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A History of Mathematics/Modern Europe/Newton to Euler

 

NEWTON TO EULER.

 

It has been seen that in France prodigious scientific progress was made during the beginning and middle of the seventeenth century. The toleration which marked the reign of Henry IV. and Louis XIII. was accompanied by intense intellectual activity. Extraordinary confidence came to be placed in the power of the human mind. The bold intellectual conquests of Descartes, Fermat, and Pascal enriched mathematics with imperishable treasures. During the early part of the reign of Louis XIV. we behold the sunset splendour of this glorious period. Then followed a night of mental effeminacy. This lack of great scientific thinkers during the reign of Louis XIV. may be due to the simple fact that no great minds were born; but, according to Buckle, it was due to the paternalism, to the spirit of dependence and subordination, and to the lack of toleration, which marked the policy of Louis XIV.

In the absence of great French thinkers, Louis XIV. surrounded himself by eminent foreigners. Römer from Denmark, Huygens from Holland, Dominic Cassini from Italy, were the mathematicians and astronomers adorning his court. They were in possession of a brilliant reputation before going to Paris. Simply because they performed scientific work in Paris, that work belongs no more to France than the discoveries of Descartes belong to Holland, or those of Lagrange to Germany, or those of Euler and Poncelet to Russia. We must look to other countries than France for the great scientific men of the latter part of the seventeenth century.

About the time when Louis XIV. assumed the direction of the French government Charles II. became king of England. At this time England was extending her commerce and navigation, and advancing considerably in material prosperity. A strong intellectual movement took place, which was unwittingly supported by the king. The age of poetry was soon followed by an age of science and philosophy. In two successive centuries England produced Shakespeare and Newton!

Germany still continued in a state of national degradation. The Thirty Years' War had dismembered the empire and brutalised the people. Yet this darkest period of Germany's history produced Leibniz, one of the greatest geniuses of modern times.

There are certain focal points in history toward which the lines of past progress converge, and from which radiate the advances of the future. Such was the age of Newton and Leibniz in the history of mathematics. During fifty years preceding this era several of the brightest and acutest mathematicians bent the force of their genius in a direction which finally led to the discovery of the infinitesimal calculus by Newton and Leibniz. Cavalieri, Roberval, Fermat, Descartes, Wallis, and others had each contributed to the new geometry. So great was the advance made, and so near was their approach toward the invention of the infinitesimal analysis, that both Lagrange and Laplace pronounced their countryman, Fermat, to be the true inventor of it. The differential calculus, therefore, was not so much an individual discovery as the grand result of a succession of discoveries by different minds. Indeed, no great discovery ever flashed upon the mind at once, and though those of Newton will influence mankind to the end of the world, yet it must be admitted that Pope's lines are only a "poetic fancy":—

"Nature and Nature's laws lay hid in night;
God said, 'Let Newton be,' and all was light."

Isaac Newton (1642-1727) was born at Woolsthorpe, in Lincolnshire, the same year in which Galileo died. At his birth he was so small and weak that his life was despaired of. His mother sent him at an early age to a village school, and in his twelfth year to the public school at Grantham. At first he seems to have been very inattentive to his studies and very low in the school; but when, one day, the little Isaac received a severe kick upon his stomach from a boy who was above him, he laboured hard till he ranked higher in school than his antagonist. From that time he continued to rise until he was the head boy.[33] At Grantham, Isaac showed a decided taste for mechanical inventions. He constructed a water-clock, a wind-mill, a carriage moved by the person who sat in it, and other toys. When he had attained his fifteenth year his mother took him home to assist her in the management of the farm, but his great dislike for farm-work and his irresistible passion for study, induced her to send him back to Grantham, where he remained till his eighteenth year, when he entered Trinity College, Cambridge (1660). Cambridge was the real birthplace of Newton's genius. Some idea of his strong intuitive powers may be drawn from the fact that he regarded the theorems of ancient geometry as self-evident truths, and that, without any preliminary study, he made himself master of Descartes' Geometry. He afterwards regarded this neglect of elementary geometry a mistake in his mathematical studies, and he expressed to Dr. Pemberton his regret that "he had applied himself to the works of Descartes and other algebraic writers before he had considered the Elements of Euclid with that attention which so excellent a writer deserves," Besides Descartes' Geometry, he studied Oughtred's Clavis, Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Barrow's Lectures, and the works of Wallis. He was particularly delighted with Wallis' Arithmetic of Infinites, a treatise fraught with rich and varied suggestions. Newton had the good fortune of having for a teacher and fast friend the celebrated Dr. Barrow, who had been elected professor of Greek in 1660, and was made Lucasian professor of mathematics in 1663. The mathematics of Barrow and of Wallis were the starting-points from which Newton, with a higher power than his masters', moved onward into wider fields. Wallis had effected the quadrature of curves whose ordinates are expressed by any integral and positive power of . We have seen how Wallis attempted but failed to interpolate between the areas thus calculated, the areas of other curves, such as that of the circle; how Newton attacked the problem, effected the interpolation, and discovered the Binomial Theorem, which afforded a much easier and direct access to the quadrature of curves than did the method of interpolation; for even though the binomial expression for the ordinate be raised to a fractional or negative power, the binomial could at once be expanded into a series, and the quadrature of each separate term of that series could be effected by the method of Wallis. Newton introduced the system of literal indices.

Newton's study of quadratures soon led him to another and most profound invention. He himself says that in 1665 and 1666 he conceived the method of fluxions and applied them to the quadrature of curves. Newton did not communicate the invention to any of his friends till 1669, when he placed in the hands of Barrow a tract, entitled De Analysi per Æquationes Numero Terminorum Infinitas, which was sent by Barrow to Collins, who greatly admired it. In this treatise the principle of fluxions, though distinctly pointed out, is only partially developed and explained. Supposing the abscissa to increase uniformly in proportion to the time, he looked upon the area of a curve as a nascent quantity increasing by continued fluxion in the proportion of the length of the ordinate. The expression which was obtained for the fluxion he expanded into a finite or infinite series of monomial terms, to which Wallis' rule was applicable. Barrow urged Newton to publish this treatise; "but the modesty of the author, of which the excess, if not culpable, was certainly in the present instance very unfortunate, prevented his compliance."[26] Had this tract been published then, instead of forty-two years later, there would probably have been no occasion for that long and deplorable controversy between Newton and Leibniz.

For a long time Newton's method remained unknown, except to his friends and their correspondents. In a letter to Collins, dated December 10th, 1672, Newton states the fact of his invention with one example, and then says: "This is one particular, or rather corollary, of a general method, which extends itself, without any troublesome calculation, not only to the drawing of tangents to any curve lines, whether geometrical or mechanical, or anyhow respecting right lines or other curves, but also to the resolving other abstruser kinds of problems about the crookedness, areas, lengths, centres of gravity of curves, etc.; nor is it (as Hudden's method of Maximis and Minimis) limited to equations which are free from surd quantities. This method I have interwoven with that other of working in equations, by reducing them to infinite series."

These last words relate to a treatise he composed in the year 1671, entitled Method of Fluxions, in which he aimed to represent his method as an independent calculus and as a complete system. This tract was intended as an introduction to an edition of Kinckhuysen's Algebra, which he had undertaken to publish. "But the fear of being involved in disputes about this new discovery, or perhaps the wish to render it more complete, or to have the sole advantage of employing it in his physical researches, induced him to abandon this design."[33]

Excepting two papers on optics, all of his works appear to have been published only after the most pressing solicitations of his friends and against his own wishes.[34] His researches on light were severely criticised, and he wrote in 1675: "I was so persecuted with discussions arising out of my theory of light that I blamed my own imprudence for parting with so substantial a blessing as my quiet to run after a shadow."

The Method of Fluxions, translated by J. Colson from Newton's Latin, was first published in 1736, or sixty-five years after it was written. In it he explains first the expansion into series of fractional and irrational quantities,—a subject which, in his first years of study, received the most careful attention. He then proceeds to the solution of the two following mechanical problems, which constitute the pillars, so to speak, of the abstract calculus:—

"I. The length of the space described being continually (i.e. at all times) given; to find the velocity of the motion at any time proposed.

"II. The velocity of the motion being continually given; to find the length of the space described at any time proposed."

Preparatory to the solution, Newton says: "Thus, in the equation , if y represents the length of the space at any time described, which (time) another space x, by increasing with an uniform celerity , measures and exhibits as described: then will represent the celerity by which the space y, at the same moment of time, proceeds to be described; and contrarywise."

"But whereas we need not consider the time here, any farther than it is expounded and measured by an equable local motion; and besides, whereas only quantities of the same kind can be compared together, and also their velocities of increase and decrease; therefore, in what follows I shall have no regard to time formally considered, but I shall suppose some one of the quantities proposed, being of the same kind, to be increased by an equable fluxion, to which the rest may be referred, as it were to time; and, therefore, by way of analogy, it may not improperly receive the name of time." In this statement of Newton there is contained a satisfactory answer to the objection which has been raised against his method, that it introduces into analysis the foreign idea of motion. A quantity thus increasing by uniform fluxion, is what we now call an independent variable.

Newton continues: "Now those quantities which I consider as gradually and indefinitely increasing, I shall hereafter call fluents, or flowing quantities, and shall represent them by the final letters of the alphabet, v, x, y, and z;…and the velocities by which every fluent is increased by its generating motion (which I may call fluxions, or simply velocities, or celerities), I shall represent by the same letters pointed, thus, , , , . That is, for the celerity of the quantity v; I shall put , and so for the celerities of the other quantities x, y, and z, I shall put , , and , respectively." It must here be observed that Newton does not take the fluxions themselves infinitely small. The "moments of fluxions," a term introduced further on, are infinitely small quantities. These "moments," as defined and used in the Method of Fluxions, are substantially the differentials of Leibniz. De Morgan points out that no small amount of confusion has arisen from the use of the word fluxion and the notation by all the English writers previous to 1704, excepting Newton and Cheyne, in the sense of an infinitely small increment.[35] Strange to say, even in the Commercium Epistolicum the words moment and fluent appear to be used as synonymous.

After showing by examples how to solve the first problem, Newton proceeds to the demonstration of his solution:—

"The moments of flowing quantities (that is, their indefinitely small parts, by the accession of which, in infinitely small portions of time, they are continually increased) are as the velocities of their flowing or increasing.

"Wherefore, if the moment of any one (as x) be represented by the product of its celerity into an infinitely small quantity 0 (i.e. by ), the moments of the others, v, y, z, will be represented by , , ; because , , , and are to each other as {{nowrap|, , , and .

"Now since the moments, as and , are the indefinitely little accessions of the flowing quantities x and y, by which those quantities are increased through the several indefinitely little intervals of time, it follows that those quantities, x and y, after any indefinitely small interval of time, become and , and therefore the equation, which at all times indifferently expresses the relation of the flowing quantities, will as well express the relation between and , as between x and y; so that and may be substituted in the same equation for those quantities, instead of x and y. Thus let any equation be given, and substitute for x, and for y, and there will arise

"Now, by supposition, , which therefore, being expunged and the remaining terms being divided by 0, there will remain

But whereas zero is supposed to be infinitely little, that it may represent the moments of quantities, the terms that are multiplied by it will be nothing in respect of the rest (termini in eam ducti pro nihilo possunt haberi cum aliis collati); therefore I reject them, and there remains

as above in Example I." Newton here uses infinitesimals.

Much greater than in the first problem were the difficulties encountered in the solution of the second problem, involving, as it does, inverse operations which have been taxing the skill of the best analysts since his time. Newton gives first a special solution to the second problem in which he resorts to a rule for which he has given no proof.

In the general solution of his second problem, Newton assumed homogeneity with respect to the fluxions and then considered three cases: (1) when the equation contains two fluxions of quantities and but one of the fluents; (2) when the equation involves both the fluents as well as both the fluxions; (3) when the equation contains the fluents and the fluxions of three or more quantities. The first case is the easiest since it requires simply the integration of , to which his "special solution" is applicable. The second case demanded nothing less than the general solution of a differential equation of the first order. Those who know what efforts were afterwards needed for the complete exploration of this field in analysis, will not depreciate Newton's work even though he resorted to solutions in form of infinite series. Newton's third case comes now under the solution of partial differential equations. He took the equation and succeeded in finding a particular integral of it.

The rest of the treatise is devoted to the determination of maxima and minima, the radius of curvature of curves, and other geometrical applications of his fluxionary calculus. All this was done previous to the year 1672.

It must be observed that in the Method of Fluxions (as well as in his De Analysi and all earlier papers) the method employed by Newton is strictly infinitesimal, and in substance like that of Leibniz. Thus, the original conception of the calculus in England, as well as on the Continent, was based on infinitesimals. The fundamental principles of the fluxionary calculus were first given to the world in the Principia; but its peculiar notation did not appear until published in the second volume of Wallis' Algebra in 1693. The exposition given in the Algebra was substantially a contribution of Newton; it rests on infinitesimals. In the first edition of the Principia (1687) the description of fluxions is likewise founded on infinitesimals, but in the second (1713) the foundation is somewhat altered. In Book II. Lemma II. of the first edition we read: "Cave tamen intellexeris particulas finitas. Momenta quam primum finitæ sunt magnitudinis, desinunt esse momenta. Finiri enim repitgnat aliquatenus perpetuo eorum incremento vel decremento. Intelligenda sunt principia jamjam nascentia finitorum magnitudinum." In the second edition the two sentences which we print in italics are replaced by the following: "Particulæ finitæ non sunt momenta sed quantitates ipsæ ex momentis genitæ." Through the difficulty of the phrases in both extracts, this much distinctly appears, that in the first, moments are infinitely small quantities. What else they are in the second is not clear.[35] In the Quadrature of Curves of 1704, the infinitely small quantity is completely abandoned. It has been shown that in the Method of Fluxions Newton rejected terms involving the quantity 0, because they are infinitely small compared with other terms. This reasoning is evidently erroneous; for as long as is a quantity, though ever so small, this rejection cannot be made without affecting the result. Newton seems to have felt this, for in the Quadrature of Curves he remarked that "in mathematics the minutest errors are not to be neglected" (errores quam minimi in rebus mathematicis non sunt contemnendi).

The early distinction between the system of Newton and Leibniz lies in this, that Newton, holding to the conception of velocity or fluxion, used the infinitely small increment as a means of determining it, while with Leibniz the relation of the infinitely small increments is itself the object of determination. The difference between the two rests mainly upon a difference in the mode of generating quantities.[35]

We give Newton's statement of the method of fluxions or rates, as given in the introduction to his Quadrature of Curves. "I consider mathematical quantities in this place not as consisting of very small parts, but as described by a continued motion. Lines are described, and thereby generated, not by the apposition of parts, but by the continued motion of points; superficies by the motion of lines; solids by the motion of superficies; angles by the rotation of the sides; portions of time by continual flux: and so on in other quantities. These geneses really take place in the nature of things, and are daily seen in the motion of bodies.…

"Fluxions are, as near as we please (quam proxime), as the increments of fluents generated in times, equal and as small as possible, and to speak accurately, they are in the prime ratio of nascent increments; yet they can be expressed by any lines whatever, which are proportional to them."

Newton exemplifies this last assertion by the problem of tangency: Let AB be the abscissa, BC the ordinate, VCH the tangent, Ec the increment of the ordinate, which produced meets VH at T, and Cc the increment of the curve. The right line Cc being produced to K, there are formed three small triangles, the rectilinear CEc, the mixtilinear CEc, and the rectilinear CET, Of these, the first is evidently the smallest, and the last the greatest. Now suppose the ordinate bc to move into the place BC, so that the A History of Mathematics (1893) page 210.pngpoint c exactly coincides with the point C; CK, and therefore the curve Cc, is coincident with the tangent CH, Ec is absolutely equal to ET, and the mixtilinear evanescent triangle CEc is, in the last form, similar to the triangle CET; and its evanescent sides CE, Ec, Cc, will be proportional to CE, ET, and CT, the sides of the triangle CET. Hence it follows that the fluxions of the lines AB, BC, AC, being in the last ratio of their evanescent increments, are proportional to the sides of the triangle CET, or, which is all one, of the triangle VBC similar thereunto. As long as the points C and c are distant from each other by an interval, however small, the line CK will stand apart by a small angle from the tangent CH, But when CK coincides with CH, and the lines CE, Ec, cC reach their ultimate ratios, then the points C and c accurately coincide and are one and the same. Newton then adds that "in mathematics the minutest errors are not to be neglected." This is plainly a rejection of the postulates of Leibniz. The doctrine of infinitely small quantities is here renounced in a manner which would lead one to suppose that Newton had never held it himself. Thus it appears that Newton's doctrine was different in different periods. Though, in the above reasoning, the Charybdis of infinitesimals is safely avoided, the dangers of a Scylla stare us in the face. We are required to believe that a point may be considered a triangle, or that a triangle can be inscribed in a point; nay, that three dissimilar triangles become similar and equal when they have reached their ultimate form in one and the same point.

In the introduction to the Quadrature of Curves the fluxion of is determined as follows:—

"In the same time that x, by flowing, becomes , the power becomes , i.e. by the method of infinite series

etc.,

and the increments

0 and etc.,

are to one another as

1 to etc.

"Let now the increments vanish, and their last proportion will be 1 to : hence the fluxion of the quantity x is to the fluxion of the quantity as .

"The fluxion of lines, straight or curved, in all cases whatever, as also the fluxions of superficies, angles, and other quantities, can be obtained in the same manner by the method of prime and ultimate ratios. But to establish in this way the analysis of infinite quantities, and to investigate prime and ultimate ratios of finite quantities, nascent or evanescent, is in harmony with the geometry of the ancients; and I have endeavoured to show that, in the method of fluxions, it is not necessary to introduce into geometry infinitely small quantities." This mode of differentiating does not remove all the difficulties connected with the subject. When 0 becomes nothing, then we get the ratio , which needs further elucidation. Indeed, the method of Newton, as delivered by himself, is encumbered with difficulties and objections. Among the ablest admirers of Newton, there have been obstinate disputes respecting his explanation of his method of "prime and ultimate ratios."

The so-called "method of limits" is frequently attributed to Newton, but the pure method of limits was never adopted by him as his method of constructing the calculus. All he did was to establish in his Principia certain principles which are applicable to that method, but which he used for a different purpose. The first lemma of the first book has been made the foundation of the method of limits:—

"Quantities and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal."

In this, as well as in the lemmas following this, there are obscurities and difficulties. Newton appears to teach that a variable quantity and its limit will ultimately coincide and be equal. But it is now generally agreed that in the clearest statements which have been made of the theory of limits, the variable does not actually reach its limit, though the variable may approach it as near as we please.

The full title of Newton's Principia is Philosophiæ Naturalis Principia Mathematica. It was printed in 1687 under the direction, and at the expense, of Dr. Edmund Halley. A second edition was brought out in 1713 with many alterations and improvements, and accompanied by a preface from Mr. Cotes. It was sold out in a few months, but a pirated edition published in Amsterdam supplied the demand.[34] The third and last edition which appeared in England during Newton's lifetime was published in 1726 by Henry Pemberton. The Principia consists of three books, of which the first two, constituting the great bulk of the work, treat of the mathematical principles of natural philosophy, namely, the laws and conditions of motions and forces. In the third book is drawn up the constitution of the universe as deduced from the foregoing principles. The great principle underlying this memorable work is that of universal gravitation. The first book was completed on April 28, 1686. After the remarkably short period of three months, the second book was finished. The third book is the result of the next nine or ten months' labours. It is only a sketch of a much more extended elaboration of the subject which he had planned, but which was never brought to completion.

The law of gravitation is enunciated in the first book. Its discovery envelops the name of Newton in a halo of perpetual glory. The current version of the discovery is as follows: it was conjectured by Hooke, Huygens, Halley, Wren, Newton, and others, that, if Kepler's third law was true (its absolute accuracy was doubted at that time), then the attraction between the earth and other members of the solar system varied inversely as the square of the distance. But the proof of the truth or falsity of the guess was wanting. In 1666 Newton reasoned, in substance, that if g represent the acceleration of gravity on the surface of the earth, r be the earth's radius, R the distance of the moon from the earth, T the time of lunar revolution, and a a degree at the equator, then, if the law is true,

, or .

The data at Newton's command gave seconds, but a only 60 instead of English miles. This wrong value of a rendered the calculated value of g smaller than its true value, as known from actual measurement. It looked as though the law of inverse squares were not the true law, and Newton laid the calculation aside. In 1684 he casually ascertained at a meeting of the Royal Society that Jean Picard had measured an arc of the meridian, and obtained a more accurate value for the earth's radius. Taking the corrected value for a, he found a figure for g which corresponded to the known value. Thus the law of inverse squares was verified. In a scholium in the Principia, Newton acknowledged his indebtedness to Huygens for the laws on centrifugal force employed in his calculation.

The perusal by the astronomer Adams of a great mass of unpublished letters and manuscripts of Newton forming the Portsmouth collection (which remained private property until 1872, when its owner placed it in the hands of the University of Cambridge) seems to indicate that the difficulties encountered by Newton in the above calculation were of a different nature. According to Adams, Newton's numerical verification was fairly complete in 1666, but Newton had not been able to determine what the attraction of a spherical shell upon an external point would be. His letters to Halley show that he did not suppose the earth to attract as though all its mass were concentrated into a point at the centre. He could not have asserted, therefore, that the assumed law of gravity was verified by the figures, though for long distances he might have claimed that it yielded close approximations. When Halley visited Newton in 1684, he requested Newton to determine what the orbit of a planet would be if the law of attraction were that of inverse squares. Newton had solved a similar problem for Hooke in 1679, and replied at once that it was an ellipse. After Halley's visit, Newton, with Picard's new value for the earth's radius, reviewed his early calculation, and was able to show that if the distances between the bodies in the solar system were so great that the bodies might be considered as points, then their motions were in accordance with the assumed law of gravitation. In 1685 he completed his discovery by showing that a sphere whose density at any point depends only on the distance from the centre attracts an external point as though its whole mass were concentrated at the centre.[34]

Newton's unpublished manuscripts in the Portsmouth collection show that he had worked out, by means of fluxions and fluents, his lunar calculations to a higher degree of approximation than that given in the Principia, but that he was unable to interpret his results geometrically. The papers in that collection throw light upon the mode by which Newton arrived at some of the results in the Principia, as, for instance, the famous construction in Book II., Prop. 25, which is unproved in the Principia, but is demonstrated by him twice in a draft of a letter to David Gregory, of Oxford.[34]

It is chiefly upon the Principia that the fame of Newton rests. Brewster calls it "the brightest page in the records of human reason." Let us listen, for a moment, to the comments of Laplace, the foremost among those followers of Newton who grappled with the subtle problems of the motions of planets under the influence of gravitation: "Newton has well established the existence of the principle which he had the merit of discovering, but the development of its consequences and advantages has been the work of the successors of this great mathematician. The imperfection of the infinitesimal calculus, when first discovered, did not allow him completely to resolve the difficult problems which the theory of the universe offers; and he was oftentimes forced to give mere hints, which were always uncertain till confirmed by rigorous analysis. Notwithstanding these unavoidable defects, the importance and the generality of his discoveries respecting the system of the universe, and the most interesting points of natural philosophy, the great number of profound and original views, which have been the origin of the most brilliant discoveries of the mathematicians of the last century, which were all presented with much elegance, will insure to the Principia a lasting pre-eminence over all other productions of the human mind."

Newton's Arithmetica Universalis, consisting of algebraical lectures delivered by him during the first nine years he was professor at Cambridge, were published in 1707, or more than thirty years after they were written. This work was published by Mr. Whiston. We are not accurately informed how Mr. Whiston came in possession of it, but according to some authorities its publication was a breach of confidence on his part.

The Arithmetica Universalis contains new and important results on the theory of equations. His theorem on the sums of powers of roots is well known. Newton showed that in equations with real coefficients, imaginary roots always occur in pairs. His inventive genius is grandly displayed in his rule for determining the inferior limit of the number of imaginary roots, and the superior limits for the number of positive and negative roots. Though less expeditious than Descartes', Newton's rule always gives as close, and generally closer, limits to the number of positive and negative roots. Newton did not prove his rule. It awaited demonstration for a century and a half, until, at last, Sylvester established a remarkable general theorem which includes Newton's rule as a special case.

The treatise on Method of Fluxions contains Newton's method of approximating to the roots of numerical equations. This is simply the method of Vieta improved. The same treatise contains "Newton's parallelogram," which enabled him, in an equation, , to find a series in powers of x equal to the variable y. The great utility of this rule lay in its determining the form of the series; for, as soon as the law was known by which the exponents in the series vary, then the expansion could be effected by the method of indeterminate coefficients. The rule is still used in determining the infinite branches to curves, or their figure at multiple points. Newton gave no proof for it, nor any clue as to how he discovered it. The proof was supplied half a century later, by Kaestner and Cramer, independently.[37]

In 1704 was published, as an appendix to the Opticks, the Enumeratio linearum tertii ordinis, which contains theorems on the theory of curves. Newton divides cubics into seventy-two species, arranged in larger groups, for which his commentators have supplied the names "genera" and "classes," recognising fourteen of the former and seven (or four) of the latter. He overlooked six species demanded by his principles of classification, and afterwards added by Stirling, Murdoch, and Cramer. He enunciates the remarkable theorem that the five species which he names "divergent parabolas" give by their projection every cubic curve whatever. As a rule, the tract contains no proofs. It has been the subject of frequent conjecture how Newton deduced his results. Recently we have gotten at the facts, since much of the analysis used by Newton and a few additional theorems have been discovered among the Portsmouth papers. An account of the four holograph manuscripts on this subject has been published by W. W. Rouse Ball, in the Transactions of the London Mathematical Society (vol. xx., pp. 104–143). It is interesting to observe how Newton begins his research on the classification of cubic curves by the algebraic method, but, finding it laborious, attacks the problem geometrically, and afterwards returns again to analysis.[36]

Space does not permit us to do more than merely mention Newton's prolonged researches in other departments of science. He conducted a long series of experiments in optics and is the author of the corpuscular theory of light. The last of a number of papers on optics, which he contributed to the Royal Society, 1687, elaborates the theory of "fits." He explained the decomposition of light and the theory of the rainbow. By him were invented the reflecting telescope and the sextant (afterwards re-discovered by Thomas Godfrey of Philadelphia[2] and by John Hadley). He deduced a theoretical expression for the velocity of sound in air, engaged in experiments on chemistry, elasticity, magnetism, and the law of cooling, and entered upon geological speculations.

During the two years following the close of 1692, Newton suffered from insomnia and nervous irritability. Some thought that he laboured under temporary mental aberration. Though he recovered his tranquillity and strength of mind, the time of great discoveries was over; he would study out questions propounded to him, but no longer did he by his own accord enter upon new fields of research. The most noted investigation after his sickness was the testing of his lunar theory by the observations of Flamsteed, the astronomer royal. In 1695 he was appointed warden, and in 1699 master, of the mint, which office he held until his death. His body was interred in Westminster Abbey, where in 1731 a magnificent monument was erected, bearing an inscription ending with, "Sibi gratulentur mortales tale tantumque exstitisse humani generis decus." It is not true that the Binomial Theorem is also engraved on it.

We pass to Leibniz, the second and independent inventor of the calculus. Gottfried Wilhelm Leibniz (1646–1716) was born in Leipzig. No period in the history of any civilised nation could have been less favourable for literary and scientific pursuits than the middle of the seventeenth century in Germany. Yet circumstances seem to have happily combined to bestow on the youthful genius an education hardly otherwise obtainable during this darkest period of German history. He was brought early in contact with the best of the culture then existing. In his fifteenth year he entered the University of Leipzig. Though law was his principal study, he applied himself with great diligence to every branch of knowledge. Instruction in German universities was then very low. The higher mathematics was not taught at all. We are told that a certain John Kuhn lectured on Euclid's Elements, but that his lectures were so obscure that none except Leibniz could understand them. Later on, Leibniz attended, for a half-year, at Jena, the lectures of Erhard Weigel, a philosopher and mathematician of local reputation. In 1666 Leibniz published a treatise, De Arte Combinatoria, in which he does not pass beyond the rudiments of mathematics. Other theses written by him at this time were metaphysical and juristical in character. A fortunate circumstance led Leibniz abroad. In 1672 he was sent by Baron Boineburg on a political mission to Paris. He there formed the acquaintance of the most distinguished men of the age. Among these was Huygens, who presented a copy of his work on the oscillation of the pendulum to Leibniz, and first led the gifted young German to the study of higher mathematics. In 1673 Leibniz went to London, and remained there from January till March. He there became incidentally acquainted with the mathematician Pell, to whom he explained a method he had found on the summation of series of numbers by their differences. Pell told him that a similar formula had been published by Mouton as early as 1670, and then called his attention to Mercator's work on the rectification of the parabola. While in London, Leibniz exhibited to the Royal Society his arithmetical machine, which was similar to Pascal's, but more efficient and perfect. After his return to Paris, he had the leisure to study mathematics more systematically. With indomitable energy he set about removing his ignorance of higher mathematics. Huygens was his principal master. He studied the geometric works of Descartes, Honorarius Fabri, Gregory St. Vincent, and Pascal. A careful study of infinite series led him to the discovery of the following expression for the ratio of the circumference to the diameter of the circle:—

etc.

This elegant series was found in the same way as Mercator's on the hyperbola. Huygens was highly pleased with it and urged him on to new investigations. Leibniz entered into a detailed study of the quadrature of curves and thereby became intimately acquainted with the higher mathematics. Among the papers of Leibniz is still found a manuscript on quadratures, written before he left Paris in 1676, but which was never printed by him. The more important parts of it were embodied in articles published later in the Acta Eruditorum.

In the study of Cartesian geometry the attention of Leibniz was drawn early to the direct and inverse problems of tangents. The direct problem had been solved by Descartes for the simplest curves only; while the inverse had completely transcended the power of his analysis. Leibniz investigated both problems for any curve; he constructed what he called the triangulum characteristicum—an infinitely small triangle between the infinitely small part of the curve coinciding with the tangent, and the differences of the ordinates and abscissas. A curve is here considered to be a polygon. The triangulum characteristicum is similar to the triangle formed by the tangent, the ordinate of the point of contact, and the sub-tangent, as well as to that between the ordinate, normal, and sub-normal. It was first employed by Barrow in England, but appears to have been reinvented by Leibniz. From it Leibniz observed the connection existing between the direct and inverse problems of tangents. He saw also that the latter could be carried back to the quadrature of curves. All these results are contained in a manuscript of Leibniz, written in 1673. One mode used by him in effecting quadratures was as follows: The rectangle formed by a sub-tangent p and an element a (i.e. infinitely small part of the abscissa) is equal to the rectangle formed by the ordinate y and the element l of that ordinate; or in symbols, . But the summation of these rectangles from zero on gives a right triangle equal to half the square of the ordinate. Thus, using Cavalieri's notation, he gets

(omn. meaning omnia, all).

But ; hence

.

This equation is especially interesting, since it is here that Leibniz first introduces a new notation. He says: "It will be useful to write for omn., as for , that is, the sum of the l's"; he then writes the equation thus:—

.

From this he deduced the simplest integrals, such as

.

Since the symbol of summation raises the dimensions, he concluded that the opposite calculus, or that of differences d, would lower them. Thus, if , then . The symbol d was at first placed by Leibniz in the denominator, because the lowering of the power of a term was brought about in ordinary calculation by division. The manuscript giving the above is dated October 29th, 1675.[39] This, then, was the memorable day on which the notation of the new calculus came to be,—a notation which contributed enormously to the rapid growth and perfect development of the calculus.

Leibniz proceeded to apply his new calculus to the solution of certain problems then grouped together under the name of the Inverse Problems of Tangents. He found the cubical parabola to be the solution to the following: To find the curve in which the sub-normal is reciprocally proportional to the ordinate. The correctness of his solution was tested by him by applying to the result Sluze's method of tangents and reasoning backwards to the original supposition. In the solution of the third problem he changes his notation from to the now usual notation dx. It is worthy of remark that in these investigations, Leibniz nowhere explains the significance of dx and dy, except at one place in a marginal note: "Idem est dx et , id est, differentia inter duas x proximas." Nor does he use the term differential, but always difference. Not till ten years later, in the Acta Eruditorum, did he give further explanations of these symbols. What he aimed at principally was to determine the change an expression undergoes when the symbol or d is placed before it. It may be a consolation to students wrestling with the elements of the differential calculus to know that it required Leibniz considerable thought and attention[39] to determine whether dx dy is the same as d(xy), and the same as . After considering these questions at the close of one of his manuscripts, he concluded that the expressions were not the same, though he could not give the true value for each. Ten days later, in a manuscript dated November 21, 1675, he found the equation , giving an expression for d(xy), which he observed to be true for all curves. He succeeded also in eliminating dx from a differential equation, so that it contained only dy, and thereby led to the solution of the problem under consideration. "Behold, a most elegant way by which the problems of the inverse methods of tangents are solved, or at least are reduced to quadratures!" Thus he saw clearly that the inverse problems of tangents could be solved by quadratures, or, in other words, by the integral calculus. In course of a half-year he discovered that the direct problem of tangents, too, yielded to the power of his new calculus, and that thereby a more general solution than that of Descartes could be obtained. He succeeded in solving all the special problems of this kind, which had been left unsolved by Descartes. Of these we mention only the celebrated problem proposed to Descartes by De Beaune, viz. to find the curve whose ordinate is to its sub-tangent as a given line is to that part of the ordinate which lies between the curve and a line drawn from the vertex of the curve at a given inclination to the axis.

Such was, in brief, the progress in the evolution of the new calculus made by Leibniz during his stay in Paris. Before his departure, in October, 1676, he found himself in possession of the most elementary rules and formulæ of the infinitesimal calculus.

From Paris, Leibniz returned to Hanover by way of London and Amsterdam. In London he met Collins, who showed him a part of his scientific correspondence. Of this we shall speak later. In Amsterdam he discussed mathematics with Sluze, and became satisfied that his own method of constructing tangents not only accomplished all that Sluze's did, but even more, since it could be extended to three variables, by which tangent planes to surfaces could be found; and especially, since neither irrationals nor fractions prevented the immediate application of his method.

In a paper of July 11, 1677, Leibniz gave correct rules for the differentiation of sums, products, quotients, powers, and roots. He had given the differentials of a few negative and fractional powers, as early as November, 1676, but had made some mistakes. For he had given the erroneous value , and in another place the value ; for occurs in one place the wrong value, , while a few lines lower is given , its correct value.

In 1682 was founded in Berlin the Acta Eruditorum, a journal usually known by the name of Leipzig Acts. It was a partial imitation of the French Journal des Savans (founded in 1665), and the literary and scientific review published in Germany. Leibniz was a frequent contributor. Tschirnhaus, who had studied mathematics in Paris with Leibniz, and who was familiar with the new analysis of Leibniz, published in the Acta Eroditorum a paper on quadratures, which consists principally of subject-matter communicated by Leibniz to Tschirnhaus during a controversy which they had had on this subject. Fearing that Tschirnhaus might claim as his own and publish the notation and rules of the differential calculus, Leibniz decided, at last, to make public the fruits of his inventions. In 1684, or nine years after the new calculus first dawned upon the mind of Leibniz, and nineteen years after Newton first worked at fluxions, and three years before the publication of Newton's Principia, Leibniz published, in the Leipzig Acts, his first paper on the differential calculus. He was unwilling to give to the world all his treasures, but chose those parts of his work which were most abstruse and least perspicuous. This epoch-making paper of only six pages bears the title: "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus." The rules of calculation are briefly stated without proof, and the meaning of dx and dy is not made clear. It has been inferred from this that Leibniz himself had no definite and settled ideas on this subject. Are dy and dx finite or infinitesimal quantities? At first they appear, indeed, to have been taken as finite, when he says: "We now call any line selected at random dx, then we designate the line which is to dx as y is to the sub-tangent, by dy, which is the difference of y." Leibniz then ascertains, by his calculus, in what way a ray of light passing through two differently refracting media, can travel easiest from one point to another; and then closes his article by giving his solution, in a few words, of De Beaune's problem. Two years later (1686) Leibniz published in the Acta Eruditorum a paper containing the rudiments of the integral calculus. The quantities dx and dy are there treated as infinitely small. He showed that by the use of his notation, the properties of curves could be fully expressed by equations. Thus the equation,

characterises the cycloid.[38]

The great invention of Leibniz, now made public by his articles in the Leipzig Acts, made little impression upon the mass of mathematicians. In Germany no one comprehended the new calculus except Tschirnhaus, who remained indifferent to it. The author's statements were too short and succinct to make the calculus generally understood. The first to recognise its importance and to take up the study of it were two foreigners,—the Scotchman Thomas Craige, and the Swiss James Bernoulli. The latter wrote Leibniz a letter in 1687, wishing to be initiated into the mysteries of the new analysis. Leibniz was then travelling abroad, so that this letter remained unanswered till 1790. James Bernoulli succeeded, meanwhile, by close application, in uncovering the secrets of the differential calculus without assistance. He and his brother John proved to be mathematicians of exceptional power. They applied themselves to the new science with a success and to an extent which made Leibniz declare that it was as much theirs as his. Leibniz carried on an extensive correspondence with them, as well as with other mathematicians. In a letter to John Bernoulli he suggests, among other things, that the integral calculus be improved by reducing integrals back to certain fundamental irreducible forms. The integration of logarithmic expressions was then studied. The writings of Leibniz contain many innovations, and anticipations of since prominent methods. Thus he made use of variable parameters, laid the foundation of analysis in situ, introduced the first notion of determinants in his effort to simplify the expression arising in the elimination of the unknown quantities from a set of linear equations. He resorted to the device of breaking up certain fractions into the sum of other fractions for the purpose of easier integration; he explicitly assumed the principle of continuity; he gave the first instance of a "singular solution," and laid the foundation to the theory of envelopes in two papers, one of which contains for the first time the terms co-ordinate and axes of co-ordinates. He wrote on osculating curves, but his paper contained the error (pointed out by John Bernoulli, but not admitted by him) that an osculating circle will necessarily cut a curve in four consecutive points. Well known is his theorem on the nth differential coefficient of the product of two functions of a variable. Of his many papers on mechanics, some are valuable, while others contain grave errors.

Before tracing the further development of the calculus we shall sketch the history of that long and bitter controversy between English and Continental mathematicians on the invention of the calculus. The question was, did Leibniz invent it independently of Newton, or was he a plagiarist?

We must begin with the early correspondence between the parties appearing in this dispute. Newton had begun using his notation of fluxions in 1666.[41] In 1669 Barrow sent Collins Newton's tract, De Analysi per Equationes, etc.

The first visit of Leibniz to London extended from the 11th of January until March, 1673. He was in the habit of committing to writing important scientific communications received from others. In 1890 Gerhardt discovered in the royal library at Hanover a sheet of manuscript with notes taken by Leibniz during this journey.[40] They are headed "Observata Philosophica in itinere Anglicano sub initium anni 1673." The sheet is divided by horizontal lines into sections. The sections given to Chymica, Mechanica, Magnetica, Botanica, Anatomica, Medica, Miscellanea, contain extensive memoranda, while those devoted to mathematics have very few notes. Under Geometrica he says only this: "Tangentes omnium figurarum. Figurarum geometricarum explicatio per motum puncti in moto lati." We suspect from this that Leibniz had read Barrow's lectures. Newton is referred to only under Optica. Evidently Leibniz did not obtain a knowledge of fluxions during this visit to London, nor is it claimed that he did by his opponents.

Various letters of Newton, Collins, and others, up to the beginning of 1676, state that Newton invented a method by which tangents could be drawn without the necessity of freeing their equations from irrational terms. Leibniz announced in 1674 to Oldenburg, then secretary of the Royal Society, that he possessed very general analytical methods, by which he had found theorems of great importance on the quadrature of the circle by means of series. In answer, Oldenburg stated Newton and James Gregory had also discovered methods of quadratures, which extended to the circle. Leibniz desired to have these methods communicated to him; and Newton, at the request of Oldenburg and Collins, wrote to the former the celebrated letters of June 13 and October 24, 1676. The first contained the Binomial Theorem and a variety of other matters relating to infinite series and quadratures; but nothing directly on the method of fluxions. Leibniz in reply speaks in the highest terms of what Newton had done, and requests further explanation. Newton in his second letter just mentioned explains the way in which he found the Binomial Theorem, and also communicates his method of fluxions and fluents in form of an anagram in which all the letters in the sentence communicated were placed in alphabetical order. Thus Newton says that his method of drawing tangents was

6 a cc d œ 13 e ff 7 i 3 1 9n 4 o 4 q rr 4 s 9 t 12 v x.

The sentence was, "Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire, et vice versa." ("Having any given equation involving never so many flowing quantities, to find the fluxions, and vice versa.") Surely this anagram afforded no hint. Leibniz wrote a reply to Collins, in which, without any desire of concealment, he explained the principle, notation, and the use of the differential calculus.

The death of Oldenburg brought this correspondence to a close. Nothing material happened till 1684, when Leibniz published his first paper on the differential calculus in the Leipzig Acts, so that while Newton's claim to the priority of invention must be admitted by all, it must also be granted that Leibniz was the first to give the full benefit of the calculus to the world. Thus, while Newton's invention remained a secret, communicated only to a few friends, the calculus of Leibniz was spreading over the Continent. No rivalry or hostility existed, as yet, between the illustrious scientists. Newton expressed a very favourable opinion of Leibniz's inventions, known to him through the above correspondence with Oldenburg, in the following celebrated scholium (Principia, first edition, 1687, Book II., Prop. 7, scholium):—

"In letters which went between me and that most excellent geometer, G. G. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, and when I concealed it in transposed letters involving this sentence (Data æquatione, etc., above cited), that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine, except in his forms of words and symbols."

As regards this passage, we shall see that Newton was afterwards weak enough, as De Morgan says: "First, to deny the plain and obvious meaning, and secondly, to omit it entirely from the third edition of the Principia." On the Continent, great progress was made in the calculus by Leibniz and his coadjutors, the brothers James and John Bernoulli, and Marquis de l'Hospital. In 1695 Wallis informed Newton by letter that "he had heard that his notions of fluxions passed in Holland with great applause by the name of 'Leibniz's Calculus Differentialis.'" Accordingly Wallis stated in the preface to a volume of his works that the calculus differentialis was Newton's method of fluxions which had been communicated to Leibniz in the Oldenburg letters. A review of Wallis' works^ in the Leipzig Acts for 1696, reminded the reader of Newton's own admission in the scholium above cited.

For fifteen years Leibniz had enjoyed unchallenged the honour of being the inventor of his calculus. But in 1699 Fato de Duillier, a Swiss, who had settled in England, stated in a mathematical paper, presented to the Royal Society, his conviction that Newton was the first inventor; adding that, whether Leibniz, the second inventor, had borrowed anything from the other, he would leave to the judgment of those who had seen the letters and manuscripts of Newton. This was the first distinct insinuation of plagiarism. It would seem that the English mathematicians had for some time been cherishing suspicions unfavourable to Leibniz. A feeling had doubtless long prevailed that Leibniz, during his second visit to London in 1676, had or might have seen among the papers of Collins, Newton's Analysis per æquationes, etc., which contained applications of the fluxionary method, but no systematic development or explanation of it. Leibniz certainly did see at least part of this tract. During the week spent in London, he took note of whatever interested him among the letters and papers of Collins. His memoranda discovered by Gerhardt in 1849 in the Hanover library fill two sheets.[40] The one bearing on our question is headed "Excerpta ex tractatu Newtoni Msc. de Analysi per æquationes numero terminorum infinitas." The notes are very brief, excepting those De Resolutione æquationum affectarum, of which there is an almost complete copy. This part was evidently new to him. If he examined Newton's entire tract, the other parts did not particularly impress him. From it he seems to have gained nothing pertaining to the infinitesimal calculus. By the previous introduction of his own algorithm he had made greater progress than by what came to his knowledge in London. Nothing mathematical that he had received engaged his thoughts in the immediate future, for on his way back to Holland he composed a lengthy dialogue on mechanical subjects.

Duillier's insinuations lighted up a flame of discord which a whole century was hardly sufficient to extinguish. Leibniz, who had never contested the priority of Newton's discovery, and who appeared to be quite satisfied with Newton's admission in his scholium, now appears for the first time in the controversy. He made an animated reply in the Leipzig Acts, and complained to the Royal Society of the injustice done him.

Here the affair rested for some time. In the Quadrature of Curves, published 1704, for the first time, a formal exposition of the method and notation of fluxions was made public. In 1705 appeared an unfavourable review of this in the Leipzig Acts, stating that Newton uses and always has used fluxions for the differences of Leibniz. This was considered by Newton's friends an imputation of plagiarism on the part of their chief, but this interpretation was always strenuously resisted by Leibniz. Keill, professor of astronomy at Oxford, undertook with more zeal than judgment the defence of Newton. In a paper inserted in the Philosophical Transactions of 1708, he claimed that Newton was the first inventor of fluxions and "that the same calculus was afterward published by Leibniz, the name and the mode of notation being changed." Leibniz complained to the secretary of the Royal Society of bad treatment and requested the interference of that body to induce Keill to disavow the intention of imputing fraud. Keill was not made to retract his accusation; on the contrary, was authorised by Newton and the Royal Society to explain and defend his statement. This he did in a long letter. Leibniz thereupon complained that the charge was now more open than before, and appealed for justice to the Royal Society and to Newton himself. The Royal Society, thus appealed to as a judge, appointed a committee which collected and reported upon a large mass of documents—mostly letters from and to Newton, Leibniz, Wallis, Collins, etc. This report, called the Commercium Epistolicum, appeared in the year 1712 and again in 1725, with a Recensio prefixed, and additional notes by Keill. The final conclusion in the Commercium Epistolicum was that Newton was the first inventor. But this was not to the point. The question was not whether Newton was the first inventor, but whether Leibniz had stolen the method. The committee had not formally ventured to assert their belief that Leibniz was a plagiarist. Yet there runs throughout the document a desire of proving Leibniz guilty of more than they meant positively to affirm. Leibniz protested only in private letters against the proceeding of the Royal Society, declaring that he would not answer an argument so weak. John Bernoulli, in a letter to Leibniz, which was published later in an anonymous tract, is as decidedly unfair towards Newton as the friends of the latter had been towards Leibniz. Keill replied, and then Newton and Leibniz appear as mutual accusers in several letters addressed to third parties. In a letter to Conti, April 9, 1716, Leibniz again reminded Newton of the admission he had made in the scholium, which he was now desirous of disavowing; Leibniz also states that he always believed Newton, but that, seeing him connive at accusations which he must have known to be false, it was natural that he (Leibniz) should begin to doubt. Newton did not reply to this letter, but circulated some remarks among his friends which he published immediately after hearing of the death of Leibniz, November 14, 1716. This paper of Newton gives the following explanation pertaining to the scholium in question: "He [Leibniz] pretends that in my book of principles I allowed him the invention of the calculus differentialis, independently of my own; and that to attribute this invention to myself is contrary to my knowledge there avowed. But in the paragraph there referred unto I do not find one word to this purpose." In the third edition of the Principia, 1725, Newton omitted the scholium and substituted in its place another, in which the name of Leibniz does not appear.

National pride and party feeling long prevented the adoption of impartial opinions in England, but now it is generally admitted by nearly all familiar with the matter, that Leibniz really was an independent inventor. Perhaps the most telling evidence to show that Leibniz was an independent inventor is found in the study of his mathematical papers (collected and edited by C. I. Gerhardt, in six volumes, Berlin, 1849–1860), which point out a gradual and natural evolution of the rules of the calculus in his own mind. "There was throughout the whole dispute," says De Morgan, "a confusion between the knowledge of fluxions or differentials and that of a calculus of fluxions or differentials; that is, a digested method with general rules."

This controversy is to be regretted on account of the long and bitter alienation which it produced between English and Continental mathematicians. It stopped almost completely all interchange of ideas on scientific subjects. The English adhered closely to Newton's methods and, until about 1820, remained, in most cases, ignorant of the brilliant mathematical discoveries that were being made on the Continent. The loss in point of scientific advantage was almost entirely on the side of Britain. The only way in which this dispute may be said, in a small measure, to have furthered the progress of mathematics, is through the challenge problems by which each side attempted to annoy its adversaries.

The recurring practice of issuing challenge problems was inaugurated at this time by Leibniz. They were, at first, not intended as defiances, but merely as exercises in the new calculus. Such was the problem of the isochronous curve (to find the curve along which a body falls with uniform velocity), proposed by him to the Cartesians in 1687, and solved by James Bernoulli, himself, and John Bernoulli. James Bernoulli proposed in the Leipzig Journal the question to find the curve (the catenary) formed by a chain of uniform weight suspended freely from its ends. It was resolved by Huygens, Leibniz, and himself. In 1697 John Bernoulli challenged the best mathematicians in Europe to solve the difficult problem, to find the curve (the cycloid) along which a body falls from one point to another in the shortest possible time. Leibniz solved it the day he received it. Newton, de l'Hospital, and the two Bernoullis gave solutions. Newton's appeared anonymously in the Philosophical Transactions, but John Bernoulli recognised in it his powerful mind, "tanquam," he says, "ex ungue leonem." The problem of orthogonal trajectories (a system of curves described by a known law being given, to describe a curve which shall cut them all at right angles) had been long proposed in the Acta Eruditorum, but failed at first to receive much attention. It was again proposed in 1716 by Leibniz, to feel the pulse of the English mathematicians.

This may be considered as the first defiance problem professedly aimed at the English. Newton solved it the same evening on which it was delivered to him, although he was much fatigued by the day's work at the mint. His solution, as published, was a general plan of an investigation rather than an actual solution, and was, on that account, criticised by Bernoulli as being of no value. Brook Taylor undertook the defence of it, but ended by using very reprehensible language. Bernoulli was not to be outdone in incivility, and made a bitter reply. Not long afterwards Taylor sent an open defiance to Continental mathematicians of a problem on the integration of a fluxion of complicated form which was known to very few geometers in England and supposed to be beyond the power of their adversaries. The selection was injudicious, for Bernoulli had long before explained the method of this and similar integrations. It served only to display the skill and augment the triumph of the followers of Leibniz. The last and most unskilful challenge was by John Keill. The problem was to find the path of a projectile in a medium which resists proportionally to the square of the velocity. Without first making sure that he himself could solve it, Keill boldly challenged Bernoulli to produce a solution. The latter resolved the question in very short time, not only for a resistance proportional to the square, but to any power of the velocity. Suspecting the weakness of the adversary, he repeatedly offered to send his solution to a confidential person in London, provided Keill would do the same. Keill never made a reply, and Bernoulli abused him and cruelly exulted over him.[26]

The explanations of the fundamental principles of the calculus, as given by Newton and Leibniz, lacked clearness and rigour. For that reason it met with opposition from several quarters. In 1694 Bernard Nieuwentyt of Holland denied the existence of differentials of higher orders and objected to the practice of neglecting infinitely small quantities. These objections Leibniz was not able to meet satisfactorily. In his reply he said the value of in geometry could be expressed as the ratio of finite quantities. In the interpretation of dx and dy Leibniz vacillated. At one time they appear in his writings as finite lines; then they are called infinitely small quantities, and again, quantitates inassignabiles, which spring from quantitates assignabiles by the law of continuity. In this last presentation Leibniz approached nearest to Newton.

In England the principles of fluxions were boldly attacked by Bishop Berkeley, the eminent metaphysician, who argued with great acuteness, contending, among other things, that the fundamental idea of supposing a finite ratio to exist between terms absolutely evanescent—"the ghosts of departed quantities," as he called them—was absurd and unintelligible. The reply made by Jurin failed to remove all the objections. Berkeley was the first to point out what was again shown later by Lazare Carnot, that correct answers were reached by a "compensation of errors." Berkeley's attack was not devoid of good results, for it was the immediate cause of the work on fluxions by Maclaurin. In France Michel Rolle rejected the differential calculus and had a controversy with Varignon on the subject.

Among the most vigorous promoters of the calculus on the Continent were the Bernoullis. They and Euler made Basel in Switzerland famous as the cradle of great mathematicians. The family of Bernoullis furnished in course of a century eight members who distinguished themselves in mathematics. We subjoin the following genealogical table:—

Nicolaus Bernoulli, the Father
Jacob, 1654–1705 Nicolaus Johann, 1667–1748
| |
Nicolaus, 1687–1759 Nicolaus, 1695–1726
Daniel, 1700–1782
Johann, 1710–1790
Daniel Johann, 1744–1807 Jacob, 1758–1789

Most celebrated were the two brothers Jacob (James) and Johann (John), and Daniel, the son of John. James and John were staunch friends of Leibniz and worked hand in hand with him. James Bernoulli (1654–1705) was born in Basel. Becoming interested in the calculus, he mastered it without aid from a teacher. From 1687 until his death he occupied the mathematical chair at the University of Basel. He was the first to give a solution to Leibniz's problem of the isochronous curve. In his solution, published in the Acta Eruditorum, 1690, we meet for the first time with the word integral. Leibniz had called the integral calculus calculus summatorius, but in 1696 the term calculus integralis was agreed upon between Leibniz and John Bernoulli. James proposed the problem of the catenary, then proved the correctness of Leibniz's construction of this curve, and solved the more complicated problems, supposing the string to be (1) of variable density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre. Of these problems he published answers without explanations, while his brother John gave in addition their theory. He determined the shape of the "elastic curve" formed by an elastic plate or rod fixed at one end and bent by a weight applied to the other end; of the "lintearia," a flexible rectangular plate with two sides fixed horizontally at the same height, filled with a liquid; of the "volaria," a rectangular sail filled with wind. He studied the loxodromic and logarithmic spirals, in the last of which he took particular delight from its remarkable property of reproducing itself under a variety of conditions. Following the example of Archimedes, he willed that the curve be engraved upon his tombstone with the inscription "eadem mutata resurgo." In 1696 he proposed the famous problem of isoperimetrical figures, and in 1701 published his own solution. He wrote a work on Ars Conjectandi, which is a development of the calculus of probabilities and contains the investigation now called "Bernoulli's theorem" and the so-called "numbers of Bernoulli," which are in fact (though not so considered by him) the coefficients of in the expansion of . Of his collected works, in three volumes, one was printed in 1713, the other two in 1744.

John Bernoulli (1667–1748) was initiated into mathematics by his brother. He afterwards visited France, where he met Malebranche, Cassini, De Lahire, Varignon, and de l'Hospital. For ten years he occupied the mathematical chair at Gröningen and then succeeded his brother at Basel. He was one of the most enthusiastic teachers and most successful original investigators of his time. He was a member of almost every learned society in Europe. His controversies were almost as numerous as his discoveries. He was ardent in his friendships, but unfair, mean, and violent toward all who incurred his dislike—even his own brother and son. He had a bitter dispute with James on the isoperimetrical problem. James convicted him of several paralogisms. After his brother's death he attempted to substitute a disguised solution of the former for an incorrect one of his own. John admired the merits of Leibniz and Euler, but was blind to those of Newton. He immensely enriched the integral calculus by his labours. Among his discoveries are the exponential calculus, the line of swiftest descent, and its beautiful relation to the path described by a ray passing through strata of variable density. He treated trigonometry by the analytical method, studied caustic curves and trajectories. Several times he was given prizes by the Academy of Science in Paris.

Of his sons, Nicholas and Daniel were appointed professors of mathematics at the same time in the Academy of St. Petersburg. The former soon died in the prime of life; the latter returned to Basel in 1733, where he assumed the chair of experimental philosophy. His first mathematical publication was the solution of a differential equation proposed by Riccati. He wrote a work on hydrodynamics. His investigations on probability are remarkable for their boldness and originality. He proposed the theory of moral expectation, which he thought would give results more in accordance with our ordinary notions than the theory of mathematical probability. His "moral expectation" has become classic, but no one ever makes use of it. He applies the theory of probability to insurance; to determine the mortality caused by small-pox at various stages of life; to determine the number of survivors at a given age from a given number of births; to determine how much inoculation lengthens the average duration of life. He showed how the differential calculus could be used in the theory of probability. He and Euler enjoyed the honour of having gained or shared no less than ten prizes from the Academy of Sciences in Paris.

Johann Bernoulli (born 1710) succeeded his father in the professorship of mathematics at Basel. He captured three prizes (on the capstan, the propagation of light, and the magnet) from the Academy of Sciences at Paris. Nicolaus Bernoulli (born 1687) held for a time the mathematical chair at Padua which Galileo had once filled. Johann Bernoulli (born 1744) at the age of nineteen was appointed astronomer royal at Berlin, and afterwards director of the mathematical department of the Academy. His brother Jacob took upon himself the duties of the chair of experimental physics at Basel, previously performed by his uncle Jacob, and later was appointed mathematical professor in the Academy at St Petersburg.

Brief mention will now be made of some other mathematicians belonging to the period of Newton, Leibniz, and the elder Bernoullis.

Guillaume François Antoine l'Hospital (1661–1704), a pupil of John Bernoulli, has already been mentioned as taking part in the challenges issued by Leibniz and the Bernoullis. He helped powerfully in making the calculus of Leibniz better known to the mass of mathematicians by the publication of a treatise thereon in 1696. This contains for the first time the method of finding the limiting value of a fraction whose two terms tend toward zero at the same time.

Another zealous French advocate of the calculus was Pierre Varignon (1654–1722). Joseph Saurin (1659–1737) solved the delicate problem of how to determine the tangents at the multiple points of algebraic curves. François Nicole (1683–1758) in 1717 issued the first systematic treatise on finite differences, in which he finds the sums of a considerable number of interesting series. He wrote also on roulettes, particularly spherical epicycloids, and their rectification. Also interested in finite differences was Pierre Raymond de Montmort (1678–1719). His chief writings, on the theory of probability, served to stimulate his more distinguished successor, De Moivre. Jean Paul de Gua (1713–1785) gave the demonstration of Descartes' rule of signs, now given in books. This skilful geometer wrote in 1740 a work on analytical geometry, the object of which was to show that most investigations on curves could be carried on with the analysis of Descartes quite as easily as with the calculus. He shows how to find the tangents, asymptotes, and various singular points of curves of all degrees, and proved by perspective that several of these points can be at infinity. A mathematician who clung to the methods of the ancients was Philippe de Lahire (1640–1718), a pupil of Desargues. His work on conic sections is purely synthetic, but differs from ancient treatises in deducing the properties of conies from those of the circle in the same manner as did Desargues and Pascal. His innovations stand in close relation with modern synthetic geometry. He wrote on roulettes, on graphical methods, epicycloids, conchoids, and on magic squares. Michel Rolle (1652–1719) is the author of a theorem named after him.

Of Italian mathematicians, Riccati and Fagnano must not remain unmentioned. Jacopo Francesco, Count Riccati (1676–1754) is best known in connection with his problem, called Riccati's equation, published in the Acta Eruditorum in 1724. He succeeded in integrating this differential equation for some special cases. A geometrician of remarkable power was Giulio Carlo, Count de Fagnano (1682–1766). He discovered the following formula, , in which he anticipated Euler in the use of imaginary exponents and logarithms. His studies on the rectification of the ellipse and hyperbola are the starting-points of the theory of elliptic functions. He showed, for instance, that two arcs of an ellipse can be found in an indefinite number of ways, whose difference is expressible by a right line.

In Germany the only noted contemporary of Leibniz is Ehrenfried Walter Tschirnhausen (1631–1708), who discovered the caustic of reflection, experimented on metallic reflectors and large burning-glasses, and gave us a method of transforming equations named after him. Believing that the most simple methods (like those of the ancients) are the most correct, he concluded that in the researches relating to the properties of curves the calculus might as well be dispensed with.

After the death of Leibniz there was in Germany not a single mathematician of note. Christian Wolf (1679–1754), professor at Halle, was ambitious to figure as successor of Leibniz, but he "forced the ingenious ideas of Leibniz into a pedantic scholasticism, and had the unenviable reputation of having presented the elements of the arithmetic, algebra, and analysis developed since the time of the Renaissance in the form of Euclid,—of course only in outward form, for into the spirit of them he was quite unable to penetrate."[16]

The contemporaries and immediate successors of Newton in Great Britain were men of no mean merit. We have reference to Cotes, Taylor, Maclaurin, and De Moivre. We are told that at the death of Roger Cotes (1682–1716), Newton exclaimed, "If Cotes had lived, we might have known something." It was at the request of Dr. Bentley that Cotes undertook the publication of the second edition of Newton's Principia. His mathematical papers were published after his death by Robert Smith, his successor in the Plumbian professorship at Trinity College. The title of the work, Harmonia Mensurarum, was suggested by the following theorem contained in it: If on each radius vector, through a fixed point O, there be taken a point R, such that the reciprocal of OR be the arithmetic mean of the reciprocals of then the locus of R will be a straight line. In this work progress was made in the application of logarithms and the properties of the circle to the calculus of fluents. To Cotes we owe a theorem in trigonometry which depends on the forming of factors of . Chief among the admirers of Newton were Taylor and Maclaurin. The quarrel between English and Continental mathematicians caused them to work quite independently of their great contemporaries across the Channel.

Brook Taylor (1685–1731) was interested in many branches of learning, and in the latter part of his life engaged mainly in religious and philosophic speculations. His principal work, Methodus incrementorum directa et inversa, London, 1715–1717, added a new branch to mathematics, now called "finite differences." He made many important applications of it, particularly to the study of the form of movement of vibrating strings, first reduced to mechanical principles by him. This work contains also "Taylor's theorem," the importance of which was not recognised by analysts for over fifty years, until Lagrange pointed out its power. His proof of it does not consider the question of convergency, and is quite worthless. The first rigorous proof was given a century later by Cauchy. Taylor's work contains the first correct explanation of astronomical refraction. He wrote also a work on linear perspective, a treatise which, like his other writings, suffers for want of fulness and clearness of expression. At the age of twenty-three he gave a remarkable solution of the problem of the centre of oscillation, published in 1714. His claim to priority was unjustly disputed by John Bernoulli.

Colin Maclaurin (1698–1746) was elected professor of mathematics at Aberdeen at the age of nineteen by competitive examination, and in 1725 succeeded James Gregory at the University of Edinburgh. He enjoyed the friendship of Newton, and, inspired by Newton's discoveries, he published in 1719 his Geometria Organica, containing a new and remarkable mode of generating conics, known by his name. A second tract, De Linearum geometricarum Proprietatibus, 1720, is remarkable for the elegance of its demonstrations. It is based upon two theorems: the first is the theorem of Cotes; the second is Maclaurin's: If through any point O a line be drawn meeting the curve in n points, and at these points tangents be drawn, and if any other line through O cut the curve in etc., and the system of n tangents in etc., then . This and Cotes' theorem are generalisations of theorems of Newton. Maclaurin uses these in his treatment of curves of the second and third degree, culminating in the remarkable theorem that if a quadrangle has its vertices and the two points of intersection of its opposite sides upon a curve of the third degree, then the tangents drawn at two opposite vertices cut each other on the curve. He deduced independently Pascal's theorem on the hexagram. The following is his extension of this theorem (Phil. Trans., 1735): If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, etc., respectively, then the free summit moves on a curve of the degree , which reduces to when the fixed points all lie on a straight line. Maclaurin wrote on pedal curves. He is the author of an Algebra. The object of his treatise on Fluxions was to found the doctrine of fluxions on geometric demonstrations after the manner of the ancients, and thus, by rigorous exposition, answer such attacks as Berkeley's that the doctrine rested on false reasoning. The Fluxions contained for the first time the correct way of distinguishing between maxima and minima, and explained their use in the theory of multiple points. "Maclaurin's theorem" was previously given by James Stirling, and is but a particular case of "Taylor's theorem." Appended to the treatise on Fluxions is the solution of a number of beautiful geometric, mechanical, and astronomical problems, in which he employs ancient methods with such consummate skill as to induce Clairaut to abandon analytic methods and to attack the problem of the figure of the earth by pure geometry. His solutions commanded the liveliest admiration of Lagrange. Maclaurin investigated the attraction of the ellipsoid of revolution, and showed that a homogeneous liquid mass revolving uniformly around an axis under the action of gravity must assume the form of an ellipsoid of revolution. Newton had given this theorem without proof. Notwithstanding the genius of Maclaurin, his influence on the progress of mathematics in Great Britain was unfortunate; for, by his example, he induced his countrymen to neglect analysis and to be indifferent to the wonderful progress in the higher analysis made on the Continent.

It remains for us to speak of Abraham de Moivre (1667–1754), who was of French descent, but was compelled to leave France at the age of eighteen, on the Revocation of the Edict of Nantes. He settled in London, where he gave lessons in mathematics. He lived to the advanced age of eighty-seven and sank into a state of almost total lethargy. His subsistence was latterly dependent on the solution of questions on games of chance and problems on probabilities, which he was in the habit of giving at a tavern in St. Martin's Lane. Shortly before his death he declared that it was necessary for him to sleep ten or twenty minutes longer every day. The day after he had reached the total of over twenty-three hours, he slept exactly twenty-four hours and then passed away in his sleep. De Moivre enjoyed the friendship of Newton and Halley. His power as a mathematician lay in analytic rather than geometric investigation. He revolutionised higher trigonometry by the discovery of the theorem known by his name and by extending the theorems on the multiplication and division of sectors from the circle to the hyperbola. His work on the theory of probability surpasses anything done by any other mathematician except Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Bernoulli's theorem by the aid of Stirling's theorem.[42] His chief works are the Doctrine of Chances, 1716, the Miscellanea Analytica, 1730, and his papers in the Philosophical Transactions.