of true physical Astronomy. Born in 1571, he soon began to render himself eminent by his works ; which were so numerous, that an abstract, or even a simple catalogue 263 catalogue of them, would carry us too far. From the various proofs he gave of his genius I shall select his discovery of the laws, which the planets obey in their motions : a discovery, to which he was led by combining with profound sagacity his own ob- servations with those of Tycho. The ancients made the planets revolve in perfect circles, the centre of which they supposed at first to be occupied by the Earth. But they soon found themselves obliged to remove the Earth to a greater or less distance from the centre of this revolution, that they might account for the changes observed in the diameters of the planets, whence it was to be in- ferred, that their distances from the Earth likewise changed, -Tycho, while he left the Earth fixed in the centre of the mundane system, had at least avowed, that Mercury, Venus, Mars, Jupiter, and Saturn, revolved round the Sun, as I have already said. The numerous observations he made on the motions of Mais in particular furnished Kepler with the means of convincing himself by immense calcu- lations, that all these motions could not be explained by the supposition of a circular orbit, in whatever situation the Sun might be supposed to be placed. He tried several other orbits in vain, till at length he found, that the common ellipsis, by placing the Sun in one of it's foci, agreed with the results of his cal- culations. This was the first step toward the grand discovery \ve have announced. Having afterward determined the dimensions of the elliptical orbit of Mars ; and comparing together the times, which the planet took in making a complete revolution from one one of the extremities of the line of the apsides, or the greater axis of the ellipse, and a given part of this revolution ; Kepler found, that these two times were always to each other as the whole of the area of the ellipsis to the area of the elliptical sector, comprised between the arc described by the planet and the radius vector drawn from each end of it to the Sun. The same proportion was also verified in the other planets; and it was afterward found, that it equally takes place in the revolution of the satellites round their primary planets. Thus it is become a fundamental principle of physical astronomy ; and is commonly called the first law of Kepler, or the law of the proportionality of the areas to the times. This important discovery led to another not less remarkable. Kepler suspected, that an analogy ex- isted between the times of the revolutions of the planets and the dimensions of their orbits ; and this he undertook to ascertain. Hence arose new calcu- lations, the extent of which may be imagined, if we consider that Kepler was groping his way as it were in the dark : but he was guided by genius, and suc- ceeded in the search. The result of all his numerical calculations was, that the squares of the times of the complete revolutions of two planets were to each other, as the cubes of the transverse axes Qf the two ellipses described by the planet, or as the cubes of their mean distances from the Sun : another funda- mental principle, verified in all the planets, and in all the satellites with respect to their primaries. This is called the second law of Kepler, or the Ifftb of the times relative to the mean distances. Those 270 Those who wish to know the origifi and progress of Kepler's ideas on this subject may consult his work entitled, Astronomia nova ccelestis tradita cum Commentariis de Motibus Stellce Martis, 1609. In this may be observed a lively imagination, fertile in resources, and in some places a kind of poetical en- thusiasm, excited by the grandeur and interesting nature of the subject Though the two laws of Kepler form the base of all astronomical calculations of the motions of the planets, we shall nevertheless see hereafter, that slight modifications are requisite, to represent the alterations occasioned in the elliptical motion of a planet round the Sun, or of a satellite round it's pri- mary, by the effect of the universal and reciprocal gravitation of all the heavenly bodies toward each other. Astronomy made fresh progress by the help of the telescope, which was invented about the beginning of the seventeenth century, as I shall relate more particularly hereafter : a happy supplement to the imperfection of the naked eye in observing remote objects. Galileo is one of the first, by whom this instru-* ment was used. He began with attentively observ* ing the Moon, on the surface of which he perceived Various inequalities, some parts prominent, and others sunk and dark. Hence he inferred, that this satel- lite was interspersed with mountains, lakes, and rivers, and that it formed an opake body, similar to the Earth. In every part of the sky he discovered an immense number of small stars, invisible to the naked 271 naked eye. A false idea had been entertained of the spots in the Sun, which were considered as a sort of temporary scum. Galileo observed, that they ad- hered to the body of the Sun, and appeared and dis- appeared in consequence of a rotatory motion, by which that body is carried round. Copernicus had predicted, that at some future time Venus would be found to have phases nearly similar to those pf the Moon ; and Galileo proved the truth of his prediction. But what gave him the greatest pleasure and astonish- ment, was his gradual discovery, that Jupiter is sur- rounded by four satellites, which revolve round this planet as the Moon does round the Earth. He called them the Medicean stars, in gratitude for the marks of esteem and respect he had received from the house of Medici : but this appellation met with little success in the World, and the simple name of the satellites of Jupiter has prevailed. The system of Copernicus, already so plausible, acquired a probability almost equivalent to demon- stration, by the observations and reasonings of Ga- lileo. Most of the objections made to this system were frivolous enough. It was said, for instance, that the Earth having the Moon for a satellite, it was not to be supposed, that the Earth itself was a satel- lite, or revolved round the Sun. To this Galileo gave an irrefragable answer, that Jupiter had four satellites, and notwithstanding revolved round the Sun, according to the observations and calculations of Tycho : and he added, that, the Moon being a body similar to the Earth, there was no reason to suppose they might not both have similar motions in celestial 272 celestial space. But the strongest probability in favour of the copernican system, and that on which Galileo insisted the most, was the simple and natural expla- nation it gave of the stationary, direct, and retro- grade appearances of the planets ; while in this re- spect the system of Ptolemy, and even that of Tycho, exhibited a complication of motions, impossible to be reconciled with the laws of mechanics and sound philosophy. Supported by all these considerations, Galileo had the courage, from the year 1615, openly to maintain the system of Copernicus. But this courage brought upon him the animadversions of the Inquisition, and he was obliged to retract, in order to avoid imprison- ment Twenty years afterward, imagining the truth to have become more mature, he declared himself anew, thoijgh in a more secret way, for this sys- tem, without which he saw clearly physical astronomy could not be supported. The Inquisition, by which he was constantly watched, now kept no measures with him : he was obliged to appear before it's tribunal, and sentenced to pass the remainder of his days in a dungeon. At the expiration of a year, however, he was released from his imprisonment, on condition, that he should not again relapse, and that he should never quit the territory of Florence ; where in fact he remained under the eye of the Inquisition, till the day of his death : a too notorious instance of the innumerable crimes, which an absurd and fanatic tri- bunal has committed against human reason, and which it has at length expiated in our days with igno- miny. In 273 In spite of the inquisitors, or of the passages in the Bible, which were incessantly brought forward as objections to the Earth's motion, the system of Copernicus continued to gain ground, and grow stronger every clay. It ought not to be omitted however, that one difficulty was started, to which neither Copernicus nor Galileo could give a peremp- tory answer ; but a complete solution of which they predicted would one day be found. This was, that, supposing the Earth arrived successively at the two extremities of one of the diameters of it's orbit, we ought to find a parallax or change of position in the stars ; and this was not to be observed. For more than a century astronomers made every ex- ertion, to clear up this doubt : some perceived a very trifling parallax in the stars, while others found none; and others again, discovered motions directly contrary to those, which should have resulted from a parallax. The certain conclusion from all these uncertainties was, that the stars are placed at such distances from us, as may be considered as infinite with respect to the radius of the terrestrial orbit, though this radius is at least ninety five millions of miles. It will be seen hereafter, that this question was completely solved before the middle of the eighteenth century, so that now the doctrine of the Earth's motion is established on irrefragable evidence. Italy was not the only country, where the use of the telescope contributed to the progress of astro- nomy. In 1631 Gassendi, a french philosopher, saw Mercury on the Sun's disk, which was the first T obser- 274 observation of the kind. Horrox, an english astrono- mer, also made a similar observation of Venus in 1639, John Baptist Morin, who was a long time professor of mathematics in the College of France, composed several works, which do no honour to his memory : yet, on the other hand, it must not be forgotten, that he was the first who pointed out the manner of solving the celebrated problem of the longitude by means of astro- nomical observations ; and that, in order to make these observations with greater accuracy, he proposed the application of a telescope to the quadrant; an idea, which has been erroneously ascribed to later astronomers. Hevelius, who was born in 1611, and died in 1688, rendered himself eminent for his numerous and delicate observations of the spots on the Sun, the motions of comets, c. To him likewise we are indebted for the first accurate description of the spots on the Moon, Riccioli, a Jesuit, left a great work after the ex- ample of Ptolemy, entitled The new Almagest; in which he has collected together all the astronomical theories known in his time, with his own observations and remarks. He was much assisted by his friend Grimakli ; who, independently of this work, pub- lished a Selenography, in which the spots on the Moon are distinguished by the names of philosophers. This nomenclature was adopted at first with applause, and is still retained, though -time has introduced into it some corrections. Mouton a canon of Lyons, determined with dex- terity and success the apparent diameters of the Sun and Moon, bynneans of the telescope and a simple pendulum ; 275 pendulum ; and to him we owe the first idea of con- necting together, by interpolation, the observations made of one object at different times. He likewise calculated a table of logarithms of sines and tangents for every second as far as four degrees, which was printed in the edition of Gardiner published at Avig- non by the Jesuits Pezenas and Dumas, in 1770. After the discovery of the satellites of Jupiter, this branch of astronomy remained almost station- ary for more than forty years, either because it de- manded extreme attention in the observer, or be- cause the telescope was not yet brought to sufficient perfection. Galileo had imagined in 1615, that he saw two satellites very near the body of Saturn. They appeared immovable for three years, always retaining the same figure ; but at length they ceased to be vi- sible altogether, and it was supposed, that Galileo had been deceived by some optical illusion. In 1655, Huygens having himself accomplished the construction of two excellent telescopes, one twelve feet in length, the other twenty-four, he dis- covered one of the satellites of Saturn ; which is that now called the fourth. He determined it's distance from Saturn, the position of it's orbit, the period of it's revolution, &c., with such clearness and accuracy, as left no doubt of the existence and motion of this new secondary planet. Astronomers at that time were so fixed in the prejudice, that the number of satellites could not exceed that of the primary planets, as to lead Huygens, after this discovery, which made the number of satellites and of planets T 2 equal, 276 equal *, to observe, in the epistle dedicatory of his Systema Saturnimn to the grand duke of Tuscany, that the number of satellites was complete, and we must not hope to see any new ones in future. Let us forgive a great man, who enriched the accu- rate sciences with so many immortal discoveries, this metaphysical errour. Perhaps indeed it may be ascribed to the high opinion he entertained of his telescopes ; for their having enabled him to see phenomena hi the heavens, which no one before had remarked, might have induced him to think, that none of the bodies in our planetary system had escaped his view. f The discovery of this satellite gradually led Huy- gens, as he informs us, to the knowledge of the ring, by which Saturn is environed. Several astronomers after Galileo hid observed Saturn under 'different irre- gular and variable forms, for which they could give no satisfactory reason. Huygens perceived with his telescopes, and soon demonstrated, that Saturn was a round body, encircled by a flat ring every where de- tached from it, which, being seen obliquely from -the Earth, ought, conformably to the rules of optics, to appear in the form of an ellipsis, more or less open ac- cording as our eye is more or less elevated above it's plane, the inclination of which to the ecliptic is about thirty degrees. Hence followed a simple and natural explanation of all the appearances of Saturn. * On the one hand there were six primary planets, Mercury, Venus, the Earth, Mars, Jupiter, and Saturn: on the other six satellites, the Moon, the four satellites of Jupiter, and one of Saturn, The 277 The ring becomes entirely invisible to our eyes, when it's thickness is too small to reflect the solar rays toward us in sufficient quantity to be perceived. Huygens found, that the extreme semidiameter of the ring is to that of Saturn as nine to four; and that it's breadth is equal to the space between it's inner surface and the body of the planet. This system, attacked at first by ignorance or envy, is at present a fundamental truth in astronomy. At this period two grand establishments for the promotion of science were formed, the Royal Society at London, in 1660, and the Royal Academy of Sciences at Paris, in 1666. These two illustrious bodies have produced men of the first order in every branch of science : at their commencement they were chiefly useful in astronomy, which has more need of being encouraged by the attention and remuneration of princes, than all the other sciences. One of the first cares of Lewis XIV, or rather of his great minister Colbert, in founding the Academy of Sciences, was not only to introduce into it the men of learning in France> but likewise to attract to it foreigners of the greatest celebrity, and such as were most capable of contributing to the splendour of the establishment, and the progress of the sciences. Among the former we may remark Claude Perrault, Mariotte, Pecquet, Auzout, Picard, Richer, &e. ; among the latter, John Dominic Cassini, Roemer, &c. John Dominic Cassini, who had already acquired a great name in science, by his meridian line at the church of St. Petrona at Bologna; by tables of the T 3 Sun 278 Sun and of the satellites of Jupiter ; and by other astronomical labours, as well as by the hydraulic works in which the popes had employed him, before he came to settle in France, found himself at perfect liberty in this country to indulge his genius and in- clination, which led him to astronomy, In this science he made a great many important discoveries ; the most striking of which was that of four more sa- tellites of Saturp, which are in the order of their dis- tances the 1st, 2d, 3d, and 5th; so that with the 4th, discovered by Huygens, this planet had now five sa- tellites, fully known. The hypothesis of the elliptical motion of the pla- nets, which Kepler had advanced, was not perfectly comprehended by every astronomer. Cassini comr bated it on an unfounded supposition. He imagined that Kepler, while he placed the Sun in one of the foci of the common ellipsis, made the other focus the centre of the mean motions, or the vertex of the areas proportional to the times ; which gave results by no means accordant with observation. To correct this fault, Cassini substituted instead of the common ellipsis another curve, which he called an ellipsis likewise, in which the product of two lines, drawn from two fixed pqints to one and the same point in the curve, forms every where an invariable quantity : whereas in the common ellipsis it is the sum of two lines drawn from the t\yo foci, which is an invariable quantity. But Kepler did not really fall into the mis- take ascribed to him by Cassini : he places the centre of the mean motions at the focus which the Sun oc- cupies, and in this case all the observation^ are very easily 279 easily explained. Cassini's curve has not the same advantage : and besides, when the two foci are very distant from each other, it takes a course, which it is physically impossible any planet should follow. Auzout was also an excellent observer; he had a per- fect knowledge of astronomical instruments, and im- proved and extended the use of the micrometer, which was first invented by Huygens. It is said, that, when he presented to Lewis XIV his observations on the co- met of 1644, he first suggested the idea of building an observatory, and furnishing it with instruments. The observatory at Paris, begun in 1667, was finished in 1672, six years after the foundation of the Academy of Sciences. England closely followed the example, snd the observatory at Greenwich was built in 1676. There are certain speculative sciences, as geometry, algebra, rational mechanics, &c., in which great pro- gress can be made only by men of a sedentary life, medi- tating in the silent retreat of the closet There are others, in which we must pass from theoretical study to practical application, make experiments, and traverse various countries : such are physics, natural history, and particularly astronomy, which frequently re- quires comparative observations made in different places very remote from each other. Abbe* Picard, distinguished by his dexterity and skill in selecting and managing the proper instru* ments for observations, performed several useful la* bours; and among them the project,* that had been frequently attempted, of measuring the Earth, with a precision on which geography and navigation might establish certain bases : for the measures of the Greeks T 4 and 80 and arabs, and even those of some modern philosophers, wanted this stamp, or at least had nothing to warrant their accuracy. He measured the arc of the heavens comprised between Amiens in Picardy, and Malvoi- sine on the confines of the Gatinois and Hurepoix: then by a comparison of this measure with that of the correspondent terrestrial arc, determined by means of a series of triangles connected with each other, and of which the first was constructed on a known base, he concluded, that the length of a degree of the Earth was 57060 toises nearly. Hence it followed, that the entire length of a great circle of the globe was 20541600 toises. In 1672 Richer was sent to Cayenne, which is within five degrees of the equator, to make various astronomical observations there. He was particularly directed, to observe the planet Mars, which Picard, then in Denmark, and Cassini and Roerner in Pro-* vence, were observing at the same time ; that from all these observations, made in such distant places, they might be able to deduce the parallax of this planet, to which the attention of most astronomers was turned, as they hoped it would throw great light on the theory of parallaxes. As soon as Richer was preparing to begin his obser- vations, he made one that had not been foreseen, ancl which was much more important than all that* had been proposed. As a measure of time he carried with him a pendulum, which swung seconds accurately at Paris: but when he came to make use of it at Cayenne, he found, that the pendulum performed it's oscillations too slowly ; and that to make it swing seconds with % precision^ 281 precision, it was necessary to shorten it about a line and a quarter. This singular observation being trans- mitted to Paris, Huygens immediately discovered the physical reason of it; which was, that, In conse- quence of the rotatory motion of the Earth round it's axis, the centrifugal force was greater toward the equator than in the latitude of Paris, and conse- quently must occasion a greater diminution of the natural and primitive gravity there : whence it fol- lowed, by a farther consequence, founded on the theory of the motion of pendulums, that the pendulum swinging seconds at Cayenne must be shorter, than the pendulum swinging seconds at Paris. Huygens gave likewise a calculation of the progressive flatten- ing of the Earth in proceeding from the equator to- ward the poles. Some years after Newton likewise found a flatten- ing in the same direction, but somewhat greater than that of Huygens, because these two illustrious geome- tricians set out with suppositions respecting the na- ture of primitive gravity a little different from each other. Huygens considered it as constant, and di- rected to the centre : Newton as the result of all the reciprocal attractions of the molecules of the terres- trial globe, which leaves the centre a little on one side. In this grand problem we find experiment preceded and enlightened theory, and France had the honour of furnishing the data, which were to be employed in it's solution. We shall hereafter see the vast and expensive undertaking which she executed, to de- termine the true dimensions of the Earth, We We may likewise reckon in the number of our astronomical discoveries that of the propagation of light, which was made about the same time. Roemer. the author of this discovery, who was born in 1644 and died in 17 JO, was indeed a dane by birth : but at the time he made it he was fixed in France by the muni- ficence of Lewis XIV, and was one of the first mem- bers of our Academy of Sciences. After the satellites of Jupiter had become known, astronomers sought diligently to determine their motions; and Dominic Cassini had gone so far as to construct tables, which accurately represented their revolutions, and their eclipses occasioned by the shadow of Jupiter. But Roemer, who assiduously observed the first satellite, perceived, that in eclipses it emerged from the shadow at certain times a few minutes later, and at others a few minutes sooner, than it ought to have done ac- cording to the tables. He also, on comparing these times together, found, that the satellite emerged from the shadow later, when the Earth was carried by it's annual motion to a distance from Jupiter, and sooner, when it approached the planet. Hence he formed this ingenious conjecture, which was soon converted into demonstration, that the motion of light is not instantaneous, as des Cartes had thought, and as was then still believed, but that it took up a certain space of time to arrive from the luminous body at the eye of the observer. According to his first calcu- lations, it was about seven minutes in traversing the radius of the Earth's orbit : but he afterward found, that the velocity of the luminous atoms was a little greater 283 greater than this. There is not a more remarkable phenomena in the natural philosophy of the heavens, or one more essential as an element in astronomical theories ; hence it must secure immortality to the name of Roemer. England produced at all times astronomers of the first order. Here we shall remark among others Hooke, Flamsteed, and Halley. Hooke, who was born in 1635, and died in 1702, was not merely a great observer in every branch of astronomy : we are indebted to him for the first idea of the system of universal gravitation, that can be called at all explicit. , He made the three following suppositions. 1st: All the celestial bodies have not only an attraction or gravitation toward their own centre, but they v mutually attract each other in their sphere of activity. 2dly : All bodies, which have a simple and direct motion, would continue to move in a right line, if some force were not incessantly turn/- ing them out of it, and compelling them to describe a circle, an ellipsis, or some other more complicated curve. 3dly : Attraction is so much $he more pow- erful, as the attracting body is more near. All these bases enter into Newton's system ; but what charac- terizes the discovery of the latter is the law of attrac- tion, which he found out, and which was unknown to Hooke, Flamsteed was born in 1646, and died in 1720. As soon as the observatory at Greenwich was built, he was appointed to the management of it by Charles JI, and began to make there that numerous series of observations of all kinds, recorded in his Historia Cekstis, 284 Celesfis, and in the Philosophical Transactions of the Royal Society. The principal services he rendered to astronomy were his prolegomena on the history of this science, and his catalogue of the fixed stars vi- sible in our climates, which was more complete than any other then known. Halley, who was born in 1 656, and died in 1742, was profoundly skilled in geometry : but led by the prevalence of his inclination for astronomy, he en- riched this science with a great number of obser- vations and researches, which were so much the more Valuable and accurate, as they had always geometry for their guide. Almost at his entrance on his career he undertook a long voyage, to make a catalogue of the stars in the southern hemisphere. As the ancients knew little more than the northern half of the globe, and those of the moderns, who had visited the southern, were tempted by prospects with which astronomy had no Concern, the stars of the south, particularly those near the pole, remained either wholly unknown, or inaccurately marked on the ce- lestial globe. To supply this desideratum, this de- ficient or imperfect part in the catalogues of Ptolemy and Tyeho, and to make observations corresponding with those of Hevelius and Flamsteed in Europe, Halley repaired in 1676 to the island of St. Helena, the southernmost of the english possessions, lying in the latitude of sixteen degrees south, and there com- pletely executed his design. The catalogue of the southern stars, drawn up from his observations, com' prises the description of a considerable continent in the vast regions of astronomy. Halley reaped many other . 285 other observations from his voyage, particularly that of the transit of Mercury over the Sun's disk, which occurred on the 3d of november, 1677. This was the fourth time the phenomenon had been seen since the invention of telescopes, before which .it was out of the question. Halley was acquainted, either psrsonally or by letter, with all the astronomers of Europe. In 1679 he paid a visit to Hevelius at Dantzic ; and the fol- lowing year he set off on a tour to France and Italy. Being half way between Calais and Paris he perceived for the first time the famous comet of 1680, so ter- rible in the eyes of the vulgar on account of it's brightness and magnitude. This suggested to him the idea of writing a short treatise on comets, which will be mentioned in it's proper place. I shall add by the way, that this same comet pro- duced Bayle's celebrated work, entitled Thoughts on the Comet ; in which that great philosopher combats with all the force of argument and reason the super- stitious errours, that still existed respecting the cause and effects of the appearance of comets. At every step a science makes, the arts connected with it, particularly those which are useful to society, are proportionally enlarged. Navigation and dialling, therefore, could not but feel the auspicious influence of the general movement, which then took place ia astronomv. 9) By confining the use of plain charts to small extents of the terraqueous globe, the inconvenience attend- ing them, which was that of expressing by equal lines the degrees of two parallel circles terminating the chart on 286 on the north and on the south, might be avoided, and the suitable proportion given to the represen- tations of these degrees. Gerard Mercator, a flemish geographer, made this remark, which however is ex- tremely simple and elementary. Edward Wright, of whom we have some astronomical observations among those of Horrox, developed Mercator's idea, or rather saw the question in a new point of view. Having remarked, that the radius of a parallel, as we proceed from the equator toward the poles, diminishes in the same ratio as the secant of the latitude increases, he proposed the construction of charts after this prin- ciple. These are called reduced charts [or M creator's, or Wright's charts]. The invention is very ingenious, and was brought into use among seamen about the year 1630. Tables were afterward calculated, to render both the theory and practice more perfect. The loxodromic or rhumb line, which is the course a vessel makes in sailing on one point of the compass, is on the globe a curve of double curvature : on the reduced chart it is a com- mon curve, the length of which is so much the more easy to calculate, because in practice the problem is simplified still more. No vessel ever keeps to the same rhumb line during a long voyage ; for all seas are interrupted by islands or continents ; and besides, it's course is frequently changed, either to seek a fair wind, or to avoid shoals, c. Thus the whole course of a vessel is composed of several parts of different rhumbs ; and each of these parts, taken separately, may be confounded with a simple straight line in most cases, without any sensible crrour. Navigation derived derived another assistance from astronomy, by adapt- ing the use of several instruments for directing the course of a ship by observations of the heavenly bodies : but it is obvious, that, from the continual motion of the vessel, observations at sea must long have been very imperfect. We have seen, that the ancients were much oc- cupied with the construction of dials ; and that tiiey delineated them of all kinds, on various surfaces, plane, cylindrical, conical, spherical, &c. Vitruvius, who entered with considerable minuteness into this subject, has not explained the theory of gnomonics, at least with the requisite perspicuity and method. We do not begin to find this theory sufficiently elu- cidated, before we come down to the writers of the sixteenth century. Munster, who was born in 1489, and died in T552, and Oronce Fine*, who was a few years younger, are supposed to be the first, by whom treatises on the subject were published. Maurolicus wrote on it a work much esteemed, in which practice and theory are combined. The treatise on Gnomonics published by father Clavius, in 1581, is also men- tioned with much commendation. So many works of the kind have since been composed, that a simple enumeration of them would be as tiresome, as useless. CHAP. 288 CHAP. VI. Progress of Optics *. SOME writers, who have never invented any thing themselves, but find every thing in the ancients after it has been invented, ascribe to these the principal modern discoveries in optics, and the construction of the instruments dependent on them. I am willing to believe, that in this they are sincere, and not prompted by any sentiment resembling that mean envy, which is always exalting the dead at the ex- pense of the living. But here their endeavours are in vain. We see by the oldest book on optics ex- isting, which is commonly ascribed to Euclid, that in this branch of mathematics the ancients had only ge- neral and vague notions, some of which were even erroneous. For instance, they knew, that light is .propagated in a direct line, when it meets with no obstacle in it's way ; and that, when it falls on a well polished plane surface, it is reflected at an angle equal to the angle of it's incidence : but they were ignorant of the law according to which an opake body is illumined in proportion as it is more or less near the luminous body; they deceived themselves * In the general term optics are here comprised, as usual, optics properly so called, or the science of the direft transmission of light; catoptrics, or the science of reflected light ; and dioptrics, or the science of refracted light. in 289 in making the apparent magnitude of an object de- pend solely on the angle under which it is viewed ; they were mistaken in saying, that the place of the image formed by reflected rays is placed at their in- tersection with the perpendicular drawn from the ob- ject to the reflecting surface ; finally, even in the time of Ptolemy, they were acquainted only with the general phenomena of the refraction of light, and never suspected, that, when a ray passes out of one medium into another, there exists a relation between the two directions of this ray, which is subjected to an invariable law. It is certain, that optics did not begin to acquire any progress, and form a real body of science, till near the end of the fifteenth century, One of the first who prepared or commenced this progress was the celebrated John Baptista Porta, a neapolitan gentleman, the inventor of the camera obscura, who was born in 1445, and died in 1515. In his work entitled Magla naturalis he says, if we make a small hole in the window shutter of a room, from which the light is every where else completely excluded, we shall see the external objects depict themselves on the wall, or on a piece of paper, in their natural colours : and he adds, that, by placing a small convex lens at the aperture, the objects will appear so distinct, as to be recognised instantaneously. From these assertions, verified by experience, there was but one step to the explanation of the mecha- nism of vision. This Porta did not completely make: he only remarked, that the bottom of the eye might be considered as a caniera obscura, without pursuing u or 2.90 or entering into any explanation of this just and happy idea. Maurolicus treated on the general theory of optics in two works ; one entitled Theoremata Lucis 8$ Umbra;; the other, Diaphanorum Paries sen Libn ires. These works contain several curious researches on the measure and comparison of the effects of light ; on the different degrees of brightness, which an opake object receives from a luminous body, ac- cording as it is more or less distant ; &c. If Maurolicus have not always hit the truth, at least he has pointed out the way to his successors, and spared them many futile attempts. He has very justly explained a well known phenomenon, on which the ancients, and in particular Aristotle, had given us nothing but reveries : this is, that the rays of the Sun, when they pass through a small hole of any shape, a triangular one for example, always form a luminous circle on a piece of paper parallel to the hole, and placed at some distance. In the first place Maurolicus observed, that, when the paper is placed very near to the hole, this must be depicted on it in it's actual figure ; but as the paper is removed farther off, the similitude gradually disappears, till at length the image becomes circular. In fact, each point of the hole may be considered as the vertex, of two op- posite cones, one of which has for it's base the Sun, the other a luminous circle cast on the paper by the decussation of the rays at the vertex ; and as the number of points in the hole is infinite, so is that of the cones. NOAV the circles formed on the paper by the the bases of the secondary cones partly cover each other, leaving at the circumference indentations, which are continually decreasing in proportion as the paper is carried farther from the hole, so that at last they become imperceptible, and the outline of the image on the paper appears to form a complete circle. All this is agreeable to experience. We are likewise indebted to Maurolicus for some just remarks, though not very profound, on the theory of the rainbow, and on that of vision. The person, who at this time came nearest to the true explanation of the phenomenon of the rainbow, was Antonio de Dominis, archbishop of Spalatro. Every one knows, that this appearance is visible only when the Sun shines during rain, and that the spec- tator must be in a certain position with respect to the rain and the Sun. The drops of rain had been comparefl to little spheres of glass; and it had been supposed, that these spheres reflected the solar rays toward the eye of the spectator : but this did not explain the colours of the rainbow, for the rays of light are separated from each other only by refraction. Antonio de Dominis employed both reflection and refraction at the same time, and he accounted pretty accurately for the superiour arch of the rainbow, but was less happy with respect to the inferiour, He v published his ideas on this subject in' 1611, hi a work entitled De Radils Visas 8$ Lucis. On reading this book we perceive, that the author had a real talent for the sciences, and we regret, that he did not make them his sole study. Some the- ological opinions, a little too bold, which he had the u 2 impru imprudence to make public, brought on him a per- secution, from which he was unable to extricate him- self but by taking refuge in England in 1616. With- out entirely embracing the principles of the .refor- mation, he rendered himself very useful and agree- able to James I, by combating several of the pre- tensions of the popes. To him we are indebted for the first edition of the History of the Council of Trent by Fra Paolo, which he printed at London in 1617. Soon after he published his great work Of the Ecclesiastical Commonwealth ; which afforded the Italians fresh occasion for reviling him furiously, and gave him more reason to be on his guard. Yet, ac- cording to some historians, remorse of conscience, according to others, altercations he had with the jprotestants in which interest was concerned, inspired him with the resolution of quitting England, and returning to Italy, where Gregory XV, who esteemed his talents, promised him perfect security, and indeed every thing he could desire. Accordingly he first publicly abjured in a chapel in London those opinions, which had offended the court of Rome, and then repaired to Italy. About two years he had remained at Rome in perfect tranquillity ; when unfortunately he afforded the malice of his enemies, by whom he was still watched, an opportunity of ruining him. In 1625 he was imprisoned in the Castle of St. Angelo, by order of pope Urban VIII, where in a few days he died of poison, as was commonly sup- posed. The inquisition ordered his corpse to be dis- interred, and burned with his works. The The comparison which Baptista Porta made of the camera obscura with the eye was very jitst, and it was by pursuing- it, that Kepler explained the nature of vision with accuracy, At first a general idea of it was conceived, by regarding the pupil as the aperture in the camera obscwra, the crystalline liis- m our as the convex lens applied to the aperture, and the retina as the paper on which the objects were de- picted : but if we proceed to the particulars of the means by which this mechanism is effected, there are several elements to be combined. The rays issuing from the luminous body fall at first on the cornea; thence they enter the aqueous humour, where they undergo a refraction, which begins to make them converge ; from this they pass through the opening of the pupil to the crystalline, the lenticular figure """of which increases their convergence; and from the crystalline they proceed to the vitreous humour, where a fresh refraction and convergence take place, Lastly, after all these refractions, they unite in one k point of the retina, where they strike on the optic nerve, and thereby excite the sensation of vision. Kepler traced and explained the course of the rays. One difficulty perplexed him a long while, which was, why objects, since they are depicted on the bottom of the eye in an inverted position, nevertheless appear in their natural situation. For this he imagined some plausible reasons. The most natural explanation of it that can be given is, that the impression, produced by a ray issuing from any point of *ain object must be referred back [doit It re rappvrtie] in an immediately opposite direction, ami that consequently we must u 3 see 294 see the upper part of an object above, and the lower part of it below. It is the same with the ray as with a stick, which being pushed in the direction of it's length, is repelled in the opposite direction, Des Cartes, in his Meteors and Dioptrics, pub- lished in 1637, explained the phenomenon of the rainbow, and the nature of vision, according to the principles of Antony de Dominis and Kepler, with- out quoting either the one or the other : an omission the more to be reprobated, as he was sufficiently rich in his own stores. It has been observed in excuse for him with respect to the archbishop of Spalatro, that he corrected his explanation with regard to the inte- riour rainbow ; this may diminish, but it does not entirely efface the injustice. The knowledge of the laws of the refraction of light is some years posteriour to the book of Antony de Dominis. According to Huygens we are indebted for it to Wiilebrord Snell. If we thrust part of a straight stick into water obliquely, the stick appears to be broken at the surface of the water, the part immersed approaching a vertical line drawn through the point where it enters the fluid. Hence Snell formed at once the general conclusion, that a ray of light, passing out of one medium into another more dense, must approach the perpendicular at the sur- face separating the mediums ; and on the contrary, that it must diverge from the perpendicular, when it passes out of a denser medium into one less dense. Experience confirmed these remarks. But the grand point of the question was, to discover the reciprocal relations 295 relations of the angles, which the incident and re- fracted rays form with the vertical line. Snell attained this by a series of numerous and de- licate experiments. He found, that, by prolonging the incident and refracted rays on each side of the point of entrance, and drawing any vertical line, the parts of the two rays comprised between the point of entrance and this vertical line always preserve a constant ratio to each other, whatever the obliquity may be, This ratio however is not the same for two other mediums : but it generally follows the ratio of the densities. Snell cjid not perceive, that his proposition amounted to the same thing as saying in other terms, that when a ray of light passes out of one medium into another, the sines of the angles it forms in. the two mediums with the vertical line always preserve a constant ratio to each other, This is the fundamental law of the refraction of light. Snell's work that contained it was never printed. In 1637 des Cartes published it in his Dioptrics, in the second form here mentioned, without quoting Snell, and some frencli geometricians thought him the in- ventor of it : but Huygens affirms^ that des Cartes had seen SnelFs manuscripts in Holland. If it were so, this is another act, which reflects no honour 04 the memory of the french philosopher. When the laws of the refraction of light were dis* covered, and attempts were made to explain them, they occasioned much perplexity, for they were en- tirely contrary to those of solids. A musket ball for example, or in general any solid body, striking still water in an oblique direction to it's surface, diverge* u 4 from 296 from the vertical line on plunging into it, while, on the contrary, the ray of light under similar circum- stances approaches it. Now the effect in the first case is very natural, and easy to comprehend : for the ball, in passing ut of the air into water, which has more density, must experience greater resistance, and consequently must be repelled a little toward the surface of the water, or diverge from a vertical line drawn through the point of entrance. But why is it not the same with the ray of light ? To account for this difference des Cartes advanced this strange paradox, that a ray of light finds less difficulty in traversing a dense than a rare medium. And the corrections and annotations of his followers all come at bottom to the same thing. Fermat made some objections to the proposition of des Cartes, which, without being conclusive, ren- dered it at least very doubtful. He likewise attempted to solve the question in a different way. The an- cients had supposed, that a ray of light, moving always in one and the same plane, being obliged to strike on a polished and immovable surface, in order to proceed from one given point to another, became reflected at an angle equal to that of it's incidence, which rendered the whole of it's course a minimum. Fermat thought, that the ray in refrac- tion, passing from one medium to another, must make the whole of it's course in a minimum of time ; and by this he found, that the ray, in passing out of a rare medium into a denser, ought to approach the perpendicular, and vice versa. But philosophers, little satisfied with this shift, which they considered as a-s a mere geometrical trick, asked why Fermat made the refraction and reflection of light depend on dif- ferent principles. This desire of uniformity in the explanations was the subject of a very ingenious paper, which Leibnitz published in the Leipsic Transactions for 1682, under the following title : The Principle of Optics, Catop- trics, and Dioptrics, one and the same. The suppo- sition on which lie founded this one principle was, that a ray of light, in passing* from one given point to another, either directly, or by reflection, or by refraction, must in all cases pursue the readiest course. What remains to be determined then is the readiest way, in all the three cases. When the motion is direct, or performed in the same medium, it is evident, that the shortest way, or a simple right line drawn from one point to the other, is the readiest. In reflected motion that way is still the readiest, which is the shortest, or the least sum of two lines, drawn from the point of reflection to the two given points ; whence it follows, that the angle of reflection must be equal to the angle of in- cidence. Lastly, in refracted motion, where the two portions of the way are not uniform, the facility in each is so much the greater, as the product of the space passed through multiplied by the resistance of the medium, is less : consequently the facility of the whole way is as the sum of the products of the two mediums multiplied by the corresponding spaces. Hence, on reducing this sum to a minimum, it will be found, that the sines of incidence and refraction, arc in a constant ratio, which is the inverse ratio of the the resistances of the mediums. This third case we see includes the two others, supposing for this that the densities of the two mediums become equal. The whole of this theory is certainly very plausible, and has greatly the advantage over those of des Cartes and Fermat. Yet as it is founded, like that of Fermat, on the metaphysical principle of final causes, it must be confessed a direct solution would be preferable. The system of attraction, or rather the law of universal gravitation, demonstrated by all the phenomena we know, gives this solution in the most precise and satisfactory manner, and free from every difficulty. Reflection and refraction are not the only deviations* to which the motion of light is subject, There is. still another, that of inflection, deflection, or dif- fraction, by which a ray of light, when it passes close by an opake body, is made to change it's direction. In fact, if you admit the light through a small hole into a camera obscura, you will see, on exposing to* it a slender body, such as a hair, a pin, or a straw, that the shadow of this body is considerably larger than it would be, if the rays, which pass by it's edge, fol- lowed their original rectilinear course. You will see likewise, that the shadows are skirted by three bands or fringes of light, parallel to each other ; and that, if the hole be enlarged, these fringes will dilate and mingle together, so as to be no longer distinguishable. Grimaldi, who has been already mentioned, was the first that remarked this phenomenon, as well as the dilatation of the fasciculus of solar rays by the prism ; as may be seen in his work entitled Physicomathesis de 299 de Lumine, c., published in 1665. Long after Newton discussed this subject completely in his Optics, and freed it from some erroneous physical explanations, which Grimaldi had introduced into it Among these earliest opticians must be mentioned, with praise, father Kircher, a man of very extensive knowledge in various sciences, who was born ia 1602, and died in 1680. To him is ascribed the in- vention of the magic lantern. James Gregory, who was born at Aberdeen -ia 1638, and died in 1675, contributed to the progress of optics by his Optka promota, which contains several curious propositions on the theory of optics, and hints for improving the construction of instru- ments dependent on this science. He is chiefly known as an optician by his reflecting telescope ; and ke was likewise a good geometrician. Barrow's Lectioncs opticce, which appeared in 1669, are remarkable for a number of elegant propositions, exhibited and demonstrated in the simplest and most methodical order. This excellence particularly dis- tinguishes his determination of the foci of different sorts of dioptric glasses, which he has reduced into very neat general formulae. Newton had laid the foundation of his optics in some papers printed in the Philosophical Transactions for 1671, 1072, &c. One of the principle disco* veries he made at this time was the different refrangi- biiity of the rays of light. We shall return to him as an optician, when we come to the year ] 706, ia which he published his Treatise oa optics complete. In 30O In 1678 Huygens communicated to the Academy of Sciences at Paris, of which he was a member, a Treatise on Light, which was not printed t?ll 1690. The principal object of this was, physically and ma- themat really to explain the laws of the motion- of fight, whether in a right line, or reflected, er re- fracted. Among other elegant researches contained in this work, the author demonstrates, that a globwle of light, traversing a medium composed of strata of different densities, must describe a curve, the funda- mental property of which he instructs us how to de- termine in each ease. For example, when the me- dium is composed of horizontal strata, and the velo- city of the globule increases or diminishes in the same ratio as the density of the strata, we find tfee curve must be an arc of the cycloid. Huygens had likewise composed at different times several other works relative to optics, which did not appear till after his decease. Of this number is his Dissertation on Coronas," Parhelia, and Paraselenes, of which I conceive it incumbent on nie to speak a few words. The reader knows that corona?, or haloes, are cir- cular rings of light, which are sometimes seen in the day surrounding the Sun, and in the night surround- ing the Moon; that parhelia are mock Suns, or the appearances of Suns round the true one; and that paraselenes are similar mock Moons. These pheno- mena have been seen in all ages ; but it is not much above fourscore years, since they were begun to be observed with accuracy: for Aristotle, and Cardan who 301 lived eighteen centuries after him, assert, that more than two parhelia are never seen at once; while in reality, by proper attention, a much greater num- ber is 'frequently observed. For instance, live Suns were seen at Rome on the 29th of march, 16&9 ; and seven at DantEic, on the 20th of February, .1661; c. Now is it possible, says Huygens, tliat in so few years six or seven parhelia have appeared, eack consisting of more than two Suns, and that the same phenomenon never occurred before ? No doubt for- merly the two lateral parhelia, which are in fact the largest, were alone considered as true parhelia, and no attention was paid to the others, as being more faint and languid. Des Cartes undertook to explain all these appearances ; but his explanation was somewhat vague, and in certain respects even erroneous. Huy- gens corrected it, and by an accurate application of the principles of catoptrics and dioptrics, which were become better known, he perfectly accounted for all the circumstances attending parhelia. For parase- lenes the theory is the same. I have already spoken m general terms of the uti- lity of the telescope in astronomy, but this is a pro- per place to give a more perfect account of it, and likewise to say something of the microscope, another instrument of the same kind, which has rendered no less service to natural history and physics, than the telescope to astronomy. It is the common opinion, that we owe the first in- vention of the telescope to James Metius; and it is placed at the beginning of the seventeenth century. Such top is the sentiment of des Cartes, who wrote in 302 in Holland about thirty years after the discovery. On this suhject he expresses himself as follows at the beginning of his Dioptrics; and though the passage is somewhat long, perhaps the reader will see it here with pleasure. c The whole conduct of our life depends on our senses, among which that of sight being the most noble, and the most universal in it's application, it is unquestionable, that those inventions, which increase it's power, are of all the most useful. And it is not easy to find one, that shall increase it more than those wonderful telescopes, which, though their date is so recent, have already discovered new stars in the firmament, and other new objects upon Earth, in greater number than those we had seen before : so that extending our view much farther, than the ima- gination of our forefathers had been able to reach, they seem to have opened to us a path, by which we may attain a much greater and more perfect know- ledge of nature, than they possessed. It is about thirty years since James Metius, of the town of Alcmaer in Holland, a man who had never studied, though he had a father and a brother professors of mathematics, but who took particular delight in making mirrors and burning glasses, forming them in winter even of ice, as experience has shown may be done, having on this account glasses of various forms, fortunately thought of looking through two ; one of which was a little thicker at the centre than at the edges, the other on the contrary much thicker at the edges than in the centre ; and he applied them so happily to the two extremities of a tube, that the 8 first 303 first of the telescopes of which we speak was com- posed ; and it is wholly after the pattern of this, that all the others we have since seen were made, &c.' Others relate, that the children of a spectacle maker of Middleburg in Zealand, with whose name we are unacquainted, playing in their father's shop, remarked, that when they put two spectacle glasses one before the other, and looked through them both at the weathercock of a neighbouring steeple, it ap- peared larger than usual. The father, struck with this singularity, thought of adjusting two glasses on a board, by means of brass rings, which might be brought nearer to each other, or farther off, at pleasure. Thus he was enabled to see better, and at a greater distance; and at length proceeded to place the glasses in a tube, and thus formed a telescope. There are still other opinions on the origin of this instrument, which I shall not recite : but I shall observe, that the tes- timony of such a man as des Cartes in favour of James Metius ought to have very great weight. The pretensions of the Italians, who have wished to ascribe the invention of telescopes to Galileo, can- not be supported : for Galileo himself says, that, being at Venice when the first rumour of this disco- very was spread there, he waited for letters from Paris, to assure himself of the wonders, which Fame re- ported ; and that, after having received a confirma- tion of them, he sought for the construction of the instrument in the laws of refraction, and thus dis- covered it. Being in possession of the principle, he by degrees formed a telescope, which magnified the diameter of an object about thirty times ; and with this 304 he discovered the satellites of Jupiter, the spots on the Sun, c. Thus he merely divined the me- chanism of the telescope, from the description that was sent him of it's effects ; and this part of the discovery does him sufficient honour, without en- deavouring to exaggerate it. The telescope of Galileo, otherwise called the dutch telescope, is composed of a convex object glass, and a concave eye glass, or planoconcave glass, placed between the object glass and it's focus, in such a manner, that the axes of the two glasses are in the same line, and their foci coincide in the same point. The rays, which the object glass tends to unite, be- come parallel as they issue from the eye glass, and form at the common focus a perceptible image, which represents the object in it's natural position. The field of telescopes of this sort is very small ; and the longer the tube, the smaller the field. This incon- venience has occasioned their disuse in astronomy, where a certain extent of field is required, and at tlie same time long tubes. They are not now employed therefore, except for short distances. Some years after the invention of this telescope, Kepler proposed another, which was imperceptibly adopted by all astronomers, and is called the astro- nomical telescope. This has a convex object glass, and for it's eye glass a lens convex on one or both sides, placed so that it's focus coincides with that of the object glass, and that this common focus falls betwen the two glasses. It shows objects inverted ; but it has. the advantage of an extensive field and a long tube. There 305 There is a third sort of telescope, the perspective glass commonly employed for terrestrial objects, which is nothing but the preceding, with the addition of two glasses for bringing the object to it's natural erect position. All these telescopes are purely dioptrical, because the simple refraction of light alone is employed in them. There are others more complicated, in which both reflection and refraction are combined, and which for this reason are called catadioptric, or rt- facting telescopes. Such are the gregorian telescope, and the newtonian, descriptions of which may be seen at full length in books on optics. The microscope is an instrument founded on the same principles as the telescope. We know neither the precise time of it's invention, nor the name of the inventor; though it is commonly supposed, that it was Cornelius Drebbel, and that the first ap- peared about 161 8, or 1620. There have been long disputes on this subject, into which I shall not enter. Some writers have greatly depreciated the merit of Drebbel : but the truth is, he received an excellent education at Alcmaer, his native place, and was >vell versed in all the physical knowledge qf his time. There are several sorts of microscopes ; but the most simple of all is a glass convex on both sides, and commonly called a convex len$. By placing it so, that it's focus falls on the part we would inspect, the rays, which are parallel as they issue from the lens^ form a strong image of the object. Sometimes, h> stead of a lens, a small sphere of glass is employed, which is easily formed by melting a bit of glass in the x flame 306 flame of a wick moistened with spirit of wine, to .avoid smoke, which would render the globules opake by mingling with the glass in fusion. A simple microscope may be made likewise by a globe of glass filled \yith water. The second sort of microscope is very similar tq the astronomical telescope. It is composed of two convex lenses : that which forms the object glass is of a very short focal distance, and the object is placed a little beyond the focus, to carry the image farther oft] and magnify it proportionally ; and the eye glass is so ordered, that it's focus falls on the place where this image is formed, that it may be seen distinctly. Sometimes in this kind of microscope an eye glass is placed nearly in the middle between the object glass and the image, in order that this} image may be formed much nearer the object glass, and the tube of the microscope become consequently shorter. By this method too the field of the microscope is en- larged. Finally catadioptric microscopes likewise are constructed. See on this subject la Caille's Lectures on Optics, Smith's Optics, Euler's Dioptrics, &c. Before we quit the subject of Optics, we have to say a few words on perspective, which belongs at least in part to this science. There can be no doubt, as I have remarked, that the ancients were acquainted with linear perspective, and even with aerial But it appears, that the principles of perspective, and the whole of it's various parts, were not begun to be re- duced into a system before the sixteenth century. A great number of authors are quotec^ who have written cm this subject : as, among others, in Italy, Luke de Borgo, 307 Borgo, John Baptist Albert!; in Germany, Albert Durer ; in France, John Cousin, &c. Most of their works do not rise above mediocrity : but we should distinguish from the herd Guido Ubaldi, who was born in 1553, and died in 1617. He published in 3600 a very good treatise on perspective, consonant with the general and certain principles of geometry and optics. END OF THE THIRD PERIOD, JPZRIOB t 308 PERIOD THE FOURTH. PROGRESS OF MATHEMATICS, FROM THE DISCOVERS OF FLJJXIQNS TO THE PRESENT DAY. As the progress made by mathematics in this fourth period is owing in gjreat measure to the method of fluxions^ otherwise called the analysis of infinites^ I shall begin with the history of this modern analysis, and pursue it without interruption down to the present day. After this I shall take up the other branches of mathematics in succession, still following the same plan. The method of fluxions having been developed by degrees, and by the solutipn of different problems, some of which belong to pure geometry, others to mechanics, astronomy, &c., it will be impossible for me to keep these problems separate: but thi wilj cause no confusion, as they have all the same object, the progress of the art by which they were solved. I shall reserve for each branch of the mathematics such problems as belong tq it, when they have not immediately concurred to this end. All the facts I shall relate have been taken by me from the original authorities, namely, from the lite- rary 309 rary journals of the times, the memoirs of academies, tracts published separately, the collections of the works of Leibnitz, Newton^ the Bernoullis, &c. It would take up too, much room to quote on every oc- casion all the writings on which my narrative is founded, and which I have attentively perused ; this therefore I shall do only when it appears to me ne- cessary: but 1 shall take care to point out with accuracy the date of every discovery. CHA*< 510 CHAP. I. The Discovery of the Analysis of Tnfinites : Leibnitz jirst published it's Elements; Newton employed a similar Method in his PRINCIPIA MATHEMATICA, OF all the grand conceptions, that do honour td the human mind, the analysis of infinites is perhaps the most remarkable, whether we consider it simply as an invention, or contemplate the variety and im- portance of it's uses. Almost at it's origin it gave an impulse to geometry, which spread by degrees to the other branches of mathematics, and was accele- rated with great rapidity, as the art rose to per- fection. Problems unknown or unconquerable by the ancient methods submitted without resistance to the new analysis: theories, which appeared isolated and independent of each other, were brought into one point of view by the generality and uniformity of it's means : and a regular and magnificent edifice arose on a solid foundation, which preserves all it's parts in due proportion, and perfect equilibrium. If the two greatest geometricians of antiquity, Archimedes and Apollonius, could now revive, they would be struck with wonder and astonishment, on contemplating the progress, which the accurate sciences have made from, their time to ours, through barbarous ages that so often iutercupted the course of genius. Let 311 Let not the human intellect, however, hence as- urne too lofty an opinion of it's powers, for which it lias no reasonable foundation. If iri this mass of knowledge, accumulated in time, we could separate the product of memory, and determine the sole part absolutely due to the native sagacity of each inventor, we should find a very great number of small portions. Every thing obeys the law of continuity, in the in- tellectual world as well as in the succession of phy- sical beings. From one truth we creep as it were to the next. Geniiis may shorten the train of principles and consequences, but it does not destroy it, and never proceeds by skips. Sometimes an idea, con- fined in appearance to a fixed and determinate space, is gradually enlarged by reflection, and forms the nucleus of a body of science, which scorns all bounds-. We have here a grand example of this. The method of drawing tangents to curve lines by the new-analysis is the cornerstone of the vast edifice of science in it's present state : as a brook trifling at it's source, in- creasing gradually by the waters it receives, becomes at length a majestic river, The ancients drew tangents to the conic sections, and to the other geometrical curves of their in- vention, by particular methods, derived in each case from the individual properties of the curve in question. Archimedes determined in a similar manner the tan- gents of the spiral, a mechanical curve. Among the moderns, des Cartes, Fermat, Roberval, Barrow, Sluze, and others, had invented uniform methods, of more or less simplicity, for drawing tangents to geome- trical curves, which was a great step: but it was x 4 312 previously necessary, that the equations of the curves should be freed from radical quantities, if they contained any ; and this operation sometimes re- quired immense, if not absolutely impracticable calcu- lations. The tangent of the cycloid, a modern me- chanical curve, had been determined only by some artifices founded on it's nature, and from which we could derive no light in other cases. A general me- thod, applicable indifferently to curves of all kinds, geometrical or mechanical, without the necessity of making their radical quantities disappear in any case, remained to be discovered. This sublime discovery, the first step in the method of fluxions, was published by Leibnitz in the Leipsic Transactions for the month of October, 1 684. The evermemorable paper that contained it is entitled : ' A New Method for Maxima and Minima, and like- wise for Tangents, which is affected neither by Frac- tions nor irrational Quantities ; and a peculiar Kind of Calculus for them.' In this we find the method of differencing alMdnds of quantities, rational, frac- tional, or radical, and the application of these calculi to a very complicated case, which' points out the mode for all cases. The author afterward resolves a problem de maxlmh et mimmis, the object of which is to find the path, in which an atom of light must traverse two different mediums, in order to pass from one point to another with most facility. The result of the solution is, that the sines of the angles of incidence and refraction must be to each other in the inverse ratio of the resistances of the two mediums. Lastly lie applies his new calculus to a problem, which 313 which Beaune had formerly proposed to des Cartes, from whom he obtained only an imperfect solution of it. This was to find a curve, the subtangent of which should be every where the same. Leibnitz showed in a couple of lines, the required curve to be such, that, if the abscisses form an arithmetical pro- gression, the ordinates form a geometrical progression : a property in which we recognize the common loga- rithmic curve. In two small tracts on the quadratures of curves, which appeared in 16*85, he published the first ideas of the calculus mmmatorius, or inverse method of fluxions. These are farther developed in another tract, entitled, ' Of recondite Geometry, and the Analysis of Indivisibles, and Infinites,' published the following year. In this Leibnitz gives the funda- mental rule of the integral calculus ; and explains in what the problems of the inverse method of tangents consist, which have since been varied in so many ways. Barrow had laboriously demonstrated, that in everj curve, the sum of the products of the infinitely small intervals of the ordinates multiplied by the subper- pendiculars of the curve is equal to half the square of the extreme ordinate. Leibnitz obtained the same result with the utmost ease by means of the integral calculus : and he observes generally, that all the problems of quadratures, before given by geome- tricians, might be resolved without any difficulty by his method. While Leibnitz was in possession of all these trea- sures, Newton had yet published nothing, from which the world could learn, that he on his part had arrived at 314 at similar results. But toward the end of the year 1686, his Philosophies naturalis Principia mathe-matica issued from the press : a vast and profound work, in which he proposed to explain by observation and cal- culation the principal phenomena of nature, particu- larly the motions of the heavenly bodies, Of this work I shall speak at large, when I come to the subject of physical astronomy. Here I shall content myself with observing, that the key of the most difficult problems resolved in it is the method of fluxions, or analysis of infinites, but exhibited in a form which disguised it, and rendered the author difficult to follow. Accordingly at first it had not all the success it deserved: it was charged with ob- scurity, with demonstrations derived from sources too remote, and an affected use of the synthetic method of the ancients, while analysis would much better have made known the spirit and progress of the in- vention. The extreme conciseness of some parts led to the conclusion, either that Newton, gifted with extraordinary sagacity, had presumed a little too much on the penetration of his readers; or that, through a weakness from which the greatest men arc not always exempt, he had endeavoured to extort that admiration, which the vulgar readily bestow on things above their comprehension. Be this as it may, the great celebrity of the Principia can scarcely be dated farther back than the beginning of the last century, when the analysis of infinites, having made great progress, enabled geometricians to comprehend it. Then it was perceived beyond all question, that theorems and problems involved in. a complex syn- 1 thesis 315 thesis had1>een originally discovered by analysis : but at the same time mathematicians did Newton the justice to acknowledge, that, at the period when his book was published, he was master of the method of fluxions to a high degree, at least with respect to that part which concerns the quadratures of curves. I shall hereafter examine his claim to the invention of this method, in competition with Leibnitz, when I come to the circumstances under which the dispute commenced; at present I shall pursue the progress pf the science. 316 CHAP. II. Leibnitz continues to extend his new Analysis : he is seconded by the two Bernoullis. Various problems proposed and resolved. The Marquis de VHopitaVs Analysis of Infinites. AT the time when Leibnitz was most intent on improving the new analysis, he was a little diverted from it at first by a dispute, which he had with the cartesians on the measure of active forces ; but at length he discovered the secret of rendering the dis- pute conducive to the success of his design. In the Lei psic Transactions for 1686, he had asserted, that des Cartes and his followers were mistaken, in measuring the forces of bodies in motion by the simple product of their mass and velocity ; and that it ought to be measured by the product of the mass multiplied by the square of the velocity* His argu- ment is reducible to the following simple reasoning. By general confession it is agreed, that the same force is required to lift a pound weight to the height of four feet, as to raise four pounds a single foot : but the velocities acquired by a body falling from the hight of four feet, and by the same body falling only one foot, will be in the ratio of two to one ; conse- quently their forces, according to the cartesian prin- ciple, would be as two to four, instead of being equal. The cartesians answered, that regard must be had to the sir the differences of the times of the fall in the two cases. Leibnitz replied, that the consideration of the time ought to be left out of the question : that the force existed in itself, and that it was of little import to know how it was acquired. In a short time the disputants were bewildered in metaphysical subtleties, which displayed their acuteness, but threw no light on the point in question. At length the equality of time, which the cartesians demanded absolutely for the measure and comparison of the moving forces, suggested to Leibnitz the idea of a curious problem ; which he proposed to them as the means at least of rendering the discussion advan- tageous to geometry. This was, to find the iso- chronous curve ; that is, the curve that a heavy body must describe equally to approach or recede from a horizontal plane in equal times. But the cartesians, hitherto very prodigal tf explanations, remarks, and replies, maintained on this occasion a profound si- lence ; and the analysis of their master, which they so much vaunted, furnished them with no means of answering the challenge addressed to them. Huygens, who had taken no part in the question of active forces, deemed this problem worthy his at- tention; and in, l6$7 he made public the properties and construction of the curve, without adding the de- monstrations of jt Thjs curve is the second cubic parabola. Leibnitz, haying waited three years in vain for trje solution of the cartesians, in 1689 gave the same curve as Huygens, and demonstrated, that it answered the problem. At the same time^ to give his 318 his adversaries, as he said, their revenge, he proposed to them to find the paracentric isochronal curve, in which a body would equally approach or recede from a given point in equal times. But this problem was more difficult than the other, and the pretended po- liteness of Leibnitz could be considered only as & banter. This little warfare, and other occupations totally foreign to mathematics, robbed Leibnitz of his time, which he was desirous of dedicating entirely to the pro* gress of the new geometry. Yet, notwithstanding so many interruptions, he was incessantly publishing in, the journals hints tending to this object : and he was soon seconded by two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of. it, that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. The reader is aware I am speaking of the two brothers, James and John Bernoulli. The elder, James, who was born in 1654, and died in 1705, Avas already celebrated for different works in geometry, mechanics, and physics. The younger, who was born in 1667, and survived his brother forty three years, had been initiated into mathematics by him. The progress they made conjointly or sepa- r,ately in the new analysis was rapid. A noble emu- lation, strengthened by the ties of blood, of friend- ship, and of gratitude, directed their studies for two or three years. Covetous of nothing but knowledge, they were stimulated only by the sublime ambition of 319 of penetrating into the labyrinth of science now laid open to their curiosity ; and that unhappy rivalship, which borders on envy, did not yet disturb their en- joyments. On entering this career, in 1690, James Bernoulli gave a solution and analysis of the problem of the common isochronous curve; which he found, with Leibnitz and Huygens, was the second cubic para- jbola. Hence he took occasion to propose to geome- tricians a problem, which Galileo had formerly at- tempted in vain. This was to find the catenary curve, or that formed by a heavy, flexible, and inextensible cord, fastened at it's extremities to two fixed points. This practice of publicly proposing problems, which had long before been introduced among geo- nietricians, and which was brought into great vogue chiefly by Leibnitz and the Bernoullis, was then a powerful mean of sharpening men's faculties, and oc- casioning the sagacity of all to concur in promoting the progress of a nascent geometry. Such was the Affect produced by the problem of the catenary curve. While the solution of this problem was seeking, James Bernoulli published two memoirs, in l6yi, is which he determined, by means of the new analysis, the tangents, quadratures of the areas, and rectifica- tions of three celebrated curves; the parabolic spiral, jtlie logarithmic spiral, and the loxodromic ; to which he acldci'., by way of supplement, the mensuration of the areas of spherical triangles. These two tracts exhibited the first specimens of any length, that had yet been given of the inverse method of fluxions, to the 320 the progress of which indeed they sensibly contri- buted. The author did not confine himself to the simple theory; he pointed out some useful properties of the loxodromic curve. Leibnitz on his part published a work on the arith- metical quadrature of the conic sections which have a centre, in which he introduced some very simple analytical formulas, easily convertible into numbers. He also applied his method to some problems concern- ing the loxodromic line. In the same year the problem of the catenary curve was resolved by Huygens, Leibnitz, and John Ber- noulli. As the two brothers then generally worked together, it is supposed, that the solution of John Bernoulli is the joint performance of both. This pro-blem forms the true epoch, at which the analysis of differential equations begins to assume a fixed and certain character. At first catenaries of uniform gra- vity alone were considered : James Bernoulli ex- tended the solution to cases, in which the weight of the catenary varies from one point to another accord^ ing to a given law. Proceeding from one step to another, and from the analogy of the cases, the same geometrician determined the curve formed by a bent bow, and that of an elastic bar, firmly fixed at one end, and loaded at the other with a given weight. He more particularly bestowed his attention on the curve formed by a flexible sail swelled with the wind, hoping that this research might be of ad- vantage to navigation : and he found, that, sup- posing the wind, after having struck the sail, has nothing to resist it's escape, the curve formed by the sail sail is the common catenarian ; but if the sail/ always considered as perfectly flexible, be filled by a fluid gravitating vertically on itself, as water presses on the sides of a vessel in which it is contained, it would form a curve known by the name of linteaire,' and the nature of which is expressed by the same equation as the common elastic curve, in which the extensions are supposed to be proportional to the forces applied at each point. The identity of the two curves not being easy to recognize, James Bernoulli displayed profound sagacity in this question, as well as in some others of the same kind. While he was busied in his first meditations on the curvature of a sail, he communicated his progress from time to time by letters to his brother, who was then at Basil. It is clear, that these communications led John Bernoulli to the solution he published of the same problem in the Journal des Savans, in 1^92; which equally showed, that the curve of the sail is a catenary. By the manner in which he exhibits the facts, he himself furnishes us with a proof of the assistance he had borrowed : of course we have some reason to be a little surprised, at finding here the first appearances of that jealousy toward his former master, which he afterward too publicly displayed. The theory of curves which produce others by rolling on similar curves was a rich field of disco- very for James Bernoulli. A. D. 1692. He sup- poses, that, any curve being given, and considered as immovable, an equal and similar curve is made to roll upon it : he determines the evolute and the caustic of the epicycloid described by a point of the y moving circle; and he derives from it two other analogous curves, which he calls the antievolute and pcricaustic. All these curves display a number of pro- perties, well worthy of exciting the curiosity of geo- metricians, particularly at a time when they had yet but little experience in the new analysis. On apply- ing his methods to the logarithmic spiral, James Ber- noulli found, that this curve is it's own evolute, caustic, antievolute, and pcricaustic : a singular cha- racter, at which he was so astonished, that he could not avoid declaring with warmth, that, if it were still the custom, as in the time of Archimedes, to place mathematical figures and inscriptions on the tombs of geometricians, he would have desired a logarithmic spiral to be engraved on his tomb with these words : Eadern mutata resurgo. The cycloid possesses properties analogous to those just mentioned of the spiral. James Bernoulli made them known in a supplement to his former memoir ;. and at the same time mentioned, that his brother had separately found the same results. I ought not to omit a work of Leibnitz, on the curves which are formed by an infinite number of straight -or curved lines terlninating in a series of points subject to a given law. This tract, which does not go into the subject very minutely, contains general hints for the solution of 'Several problems, such as those of caustics, of curves cutting a series of other curves at a given angle, &c. Leibnitz sel- dom entered into minutiae : as soon as he found him- self in possession of a method, he gave it up, leaving to 333 to others the pleasure of extending it, and carrying it to perfection* Among this multitude of problems a very curious one was proposed in 1 6'92 by Viviani, a celebrated Italian geometrician, under the following title: JEnigma gcometricum de mlfo Opificio Testud'mis qua- drabills Htmisphcericce. The author feigns, that among the monuments of ancient Greece there still exists a temple of a hemispherical figure, pierced by four equal windows with such art, that the remainder of the dome was capable of being perfectly squared ; and hopes, that the illustrious analysts of the age, so he styles the geometricians skilled in the new calculus, will easily divine this enigma. His hope was not disappointed. The very day on which Leibnitz and James Bernoulli received Vivi- ani 's challenge, they solved the problem : and other geometrical analysts would no doubt have solved it also, had it reached them in time. Viviani was pro- foundly skilled in the ancient geometry; and had particularly distinguished himself by divining or re- storing the five books of conic sections of the an- cient Aristeus, which are lost: but when the geo- metry of infinites appeared, he was too much ad- vanced in years, to study and make himself master of it. He was however a truly modest man, and had no intention to perplex ' the illustrious analysts.' At the same time it must be acknowledged, that his own solution, founded on the synthetical method of the ancients, is eminent for it's simplicity and ele- gance. He demonstrated, that the question might be solved by placing parallel to the base of the Y 2 hem is- hemispherical dome two right cylinders, the axes of which should pass through the centres of two radii constituting a diameter of the circle of the base, and piercing the dome each way. A problem belonging to the method of maxima and minima long employed the two Bernoullis with- out success. This was, to find the day of the shortest twilight for a place of which the latitude is given. This question, treated in the analytical way, leads to an equation of the fourth order, in which it is em- barrassing to separate the useful roots from those which ought to be rejected: but by employing the synthetic method, each of them separately obtained a very simple analogy, and very convenient for astro- nomical computation. The place of mathematical professor in the univer- sity of Basil, which was occupied by James Bernoulli, procured his pupils and the public an excellent trea- tise on the summation of series. The first part had appeared in 1689, the second was published in 1692. Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters ; and the periodical publications became a kind of learned am- phitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de 1'Hopital combated with bloodless weapons ; the honour of France being ably supported by the marauis for several years. The following problem, proposed by John Bernoulli, in 1693, contributed greatly to the progress of tlie methods for summing up differences. To find a curve such that the tangents terminating at the axis shall be 3*25 be in a given ratio with the parts of the axis com- prised between the curve and these tangents. 7 This was resolved by Iluygens, Leibnitz, James Bernoulli, and the marquis de 1'Hopital. On this occasion Iluygens passed on the new me- thods an encomium so much the more honourable, as this great man, having made several sublime disco- veries without them, might have been dispensed from proclaiming their advantages. He confessed, that he beheld * with surprise and admiration the extent and fertility of this art; that, wherever he turned his eyes, it presented new uses to his view ; and that it's progress would be as unbounded as it's speculations/ How unfortunate, that science was bereft of him at an age, when with this new instrument he might still have rendered it so many important services ! Tschirnhausen had some years before made known the celebrated curves designated by the name of caus- tic, which are formed by the concurrence of the rays of light, either reflected or refracted by some other curve. By the aid of common geometry alone Tschirnhausen had discovered several elegant pro- perties of them : as, for instance, that they are equal to right lines, when they are produced by geometrical curves. The geometry of infinites greatly facilitated all these researches, and in 1693 James Bernoulli carried them very far, particularly the theory of caustics by refraction. The copiousness of the subject, and the limits of this essay, compel me to pass over in silence several other papers, which James Bernoulli gave to the public in the same year, on the different subjects of y 3 geometry, 326 geometry, mechanics, hydraulics, &c. I equally omit the reflections of Leibnitz on the mode of re^ solving problems of quadratures by the construction of certain curves, which he describes by motions sub- jected to given laws. The description of the tractrix is an example of these motions : and it was in fact on occasion of this curve, the nature of which Claude Perrault had inquired of him, that Leibnitz made those remarks, in which his usual subtilty is ob- servable. The same thing may be said of a new applica- tion, which Leibnitz made of his differential calculus, for the construction of curves from a condition of the tangents. About the same time also other geome^ tricians published various works, or solutions of pro- blems, which it would be tedious to enumerate. Geometricians seemed .to have forgotten the pro- blem of the paracentric isochronal curve, which Leibnitz had proposed in 1 689, and the solution of which he still kept secret. The cause of this appa-^ rent forgetfulness was no doubt the difficulty of sepa- rating the indeterminates from the equation found ^vhen the curve is referred to perpendicular coordi- nates. In 1694 James Bernoulli surmounted this difficulty, by taking for ordinates parallel right lines, and for abscisses the chords of an infinite number of circles, all of Avhich have for their centre the given point. Thus he obtained a separate equation, which he constructed at first by the rectification of the elastic curve, and afterward by that of an algebraic curve. Soon after John Bernoulli likewise resolved the same problem. He gave a complete and minute analysis of it, on which I should bestow much praise, -2 327 if he had not saved me the trouble, and at the same time had refrained from unjustly criticising the con- structions of his brother, to which even this analysis may be referred in substance. At the same time Leibnitz published his own solution, which does not essentially differ from that of the Bernoullis, but which is accompanied with reflections conducive to the progress of geometry. In the Commercium epistoticwn of Leibnitz and John Bernoulli, which was not published till 174-5, we learn, that in the year 1 0,4 each of them had separately discovered that branch of the new analysis, which is called the exponential calculus. Leibnitz has the priority in point of date: but John Bernoulli made the discovery himself; and in 1697 he pub- lished the rules and use of this calculus, whence he is commonly taken for the original 1 and even sole in- ventor of it. In this same Epistolary Correspondence we find an important remark by Leibnitz in the year 1695, on the analogy that subsists between the powers of a polynomial composed of variable terms, and the fluxions (of the same order) of the product of these terms. From this John Bernoulli deduced a method for summing up the differential formulae of every order, in certain cases. Among the most curious problems of this time must be reckoned that of the curve of equilibration in draw-bridges, solved by the marquis de 1'Hopital in 1695. It chiefly merited the attention of geometri- cians on account of the practical utility expected from it. John Bernoulli observed, that the required y 4 curve. curve, of which the marquis de PHopital had given the general equation, was an epicycloid, or that it could he generated by a point fixed in a circle re- volving round another circle. Leibnitz and James Bernoulli likewise gave solutions of this problem. About the same period we find an excellent tract by James Bernoulli concerning the elastic curve, iso- chronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already ; but here he has given them with additions, corrections, and improvements. His scientific dis- cussions are interspersed with some historical circum- stances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother ; and exhorts him to moderate his pre- tensions ; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy ; and to acknowledge, that, * as quantities in geometry increase by degrees, so every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived. This memoir concluded with an invitation to ma^ thematicians, to sum up a very general differential equation, of great use in analysis. The solution which James Bernoulli had found of this problem, as well as those which Leibnitz and John Bernoulli gave of it, were published in the Leipsic Transactions. In 1696 a great number of works appeared, which gave a new turn to the analysis of iuimjtes. Such were 329 were the Memoir on the quadratures of spheroidal surfaces by James Bernoulli, in which we find pro- blems analogous to those of Viviani, but more ge- neral, and more complicated; several elegant theo- rems on the same quadratures by John Bernoulli; the third part of James Bernoulli's Treatise on Series; and above all the celebrated work of the marquis de FHopital, entitled : * The Analysis of Infinites, for the understanding of curve Lines/ to which I shall spare a few moments. Such a work had long been a desideratum. ' Hi- therto/ says Fontenelle, in his eulogy on the mar- quis-, ( the new geometry had been only a kind of mystery, a cabbalistic science, confined to five or six persons. Frequently solutions were given in the public journals, while the method, by which they had been obtained, was concealed : and even when it was exhibited, it was but a faint gleam of, the science breaking out from those clouds, which quickly closed upon it again. The public, or, to speak more pro- perly, the small number of those who aspired to the higher geometry, were struck with useless admiration, by which they were not enlightened i and means were found to obtain their applause, while the infor- mation, with which it should have been repaid, was withheld.' The work of the marquis de 1'Hopital, completely unveiling the science of the differential calculus, was received with universal encomiums, and still retains it's place among the classical works on the subject. But the time was not yet arrived for treating in the saine manner the inverse method of fluxions, which is 330 Is immense in it's detail, and which, notwithstanding the great progress it has made, is still far from being entirely completed. Leibnitz promised a work, which, under the title of Scientia Infinity was to comprise both the direct and inverse methods of fluxions : but this, which would have beeu of great utility at that time, never appeared. CHAP. 331 CHAP. Til. Extraordinary progress in the theory of maxima and minima. Dispute between the two Bernoulli^ on the problem of isoperimetrical figures. THE sole object of all the problems of maxima and minima, that had been resolved previous to the time at which we are arrived, A. D. 1696', had been, to find in the number of explicit functions, which con- tain but one variable quantity, or are reducible to one, those which, among their similars, could be- come maxima or minima. Des Cartes, Fermat, Sluze, Hudde, and others, had contrived particular methods for these problems : but that of fluxions had supplanted them all by it's simplicity and generality. There remained another class of problems of the same kind, but far more complicated and profound, where both the direct and inverse methods of fluxions were necessary. This consisted in finding, among the implicit or affected functions of summatory signs, those which give maxima or minima : as for instance the curve which includes the greatest area according to given conditions, or which produces by it's revo- lution the greatest solid within given limits, &c. Newton, Prin. lib. n, prop. 34, after having deter- mined, among all the truncated right cones, of the same base and the same altitude, that which, being moved in a fluid by the smallest (unknown) base, in the the direction of it's axis, experiences the least resist- ance possible (which was a problem of the ancient kind), had given, without any demonstration, a ratio, from which might be deduced the differential equation of the curve, that produces, by revolving on it's axis, the solid of least resistance : a problem re- lating to the second kind. The principle of this solution, of which Newton as usual made a mystery, is, that, when a property of a maximum or a minimum pertains to a curve, or to a finite portion of a curve, it likewise pertains to a portion infinitely small. It has some analogy with means frequently employed in geometry : as, for in- stance, when we demonstrate the equality of a sphe- rical zone with the corresponding area of the circum- scribed cylinder, by the reciprocal equality of their elements. But even had Newton formally announced this principle, the general problem would still have had it's particular difficulty in each individual case, either in finding the differential equation of the curve, or in resolving it. The sciences, therefore, have an obligation of the highest importance to John Ber- noulli, for having drawn the attention of geometri- cians to this general theory, by proposing to them in 1697 the celebrated problem of the brachystochronon, or * that curve, along .the concave side of which if a heavy body descend, it will pass in the least time possible from one point to another, the two points not being in the same vertical line.' It is certain, that, -at the period in question, this problem was more difficult than that of the solid of least resist- ance; the solution of which Newton had even left in* 333 incomplete, since he had not resolved the differential equation of the generating circle. At first view it would be imagined, that a right line, as it is the shortest path from one point to the other, must likewise be the line of swiftest descent: but the attentive geometrician will not hastily assert this, when he considers, that in a concave curve, de- scribed from one point to the other, the moving body descends at first in a direction more approach- ing to the perpendicular, and consequently acquires a greater velocity, than down an inclined plane; which greater velocity is to be set against the length of the: path, and may cause the body to arrive at the end sooner by the curve, than by the straight line, Metaphysics alone, therefore, cannot solve the ques- tion, which must be examined by the most accurate calculation. Now the result of this calculation shows, that the line sought is in reality a curve, which is, in fact, an arc of a cycloid reversed : a new and very remarkable property of this curve, which the researches of Huygens and Pascal had already rendered so celebrated. Leibnitz resolved the problem on the day on which he received it from John Bernoulli, and immediately informed him of it. They both agreed, to keep their solutions secret, and to give other geometricians a year to exercise their ingenuity on this curious question. This delay was announced in the peri- odical publications, and in a. circular paper, which John Bernoulli sent to all quarters. The year had not expired, when three other solu- tions appeared, the authors of which were Newton, 4 the 334 the marquis de 1'Hopital, and James Bernoulli. That of Newton appeared without a name in the Philo- sophical Transactions of the Royal Society at London, but John Bernoulli guessed the author ; tanquam, said he, ex uHgue konem. The marquis de 1' Hopital had great difficulty in finding his solution : yet he might have obtained it readily enough from a principle, which he employed himself when seeking the path a traveller ought to pursue, in order to go from one place to another in the shortest time possible, if he have to cross two fields, in which he experiences obstacles to his pro- gress, that occasion his pace to vary in a given ratio. For, if .the two fields be considered as the two ele- ments of a curve, situated in a vertical plane ; and if we suppose, agreeably to the laws of the descent of heavy bodies, that the velocities of a body along any curve are as the square roots of the heights from which the body has descended, we shall immediately obtain the differential equation of a cycloid. But no person at that time made the re- mark, or associated together ideas, which now appear to us so closely connected. Lastly, before the expiration of the time prescribed by his brother, James Bernoulli gave a solution, in which he demonstrated, that the curve sought is an arc of the cycloid. In the course of his investigation he had ascended to problems on isoperimetricalfi-gures, requiring still more profound speculations; and after he had resolved these, he proposed them to the public at the conclusion of his method for the curve of swiftest descent 8 All 333 All these solutions appeared at the same time, and without any possibility of one of the authors having derived any information from the others. The rivalry in glory, which had long divided the Bernoullis, was fully displayed on this occasion. At first it was a little moderated by their habits of seeing each other, at least occasionally, and by the inter- vention of some common friends ; but the younger brother having been appointed professor of mathe- matics at Groningen in 16^5, all private intercourse between them soon ceased, and they no longer cor- responded, except through the medium of perio- dical publications for the purpose of proposing to each other the most difficult problems. John Ber- noulli was the aggressor : but perhaps his brother displayed a little too much haughtiness in the first answer he made him, a sketch of which I have given. Their minds being exasperated, John Bernoulli fre- quently returned to the charge, and his quondam master was not a man to endure for a long time at- tacks unjust in themselves, independent of the gra- titude, by which they ought to have been restrained. In this frame of mind James Bernoulli, desirous at length of avenging himself jn a signal manner, which should at the same time be beneficial to geo- metry, challenged his brother by name, to resolve the following problem. ' To find, among all the isope- rimetrical curves between given limits, such a curve, that, constructing a second curve, the ordinates of which shall be functions of the ordinates or arcs of the former, the area of the second curve shall be a maximum or a minimum. ' To this leading problem he added 336 added another, more analogous to that of the line of swiftest descent. This was : ' To find among all the cycloids, which a heavy body may describe in it's de- scent from a point to a line the position of which is given, that cycloid which is described in the least possible time.' He concluded his challenge nearly in the following words : 'A person, for whom I pledge myself (Prodit NON NEMO, pro quo caveo), engages to give my brother, independently of the praise he will deserve, a prize of fifty florins, on condition, that within three months he engages to resolve these pro- blems, and within a year publishes legitimate solu- tions of them. Adding, if, at the expiration of this time, no one shall have resolved them, I will make public my own solutions.' As soon as John Bernoulli had received the dif- ferent papers, which contained solutions of his pro- blem of the brachystochronon, he thought himself at liberty to give his opinion of them, and this he failed not to do. He bestowed great praise on Leibnitz, the marquis cle 1'Hopital, and Newton. He ac- knowledged too, that his brother had very well re- solved it ; but he charged him with having spent too much time about it. No doubt lie forgot, that, in this same space of four or five months, James Bernoulli had in addition conceived the general theory, and executed the calculations, of the grand problem of isoperimetrical figures, which he proposed, and the solution of which he had ready to appear. After John had thus noticed the answers to his own problem, he proceeded to the new ones proposed to himself: and imagining, that his theory of the line of swiftest descent was alonq sufficient to solve them> 337 them, the following expressions of ingenuous vanity escaped him. c Difficult as these problems appear, I did not fail to apply to them, the instant they came to my hands, and hear with what success : in- stead of three months allowed me to sound their depth, and the remainder of the year to find their so- lution, I have employed only three minutes, to ex- amine, enter upon, and dive to the bottom of this mystery.' These fine promises were accompanied with the constructions he gave of the problems, and the consequent demand of the prize, which he said he should give to the poor, as.it cost him so little trouble to gain it. But the business was by no means so far advanced as he supposed: and unquestionably he would have spared his boast, had he foreseen, that it would have drawn on him vexations so much the more cutting, as to superiour geometrical talents *he united the frankness, or imprudence, of too publicly showing the high opinion he entertained of himself. His construction of the problem of the cycloid of swiftest descent was accurate. We see likewise, that he had fortuitously hit upon the true solution, or rather the true result, of one case of the isoperi- ineters ; but his method did not extend to the general problem. James Bernoulli, perfectly sure of his own, and finding, that the two methods did not give the same equation, when the ordinates of the second curve are functions of the arcs of the first, printed an advertisement, A. D. 1698, in which he asserted, that his brother's method was defective. He still allowed geometricians time to find the solution, and, if no one gave it, he pledged himself for three things : 1st, z to 338 to divine with precision the analysis of his brother : 2clly, whatever it might be, to point out fallacies in it : 3dly, to give the true solution of the problem in all it's parts. To this he added the following wager; that, if any person were sufficiently interested in the progress of science, to venture a prize on each of these articles, he would engage to forfeit an equal sum, if he failed in the first; double the sum, if he did not succeed in the second ; and triple the sum, if he did not accomplish the third. The singularity of this advertisement, and the weight of the writer as a geometrician, a little stag- gered John Bernoulli's confidence in his method. He revised his solution ; acknowledged, that he had made a trifling mistake, which he ascribed to too great precipitancy; sent a new result, but without as- suming a more modest tone ; and again demanded the prize proposed. To these pretensions James Bernoulli laconically answered : ' I beg my brother, to revise his last so- lution anew, to examine it carefully in every point, and then to let us know, whether it be all right ; as I must assure him, that no attention will be paid to his excuses of precipitancy, after I have published mine.' John Bernoulli, who was far from suspecting at that time the radical defect of his method, replied, that he had no occasion to revise his second solution, that it was sound, and that his time would be better employed in making new discoveries. At the time when James Bernoulli published his first Advertisement, he wrote a letter to Varignon on 1 the 339 the subject, which Varignoil ought to have inserted immediately in the Journal des Savans. I know not for what reason it's appearance was deferred, but it was not made public till four months after John Bernoulli's second solution : the editor merely an- nounced, that this second solution had not induced the author of the letter to change his opinion. The object of this was, to answer the first and second o the conditions, which James Bernoulli had imposed on himself; which were, to divine the method of his brother, and to show in what it was erroneous. He pointed out an analysis defective in itself, but which, errours in it being balanced by other errours, led in cer- tain cases to a true result : and by means of this ana- lysis he arrived at the equations of his brother, whence he concluded, that from this in all appearance they were deduced. To this letter James Bernoulli added an advertise- ment, just drawn up on occasion of his brother's se- cond solution ; in which the triumphant tone of John Bernoulli in announcing his solutions, his re- fusal to revise the last, and the pretence for this re- fusal, are ridiculed with a degree of wit and vivacity, which could scarcely be expected from a geome- trician; and at which we have the more reason to be surprised, as the author, though it is written in french, was born and resided in Switzerland. ' I never believed,' says he, ' that my brother was masted of the true solution of the isoperimetrical problem, * 1 doubt it now more than ever, from the diffi- culty he makes of revising his solutions. If it cost hint " but three minutes," as he asserts, " to examine, z enter 340 enter upon, and dive to the bottom of the whole mystery," surely the revisal would not require more. But suppose he spent double this time about it, of how many new discoveries would he be robbed by the six minutes thus employed?' When John Bernoulli received the journal in which these pieces were inserted ; he fell into a passion not to be conceived, which vented itself in a torrent of coarse and dull abuse. The journalists were so com- plaisant to him as to publish it : but we shall pass it over in consideration of his genius in the sciences. There were no other means of terminating the dis- pute, but by both of them publishing their methods, and submitting them to the judgment of the most able geometricians in Europe. John Bernoulli demanded Leibnitz for the arbiter, to whom he had sent his so- lutions, and who, no doubt having examined them without sufficient attention, had given them his ap- probation. James consented to have not only Leib- nitz for their judge, but with him Newton, the marquis de ITIopital, and all the other celebrated geometricians of the time, provided he was allowed liberty to speak, and to place the truth in it's proper light. Thus the business rested for about two years. In 1700 James Bernoulli printed at Basil a letter addressed to his brother, in which he invites him with great mo- deration, though .somrthing of a tone of conscious superiority is perceptible in it, to publish his method: and he concludes by giving the formulae of the pro- blem, without any demonstrations. These formula* were immediately inserted in the Leipsic Transac- tions, 341 tions*. John Bernoulli then perceived how far he differed from his brother as to their results : but not discovering the principle of the true solution, and still persuaded, that his own method was accurate, he gave it at large in a paper, which was sent under a seal to the Academy of Sciences at Paris, in the month of February 1701, on condition, that it should not be opened without his consent, and after his brother had published his analysis. As soon as James Bernoulli was informed of this, he had no longer any reason to keep his method se- cret. Accordingly he made it public, and maintained it by way of a thesis at Basil, in inarch 1701, with a dedication to the four illustrious geometricians, FHo- pital, Leibnitz, Newton, and Fatio de Duillicr. He likewise printed it separately at Basil, and in the Leipsic transactions for may 1701, under the follow- ing title : Analysis magni Probkmatis isoperimetrici. It was considered as a prodigy of sagacity and in- vention : and indeed, if the time be considered, it will not be too much to assert, that a more difficult problem never was resolved. The marquis de 1'Hppital wrote to Leibnitz, that he had read it with avidity, and "that he had found it very direct, and very ac- curate. This testimony Leibnitz transmitted to John Bernoulli himself, though he was much prejudiced in his favour. Com. Epist, f. II, p. 48. * The journalists suppressed the preceding part of the letter ; and through the influence of John Bernoulli it was likewise excluded from the edition of his brother's works published in 1744. I have had it reprinted in the Journal de Physique for September 1792. 342 After so much bustle it was to be expected, that John Bernoulli would either criticise the solutions of his brother, or that he would publicly avow their ac- curacy. But from this time he maintained a pro- found silence : no observations, no criticisms, ap- peared on his part ; and instead of setting his own method in opposition to that of his rival, he allowed it to rest quietly for five years in the repository of the academy. At length, in 1705, James Bernoulli died; and, soon after, this method appeared in the Memoirs of the Academy for 1 706. What must we think of this strange conduct? Can we suppose, contrary to every appearance, that a man of so ardent and impetuous a temper as John Ber- noulli was desirous of dropping a dispute, of which he had grown weary ? Is it not much more probable, that, suspecting some defect in his method, he was afraid to submit it to the judgment of his brother ; and that, this brother being dead, the shame of ap- pearing vanquished in the eyes of all Europe induced him to publish this memoir, sent in 1701; in the hope that no person would enter so deeply into the question, as to decide between the two methods, and that, with some of the learned world he should at least obtain the credit of having likewise resolved the problem ? This conjecture is strengthened by the re- flection, that Fontenelle, in his eulogy of James Ber- noulli, and forty-three years afterward Fouchi, in that of John, have both spoken of their solutions as equally accurate, and equally general. The most profound geometricians passed a very Different judgment, and the palm of victory was de- creed 343 creed to the methods of James Bernoulli. In spite of all the shifts of John, and all the specious means he employed for giving his method an appearance of truth, it was really defective, as his brother had con- stantly maintained. The radical errour of it was> that John Bernoulli considered only two elements of the curve, instead of which it is requisite to consider three, or employ an equivalent condition. In pro- blems of the same kind as that of the line of swiftest descent, where it is simply required to fulfil the con- dition of the maximum or minimum, the applying of this condition to two elements is sufficient, to find the differential equation of the curve. But when, beside the maximum or minimum, the curve must possess a farther property, that of being isoperiU metrical to another, this new condition requires that a third element of the curve shall have a certain in- clination with respect to the other two; and every determination founded simply on the first consider- ation will give false results ; except in cases, where a curve cannot satisfy one of the two conditions, without at the same time fulfilling the other. In vain John Bernoulli imagined, that he had fulfilled the condition of it's being isoperimetrical, without derogating from the maximum and minimum, by considering two elements of the curve as two small right lines, drawn to an intermediate point between the two foci of an infinitely small ellipsis : this sup- position did not introduce a new condition into the calculus ; it had no other effect than that of render- ing the differential of the absciss constant or variable. James Bernoulli had explicitly employed three ele- z 4 ments 344* merits of the curve, and by this he had obtained pre- cise, general, and complete solutions. This consideration of the three elements was then so essential, that at length John Bernoulli made it the basis of a new solution, more than thirteen years after his brother's death, confessing himself deceived in his first Mem. de VAcad. 1718. This was a tardy avowal, but at least it would have done him honour, had he at the same time acknowledged, that his new solution was in substance the same as his brother's, given in a form which considerably abridged the cal- culation ; and had he not sought with some degree of affectation, to point out certain superfluities in that of James; which, however, though useless, were no way detrimental to it's accuracy or generality. I have thought it incumbent on me to give a con^ nected account of the dispute between these two "brothers on the subject of isoperimetrical figures : and before I take leave of it, I cannot avoid expressing my astonishment, that ho other geometrician of that time, at least publicly, undertook to solve these problems ; for, though James Bernoulli challenged his brother in particular, every person was at liberty to enter the lists, and the question proposed united every circumstance capable of exciting emulation povelty of the subject, great difficulties to be sur- mounted, and an addition to the treasures of geometry. CHAP, 345 CHAP. IV. Solutions of various problems. Leibnitz invents thz method of differencing cle curva in curvam. Justi- fication of the marquis de rHopitaL Newton's works. Account of some other geometricians. THE dispute, of which I have just given an account, lias led me to anticipate a little on the order of time, and to pass by several interesting and remarkable problems, to which I shall now return. When James Bernoulli proposed the problem cf isoperimetrical figures in 16*97, he added to it that of the cycloid of swiftest descent to a line of a given position, in order to complete in some degree the theory of the brachistochronon. He demonstrated, that the cycloid sought is that which cuts the given line at right angles : and he taught generally how to find among similar curves, terminating at a line the position of which is given, that which possesses some property of a maximum or a minimum. , John Bernoulli had arrived at similar results by a method a little indirect, but very ingenious, which ga^e birth to a signal enlargement of the infinitesimal analysis. In this research he employed tlie consi- deration of the synchronous curve, or that which cuts a series of similar curves placed in similar positions, so that the arcs of the latter, included between a given point and the synchronous curve, shall 346 shall be passed through in equal times by a heavy body. He demonstrated, that among all the cycloids thus intersected, that which is cut perpendicularly is passed through in less time than any other termi- nating equally at the synchronous curve. The only question therefore was, how to draw a tangent to the synchronous curve of the cycloids in a given direc- tion : and to solve the problem generally, it was re- quisite, that the solution should depend, not on the properties of the cycloid alone, but on principles ap- plicable to every other series of similar curves simi- larly placed. John Bernoulli determined by a geo- metrical construction the synchronous curve corre- sponding to the cycloid of the shortest time : but he was unable to discover the analytical expression of the subtangent of synchronal curves for all kinds of similar curves. Having long sought the solution of this problem in vain, he proposed it to Leibnitz, who resolved it very readily, and on this occasion invented the celebrated method of differencing de curca in curvam. On the receipt of the letter which contained this method, John Bernoulli was transported with joy and admiration, and complained in a friendly manner, that * the god of geometry had admitted Leibnitz farther than him into his sanctuary.' This first sentiment was just: but we find with regret, that, after the death of Leibnitz, he endeavoured to make himself considered as the eoinventor of this method, though in fact he could claim no merit further than that of having made some very ingenious applications of it, as may be seen in the second volume of his works. Leibnitz 347 Leibnitz never published it bimself ; and it did not appear under his name till 1745, when his Corre- spondence with John Bernoulli was published. We find by the posthumous works of James Ber- noulli, that he had likewise discovered a similar me- thod, and employed it for the solution of the pro- blems which his brother had proposed to him during the course of their dispute on isoperimetrical figures : but he had contented himself with indicating his solutions by means of anagrams, wishing to avoid every thing that could lead to a diversion from the business of the isoperimetrical problem, till this was brought to a conclusion. These incidental problems related to the method of maxima and minima. I shall mention but one, which will be sufficient to give a general idea of them all. John Bernoulli demanded, * which of all the semiellipses, that'can be described in the same vertical plane, and on the same hori- zontal axis, would be passed through in the least pos- sible time by a heavy body, the motion of which should commence from one of the extremities of the given axis.' An innumerable multitude of curious and difficult researches beside these occupied the attention of geo- metricians at this time. A. D. 1699, 1700, 1701, c. These were the quadrature of certain cycloidal spaces; the indefinite section of circular arcs ; the curve of equal pressure ; the transformation of curves into others of the same length ; new methods of approxi- mation for the quadratures and rectifications of curves j how to find certain curves from the given ratios of their branches ; &c. Leibnitz, the two Bernoullis, 348 Bernoullis, the marquis de I'Hopital, &c., appeared continually in the lists. These researches were not all of equal use, but every one contributed more or less to the progress of geometry. I should never have finished, were I to attempt to enter into them at any length : but I shall notice a work of John Ber- noulli, because it attacks the memory of an illustrious frenchman, whom it is my duty to defend, as far as I am able. The marquis de I'Hopital had given in his work on the analysis of Infinites a very ingenious rule, for finding the value of a fraction, the numerator and denominator of which should vanish at the same time, when a certain determinate value is given to the variable quantity that enters into it. No person thought proper to dispute his title to this while he lived ; but about a month after his death, John Ber- noulli, remarking that this rule was incomplete, made a necessary addition to it, and thence took oc- casion to declare himself it's author. Several of the marquis de I'HopitaTs friends complained loudly and with warmth of a claim, which ought to have been made sooner, if it had not been without foundation. Instead of retracting his assertion, John Bernoulli went much farther; and by degrees he claimed as his own every thing of most importance in the Ana- lysis of Infinites. The reader will indulge me in a brief examination of his pretensions. In 169% John Bernoulli came to Paris. He was received with great distinction by the marquis de I'Hopital, who soon after carried him to his country seat at Ourques in Touraine, where they spent four months 349 months in studying together the new geometry. Every attention, and every substantial mark of ac- knowledgment, were lavished on the learned foreigner. Soon after, the marquis cle 1'Hopital found himself enabled, by persevering and excessive labour which totally ruined his health, to solve the grand problems, that were proposed to each other by the geometri- cians of the time. From the year 1693 he made one in the lists of mathematical science, in which he dis- tinguished himself till his death. At this period he was ranked among the first geometricians of Europe ; and it is particularly to be observed, that John Ber- noulli was one of his most zealous panegyrists. Per- haps he was exalted too high during his lifetime : but the accusation brought against him by John Ber- noulli after his death forms too weighty a counter- poise, and justice ought to restore the true balance. Now I will boldly ask, is it probable, that a geo- metrician, who had given so many proofs of profound knowledge before his publication of the Analysis of Infinites, who had resblved, for example, the curve of equilibration in drawbridges, was a mere editor of all the difficult parts of that work ? Can we suppose him possessed of so little delicacy, as to ask or ac- cept such humiliating assistance ? Do we not know on the contrary, that he had great loftiness of mind? The extracts of letters, which John Bernoulli has brought forward, are far from proving what he has asserted. Act. Leips. 172,1. It is true we find from them, that John Bernoulli had composed lessons in geometry for the marquis cle FHopital, but by no means that these lessons v were the Analysis of Infi- nites : 350 nites : the pupil, a man of genius, had become mas- ter of his art, and soared on his own wings. We sec too in these extracts, that the marquis, while at work on his book, solicited from John Bernoulli,- with the confidence of friendship, explanations relative to cer- tain questions, which are treated in it: but we have not the answers of John Bernoulli, and we know not whether these explanations were furnished by him, or found by the marquis himself after farther reflecting on them. Amid all these uncertainties, it is most equitable and prudent, to adhere to the general declaration made by the marquis in his preface, that he was greatly indebted to John Bernoulli \aux lumtires de J. -B.] ; and to presume, that if he had any obliga- tions to him of a particular nature, he would not have ventured to mask them in the expressions of vague and general acknowledgment If, notwith- standing all these reasons, any one should think pro- per to credit John Bernoulli on his bare word, when he gives himself out for the author of the Analysis of infinites, the code of morality at least will never absolve him, for having disturbed the ashes of a ge- nerous benefactor, in order to gratify a paltry love of self, so much the less excusable, as he possessed suf- cient scientific wealth besides. This example, how- ever, may afford a striking lesson to those ambitious men, who are desirous of posting too hastily to the goal of reputation : it, warns them, to reject offici- ous services, more frequently the offers of vanity than of kindness ; and to bear this strongly in mind, that true and solid glory is never to be obtained but by our own exertions. After After the appearance of the Principia, no discovery of importance in geometry had been published by the english, if we except the solution of the pro- blem of the line of swiftest descent. Toward the end of the year 1704, Newton gave to the World in one volume his Optics in english, an enumeration of lines of the third order, and a treatise on the quadratures of curves, both in latin. His optics are foreign to the purpose here. The enumeration of lines of the third order is a profound and original work, though it rests simply on the common analysis, and the theory of series, which Newton had carried to a great length. It contains little more than enunciations and results, and has since been commented upon by se- veral learned geometricians, to whom it has afforded an ample harvest of very curious researches. His other work, the treatise on quadratures, belongs to the new geometry. The particular object of this treatise is the reso- lution of differential formuke of the first order, or of a single variable quantity ; on which depends the pre- cise, or at least the approximate, quadrature of curves. With great address Newton forms series, by means of which he refers the resolution of certain compli- cated formula? to those of more simple ones; and these series, suffering an interruption in certain cases, then give the fluents in finite terms. The develop- ment of this theory affords a long chain of Very elegant propositions, where among other curious pro- blems we remark the method of resolving rational fractions, which was at that time difficult, particu- larly when the roots are equal. Such an important and 352 , :; ;, and happy beginning makes us regret, that the author has given only the first principles of the analysis of differential equations. It is true he teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus : but he, does not inform us, how to solve the inverse problem ; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem. Newton's opponents have argued from his treatise on quadratures, that, when this work appeared, the au- thor was perfectly acquainted only with that branch of the inverse method of fluxions which relates to quadratures, and not with the- resolution of differential equations. Newton almost entirely melted down the treatise of Quadratures info, another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinites, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several 4 times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher o geometry. In 1730, nine years after Newton's death, Dr. Pembcrton gave it to the World in engiish. In 1740 it was translated into french, and a preface Was prefixed to it, in which the merits of Leibnitz are depreciated so excessively, and in such a decided tone as might impose on some readers, if the writer of this preface [Buffon] had not sufficiently blunted his own criticisms, by betraying how little knowledge of the subject he possessed. Notwithstanding his; public and frequently repeated efforts, he was' never able to penetrate to any depth in the higher geo- metry ; and the anecdote of the strange meaning he had affixed to the latin words de testitudine quadra- bili of Viviani, oii which he had written a little dis- sertation, that was fortunately left out of that pre- face by the advice of one of his friends, is still re- - inembered. To posterity he is already known only by his Natural History, in which the philosopher, while he condemns some wanderings of the imagina- tion, cannot avoid admiring several grand and just ideas, as well as the loftiness and elegance of the diction. In 1711 another work of Newton's appeared, 1m Method of Fluxions, the foundation of which he had already laid in his Principia under a different form. The object of this Method is. to find the linear coefficients of an equation that satisfies as many conditions as there are coefficients, or to con- A a struct 3,54 struct a curve of the parabolic kind passing through aay number of given points. From this arises an easy and commodious method of finding by approxir mation the quadratures of curves, of which a certain number of ordinates can be determined. In this work, however, Newton has employed only common simple algebra; and it is by mistake, that some of his admirers, actuated by a little too much zeal, have imagined they find in it the first elements of the in- tegral calculus with finite differences, so celebrated in our days. In the beginning of the last century considerable progress in the new geometry was made in Italy. Tiiifi was principally owing to the work, which Ga- briel Manfredi published in 170? under the following title : De Constructions JEquationum differentia-limn primi Gradus. The author displays much address in subjecting certain differential equations to conditions whidi render them integral. In his method of sepa- rating the indetermmate quantities in homogeneous differential equations of the first order he has shown a similarity of genius and doctrine with John Ber- noulli. The loss Germany had sustained in geometry by the death of James Bernoulli was repaired in some measure by the scholars of that celebrated man, as James Hermann, his countryman, Nicholas Ber- noulli, his nephew, and others, whom it would be too tedious to enumerate. Hermann first made himself known by a method of- finding the oscillatory radii in polar curves ; and - bition, we need not be surprised at the warmth, with which Leibnitz and Newton disputed the discovery of the new geometry. These two illustrious rivals^., or rather Germany and England, contended in some respects for the empire of .science. The first spark of this war was excited by Nicholas Facio de Duillier, a genevese retired to England; tbe same who afterward exhibited a strange instance of madness, by attempting publicly to resuscitate a dead body in St. Paul's church, but who was at that time in his sound senses, and enjoyed some reputation among geometricians. Urged on the one hand by the english, and on the other by personal resentment against Leibnitz, from whom he professed not to have received the marks of esteem he conceived to be liis due, he thought proper to say, in a little tract ' on the curve of swiftest descent, and the solid of least resistance/ which appeared in 1(>99> that Newton was the first inventor of the new calculus; that he said this for the sake of truth and his own consci- ence; and that he left to others the task of deter- A a 4 mining 36*0 joining what Leibnitz, the second inventor, had bor- fowed from the english geometrician. Leibnitz, justly feeling himself hurt by this priority of invention ascribed to Newton, and the conse- quence maliciously insinuated, answered with great moderation, that Facio no doubt spoke solely on his pwn authority; that he could not believe it was with Newton's approbation ; that he would not enter into, any dispute with that celebrated man, for whom he bad the profoundest veneration, as he had shown on all occasions ; that, when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia, that neither had borrowed anything from the other; that, when he published his differential calculus in 1684, he had been piaster of it about eight years ; that about the same time, it was true, Newton had informed him, but without any explanation, of his knowing how to draw tangents by a general method, which was not impeded by irrational quantities ; but that he could not judge whether this method were the differential calculus, since Huygens, who at that time was unacquainted with this calculus, equally affirmed himself to be in possession of a method, which had the same ad- vantages ; that the first work of an english writer, in which the different ial calculus was explained in a positive manner, was the preface to Wallis ? s Algebra, not published till 1693; that, relying on all these circumstances, he appealed entirely to the testimony and candour of Newton, &c. The assertion of Facio, being altogether destitute of proof, was forgotten for several years, In 1 708, Keii. 361 JCeil, perhaps excited by Newton, or at least secure .of not being disavowed by him, renewed the same ac- .cusation. Leibnitz observed, that Keil, whom he notwithstanding termed ti /e^r/rerf man, was too young, to pass a decided judgment on things, that had oc- curred several years before ; and he repeated, what he had before said, that he rested himself on. the candour arid justice of Newton himself. Keil re- turned to die charge; and in 17 11, in a letter to sir Hans Sloane, secretary to the Royal Society, he was not contented with saying, that Newton was the first inventor ; but plainly intimated, that Leib- nitz, after having taken his method from Newton's writings, had appropriated it to himself, merely em- ploying a different notation ; which was charging Jrim in other words with plagiarism. Leibnitz, indignant at such an accusation, com- plained loudly to the Royal Society ; and openly re- quired it to suppress the clamours of an inconsiderate man, who attacked his fame and his honour. The Royal Society appointed a committee, to examine all the writings that related to this question ; and in 1712 it published these writings, with the report of the committee, under the following title: Commercium epistolicum de Analyst promota. Without being ab- solutely affirmative, the conclusion of the report is, that Keil had not calumniated Leibnitz. The work was dispersed over all Europe with profusion. Newton was at that time president of the Royal Society, where he enjoyed the highest respect, and most ample power : perhaps therefore delicacy should Jiave induced him, to lay the cause before another tribunal. 362 tribunal. It is true Fontenelle lias said, in his eulogy of Leibnitz, that ' Newton did not appear in the business, but left the care of his glory to his country- men, who were sufficiently zealous.' Leibnitz, how- ever, was dead, and Newton living, when he said this: and no doubt he had been deceived by false docu- ments, for in the course of the dispute Newton wrote two very sharp letters against Leibnitz, in which we may be a little surprised to perceive too much art and ingenuity employed, for the purpose of revoking or weakening the testimonies of high esteem, which he had formerly expressed for him on different occasions, particularly in the celebrated scholia to the 7th pro- position of the 2d book of the Principla. It appears, that the Royal Society, while it hastened the publication of the documents that made against Leibnitz, without waiting for those which he promised in his defence, was sensible that it would be accused of partiality or precipitation : for it took care soon after to declare, that it had no intention of passing judgment on the cause, but left all the world at liberty to discuss it, and give it's opinion. I beg leave, therefore, to go into this examination, to which I will pay all the attention in my power. To me Leibnitz and Newton are both indifferent : I have re- ceived from neither of them, if I may use an ex- pression of Tacitus, either benefit, or injury. The sublime genius of both demands profound homage ; but it is incumbent on us, still more to respect the truth. Newton, gifted by nature with superiour intellect, and bora at a time when Harriot, Wren, Wallis, Barrow, 363 Barrow, and others, had already rendered the mathe- matical sciences flourishing in England, enjoyed like- wise the advantage of receiving lessons from Barrow in his early youth at Cambridge. The whole hent of his genius was toward studies of this kind, and the success he obtained was prodigious. Fontenelle has applied to him, what Lucan said of the Nile : that mankind had not been permitted to see his feeble be- ginnings. It is affirmed, that he had laid the foun- dations of the grand theories, by which he afterward obtained so much fame, at the age of twenty- five. Leibnitz, who was four years younger, found but moderate assistance in his studies in Germany. He formed himself alone. His vast and devouring ge- nius, aided by an extraordinary memory, took in every branch of human knowledge ; literature, his- tory, poetry, the law of nations, the mathematical sciences, natural philosophy, &c. This multiplicity of pursuits necessarily checked the rapidity of his progress in each ; and accordingly he did not appear as a great mathematician till seven or eight years after Newton. Both these great men were in possession of the new analysis long before they made it known to the world. If priority of publication determined priority of dis- covery, Leibnitz would have completely gained his cause : but this is not sufficient on the present occa- sion, to enable us to pronounce with confidence. The inventor may have long kept the secret to him- self ; he may have allowed some hints to escape him, on which another may have seized. If possible, (therefore, let us trace it to it's source, and endeavour to 36* to discover the beneficent being, who, to adopt the fine simile of Fontenelle, like Prometheus in the fable stole fire from the gods, to impart it to man- kind. The Commercium eplstoUcum contains in the first place, to date from the year 1669, several analytical discoveries of Newton. In the piece entitled, T)t Analyst per JE qua t tones Numero Term wor um infinitas, beside the method for resolving equations by approx- imation, which has nothing to do here, Newton teaches how to square curves, the ordinates of which are expressed by monomials, or sums of monomials ; and when the ordinates contain complex radicals, he reduces the question to the former case, by evolving the ordinate into an infinite series of simple terms, by means of the binomial theorem, which no one had before clone. Sluze and Gregory had each separately found a method for tangents ; and Newton, in a letter to Collins, dated december the 10th, 1672, proves, that he had likewise found one : he applies it to an example without adding the demonstration ; and he afterward says, that it is only a corollary of another general method, which he has for drawing tangents, squaring curves, finding their lengths and centres of gravity, &c., without being stopped by the radical quantities, as Hudde was in his method for maxima and minima. In these two writings of Newton the english have clearly perceived the method of fluxions; after it had been made known throughout all Europe, however, by the writings of Leibnitz and the two Bernoullis : but the geometricians of other countries have not seen 365 'seen with exactly the same eyes. While they agrej that the evolution of radicals into series is a con- siderable step made by Newton, they immediately perceive, without the assistance of any subsequent and conjectural light, that the methods of Fermat, Wallis, and Barrow, might have been employed to find the results concerning quadratures, which Newton contents himself with enunciating; since, after the evolution of the radicals, if there be any, nothing more is necessary but to sum up the monomial quan- tities. They confess, that the two pieces in question contain a vague indication of the method of fluxions, if you will: an indication perhaps sufficient to show, that Newton was then in possession of the first prin- ciples of this method ; but too obscure, to make the reader at all acquainted with it. What renders this conjecture very probable is, that Mr. Oldenburg, secretary to the Ptoyal Society, send- ing a copy of Stuze's Method for Tangents, which had been printed at London, to the author, on the I Oth of July, 1673, encloses him an extract from a letter of Newton's ; in which, after having observed, that this method justly belongs to Sluze, Newton goes on thus : ' as to the methods,' (he is speaking of that of Sluze and his own) * they are the same, though I believe they are derived from different principles. I know not, however, whether the principles of Mr. Sluze be as fertile as mine, which extend to the ad- fected equations of irrational terms, without it's being necessary to change their form.' If he had then possessed the method of iluxions in such an advanced state, .as .lias since been pretended, would he have spoken 366 spoken with so much reserve, instead of saying plainly, that the method of Sluze and that of fluxions were different ? Will it be supposed, that he ex- pressed himself thus out of modesty? Surely the truth may be spoken without any infringement of the laws of modesty, even when it is to our own ad- ' vantage. All these considerations appear to me to evince, that, if the piece De Analyst per JEquationes, and the letter of 1672, contain the method of fluxions, it was at least enveloped in great darkness. But whether it were or not, I shall proceed to demonstrate, that Leibnitz either had no knowledge of these two pieces before he discovered his differential calculus, or derived no information from them. This is a grand point, which his defenders have not sufficiently established, and on which I hope to leave no doubt remaining. In 167$ Leibnitz quitted the universities of Ger- many, and came to France, where he was chiefly oc- cupied in the study of the law of nations and history. He was already initiated into mathematics, however, as in 1666 he had published a little tract on some properties of numbers. In the beginning of 16/3 he went to London, where he saw Oldenburg, with whom he commenced an epistolary correspondence. In one of his letters to Oldenburg, written even while he was at London, Leibnitz says, that, having disco- vered a method of summing up certain series by means of 'their differences, this method was shown to him already published in a book by Mouton, canon of St. Paul's at Lyons, ' On the Diameters of the Sun aud 367 and Moon :' that he then invented another method, which he explains, of forming the differences, and thence deducing the sums of the series : that he is capable of summing up a series of fractions, of which the numerators are unity, and the denominators either the terms of the series of natural numbers, those of the series of triangular numbers, or those of the series of pyramidal numbers, &c. All these researches are ingenious, and seem to have at least a remote relation to the calculation of differences. The english have never asserted, and at any rate there exists not the least proof^ that Leibnitz had seen the two pieces by Newton abovementioned during this first visit to England. o After staying some months in London, Leibnitz returned to Paris, where he formed an acquaintance with Huygens, who laid open to him the sanctuary of the profoundest geometry. He soon found the approximate quadrature of the circle by a series ana-- logous to that, which Mercator had given for. the approximate quadrature of the hyperbola. This se- ries he communicated to Huygens, by whom it was highly applauded ; and to Oldenburg, who answered him, that Newton had already invented similar things, not only for the circle, but for other curves, of which he sent him sketches. In fact the theory of series was already far advanced in England at that time ; and though Leibnitz had likewise penetrated deeply into it, he always acknowledged, that the english, and Newton particularly, had preceded and surpassed him in that branch of analysis : but this is not the differential calculus, arid the english have shown too 9 evident Evident a partiality in their endeavours to conned these two objects together. Let us hear and examine the history, which Leibnitz gives of his discovery of the differential calculus. lie relates, that, on combining his old remarks on the differences of numbers with his recent meditations on geometry, he hit upon this calculus about the year l6?6; that he made astonishing applications of it to geometry ; that being obliged to return to Han- over about the same time, he could not entirely follow tke thread of his meditations ; that endeavouring ne- vertheless to bring forward [a fair e va loir] his new discovery^ he went by the way of England and Hal- land ; that he staid some days at London, where he became acquainted with Collins, who showed him se- veral letters from Gregory, Newton, and other mathe- maticians, which turned chiefly oil series; According to this- account, it would appear, that Leibnitz, wishing to spread abroad his new discovery, must then have made known in England the dif- ferential calculus. Let us add; that in a letter front Collins to Newtoiij dated the 5th of march 1677, it is said, that Leibnitz, having spent a week in London in October 1676, had put into Callings hands some papers *, of which extracts or copies should be sent to Newton immediately. Collins says nothing of the nature of these papers, and we find no trace of them in the Commercium epistolicum. But if the account given by Leibnitz be just, or if his memory did not * This passage, and several other large fragments of the same- letter, were suppressed in the Commercium ephtolkum. See it complete in Wallis's Works, Vol. III. p. 646. deceive 369 deceive him > when he said he was in possession of the differential calculus before his second visit to England, no doubt some private reason then occurred, to in- duce him to keep his discovery secret, contrary to the design he had first formed of bringing it forward : for in this very letter Collins mentions another from Leibnitz to Oldenburg, written from Amsterdam the 28th of November 1676, in which Leibnitz proposes the construction of tables of formulae tending to im- prove the method of Sluze, instead of explaining the differential calculus, or at least pointing it out as much more expeditious and more convenient. The english therefore are justified in saying, "that Leibnitz, when he passed through London in 1676, did not teach them the differential calculus : but they ought to acknowledge, that the same letter con- clusively proves, that he likewise learned nothing from them on the subject. In fact, ifj as has 'been asserted since, a knowledge of the method of fluxions had then been imparted to him, must he not have been out of his senses, to propose a month after to the secretary of the Royal Society, a man of great skill in those matters, means of improving the method of Sluze itself, without saying a single word of another method much more simple, which had just been taught him in England ? Thus I believe I may decidedly conclude, either that Leibnitz did not see the work De Analysi and Newton's letter, when he was in England in October 1676; or, if he did see them, that he derived no as- sistance from them, any more than the learned geome- tricians of England, who had had all that time to me- B B ditate 370 ditate on them, and besides were at hand to apply td the author for every necessary illustration. The english have never formally declared, that he had seen the book7> Analysi^ they contented themselves with positively asserting, that he had seen the letter of 1672. But supposing this to be true, we can draw no inference against Leibnitz from it : for beside that the letter contains only results, without any de- monstration, it is not very clear, that it indicates a method essentially different from that of Sluze, as the reader may have remarked from Newton's words al- ready quoted. In the whole of this business there are but three pieces truly decisive : 1st, a letter from Newton to Oldenburg, dated the 24th of October, 1676, which was communicated to Leibnitz the year following : d, the answer which Leibnitz returned to Oldenburg respecting this letter, June the 21st, 1677 : 3d, the scholia to Newton's Principia already quoted, pub- lished toward the end of 1686. Let us briefly analyze these three pieces. Newton's letter, exclusive of different researches concerning series, which are here to be left out of the question, contains several theorems, that have the method of fluxions for their basis, but the author keeps his demonstrations secret. He contents him- self with saying, that he has deduced them from the solution of a general problem, which he expresses enigmatically by transposing the letters, and the sense of which, as explained after the business was known, is : ' an equation containing flowing quan- tities being given, to find the fluxions, and inversely.' What 371 What light could Leibnitz derive from such an ana- gram ? All we can conclude from such a letter is, that, at the time when it was written, Newton was in possession of the method of fluxions ; by which, however, is to be understood simply the method of tangents and of quadratures ; for the method of re- solving differential equations was then out of the question, this not being invented till long after, as has been said above. Leibnitz, in his letter to Oldenburg, begins with saying, that he, as well as Newton, had found Sluze's method for tangents to be imperfect. Then he explains openly and without mystery that of the differential calculus, affirming, that he had long em- ployed it for drawing the tangents of curve lines, Here then we have the clear and positive solution of the problem, the possession of which Newton so care- fully endeavoured to reserve to himself. The scholia to the Pnncipia say as follows : ' In a correspondence in which I was engaged with the very learned geometrician Mr. Leibnitz ten years ago*, having informed him, that I was acquainted with a method of determining the maxima and mi- nima, drawing tangents, and doing other similar things, which succeeded equally in rational equations and radical quantities, and having concealed this method by transposing the letters of the words, which signified : an equation containing any number of flow- ing quantities being given, to find the fluxions, and mvcrsely : that celebrated gentleman answered, that . * Through the medium of Oldenburg* BBS he 372 he had found a similar method ; and this, which he communicated to me, differed from mine only in the enunciation and notation.' To this the edition of 1714 adds: e and in the idea of the generation of quantities.' Is it possible, to say more expressly, that Leibnitz separately invented the method of fluxions, and that he -had communicated it frankly, without involving' himself in mystery like Newton ? From these three pieces therefore it is clear, that, if Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of december 1672, Leibnitz equally invented it on his part, without borrowing any tiling from his rival. These two great men by the strength of their genius arrived at the same discovery through different paths: one, by considering fluxions as the simple relations of quantities, which rise or vanish at the same instant; the other, by reflecting, that, in a series of quantities which increase or decrease, the difference between two consecutive terms may become infinitely small, that is to say, less than any determinate finite magnitude. This opinion, at present universally received except in England, was that of Newton himself, when he first published his Principui, as we see from the ex- tract above given. At that time the truth was near it's source, and not yet altered by the passions. In vain did Newton afterward change his language, led away by the flattery of his countrymen and disciples; in vain did he pretend, that the glory of a discovery belongs entirely to the first inventor, and that se- cond inventors ought not to be admitted to share it In the first place, without discussing his pretended priority, it was replied, that two men, who separately make 373 make the same important discovery, have an equal claim to admiration ; and that he, who first makes it public, has the first claim to the public gratitude. It was then proved to him, that even his own principle did not justly apply here. The design of stripping Leibnitz, and making him pass for a plagiary, was carried so far in England, that during the height of the dispute it was said (and Newton himself was not ashamed to support the ob- jection), that the differential calculus of Leibnitz was nothing more than the method of Barrow. What are you thinking of, answered Leibnitz, to bring such a charge against me? Will you have the differential calculus to be nothing but the method of Barrow, when I claim it ? and at the same time say it was invented by Mr. Newton, when you wish to rob me of it ? Can you be so blinded by passion, as not to perceive this manifest contradiction ? If the diffe- rential calculus were really the method of Barrow (which you well know it is not), who would most deserve to be called a plagiary? Mr. Newton, who was the pupil and friend of Barrow, and had oppor- tunities of gathering from his- conversation ideas, which are not in his works ? or I, who could be in- structed only by his works, and never had any ac- quaintance with the author ? John Bernoulli, who in concert with his brother learned the analysis of infinites from the writings of Leibnitz, opposed to the Commercium epistolicum a letter, where he advances, not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that BBS Newton 374 Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany. His reasons are in substance, Jst, the Commercium epistolicum ex- hibits no vestige of Newton's having employed dotted letters, to denote fluxions, in the writings alleged ; 2dly, in the Principia, where the author had so fre- quently occasion for employing this calculus, and giving it's algorithm, he has not done it ; he proceeds every where by means of lines and figures, with- out any determinate analysis, and simply in the manner of Huygens, Roberval, Cavalleri, c. : 3dly, the dotted letters first began to appear in the third volume of Wallis's Works, several years after the diffe- rential calculus was every where known : 4thly, the true method of differencing differences, or of taking the fluxions of fluxions, was unknown to Newton, since even in his treatise on quadratures, not pub- lished till 1704, the rule he gives at the end for de- termining the fluxions of all orders, by considering these fluxions as the terms of the power of a bino- mial formed of a variable quantity, and it's first fluxion, and treating this first fluxion as constant, is false, except simply for the term which answers to the first fluxion : 5thly, at the same period of 1704, Newton was riot versed in the integral calculus of differential equations, which Leibnitz and the two Bernoullis had already carried so far; otherwise he would not have failed to treat this part of the analysis of infinites, the most difficult, and at least as worthy of being 375 , being promulgated and carried to perfection as the quadratures, on which he enlarged so much. To this letter the english answered, that the nota- tion did not constitute the method : that the princi- ples of the calculus of fluxions were contained in Newton's great work, and in his letters : that thq rule in the treatise on quadratures for finding the fluxions of all orders was true, suppressing the deno- minators of the terms of the series, and gave by con- sequence quantities proportional to the true fluxions. I do not rind, that they gave any answer to the last objection. The partizans of Leibnitz replied, that the advan- tages of an analytic method depend in great measure on the simplicity of the algorithm : that the Charac- teristic of Leibnitz had already occasioned the new analysis to make immense progress, at a time when scarcely any one had heard of Newton's book : that it was in vain to endeavour to deny or palliate the erroneousness of Newton's rule for finding the flux- ions of all orders : and that it could not be said, that the terms of a series of fractions were proportional tp the terms of another series of fractions, when the corresponding terms had different denominators, as was the case here. Such were nearly the reasons alleged and contested between the two parties for more than four years. The death of Leibnitz, which happened in 1716, it may be supposed, should have put an end to the dis- pute : but the english, pursuing even the manes of that great maia, published in 172,6 an edition of the B $ 4? . 376 Principia, in which the scholium relating to Leibnitz was omitted. This was confessing his discovery in a very authentic and awkward manner. Must they not be aware, that the chimerical design of annihi- lating the testimony, which an honourable emulation had formerly rendered to truth, would be ascribed to national prejudice, or to a sentiment even still more unjust ? In later times there have been geometricians, who, without taking a decided part between Newton and Leibnitz, have objected to the latter, that the meta- physics of his method were obscure, or even defec- tive ; that there are no quantities infinitely small ; and that there remain doubts concerning the accu- racy of a method, into which such quantities are in- troduced. But Leibnitz might answer : c I have proposed the existence of infinitely small quantities only as sub- sidiary, or as a simple hypothesis, serving to abridge the calculus and reasonings on which it is founded, I have no need of the existence of infinitely small quantities : it is enough for my purpose, as J I have said in several of my works, that my differ- ences are less than any finite quantity you please to assign ; and that consequently the errour, which may result from my supposition, is less than any deter- in inable errour, which is the same as absolutely no^ thing. The manner, in which Archimedes demon- strates the proportion of the sphere to the cylinder, has a similar principle for it's basis. Mr. de Fonte- uelle, who however meant me well, was wrong, when he 377 he contented himself with saying at the beginning of his G&om6trie de rinjini, that, after having at first admitted infinitely small quantities, I had at length receded so far, as to reduce the infinites of different orders to mere incomp arables^ in the sense in which a grain of sand would be incomparable to the globe of the Earth. He should have added, that I use this similitude only for the purpose of presenting a general and sensible idea of my differences to the imagination of certain readers ; and that in the paper to which he alludes, I conclude with remarking ex- pressly, that, instead of an infinitely large or small quantity, quantities should be assumed as large, or as small, as is necessary to render the errour less than. 3ny given errour. The metaphysics of my calcula- tion, therefore, are perfectly conformable to those of the method of exhaustion of the ancients, the cer- tainty of which has never been questioned by any one ; and whatever may have been urged, in this re- spect my rival has no real advantage over me.' Lastly it has been said, that, notwithstanding Newton's affectation of employing synthesis alone in his Principia, at present it cannot be doubted, that }ie discovered a great number of propositions by the analytical method of fluxions : that this application of it to a multitude of objects so important implies a long series of meditations : and that at least, accord- ing to all appearance, he must have been in posses- sion of the method of fluxions before Leibnitz, for he must have spent many years in writing his book. Let us examine the consequences to be drawn from this induction. Perhaps 37S Perhaps there never existed a man more highly en- dowed than Newton with that strength of intellect and vigour of mind, which are capable of conceiv- ing, pursuing, and executing a vast design. Leib- nitz has published no single work, that can be com- pared with the Principia for importance and connec- tion of subject. Too much carried away by the vivacity of his genius, and the multitude and variety of his occupations, journeys, and literary correspond- ence with most of the learned in all parts of the World, he could not confine himself to delving long at one subject, or pursuing in detail all the consequences of a grand principle : but the Collection of his Works, and his Epistolary Correspondence with John Bernoulli, bear in every part the strongest marks of invention. Every where he disseminates new ideas, and germes of theories, the development of which would sometimes produce whole treatises. He lias the ad- vantage over Newton of having invented and carried to a great length the integral calculus of differential equations. If he be not equally profound with the english geometrician, he appears to surpass him in that rapid penetration and sharpness of intellect, which seize on the mqst subtile and striking questions in any subject. The one has left us a greater mass of geometrical truths : the other more accelerated the progress of science in his time, by the simple and commodious notation of his calculus, the applications, he made of it himself, or enabled others of the learned to make, the encouragements he gave them, ami the new paths he was continually opening to their 379 their meditations. To conclude, whatever length of time the completion of the Principia may have re- quired, we ought not to forget, that this work did not appear till two or three years after Leibnitz had published his differential calculus, and some sketches of the integral. CHAP* CHAP. VI. Continuation of the dispute. War of problems between John Bernoulli and the english. Miscellaneous articles* IN this long dispute the mutual respect, which the laws of decorum exact from all men, was too fre- quently forgotten : yet it had at least the advantage of exciting a very active emulation among the greatest geometricians of the time. It produced challenges of very difficult problems, the solutions of which gave occasion to new theories, and consider- ably extended the domains of geometry. Some time before his death, Leibnitz, wishing ta feel the pulse of the englixh, as he expressed him- self, caused to be proposed to them the celebrated problem of orthogonal trajectories, which con- sisted in finding the curve, that cuts a series of given curves, at a constant angle, or at an angle varying according to a given law. It is said, that Newton, returning home at four o'clock greatly fatigued, received the problem, and did not go to bed till he had solved it. His method may be re- duced to the few following words. ' The nature of the curves to be intersected gives their tangents at the points of intersection : the angles of intersection give the perpendiculars of the intersected curves : two adjacent perpendiculars give by their points of con- 381 concurrence the centre of curvature of the intersecting curve. Let the axis of the abscisses be conveniently placed, and assume the first fluxion of the absciss as unity : the position of the perpendicular will give the first fluxion of the ordinate of the required curve, and the curvature of this same curve will give the second fluxion of the ordinate : thus the problem will always be reduced to an equation. As to the resolu- tion of the equation,' he adds, < this belongs to ano- ther method.' The english already triumphed : but John Ber- noulli, taking up the cause of Leibnitz, who was just dead, laughed at this scheme of a solu- tion. He maintained, that nothing was more easy, than to come at the equation of the trajectory : that several particular questions of the kind had been handled with success, even long before: that the essential part of the business was to resolve the dif- ferential equation of the trajectory, when it could be done, either precisely, or by the quadratures of curves : and that this resolution, far from being foreign to the problem, was the necessary completion of it : whence he concluded, that Newton, having given no method for it, had only eluded the difficulties of the question, and by no means surmounted them. Nicholas, the son of John Bernoulli, resolved in a very elegant manner the particular case, in which the intersected curves are hyperbolas with the same centre and the same vertex. His cousin Nicholas Bernoulli and Hermann treated the question more generally by methods which came to the same thing, though they had no communication with each other on the subject. These methods easily applied to all cases 3S2 cases where the intersected curves are geometrical, and even to some transcendental curves. Hermann, desirous of extending the formulae farther than they would bear, fell into some mistakes, which were pointed out by the Bernoullis. However, they all agreed in considering Newton's solution as insuffi- cient, and of no use. It appears, that Newton from this time entirely quitted the field : but some of his friends or disciples continued the war with ardour. Dr. Taylor distin- guished himself in it the most. Without stopping to develop Newton's solution, he gave one of his own in the Philosophical Transactions for 1717, which answered the question as proposed by Leibnitz in it's full extent. Had he contented himself with this, he would have merited only praise : but, hurried on by his resentment against John Bernoulli, who had spoken a little slightingly of him on another occa- sion, he prefixed to his solution some insulting re-* flections on the partizans of Leibnitz, having princi- pally in view John Bernoulli their chief. Among other things he said, that, if they did not perceive how Newton's solution led to the equations of the problem, it must be attributed to their ignorance : illorum imperitice tribuendum. The man, to whom this strange insult was addressed, was by no means inclined to forbearance ; and he avenged himself in a manner the most humiliating to Taylor's vanity. In a dissertation on orthogonal trajectories, in the Leipsic Transactions for 1718, composed by John Bernoulli and his son Nicholas jointly, it was allowed, that Dr. Taylor's solution was accurate, and even evinced 3S3 evinced some sagacity ; but then it was shown, that it was far from being sufficiently general, and that there existed a great number of resolvable cases, to which it could not be applied. At the same time John Bernoulli gave another method, which to the advantage of being incomparably more simple, added that of embracing all the geometrical curves, all the mechanical curves completely similar, and lastly a great number of mechanical curves incompletely simi- lar. These discoveries were the product of a pro- found, new, and delicate analysis. The author had in his hands an instrument, which he managed with dexterity, the method of differencing de curva in curvam. His victory was unequivocal : and Dr. Taylor, notwithstanding the tone of self-sufficiency he at first assumed, was forced to acknowledge here a superiour. I shall observe by the by, that the authors of this dissertation mention on the same subject a little piece of Nicholas Bernoulli the nephew's, in which we find for the first time the celebrated theorem of con- dition, on which depends the reality of differential equations of the first order with three variable quan- tities : a theorem, which some modern geometricians have endeavoured to arrogate to themselves. While the question of trajectories was agitating, Dr. Taylor proposed various problems on the integra- tion of rational fractions, at that time new and very difficult. John Bernoulli, who had already made some attempts in this way, in the Memoirs of the Academy of Sciences for 1702, easily solved all these problems, in the Leipsic Transactions for 1719; and from the 8 results 384 * results he obtained he formed a series of curious the-* orems, the development and demonstration of which were useful exercises for his son and nephew. We ought not to omit here for the honour of Eng- land, that Roger Cotes, professor of mathematics at Cambridge, had treated the same subject, and re- duced the integration of rational fractions to general and very commodious formulae, in his celebrated work entitled Harmonia Mensiirarum : but this was not published till six years after his death, which hap- pened in 17l6 % ; no doubt therefore Taylor and the Bernoullis were unacquainted with it's contents. In the same work of Cotes, there are several other very useful discoveries, such as his method of estimating errours in mixed mathematics, his remarks on the differential method of Newton, his celebrated the- orem for the resolution of certain equations, &c. Cotes was but thirty-four years of age when he died. Newton esteemed him highly, and often said, ' if Mr. Cotes had lived, he would have taught us all something.' The animosity between Dr. Taylor and John Ber- noulli increased daily. In 1715 the Dr. published his Methodus Tncrementorum directa et inrersa ; a pro- found work, but a little obscure, in which he treated several problems, that had been already resolved, without quoting any person. In 1716 a letter ap- peared in the Leipsic Transactions, commending John Bernoulli, and openly treating Taylor as a pla- giary. Of this he complained with acrimony -, and at the same time retorted the accusation, by showing, that John Bernoulli, in his last solution of the isope- rimetrical 385 rimetrical problem, had only travestied the solution of his brother, and that all the simplifications he had made in it had not changed it's nature. From that time John Bernoulli kept no measures with him ; and he published under the name of one Burcard, a schoolmaster at Basil, an answer to Taylor, which was full of insult and ridicule, among which however AVC meet with some useful truths. The problem of orthogonal trajectories gave birth to that of reciprocal trajectories, which was proposed at the end of the dissertation of the Bernoullis. They required the curves, which, being constructed in two contrary directions on one axis of a given po- sition, and then moving parallel to themselves with unequal velocities, should constantly intersect each other at a given angle. This was a fresh subject of analytical difficulties to be surmounted, and for extend- ing the science. It was a long time agitated between John Bernoulli and an anonymous englishman, who was afterward known to be Dr. Pemberton, the par- ticular friend of Newton. We are again obliged to say, that John Bernoulli retained his superiority here, by the simplicity and elegance of his solutions. The cnglish geometricians had formed a league against John Bernoulli, and attacked him on subjects of every kind. Alone, says Fontenelle, like another Horatius Codes, he sustained, on the bridge, the efforts .of their whole army. Keill, a soldier more bold than puissant, imagined he had found an opportunity of .perplexing him. The theory of the resistance of mediums to the motion of bodies passing through them formed a considerable part of the Prindpia. c c Newton Newton "had determined the curve described by a projectile in a medium resisting in the ratio of the simple velocity : but he had not touched upon the case, at that time more difficult, where the resistance of the medium is as the square of the velocity. This case Keill proposed to John Bernoulli, who not only resolved it in a very short time, but extended the solution to the general hypothesis, in which the resistance of the medium should be as any power of the velocity of the projectile. When he had disco- vered this theory, he offered repeatedly to send it to a confidential person in London, on condition, that Keill would give up his solution likewise : but Keill, though strongly urged, maintained a profound si- lence. The reason of this it is not difficult to con- jecture: he had not resolved his own problem. When he proposed it, he expected, that no one would discover what had escaped the sagacity of Newton. In this conjecture he was cruelly mistaken : and his challenge, which was something more than indiscreet, drew on him a reprimand from the swiss geometrician, that was so much the more poignant, as the only mode of answering it satisfactorily was by a solution of the problem, which he could neither effect by his own 'skill, nor by the assistance of hrs friends. John Bernoulli's triumph was complete. In the first intoxication of victory he indulged himself in sarcasms and jests against his rivals, not com- mendable for their elegance, but certainly pardonable in a man of a frank and honest* disposition insidi- ously attacked, and who had to avenge affronts, offered offered not only to himself, but also to an illustrious friend, whose loss he still lamented. These learned contests drew the attention of all geometricians ; and notwithstanding the acrimony infused into them by the passions, they stimulated men's minds, and produced new proselytes to mathe- matics on all sides. I shall now step back a little, and resume some other subjects, which I have been obliged to leave in arrear. In 1711 appeared the ' Analysis of Games of Chance,' by Remond de Montmort: a work abound- ing with acute and profound ideas, the object of which is, to subject probabilities to calculation; to estimate chances ; to regulate wagers, &c. It does not properly belong to the new geometry, yet it con- tributed to it's progress, by stimulating the spirit of combination in general, and by the extent which the Author gave to the theory of series, a happy supple- ment to the imperfection of the rigorous methods in all the branches of mathematics. Three years afterward de Moivre published a little treatise on the same subject, entitled, Mensura Sortis, chiefly remarkable for containing the elements of the theory of recurrent series, and some very ingeni- ous applications of it. This Essay, gradually in- creased by the reflections of the author, has grown up into a considerable work, admired by all geome- tricians. The best edition is that of 1738, in english, under the title of the Doctrine of Chances. De Moivre was a french geometrician, whom the revoca- tion of the edict of Nantes had obliged to quit his c c 2 country, 388 country, and who had fled to London. Born with superiour talents for geometry, the narrowness of his fortune obliged him to teach mathematics for a liveli- hood. Newton had the highest esteem for him. It is reported, that during the last ten or twelve years of Newton's life, when any person came to ask him for an explanation of any part of his works, he used to say : ' Go to Mr. de Moivre ; he knows all these things better than I do.' Nicholas Bernoulli, the nephew, came to Paris in 1711. His great reputation, and mild and easy manners, gained him many illustrious friends. Among the number of these was Montmort, with whom he formed a strict intimacy, in consequence of the simi- larity of their dispositions, and ta.ste for the analysis of probabilities. They spent three whole months to- gether in the country, solely employed jn resolving the most difficult problems on this subject. All these new researches, and the elucidations arising from ' O them, produced a second edition of Montmort's book in 171 4-, much superiour to the first. I have already slightly mentioned Dr. Taylor's Methodus Increment or um^ but this work, celebrated even in the present day, deserves more particular notice. The author gives the name of increments or decre- ments of variable quantities to the differences, whe- ther finite or infinitely small, of two consecutive terms in a series formed according to a given law. When these differences are infinitely small, their cal- culus, either direct or inverse, belongs to the leib- analysis, or the method of fluxions ; and Dr. Taylor 38.9 Taylor resolves a great number of problems of this kind. But -when the differences are finite, the me- thod of finding the relations they bear to the quan- tities that produce them forms a new kind of cal- culus, the first principles of which were given by Dr. Taylor; and in this respect his book has the merit of originality. In this manner he has summed up some very curious series. The extreme conciseness, or rather obscurity, with which this work is written, long retarded the success which was due to it. Nicole, however, a very dis- tinguished french geometrician, was able to understand it: he very clearly unfolded the method for resolving finite differences, and added several new series of Jiis own invention. The two excellent papers, which he published on this subject in the Memoirs of the Aca- demy of Sciences for 1717 and 1728, may be considered as the first methodical and luminous elementary treatise on the integral calculus with finite differences, that ever appeared. Several other works of the time might be men- tioned, but I must be brief. I would request the reader, therefore, to consult the periodical publica- tions of Germany, France, England, Italy, &c., with those by the different academies, where he will find a number of valuable papers on every branch of mathematics. It has been observed, that the Royal Society of London and the Academy of sciences at Paris arose nearly at the same time, or about the year 1660. The Academy of Berlin, the establishment of which was projected in 1700, took a regular and legal form c c 3 in 390 in 1710, under the auspices of Frederic, elector of Brandenburg, and the first king of Prussia, and Leibnitz was appointed perpetual president. The In- stitute of Bologna was founded in 1713, through the means of the celebrated count de Marsigli, to whom natural history is so much indebted. In 1726 Ca- tharine I, the widow of Peter the Great, founded the Academy of Petersburg. Several other learned so^ cieties have since been formed, which it would take up too much room to particularize. All these esta- blishments have been of extreme utility to the pro- gress of the sciences, CHAP. 392 CHAP. VII. Continuation of the progress of geometry. Solution* of various problems. Two very curious problems, proposed by Hermann in the Leipsic Transactions for 1719, employed, the geometricians for some time, with much utility. The iirst consisted in finding a curve, the area of which should be equal to a certain proposed function of the coordinates. The second, far more difficult, was to determine an algebraic curve, such that the inde- finite expression of it's length should include the quadrature of a given algebraic curve, plus or minus a given number of algebraic quantities. Nicholas Bernoulli, the son, resolved the first: Act. Lips. 1720. As to the second he confessed, though he wrote under his father's eye, that he could not resolve it without certain suppositions, which restrictexl it's ge- nerality. Hermann gave the general solution, by a very ingenious method, founded on the theory of evolutes, ib. 1723; and on this occasion he had the advantage over the Bcrnoullis. The next year John Bernoulli returned to the same question, and treated it in a more direct and ana- lytical manner, giving it at the same time still greater extent. One general observation may be made on a.11 the problems thus depending on the analysis of infinites. C c 4 Commonly 392 Commonly they can be formed into equations with tolerable ease: but the principal diffieijlty is to resolve these equations, which is frequently such, as to elude all the powers of analysis. Accordingly the greatest geometricians have turned their .attention to the im- provement of the integral calculus, or the resolution of differential equations of all orders. With this view count James Riccati, having fallen on a differential equation of the first order with two variable quantities, apparently very simple, but which notwithstanding he was unable to resolve generally, proposed the question to all geometricians, in the Leipsic Transactions for 17-5. No one could com- pletely accomplish the object : but a great number of cases were pointed out, in which 'the indeterminates are separable, and in which the equation may coi> sequently be resolved by the quadratures of curves. The authors of these elegant discoveries were Riccati himself, Nicholas Bernoulli the nephew, Nicholas Bernoulli the son, his brother Daniel, and Goldebach. They all obtained the same results by different me- thods. The equation in question is commonly called Riccati's, though it had been already considered by James Bernoulli, who had resolved certain particular cases of it. It is much the same in the analysis of in- finites as the quadrature of the circle is- in elementary geometry. When an equation is reduced to this, the problem is considered as resolved. If the equation do not fall under separable cases, we have no other resource, but to resolve it by the methods of approx- imation. The 593 The celebrated. Euler, who was born in 1707, and died in 1783, a man destined to produce a revolution in the analytical science, made himself known at this time by various researches ; and among others by a very elegant solution of the problem of reciprocal trajectories, which appeared in the Leipsic Trans- actions for 1727, and which he afterward extended and improved. He had acquired his first knowledge of mathematics under John Bernoulli, who, at the end of his own solution of the problem just mentioned, predicted what such a pupil would one day become. At the foundation of the Academy of Petersburg, the example that Ptolemy Philadelphia had given at the Museum at Alexandria was revived : a colony of geometricians, astronomers, natural philosophers, c., was invited from all parts of Europe. Among them were Nicholas Bernoulli and his brother Daniel, Euler, Leu tm ami, Bulfinger, c. : and beside these resident members, the academy had several illustrious foreign associates, as John Bernoulli, Wolf, Poleni, Miehelotti, c. All these men, ardent, laborious, and eminent for their genius, were eager to 'enrich the collections of the academy. In" the first volume of it's Memoirs, published in 1726, we find two or three excellent papers by Ni- cholas Bernoulli, who was unhappily cut off almost at his entrance on his career. The two persons, who contributed most to the glory of geometry in this establishment, both at it's commencement, and during it's progress, were Daniel Bernoulli and Euler. Most of the problems, that had been attempted at the first effervescence of the new geometry, had for their 394 their object particular theories, to which all the ex- tent they were susceptible of had not been given. Daniel Bernoulli and Euler generalized many of these old problems, such as that of the catenary curves, and isoperimetrical figures : they likewise treated others, that were absolutely new and very difficult, as, for example, the determination of the oscillatory motions of a heavy chain suspended vertically, the investigation of the tones given by an elastic slip of metal when struck, the motions resulting from the eccentric percussions of bodies, &c. All these questions required great natural sagacity, and a profound knowledge of analysis. Our two geometricians resolved them each separately ; and we ought not to forget the rare example of moderation and honour which they then exhibited, and from which they never once deviated. They mutually proposed problems to each other, and worked on the same subjects, without their rivalry of talents, or difference of opinion on certain points connected with physics, diminishing that strict friendship, which they had contracted in their youth. Each frankly and without hesitation did justice to the other : in the science of analysis Daniel Bernoulli struck to Euler, whom he called his admiral ; but in questions that required more acuteness of intellect than profound geometry, Daniel Bernoulli took the lead : in fact he had a peculiar talent for applying geometry to phy- sics, and subjecting phenomena, which were known only in a general and vague manner, to precise cal- culation. To To Pascal has been attributed the project of making men submit to the yoke of religion, by the force of eloquence and reasoning. It seems as if Euler had been in like manner desirous of rendering analysis paramount over all parts of mathematics. We find him continually busied in improving this grand in strument, and showing the art of handling it with dexterity. Scarcely had lie attained the age of twenty- one, when he gave a new and general method of re- solving whole classes of differential equations of the second order, subject to certain conditions. Mem. of the Ac. of Petersburg for 1728. This had been ac- complished before only in certain particular cases ; and then rather by the sagacity of the analyst, than by uniform and determinate methods. In Italy, Gabriel Manfred i published, from time to time, ingenious papers on geometry and analysis, in the Commentaries of the Institute of Bologna, and in the journals. Another geometrician of the same nation, the count de Fagnani, opened a field of new problems of a very attractive kind. He taught how to determine arcs of the ellipsis, or hyperbola, the difference of which is an algebraic quantity. Leibnitz and John Ber- noulli, who had attempted this research, judged, that it could not be subjected to the new calculi. They had merely resolved the question for the parabola, and by employing the common algebraic calculus. It is likewise resolved by the same means in the mar- quis de 1'Hopital's treatise on Conic Sections. Fag- nani very dexterously applied t'he integral calculus to the arc$ -of the ellipsis and hyperbola, thus including the $96 the parabola as a particular case. His method con* sists in transforming the differential polynomial, which represents the elementary, elliptical, or hyperbolical arc, into another polynomial negatively similar; from which an algebraic quantity results by subtraction, and the subsequent resolution. The glory of having explored this nook of geometry, if I may so say, has placed Fagnani among the most subtile analysts. A long time after this, in 1756, Euler having con- sidered the same subject, not only resolved the pro- . blems of Fagnani in a new manner, but attained a method of resolving a very extensive class of separate differential equations, the two sides of which, though uot resolvable separately, form a whole perfectly re- solvable. How to resolve equations of this kind, when the two sides depend at the same time on arcs of the circle, or on logarithms, was already known; but the new solutions of Euler were much more ex- tensive : tliey form a new, pleasing, and very useful branch of the integral calculus, in which the author displays all the resources of genius, and the pro foundest knowledge of analysis. Viviani's problem of the quadrature of the hemi- spherical vault, long after, gave rise to another of a similar nature, proposed by a geometrician in other respects little known, Ernest von OrTenbiiig, in the Leipsic Transactions for 1718. This was, to pierce a hemispherical vault with any number of oval win- dows, with the condition that their circumferences should be expressed by algebraic quantities : in other words, it was required, to determine on the surface of a sphere curves algebraically rectifiable. It is ob- vious 397 vlous at first sight, that the curves sought could not be formed by the intersection of the sphere by a plane, for all these intersections, in whatever direc- tion made, must be mere circles : but they belong to the class of curves of double curvature. This pro- blem, though curious and difficult, remained a long time unattempted, and it is not known, whether it were resolved by the author. Hermann, in a paper on the rectification of sphe- rical epicycloids, in the Petersburg Transactions for 1726, imagined, that these curves generally satisfied the question of OfFenburg, or were algebraically reo tifiable : but this they are only in certain particular cases, the rectification of spherical epicycloids de- pending in general on the quadrature of the hyper- bola. In the Memoirs of the Academy of Paris for 1732, John Bernoulli pointed out Hermann's mis- take ; and, not content with assigning the true al- gebraic and rectifiable epicycloid, he directly and a priori resolved the problem of Offenburg ; that is, he gave the general method of determining the rectifiable curves, that may be traced on the surface of a sphere. He then proposed the same research to Maupertuis, as the most eminent french geometrician of the time, offering to send his own solution, if it were desired. The offer was accepted ; and while the solution of Bernoulli was on the road, Maupertuis likewise re- solved the problem : at least he declares he did ; add- ing, that he took great care to authenticate his dis- covery : a precaution 'so much the more necessary, as the two solutions are ia substance the same. At 393 At the same time Nicole gave the method of find- ing the general expression of the rectification of spherical epicycloids, and afterward determining the cases, in which these curves become algebraic and rectifiable. Clairaut, at that time only one and twenty years of age, yet who had already acquired reputation by his ' Inquiries concerning Curves of double Curva- ture/ treated the question in the same view as Nicole: but his method, in the Memoirs of the Academy for 1734, bears a peculiar stamp of elegance, which al- ways distinguished his different works. Geometry in a short time after made another very important acquisition. Clairaut considered a class of problems already sketched out by Newton and the Bernoullis; which was, to find curves, the property of which consists in a certain relation between their branches, expressed by a given equation. In this there would be no difficulty, were we permitted to employ the branches of two curves : but here the branches must belong to one and the same curve, and then the calculus is of a new and delicate kind. In this research Clairaut made one observation particularly worthy of attention. There arc questions of this kind, which adniit two solutions, one direct, and independent of the inverse method of fluxions, the other founded on this method. The second, in which we suppose care has been taken to introduce an arbitrary constant quantity, seems as if it must include the first, by giving to the constant quantity aU the values of which it is susceptible. But this is not the the case ; for whatever value be given to the constant quantity, we never fall into the first solution. This kind of paradox in the integral calculus, remarked by Clairaut, was at the same time observed by Euler, as we see in his Mechanics, which appeared in 1736, as well as in the Memoirs of the Academy of Sciences at Paris for 1734. This was the germe of the cele- brated theory of particular integrals, which Euler and several other learned geometricians have com- pletely explained. It does not appear, that Clairaut followed up his ideas on this subject CHAF, 400 CHAP. VIII. Problem of isochronous cunes in resisting mediums. General reflections on problems of pure theory. Al- gebra of sines and cosines. Utility of methods of and in particular of infinite series. IHE problem of isochronous curves is remarkable in the history of geometry, as well for it's singular 'na- ture, as for the difficulties that were to be surmounted in it's solution. It consists, as is well known, in finding a curve of such a nature, that a heavy body, descending along it's concavity, shall always arrive at the lowest point in the same space of time, from whatever point of the curve it may begin to descend. Huygens, examining the properties of the cycloid, found it that of being the isochronous curve in vacuo. Newton shows in his Principia, that the same curve is likewise isochronous, when the descending body, subject to the action of a uniform gravity, ex- periences every instant from the air, or whatever medium it moves in, a resistance proportional to it's velocity. Euler and John Bernoulli both separately determined the isochronous curve in a medium re- sisting as the square of the velocity. Mem. of the Ac. of Petersb. 1729; of Paris, J730. These three cases form three different problems, for each of which different methods are employed. In the former two, when the body, after having de- scended, 401 scended, reascends through the second branch of the cycloid, it passes through the ascending arc in the same time as through the descending; so that all the oscillations, each of which consists of a descent and an ascent, are performed in the same time. But in the hypothesis of the resistance being as the square of the velocity, the isochronous descending arc is not the same as the isochronous' ascending arc, so that they must be sought out separately. They are found, however, exactly in the same manner, and conse- quently it is sufficient to consider either of them. Fontaine made a great step in this theory. He invented a method of a truly original cast, by which alone he resolved the three cases proposed ; and he even added a fourth, in which the resistance shall be as the square of the velocity plus the product of the velocity by a constant coefficient. It is very remark- able, that the isochronous curve is the same in the fourth case as in the third. The spirit of this method is, to consider the variable quantities both with re- spect to the difference of two proximate arcs, and with respect to the elements of one and the sarne arc : the author employs the differentials of Leibnitz for the variations of the first kind, and the fluxions of Newton for those of the second. Dr. Taylor had given an open- ing to this Jluxio-differential method ; and Fontaine also resembled him in the defect of being obscure: but they were both profound geometricians. Euler, who, not content with enriching geometry from his own stores, has sometimes remoulded the works of others, and always for their advantage, de- veloped the method of Fontaine, and placed it in a i> D clearer 402 clearer light, in the Petersburg Memoirs for 1764, bestowing on it at the same time all the praises it deserves. He goes over each of the cases already re- solved ; and adds another, which includes them all : that in which the resistance is composed of three terms, the square of the velocity, the product of the velocity by a given coefficient, and a constant quan- tity. The method of Fontaine extends no farther : be- sides, as it gives the isochronous curve independantly of the consideration of the time, the expression of the time, which the body employs in passing through any arc of the curve, remained still to be determined. Euler has solved this new problem, which depended on the resolution of a very complicated differential equation. Fontaine imagined he had so completely exhausted the theory of isochronous curves, that in the collec- tion of his works, published in 1764, he says, when speaking of his solution of 1734, f the problem ceased to be a subject of discussion soon after it's appearance.' Happily, however, it continued a subject of discussion still. It was not enough, to have found the iso- chronous curves on certain hypotheses of accelerating forces : it was necessary, by inverting the problem, to point out the means of determining, what are the hypotheses of accelerating forces, that admit isochro- nism. Two great geometricians have made this dis- covery, and thereby opened a new store of problems on the subject. Memoirs of the Ac. of Berlin, 1765. When the medium is rare, or of little resistance, the investigation of the isochronous curve is more easy. Euler has resolved several cases of this nature with 403 with great simplicity and elegance in his Mechanics, to whatever powers of the velocity the resistance may be proportional. The enemies of geometry, or even those who are imperfectly acquainted with it, consider all these difficult theoretical problems as mere amusements, consuming that time a'nd reflection which might be better employed. But they do not consider, that nothing is more capable of arousing and unfolding all the powers of the human intellect ; that the mind, to use an .expression of Fontenelle's, has it's wants, as well as the body ; and finally, that a speculation, which appears sterile at first sight, ultimately finds if s application, or sometimes, when least expected, gives rise to new views respecting objects of public utility. Let us give genius a free wing : let the geo- metrician seek and contemplate intellectual truths, while the poet depicts the passions of the heart, or the beauties of nature. The more attraction those who cultivate the sciences find in this faculty of invention, the more does it in- cline them to avoid working on the dead matter, which would lead to no new and useful result. Thus, when such a one has surmounted the analytical difficulties of an abstract problem, he rarely completes the cal- culation : commonly he contents himself with point- ing it out clearly ; or, in cases that require it, re- duces it to certain formulas, which evade a strict ana- lysis, as the quadrature of the circle, the equation of Riccati, &c. ; and when we come to these, the pro- blem is considered as resolved. , But when we would apply analytical formulas to the physico-mathematical p D 2 sciences, 404 sciences, where all the quantities must be ultimately expressed in numbers, we cannot dispense with a completion of the analytical calculation ; and then we have frequent occasion for methods that serve to abridge it, whether it terminate in algebraical results, or contain expressions of which the values can be given only by approximation. The algebra of sines and cosines, for which we are principally indebted to Euler, is one of the means^ of abbreviation, to which all parts of mathematics, and physical astronomy in particular, have inesti- mable obligations. By the combination of arcs, sines, and cosines, we obtain formulae, which in many cases readily submit to the methods of resolution : and thus w r e are enabled to solve a number of pro- blems, which we should be forced to abandon from the length or difficulty of the computations, if we were to employ the arcs, sines, and cosines, in their ordinary form, or even in the exponential one. In defect of strict solutions \ve are obliged to recur to methods of approximation, and to these in great measure we owe the success of practical mathematics. The theory of infinite series is the chief basis of all these methods. Frequently there is much difficulty, and much art is required, to form the series proper to resolve promptly, and sufficiently near t*o the truth, the questions which require them. The english, as Newton, Wallis, Stirling, &c., greatly cultivated this beautiful part of analysis : but no person has carried it so far as Euler; no 'one has summed up so many curious series, or applied this mean to the so- lution 405 Jution of such a multitude of delicate and important problems. The collections of the academies of Petersburg!! and Berlin, and of his works, abound with discoveries of this kind, which are considered as the chief testimonies of his genius. D D 3 CHAP. 406 CHAP. IX. Continuation* Progress of the methods for resolving differential equations. New step in the problem of isoperimetrical figures. The integral calculus with partial differences. 1 HE knowledge of some general property, capable of directing the method of resolution, was still want- ing to the theory of differential equations. Since the problem of catenary curves, whence this theory began to assume a solid form, a great number of differential equations of all orders had been resolved ; but for each particular case a particular method was employed ; and frequently the object was attained only by a kind of tentative process, which might lead us to admire the genius and sagacity of the analyst, but after all gave no opening to problems of another kind. Geometricians therefore wanted some sign, or mark, by which they might know, whe- ther an equation were directly resolvable in the state in which it offered itself to them, or required some preparation to render it so. That many useless attempts must be spared by such a knowledge is obvious; and the honour of having made this noble discovery for differential equations of the first order is shared between Ger- many and France. Euler, Fontaine, and Clairaut, each separately arrived at it about the same time, or at 407 at least without having derived any assistance from each other. Justice however forbids me to conceal, that Euler took the first step. In his Mechanics, published in 1736, he employs an equation dependent on this theory; but the demonstration did not make it's appearance till the year 1740, when it was given in the Memoirs of the Academy of Petersburg for 1734-. Now the researches of Fontaine and Clairaut bear date in 1739, so that these mathematicians could not have seen those of Euler. Euler having afterward discovered the conditions, under which differential equations of higher orders are resolvable, transmitted them to Condorcet, but without adding the demonstrations. The french geometrician not only discovered these in a very di- rect and simple way, but gave a farther extension to the theory. This was the first essay of great talents for analysis, to which it is to be for ever regretted, that the author did not entirely devote himself, both for his own happiness, and for the advancement of science. All the World knows, that Condorcet, having involved himself in the political dissensions of the french revolution, was driven, to commit suicide, as the only means of avoiding the scaffold. The isoperimetrical problem, which had been so much agitated between James and John Bernoulli, reappeared again occasionally on the stage, in con- sequence either of new applications of it, or attempts to simplify it's general solutions. Among those, who turned their attention to it, Euler is principally to be distinguished. Passing over his first essays, printed among the Memoirs of the Academy of Petersburgb, I D 4 shall 408 shall come at once to his celebrated book, Afcthodus inveniendi Lineas curcas, maximi viinimvce Proprktate gaudentes, published in 1744. In this work tbe author distinguishes two sorts of maxima and minima, the one absolute, the other re- lative. The maxima oraninima are absolute, when a curve possesses, without restriction, a certain property of a maximum or minimum, among all the curves corresponding to the same absciss. Such is the curve of swiftest descent. The maxima or minima are rela- tive, when the curve, which is to possess a certain pro- perty of a maximum or minimum, must likewise fulfil another condition ; such, for instance, as of being equal in length to all the curves terminated with it by two given points ; as the circle, which has the property of including the greatest area of all curves of equal circumference. The methods employed by Euler for resolving all these problems are very simple, and as general as can be required. He first invented and demonstrated a theorem of the highest importance on this subject; which was, that problems of the second class might always be reduced to the first, by multiplying the two terms expressing the two condi- tions of the curve by constant coefficients, adding together the 'two products, and supposing, that the sum of them forms a maximum or minimum. Euler's work contains a number of curious applications, in which we every where perceive the prcfoundest science in the calculations, and the greatest elegance in the solutions. Considered in this view, it has all the perfection possible in the present state of ana- lysis ; but the general solutions have been still far- ther simplified, and subjected to easy calculations, by 409 by means of the method of variations, which Euler himself afterward adopted, and which he has de- veloped in several particular memoirs, as well as in an appendix to the third volume of his treatise on the Integral Calculus. Finally, he has reduced this kind of calculus to the common integral calculus. o About the middle of the last century a new disco- very was made in analysis, the extent and applica- tions of which are without bounds. For this we are indebted, at least in part, to the illustrious d'Alem- bert, one of the men who have done most honour to France as a geometrician of the first order, and it may be added as author of the elegant preface to the Encyclopedic. I speak of that branch of the integral calculus, which is at present called the integral cal- culus with partial differences. The nature of my work does not allow me here to give a very clear idea of it to the reader; ac- cordingly I must content myself with saying, that the object of this species of calculus is, to find a function of several variable quantities, when we know the relation of the coefficients, which affee~l the differentials of the variable quantities of which this function is composed. Let us suppose, for instance, a differential equation of the first order, with three variable quantities : in the problems of the common integral calculus, the differential coefficients are given directly by the conditions of the question ; and then the business is, to resolve the equation, either exactly, when this can be done, or by multiplying it by a factor, or by separating the indeterminate quan- tities, or lastly by the methods of approximation ; by any 410 any one of these means we arrive at a finite equation, which includes an arbitrary constant quantity. But if, in the differential equation proposed, the differen- tial coefficients be originally given, the method, that must be employed to find the finite equation, belongs to the integral calculus with partial differences. This equation includes an arbitrary function of one of the three variable quantities, and may contain likewise an arbitrary constant quantity comprised in the func- tion. There would be arbitrary functions of two va- riable quantities, if the primitive differential equa- tion were of the second order. In general, the ope- rations of the integral calculus with partial differ- ences bring out arbitrary functions, in the same manner, and in like number, as those of the com- mon integral calculus do arbitrary constant quan- tities. Some vestiges of this new kind of calculus may be found in a paper of Euler's, already mentioned, in the Petersburg!! Transactions for 1734. The work of d'Alembert, ' On the general Cause of Winds/ contains somewhat fuller notions of it ; and this geo- metrician was the first, who employed it in an explicit manner, though subjected a little too much to the common integral calculus, in the general solution of the problem of vibrating cords. Dr. Taylor, in his Metkodus Incrementorum, had determined the curve formed by a vibrating cord, stretched by a given weight, supposing, 1st, that the cord, in it's greatest excursions, moves but little out of the rectilinear direction of the axis; 2dly, that all it's points arrive at the axis in the same time. He 411 lie found this curve to be a very elongated trochoid ; and then he assigned the length of the simple pendu- lum, that performs it's oscillations in the same time, as the vibrating cord. This was a new and original problem at that time, and several other geometricians afterward treated it according to the same data. The first supposition, that the excursions of the cord on either side of the axis always remain very small, is sufficiently conformable to the nature of things : and besides it is the only one, which admits of calculation, even in the present state of analysis. As to the second, that all the points of the cord ar- rive at the axis in the same time, it is absolutely pre- carious, and the problem required to be freed from this limitation. D'Alembert invented a solution in- dependant of this. In the Memoirs of the Academy of Berlin for 174-7 he has directly and a priori deter- mined the curve, which a vibrating cord forms at every instant, without making any other supposition, but that it moves only a little from the axis in it's greatest deviations. The nature of this curve is ex- pressed at first by an equation of the second order, one member of which is the second differential of the ordinate, taken by making the time alone vary, and supposing it's differential constant : the other mem- ber is the second differential of the ordinate, taken by making the absciss alone vary, and supposing it's differential constant. Hence, by satisfying these two conditions successively, we arrive at a finite equation of such a nature, that the ordinate has for it's value the assemblage of two arbitrary functions, one the sum of the absciss and the time, the other their dif- ference. 412 ference. It is obvious, that, by means of this equa- tion, if any two of the three variable quantities, the ordinate, the absciss, and the time, be given, we can determine the third, and all the circumstances of the motion of the cord. Euler, struck with the beauty of this problem, employed himself on it a very long time, and resumed it more than once in the Memoirs of the Academies of Berlin, Petersburgh, and Turin. Notwithstanding the conformity between the results of the two great geometricians just mentioned, they had a long dis- pute on the extent that might be given to the arbi- trary functions, which enter into the equation of the vibrating cord. D'Alembert would have the initial curvature of the cord subjected to the law of conti- nuity : Euler considered it as altogether arbitrary, and introduced discontinuous functions into the cal- culation. Other geometricians have thought, that this discontinuity of the functions might be admitted, but that it should be subjected to some law, and that it was necessary, that three consecutive points of the initial curvature should always belong to a continued curve. No person yet, however, appears to have given demonstrative arguments for his opinion, at which we need not be surprised. The question is connected with metaphysical ideas ; and the problems of mechanics, or of pure analysis, to which this new kind of calculus has been applied, have not yet fur- nished any means of discerning which of the opinions affords results conformable, or contrary, to truths already discovered and universally acknowledged. The 413 The celebrated Daniel Bernoulli, without taking any part in the dispute, bestowed the highest enco- miums on the calculations of Euler and d'Alembert : but at the same time he undertook to show, that the vibrating cord always forms either a simple trochoid, such as Taylpr's theory gives, or an assemblage of these frochoids ; and that all the curves determined by d'Alembert and Euler were inadmissible, and in- applicable to nature, except as far as they were re- ducible to such a form. This discussion gave him occasion to investigate the physical formation of sound, which was then very imperfectly known. He explains, for instance, with all the clearness possible, how a cord set in vibration, or generally any sonorous body whatever, may emit at once several different sounds composing one in- dividual whole. But while we admire his address in simplifying this subject, and bringing experience in support of his reasoning, geometricians agree, that his solution is less general and perfect than those of his two rivals. In fact, the latter, whatever extent may be attributed to them, are founded on a kind of calculus, which is beyond the reach of dispute, and contain the general solution of Daniel Bernoulli as a particular case. I say as much with respect to the problem of the propagation of sound, which is of the same nature with that of vibrating cords, and to which Euler and Daniel Bernoulli have equally ap- plied each his particular method. ' The different points of view, under which Euler has considered and exhibited the integral calculus with partial differences, have settled it's true nature, and 414 and made known the applications of which it is susceptible in a number of pbysico-mathematical pro- blems. Lastly, he lias thoroughly developed the method, and given it's algorithm, in an excellent paper in the Petersburg!} Transactions for 1/62, en- titled, Investigatio Functionum ex data Differentialium Conditions. Hence some geometricians have con- sidered Euler as the principal, if not the sole inventor of this calculus : but it ought not to be forgotten, that d'Alembert first made an important and original application of it, which furnished Euler with hints, as he himself confesses. Were I allowed to give my opinion, I should say, that these two illustrious men have nearly an equal right to the glory of this beautiful discovery. In proportion as this new calculus has been in- vestigated, and it's utility perceived, it has been cul- tivated with so much the more ardour, as the field it offers to research is immense, Some geometricians of our own time have already obtained brilliant suc- cess in it ; and farther efforts will be crowned with fresh laurels. If the career become daily narrower, and if the steps of the competitors appear shorter and less profound as they advance, those who are real judges of the subject will know how to proportion their esteem to the obstacles surmounted, or the utility of the discovery; and this esteem is the noblest reward, for which he who merits it can contend. CHAP. 415 CHAP. X. Of some works on analysis. UNWILLING to interrupt the history of the new calculi, I have abstained from reviewing particular works, great numbers of which appeared in this fourth period, on the analysis of finite quantities as well as of infinites : but I shall now take a succinct view of the chief of those, that relate to the analysis of in- finites, either as direct treatises on it, or as introduc- tory to the subject, confining myself however to de- ceased authors. It has been already observed, that the marquis de 1'Hopital's Analysis of Infinites was the first work, in. which the differential calculus was explained at large. It was long styled the young geometrician's Breviary. To the general idea before given of it I shall here add, that, independent of the theory of tangents, and of maxima and minima, which constituted the prin- cipal object of the differential calculus, the author has resolved a number of other problems, at that time difficult as well as interesting. Some of these pro- blems were new : of others the solutions had been given without analysis, and without demonstration* The marquis unfolded all these mysteries, and thus rendered science one of the most important services it ever received. In sections vr and vn, for in- stance, he explains in the clearest and most complete 6 manner 416 manner the whole theory of caustics by reflection and by refraction ; those celebrated curves, which Tschirnhausen had pointed out to geometricians, and of which James Bernoulli had contented himself with announcing the principal properties. Section VJIT is occupied by the investigation of right or curve lines touching an infinite number of given right lines or curves : a subject curious in itself, and including questions applicable to the science of projectiles. In section ix the author exhibits the celebrated rule for finding the value of a fraction, the numerator and denominator of which both vanish at the same time. The xth and last section presents the differential cal- culus in a new point of view, whence the marquis de- duces the methods of des Cartes and Hudde for tan- gents. This subject, treated with the same precision and perspicuity as the rest, can be of no use in the present day, except to exercise the young geo- metrician. The marquis de FHopital left behind him a work on the general theory and peculiar properties of the Conic Sections, which was published in 1707. Though the cartesian analysis is alone employed in this, it deserves to" be distinguished, both for the in- trinsic value of it's contents, and because it has opened the way for some problems, in which the analysis of infinites was necessary. It still holds a place among the small number of classical works. This was soon followed by another book of still greater utility, at least in France; the Analysis de- monstrated of farther Reyneau, which made it's first appearance in 1708, In this the author proposed to himself 417 himself two objects : first to demonstrate and eluci- date several methods of pure algebra ; secondly, to owe in a similar manner the elements of the diffe- o rential and integral calculus. The differential cal- culus being sufficiently made known by the marquis de 1'Hopital's work ; he enlarges but little on it, em- ploying himself chiefly in unfolding the elements of the integral calculus, which was then in it's infancy. For a long time it was the the only guide among us for beginners, in which they could acquire a know- ledge of the new calculus, and was called the Euclid of the higher geometry. But, while the author has retained the esteem justly his due, his book lias been forgotten ; other works, more learned and more com- plete, produced by the progress of science, having supplanted it's use. The method of infinites, which the marquis de rilopital and father Reyneau had adopted, was not free from some difficulties, which these authors had wholly eluded, or not sufficiently explained. It was only by dint of bringing it forward, applying it to new uses, and pointing out, as opportunity offered^ the conformity of it's results with those of the ancient methods, that it wds at length universally admitted, as no less certain and exact than all the other geo- metrical theories. It left some doubts, however, in the mind of those who did not enter sufficiently into it's true principles. On this subject the reader will excuse a slight anecdote relating to myself. When I began to study the marquis de ITIopitars book, I found it difficult to conceive, that a quantity infi- nitely small in comparison with -a finite quantity E might 418 might be absolutely neglected, without any errour. I communicated my difficulty to a celebrated geo- metrician, Fontaine, who said to me : * admit in- finitely small quantities as an hypothesis, study the practice of the calculus, and you will become a be- liever.' I found his words true, and was convinced, that the metaphysics of the analysis of infinites are the same ay those of the method of exhaustion of the ancient geometricians. The same objection to the inaccuracy of the new calculus lias been frequently repeated. In 1734 a letter appeared in England, entitled the Analyst, the author of which, a man of eminent merit in other respects, represented the method of fluxions as full of mysteries, and founded on false reasoning. There was no way of annihilating for ever these strange imputations, but by establishing the theory on prin- ciples so certain and evident, that no reasonable and intelligent man could refuse his assent to them. This difficult and necessary task was undertaken by Mac- laurin. In 1742 this gentleman published his Trea- tise on Fluxions, in which the principles of this cal- culus are demonstrated with the utmost rigour, and after the manner of the ancient geometricians, who have never been accused of laxity in the choice and soundness of their proofs. This synthetic method is a little prolix, and sometimes fatiguing to follow ; but it -affords the mind a clearness and satisfaction, which cannot be purchased too dearly. After having firmly established his foundations, Maclaurin gratifies the curiosity of the reader' with a number of beautiful problems in geometry, mechanics, and astronomy, ;> some 419 some of which are new, and all resolved with ar^ elegance remarkable for the choice of the means em- ployed These advantages place Maclaurin's book among those productions of genius, which do honour to their author, and to Scotland his country. It fcas been translated into french, and many of the ma- thematicians, who have since acquired celebrity in France, had it for the gjuide of their studies in the new geometry. While bestowing on this excellent work all the praises it deserves ; and acknowledging, that Mac- laurin contributed more than any person to feed the sacred flame of the ancient geometry among the cnglish, who make it a particular point of honour, to preserve it carefully ; it must not be dissembled, that, even at the period when the treatise on fluxions appeared, the analytical part of it was incomplete iu several respects. Analysis, however, though we ought not to entertain an exclusive predilection for it, is the true key to all the grand problems of mechanics and physical astronomy, of which we should in vain atteaapt a solution by synthesis. It was still to be wished therefore, that all the discoveries, with which geometricians had enriched and continued to enrich the science of analysis, should be collected into a system, for general use. The honour of doing this was reserved for Euler lie not only extended and improved-all the branches of analysis, in the numerous papers of his to be found among those of the academies of Petersburg and Berlin, and in several other collections, but'publishcd separate works on the subject, particularly adapted * E 2 to 420 to the instruction of readers of every class. One of the first and most important was his ' Method of finding Curves possessing the Property of Maxima or Minima,' of which a sufficient idea has already heen given. Subjoined to this treatise we find a learned theory of the curvature of elastic lamina? ; and an essay, in which the author determines, by the method of maxima and minima, the motion of projectiles in an unresisting medium ; the first important applica- tion of this method to the class of mechanical pro- blems capable of being solved by the theory of final causes. The ' Introduction to the Analysis of Infinites/ a more elementary work by the same author, pub- lished in 1748, contains in two books such instruc- tions in Analysis and pure geometry, as are necessary for those who would understand perfectly the diffe- rential and integral calculi. In the first book Euler explains every thing, that concerns algebraical or transcendent functions, their expansion into series, the theory of logarithms, that of the multiplication of angles, the summation of several very curious and useful series, the decomposition of equations into trinomial factors, &c. In the second he begins with establishing the general principles of the theory of geometrical curves, and their division into orders, classes, and genera: he applies these principles in detail to the conic sections, all the properties of which are here deduced from their general equation : and lie concludes with a very elegant theory of the surfaces of geometrical bodies. He teaches us how to find I he equations of these surfaces, by referring them them to three coordinates perpendicular to each other ; and he divides them into orders, classes, and genera, as he had done for the simple curves traced on a plane, &c. All these subjects are treated with a perspicuity and method, which facilitate their study to such a degree, that any tolerably intelligent reader may pursue them by hifnself, without any assistance. Finally Euler has collected into five or six volumes in 4to the complete science of the differential and integral calculi. All the stores of the art before known, and a great number of theories absolutely new, are here exhibited and unfolded in the most luminous and instructive manner, and under that original and commodious form, which the author has given to all parts of the higher mathematics. These different treatises together compose the most beautiful and ample body of analytical science, that the human mind ever produced. Every geometrician, who has had it in his power to read these works, has derived information from them ; and some have even arro- gated to themselves the honour of methods, which they contain. If father Reyneau were styled for a time, and by hyperbole, the Euclid of the higher geometry, it may be truly said, that Euler was this Euclid ; and it may be added, that in genius and copiousness he was far superiour to the ancient. Among the benefactors of the new geometry, I must not forget, to mention Cramer with distinction. His < Introduction to the Analysis of algebraical Curves' is the most complete treatise, that exists 011 the subject. The author leaves nothing to be wished respecting the theory of the infinite branches of E 3 curves, curves, their multiple points, and all the symptoms in general which serve to characterise them. He was contemporary with Daniel Bernoulli and Euler; like them a pupil of John Bernoulli ; and to all these great men he approached very near. We have an excellent Commentary of his on the works of James Bernoulli. In 176S, fathers le Seur and Jacquier, of the con- gregation of minims, published a ( Treatise on the integral Calculus,' a work somewhat prolix, and oc- casionally defective in method, vet containing several t/ ' / O new and interesting things, as for instance a very clear explanation of Newton's treatise on Quadratures. The art of exterminating the unknown quantities, or reducing the equations of a problem to the least number possible, is an essential part of analysis. To this several geometricians turned their attention ; among whom Cramer had already extended and sim- plified it greatly. But Bezout has made it the sub- ject of a learned treatise, in which he has carried it much farther, than any person had done before him. In the year 1801 the sciences lost Cousin, who had published several works, particularly a treatise on the integral Calculus. This treatise is charged with a little obscurity and want of order : but it is allowed to be very learned, and to contain several new things, chiefly on the solution of equations with partial dif- ferences, CHAP, 423 * CHAP. XI. Progress of Mechanics. THE science of mechanics is founded on a small number of general principles, and when these are once discovered, all the applications, that can be made of them, belong properly to geometry. But these ap- plications, particularly in the problems that relate to motion, frequently require much sagacity, and con- stitute a particular science, which the moderns have carried very far, through the assistance of the analysis of infinites. After the foundations ,of statics were laid by Ar- chimedes, it was not difficult, to discover the con- ditions of equilibrium in every particular case ; and these had guided the genius of invention in a number of machines, but they were not yet reduced to a general and uniform principle. Varignon undertook and accomplished this plan of combining them, by means of the theory of compound motions. lie gave some sketches of this in 1687, in his ' Project of a new System of Mechanics;' and he in some degree exhausted all the combinations of the equilibrium of machines, in his * General Mechanics,' not published till 1 7 ( 25, after his decease. This work, already quoted, is very prolix and tiresome to read, but it is to be commended for the perspicuity of it's detail, K 4 111 424 In the second volume of this work Varignon has given the first notions of the celebrated principle of virtual velocities, from a letter written to him by John Bernoulli in 1717. What is called the virtual velocity of a body is the infinitely small space, which the body, excited to move, tends to pass through in one ipstant of time; and the principle in question, ap- plied to' the equilibrium, may be' thus generally ex- pressed : l Let there be any system whatever of small bodies, impelled or drawn by any powers, and balan- cing each other: and let as light motion be impressed on this system, so that each body shall pass through an infinitely small space, which expresses it's virtual velocity : then the sum of the products of the powers, each multiplied by the small space, which the body to which it is applied passes through, will always be equal to nought, the motions in one direction being subtracted from those in the opposite direction.' Va- rignon applies this principle to the equilibrium of all the simple machines. In 1695 la Hire published a Treatise on Mechanics, the general object of which, like that of Varignon 's,- is the equilibrium of machines. Beside this, it con- tains various applications of machines to the arts, in which the author was well versed. Subjoined to this work is a treatise ' on Epicycloids, and their use in Mechanics.' La Hire demonstrates, that the teeth of wheels, intended to communicate motion by means of cogs [_e-ngrenagcs], should have the figure of epi- cycloids, the properties and dimensions of which he determines. This theory is very beautiful, and must rfp the author great honour; but Leibnitz, in his let- 425 ters to John Bernoulli, asserts, that it belongs to Roemer, who communicated it to him twenty years before the book of la Hire appeared. Could we suspect Leibnitz of any partiality in fa- vour of Roemer, we should soon be checked by the little probability, that such a discovery was made by la Hire, a geometrician of very moderate science. In fact no other trait of genius is to be observed in his Mechanics', in which, on the contrary, we find a gross blunder, on the subject of the isochronism of the cycloid, perhaps not the only one there. The author intending to demonstrate, prop, cxx, that a heavy body, descending along a reversed cycloid, al- ways arrives at the lowest point in the same length of time, from whatever point it began to descend, em- ploys a mode of reasoning, from which he concludes, that the time of the descent through the half of the reversed cycloid is double the time of the fall through the vertical diameter of the generating circle. But this proposition is false : for it is known by the in- contestable demonstrations of Huygens, and it may be proved in several other ways, that the former time is to the latter, as half the circumference of the circle is to it's diameter. The blunder of la Hire comes from his having assumed as. a principle, that, if we have a series of any proportions whatever, the sum of all the first antecedents is to that of all the first con- sequents, as the sum of all the second antecedents to that of the second consequents : which is true only in the single case where all the proportions, being in other respects what they may, are composed of equal ratios. A great number of elementary treatises on statics liave been published beside these ; but my plan does not 426 not allow me to analyse them all, and a simple cata^- Jogue of them would be useless. Accordingly I shall content myself with quoting the Mechanics of Camus, a work valuable for- the strictness and per- spicuity of it's demonstrations. Among other things the author gives the whole theory of toothed wheels with much accuracy and method. He was not a very profound geometrician ; but he had a very accu- rate judgment, and was well versed in the synthetic method of the ancients, which justly stood very high in his estimation. In this way he solved the problem of placing in equilibrio, between two inclined planes, a rod having a weight applied to any part of it's length. It is true this problem is very easy in the analytical method, but it leads to a calculation of some length. The synthetic solution of Camus merits attention for it's simplicity and elegance, an advantage which synthesis sometimes enjoys over analysis, and which should not be neglected when opportunity offers. A description of the machines invented within little more than a century, even were it confined to the most ingenious or most useful, would fill of itself an extensive work. Were it compatible with my design, I should by no means pass over the steam-engine, which ought to be placed in the first rank of the pro- ductions of mechanical genius. I shall merely ob- serve, that the moving power of this machine is the vapour of water, alternately expanded and con- densed ; and that it's motion is effected by mecha- nical means, nearly of the same nature as those em- ployed in clocks and watches. It appears, that the force 427 force of steam began to be made known by the expe- riments of the marquis of Worcester, about the year 1660. Afterward Papin, a french physician, having farther investigated the nature of this agent by his celebrated digester, constructed, in 1698, the first steam engine ever seen. This was very imperfect; but it gave birth to that of captain Savery, which was far superiour, and has itself been succeeded by others still more perfect. At present steam-engines are employed in every country in Europe for various purposes. I return to the general theory of me- chanics. Since the principle of the parallelogram of powers had been begun to be applied to statics, no one had thought of examining it's foundation very rigorously. All geometricians had at once agreed in admitting, that, if a body be acted upon at once by two powers capable of impelling it separately, in the same length of time, along the two sides of a parallelogram, it would be moved through the diagonal by their joint action. The same law was afterward extended to the simple powers of pressure; and it was concluded, that two powers of this kind being represented by the sides of a parallelogram, their result would be repre- sented by the diagonal. But Daniel Bernoulli, not finding sufficient connection and evidence in the transition from one case to the other, demonstrated the second proposition in a direct manner, and inde- pendently of all consideration of compound motion. Mem. of the Ac. of Petersburg for 1726. Many other geometricians, in particular d'Alembert, equally demonstrated it by various methods, more or less simple. 428' simple. Unfortunately all these demonstrations arc too long and perplexing, to find a convenient place in elementary treatises on statics ; but at least they exist among the writings of geometricians, as nu- merous guarantees of a truth, which is proved how- ever by means more simple, and better adapted to the wants -of beginners. I have already spoken of the problems of the ca- tenary curve, the sail distended by the wind, the elastic curve, &c., when giving an account of the progress of the analysis of infinites, to which they directly contributed. These problems, and several others of the same nature, were again resolved by Daniel Bernoulli, Euler, Hermann, and others, but with new additions, and fresh difficulties, which en- Jianced the honour of success, and enlarged the do- mains of science. The general theory of various motions offered a new and extensive field to the researches of geometricians furnished with the analysis of infinites. Galileo had made known the properties of rectilinear motion uni- formly accelerated : Huygens had considered curvili- near motion ; and had risen by degrees to the beau- tiful theory of central forces, in the circle, which is equally applicable to motion in any curve, by consi- dering all curves as infinite series of small arcs of a circle, agreeably to the idea, which he himself had employed in his general theory of evolutes. The laws of the communication of motion, like- wise, sketched by des Cartes, and farther pursued by Wallls, Huygens, and Wren, had made a new and very considerable step, by means of the solution which 429 \vliich Httygens gave of the celebrated problem, of centres of oscillation. All these acquisitions, at first separate and in some measure independent of each other, having been re- duced to a small number of simple, commodious, and general formulas, by means of the analysis of infinites, mechanics took a flight, which nothing but the difficulties still arising from the imperfection of it's instruments could check, and of which I will endeavour to impart some idea. The problems relating to motion may all be reduced to two classes. The first comprises the general pro- perties of the motion of a single body, acted upon by any given powers : the second, the motions which result from the action and reaction, that several bodies exert on each other, in any given manner. In the motion of a single body, we observe, that, matter being of itself indifferent to rest or motion, a body set in motion must uniformly persevere in it; and that it's velocity cannot increase or diminish, unless by the continual action of a constant or varia- ble power. Hence arise two principles ; that of the vis inertia, and that of compound motion ; on which is founded the whole theory of motion, rectilinear or curvilinear, constant or variable according; to a ffiveii 7 O O law. By virtue of the vis inertire, motion at every instant is essentially rectilinear and uniform, all re- sistance, and every kind of obstacle, being put out of consideration. By the nature of compound mo- tion, a body exposed to the action of any given number of forces, all tending at the same time to change the quantity and direction of it's motion, takes 430 takes such a path through space, that in the last in- stant it reaches the same point, at which it .would have arrived, had it successively and freely obeyed each of the forces proposed. On applying the first of these principles to recti* linear motion uniformly accelerated, we perceive, 1st, that, in this motion, the velocity increasing by equal degrees, or proportionally to the time, the accele- rating force must be constant, or incessantly give equal impulses to the moving body ; and that, con- sequently, the final velocity is as the product of the accelerating force multiplied by the time : 2dly, each elementary portion of space passed through being as the product of the corresponding velocity multiplied by the element of the time, the whole of the space passed through is as the product of the accelerating- force multiplied by the square of the time. Now these two properties equally take place for each ele- mentary portion of any variable motion whatever : for there is no reason why we should not generally consider the accelerating force, though varying from one instant to another, as constant during each sin- gle instant, or as undergoing it's changes only at the commencement of each elementary portion of the time. Thus, in every rectilinear motion, variable according to any given law, the increment of the velocity is as the product of the accelerating force, multiplied by the element of the time; and the dif- ferential second of the space passed through is as the product of the accelerating force multiplied by the square of the element of the time. Now 431 Xow if to this principle we add that of compound motion, we shall arrive at the knowledge of all curvi- linear motion. In fact, whatever be the forces ap- plied to a body describing any curve whatever, we can at each instant reduce these forces to two, the one the tangent to the element of the curve, the other the perpendicular to it The first produces an instantaneous rectilinear motion, to which the principle of the vis inertias applies : the second is expressed by the square of the actual velocity of the body, divided by the radius of curvature, agreeably to the theory of central forces in the circle ; which equally reduces to the same principle the motion in the direction of the radius of curvature. Such are the means that were long employed to fletermine the motions of isolated bodies, acted upon by accelerating forces, of whatever magnitude or di- rection. Newton followed this method : he only clothed his solutions in a synthesis, which frequently conceals the greatest difficulties under the appearance of elegance and simplicity. In 1716 Hermann published a treatise De Phoro- mwia, in which he undertook to explain all that re- gards the mechanics both of solids and fluids ; that is to say, statics, the science of the motion of solids, hydrostatics, and hydraulics. This multiplicity of objects did not allow him to explain them as fully and clearly as was requisite. Besides, like Newton, he affects to employ the synthetic method as much as possible, which frequently interrupts the unity and connection of his problems. To this may be added, that in some places he is mistaken, The 432 The Mechanics of Euler, published in 1735, eon-* tain the whole theory of rectilinear or curvilinear motion in an isolated body, acted upon by any accele- rating forces whatever, either in vacuo, or in a re- sisting medium. The author has every where fol- lowed the analytical method ; which, by reducing all the branches of this theory to uniformity, greatly facilitates our understanding it, as Euler manages this method with an elegance and sagacity, of which be- fore him there was no example. He not only re- solves a number of difficult problems, some of which were then new, but he even improves the analysis it- self by new and delicate solutions, to which his sub- ject gives occasion. As to the principles of me- chanics for putting the problems into equations, he employs those mentioned above. Though this manner of laying the foundation of the calculation was sufficiently commodious, the same end might be attained by means still more simple, This was, to resolve at every instant the forces and the motions into other forces and other motions, parallel to fixed lines of a given position in space. Nothing more then is necessary, but to apply the equations of the principle of the vis inertias to these forces and motions ; in which case there is no need of recurring to the theorem of Huygens. This simple and happy idea, of which Maclaurin first made use in his Treatise on Fluxions, has thrown new light on mechanics, and singularly facilitated the solution of various problems. When the body moves constantly in one plane, two fixed axes only are to be taken, wn*cn are Supposed to be perpendicular to each other, 433 for the sake of greater simplicity : but when we arc obliged by the nature of the forces, to change the path continually in all directions, and to describe a curve of double curvature, three fixed axes are to be employed, perpendicular to each other, or form- ing the edges of a rightangled parallelepiped. The problems of the communication of motion, commonly called dynamic problems, required new principles. These, for instance, consist in deter- mining the motions, that result from the mutual percussion of several bodies ; the centre of oscilla- tion of a compound pendulum; the motions of se- veral bodies strung upon a rod, which has a rotatory movement round a fixed point; c. Now it is evi- dent, that in all cases of this sort the motion is not the same as if the bodies were isolated and at liberty, but that there must be a distribution of the forces among all the bodies forming one whole, so that the motion lost by some of them is gained by others. The motion lost or gained is always esti- mated by the product of the mass multiplied by the velocity lost or received, whether the communication or loss of motion be produced every instant by finite de- grees, as in the shock of hard bodies, or whether the velocity change at each instant only by degrees infi- nitely small, as in the motions of several bodies strung on a movable rod; and generally in all cases in which the forces act in the manner of gravitation. When Huygens gave his solution of the problem of centres of oscillation, some unskilful geometri- cians attacked it in the reviews. James Bernoulli defended it in the Leipsic'Transactipns for l6S6, and F f under 434* undertook to give a direct demonstration of it by means of the principle of the lever. At first he con- sidered only two equal weights, fastened to an inflex- ible rod devoid of gravity, which was movable round a. horizontal axis. Having then observed, that the velocity of the weight nearest the axis of rotation must necessarily be less, and that of the other on the contrary greater, than if each acted upon the rod separately, he concludes, that the force lost and the force gained balance each other, and that conse- quently the product of the quantity of matter in one multiplied by the velocity it loses, and that of the other multiplied by the velocity it gains, must be in- versely proportionate to the arms of the lever. The substance of this luminous reasoning is accu- rate: only James Bernoulli mistook at setting out, in considering the velocities of the two bodies as finite; instead of which he ought to have considered the elementary velocities, and compared them with the similar velocities produced every instant by the action of gravitation. The marquis de FHopital re- marked this errour, and in correcting it he found the , centre of oscillation of the two weights, without de- parting in other respects from the principle of Ber- noulli. Desirous then of proceeding to a third weight, he united the former two at their centre of oscillation, and combined this new weight, with the third, as he had combined together the former two; ancl so on. But tliejiiiion proposed was a little precarious, and could not be admitted without a demonstration. The paper of the marquis therefore produced no other advantage, than that of inducing James Bernoulli to revise 435 revise his former solution, to improve it, and extend it; to any number of bodies, All this was successively done. James Bernoulli began with remoulding his first, and sketching a general solution: and at length he resolved the problem completely, whatever were the number and position of the bodies composing the whole. His method consists in resolving the motion of each body, at any given instant, into two other motions ; one, that which the body actually takes ; the other, that which is destroyed ; and in forming equations, which express the condition of equilibrium between the motions lost. By these means the pro- blem is brought under the common laws of statics. The author applies his principle to several examples : and he demonstrates strictly, as well as in the most evident manner, the proposition which Huygens em- ployed as the basis of his solution. At the conclusion of this memorable paper, which is among those of the Academy of Paris for 1703, he shows by the same principles, that the centre of oscillation and the centre of percussion exist in the same point. This solution of the problem of centres of oscillation seemed to leave nothing to be desired ; yet in 1714 it was brought forward again by John Bernoulli and Dr. Taylor, who gave solutions of it that were fun- damentally the same. Their similarity occasioned a warm dispute between them, each accusing the other of plagiarism. .In this new mode of treating the question it is supposed, that, instead of the ele- mentary weights of which the pendulum is composed, other weights are substituted in one and the same point, such that their motions of angular acceleration, F F 2 and 436 and their motions with respect to the axis of rotation, shall be the same, and the new pendulum oscillate as the former. While confessing, that this solution, merits praise, all geometricians at present agree, that it is neither so luminous, nor simple, as that of James Bernoulli, founded immediately on the laws of equi- librium. It has been seen, that Leibnitz estimated the mo- menta of bodies in motion by the product of the quantity of matter multiplied by the square of the velocity. John Bernoulli, having adopted this opi- nion, gave to the principle of II uy gens, for the pro- blem of centres of oscillation, the name of c the prin- ciple of conservation of the vires m\:' and this it has retained, because in fact, in the motions of a system of heavy bodies, the sum of the products of the masses by the squares of the velocities remains the same, when the bodies descend conjointly, and whem they afterward ascend separately with the velocities they acquired by their descent. Huygens himself had briefly made this remark, in a letter on the first paper of James Bernoulli and on that of the marquis cle THopitaL This law equally holds in the shock of perfectly elastic bodies, and in all the motions of bodies acting on each other by pressure: it neces- sarily follows from the nature of these movements, and is independent of every hypothesis respecting the. measure of vires viva;. Accordingly the geome- tricians of the last century employed it with success in many problems in dynamics. But as it gives only a single equation, from which the velocity or the must be afterward expunged, this second object was 437 ivas attained by different means. John Bernoulli employed for this the principle of tensions', Euler, that of pressures \ Daniel Bernoulli, the virtual power, which d system of bodies has of reestablishing itself in it's former state; and in certain cases both Euler and Daniel Bernoulli, the constant quantity of circula- tory motion round a fixed point. When at length all the differential equations of the problem were established, nothing more remained but the difficulty of resolving them ; a new rock, on which analysts of moderate talents sometimes split. In 1743 d'Alembert had the happy thought of ge- iieralizing the principle, which James Bernoulli had employed for resolving the problem of centres of oscillation. He showed, that, in whatever manner the bodies of one svstem act on each other, their y * motions may always be resolved, at every instant, Into two sorts of motion \ those of the one being de- stroyed the instant following, and those of the other retained ; and that the motions retained are neces- sarily known from the conditions of the equilibrium between the motions destroyed. This general principle applies to all the problems of dynamics, and at least teduces ail their difficulties to those of the problems of simple statics. It also renders that of the conservation of vires vivce useless. By means of it d'Alembert has resolved a number of very beautiful and very difficult problems, some of which were absolutely new, as, for example, that of the precession of the equinoxes. His Treatise on Dynamics, therefore, published in 3749, must be considered as an original work. In Vain may it be objected, that James Bernoulli had F 3 traced 43S traced out the road: it was equally traced out for other geometricians, who preceded d'Alembert, yefc none of whom perceived it during the space of fprfy years. Dynamics having thus gradually attained a high degree of perfection, were still farther enriched in. 1765 by an important discovery, fertile in corollaries. Jn a short paper, entitled, Specimen Theories Turbinurm Segner observed, that, if a body of any size and figure, after rotatory or gyratory motions in. all directions have been given to it, be left entirely to itself, it will always have three principal axes of rotation ; that is to say, that all the rotatory motions, by which it is affected, may be constantly reduced to three, which are performed round three axes perpendicular to each other, passing through the centre of gravity or of inertias of the body, and always preserving the same position in absolute space, while the centre of gravity is at rest, or moves uniformly in a right line. The position of these three axes is determined by an equa- tion of the third order, the three real roots of which relate to each of them. This theory, which it's author had not sufficiently developed, Albert, the worthy son of the great Euler, treated at length in his paper on the stowage of ships, which shared the prize of the Academy of Sciences of Paris in l?6l : as did likewise his father, according to the same method, in the Memoirs of the Academy of Berlin for 1759, and in his work entitled Theoria Motus Corporum rigidorum, 1 765. Lastly d'Alembert showed in the fourth volume of his Mathematical Opitscula, published in 17,68, that the solution of the 1 problem 439 problem was deducible from the formula?, which he had given in a ' Memoir for determining the motion of a Body of any Figure, acted upon by any Forces whatever/ printed in the first volume of his Opuscula, in 1761. The knowledge of these motions of free rotation round three principal axes easily leads to the deter- mination of the motion round any variable axis what- ever. Hence, if we now suppose the body to be acted upon by any given accelerating forces, we shall begin with determining the rectilinear or curvilinear motion of the centre of gravity, abstractedly from all rotatory motion : then combining this progressive motion with the rotatory motion of a given point of the body round a variable axis, we shall know at every instant the compound motion of this point in absolute space. In this manner Euler has resolved several new pro- blems of dynamics. F 4 45HAPt 440 CHAP. XIT. Progress of Hydrodynamics. THE principle of equal pressure, applied to the ge- neral laws of hydrostatics, \^as sufficient to explain all the particular cases of equilibrium, which are referrible to it. But the science of the motion of fluids, at least as to it's theoretical part, was still confined to the single proposition of Torricclli ; that is, to the knowledge of the issue of fluids through infinitely small, or very minute apertures. Newton undertook to- solve this problem in his Principia, A. D. 1686, without confining himself to this supposition. He considers a vertical cylindrical vessel, perforated through the bottom by a hole of any diameter, from which the water issues, while the vessel is constantly receiving at the top as much as escapes at the bottom, in such a manner, that the water flowing in may be considered as forming a stra- tum of uniform thickness, suddenly spread on the surface of the water of the cylinder, which thus re- mains constantly filled to the same height, He then conceives the water in the cylinder to be divided into two parts, the one central, and freely movable, which he calls the cataract ; the other adjacent, and im- movable, which is confined externally by the sides of the vessel. He supposes, that the velocity of any horizontal section of the cataract is owing to the cor- respondent 441. respondent height of the water of the cylinder, in- cluding in this height the thickness of the stratum that replaces what is left : and as, at the same time, to preserve the continuity of the cataract, the velo- cities of it's different horizontal sections must be ia the inverse ratio of their surfaces, it is demonstrated by calculation, that the cataract must assume the form of a solid produced by the revolution of an hy- perbola of the fourth order round the vertical line, which passes through the centre of the orifice. Hence we know the quantity of water, that flows out in a given time. As Newton did not at first notice the diminution of the quantity of water flowing out, that must be occasioned by the contraction of the stream at it's effluence from the aperture, he had concluded, that the velocity at this effluence js simply owing to half the height of the water in the cylinder; which is contrary to the experience of jets-d'eau. In the second edition of his work he corrected this mistake; but his general theory remained no less vague, pre- carious, and even fundamentally false ; for experience and the laws of hydrostatics have demonstrated, that the formation and figure of the newtonian cataract are physically impossible. See John Bernoulli's Works, vol. IV, p. 484. Many authors, as Varignon, Guglielmini, and others, giving greater extent to the theorem of Tor- ricelli than it admits, produced only hypothetical and uncertain propositions on the theory of running water, or the motion of water flowing through chan- nels, which were sometimes palpably contradicted by expe- experience. This defect in Guglielmini's Treatise on Mivers is compensated by some excellent physical re- marks on the course of waters. Let me add, the difficulty of the problem should lead us to excuse all these useless or barren attempts, I do not here speak of th published in 1714, exhibited the truth in it's proper light, and pointed out the parallogisms, into which the chevalier had fallen. Bernoulli de- tected another grand mistake of Renau, respecting the angle of lee way in traverse sailing. Though Bernoulli did not give solutions sufficiently general of most of the problems connected with his subject, he nevertheless rendered very great service to the art of navigation, in laying down with accuracy the prin- ciples then admitted, on which questions of this nature must be founded. The most difficult problems in the working of vessels had been attempted from the beginning, with- out the conditions essential to their equilibrium hav- ing been much investigated : yet on these conditions the safety of navigation depends, as well as all the advantages, that can render it speedy and easy. Ac- cordingly the mathematicians turned back again, and resumed the naval science in some respect from it's elements. It had long been known, that for a solid body, floating on a fluid, to remain in equilibrium, it was necessary, 1st, that it's absolute weight, and that of the fluid displaced, should be equal : 2dly, that the centre of gravity of the body itself, and that of the immersed part of it, considered as homogeneous, should be in the same vertical line. But this is not sufficient, to form a solid and permanent equilibrium. Daniel Bernoulli showed in addition, that, with re- gard to the various situations, which the two centres of gravity might respectively have in the vertical G G 2 line, 452 line, there are various states of equilibrium, more or less stable. When the centre of gravity of all the matters that constitute the burden of a vessel is placed below the centre of gravity of the ship's bottom, or the part immersed, the -equilibrium is always stable, or has a tendency to restore itself, if it have been de- ranged by any external cause, as the agitation of the waves, the inequality of the impulse of the wind, &c. ; and the vessel returns to her former position with so much the more energy, in proportion as it's centre of gravity is placed lower. But when the two centres of gravity are confounded together, or when that of the ship is higher than that of the bottom, the equi- librium is versable, and the more so, the higher it is. Daniel Bernoulli gave formulas for estimating the de- gree of stiffness of a vessel in all cases-: it also appears, that Euler had separately found similar results at the same time, which he develops and demonstrates in his Scientia Navalis, published in 1749. Bouguer explains the same theory at length, in a new and very simple manner, in his Treatise on a Ship, published in 1746... Under the term of metacentre he shows the limit, beneath which the centre of gravity of the whole burden of a vessel ought to be placed ; and he examines the best position for the masts, the extent that should be given to the sails, and the various motions of rolling and pitching that may take place, according to the changes of the velic point, that is, the point in which we may conceive all the effort of the wind upon the sails to be united. The practical skill, which he combined with profound theoretical 453 theoretical knowledge, enabled him to throw such light on this suhject, as may be of great utility to the mariner, Bouguer treated still more particularly On the Working of Ships, in another performance, published under this title in 1757. But unfortunately there is a radical defect in his researches, which considerably diminishes their practical utility : they are founded for the most part on the common theory of the re- sistance of fluids, which can be employed only with the restrictions I have mentioned. The Academy of Sciences at Paris, attentive to every means of improving navigation, proposed as the subjects of it's prizes several questions relating to this important object, as for instance the best mode of masting vessels, with respect to the situation, number, and height of the masts : the most perfect figure and construction of anchors : the correction and improvement of the capstan : the most advan- tageous rules for the stowing a ship's hold, whether for diminishing the motions of rolling and pitching, increasing the velocity of the ship's way, or render- ing her more quick in answering the helm : &c. The pieces, to which the prizes were adjudged, have been productive of advantages, with which the learned and experienced seaman will be eager to become ac- quainted. G G 3 CHAP. 454 CHAP. XII I. Progress of Astronomy. THE progress, that astronomy has made within the last hundred years, would appear astonishing if we did not reflect on the assistance it has received from na- tural philosophy, mechanics, and geometry; either in improving it's ancient instruments, inventing new, rendering it's observations more accurate, or appre- ciating and removing all the causes of variation, whether real or apparent, by which these observations are liable to be affected. Every thing has conspired to give new fife to this science, and to connect all it's parts more closely to- gether, by a more intimate knowledge 0f their mu- tual relations. Several new phenomena of the hea- vens have been discovered : the theory of the planets, both primary and secondary, has been improved, and tables of their motions have been constructed, far superior to any that existed before: a great number of comets have been carefully observed : &c. The geometricians on their part have exerted themselves, to assign the physical causes of the celestial motions with accuracy ; and their calculi have been of infi- nite use to practical astronomy itself, by the advan- tage they enjoy of connecting together the observa- tions of any phenomenon, and of subjecting to the law 455 law of continuity those isolated facts, which these observations make known. It is obviously impossible here, to enter at large into so many labours, which would require a separate history. Not to deviate from my plan, I must con- fine myself to the discoveries, that particularly cha- racterize the astronomy of this fourth period. I shall divide the present chapter into two sections; the first will comprise practical astronomy, that is to say, the knowledge of the celestial motions founded directly on observations, or on deductions from ob- servations; the second, physical astronomy, or the explanation of the celestial motions by applying geo- metry to the laws, by which these motions are regu- lated. SECTION 456 SECTION I. Practical Astronomy. MODERN astronomers, enjoying the advantage of excellent instruments, have not only improved all the ancient theories of the celestial motions, but have established several others of the highest importance, some of them absolutely new, and the rest merely conjectural before. Among these are to be distin- guished the libration of the Moon, the aberrations of the fixed stars, the nutation of the Earth's axis, the catalogues of the fixed stars, the figure of the Earth, and the general laws of the motion of comets. As soon as men began to contemplate the Moon, it was perceived, that it constantly presented the same face to the Earth, for it's spots were always the same, and arranged in the same manner. The ancient astronomers stopped at this general observation ; but an attentive examination of the lunar spots showed Galileo, that the Moon had a libratory motion on it's centre, in consequence of which certain spots toward the edges disappeared for a time, then reappeared, and thus continued to disappear and reappear alter- nately. This motion is what is called the. libration of the Moon. Galileo explained it generally by a rota- tory motion about an axis, which he ascribed to the Moon while at the same time it was revolving round the Earth ; but he determined neither the exact posi- tion 457 tion of this axis, nor the precise quantities of the libratory motions it ought to produce, both in lati- tude and longitude. As the primary planets, which revolve on their axes, present to us different spots, and these appear- ances are even the marks, by which their rotatory motions have been discovered ; des Cartes, seeing nothing similar in the Moon, maintained, that it has no rotatory motion. To account for the constant ap- pearance of the same spots, he supposed the lunar globe to consist of two hemispheres of unequal gra- vity, separated by the circle perpendicular to a line drawn from the Earth to the centre of the Moon ; and he concluded, that, both these hemispheres be- ing subjected to the action of the centrifugal force, which is produced by the revolution of the Moon round the Earth, the heaviest, or most massive, having the greatest centrifugal force, must constantly keep itself farther from us. As to the libratory mo- tion, this, according to the same author, is produced by a slight libration of the circle, that forms the base of the two hemispheres. It is unnecessary to remark how hypothetical all this explanation is. The pretended inequality in the gravity or mass of the two hemispheres is destitute of all probability ; and besides, it gives only a vague and insufficient reason for the phenomena of the libration. The celebrated Dominic Cassini, and his son James, worthy of such a father, were the first who gave a complete and accurate explanation of this motion of the Moon, which is conformable to observation, and has consequently been adopted by all astronomers, It 458 It is given by James Cassini in the Memoirs of tf*e Academy of Sciences at Paris for the year 1721, and in his Elements of Astronomy, published in 1?40. According to him the libration of the Moon is pro- duced by the combination of two motions, one of which is the revolution of this planet round the Earth, the other is a rotatory motion of the Moon round an axis, subject to the following conditions : 1st, The axis of rotation of the Moon is inclined to the plane of the ecliptic in an angle of 87 degrees and half, and to the plane of the lunar orbit in an angle of 82 degrees and half: so that the plane of the Moon's equator makes an angle of two degrees and a half with the plane of the ecliptic,, and an angle of seven degrees and a half with the plane of the lunar orbit. 2dly, The poles of the Moon are in the circumfer- ence of the great circle, which is formed by cutting it's globe at every instant by a plane parallel to that great circle in the heavens, which passes through the poles of the ecliptic, and those of the lunar orbit. This circle may be called the colure of the Moon, for the same reason as the great circle, which passes through the poles of the ecliptic and those of the equinoctial line, is called the colure of the solstices. 3dly, The globe of the Moon turns round it's axis according to the order of the signs, or from west to east, in the space of 27 days 5 hours ; a period equal to that of the Moon's return to the node of it's orbit with the ecliptic. This motion is analogous to the revolution which the Earth makes round it's axis in the 459 the order of the signs, returning to the same colure in the space of 23 hours 56 minutes. The geacral result of these suppositions is, that, if we prolong in imagination the axis of the globe of the Moon into the heavens, the extremities of this axis will appear to us, to describe round the poles of the ecliptic, from which they are two degrees and li'il- distant, two polar circles, from east to west, in IB years and 7 mouths, or in the same time and di- rection as the nodes of the Moon. This motion, we see, is similar to that, by which the poles of the Earth make their revolution round the poles of the idiotic from east to west, in two circles distant from tliera 9.3 degrees and a half, and in a period of about 00 years : which produces the apparent motion of the stars from west to east in the same time, and in consequence he precession of the equinoxes. A full explanation of ch'e phenomena of the Moon's libra- tion does not belong to this work, but must be sought in treatises on astronomy. A discovery still greater, both with respect to it's difficulty, and to it's influence on all parts of astro- nomy, is that of the causes, which produce the ap- parent aberration of the fixed stars. For this we are indebted to Bradley, the english Hipparchus. Among the reasons, that were alleged against th$ system of Copernicus, at the time when it was first brought forward, was this, as we have already ob- served, that, if the Earth really revolved round the Sun, it ought to occasion a parallax in the star$, which was termed the parallax of the great orb, when it passes from one point of it's orbit to that diametri- cal 460 cally opposite. This was a solid objection, to which Copernicus and Galileo could answer only by con- jectures, that did not obtain a complete assent. Succeeding* astronomers, persuaded of the exist- ence of this parallax, employed every means of detecting it's quantity. Some imagined they had determined it; and ventured to say, that it was four or five seconds. A greater number, supported by more accurate observations, found it altogether imperceptible; and this opinion prevailed. But it did not subvert the copernican system : it only led to the conclusion, that the distance from the Earth to the stars was so immensely great, that' it must be considered as infinite with regard to the di- ameter of the terrestrial orbit. Still certain per- ceptible motions observed in the. stars, contrary for the most part to those, which the parallax of the great orb and the precession of the equinoxes should have produced, remained to be accounted for. These irregular motions were designated under the general denomination of the apparent aberrations of the jived stars. Not knowing to what to ascribe them, astro- nomers took every precaution to avoid the errours, which they might have occasioned in determining the motions of the planets with regard to the fixed stars. Mr. Molyneux, an astronomer of Ireland, under- took in 1725, to determine these motions of aberra- tion. He observed them at Kew, near London, with an excellent sector made by Graham, but he was un- able to reduce them to general laws. Bradley 461 Bradley was more fortunate; being an excellent observer, as well as a learned geometrician he pur- sued the same research, at the same place, with a constancy, which led him at length to a perfect know- ledge of all these singular phenomena. He perceived, that certain stars appeared, in the space of a year, to have a kind of libration in longitude, without changing their latitude in the least ; and that others varied their latitude alone ; while a still greater number appeared to describe in the heavens, during the same period, a small ellipsis, more or less prolate. This period of a year, to which all these motions answered, though so different in other respects, was a certain indication, that they had some connection with the motion of the Earth in it's orbit round the Sun : but this was no more than a general hint, in- sufficient to give a precise and complete reason for the phenomena. Bradley made a new step, which decided the question. He conceived the happy idea, that the apparent aberration of the fixed stars was produced by the progressive motion of light, com- bined with the armual motion of the Earth : and he arrived at this by reasoning with himself as follows. The theory of Roemer teaches, me, that the velo- city of light is not instantaneous, and that it has a definite ratio, of about 10000 to 1, to the velocity of the Earth in it's orbit. Consequently a ray of light, issuing from a star, and conveying the impression of that star to my eye, does not arrive, till the Earth has undergone a sensible change of place since the instant when the ray departed: accordingly, when my eye receives the impression, it must refer the star to to a place different from that, to -which it would have referred it, had I remained still in the same place. An observer on the Earth, therefore, does not see the stars in their real places in the heavens, and must ascribe to them different motions, which depend on the different positions they have with respect to him/ Furnished with this key, Bradley explained all the, apparent aberrations of the fixed stars in an exact and precise manner, conformable to his own observa- tions and those of all other astronomers. Thus all these uncertainties were removed ; and to the proofs already adduced in support of the copeniican system he added a new one, which may be called a mathe- matical demonstration. Not contented with having laid the foundation of this theory by observations, he reduced it into trigo- nometrical formulae, the results of which he pub- lished, without demonstrations, in the Philosophical Transactions for 17 17. The novelty and interesting nature of the subject at- tracted the attention of all astro 13 omers and geometri- cians. Clairaut gave the demonstrations, which Bradley had suppressed ; and he added to them se- veral other easy and commodious theorems ; an im- portant service, which contributed not a little to ac- celerate the progress of this new branch of astro- nomy. About ten years after, the same geometrician ap- plied the theory of aberration to the motions of the planets and comets ; to which, it is evident, it must equally apply. The time, which light takes in coming from a planet or a comet to the Earth, neces- sarily 463 sarily produces some apparent change in the situation of the comet or planet. The problem therefore is of the same nature as for the stars ; with this difference however, that, the stars being fixed, while the planets and comets have motions to be taken into the ac- count, the formulas of aberration for these are of course more complex. To this must be added the difficulty in the calculation, which arises from the eccentricity of the orbits of the planets and comets. Modern astronomy is indebted to Bradley for ano- ther discovery not less remarkable, that of the nuta- tion of the Earth's axis, to which geometry smoothed his way, by pointing out the observations he must make to arrive at it. Having the general knowledge, that the inequali- ties of the attractions of the Moon and the Sun, on different parts of the terrestrial spheroid, must occa- sion different motions in it's axis with regard to the plane of the ecliptic, Bradley applied himself to dis- cover and unravel these motions, by a long series of laborious and delicate observations, made in thosQ positions of the Sun and Moon, which were best cal- culated to show the effects he sought. Accordingly he found, 1st, that the axis of the Earth has a conical motion, by which it's extremities describe round the poles of the ecliptic, and contrary to the order of the signs, a complete circle in 25000 years, or about an arc of 50 seconds in a year, which produces the precession of the equinoxes: Sclly, that this axis has a libratory motion with regard to the plane of the ecliptic, or an alternate preponderatiou of each extremity, by which it is inclined about 18 5 seconds 464 seconds during one revolution of the lunar nodes ; which make their circuit, contrary to the order of the signs, in a period of about nineteen years : after which period the axis returns to it's former position, to incline again new. These observations, which are consistent with the newtonian theory of attraction, constitute an additional demonstration of it, as I shall more particularly observe farther on. Since these discoveries, the nutation of the Earth's axis forms as essential a part of astronomical calcula- tions, as the precession of the equinoxes, the quan- tity of which was pretty nearly known before. As the fixed stars are points, to which the mo- tions of the planets are referred, astronomers at all times have bestowed great pains on augmenting their number, and fixing their respective positions. These are the two chief objects of catalogues of the stars. It has been seen, that Hipparchus had made an accu- rate enumeration of the stars known in his time ; and Ptolemy and the arabian astronomers afterward im- proved his labours. In the preceding period notice was taken of Flamsteed's catalogue of the stars visi- ble in our climates, and of that drawn up for the stars of the southern hemisphere from the observa- tions made by Halley at St. Helena. La Caiile, one of the best and most indefatigable astronomers that ever lived, after having calculated the positions of a great number of stars in France, undertook a voyage to the Cape of Good Hope in 1751, in order to en- large and improve the catalogue of southern stars. I shall not enter into a minute account of the means he employed, and the precautions he adopted, to execute this 465 this grand work so Useful to astronomy, and at pro sent one of it's grand foundations : but I shall ob- serve, that he brought to Europe an accurate cata- logue, carefully verified, of more than 9800 stars, included between the south pole and the tropic of Capricorn* During the course of these principal observations, la Caille occasionally made others on different inte- resting points of astronomy, as on refractions, the elevation of the pole, the length of the pendulum, and the longitude of the Cape of Good Hope, on which the opinions of the ablest geographers differed more than three degrees. He took particular pains to observe the meridian altitudes of Mars, Venus, and the Moon, which enabled him to determine their parallaxes with precision, by comparing his observa- tions with those made at the same time in France, England, Sweden, and Prussia. Lastly he measured a degree of the Earth, of which I shall have occa- sion to speak more at large in the succeeding article. The question of the figure of the Earth is of the highest importance to Astronomy and Navigation. Hence attempts have been made in all ages to solve it ; but it is only since the measure taken by Picard, that we have begun to obtain results, on the accu- racy of which dependance may reasonably be placed. This astronomer found in 1669, that the length of a degree of the meridian was 57060 toises [121569 yards], in the latitude of 49 23' north. Though this determination was considered as incomparably more accurate than any that had preceded it, some- ii a thing 466 tiling still remained to be desired, both from thf want of. precision in some of the elements on which it was founded, and because it was not sufficient, to give a complete idea of the figure and dimensions of the terrestrial globe. To calculate the length of the degree, Pieard had employed thirteen triangles, in a space of thirty two leagues. Now some sensible errours might have crept into the trigonometrical re- solutions of so many triangles : and on the other hand the best instruments then known could not give the value of the corresponding celestial arc within four seconds, which four seconds would amount to near seventy toises on the. Earth. Lastly, a single degree could not make known, whether the Earth be spherical, or deviate from this figure. These considerations being laid before the french government, always favourably inclined toward the sciences, it issued orders, not only that Picard's mea-. sure should be verified, but that the meridian line should be continued from it to Dunkirk in the north, and to Colioure in the south ; which includes a space of about eight degrees. The northern part was en- trusted to la Hire; and the southern to Dominic Cassini, who was afterward assisted by his son James. From these operations, begun in 1683 and finished in 1701, it appeared, that the mean length of the degree in Fiance is .57061 toises, about a toise more than Pieard assigned it. The mathematicians employed on these measures, persuaded by the experiment of the shortening of the pendulum at "Cayenne, and by the theory of Huygens 467 Huygens and Newton, that the Earth was a spheroid flattened toward the poles; but misled by a false ap- plication of -geometry, which led them to imagine, that in such a spheroid the degrees of the Earth must diminish in length in proceeding from south to north ; were not perhaps sufficiently on their guard against the sources of illusion, to which this prejudice might give rise. Whether from this cause, or from the want of accuracy in their instruments, or from some negligences almost inevitable in so long a series of observations, they found, that the degrees did in fact diminish from south to north : and this result they hastened to pub- lish with the more confidence, as they imagined thereby they confirmed the flattening of the Earth, 'which was considered as highly probable. The question appeared to be completely resolved, and philosophers remained for some years secure hi the conviction, that the observations agreed with the theory* at least as to the general consequence* But at length some geometricians interrupted this tran- quillity. They demonstrated* that this pretended agreement of observation with theory was founded on a geometrical parallogism ; and that, in a spheroid flattened toward the poles, the degrees of latitude ought to increase from south to north, and on the contrary diminish in a prolate spheroid. In fact, it does not require the assistance of a geometrical figure to convince us, that, in an oblate spheroid, the terrestrial meridian being more curved near the equator, than round the pole, the length of H H 2 the 468 the terrestrial arc of a degree, corresponding to the ce- lestial arc of a degree, must increase in proportion as the curve of the terrestrial meridian diminishes, or in proportion as it approaches the pole. In a prolate spheroid the contrary must take place. The truth of this reasoning, so simple and conclusive, could not fail soon to strike every mind. The authors of the new measures then found themselves greatly em- barrassed. On the one hand, unable to reject the demonstrations brought against them ; and on the other, unwilling to give up observations, which they considered as very certain ; they were at length re- duced to assert, that the Earth was elongated toward the poles. New measures, taken likewise in France in 1733 and 1736, seemed to confirm the opinion, that the length of the degree diminished from south to north. Thus for about forty years the Earth was considered, in France at least, as a prolate spheroid, in spite of Huygens and Newton. Still however geometricians were not convinced. From time to time they renewed their protestations against a system, which they could not reconcile with the laws of hydrostatics. They maintained, that, even supposing the observations made in France to be as accurate as possible, the differences between the de- grees were too small to be perfectly ascertained; and that well marked and adequate differences could be obtained only by the comparison of degrees measured in places vefy remote from each other in their meridian distance. The french government listened to objections so well founded : and the count de 469 de Maurepas, then minister of the academy of sci- ences, sent a company of mathematicians to measure a degree of the meridian in Peru, near the equator, and another to perform a similar operation in Lap- land, within the polar circle. Godin, Bouguer, and la Condamine set out on their voyage for the former purpose in 1735 ; and the year following Maupertuis, Ciairaut, Camus, and le Mon- nier repaired to Lapland, being joined by Celsius, the celebrated professor of astronomy at L T psal. The former met with impediments and delays of every kind in their operations, and could not return to France, till near seven years had expired ; the others found every thing as they could wish ; their work was begun and finished in a short time ; and they returned to their country, after being absent only fifteen or sixteen months. Perhaps it would have been proper, to have waited for the return of the academicians from Peru, before an account was given of operations undertaken for the same purpose ; as was the opinion of the more moderate and equitable of the party. But Maupertuis, the leader of the northern expedition, a man eager to appear upon the scene, negatived a proposal so contrary to his views. He made it his first business to proclaim every where, in the academy, to the public, and among the great, with whom he had considerable intercourse, the result of an operation, all the glory of which he in some degree appropriated to himself, though he had but a very moderate share in it as a fellow labourer. This result was, that the length of a degree of the meridian, under the polar circle, H H 3 amounted 470 amounted to 57438 toises [122365 yards] nearly, On comparing this with the length of the degree iu France, it appears incontestible, that the degrees increase in length toward the north, and that con- sequently the Earth is a spheroid flattened at the poles. It was found likewise, that the axis of revo* Jution of this spheroid, and the diameter of it's equator, are to each other nearly as 177 to 178. These conclusions were enthusiastically adopted by a numerous party, Maupertuis was extolled, as if he had benefited mankind by discovering a new and ex- traordinary truth. Jn certain places he was called by no other name but that of rapplatissenr de la Terre, * the flattener of the Earth.' He had a portrait of himself painted, in which Lapland was the scene, and his hand was placed on the terrestrial globe, as if to compress it into a spheroidal form. Voltaire, who was at that time his friend, wrote four bad verses to he placed under the plate engraved from this painting, which were admired at the time, but have Jnce been deservedly forgotten *. The partisans of the opinion of the prolate figure of the Earth saw with vexation ihe progress of a m, v/hich in a moment subverted the whole of * Ge globe mal connu, qu'il a su mesurefj Pevient un monument ou sa gloire sc fonde ; Son sort est de fixer la figure du monde, De lui plaire et de 1'eclairir. t This globe, the form of which was not well known, till he con. frived to measure it, becomes the monument of his glory, who wa$ Destined to please and enlighten that World, of which he ascertaine4 ffcc figure/ 471 the edifice, that had been raised by them so slowlv, and with so much pains. Still persuaded by the ob- servations made in France, that the length of the terrestrial degree continued diminishing from the equator to the poles, they started doubts respecting the accuracy of the measure taken in the north. They asserted, that it had been executed carelessly, and that the severity of the climate itself might have prevented the employment of all the scrupulous nice- ness and precision necessary. This accusation was repelled with warmth. The polemical tracts on each side increased ; and the love of self was soon more observable in them than the love of truth. A zealous defender of the prolate figure, imagining he had com- pletely refuted the opposite party, would not send his manuscript to the press however, till he had shown it to Fontenelle, whose authority at that time was of great weight. Fontenelle read the work, returned it to the author, and advised liirn to publish it. The author, a little doubtful of the opinion of the judge, said to him after a moment's silence: * you give me advice, sir, which you did not follow yourself: a great deal has been written against you, yet you never answered it.' < O sir,' answered wittily the prudent secretary of the academy, ' I was not so certain of being in the right as you are.' In this contest the system of the flattening of the poles daily gained ground, from the double advant- age it enjoyed of being founded on observations and on the theory of centrifugal forces. Even the Cas- sinis, the authors of the opposite system, began to waver, and at length acknowledged the necessity of H H 4 verifying 472 verifying the french measures with more accurate instruments, than those which they had employed. In 1739 and 1740, Cassini de Thury, the son of James Cassini, made this verification in concert with the abbe" de la Caille; on which occasion the best instruments were employed, and every possible pre- caution was taken, to ensure accuracy. They found, that the degrees for the most part increased from south to north, while a small number only appeared to diminish. The inference from this was in favour of the flattening of the poles, and nothing more was required, but to give it publicity in an au<* thentic form. Cassini de Thury, with his father's consent, had the magnanimity to declare in a public meeting of the academy of sciences, that some errours had crept into the former measures of the degree in France; and to conclude, that the new measures agreed with those of the north in proving the Earth to be an oblate spheroid. He published an account of the business in a book entitled, ' The Meridian of the Royal Observatory verified, &c.' From this time the Earth took, with the common consent of astro- nomers, and to the great satisfaction of geometricians, the oblate figure which had so long been disputed. IVIaupertuis, who was still held out as the author of this revolution, might have enjoyed an unblemished triumph, if his restless and jealous disposition had not continually held before his eyes the speedy arrival of the academicians from Peru, with whom the whole question would require to be discussed anew. Dis- interested men of science, without disputing the flat- ness, of the poles, awaited their return, in order to obtain 475 obtain a more perfect knowledge of the form and dimensions of the terrestrial globe. It was well known, that Godin and Bouguer were aftronomers of the first rank ; that Bouguer was a great geo- metrician -, and that Condamine, though inferiour to his colleagues in science, had surmounted by his zeal and activity a number of obstacles, which opposed the success of their operations. Accordingly there was every reason to presume, that their labours would throw new light on the subject. The friends of Maupertuis took every means in their power, to destroy or weaken these well founded hopes. They ceased not to repeat, that the problem was already solved, and that the peruvian measures would teach us nothing new, or at most confirm a truth already known. Even the arms of ridicule was employed against them beforehand. Naturally satirical, Maupertuis said in those frivolous com- panies, where a jest, good or bad, passes for an ar- gument : ' when these peruvians arrive, they will be more perplexed with the figure they will make them- selves, than with that of the Earth.' All this however was to little purpose ; in spite of sarcasms and intrigue the measures taken in Peru met with the reception they deserved, Bouguer, in his book On the Figure of the Earth, 1749, re- lated the essential precautions taken by his colleagues and himself, as well for the verification and perfect accuracy of their instruments, as for the most proper selection and use of observations : he discussed se- veral points of astronomy, not before sufficiently elu- cidated ; he made the important remark, that the elliptical 474 elliptical figure did not exactly agree with every point of the meridians of the Earth : he started other hy- potheses, more conformable to the truth in a great number of cases, c. So many nice researches stamped on the operations in Peru a character of evidence and certainty, which made them be con- sidered as the most perfect in their kind, that had ever yet been executed. Time has confirmed this favourable opinion, while the measures taken in the north are far from being so highly estimated. Still however the conclusion was, that the Earth becomes flatter toward the poles. The length of a degree at the equator is 56753 toises [120915 yards] : whence it follows, on comparing this with the length of the degree in France, that the axes of the Earth are to each other nearly as 178 to 1/9. Our unfortunate planet would seem destined to torment man in every view. Scarcely had it regained it's oblate figure, when the regularity of it's compo- sition was disputed, which had never before been called in question ; for if the observations in Peru refuted in certain cases the elliptical figure of the meridians, the Earth at least was always considered as a solid of revolution. New observations threw doubts on this opinion, though so natural, and apparently a necessary consequence of the uniform rotation of the Earth round it's axis. La Caille, in his visit to the Cape of Good Hope in 1750, having measured the length of a degree in the latitude of 33 18' south, found it to be 5/037 toises [121520 yards] ; which being greater than that of a degree at the equator, and less than that of a degree 475 a degree at the polar circle, indicates a flattening of the Earth it is true, but less than would be inferred from the measure of a degree in France, which seems to prove this flattening to be irregular. The Jesuits Boscovich and la Maire established this ip'egularity, in a manner that would be still more de- cisive, were it absolutely incontestable. By measures of several degrees of the meridian, taken in Italy in latitudes equal to those taken in France, they found the lengths differ very sensibly from the french measures. Still more: supposing all the meridians of the Earth to be equal and similar, they could not reconcile their own measures either with each other, or with the operations in the north and in Peru. Hence they concluded, that the hypothesis- of the similariryofmcridiansmust.be given up. Thus se- veral astronomical theories must fall to the ground ; the Earth being no longer a solid of revolution, the direction of the plumbline will no longer indicate a perpendicular to the surface of the Earth, or to the plane of the meridian : the observation of the distances of stars from the zenith will not give the true measure of degrees in the heavens, and consequently not of the corresponding degrees on the Earth ; &c. These troublesome consequences did not ,stop the authors of the new system. Why, said they, is it .essential, that the Earth should have a regular figure? If at the beginning it had been a homogeneal fluid mass, the mutual attraction of it's parts, combined with it's rotatory motion round it's axis, would have .caused it to assume the figure of an oblate elliptical spheroid : 47o spheroid : or if it had been composed at first of fluids of different densities, these fluids, endeavouring to place themselves in equilibrium, would have finally arranged themselves in a regular order, and the me- ridians would still have been similar. But why must the Earth have been originally fluid in either manner? And even if it were, why must it have retained it's primitive form ? In the present state of things, one part of it's surface is solid, and composed of sub- stances of different densities, distributed confusedly, and without any order for which a cause can be as- signed. It's surface has been turned topsy turvy; land has been changed into water, the Earth has sunk down in some places, and been raised up in others ; and must not all these revolutions have altered con- siderably it's primitive form, whatever we may sup- pose this to have been ? Is it not highly probable, that they have not only affected the surface of the Earth, but have extended to the interiour of the globe? To conclude, if observations imperiously re- quire it, we must acknowledge the meridians of the Earth to be neither equal nor similar. To these arguments are opposed others, which destroy their effect ; and, if they be not absolutely .conclusive, are at least fully sufficient, to convert as- sertions too affirmative into simple doubts. I will begin with those deduced from natural philosophy. In the first place it is certain, that the globe of the Earth is nearly spherical, or at least may be considered as a very oblate elliptical spheroid. As proofs of this may be quoted the elevations of the pole, which are found to be equal in equal latitudes under different meridians ; 477 meridians ; the rules of navigation founded on these principles, which are so much the more certain a they are observed with more care ; the constant and uniform rotation of the Earth round it's axis ; the regularity of the shadow of the Earth in lunar eclipses; &c. It may be added, that the surface of the globe is for the greater part fluid, and consequently homo- geneal ; that the solid matter, which forms the rest of it's surface, almost every where differs but little in gravity from water; and that consequently the figure of the Earth must be nearly the same, as it would have been on the hypothesis of a complete original fluidity. The inequalities observable on the surface of the globe, the depths of the sea, and the heights of the loftiest mountains, are trifling compared with the radius of the Earth ; the greatest difference not being equal to the tenth of a line on a globe of two feet diameter. The largest mountains are but very small bodies compared with the whole mass of the Earth; and in fact it has been observed in Peru, that mountains nearly three miles high changed the di- rection of the pendulum about seven seconds only. Now a hemispherical mountain of this height ought to change the direction of the pendulum about one minute 18 seconds; whence it follows, that moun- tains contain very little solid matter in proportion to the rest of the globe ; and this consequence is sup- ported by other observations, which have disclosed vast cavities in these mountains. These inequalities, which appear to us so great, though in fact so little, have been produced by the catastrophes which the Earth has suffered, and the effect of which we may suppose 47* suppose not to have extended much below the surface and the uppermost strata. We find no argument in natural philosophy, there- fore, to prove the dissimilarity of the meridians : let us see whether observations will teach us any thing farther. The irregularity resulting from the measure of la Caille is not very great, and it may be explained, without much violence, on the supposition of a si- milarity of meridians. To the measures taken in Italy more weight is attached. But to appreciate the consequences deduced from them, it must be observed, that the difference between the degree measured in France, and the degree measured iti Italy in the same latitude, is only 7o toises [149 yards], about 35 [74 yards] for each degree. Now is this difference too great, to be ascribed to some errour in the observations, however accurate we may suppose them? A mistake of two seconds in the measure of the celestial arc alone would occasion an errour of 32 toises in the length of the terrestrial degree ; and who will answer, that the astronomical and geometrical operations were exempt from such an errour as this ? Thus it appears, that at the time when certain persons argued from the foundations I have just mentioned, nothing obliged us to suppose, that the meridians did not follow a constant and uniform law. To decide the question completely, it would be necessary to measure several degrees of the same meridian in very distant latitudes, and several degrees of meridians corresponding to the same latitude in very different longitudes. 479 longitudes. A great number of degrees of meridian* in very unequal latitudes have been measured under the auspices of european governments, particularly that of France. The excellent operations lately ex- ecuted with this view are well known. All these measures have completely fulfilled the object pro- posed ; and it is now to be wished, that a great number of terrestrial arcs, in very different latitudes and longitudes, should be compared together. This may be accomplished without difficulty, and at little expense, by means of calculations on the length of the pendulum swinging seconds at each place. These determinations possess the advantage of being capable of repetition at all times, by astronomers of all countries ; while direct measures of degrees on the Earth require a considerable apparatus and vast expense, to which governments, who alone are able to carry them into execution, have not always the means or the will of devoting the sums neces- sary. To this may be added, that it is sometimes very hazardous to recommence these great opera* tions, which cannot be verified when required : for, when an operation is repeated, if the second agree with the first, the suspicious or malignant may say, that the results have been made to agree; and if they differ, disputes concerning which is to be pre- ferred may arise, in which the truth may be difficult to discern. Lastly, some countries are not calcu* lated for these operations ; but observations with the pendulum may always be rriade, It is at present universally agreed, that comets arc solid and opake bodies like the planets ; and that 9 they 480 they all describe ellipses, which have the Sun in one of their foci. There is this difference however, the planets move from west to east in a spherical band about 16 degrees broad, and for the most part de- scribe orbits, that do not deviate far from a circle; while the comets traverse through space in all directions, and sometimes describe ellipses so elongated, that they may be taken for parabolas. But it is obvious, that these differences in their directions and orbits are foreign to the bodies themselves, and cannot establish any real distinctions between planets and comets ; but serve merely to form two separate denominations, which simplify and abridge discourse. The ancient opinion, that comets are only collec- tions of matter liable to dissipation, had taken such deep root, that even in the last century there were astronomers of repute, who endeavoured to support or revive it. La Hire, for instance, could not bring himself, to place comets in the same rank with the planets. In the Memoirs of the Academy of Paris for 1702 he thus expresses himself on the subject. ' If the comets were planets, which became visible to our Earth only when they approached very near it, unquestionably they must appear to increase gradu- ally, in the same manner in which we commonly see them vanish and disappear, both witji respect to their movement, which becomes slower toward the end of their appearance, and to the diminution of their light, which likewise becomes fainter nearly in the same proportion. But we almost always begin to sec comets when they are in their brightest lustre, and when the path they pass through in a given time is apparently 481 apparently greatest : whence we may be induced to believe, that they are merely fires, which are suddenly kindled, and gradually dissipated, their velocity at the same time diminishing, &c.' This conjecture can only be ascribed to the imper- fection of men's knowledge of the motions of comets at that time. Astronomers, particularly occupied with the motion of the planets, were not sufficiently attentive to explore all parts of the celestial regions, and suffered several comets to escape them unob- served : others they did not perceive, till long after they had become visible: they would have also the light of comets to be similar to that of planets, which was a gratuitous supposition : they did not consider, that, as the Earth has a dense atmosphere, very different from those of the Moon and of some of the other planets, comets too may have atmospheres more or less ex- tensive and of greater or less density, which occasion their appearances to vary in several ways. All these causes of illusion have at length been gradually dis- sipated, by greater assiduity in exploring the hea- vens ; and by the particular researches that have been made, with the assistance of the most excellent instruments, into the nature of comets, and all the circumstances attending them. I cannot do more here, than point out the objects and progress of the history of comets. Those who would enter deeply into this interesting branch of astronomy, will find abundant satisfaction in the excellent work, which Pingre*, one of our most ce- lebrated astronomers, published on this subject in 1783, In this nothing is omitted :^. their history and i r natural 482 Natural philosophy, observations, probabilities, con- jectures, and every thing concerning them, are re- ported and analyzed with the most scrupulous accu- racy. It is impossible to determine the number of comets, which have appeared since men began to observe the heavens ; but there is reason, to believe it to be considerable. Pingre has remarked about 380 from the birth of Christ to the year 1783, the appearance of which he deems sufficiently probable. There are several others, which we can quote only by conjec- ture. If to these comets, known or suspected, we add all those that may have passed without notice, either from their apparent minuteness, their proxi- mity to the Sun, the brightness of the Moon, the inability of seeing them on account of the weather, or because they may have been invisible above our horizon, the number of comets may be considered as immense. It is to be observed however, that, among the comets which have been seen, there may be se- veral, that were really the same returning at different periods. The ancients have transmitted to us no means of following the motion of comets ; but several attempts have been made by the moderns, to solve this intri- cate problem. Since it has been known, that comets, 3 well as the planets, describe ellipses round the sun, astronomers have endeavoured to determine the di- mensions of these ellipses from a certain number of accurate observations. Their great eccentricity allows them to be considered as parabolas, at least in a part of their extent ; which simplifies the problem, as the equation 483 equation of the parabola is less complicated than that of the ellipsis. Sometimes a portion of the cometary orbit may even be considered as a simple right line. These suppositions facilitate the inquiry into the ap- proximate motion of the comet ; but afterward they frequently require to be rectified by calculations 'founded on the curve, which the comet really de- scribes. Notwithstanding all the care, with which modern astronomers have observed the motion of comets, only one can yet be mentioned, the periodical return of which is known. This bears the name of H alley, the great astronomer, who first determined it's mo- tion. In a small treatise on Comets, which hef published in ] 705, he proved to demonstration the similarity of the motion of comets with that of the planets. Having calculated with extreme care, by a method of New- ton's, and after the best observations, a general table of the motion of comets in a parabolic orbit ; and having afterward applied this table to the motions of several comets; he found, that one which had ap- peared in the years 1531 and 1667, arfd which he himself observed with the greatest attention in 1685, had shown itself under such similar circumstances in it's motion, with respect to the figure, magnitude, and position of it's orbit, as to leave him no doubt of it's being the same. It is true, there were consider- able differences in the periods of it's revolutions; but this difficulty did not stop Halley. Already instructed by the theory of the mutual gravitation of the i i 2 planets, 484 planets, that these bodies disturb the motions of each other : that the motion of Saturn, for instance, is affected by the other planets, and particularly by Jupiter, so that it can only be determined within a few days ; he thought, that the motion of the comet might in like manner have been affected by tbe at- traction of the planets, near which it had passed, and by that of Jupiter in particular. By calculations, which however he gave only for approximations, sus- ceptible of a latitude of some months, he predicted, that this comet would reappear about the end of the year 1758, or the beginning of 1759; a prediction, which was verified by the event. The comet was seen in Saxony in the month of december, 1/58; and passed it's perihelion on the 15th of march, 1759. This comet, therefore, describes, like the planets, an ellipsis round the Sun ; with this differ- ence only, that it's orbit is very eccentric, while those of the planets approach near to a circle, that of Mercury excepted, the eccentricity of which is con- siderable. The same astronomer conjectured, that the comet of 1661 had already appeared in 1532; that it's period was 18 or 129 years ; and that it might reap- pear about 1789 or 1790. But he was not so for- tunate on this occasion as on the former, the cornet not having been seen. He thought likewise, that the great comet of lo'SO was the same as had appeared at the death of Julius Cassar ; and fixed it's period, but with great modesty and circumspection, to 575 years, or there- about ; 485 about ; so that posterity will decide, whether he were mistaken. Pingre' imagines, that the comet of 1556 may pro- bably have been the same as that of 126'4 ; that it makes it's revolution in about 292 years ; and that it will be seen again in 1848. There are some other comets, the return of which have been foretold ; but all these predictions are very vague and uncertain. If the ancients had left us any tolerably accurate ob- servations of comets, we should be better acquainted with their motions. The moderns however observe them with care, and thus lay the foundations of an edifice, which posterity alone can complete. It has been supposed, that comets from time to time fall into the San ; and they have been consi- dered as the means of repairing the loss of substance, which the Sun suffers from the prodigious quantity of luminous rays, that it emits on all sides into the regions of space. In this there is nothing impossible. A comet having been launched in a certain direction, and at the same time being continually attracted by the Sun, would describe round it a precise ellipsis, one of the foci of which the Sun would occupy, did these two bodies alone exist in the universe. But in the actual state of things, the comet, beside it's principal tendency toward the Sun, is acted upon by the attractions of several other of the celestial bo- dies, whether planets or fixed stars: and it may happen, that all these forces combine together in such a manner, as to precipitate the comet into the Sun, or make it graze it's surface. This precise i i 3 com- 436 combination must very rarely occur : but it is within the sphere of probabilities, and unquestionably among the immense number of comets some must have ex- perienced this fate. According to some calculations, the comet of 1680 passed so near the Sun, that, at the moment of it's perihelion, it was about a third of the Sun's semidiameter only from it's surface. Perhaps it will at length fall into the Sun. But this event, if it happen, is very remote, and we need not be alarmed at it. In a general vie\r, any comet what- ever falling into the Sun cannot derange it from it's place, so as to cause us to fear the destruction of our planetary system. The very probable opinion, that the Moon, Venus, Mars, &c., which are solid and opake bodies like the Earth, have inhabitants like it, have led to the idea, that it may be the same with respect to the comets. But this system it is difficult to admit. Comets must be subject to vicissitudes of heat and cold, light and darkness, which scarcely appear compatible >vith any constitution we can conceive of animals. Newton, having calculate4 'the degree of heat, which the comet of lo'SQ must have experienced in it's perihelion, estimated it to be two thousand times greater than that of redhot iron : on the other hand, the comet must have received a proportionably immense increase of light from the Sun. Now, supposing the term of it's periodical revolution to be .575 years, we shall find, that the diameter of the Sun must be seen from the comet at it's perihelion under an angle of 73 degrees, and at is ^phelipn under 487 under an angle of 14 seconds only. Hence it fol- lows, that from the perihelion to the aphelion it must pass from prodigious heat to extreme cold, and from intense light to profound darkness. But how could these alternations be supported by living beings, unless they were of an extraordinary nature, of which our terrestrial anirnals can furnish us with no idea ? J I 4 SECT>1 488 SECT. II. Physical Astronomy* AT present physical astronomy rests entirely on the general law of the mutual attraction, which all parts qf matter exercise on each other. The name of system is commonly given to this law ; but it is a very improper term, as universal gravitation is now a demonstrated truth, though we may be allowed to employ it for the sake of brevity, and to avoid cir- cumlocution. I shall begin with a concise account of the man- ner, in which the celestial motions were formerly ex- plained-; I shall then make known the means, by which Newton was led to the discovery of the grand engine of attraction ; and I shall afterward relate the principal applications made of it. The ancients rarely employed the test of experi- ment in natqral philosophy, where however it is in- dispensable: for, the means by which nature acts being almost always unknown to us, we have no re- source but that of studying and comparing their effects. Swayed by the spirit of system, in the worst sense of the word, and more eager to display their conjectures and opinions, than ambitious of the solid glory to be obtained by first instructing them- selves through the medium of an uninterrupted and critical observation of phenomena ; into their explana- tion? 489 tions of tliese they introduced substantial forms, .oc- cult qualities, &c. ; viz. words devoid of meaning, and contrived to give free scope to all the freaks of imagination. Des Cartes felt, that such a mode of philosophizing was only a perpetual source of false reasoning and erroneous consequences. He was desirous of ex- plaining every thing by means of matter and motion, without admitting in bodies any other properties, than those which are essential to them. With this view he laid down as a principle, that all bodies are composed of the same elements : that their constitu- tion, both internal and external, depends solely on certain simple figures of their constituent parts : and that, when these primordial figures are once known, nothing more is necessary, but to extend and follow their combinations, in the different accidents of rest and motion, to which bodies are subject. This commencement was rational, and indicated views, which might have led to very useful truths, if directed by experience. But soon embarrassed by the number and variety of phenomena to be explained, as well as dazzled by some imperfect experiments, and fan^ eying himself capable of divining others by the mere strength of his genius, des Cartes admitted in the constituent parts of matter arbitrary magnitudes and configurations, with motions and situations for which no cause existed, except the necessities of his system. He imagined invisible fluids, of extreme tenuity,^ agitated by secret motions, penetrating the pores of bodies without experiencing any resistance, and Always obedient, if I may so express myself, to the different 490 different orders, which he issued to them as circum- stances required. Lastly from suppositions to stippo- . sitions he preceded to the conception of those famed vortices, or immense . currents of ethereal matter, which he made to carry the planets, as a river car- ries a boat. His disciples were neither less daring, Bor more fortunate, than himself. Obliged to give up several essential points of his system, they substi- tuted instead of them, on every occasion, new hypo- theses, as fragile and precarious as those of their mas- ter. Notwithstanding so many exertions made to prop it up, this vast edifice has almost wholly crum- bled to pieces. Newton, wisely rejecting the illusions of imagina^ tion, studied Nature in herself and at length divined her secret by dint of research and meditation. Pro- found skill in geometry, and the theory of central forces discovered by Huygens, enabled the learned englishman to discover the law of the force, which retains the Moon in it's orbit round the Earth, or which occasions this satellite to gravitate constantly toward it's primary planet. He then extended this law to all the bodies of our planetary system. Such was nearly the gradation of his ideas on this vast sub- ject. We see that a cannon ball, impelled by the explo- sion of a charge of powder, falls at a distance so much the greater, as the impulse of the explosion is more powerful. Farther, the theory of Huygens in- forms us, that, if the ball, acted upon by a uniform gravitation constantly directed toward the centre of Earth, were projected horizontally with a velocity 3 equai 491 equal to what It would acquire, by falling freely through a right line equal iri height to half the radius of the Earth, it would revolve round the globe with- out ceasing, supposing it met with no resistance; and at every revolution would pass through the point from which it set out. The same reasoning holds good, only making the necessary changes in the proportions, if the ball, instead of setting oif from a point on the surface of the Earth, should be projected from a point above the surface, and distant from it a mile, two miles, or any other height. We may carry it therefore in idea as far as the Moon, or suppose it to be the Moon itself, which in fact revolves circularly round the Earth: and then, by the velocity with which the Moon moves, we shall find the ratio of the force, that retains it in it's orbit, or which is conti- nually occasioning it's deflection from the rectilinear path, to the gravity which occasions bodies here be- low to fall to the ground. Now according to astronomical and geometrical observations, the radius of the Earth is equal to 4000 english miles * : the mean distance of the Moon from the Earth, or the mean radius of the lunar orbit, is equal to sixty times the radius of the Earth : and the Moon performs it's revolution round the Earth in 27 days, 7 hours, 43 minutes. From these data we find, 1st, the whole circumference of the lunar orbit, and the length of the arc, which the * In these calculations J omit small quantities, which would lengthen them uselessly, for all, that is necessary here, is to sho\y fhe grounds of the method. Moon 492 Moon passes through in a given time, in one minute for instance: 2dly, the centripetal force of the Moon, or the quantity of the force with which it is attracted toward the Earth, in one minute ; this be- ing evidently a third proportional to the diameter of the lunar orbit, and the arc which it describes in a minute. The result of all these calculations is, that the quantity of the Moon's deviation from the tangent, or approach to the Earth, in one minute, is about \6 feet. And as we know by experience, that heavy bodies at the surface of the Earth fall sixteen feet in a second, or 3600 times 16 feet in a minute, we per- ceive, that from the Earth to the Moon gravitation is not uniform, but that it has diminished in the ratio of 3600 to 1, or of the square of sixty to the square of one, that is as the- square of the distance: between the Earth and Moon to the square of the Earth's radius. This is the first instance of that cele- brated law of the gravitation of the celestial bodies in the inverse ratio of the squares of their distances. Before I proceed any farther, I cannot refrain from pointing out a new and striking mark of the slowness, with which human knowledge advances. In 1673, fifteen years before Newton's book ap- peared, Huygens had given the properties of the cen- trifugal and centripetal forces in the circle in thirteen propositions. Had he applied this theory to the rota- tory motion of the Earth round it's axis, and to the motion of the Moon round the Earth, he would have discovered the law of the Moon's gravitation toward the Earth. In fact, according to propositions ii and 493. ii and in combined together, the centrifugal force of the Moon is to the centrifugal force at the surface of the Earth, as the- square of the space, which the Moon traverses in a minute, divided by 60, is to the square of the space, which a point on the surface of the Earth traverses in a minute, divided by ne : and according to proposition v, combined with the or- dinary theory of the fall of heavy bodies, the cen- trifugal force of a point on the surface of the Earth is to the force of gravity at this surface, as 1 to 289- Now if we multiply these two proportions term by term, and perform the calculations pointed out, we shall find the centrifugal force of the Moon to be to the force of gravity at the Earth's surface, as 1 to 3600 ; which is the conclusion of Newton. But Huygens did not make this application, and the honour of having discovered the law of the gravitation of the planets, and confirming it by calculation, be- longs to the english geometrician. When Newton had discovered the law of the gra- vitation of the Moon toward the Earth, it was not difficult for him equally to determine the tendencies of the primary planets toward the Sun, and those of the satellites toward their primary planets. Here the elements, of the calculations were furnished by the laws of Kepler. The primary planets describe ellipses round the Sun, which occupies one of their foci ; and in a similar manner the satellites describe ellipses round their primaries. Now by the first law of Kepler, the times employed in passing through the parts of any orbit are as the areas included between the longer axis 4^4 axis of the ellipsis, the radius vector, and the arc passed through : whence it may be concluded, that the primary planet is impelled toward the Sun, or the satellite toward it's primary, by a force inversely pro- portional to the square of the distance of the revolv- ing body from the centre to which it tends. Thus were furnished the means of comparing the gravi- tations of any planet in any two points of it's orbit. But this was not sufficient: it was necessary, like- wise, to know how to compare the gravitations of two different planets ; for it might happen, that in different planets gravitation might not follow the in- verse ratio of the squares of the distances, which would have deprived the principle of it's generality, and of it's most essential advantages. The second law of Kepler completes this theory, and reduces all the gravitations to uniformity : it proves, that all the primary planets are impelled to- ward the Sun by one and the same force, varying in the inverse ratio of the squares of the distances. Thus, for example, the tendency of Mars toward the Sun is to that of Jupiter, as the square of Jupiter's distance from the Sun is to the square of that of Mars. It is the same also with the satellites in respect to their primary planets. Gravitation is reciprocal between all the bodies in the universe. As the primary planets gravitate to- ward the Sun, and the satellites toward their pri- mary planets, the Sun in it's turn gravitates toward the planets, and these toward their satellites. A stone, that falls on the surface of the Rarth, is at- tracted by the terrestrial globe, and attracts this gilobe in 495 in it's turn. The attraction exerted by every body is proportional to it's mass : for there is no reason why the attractive power should exist in one par- ticle of a body, rather than in another : it is com- mon to all, and the sum of the attraction is pro- portional to the mass. If two bodies, therefore, be in free space they will move toward each other, passing through spaces inversely proportional to their masses. Hence we see, in our example, that, in con- sequence of the enormous disproportion of masses, the tendency of the globe of the Earth toward the stone must appear nothing, in comparison of that of the stone toward the globe. As to the dimi- nution, which the power of gravitation undergoes, in proportion as the distance increases, this can be- come sensible only when the distance is very great Hence two bodies falling to the surface of the Earth from different heights, but both moderate, are acted upon by gravitations apparently equal ; and the two heights passed through are proportional to the squares of the times, as Galileo first found. But this law? does not hold, when the two heights differ consider- ably; as for instance, if one were 100 feet, and the other equal to the radius of the lunar orbit ; for from the Earth to the Moon gravitation diminishes in the ratio of 3600 to 1.' From the reciprocal attraction which two planets, as the Earth and Moon, exert on each other, it fol- lows, that the Earth must approach the Moon, at the same time as the Moon approaches the Earth, so that the motion of the Moon is performed round a movable point. It does not, however, on this account 496 follow any other laws, than if the Earth were lixed : for if we seek generally the curves described by two bodies, which, in consequence of their mutual at- tractions, traverse paths toward each other inversely proportional to their masses, and which are projected through space in any given directions, we shall find, that these bodies describe four curves similar to eacli other ; namely, each one round the other body con- sidered as . immovable, and each one round their common centre of gravity ; which besides may be either at rest, or move uniformly in a right line. If in the heavens there were only two bodies, turn- ing round each other by virtue of an original im- pulsion, and of the newtonian attraction, constantly acting, they would move in a manner strictly con- formable to the laws of Kepler. But if there be more than two bodies, which is actually the case in nature, the elliptical motion of the former two will be altered every instant by the attractions of the others. Of these inequalities I shall speak hereafter ; but I shall first consider some particular applications, that have been made of the principle of attraction, to problems of another kind. Among these problems, the first that presents itself is the question concerning the figure of the Earth, as far as it depends on the laws of hydrostatics. Huygens, as has been mentioned, had explained the experiment of Richer at Cayenne, by the combination of the centrifugal force with a constant primitive gra- vitation always directed toward the centre of the Earth. Instead of this gravitation Newton substituted the result of all the particular attractions, which the 8 molecules 497 molecules of the terrestrial globe exert on each other. At present there can be no doubt, which of these two laws of gravitation is to be preferred : the prin- ciple of Newton is avowed by nature; let us examine the use he has made of it, and the great scope he gave to his theory. Newton tacitly supposes, without offering any de- monstration of it, that the Earth, originally fluid and homogeneous, has acquired the figure of an oblate el- liptical spheroid, in consequence of the mutual at- traction of it's parts, and the centrifugal force. He calculates the weight of the central equatorial column, and of the central polar. From the weight of the former he subtracts the sum of the centrifugal forces of all the molecules that compose it, and makes the remainder equal to the weight of the polar column : whence he deduces the ratio of the equa- torial diameter to the polar to be nearly as 230 to ooo **^y. Beside the difference of the hypotheses, which Huygens and Newton had adopted respecting the nature of primitive gravity, they determined the figure of the Earth by different methods^ Huygens set out with this principle, that the result of the pri- mitive gravity and the centrifugal force must be every where perpendicular to the surface of the fluid : Newton, on the contrary, conceived, that the columns in the direction of the axes of the spheroid must re- ciprocally counterpoise each other. These two prin- ciples appear equally necessary at the same time ; one, to establish an equilibrium at the surface of the fluid; the other, in the interiour of the n\ass. Hence K K Bouguer Bouguer and Maupertuis took occasion, to inves- tigate by each method the nature of the meridian on different hypotheses of gravitation directed toward one centre or toward several ; and they rejected all the cases, where the two methods did not agree in giving the same curve to the meridian, which frequently happened. See the Mem. of the Ac. of Paris, for 1734. But all these problems, little difficult in them- selves, were in fact mere geometrical amusements. The nature of gravitation is fixed : and every other principle than that of an attraction inversely pro- portional to the squares of the distances is foreign to the true question. The fundamental proposition of Newton, that the Earth is an oblate elliptical spheroid, required to be demonstrated. This was accordingly done by Stir- ling, in the case where, the fluid being perfectly ho- xnogeneal, the oblateness is supposed to be very small. Phil. Trans. 1736 and 1737. Clairaut likewise de- monstrated it on the same supposition of a very small oblateness, not only when the fluid is homogeneous, but even when it is composed of strata of different densities. In the latter case however he misled him- self by considering the strata as similar ; which can- not be the case when they are fluid, as he remarked himself in his Theory of' the Figure of the Earth) published in 1743. Maclaurm was the first who demonstrated this elegant theorem, that, in whatever manner a homo- geneous fluid mass, the particles of which attract each other in the inverse ratio of the squares of the dis- tances, while at the same time it revolves round an 8 axis, 499 axis, assumed the figure of an oblate or prolate ellip- tical spheroid, of whatever proportions, it will remain in equilibrium, or preserve it's figure. He does not content himself with establishing the equilibrium of the central columns, both in the directions of the axes of the spheroid, and in all other directions : but he shows in addition, that any given point, taken in the interiour of the spheroid, is in equilibrium, or equally pressed upon in all directions ; which is in some measure a superabundant proof. He extends this proposition to the case, where the particles of the Earth, beside their reciprocal attractions and cen- trifugal forces, are likewise attracted by the Sunj and by the Moon. He gives a great number of other very remarkable theorems on the attractions of ellip- soidal spheroids, which have circles or ellipses for their equator: and he applies the whole of this theory to the figure of the planets, and the pheno- mena of the tides. The method he adopts, to de* monstrate his chief propositions, is purely synthetical; and in the opinion of geometricians is esteemed a masterpiece of invention and sagacity, equal to any thing, however admirable, that we have of Archimedes or Apollonius. See his Treatise on Fluxions, VoL II, chap. 14. On restricting this theory to the particular case, where the Earth, originally fluid and homogeneous, forms an oblate elliptical spheroid in consequence of attraction and the centrifugal force, we find the two axes of this spheroid to be to each other in the ratio of 230 to 22$, as Newton had concluded from his suppositions, which are thereby verified. K K % Clairaut, 500 Clairaut, who had many reasons for investigating the same question, since he had been concerned in the operations in the north, and had already demon- strated Newton's suppositions in part, composed an. entire work on this subject, in which he treated the question at length, according to the laws of hydro- statics. As the problems of Bouguer and Maupertuis had drawn the attention of geometricians, Clairaut deemed it incumbent on himself, to consider these likewise. He demonstrated, that under an infinite number of hypotheses of gravitation the fluid would not be in equilibrium, though the central columns reciprocally balanced each other, and the direction of gravita- tion were perpendicular to the surface of the fluid : he gives a general method for discriminating the hy- potheses of gravitation, which admit the equilibrium, and for determining the figure which the fluid ought to assume, and shows, that, when the gravitation is the result of the attractions of all the parts and the centri- fugal force, it is sufficient that one of the principles, either that of Huygens or that of Newton, be ob- served, for the other to be so likewise, and the planet to be in equilibrium. Coming then to the true state of the question, founded on the newtonian system of attraction, Clairaut first determines the figure of the Earth on the hypothesis of the homo- geneity of it's particle^; and here he departs from his own method to follow that of Maclaurin, which he deems preferable. Afterwards, without borrowing far- ther from any person, he proceeds to other very pro- found investigations. He explains the manner of de- termining 501 terrain ing the variations of gravity from the equator to the pole, in a spheroid composed of strata, the densities and ellipticities of which follow any given law from the centre to the superficies : he determines the figure which the Earth would have, if, supposing it entirely fluid, it were a mass of an jnfinite number of fluids of different densities: he then compares his theory with observations ; and in this comparison he examines the errours that must be ascribed to obser- vations, in order that the dimensions of the terrestrial spheroid may be nearly such as the theory demands. So many new and useful views have placed this work of Clairaut among those productions of genius, which do honour to the sciences. Still in this intricate and copious subject there re- mained many important points to be elucidated, both respecting the law of the densities of the terrestrial spheroid, and the conditions of equilibrium to which this law is subjected, according to the different cases. D'Alembert published a great number of excellent memoirs on this subject, in his Essay on the Resist- ance of Fluids, 1752 ; his Inquiries Into the Mundane System, 1754; and his Mathematical Opuscula, 1768. I regret, that they do not admit of having an ab- stract of them inserted here ; and must content my- self with observing, that the author has given a method, long wished for by geometricians, for de- Jtermining the attraction of the terrestrial spheroid on an infinite number of hypotheses beside that of the elliptical figure. He supposes, that the radius of the terrestrial spheroid is represented by an expression, which includes a constant quantity plus the series of K K 3 all 502 all the powers of the sines of latitude; and he finds the attraction, which such a spheroid exerts on a particle placed on it's surface. This includes as a particular case the common supposition, into which the square of the sine of the latitude is the only one of these powers that enters. From this important problem and it's consequences a new treatise on the figure of the Earth is formec). In this the author supposes, that the meridians are equal and similar: but by a farther effort he has likewise accomplished the determination of the attraction of a spheroid, which is not a solid of revolution ; which would be of use, if the terrestrial globe had in fact an irregular figure. Since the invention of the telescope, it has been gradually discovered, by observing the spots on their disks, that the other planets have a rotatory motion jound their axes like the Earth : whence it is to be concluded, that they too are of an oblate figure ; and more or less so, according as their rotation is more or less rapid. The Earth turns uniformly round it's axis in 24 hours : but on account of the inequality of it's annual elliptical orbit, and the oblique po- sition of it's equator with respect to the ecliptic, it's days, or the intervals of time which the Sun's ap- parent motion occupies in returning to the same meridian, are unequal, being sometimes longer, some- times shorter. Their mean duration is 23 hours 56 minutes. The Sun makes one revolution round it's axis, in twenty five days and half; Venus, in 23 hours, 20 minutes; Mars, in 24 hours, 40 minutes; Jupiter, in 9 hours, 56 minutes; and Saturn, in 10 hours, 503 hours, 16 minutes. As to Mercury, it's smallness and proximity to the Sun prevent us from discovering, whether it have any rotatory motion ; but no doubt in this it resembles the other planets. The fixed stars, which are Suns similar to our own* and round which planets and comets probably revolve as in our system, have likewise, according to all ap- pearance, rotatory motions. Besides, the axis of ro- tation of a star may change it's position in the heavens, either by the attraction and disposition of the planets with which it is surrounded, or by the attraction of some large comets belonging to the neighbouring systems. These hypotheses, which are very admissible, serve to explain, why we sometimes perceive certain stars appear or disappear, and why some vary in their magnitude and lustre. When a star presents to us the plane of it's equator, we see it of a circular figure, and with it's greatest brightness, as if it were perfectly spherical. But if a star be con- siderably flattened, and the plane of it's equator be- come inclined with respect to us, it will diminish in size and lustre; and may even disappear entirely from our sight, when, exhibiting to us it's edge, we no longer receive from it a sufficient quantity of light to be perceptible. By a contrary movement of the plane of their equator we may be enabled to see new stars, which will afterward disappear, as they return to their former state. Such was the great star seen in the constellation of Cassiopeia in 1572. The motion, by which the planets revolve about? their axes, does not follow the same laws, as that by which they are, carried round the Sun. The latter is K K 4- 504 more slow, in proportion as the planet is farther dis- tant from the Sun : while Jupiter and Saturn, which are more remote than Venus, the Earth, or Mars, re- volve round their axes in a much shorter period than these planets. Yet these two motions may be pro- duced by one and the same cause. It is sufficient, that the planets should not have been originally pro- jected through space by forces, the directions of which passed through their centres of gravity, or the centres of their mass. For on this hypothesis a planet receives two motions, one of rotation round it's axis, the other of revolution round a centre : the velocity of the latter is independant of the direction of the force with respect to it's centre of gravity, and would always be the same with the same force; but the planet will turn round it's axis with the more quick- ness, in proportion as the direction of the force passes farther from it's centre of gravity. Thus a cannon ball, issuing from a gun, has a rotatory motion, when the force resulting from the impulse of the powder, the friction, and some shocks against the sides of the orifice, does not pass through it's centre of gravity, which must generally be the case. For this explanation of the double motion of the planets we are indebted to John Bernoulli ; and I cannot take leave of this great geometrician, without rendering him farther homage. I have not con- cealed some weaknesses, by which he paid tribute to human nature ; but posterity now sees in him only the man of genius, and the worthy rival of his brother Jarnes. No doubt the reader expects me .here to draw a comparison between them , and accordingly I shall 505 I shall give this parallel in few words, agreeably, I believe, to the opinion generally received among geometricians. Extent, strength, and profundity characterize the genius of James Bernoulli : in John we find more flexibility, and that turn of mind which applies in- differently to all objects. The former published a greater number of truly original works, which belong exclusively to himself; as the theory of spiral lines f the problem of the elastic curve, that of isoperi- meters, which occupies so great a place in the history of geometry, the principle from which was afterward derived the solution of problems in dynamics, the treatise de Arte Conjectandi, &c. The latter was fond of uncommon and curious questions in every branch of mathematics : he had a peculiar art of proposing and resolving new problems : whatever object was offered to his investigation, he entered into it with extreme readiness, and never treated any one without placing it in the most perspicuous light, and making some important discovery in it. To conclude, James Bernoulli became what he was of himself, and died at the age of fifty-: John was initiated into mathe- matics by his brother, and lived fourscore years. In this he had immense advantage : for if all the facul- ties of the human mind be enfeebled by age, this loss is compensated in the mathematical sciences, which are the fruits of study and reasoning, by the mass of knowledge acquired ; and by a long practice in geome- trical methods, which enables us to discern that which is most proper for the solution of a problem; so that >ve are often saved many useless attempts, and the powers 506 powers of the mind are less exhausted. All things considered, I compare James Bernoulli to Newton, John to Leibnitz. I now resume the explanation of the great phe- nomena of nature by the principle of attraction. Of this number is the alternate motion of the ebb and flow of the sea. Every person knows, that in large and deep seas the waters alternately rise and fall in about the space of six hours ; so that in twenty- four hours there are two tides, each consisting of an ebb. and flood. The strength of the flood pushes back the water of rivers,, which run into the sea; and during the ebb they resume their ordinary course. It is only in the ocean, that the flux and reflux are very perceptible : they are scarcely, if at all, to be observed in lakes, gulfs, rivers, and in general all bodies of water of little ex- tent compared with the ocean. Sometimes, however* the water in mediterranean seas, being forced into narrow places, exhibit ebbing and flowing motions, Such are perceptible at the entrance of the gulf of Venice, for example; though they are very trifling, or scarcely observable, on the greater part of the coasts of the Mediterranean. The Cartesians pretended, that the waters of the sea rose in consequence of a pressure, which the Moon, when in the meridian, exerted on the portion of the atmosphere placed between it and the sea; and that they afterward fell by their own weight, when the Moon went down. But for such a pressure to take place, it is necessary that the atmosphere beneath the Moon should have something to prevent it 507 it from extending in all directions; otherwise the Moon will only take the place of a volume of air equal to it's own bulk, and leave the waters of the sea in the same state as they were. At present there is no doubt, that, gravitation be- ing reciprocal between all the bodies in the universe, the ebbing and flowing of the sea are produced by the attractions of the Sun and Moon, combined with the daily rotation of the Earth about it's axis. When the Moon is in the meridian, it attracts the waters of the sea, which consist of corpuscles detached from the rest of the globe ; and it likewise attracts the whole mass of the Earth, which must be considered as uniting entirely in the centre. Now, as the waters are nearer the Moon than the centre of the Earth is, they are attracted more than the centre, and conse- quently they must tend to leave the Earth, if the expres- sion may be allowed, and rise. Thus a flood is produced. On the contrary, in the point diametrically opposite, or at the antipodes of this place, the waters, being more remote, are less attracted than the centre of the Earth, and consequently they must also recede from this centre, or rise; which produces another flood. Thus, in both these places, the motion of flood is occasioned by the difference between the Moon's attractive force at the centre of the Earth and at the surface of the waters, and must occur at the same time at each extremity of the Earth's diameter in a line with the Moon. As to the ebb, this takes place, when, the Moon having left the meridian, it's force of attraction diminishes, and allows the natural t of the waters to depress them. All 508 All that I have said with respect to the Moon is equally applicable to the Sun : and if we calculate the actions, which these two bodies exert on the waters of the sea, whether they combine to produce the same effect, or tend partly to counteract each other, we shall obtain results conformable to the phenomena; whence we have another proof of the universality of gravitation. This interesting subject is treated with all the requisite detail in the three ex- cellent pieces of Daniel Bernoulli, Maclaurin, and Euler, who in 1740 shared the prize proposed by the Academy of Sciences at Paris for a complete solution of this problem, of which Newton only gave a sketch. It is the same with the atmosphere, as with the sea : the attractions of the Sun and Moon excite in it winds, or motions similar to those of the flux and reflux of the sea. The investigation of the general cause of winds was the subject proposed for a prize by the Academy of Berlin in 1746. D'Alembert, to whom it was awarded, found the required cause in the attractions of the Sun and Moon, modified in their effects by the height and directions of the mountains, that cover the Earth. His paper is remarkable for the solution of several very difficult new problems, and more especially for it's displaying, at this time, a considerable knowledge of the integral calculus with partial differences. The elliptical motion of two planets in one system is continually affected, as has already been observed, by the attractions of the other heavenly bodies. Here we have a new field of problems, in which geome- 1 tricians 509 tricians have reaped an ample harvest. I shall pro- ceed to give the substance of the results of their labours, beginning with the Moon, which, being our satellite, naturally first claims our attention. The attractive force of the Earth on the Moon is the most powerful this satellite experiences, in conse- quence of which it revolves round the Earth. Next to this is the attraction of the Sun, by which the elliptical motion of the Moon is perturbed, and to which we cannot avoid paying attention, if we would obtain accurate results. The other celestial bodies, it is true, likewise produce some alterations in this motion ; but they are very trifling, and are therefore neglected. In the same manner, when investigating the motions of Jupiter and Saturn, we consider only the inequalities occasioned by the mutual attractions of these two planets. Geometricians have in con- sequence proposed to themselves the following ge- neral problem, known by the name of the problem of three bodies : ' to determine the curves described by three bodies, projected through space in any given directions, with any given velocities, and exerting upon each other attractions, which are as the quotients of their masses divided by the squares of the distances.' This problem is not capable of a rigorous solution in the present imperfect state of analysis : but approx- imate solutions of it may be given, more or less per- fect, according to the sagacity of the geometrician, and his choice of the observations on which the cal- culation must be founded. In proportion as men have advanced in these the- ories, they have found, that on many occasions the attrac- 510 attractions of more than three bodies must be con* siclered. But the methods of approximation for the problem of three bodies equally apply to the other : and accordingly geometricians have employed all the elements essential to each question, without being deterred by the length of the calculations. Newton had determined several great inequalities of the Moon by the theory of gravitation : namely, 1st, the variation, the quantity of which is about 35 minutes in the octants of the Moon, that is to say, when the Moon is about 45 degrees from the Sun, or from the Earth : 2dly, the annual retrograde motion of the nodes of the lunar orbit, the quantity of which is about 19 degrees : 3dly, the principal equation or inequality of the motion of the nodes, which amounts to a degree and a half; 4thly, the variation of the inclination of the lunar orbit to the plane of the ecliptic, which is about 8 or 9 minutes, alternately in opposite directions. All these calculations are founded on the supposi- tion, that the lunar orbit is nearly an ellipsis; and even the eccentricity of this is neglected by Newton: but this supposition deviates sensibly from the truth, and gives only approximations, with which we are no longer allowed to content ourselves. There are se- veral other inequalities of the Moon, some of which Newton informs us he calculated by the same theory, but without pointing out the path he pursued; as that which arises from the equation of the centre of the Sun, and that depending on the Sun's, distance from the node of the Moon. We have reason, however, to apprehend that he was not more precise in 511 in these calculations, than in those of the inequalities before mentioned : Lastly, he contented himself with deducing simply from observations the motion of the apogee ; the equation of this motion, which is considerable ; the variations of the eccentricity, and some other inequalities. We see, by this abstract, that Newton's theory of the Moon, though a great effort of genius, was in- sufficient; and that it required, not only to be im- proved in almost all it's parts, but likewise to be com- pleted in several other respects. In J747, Euler, Clairaut, and d'Alembert began separately to turn their attention to this important problem, without communicating any thing to each other. The progress, that analysis had made in the course of sixty years, and a greater accuracy in astro- nomical observations, enabled these three geome tricians, not only to determine the inequalities, con- sidered by Newton, with more accuracy than he had done, but also to discover or confirm several others, of which he had made no mention, or which he had deduced merely from observations. The motion of the Moon's apogee however appeared at first to form an exception to the advantage, which the system of gravitation possessed, of easily account- ing for the inequalities of this satellite. The reader knows, that this point, which is the Moon's greatest distance from the Earth, is not fixed in the heavens, but answers successively to the different degrees of the Zodiac; and that it's revolution, following the order of the signs, is accomplished in a space of about nine years, at the end of which it returns nearly to the same 512 same place from which it set out. Clairaut, Euler, and d'Alembert found, each by his own separate cal- culation, that the formula for this motion gave about a moiety of it only. This difference between the theory and observation made a considerable noise. It was supposed, that, the system of attraction being over- turned in an essential point, it would all fall to pieces on a fresh examination ; and the cartesians already triumphed. Clairaut, a partisan of this system, yet a still greater lover of truth, announced in a public meeting of the Academy of Sciences, on the 15th of November 1747, that the law of the inverse ratio of the squares of the distances appeared to him in- sufficient, to account completely for the inequalities of the Moon ; and he proposed the addition of a new term, to explain the other moiety of the -motion of the apogee in particular. But on examining more attentively in 1749 his first calculations, he per- ceived, that lie had not carried far enough the ap- proximation of the series, which represented the motion of the apogee. Having employed all the necessary precision in this operation, he found the other moiety of the motion, without adding any thing to the newtonian law of attraction. Euler and d'Alembert also made on their parts the same remark. Thus attraction was reestablished with honour in the leavens, from which the cartesians had hoped to see it entirely banished. The lunar theories of these three great geometri- cians were printed, either in the collections of the academies, or separately, in the years 1752, 1753, and 1754. During 513 During the time that Euler was employed on the problem of the Moon, he composed his excellent paper on the theory of the motions of Jupiter and Sa- turn, which obtained the prize of the Academy of Sciences at Paris for 1748. This problem is of the same nature as that of the motions of the Moon. Saturn and Jupiter reciprocally disturb the elliptical motion, which each ought to have separately round the Sun. The researches of Euler on this subject are remarkable for a profound analysis, and for se- veral series of a kind absolutely new. Nevertheless, as the difficulty and immense extent of the calcula-^ tions, which such a question demanded, did not allow him to carry his theory to perfection at once, the academy of sciences proposed the same subject anew for the prize of 1750, and again postponed it to 1752 with a double prize, Euler sent a second paper, to which this prize was awarded. It is founded on a method in many re- spects new. In the former the author had been led to approximations, of the sufficiency of which some doubts might be entertained : for the number of in- equalities being as it were infinite, those, which he had determined, depended according to his method on other inequalities, which he had neglected ; and this rendered their values incomplete, and even a little uncertain. The paper of 1752 is more perfect in this respect. It separates and more fully unfolds the inequalities, which are to be discovered in suc- cession ; and thus the analytical formula?, to which it leads, are more simple, and more easily applicable to observations. The author has not treated anew L L the 514 the inequalities, which effect the line of the nodes, and the mutual inclination of the orhits of the two planets, this part of the subject having been com- pletely developed in the paper of J/48. The academy of sciences at Paris having proposed for the prize of 1754, and afterward for the double prize of ]?'56, the theory of the inequalities, which the planets may occasion in the motions of the Earth, Euler was, likewise, equally successful in this. He began by giving general formulae, for determining the altera- tions, which the primary planets mutually occasion in the motions of each other round the Sun. Not to render the question unnecessarily complicated, by introducing into it terms that might be neglected, he considers only two planets at a time ; and he deter- mines the alterations, that the elliptical motion of the one round the Sun must undergo from the attrac- tion of the other: alterations, which, being very small, would produce only infinitely small quantities of the second erder, if they were combined with those, whicli might arise from other planets. He then applies this general theory to the subject pro- posed : he analyzes successively, and in order, the alterations, which Saturn, Jupiter, Mars, and Venus produce in the motion of the Earth : and he finds, that their general effect is, to occasion the aphelion of the Earth to advance in the order of the signs, to vary the obliquity of the ecliptic, and alter the lati- tude and longitude of the Sun, &c. The action of the Moon on the Earth's orbit Euler did not take into consideration, either because he deemed it to make no part of the problem proposed by the academy, or 6 because 515 because d'Alembert had already treated this question in the second volume of his Inquiries into the Man- (Jane System, published in 1754. Clairaut, in a memoir read before the Academy of Sciences at Paris in 1757, and printed by anticipa- tion in the volume of 1754, applied his method for the problem of three bodies to the motions of the Earth. And to the perturbations considered by Euler he added the action of the Moon, which tended to make the theory complete. Mayer the celebrated astronomer, who was likewise a skilful geometrician, constructed new lunar tables, partly from Euler's theory, and partly from observa- tions, which were more accurate than any that had be- fore appeared. A. D. 1754, 1759. Clairaut also con- structed very good ones from his own theory, in 1764. Tables of this kind, requiring a number of scrupulous attentions, and a choice of the most ex- cellent observations, on which the data of the pro- blem depend, cannot be too frequently renewed or corrected. Notwithstanding the efforts of geometricians, the theory of the Moon still remained imperfect in cer- tain respects. By observations alone, dextrously combined, Clairaut and Mayer had obtained several results, which would have gone near to destroy the system of gravitation. The chief cause of these difficulties arose from attributing to the Moon an orbit movable on the plane of the ecliptic ; and making an angle with this plane, which varied from one instant to another; so that, to know the true L L 2 place place of the Moon, or it's longitude and latitude, it was necessary first to determine the intersection of the lunar orbit with the ecliptic, or the line of the nodes, and afterward the inclination of the two orbits; which led to a great number of equations, some of which were uncertain, or precarious. In 1769, Euler, contemplated the questions in a new point of view, and arrived at a more simple, clear, and accurate solution, than any before known. He determined the true place of the Moon, by referring it to three normal coordinates, two of which are in the plane of the ecliptic, and the third perpendicular to it. The values of these coordinates are determined at every instant by equations founded on eight sorts of quantities, four of them constant, and four variable. The constant quantities are the mean eccentricity of the lunar orbit, the mean inclination of this orbit to the plane of the ecliptic, the mean eccentricity of the Earth's orbit, and the ratio between the Earth's mean distance from the Sun and the Moon's mean distance from the Earth. The variable quantities are the four angles proportional to the time ; namely, the mean elongation of the Moon, the mean ano- maly of the Moon, the mean argument of the lati- tude of the Moon, and the mean anomaly of the Sun. These are the bases, on which all the equations of the inequalities of the Moon are established. By these also they are distributed into different classes, and the calculations are performed separately, so that we have no reason to fear that an errour committed in one part will affect the rest. On 517 On this theory Euler constructed newlunar tables, in which the number of equations is less, and which are more convenient for use, than those formed according to the ancient methods. This vast labour is the object of a particular work, printed at Petersburg in 1772, under the following title: Thcoria Motuum Lunce> nova Methodo pej^tractata. As the author was at this time nearly blind, three of his most illustrious scho- lars, John Albert Euler, his son, Lewis Krafft, and John Lexel, executed or verified the calculations. The Academy of Sciences at Paris, having proposed for the prize subjects of 1770 and 1772 the improve- ment of the theory of the Moon, awarded the prizes in one instance, and part of it in the other, to the two papers sent by Euler, in which his new theory was still farther simplified. The Moon is not the only satellite, of which the motion has been considered. The principle of gravi- tation has been successfully applied to the inequalities of the satellites of Jupiter likewise. After it was known, that comets are bodies per* fectly similar to the planets, and subjected to the same laws of motion round the Sun, the researches into the inequalities of the planets could not fail to be extended to the comets ; particularly as the comet of Halley offered a direct application of these new calculations. This astronomer had found, that, in consequence of the attraction of Jupiter, the period of the comet in question beginning in 1682 would be somewhat more than a year longer than it's pre- ceding period ; but the state of geometry in his time did not allow him, to make the computation with all i L 3 the 518 the exactness necessary. Besides, he had neglected the attraction of Saturn, which however bears a sensible proportion to that of Jupiter, since the quan- tity of matter in that planet is about a third of the quantity in this. The attraction of the Earth also has a perceptible influence on the motion of the comet. Every thing therefore invited the geometri- cians, who had treated the perturbations of the planets with so much success, to examine those of the comets according to the same principles. Clairaut was the first, who applied his solution of the problem of three bodies to the motion of comets, and in particular to that of Halley's. In his calcu- lations he employed the attractions of both Jupiter and Saturn. This new problem had it's peculiar dif- ficulties. In the motion of the planets the orbits are but little eccentric, and their inclination with respect to each other is small ; but in that of the comets the radius vector varies considerably, and the orbit of the comet may make a very great angle with that of the per- turbing planet. These differences necessarily alter the nature of some of the means to be employed in the two cases for arriving at converging formula;. Clairaut however surmounted the difficulties attached to the motion of comets, at least in a great measure. Having nearly completed his calculations, he announced at a public meeting of the academy of sciences, on the 14th of november, 1758, that the comet of 1682 would appear in the beginning of 175.9, and pass it's perihelion about the 15th of april. This excited -the attention and curiosity of the public. As soon as the comet was seen, which was in in the beginning of January, the news, adroitly spread through the principal societies in Paris, where Cl'iiraut had many friends, brought his name into the highest repute ; the greater part considered him as the sole author of the prediction of the comet's return ; and the voice of the learned few, who maintained the rights of Halley, was not heard. Some of Clairaut's pupils, a little too jealous of their master's honour, went so far as to say, that the solution of the pro- blem of three bodies had a particular advantage over all others, which rendered it alone easily applicable to the motion of comets. This assertion, which Clairaut had the weakness tacitly to support, was an. unpardonable injustice to- ward Euler and d'Alembert. Euler, solely occupied by the question itself, on which he composed an ex- cellent piece, that shared the prize of the Academy of Petersburg in 1762 with one of Clairaut's, took no notice of it. D'Alembert, living in the midst of the vortex of Paris, could not preserve the same in- difference. He showed, not only that Clairaut's ana* lytical solution was destitute of the exclusive advan- tage ascribed to it, but that it was even incomplete, or at least very inconvenient for use, and of little ac- curacy in certain parts of the orbit of the comet. He even carried his criticism still farther ; and, tracing the solution up to it's very principles, he pointed out essential defects in it, even with regard to the motion of the planets. At the same time he treated the pro- blem of comets by a very simple and complete me- thod, free from every objection. Too fond, how- ever, of speculative researches, and averse to the la- L L 4 borious 520 i borious task of numerical computations, he allowed the glory of rendering a great practical service to astronomy, to be ravished from him on this occasion, as he did on many others. Clairaut, much less fertile in analytical discoveries, but more dexterous in seizing the means of exciting public applause, of which he was extremely covetous, commonly directed his pursuit toward objects, of which many could appreciate the results, if not the theory. He laboured his performances with the greatest care, and seldom failed to give them all the perfection of which they were susceptible. Accord- ingly he enjoyed, even during his lifetime, a very high reputation. His gentle disposition, his polite- ness, and the extreme care he took to wound the va- nity of no one, made him greatly sought after in the World. Unfortunately for the sciences he was too compliant in this respect : engaged at supper parties, keeping late hours, and leading a way of life, which he would fain have reconciled with his ordinary labours, but could not, his health was impaired, and he died while yet young, though he was naturally of a good constitution. D'Alembert, confident of his own superiority, dis- dained the praise that is echoed by one person after another, without being felt. An excellent man, a tender and compassionate friend, a generous benefactor, he possessed all the essential virtues. The faults, with which he is reproached, arose from a fund of gayety and jocularity, to which he sometimes gave himself up, without listening to the dictates of prudence or moderation. He dismissed with a cold reception the flatterers 521 flatterers, or troublesome visitors, who came to pay their court to him. ' I had rather be uncivil,' he would say, ' than be pestered with such men/ Never asking a favour from a man in power, he re- served to himself the privilege of making them feel the keenness of his wit, when they merited it, and which he was very capable of exercising. With such principles, and such conduct, he made himself a multitude of enemies. Some men of letters, of mean and jealous dispositions, could not forgive him for endeavouring to share their labours and their laurels. They would have respected in him the great geometrician alone; but they endeavoured to pull down a rival author : and as he attained not perhaps the first rank as a writer, envy endeavoured to persuade the world, that he had not done this in-any thing else. Such reasoning, however, was flimsy sophistry ; and it would have been more just to conclude, that this transition from the thorns of the higher geometry to the flowers of literature marked the flexibility of a genius of the first order, whose principal talent was for the mathematical sciences. While the learned were employed on the problem of three bodies, d'Alembert resolved another, which required him to create a mechanism in some respects n\v. The object was, to assign the physical cause, that produces the precession of the equinoxes, and the nutation of the Earth's axis, according to the Dew toman system. Observations had taught, that the axis of the Earth a circular motion round the poles of the ecliptic, contrary contrary to the order of the signs ; and that it like- wise experiences a libration with respect to the plane of the ecliptic, which is accomplished during one re- volution of the lunar nodes. It was known also, that the globe of the Earth is not spherical, but forms an oblate spheroid, Now, if we inscribe in the ter- restrial spheroid a sphere, which has for it's diameter the axis of revolution of the spheroid, we shall per- ceive, that, on account of the reciprocal inclinations of the ecliptic and the equator, the Sun, or Moon, does not exert equal attractions on two corresponding points of the spheroidal crust, which constitutes the excess of the spheroid over the inscribed sphere. Hence it follows, that the force resulting from all the attractions of these two celestial bodies does not pass, unless accidentally, through the centre of gravity of the terrestrial spheroid ; and consequently it will oc- casion the axis of the Earth, to have a certain motion with regard to the plane of the ecliptic. This motion is composed at every instant of the mean retrograde motion of the equinoctial points, and the libration of the axis of the Earth with respect to the plane of the ecliptic. It remains then to be subjected to a precise calculation, at least as far as the imperfection of analysis will allow. Newton, employing as axioms certain propositions, of which some were not sufficiently evident in them- selves, and others deviated a little from the truth, made nevertheless such an adroit and happy combination of the forces by which he supposed the axis of the Earth must be affected, that he found the mean quantity 523 quantity of the precession of the equinox.es to be about fifty seconds a year,' as it is given by observations. But in 1 749, when d'Alembert attacked this problem, by de- monstrated instead of hypothetical methods, men of science could so much the less content themselves with Newton's solution, as, exclusive of the defects I have pointed out, the author did not know, or at least did not take into the calculation, the nutation of the Earth's axis. D'Alembert therefore rendered a service of the highest importance to physical astronomy and the newtonian system, by determining, according to the laws of strict and profound mechanics, all the forces, which affect the parallelism of the Earth's axis, and occasion in it the two kinds of motion just mentioned; that in a retrograde direction round the poles of the ecliptic, and that of libration with respect to the plane of the ecliptic. The results of his formulae agree with the observations of Bradley, and afford a new and striking proof of the universal system of gravitation. It may be added, that the manner, in which d'Alembert discovered the motion of the Earth's axis, was the germe of the general theory for deter- mining the motion of a body of any given figure, attracted by any given force, which has since been carried to the utmost degree of perfection with regard to mechanics, and in which no difficulty remains, but that of resolving the equations to which it leads. This first solution of the problem of the precession of the equinpxes was susceptible of a degree of per- fection, which the author gradually endeavoured to give it, either by more strict resolutions of the diffe- rential rential equations of the problem, or by corrections of some numerical coefficients according to new obser- vations. He had first supposed, that the meridians of the Earth are equal and similar ellipses : but afterward he examined the question on the hypothesis of dissi- milar meridians, which produced some slight differ- ences in the results. CHAF, CHAP. XIV, Progress of Optics. THE chief properties of light, it's inflexibility, re- frangibility, and heat, when concentrated in the focus of a burning glass, &c., had long been known; though of the intimate texture of this fluid, or the nature of it's component parts, men were wholly ig- norant. Newton was the first, that penetrated and revealed, this grand secret. He may be said indeed to have anatomized light and colours. Always careful to avoid the spirit of system, always guided by experi- ment, he spent thirty years in the investigation of optics ; and after having occasionally published some of the fruits of his meditations in the Philosophical Transactions, he at length collected together all his early and late opinions into one Treatise on Optics, which appeared in 1 706 ; an original work, deserving to be placed by the side- of his Princlpia. Light is not, as had formerly been supposed, a pure and homogeneal substance : it is composed of seven primordial species of luminous particles, differ- ing in colour, refrangibility, and reflexibility. These seven primitive rays are the red, orange, yellow, green, blue, indigo or purple, and violet. Newton separated them by the following experiment, now known to every one. Introducing the rays of the Sun through a very small hole into a dark room, and presenting 526 presenting obliquely to them one of the faces of a tri- angular glass prism, the axis of which is placed perpen- dicular to that of the fasciculus of rays, we perceive, that this fasciculus is broken, or changes it's course on entering into the glass; that it traverses the prism in a straight line, passes into the air again in a different direction, and proceeds to form an oblong image on a piece of paper sixteen or eighteen feet distant ; in which seven coloured bands may be plainly dis- tinguished, in the following order, reckoning from the bottom : red, orange, yellow, green, blue, indigo, and violet. The whole fasciculus, therefore, is com- posed of seven rays of different refrangibility. The red ray is the least refrangible, as it deviates least from the perpendicular, on it's issuing from the prism : and the refrangibility of the other rays increases pro- gressively to the violet, which is the other extreme. If any number of prisms be placed behind the former, and the rays traverse them all, new refractions will take place ; the image thrown on the paper will be inverted, and again resume it's position ; but the seven coloured bands will still remain, and preserve the same relative situation. Objects not luminous of themselves, or which have only a reflected light, appear to us red, orange, yellow, c., according as they reflect to us, wholly or for the greater part, red, orange, yellow, or other rays. White is formed by the concurrence of all the rays : black absorbs the rays it receives, and is seen only by the reflection of the rays from surrounding objects. In all cases a loss of rays takes place, as some remain in the interstices of the object, or are dispersed on either cither side. The rays absorbed may also produce a sensible heat: thus, for instance, in the sunshine a black hat is hotter than a white one. A ray of light passing from one medium to another is refracted, and approaches or recedes from a right line drawn through the point of entrance perpen- dicular to the surface of separation, according as the first medium is less or more dense than the second ; and the greater the difference of density between the two mediums, the more perceptible the effect : but the ratio which the sine of the angle of incidence bears to the sine of the angle of refraction, remains always the same for every degree of obliquity, and changes it's quantity only when the two comparative mediums alter. Thus if the ray pass from air into water, the two sines are in the ratio of four to three, or as twelve to nine ; if from air to glass, they are as three to two, or as twelve to eight. The seven primitive rays having different refrangi- bilities,' when we speak generally of the refraction of a beam of light, which comprises all the rays, the mean refraction is to be understood, which is nearly that of the green. Sometimes this mean refraction is all that is required ; but at others it is necessary, to pay attention to the different refrangibilities of all the rays, as will be seen when we speak of achromatic telescopes. If a ray of light, after having passed out of one medium into another more dense, as for instance from air to water, were to return back again, it would take precisely the same path. Thus, having approached the perpendicular in the former case, it would recede from 528 from it in the latter. Hence, and from the con- stancy of the ratio between the sine of incidence and the sine of refraction, it may happen, that refraction will be converted into reflection, and the contrary. For example, a ray of light passing from air into water, and almost skimming it's surface, or making the angle of incidence almost a right angle, is re- fracted at an angle of about 48 50': if therefore the ray returned from the water into the air, it would be refracted at an angle of M WHICH BORROWED LOAN DEPT. _ WNEWAIS ONLY-TEL NO. 642-34OS RETURN CIRCULATION DEPARTMENT TO ^ 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 1 -month loans may be renewed by calling 642-3405 1-year loans may be recharged by bringing the books to the Circulation DesK Renewals and recharges may be made 4 days prior to due date DUE AS STAMPED BELOW RECEIVED B r Mub^O *3&3 CIRCULATION W PT. V*liW''fcj~ AUTO ni rO ****> i V^. U! MAY /) ^30. 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