A Short Account of the History of Mathematics · W. W. Rouse Ball
Leaving for a moment the English mathematicians of the first half of the eighteenth century we come next to a number of continental writers who barely escape mediocrity, and to whom it will be necessary to devote but few words. Their writings mark the steps by which analytical geometry and the differential and integral calculus were perfected and made familiar to mathematicians. Nearly all of them were pupils of one or other of the two elder Bernoullis, and they were so nearly contemporaries that it is difficult to arrange them chronologically. The most eminent of them are Cramer, de Gua, de Montmort, Fagnano, l'Hospital, Nicole, Parent, Riccati, Saurin, and Varignon.
L'Hospital. Guillaume Franc
ois Antoine l'Hospital, Marquis de St.-Mesme, born at Paris in 1661, and died there on Feb. 2, 1704, was among the earliest pupils of John Bernoulli, who, in 1691, spent some months at l'Hospital's house in Paris for the purpose of teaching him the new calculus. It seems strange, but it is substantially true, that a knowledge of the infinitesimal calculus and the power of using it was then confined to Newton, Leibnitz, and the two elder Bernoullis--and it will be noticed that they were the only mathematicians who solved the more difficult problems then proposed as challenges. There was at that time no text-book on the subject, and the credit of putting together the first treatise which explained the principles and use of the method is due to l'Hospital; it was published in 1696 under the title Analyse des infiniment petits. This contains a partial investigation of the limiting value of the ratio of functions which for a certain value of the variable take the indeterminate form 0:0 , a problem solved by John Bernoulli in 1704. This work had a wide circulation; it brought the differential notation into general use in France, and helped to make it known in Europe. A supplement, containing a similar treatment of the integral calculus, together with additions to the differential calculus which had been made in the following half century, was published at Paris, 1754--56, by L. A. de Bougainville.
L'Hospital took part in most of the challenges issued by Leibnitz, the Bernoullis, and other continental mathematicians of the time; in particular he gave a solution of the brachistochrone, and investigated the form of the solid of least resistance of which Newton in the Principia had stated the result. He also wrote a treatise on analytical conics, which was published in 1707, and for nearly a century was deemed a standard work on the subject.
Varignon. Pierre Varignon, born at Caen in 1654, and died in Paris on Dec. 22, 1722, was an intimate friend of Newton, Leibnitz, and the Bernoullis, and, after l'Hospital, was the earliest and most powerful advocate in France of the use of the differential calculus. He realized the necessity of obtaining a test for examining the convergency of series, but the analytical difficulties were beyond his powers. He simplified the proofs of many of the leading propositions in mechanics, and in 1687 recast the treatment of the subject, basing it on the composition of forces. His works were published at Paris in 1725.
De Montmort. Nicole. Pierre Raymond de Montmort, born at Paris on Oct. 27, 1678, and died there on Oct. 7, 1719, was interested in the subject of finite differences. He determined in 1713 the sum of n terms of a finite series of the form na+n(n-1)12 a + n(n-1)(n-2)123^2a+; a theorem which seems to have been independently rediscovered by Chr. Goldbach in 1718. François Nicole, who was born at Paris on Dec. 23, 1683, and died there on Jan. 18, 1758, published his Traité du calcul des differences finies in 1717; it contains rules both for forming differences and for effecting the summation of given series. Besides this, in 1706 he wrote a work on roulettes, especially spherical epicycloids; and in 1729 and 1731 he published memoirs on Newton's essay on curves of the third degree.
Parent. Saurin. De Gua. Antoine Parent, born at Paris on Sept. 16, 1666, and died there on Sept. 26, 1716, wrote in 1700 on analytical geometry of three dimensions. His works were collected and published in three volumes at Paris in 1713. Joseph Saurin, born at Courtaison in 1659, and died at Paris on Dec. 29, 1737, was the first to show how the tangents at the multiple points of curves could be determined by analysis. Jean Paul de Gua de Malves was born at Carcassonne in 1713, and died at Paris on June 2, 1785. He published in 1740 a work on analytical geometry in which he applied it, without the aid of the differential calculus, to find the tangents, asymptotes, and various singular points of an algebraical curve; and he further shewed how singular points and isolated loops were affected by conical projection. He gave the proof of Descartes's rule of signs which is to be found in most modern works. It is not clear whether Descartes ever proved it strictly, and Newton seems to have regarded it as obvious.
Cramer. Gabriel Cramer, born at Geneva in 1704, and died at Bagnols in 1752, was professor at Geneva. The work by which he is best known is his treatise on algebraic curves published in 1750, which, as far as it goes, is fairly complete; it contains the earliest demonstration that a curve of the n th degree is in general determined if 1/2n(n + 3) points on it be given. This work is still sometimes read. Besides this, he edited the works of the two elder Bernoullis; and wrote on the physical cause of the spheroidal shape of the planets and the motion of their apses, 1730, and on Newton's treatment of cubic curves, 1746.
Riccati. Jacopo Francesco, Count Riccati, born at Venice on May 28, 1676, and died at Trèves on April 15, 1754, did a great deal to disseminate a knowledge of the Newtonian philosophy in Italy. Besides the equation known by his name, certain cases of which he succeeded in integrating, he discussed the question of the possibility of lowering the order of a given differential equation. His works were published at Trèves in four volumes in 1758. He had two sons who wrote on several minor points connected with the integral calculus and differential equations, and applied the calculus to several mechanical questions: these were Vincenzo, who was born in 1707 and died in 1775, and Giordano, who was born in 1709 and died in 1790.
Fagnano. Giulio Carlo, Count Fagnano, and Marquis de Toschi, born at Sinigaglia on Dec. 6, 1682, and died on Sept. 26, 1766, may be said to have been the first writer who directed attention to the theory of elliptic functions. Failing to rectify the ellipse or hyperbola, Fagnano attempted to determine arcs whose difference should be rectifiable. He also pointed out the remarkable analogy existing between the integrals which represent the arc of a circle and the arc of a lemniscate. Finally he proved the formula = 2i (1-i)/(1+i), where i stands for sqrt(-1) . His works were collected and published in two volumes at Pesaro in 1750.
It was inevitable that some mathematicians should object to methods of analysis founded on the infinitesimal calculus. The most prominent of these were Viviani, De la Hire, and Rolle, whose names were mentioned at the close of chapter xv.
So far no one of the school of Leibnitz and the two elder Bernoullis had shewn any exceptional ability, but by the action of a number of second-rate writers the methods and language of analytical geometry and the differential calculus had become well known by about 1740. The close of this school is marked by the appearance of Clairaut, D'Alembert, and Daniel Bernoulli. Their lives overlap the period considered in the next chapter, but, though it is difficult to draw a sharp dividing line which shall separate by a definite date the mathematicians there considered from those whose writings are discussed in this chapter, I think that on the whole the works of these three writers are best treated here.
Clairaut. Alexis Claude Clairaut was born at Paris on May 13, 1713, and died there on May 17, 1765. He belongs to the small group of children who, though of exceptional precocity, survive and maintain their powers when grown up. As early as the age of twelve he wrote a memoir on four geometrical curves; but his first important work was a treatise on tortuous curves, published when he was eighteen--a work which procured for him admission to the French Academy. In 1731 he gave a demonstration of the fact noted by Newton that all curves of the third order were projections of one of five parabolas.
In 1741 Clairaut went on a scientific expedition to measure the length of a meridian degree on the earth's surface, and on his return in 1743 he published his Théorie de la figure de la terre. This is founded on a paper by Maclaurin, wherein it had been shewn that a mass of homogeneous fluid set in rotation about a line through its centre of mass would, under the mutual attraction of its particles, take the form of a spheroid. This work of Clairaut treated of heterogeneous spheroids and contains the proof of his formula for the accelerating effect of gravity in a place of latitude l , namely, g = G 1 + (52 m - )^2l, where G is the value of equatorial gravity, m the ratio of the centrifugal force to gravity at the equator, and epsilon the ellipticity of a meridian section of the earth. In 1849 Stokes shewed that the same result was true whatever was the interior constitution or density of the earth, provided the surface was a spheroid of equilibrium of small ellipticity.
Impressed by the power of geometry as shewn in the writings of Newton and Maclaurin, Clairaut abandoned analysis, and his next work, the Théorie de la lune, published in 1752, is strictly Newtonian in character. This contains the explanation of the motion of the apse which had previously puzzled astronomers, and which Clairaut had at first deemed so inexplicable that he was on the point of publishing a new hypothesis as to the law of attraction when it occurred to him to carry the approximation to the third order, and he thereupon found that the result was in accordance with the observations. This was followed in 1754 by some lunar tables. Clairaut subsequently wrote various papers on the orbit of the moon, and on the motion of comets as affected by the perturbation of the planets, particularly on the path of Halley's comet.
His growing popularity in society hindered his scientific work: "engagé," says Bossut, "à des soupers, à des veilles, entraîné par un goût vif pour les femmes, voulant allier le plaisir à ses travaux ordinaires, il perdit le repos, la santé, enfin la vie à l'âge de cinquante-deux ans."
D'Alembert. Jean-le-Rond D'Alembert was born at Paris on November 16, 1717, and died there on October 29, 1783. He was the illegitimate child of the chevalier Destouches. Being abandoned by his mother on the steps of the little church of St. Jean-le-Rond, which then nestled under the great porch of Notre-Dame, he was taken to the parish commissary, who, following the usual practice in such cases, gave him the Christian name of Jean-le-Rond; I do not know by what authority he subsequently assumed the right to prefix de to his name. He was boarded out by the parish with the wife of a glazier in a small way of business who lived near the cathedral, and here he found a real home, though a humble one. His father appears to have looked after him, and paid for his going to a school where he obtained a fair mathematical education.
An essay written by him in 1738 on the integral calculus, and another in 1740 on "ducks and drakes" or ricochets, attracted some attention, and in the same year he was elected a member of the French Academy; this was probably due to the influence of his father. It is to his credit that he absolutely refused to leave his adopted mother, with whom he continued to live until her death in 1757. It cannot be said that she sympathised with his success, for at the height of his fame she remonstrated with him for wasting his talents on such work: "Vous ne serez jamais qu'un philosophe," said she, "et qu'est-ce qu'un philosophe? c'est un fou qui se tourmente pendant sa vie, pour qu'on parle de lui lorsqu'il n'y sera plus."
Nearly all his mathematical works were produced during the years 1743 to 1754. The first of these was his Traité de dynamique, published in 1743, in which he enunciates the principle known by his name, namely, that the "internal forces of inertia" (that is, the forces which resist acceleration) must be equal and opposite to the forces which produce the acceleration. This may be inferred from Newton's second reading of his third law of motion, but the full consequences had not been realized previously. The application of this principle enables us to obtain the differential equations of motion of any rigid system.
In 1744 D'Alembert published his Traité de l'équilibre et du mouvement des fluides, in which he applies his principle to fluids; this led to partial differential equations which he was then unable to solve. In 1745 he developed that part of the subject which dealt with the motion of air in his Théorie générale des vents, and this again led him to partial differential equations. A second edition of this in 1746 was dedicated to Frederick the Great of Prussia, and procured an invitation to Berlin and the offer of a pension; he declined the former, but subsequently, after some pressing, pocketed his pride and the latter. In 1747 he applied the differential calculus to the problem of a vibrating string, and again arrived at a partial differential equation.
His analysis had three times brought him to an equation of the form ^2 u t^2 = ^2 u x^2, and he now succeeded in shewing that it was satisfied by u = (x+t) + (x-t), where phi and psi are arbitrary functions. It may be interesting to give his solution which was published in the transactions of the Berlin Academy for 1747. He begins by saying that, if partial u/partial x be denoted by p and partial u/partial t by q , then du = pdx+qdt. But, by the given equation, partial q/partial t = partial p/partial x , and therefore pdt+qdx is also an exact differential: denote it by dv .
Therefore dv = pdt+qdx. Hence du+dv = (pdx+qdt) + (pdt+qdx) = (p+q)(dx+dt), and du-dv = (pdx+qdt) - (pdt+qdx) = (p-q)(dx-dt).
Thus u+v must be a function of x+t , and u-v must be a function of x-t . We may therefore put
u+v = 2(x+t), and u-v = 2(x-t). Hence u = (x+t) + (x-t).
D'Alembert added that the conditions of the physical problem of a vibrating string demand that, when x=0 , u should vanish for all values of t . Hence identically (t) + (-t) = 0. Assuming that both functions can be expanded in integral powers of t , this requires that they should contain only odd powers. Hence (-t) = -(t) = (-t). Therefore u = (x+t) + (x-t).
Euler now took the matter up and shewed that the equation of the form of the string was partial^2 u/partial t^2 = a^2partial^2 u/partial x^2 , and that the general integral was u = phi(x+at) + psi(x-at) , where phi and psi are arbitrary functions.
The chief remaining contributions of D'Alembert to mathematics were on physical astronomy, especially on the precession of the equinoxes, and on variations in the obliquity of the ecliptic. These were collected in his Système du monde, published in three volumes in 1754.
During the latter part of his life he was mainly occupied with the great French encyclopaedia. For this he wrote the introduction, and numerous philosophical and mathematical articles; the best are those on geometry and on probabilities. His style is brilliant, but not polished, and faithfully reflects his character, which was bold, honest, and frank. He defended a severe criticism which he had offered on some mediocre work by the remark, "j'aime mieux être incivil qu'ennuyé"; and with his dislike of sycophants and bores it is not surprising that during his life he had more enemies than friends.
Daniel Bernoulli. Daniel Bernoulli, whose name I mentioned above, and who was by far the ablest of the younger Bernoullis, was a contemporary and intimate friend of Euler, whose works are mentioned in the next chapter. Daniel Bernoulli was born on Feb. 9, 1700, and died at Bâle, where he was professor of natural philosophy, on March 17, 1782. He went to Petrograd in 1724 as professor of mathematics, but the roughness of the social life was distasteful to him, and he was not sorry when a temporary illness in 1733 allowed him to plead his health as an excuse for leaving. He then returned to Bâle, and held successively chairs of medicine, metaphysics, and natural philosophy there.
His earliest mathematical work was the Exercitationes, published in 1724, which contains a solution of the differential equation proposed by Riccati. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motions of rotation. His chief work is his Hydrodynamica, published in 1738; it resembles Lagrange's Mécanique analytique in being arranged so that all the results are consequences of a single principle, namely, in this case, the conservation of energy. This was followed by a memoir on the theory of the tides, to which, conjointly with memoirs by Euler and Maclaurin, a prize was awarded by the French Academy: these three memoirs contain all that was done on this subject between the publication of Newton's Principia and the investigations of Laplace. Bernoulli also wrote a large number of papers on various mechanical questions, especially on problems connected with vibrating strings, and the solutions given by Taylor and by D'Alembert. He is the earliest writer who attempted to formulate a kinetic theory of gases, and he applied the idea to explain the law associated with the names of Boyle and Mariotte.
I have reserved a notice of the English mathematicians who succeeded Newton, in order that the members of the English school may be all treated together. It was almost a matter of course that the English should at first have adopted the notation of Newton in the infinitesimal calculus in preference to that of Leibnitz, and consequently the English school would in any case have developed on somewhat different lines to that on the continent, where a knowledge of the infinitesimal calculus was derived solely from Leibnitz and the Bernoullis. But this separation into two distinct schools became very marked owing to the action of Leibnitz and John Bernoulli, which was naturally resented by Newton's friends; and so for forty or fifty years, to the disadvantage of both sides, the quarrel raged. The leading members of the English school were Cotes, Demoivre, Ditton, David Gregory, Halley, Maclaurin, Simpson, and Taylor. I may, however, again remind my readers that as we approach modern times the number of capable mathematicians in Britain, France, Germany, and Italy becomes very considerable, but that in a popular sketch like this book it is only the leading men whom I propose to mention.
To David Gregory, Halley, and Ditton I need devote but few words.
David Gregory. David Gregory, the nephew of the James Gregory mentioned above, born at Aberdeen on June 24, 1661, and died at Maidenhead on Oct. 10, 1708, was appointed professor at Edinburgh in 1684, and in 1691 was on Newton's recommendation elected Savilian professor at Oxford. His chief works are one on geometry, issued in 1684; one on optics, published in 1695, which contains [p. 98] the earliest suggestion of the possibility of making an achromatic combination of lenses; and one on the Newtonian geometry, physics, and astronomy, issued in 1702.
Halley. Edmund Halley, born in London in 1656, and died at Greenwich in 1742, was educated at St. Paul's School, London, and Queen's College, Oxford, in 1703 succeeded Wallis as Savilian professor, and subsequently in 1720 was appointed astronomer-royal in succession to Flamsteed, whose Historia Coelestis Britannica he edited; the first and imperfect edition was issued in 1712. Halley's name will be recollected for the generous manner in which he secured the immediate publication of Newton's Principia in 1687. Most of his original work was on astronomy and allied subjects, and lies outside the limits of this book; it may be, however, said that the work is of excellent quality, and both Lalande and Mairan speak of it in the highest terms. Halley conjecturally restored the eighth and lost book of the conics of Apollonius, and in 1710 brought out a magnificent edition of the whole work; he also edited the works of Serenus, those of Menelaus, and some of the minor works of Apollonius. He was in his turn succeeded at Greenwich as astronomer-royal by Bradley.
, born in Gloucestershire in 1692, and died in 1762, was the most distinguished astronomer of the first half of the eighteenth century. Among his more important discoveries were the explanation of astronomical aberration (1729), the cause of nutation (1748), and his empirical formula for corrections for refraction. It is perhaps not too much to say that he was the first astronomer who made the art of observing part of a methodical science.
Ditton. Humphry Ditton was born at Salisbury on May 29, 1675, and died in London in 1715 at Christ's Hospital, where he was mathematical master. He does not seem to have paid much attention to mathematics until he came to London about 1705, and his early death was a distinct loss to English science. He published in 1706 a text book on fluxions; this and another similar work by William Jones, which was issued in 1711, occupied in England much the same place that l'Hospital's treatise did in France. In 1709 Ditton issued an algebra, and in 1712 a treatise on perspective. He also wrote numerous papers in the Philosophical Transactions. He was the earliest writer to attempt to explain the phenomenon of capillarity on mathematical principles; and he invented a method for finding the longitude, which has been since used on various occasions.
Taylor. Brook Taylor, born at Edmonton on August 18, 1685, and died in London on December 29, 1731, was educated at St. John's College, Cambridge, and was among the most enthusiastic of Newton's admirers. From the year 1712 onwards he wrote numerous papers in the Philosophical Transactions, in which, among other things, he discussed the motion of projectiles, the centre of oscillation, and the forms taken by liquids when raised by capillarity. In 1719 he resigned the secretaryship of the Royal Society and abandoned the study of mathematics. His earliest work, and that by which he is generally known, is his Methodus Incrementorum Directa et Inversa, published in London in 1715. This contains [prop. 7] a proof of the well-known theorem f(x + h) = f(x) + hf'(x) + h^2|2f"(x) + , by which a function of a single variable can be expanded in powers of it. He does not consider the convergency of the series, and the proof which involves numerous assumptions is not worth reproducing. The work also includes several theorems on interpolation. Taylor was the earliest writer to deal with theorems on the change of the independent variable; he was perhaps the first to realize the possibility of a calculus of operation, and just as he denotes the nth differential coefficient of y by yn , so he uses y-1 to represent the integral of y ; lastly, he is usually recognized as the creator of the theory of finite differences.
The applications of the calculus to various questions given in the Methodus have hardly received that attention they deserve. The most important of them is the theory of the transverse vibrations of strings, a problem which had baffled previous investigators. In this investigation Taylor shews that the number of half-vibrations executed in a second is DP/LN, where L is the length of the string, N its weight, P the weight which stretches it, and D the length of a seconds pendulum. This is correct, but in arriving at it he assumes that every point of the string will pass through its position of equilibrium at the same instant, a restriction which D'Alembert subsequently shewed to be unnecessary. Taylor also found the form which the string assumes at any instant.
The Methodus also contains the earliest determination of the differential equation of the path of a ray of light when traversing a heterogeneous medium; and, assuming that the density of the air depends only on its distance from the earth's surface, Taylor obtained by means of quadratures the approximate form of the curve. The form of the catenary and the determination of the centres of oscillation and percussion are also discussed.
A treatise on perspective by Taylor, published in 1719, contains the earliest general enunciation of the principle of vanishing points; though the idea of vanishing points for horizontal and parallel lines in a picture hung in a vertical plane had been enunciated by Guido Ubaldi in his Perspectivae Libri, Pisa, 1600, and by Stevinus in his Sciagraphia, Leyden, 1608.
Cotes. Roger Cotes was born near Leicester on July 10, 1682, and died at Cambridge on June 5, 1716. He was educated at Trinity College, Cambridge, of which society he was a fellow, and in 1706 was elected to the newly-created Plumian chair of astronomy in the university of Cambridge. From 1709 to 1713 his time was mainly occupied in editing the second edition of the Principia. The remark of Newton that if only Cotes had lived "we might have known something" indicates the opinion of his abilities held by most of his contemporaries.
Cotes's writings were collected and published in 1722 under the titles Harmonia Mensurarum and Opera Miscellanea. His lectures on hydrostatics were published in 1738. A large part of the Harmonia Mensurarum is given up to the decomposition and integration of rational algebraical expressions. That part which deals with the theory of partial fractions was left unfinished, but was completed by Demoivre. Cotes's theorem in trigonometry, which depends on forming the quadratic factors of x^n - 1 , is well known. The proposition that "if from a fixed point O a line be drawn cutting a curve in Q1 , Q2 , dots , Qn , and a point P be taken on the line so that the reciprocal of OP is the arithmetic mean of the reciprocals of OQ1 , OQ2 , dots , OQn , then the locus of P will be a straight line" is also due to Cotes. The title of the book was derived from the latter theorem. The Opera Miscellanea contains a paper on the method for determining the most probable result from a number of observations. This was the earliest attempt to frame a theory of errors. It also contains essays on Newton's Methodus Differentialis, on the construction of tables by the method of differences, on the descent of a body under gravity, on the cycloidal pendulum, and on projectiles.
Demoivre. Abraham Demoivre (more correctly written as de Moivre) was born at Vitry on May 26, 1667, and died in London on November 27, 1754. His parents came to England when he was a boy, and his education and friends were alike English. His interest in the higher mathematics is said to have originated in his coming by chance across a copy of Newton's Principia. From the éloge on him delivered in 1754 before the French Academy it would seem that his work as a teacher of mathematics had led him to the house of the Earl of Devonshire at the instant when Newton, who had asked permission to present a copy of his work to the earl, was coming out. Taking up the book, and charmed by the far-reaching conclusions and the apparent simplicity of the reasoning, Demoivre thought nothing would be easier than to master the subject, but to his surprise found that to follow the argument overtaxed his powers. He, however, bought a copy, and as he had but little leisure he tore out the pages in order to carry one or two of them loose in his pocket so that he could study them in the intervals of his work as a teacher. Subsequently he joined the Royal Society, and became intimately connected with Newton, Halley, and other mathematicians of the English school. The manner of his death has a certain interest for psychologists. Shortly before it he declared that it was necessary for him to sleep some ten minutes or a quarter of an hour longer each day than the preceding one. The day after he had thus reached a total of something over twenty-three hours he slept up to the limit of twenty-four hours, and then died in his sleep.
He is best known for having, together with Lambert, created that part of trigonometry which deals with imaginary quantities. Two theorems on this part of the subject are still connected with his name, namely, that which asserts that cos nx + i sin nx is one of the values of (cos x + i sin x)^n , and that which gives the various quadratic factors of x^2n - 2px^n + 1 . His chief works, other than numerous papers in the Philosophical Transactions, were The Doctrine of Chances, published in 1718, and the Miscellanea Analytica, published in 1730. In the former the theory of recurring series was first given, and the theory of partial fractions which Cotes's premature death had left unfinished was completed, while the rule for finding the probability of a compound event was enunciated. The latter book, besides the trigonometrical propositions mentioned above, contains some theorems in astronomy, but they are treated as problems in analysis.
Maclaurin. Colin Maclaurin, who was born at Kilmodan in Argyllshire in February 1698, and died at York on June 14, 1746, was educated at the university of Glasgow; in 1717 he was elected, at the early age of nineteen, professor of mathematics at Aberdeen; and in 1725 he was appointed the deputy of the mathematical professor at Edinburgh, and ultimately succeeded him. There was some difficulty in securing a stipend for a deputy, and Newton privately wrote offering to bear the cost so as to enable the university to secure the services of Maclaurin. Maclaurin took an active part in opposing the advance of the Young Pretender in 1745; on the approach of the Highlanders he fled to York, but the exposure in the trenches at Edinburgh and the privations he endured in his escape proved fatal to him.
His chief works are his Geometria Organica, London, 1720; his De Linearum Geometricarum Proprietatibus, London, 1720; his Treatise on Fluxions, Edinburgh, 1742; his Algebra, London, 1748; and his Account of Newton's Discoveries, London, 1748.
The first section of the first part of the Geometria Organica is on conics; the second on nodal cubics; the third on other cubics and on quartics; and the fourth section is on general properties of curves. Newton had shewn that, if two angles bounded by straight lines turn round their respective summits so that the point of intersection of two of these lines moves along a straight line, the other point of intersection will describe a conic; and, if the first point move along a conic, the second will describe a quartic. Maclaurin gave an analytical discussion of the general theorem, and shewed how by this method various curves could be practically traced. This work contains an elaborate discussion on curves and their pedals, a branch of geometry which he had created in two papers published in the Philosophical Transactions for 1718 and 1719.
The second part of the work is divided into three sections and an appendix. The first section contains a proof of Cotes's theorem above alluded to; and also the analogous theorem (discovered by himself) that, if a straight line OP1P2dots drawn through a fixed point O cut a curve of the n th degree in n points P1 , P2 , dots , and if the tangents at P1 , P2 , dots cut a fixed line Ox in points A1 , A2 , dots , then the sum of the reciprocals of the distances OA1 , OA2 , dots is constant for all positions of the line OP1P2dots . These two theorems are generalizations of those given by Newton on diameters and asymptotes. Either is deducible from the other. In the second and third sections these theorems are applied to conics and cubics; most of the harmonic properties connected with a quadrilateral inscribed in a conic are determined; and in particular the theorem on an inscribed hexagon which is known by the name of Pascal is deduced. Pascal's essay was not published till 1779, and the earliest printed enunciation of his theorem was that given by Maclaurin. Amongst other propositions he shews that, if a quadrilateral be inscribed in a cubic, and if the points of intersection of the opposite sides also lie on the curve, then the tangents to the cubic at any two opposite angles of the quadrilateral will meet on the curve. In the fourth section he considers some theorems on central force. The fifth section contains some theorems on the description of curves through given points. One of these (which includes Pascal's as a particular case) is that if a polygon be deformed so that while each of its sides passes through a fixed point its angles (save one) describe respectively curves of the m th, n th, p th, degrees, then shall a remaining angle describe a curve of the degree 2mnp ; but if the given points be collinear, the resulting curve will be only of the degree mnp . This essay was reprinted with additions in the Philosophical Transactions for 1735.
The Treatise of Fluxions, published in 1742, was the first logical and systematic exposition of the method of fluxions. The cause of its publication was an attack by Berkeley on the principles of the infinitesimal calculus. In it [art. 751, p. 610] Maclaurin gave a proof of the theorem that f(x) = f(0) + xf'(0) + x^2|2f"(0) + . This was obtained in the manner given in many modern text-books by assuming that f(x) can be expanded in a form like f(x) = A0 + A1x + A2x^2 + , then, on differentiating and putting x=0 in the successive results, the values of A0 , A1 , are obtained; but he did not investigate the convergency of the series. The result had been previously given in 1730 by James Stirling in his Methodus Differentialis [p. 102], and of course is at once deducible from Taylor's theorem. Maclaurin also here enunciated [art. 350, p. 289] the important theorem that, if phi(x) be positive and decrease as x increases from x=a to x=infinity , then the series (a) + (a+1) + (a+2) + is convergent or divergent as the integral from x=a to x=infinity of phi(x) is finite or infinite. The theorem had been given by Euler in 1732, but in so awkward a form that its value escaped general attention. Maclaurin here also gave the correct theory of maxima and minima, and rules for finding and discriminating multiple points.
This treatise is, however, especially valuable for the solutions it contains of numerous problems in geometry, statics, the theory of attractions, and astronomy. To solve these Maclaurin reverted to classical methods, and so powerful did these processes seem, when used by him, that Clairaut, after reading the work, abandoned analysis, and attacked the problem of the figure of the earth again by pure geometry. At a later time this part of the book was described by Lagrange as the "chef-d'oe uvre de géométrie qu'on peut comparer à tout ce qu'Archimède nous a laissé de plus beau et de plus ingénieux." Maclaurin also determined the attraction of a homogeneous ellipsoid at an internal point, and gave some theorems on its attraction at an external point; in attacking these questions he introduced the conception of level surfaces, that is, surfaces at every point of which the resultant attraction is perpendicular to the surface. No further advance in the theory of attractions was made until Lagrange in 1773 introduced the idea of the potential. Maclaurin also shewed that a spheroid was a possible form of equilibrium of a mass of homogeneous liquid rotating about an axis passing through its centre of mass. Finally he discussed the tides; this part had been previously published (in 1740) and had received a prize from the French Academy.
Among Maclaurin's minor works is his Algebra, published in 1748, and founded on Newton's Universal Arithmetic. It contains the results of some early papers of Maclaurin; notably of two, written in 1726 and 1729, on the number of imaginary roots of an equation, suggested by Newton's theorem; and of one, written in 1729, containing the well-known rule for finding equal roots by means of the derived equation. In this book negative quantities are treated as being not less real than positive quantities. To this work a treatise, entitled De Linearum Geometricarum Proprietatibus Generalibus, was added as an appendix; besides the paper of 1720 above alluded to, it contains some additional and elegant theorems. Maclaurin also produced in 1728 an exposition of the Newtonian philosophy, which is incorporated in the posthumous work printed in 1748. Almost the last paper he wrote was one printed in the Philosophical Transactions for 1743 in which he discussed from a mathematical point of view the form of a bee's cell.
Maclaurin was one of the most able mathematicians of the eighteenth century, but his influence on the progress of British mathematics was on the whole unfortunate. By himself abandoning the use both of analysis and of the infinitesimal calculus, he induced Newton's countrymen to confine themselves to Newton's methods, and it was not until about 1820, when the differential calculus was introduced into the Cambridge curriculum, that English mathematicians made any general use of the more powerful methods of modern analysis.
Stewart. Maclaurin was succeeded in his chair at Edinburgh by his pupil Matthew Stewart, born at Rothesay in 1717 and died at Edinburgh on January 23, 1785, a mathematician of considerable power, to whom I allude in passing, for his theorems on the problem of three bodies, and for his discussion, treated by transversals and involution, of the properties of the circle and straight line.
Simpson. The last member of the English school whom I need mention here is Thomas Simpson, who was born in Leicestershire on August 20, 1710, and died on May 14, 1761. His father was a weaver, and he owed his education to his own efforts. His mathematical interests were first aroused by the solar eclipse which took place in 1724, and with the aid of a fortune-telling pedlar he mastered Cocker's Arithmetic and the elements of algebra. He then gave up his weaving and became an usher at a school, and by constant and laborious efforts improved his mathematical education, so that by 1735 he was able to solve several questions which had been recently proposed and which involved the infinitesimal calculus. He next moved to London, and in 1743 was appointed professor of mathematics at Woolwich, a post which he continued to occupy till his death.
The works published by Simpson prove him to have been a man of extraordinary natural genius and extreme industry. The most important of them are his Fluxions, 1737 and 1750, with numerous applications to physics and astronomy; his Laws of Chance and his Essays, 1740; his theory of Annuities and Reversions (a branch of mathematics that is due to James Dodson, died in 1757, who was a master at Christ's Hospital, London), with tables of the value of lives, 1742; his Dissertations, 1743, in which the figure of the earth, the force of attraction at the surface of a nearly spherical body, the theory of the tides, and the law of astronomical refraction are discussed; his Algebra, 1745; his Geometry, 1747; his Trigonometry, 1748, in which he introduced the current abbreviations for the trigonometrical functions; his Select Exercises, 1752, containing the solutions of numerous problems and a theory of gunnery; and lastly, his Miscellaneous Tracts, 1754.
The work last mentioned consists of eight memoirs, and these contain his best known investigations. The first three papers are on various problems in astronomy; the fourth is on the theory of mean observations; the fifth and sixth on problems in fluxions and algebra; the seventh contains a general solution of the isoperimetrical problem; the eighth contains a discussion of the third and ninth sections of the Principia, and their application to the lunar orbit. In this last memoir Simpson obtained a differential equation for the motion of the apse of the lunar orbit similar to that arrived at by Clairaut, but instead of solving it by successive approximations, he deduced a general solution by indeterminate coefficients. The result agrees with that given by Clairaut. Simpson solved this problem in 1747, two years later than the publication of Clairaut's memoir, but the solution was discovered independently of Clairaut's researches, of which Simpson first heard in 1748.
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End of part 24 of 28
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