A History of Mathematics/Modern Europe/Euler, Lagrange, and Laplace

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EULER, LAGRANGE, AND LAPLACE.

 

During the epoch of ninety years from 1730 to 1820 the French and Swiss cultivated mathematics with most brilliant success. No previous period had shown such an array of illustrious names. At this time Switzerland had her Euler; France, her Lagrange, Laplace, Legendre, and Monge. The mediocrity of French mathematics which marked the time of Louis XIV. was now followed by one of the very brightest periods of all history. England and Germany, on the other hand, which during the unproductive period in France had their Newton and Leibniz, could now boast of no great mathematician. France now waved the mathematical sceptre. Mathematical studies among the English and German people had sunk to the lowest ebb. Among them the direction of original research was ill-chosen. The former adhered with excessive partiality to ancient geometrical methods; the latter produced the combinatorial school, which brought forth nothing of value.

The labours of Euler, Lagrange, and Laplace lay in higher analysis, and this they developed to a wonderful degree. By them analysis came to be completely severed from geometry. During the preceding period the effort of mathematicians not only in England, but, to some extent, even on the continent, had been directed toward the solution of problems clothed in geometric garb, and the results of calculation were usually reduced to geometric form. A change now took place. Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected an elaborate structure. There are few great ideas pursued by succeeding analysts which were not suggested by Euler, or of which he did not share the honour of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, Lagrange developed the infinitesimal calculus and placed analytical mechanics into the form in which we now know it. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labours of Newton, gave a full analytical discussion of the solar system. He also wrote an epoch-marking work on Probability. Among the analytical branches created during this period are the calculus of Variations by Euler and Lagrange, Spherical Harmonics by Laplace and Legendre, and Elliptic Integrals by Legendre.

Comparing the growth of analysis at this time with the growth during the time of Gauss, Cauchy, and recent mathematicians, we observe an important difference. During the former period we witness mainly a development with reference to form. Placing almost implicit confidence in results of calculation, mathematicians did not always pause to discover rigorous proofs, and were thus led to general propositions, some of which have since been found to be true in only special cases. The Combinatorial School in Germany carried this tendency to the greatest extreme; they worshipped formalism and paid no attention to the actual contents of formulæ. But in recent times there has been added to the dexterity in the formal treatment of problems, a much-needed rigour of demonstration. A good example of this increased rigour is seen in the present use of infinite series as compared to that of Euler, and of Lagrange in his earlier works.

The ostracism of geometry, brought about by the master-minds of this period, could not last permanently. Indeed, a new geometric school sprang into existence in France before the close of this period. Lagrange would not permit a single diagram to appear in his Mécanique analytique, but thirteen years before his death, Monge published his epoch-making Géometrie descriptive.

Leonhard Euler (1707–1783) was born in Basel. His father, a minister, gave him his first instruction in mathematics and then sent him to the University of Basel, where he became a favourite pupil of John Bernoulli. In his nineteenth year he composed a dissertation on the masting of ships, which received the second prize from the French Academy of Sciences. When John Bernoulli's two sons, Daniel and Nicolaus, went to Russia, they induced Catharine I., in 1727, to invite their friend Euler to St. Petersburg, where Daniel, in 1733, was assigned to the chair of mathematics. In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded some months' time, was achieved in three days by Euler with aid of improved methods of his own. But the effort threw him into a fever and deprived him of the use of his right eye. With still superior methods this same problem was solved later by the illustrious Gauss in one hour![47] The despotism of Anne I. caused the gentle Euler to shrink from public affairs and to devote all his time to science. After his call to Berlin by Frederick the Great in 1747, the queen of Prussia, who received him kindly, wondered how so distinguished a scholar should be so timid and reticent. Euler naïvely replied, "Madam, it is because I come from a country where, when one speaks, one is hanged." In 1766 he with difficulty obtained permission to depart from Berlin to accept a call by Catharine II. to St. Petersburg. Soon after his return to Russia he became blind, but this did not stop his wonderful literary productiveness, which continued for seventeen years, until the day of his death.[45] He dictated to his servant his Anleitung zur Algebra, 1770, which, though purely elementary, is meritorious as one of the earliest attempts to put the fundamental processes on a sound basis.

Euler wrote an immense number of works, chief of which are the following: Introductio in analysin infinitorum, 1748, a work that caused a revolution in analytical mathematics, a subject which had hitherto never been presented in so general and systematic manner; Institutiones calculi differentialis, 1755, and Institutiones calculi integralis, 1768–1770, which were the most complete and accurate works on the calculus of that time, and contained not only a full summary of everything then known on this subject, but also the Beta and Gamma Functions and other original investigations; Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, 1744, which, displaying an amount of mathematical genius seldom rivalled, contained his researches on the calculus of variations (a subject afterwards improved by Lagrange), to the invention of which Euler was led by the study of isoperimetrical curves, the brachistochrone in a resisting medium, and the theory of geodesics (subjects which had previously engaged the attention of the elder Bernoullis and others); the Theoria motuum planetarum et cometarum, 1744, Theoria motus lunæ, 1753, Theoria motuum lunae, 1772, are his chief works on astronomy; Ses lettres à une princesse d'Allemagne sur quelques sujets de Physique et de Philosophie, 1770, was a work which enjoyed great popularity.

We proceed to mention the principal innovations and inventions of Euler. He treated trigonometry as a branch of analysis, introduced (simultaneously with Thomas Simpson in England) the now current abbreviations for trigonometric functions, and simplified formulæ by the simple expedient of designating the angles of a triangle by , , , and the opposite sides by , , , respectively. He pointed out the relation between trigonometric and exponential functions. In a paper of 1737 we first meet the symbol to denote 3.14159….[21] Euler laid down the rules for the transformation of co-ordinates in space, gave a methodic analytic treatment of plane curves and of surfaces of the second order. He was the first to discuss the equation of the second degree in three variables, and to classify the surfaces represented by it. By criteria analogous to those used in the classification of conics he obtained five species. He devised a method of solving bi-quadratic equations by assuming , with the hope that it would lead him to a general solution of algebraic equations. The method of elimination by solving a series of linear equations (invented independently by Bezout) and the method of elimination by symmetric functions, are due to him.[20] Far reaching are Euler's researches on logarithms. Leibniz and John Bernoulli once argued the question whether a negative number has a logarithm. Bernoulli claimed that since , we have and , and finally . Euler proved that has really an infinite number of logarithms, all of which are imaginary when is negative, and all except one when is positive. He then explained how might equal , and yet not equal .

The subject of infinite series received new life from him. To his researches on series we owe the creation of the theory of definite integrals by the development of the so-called Eulerian integrals. He warns his readers occasionally against the use of divergent series, but is nevertheless very careless himself. The rigid treatment to which infinite series are subjected now was then undreamed of. No clear notions existed as to what constitutes a convergent series. Neither Leibniz nor Jacob and John Bernoulli had entertained any serious doubt of the correctness of the expression . Guido Grandi went so far as to conclude from this that . In the treatment of series Leibniz advanced a metaphysical method of proof which held sway over the minds of the elder Bernoullis, and even of Euler.[46] The tendency of that reasoning was to justify results which seem to us now highly absurd. The looseness of treatment can best be seen from examples. The very paper in which Euler cautions against divergent series contains the proof that

as follows:

these added give zero. Euler has no hesitation to write , and no one objected to such results excepting Nicolaus Bernoulli, the nephew of John and Jacob. Strange to say, Euler finally succeeded in converting Nicolaus Bernoulli to his own erroneous views. At the present time it is difficult to believe that Euler should have confidently written , but such examples afford striking illustrations of the want of scientific basis of certain parts of analysis at that time. Euler's proof of the binomial formula for negative and fractional exponents, which has been reproduced in elementary text-books of even recent years, is faulty. A remarkable development, due to Euler, is what he named the hypergeometric series, the summation of which he observed to be dependent upon the integration of a linear differential equation of the second order, but it remained for Gauss to point out that for special values of its letters, this series represented nearly all functions then known.

Euler developed the calculus of finite differences in the first chapters of his Institutiones calculi differentialis, and then deduced the differential calculus from it. He established a theorem on homogeneous functions, known by his name, and contributed largely to the theory of differential equations, a subject which had received the attention of Newton, Leibniz, and the Bernoullis, but was still undeveloped. Clairaut, Fontaine, and Euler about the same time observed criteria of integrability, but Euler in addition showed how to employ them to determine integrating factors. The principles on which the criteria rested involved some degree of obscurity. The celebrated addition-theorem for elliptic integrals was first established by Euler. He invented a new algorithm for continued fractions, which he employed in the solution of the indeterminate equation . We now know that substantially the same solution of this equation was given 1000 years earlier, by the Hindoos. By giving the factors of the number when , he pointed out that this expression did not always represent primes, as was supposed by Fermat. He first supplied the proof to "Fermat's theorem," and to a second theorem of Fermat, which states that every prime of the form is expressible as the sum of two squares in one and only one way. A third theorem of Fermat, that , has no integral solution for values of greater than 2, was proved by Euler to be correct when . Euler discovered four theorems which taken together make out the great law of quadratic reciprocity, a law independently discovered by Legendre.[48] Euler enunciated and proved a well-known theorem, giving the relation between the number of vertices, faces, and edges of certain polyhedra, which, however, appears to have been known to Descartes. The powers of Euler were directed also towards the fascinating subject of the theory of probability, in which he solved some difficult problems.

Of no little importance are Euler's labours in analytical mechanics. Says Whewell: "The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made mechanics analytical; I mean Euler."[11] He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. Thus, on reading Virgil's lines, "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by Maupertius. He wrote also on tides and on sound.

Astronomy owes to Euler the method of the variation of arbitrary constants. By it he attacked the problem of perturbations, explaining, in case of two planets, the secular variations of eccentricities, nodes, etc. He was one of the first to take up with success the theory of the moon's motion by giving approximate solutions to the "problem of three bodies." He laid a sound basis for the calculation of tables of the moon. These researches on the moon's motion, which captured two prizes, were carried on while he was blind, with the assistance of his sons and two of his pupils.

Most of his memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years—a promise more than fulfilled, for down to 1818 the volumes usually contained one or more papers of his. It has been said that an edition of Euler's complete works would fill 16,000 quarto pages. His mode of working was, first to concentrate his powers upon a special problem, then to solve separately all problems growing out of the first. No one excelled him in dexterity of accommodating methods to special problems. It is easy to see that mathematicians could not long continue in Euler's habit of writing and publishing. The material would soon grow to such enormous proportions as to be unmanageable. We are not surprised to see almost the opposite in Lagrange, his great successor. The great Frenchman delighted in the general and abstract, rather than, like Euler, in the special and concrete. His writings are condensed and give in a nutshell what Euler narrates at great length.

Jean-le-Rond D'Alembert (1717–1783) was exposed, when an infant, by his mother in a market by the church of St. Jean-le-Rond, near the Nôtre-Dame in Paris, from which he derived his Christian name. He was brought up by the wife of a poor glazier. It is said that when he began to show signs of great talent, his mother sent for him, but received the reply, "You are only my step-mother; the glazier's wife is my mother." His father provided him with a yearly income. D'Alembert entered upon the study of law, but such was his love for mathematics, that law was soon abandoned. At the age of twenty-four his reputation as a mathematician secured for him admission to the Academy of Sciences. In 1743 appeared his Traité de dynamique, founded upon the important general principle bearing his name: The impressed forces are equivalent to the effective forces. D'Alembert's principle seems to have been recognised before him by Fontaine, and in some measure by John Bernoulli and Newton. D'Alembert gave it a clear mathematical form and made numerous applications of it. It enabled the laws of motion and the reasonings depending on them to be represented in the most general form, in analytical language. D'Alembert applied it in 1744 in a treatise on the equilibrium and motion of fluids, in 1746 to a treatise on the general causes of winds, which obtained a prize from the Berlin Academy. In both these treatises, as also in one of 1747, discussing the famous problem of vibrating chords, he was led to partial differential equations. He was a leader among the pioneers in the study of such equations. To the equation , arising in the problem of vibrating chords, he gave as the general solution,

,

and showed that there is only one arbitrary function, if be supposed to vanish for and . Daniel Bernoulli, starting with a particular integral given by Brook Taylor, showed that this differential equation is satisfied by the trigonometric series

,

and claimed this expression to be the most general solution. Euler denied its generality, on the ground that, if true, the doubtful conclusion would follow that the above series represents any arbitrary function of a variable. These doubts were dispelled by Fourier. Lagrange proceeded to find the sum of the above series, but D'Alembert rightly objected to his process, on the ground that it involved divergent series.[46]

A most beautiful result reached by D'Alembert, with aid of his principle, was the complete solution of the problem of the precession of the equinoxes, which had baffled the talents of the best minds. He sent to the French Academy in 1747, on the same day with Clairaut, a solution of the problem of three bodies. This had become a question of universal interest to mathematicians, in which each vied to outdo all others. The problem of two bodies, requiring the determination of their motion when they attract each other with forces inversely proportional to the square of the distance between them, had been completely solved by Newton. The "problem of three bodies" asks for the motion of three bodies attracting each other according to the law of gravitation. Thus far, the complete solution of this has transcended the power of analysis. The general differential equations of motion were stated by Laplace, but the difficulty arises in their integration. The "solutions" hitherto given are merely convenient methods of approximation in special cases when one body is the sun, disturbing the motion of the moon around the earth, or where a planet moves under the influence of the sun and another planet.

In the discussion of the meaning of negative quantities, of the fundamental processes of the calculus, and of the theory of probability, D'Alembert paid some attention to the philosophy of mathematics. His criticisms were not always happy. In 1754 he was made permanent secretary of the French Academy. During the last years of his life he was mainly occupied with the great French encyclopædia, which was begun by Diderot and himself. D'Alembert declined, in 1762, an invitation of Catharine II. to undertake the education of her son. Frederick the Great pressed him to go to Berlin. He made a visit, but declined a permanent residence there.

Alexis Claude Clairaut (1713–1765) was a youthful prodigy. He read l'Hospital's works on the infinitesimal calculus and on conic sections at the age of ten. In 1731 was published his Recherches sur les courbes à double courbure, which he had ready for the press when he was sixteen. It was a work of remarkable elegance and secured his admission to the Academy of Sciences when still under legal age. In 1731 he gave a proof of the theorem enunciated by Newton, that every cubic is a projection of one of five divergent parabolas. Clairaut formed the acquaintance of Maupertius, whom he accompanied on an expedition to Lapland to measure the length of a degree of the meridian. At that time the shape of the earth was a subject of serious disagreement. Newton and Huygens had concluded from theory that the earth was flattened at the poles. About 1713 Dominico Cassini measured an arc extending from Dunkirk to Perpignan and arrived at the startling result that the earth is elongated at the poles. To decide between the conflicting opinions, measurements were renewed. Maupertius earned by his work in Lapland the title of "earth flattener" by disproving the Cassinian tenet that the earth was elongated at the poles, and showing that Newton was right. On his return, in 1743, Clairaut published a work, Théorie de la figure de la Terre, which was based on the results of Maclaurin on homogeneous ellipsoids. It contains a remarkable theorem, named after Clairaut, that the sum of the fractions expressing the ellipticity and the increase of gravity at the pole is equal to times the fraction expressing the centrifugal force at the equator, the unit of force being represented by the force of gravity at the equator. This theorem is independent of any hypothesis with respect to the law of densities of the successive strata of the earth. It embodies most of Clairaut's researches. Todhunter says that "in the figure of the earth no other person has accomplished so much as Clairaut, and the subject remains at present substantially as he left it, though the form is different. The splendid analysis which Laplace supplied, adorned but did not really alter the theory which started from the creative hands of Clairaut.

In 1752 he gained a prize of the St. Petersburg Academy for his paper on Théorie de la Lune, in which for the first time modern analysis is applied to lunar motion. This contained the explanation of the motion of the lunar apsides. This motion, left unexplained by Newton, seemed to him at first inexplicable by Newton's law, and he was on the point of advancing a new hypothesis regarding gravitation, when, taking the precaution to carry his calculation to a higher degree of approximation, he reached results agreeing with observation. The motion of the moon was studied about the same time by Euler and D^Alembert. Clairaut predicted that "Halley's Comet," then expected to return, would arrive at its nearest point to the sun on April 13, 1759, a date which turned out to be one month too late. He was the first to detect singular solutions in differential equations of the first order but of higher degree than the first.

In their scientific labours there was between Clairaut and D'Alembert great rivalry, often far from friendly. The growing ambition of Clairaut to shine in society, where he was a great favourite, hindered his scientific work in the latter part of his life.

Johann Heinrich Lambert (1728–1777), born at Mühlhausen in Alsace, was the son of a poor tailor. While working at his father's trade, he acquired through his own unaided efforts a knowledge of elementary mathematics. At the age of thirty he became tutor in a Swiss family and secured leisure to continue his studies. In his travels with his pupils through Europe he became acquainted with the leading mathematicians. In 1764 he settled in Berlin, where he became member of the Academy, and enjoyed the society of Euler and Lagrange. He received a small pension, and later became editor of the Berlin Ephemeris. His many-sided scholarship reminds one of Leibniz. In his Cosmological Letters he made some remarkable prophecies regarding the stellar system. In mathematics he made several discoveries which were extended and overshadowed by his great contemporaries. His first research on pure mathematics developed in an infinite series the root of the equation . Since each equation of the form can be reduced to in two ways, one or the other of the two resulting series was always found to be convergent, and to give a value of . Lambert's results stimulated Euler, who extended the method to an equation of four terms, and particularly Lagrange, who found that a function of a root of can be expressed by the series bearing his name. In 1761 Lambert communicated to the Berlin Academy a memoir, in which he proves that is irrational. This proof is given in Note IV. of Legendre's Géometrie, where it is extended to . To the genius of Lambert we owe the introduction into trigonometry of hyperbolic functions, which he designated by , , etc. His Freye Perspective, 1759 and 1773, contains researches on descriptive geometry, and entitle him to the honour of being the forerunner of Monge. In his effort to simplify the calculation of cometary orbits, he was led geometrically to some remarkable theorems on conics, for instance this: "If in two ellipses having a common major axis we take two such arcs that their chords are equal, and that also the sums of the radii vectores, drawn respectively from the foci to the extremities of these arcs, are equal to each other, then the sectors formed in each ellipse by the arc and the two radii vectores are to each other as the square roots of the parameters of the ellipses."[13]

John Landen (1719–1790) was an English mathematician whose writings served as the starting-point of investigations by Euler, Lagrange, and Legendre. Landen's capital discovery, contained in a memoir of 1755, was that every arc of the hyperbola is immediately rectified by means of two arcs of an ellipse. In his "residual analysis" he attempted to obviate the metaphysical difficulties of fluxions by adopting a purely algebraic method. Lagrange's Calcul des Fonctions is based upon this idea. Landen showed how the algebraic expression for the roots of a cubic equation could be derived by application of the differential and integral calculus. Most of the time of this suggestive writer was spent in the pursuits of active life.

Étienne Bézout (1730–1783) was a French writer of popular mathematical school-books. In his Théorie générale des Équations Algébrique, 1779, he gave the method of elimination by linear equations (invented also by Euler). This method was first published by him in a memoir of 1764, in which he uses determinants, without, however, entering upon their theory. A beautiful theorem as to the degree of the resultant goes by his name.

Louis Arbogaste (1759–1803) of Alsace was professor of mathematics at Strasburg. His chief work, the Calcul des Dérivations, 1800, gives the method known by his name, by which the successive coefficients of a development are derived from one another when the expression is complicated. De Morgan has pointed out that the true nature of derivation is differentiation accompanied by integration. In this book for the first time are the symbols of operation separated from those of quantity. The notation for is due to him.

Maria Gaetana Agnesi (1718–1799) of Milan, distinguished as a linguist, mathematician, and philosopher, filled the mathematical chair at the University of Bologna during her father's sickness. In 1748 she published her Instituzioni Analitiche, which was translated into English in 1801. The "witch of Agnesi" or "versiera" is a plane curve containing a straight line, , and a cubic .

Joseph Louis Lagrange (1736–1813), one of the greatest mathematicians of all times, was born at Turin and died at Paris. He was of French extraction. His father, who had charge of the Sardinian military chest, was once wealthy, but lost all he had in speculation. Lagrange considered this loss his good fortune, for otherwise he might not have made mathematics the pursuit of his life. While at the college in Turin his genius did not at once take its true bent. Cicero and Virgil at first attracted him more than Archimedes and Newton. He soon came to admire the geometry of the ancients, but the perusal of a tract of Halley roused his enthusiasm for the analytical method, in the development of which he was destined to reap undying glory. He now applied himself to mathematics, and in his seventeenth year he became professor of mathematics in the royal military academy at Turin. Without assistance or guidance he entered upon a course of study which in two years placed him on a level with the greatest of his contemporaries. With aid of his pupils he established a society which subsequently developed into the Turin Academy. In the first five volumes of its transactions appear most of his earlier papers. At the age of nineteen he communicated to Euler a general method of dealing with "isoperimetrical problems," known now as the Calculus of Variations. This commanded Euler's lively admiration, and he courteously withheld for a time from publication some researches of his own on this subject, so that the youthful Lagrange might complete his investigations and claim the invention. Lagrange did quite as much as Euler towards the creation of the Calculus of Variations. As it came from Euler it lacked an analytic foundation, and this Lagrange supplied. He separated the principles of this calculus from geometric considerations by which his predecessor had derived them. Euler had assumed as fixed the limits of the integral, i.e. the extremities of the curve to be determined, but Lagrange removed this restriction and allowed all co-ordinates of the curve to vary at the same time. Euler introduced in 1766 the name "calculus of variations," and did much to improve this science along the lines marked out by Lagrange.

Another subject engaging the attention of Lagrange at Turin was the propagation of sound. In his papers on this subject in the Miscellanea Taurinensia, the young mathematician appears as the critic of Newton, and the arbiter between Euler and D'Alembert. By considering only the particles which are in a straight line, he reduced the problem to the same partial differential equation that represents the motions of vibrating strings. The general integral of this was found by D'Alembert to contain two arbitrary functions, and the question now came to be discussed whether an arbitrary function may be discontinuous. D'Alembert maintained the negative against Euler, Daniel Bernoulli, and finally Lagrange,—arguing that in order to determine the position of a point of the chord at a time , the initial position of the chord must be continuous. Lagrange settled the question in the affirmative.

By constant application during nine years, Lagrange, at the age of twenty-six, stood at the summit of European fame. But his intense studies had seriously weakened a constitution never robust, and though his physicians induced him to take rest and exercise, his nervous system never fully recovered its tone, and he was thenceforth subject to fits of melancholy.

In 1764 the French Academy proposed as the subject of a prize the theory of the libration of the moon. It demanded an explanation, on the principle of universal gravitation, why the moon always turns, with but slight variations, the same phase to the earth. Lagrange secured the prize. This success encouraged the Academy to propose as a prize the theory of the four satellites of Jupiter,—a problem of six bodies, more difficult than the one of three bodies previously solved by Clairaut, D'Alembert, and Euler. Lagrange overcame the difficulties, but the shortness of time did not permit him to exhaust the subject. Twenty-four years afterwards it was completed by Laplace. Later astronomical investigations of Lagrange are on cometary perturbations (1778 and 1783), on Kepler's problem, and on a new method of solving the problem of three bodies.

Being anxious to make the personal acquaintance of leading mathematicians, Lagrange visited Paris, where he enjoyed the stimulating delight of conversing with Clairaut, D'Alembert, Condorcet, the Abbé Marie, and others. He had planned a visit to London, but he fell dangerously ill after a dinner in Paris, and was compelled to return to Turin. In 1766 Euler left Berlin for St. Petersburg, and he pointed out Lagrange as the only man capable of filling the place. D'Alembert recommended him at the same time. Frederick the Great thereupon sent a message to Turin, expressing the wish of "the greatest king of Europe" to have "the greatest mathematician" at his court. Lagrange went to Berlin, and staid there twenty years. Finding all his colleagues married, and being assured by their wives that the marital state alone is happy, he married. The union was not a happy one. His wife soon died. Frederick the Great held him in high esteem, and frequently conversed with him on the advantages of perfect regularity of life. This led Lagrange to cultivate regular habits. He worked no longer each day than experience taught him he could without breaking down. His papers were carefully thought out before he began writing, and when he wrote he did so without a single correction.

During the twenty years in Berlin he crowded the transactions of the Berlin Academy with memoirs, and wrote also the epoch-making work called the Mécanique Analytique. He enriched algebra by researches on the solution of equations. There are two methods of solving directly algebraic equations,—that of substitution and that of combination. The former method was developed by Ferrari, Vieta, Tchirnhausen, Euler, Bézout, and Lagrange; the latter by Vandermonde and Lagrange.[20] In the method of substitution the original forms are so transformed that the determination of the roots is made to depend upon simpler functions (resolvents). In the method of combination auxiliary quantities are substituted for certain simple combinations ("types") of the unknown roots of the equation, and auxiliary equations (resolvents) are obtained for these quantities with aid of the coefficients of the given equation. Lagrange traced all known algebraic solutions of equations to the uniform principle consisting in the formation and solution of equations of lower degree whose roots are linear functions of the required roots, and of the roots of unity. He showed that the quintic cannot be reduced in this way, its resolvent being of the sixth degree. His researches on the theory of equations were continued after he left Berlin. In the Résolution des équations numériques (1798) he gave a method of approximating to the real roots of numerical equations by continued fractions. Among other things, it contains also a proof that every equation must have a root,—a theorem which appears before this to have been considered self-evident. Other proofs of this were given by Argand, Gauss, and Cauchy. In a note to the above work Lagrange uses Fermat's theorem and certain suggestions of Gauss in effecting a complete algebraic solution of any binomial equation.

While in Berlin Lagrange published several papers on the theory of numbers. In 1769 he gave a solution in integers of indeterminate equations of the second degree, which resembles the Hindoo cyclic method; he was the first to prove, in 1771, "Wilson's theorem," enunciated by an Englishman, John Wilson, and first published by Waring in his Meditationes Algebraicœ; he investigated in 1775 under what conditions and ( and having been discussed by Euler) are quadratic residues, or non-residues of odd prime numbers, ; he proved in 1770 Méziriac's theorem that every integer is equal to the sum of four, or a less number, of squares. He proved Format's theorem on , for the case , also Fermat's theorem that, if , then is not a square.

In his memoir on Pyramids, 1773, Lagrange made considerable use of determinants of the third order, and demonstrated that the square of a determinant is itself a determinant. He never, however, dealt explicitly and directly with determinants; he simply obtained accidentally identities which are now recognised as relations between determinants.

Lagrange wrote much on differential equations. Though the subject of contemplation by the greatest mathematicians (Euler, D'Alembert, Clairaut, Lagrange, Laplace), yet more than other branches of mathematics did they resist the systematic application of fixed methods and principles. Lagrange established criteria for singular solutions (Calcul des Fonctions, Lessons 14–17), which are, however, erroneous. He was the first to point out the geometrical significance of such solutions. He generalised Euler's researches on total differential equations of two variables, and of the ninth order; he gave a solution of partial differential equations of the first order (Berlin Memoirs, 1772 and 1774), and spoke of their singular solutions, extending their solution in Memoirs of 1779 and 1785 to equations of any number of variables. The discussion on partial differential equations of the second order, carried on by D'Alembert, Euler, and Lagrange, has already been referred to in our account of D'Alembert.

While in Berlin, Lagrange wrote the Méchanique Analytique," the greatest of his works (Paris, 1788). From the principle of virtual velocities he deduced, with aid of the calculus of variations, the whole system of mechanics so elegantly and harmoniously that it may fitly be called, in Sir William Rowan Hamilton's words, "a kind of scientific poem." It is a most consummate example of analytic generality. Geometrical figures are nowhere allowed. "On ne trouvera point de figures dans cet ouvrage" (Preface). The two divisions of mechanics—statics and dynamics—are in the first four sections of each carried out analogously, and each is prefaced by a historic sketch of principles. Lagrange formulated the principle of least action. In their original form, the equations of motion involve the co-ordinates , , , of the different particles or of the system. But , , , are in general not independent, and Lagrange introduced in place of them any variables , , , whatever, determining the position of the point at the time. These may be taken to be independent. The equations of motion may now assume the form

;

or when , , ,… are the partial differential coefficients with respect to , , ,… of one and the same function , then the form

.

The latter is par excellence the Lagrangian form of the equations of motion. With Lagrange originated the remark that mechanics may be regarded as a geometry of four dimensions. To him falls the honour of the introduction of the potential into dynamics.[49] Lagrange was anxious to have his Mécanique Analytique published in Paris. The work was ready for print in 1786, but not till 1788 could he find a publisher, and then only with the condition that after a few years he would purchase all the unsold copies. The work was edited by Legendre.

After the death of Frederick the Great, men of science were no longer respected in Germany, and Lagrange accepted an invitation of Louis XVI. to migrate to Paris. The French queen treated him with regard, and lodging was procured for him in the Louvre. But he was seized with a long attack of melancholy which destroyed his taste for mathematics. For two years his printed copy of the Mécanique, fresh from the press,—the work of a quarter of a century,—lay unopened on his desk. Through Lavoisier he became interested in chemistry, which he found "as easy as algebra." The disastrous crisis of the French Revolution aroused him again to activity. About this time the young and accomplished daughter of the astronomer Lemonnier took compassion on the sad, lonely Lagrange, and insisted upon marrying him. Her devotion to him constituted the one tie to life which at the approach of death he found it hard to break.

He was made one of the commissioners to establish weights and measures having units founded on nature. Lagrange strongly favoured the decimal subdivision, the general idea of which was obtained from a work of Thomas Williams, London, 1788. Such was the moderation of Lagrange's character, and such the universal respect for him, that he was retained as president of the commission on weights and measures even after it had been purified by the Jacobins by striking out the names of Lavoisier, Laplace, and others. Lagrange took alarm at the fate of Lavoisier, and planned to return to Berlin, but at the establishment of the École Normale in 1795 in Paris, he was induced to accept a professorship. Scarcely had he time to elucidate the foundations of arithmetic and algebra to young pupils, when the school was closed. His additions to the algebra of Euler were prepared at this time. In 1797 the École Polytechnique was founded, with Lagrange as one of the professors. The earliest triumph of this institution was the restoration of Lagrange to analysis. His mathematical activity burst out anew. He brought forth the Théorie des fonctions analytiques (1797), Leçons sur le calcul des fonctions, a treatise on the same lines as the preceding (1801), and the Résolution des équations numeriques (1798). In 1810 he began a thorough revision of his Mécanique analytique, but he died before its completion.

The Théorie des fonctions, the germ of which is found in a memoir of his of 1772, aimed to place the principles of the calculus upon a sound foundation by relieving the mind of the difficult conception of a limit or infinitesimal. John Landen's residual calculus, professing a similar object, was unknown to him. Lagrange attempted to prove Taylor's theorem (the power of which he was the first to point out) by simple algebra, and then to develop the entire calculus from that theorem. The principles of the calculus were in his day involved in philosophic difficulties of a serious nature. The infinitesimals of Leibniz had no satisfactory metaphysical basis. In the differential calculus of Euler they were treated as absolute zeros. In Newton's limiting ratio, the magnitudes of which it is the ratio cannot be found, for at the moment when they should be caught and equated, there is neither arc nor chord. The chord and arc were not taken by Newton as equal before vanishing, nor after vanishing, but when they vanish. "That method," said Lagrange, "has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratios of two quantities, as long as they remain finite, that ratio offers to the mind no clear and precise idea, as soon as its terms become both nothing at the same time." D'Alembert's method of limits was much the same as the method of prime and ultimate ratios. D'Alembert taught that a variable actually reached its limit. When Lagrange endeavoured to free the calculus of its metaphysical difficulties, by resorting to common algebra, he avoided the whirlpool of Charybdis only to suffer wreck against the rocks of Scylla. The algebra of his day, as handed down to him by Euler, was founded on a false view of infinity. No correct theory of infinite series had then been established. Lagrange proposed to define the differential coefficient of with respect to as the coefficient of in the expansion of by Taylor's theorem, and thus to avoid all reference to limits. But he used infinite series without ascertaining that they were convergent, and his proof that can always be expanded in a series of ascending powers of , labours under serious defects. Though Lagrange's method of developing the calculus was at first greatly applauded, its defects were fatal, and to-day his "method of derivatives," as it was called, has been generally abandoned. He introduced a notation of his own, but it was inconvenient, and was abandoned by him in the second edition of his Mécanique, in which he used infinitesimals. The primary object of the Théorie des fonctions was not attained, but its secondary results were far-reaching. It was a purely abstract mode of regarding functions, apart from geometrical or mechanical considerations. In the further development of higher analysis a function became the leading idea, and Lagrange's work may be regarded as the starting-point of the theory of functions as developed by Cauchy, Riemann, Weierstrass, and others.

In the treatment of infinite series Lagrange displayed in his earlier writings that laxity common to all mathematicians of his time, excepting Nicolaus Bernoulli II. and D'Alembert. But his later articles mark the beginning of a period of greater rigour. Thus, in the Calcul de fonctions he gives his theorem on the limits of Taylor's theorem. Lagrange's mathematical researches extended to subjects which have not been mentioned here—such as probabilities, finite differences, ascending continued fractions, elliptic integrals. Everywhere his wonderful powers of generalisation and abstraction are made manifest. In that respect he stood without a peer, but his great contemporary, Laplace, surpassed him in practical sagacity. Lagrange was content to leave the application of his general results to others, and some of the most important researches of Laplace (particularly those on the velocity of sound and on the secular acceleration of the moon) are implicitly contained in Lagrange's works.

Lagrange was an extremely modest man, eager to avoid controversy, and even timid in conversation. He spoke in tones of doubt, and his first words generally were, "Je ne sais pas." He would never allow his portrait to be taken, and the only ones that were secured were sketched without his knowledge by persons attending the meetings of the Institute.

Pierre Simon Laplace (1749–1827) was born at Beaumont-en-Auge in Normandy. Very little is known of his early life. When at the height of his fame he was loath to speak of his boyhood, spent in poverty. His father was a small farmer. Some rich neighbours who recognised the boy's talent assisted him in securing an education. As an extern he attended the military school in Beaumont, where at an early age he became teacher of mathematics. At eighteen he went to Paris, armed with letters of recommendation to D'Alembert, who was then at the height of his fame. The letters remained unnoticed, but young Laplace, undaunted, wrote the great geometer a letter on the principles of mechanics, which brought the following enthusiastic response: "You needed no introduction; you have recommended yourself; my support is your due." D'Alembert secured him a position at the École Militaire of Paris as professor of mathematics. His future was now assured, and he entered upon those profound researches which brought him the title of "the Newton of France." With wonderful mastery of analysis, Laplace attacked the pending problems in the application of the law of gravitation to celestial motions. During the succeeding fifteen years appeared most of his original contributions to astronomy. His career was one of almost uninterrupted prosperity. In 1784 he succeeded Bézout as examiner to the royal artillery, and the following year he became member of the Academy of Sciences. He was made president of the Bureau of Longitude; he aided in the introduction of the decimal system, and taught, with Lagrange, mathematics in the École Normale. When, during the Revolution, there arose a cry for the reform of everything, even of the calendar, Laplace suggested the adoption of an era beginning with the year 1250, when, according to his calculation, the major axis of the earth's orbit had been perpendicular to the equinoctial line. The year was to begin with the vernal equinox, and the zero meridian was to be located east of Paris by 185.30 degrees of the centesimal division of the quadrant, for by this meridian the beginning of his proposed era fell at midnight. But the revolutionists rejected this scheme, and made the start of the new era coincide with the beginning of the glorious French Republic.[50]

Laplace was justly admired throughout Europe as a most sagacious and profound scientist, but, unhappily for his reputation, he strove not only after greatness in science, but also after political honours. The political career of this eminent scientist was stained by servility and suppleness. After the 18th of Brumaire, the day when Napoleon was made emperor, Laplace's ardour for republican principles suddenly gave way to a great devotion to the emperor. Napoleon rewarded this devotion by giving him the post of minister of the interior, but dismissed him after six months for incapacity. Said Napoleon, "Laplace ne saisissait aucune question sous son véritable point de vue; il cherchait des subtilités partout, n'avait que des idées problematiques, et portait enfin l'esprit des infiniment petits jusque dans l'administration." Desirous to retain his allegiance, Napoleon elevated him to the Senate and bestowed various other honours upon him. Nevertheless, he cheerfully gave his voice in 1814 to the dethronement of his patron and hastened to tender his services to the Bourbons, thereby earning the title of marquis. This pettiness of his character is seen in his writings. The first edition of the Système du monde was dedicated to the Council of Five Hundred. To the third volume of the Mécanique Céleste is prefixed a note that of all the truths contained in the book, that most precious to the author was the declaration he thus made of gratitude and devotion to the peace-maker of Europe. After this outburst of affection, we are surprised to find in the editions of the Théorie analytique des probabilités, which appeared after the Restoration, that the original dedication to the emperor is suppressed.

Though supple and servile in politics, it must be said that in religion and science Laplace never misrepresented or concealed his own convictions however distasteful they might be to others. In mathematics and astronomy his genius shines with a lustre excelled by few. Three great works did he give to the scientific world,—the Mécanique Céleste, the Exposition du système du monde, and the Théorie analytique des probabilities. Besides these he contributed important memoirs to the French Academy.

We first pass in brief review his astronomical researches. In 1773 he brought out a paper in which he proved that the mean motions or mean distances of planets are invariable or merely subject to small periodic changes. This was the first and most important step in establishing the stability of the solar system.[51] To Newton and also to Euler it had seemed doubtful whether forces so numerous, so variable in position, so different in intensity, as those in the solar system, could be capable of maintaining permanently a condition of equilibrium. Newton was of the opinion that a powerful hand must intervene from time to time to repair the derangements occasioned by the mutual action of the different bodies. This paper was the beginning of a series of profound researches by Lagrange and Laplace on the limits of variation of the various elements of planetary orbits, in which the two great mathematicians alternately surpassed and supplemented each other. Laplace's first paper really grew out of researches on the theory of Jupiter and Saturn. The behaviour of these planets had been studied by Euler and Lagrange without receiving satisfactory explanation. Observation revealed the existence of a steady acceleration of the mean motions of our moon and of Jupiter and an equally strange diminution of the mean motion of Saturn. It looked as though Saturn might eventually leave the planetary system, while Jupiter would fall into the sun, and the moon upon the earth. Laplace finally succeeded in showing, in a paper of 1784–1786, that these variations (called the "great inequality") belonged to the class of ordinary periodic perturbations, depending upon the law of attraction. The cause of so influential a perturbation was found in the commensurability of the mean motion of the two planets.

In the study of the Jovian system, Laplace was enabled to determine the masses of the moons. He also discovered certain very remarkable, simple relations between the movements of those bodies, known as "Laws of Laplace." His theory of these bodies was completed in papers of 1788 and 1789. These, as well as the other papers here mentioned, were published in the Mémoirs présentés par divers savans. The year 1787 was made memorable by Laplace's announcement that the lunar acceleration depended upon the secular changes in the eccentricity of the earth's orbit. This removed all doubt then existing as to the stability of the solar system. The universal validity of the law of gravitation to explain all motion in the solar system was established. That system, as then known, was at last found to be a complete machine.

In 1796 Laplace published his Exposition du système du monde, a non-mathematical popular treatise on astronomy, ending with a sketch of the history of the science. In this work he enunciates for the first time his celebrated nebular hypothesis. A similar theory had been previously proposed by Kant in 1755, and by Swedenborg; but Laplace does not appear to have been aware of this.

Laplace conceived the idea of writing a work which should contain a complete analytical solution of the mechanical problem presented by the solar system, without deriving from observation any but indispensable data. The result was the Mécanique Céleste, which is a systematic presentation embracing all the discoveries of Newton, Clairaut, D'Alembert, Euler, Lagrange, and of Laplace himself, on celestial mechanics. The first and second volumes of this work were published in 1799; the third appeared in 1802, the fourth in 1805. Of the fifth volume, Books XI. and XII. were published in 1823; Books XIII., XIV., XV. in 1824, and Book XVI. in 1825. The first two volumes contain the general theory of the motions and figure of celestial bodies. The third and fourth volumes give special theories of celestial motions,—treating particularly of motions of comets, of our moon, and of other satellites. The fifth volume opens with a brief history of celestial mechanics, and then gives in appendices the results of the author's later researches. The Mécanique Céleste was such a master-piece, and so complete, that Laplace's successors have been able to add comparatively little. The general part of the work was translated into German by Joh. Karl Burkhardt, and appeared in Berlin, 1800–1802. Nathaniel Bowditch brought out an edition in English, with an extensive commentary, in Boston, 1829–1839. The Mécanique Céleste is not easy reading. The difficulties lie, as a rule, not so much in the subject itself as in the want of verbal explanation. A complicated chain of reasoning receives often no explanation whatever. Biot, who assisted Laplace in revising the work for the press, tells that he once asked Laplace some explanation of a passage in the book which had been written not long before, and that Laplace spent an hour endeavouring to recover the reasoning which had been carelessly suppressed with the remark, "Il est facile de voir." Notwithstanding the important researches in the work, which are due to Laplace himself, it naturally contains a great deal that is drawn from his predecessors. It is, in fact, the organised result of a century of patient toil. But Laplace frequently neglects to properly acknowledge the source from which he draws, and lets the reader infer that theorems and formulae due to a predecessor are really his own.

We are told that when Laplace presented Napoleon with a copy of the Mécanique Céleste, the latter made the remark, "M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator." Laplace is said to have replied bluntly, "Je n'avais pas besoin de cette hypothèse-la." This assertion, taken literally, is impious, but may it not have been intended to convey a meaning somewhat different from its literal one? Newton was not able to explain by his law of gravitation all questions arising in the mechanics of the heavens. Thus, being unable to show that the solar system was stable, and suspecting in fact that it was unstable, Newton expressed the opinion that the special intervention, from time to time, of a powerful hand was necessary to preserve order. Now Laplace was able to prove by the law of gravitation that the solar system is stable, and in that sense may be said to have felt no necessity for reference to the Almighty.

We now proceed to researches which belong more properly to pure mathematics. Of these the most conspicuous are on the theory of probability. Laplace has done more towards advancing this subject than any one other investigator. He published a series of papers, the main results of which were collected in his Théorie analytique des probabilités, 1812. The third edition (1820) consists of an introduction and two books. The introduction was published separately under the title, Essai philosophique sur les probabilités, and is an admirable and masterly exposition without the aid of analytical formulæ of the principles and applications of the science. The first book contains the theory of generating functions, which are applied, in the second book, to the theory of probability. Laplace gives in his work on probability his method of approximation to the values of definite integrals. The solution of linear differential equations was reduced by him to definite integrals. One of the most important parts of the work is the application of probability to the method of least squares, which is shown to give the most probable as well as the most convenient results.

The first printed statement of the principle of least squares was made in 1806 by Legendre, without demonstration. Gauss had used it still earlier, but did not publish it until 1809. The first deduction of the law of probability of error that appeared in print was given in 1808 by Robert Adrain in the Analyst, a journal published by himself in Philadelphia.[2] Proofs of this law have since been given by Gauss, Ivory, Herschel, Hagen, and others; but all proofs contain some point of difficulty. Laplace's proof is perhaps the most satisfactory.

Laplace's work on probability is very difficult reading, particularly the part on the method of least squares. The analytical processes are by no means clearly established or free from error. "No one was more sure of giving the result of analytical processes correctly, and no one ever took so little care to point out the various small considerations on which correctness depends" (De Morgan).

Of Laplace's papers on the attraction of ellipsoids, the most important is the one published in 1785, and to a great extent reprinted in the third volume of the Mécanique Céleste. It gives an exhaustive treatment of the general problem of attraction of any ellipsoid upon a particle situated outside or upon its surface. Spherical harmonics, or the so-called "Laplace's coefficients," constitute a powerful analytic engine in the theory of attraction, in electricity, and magnetism. The theory of spherical harmonics for two dimensions had been previously given by Legendre. Laplace failed to make due acknowledgment of this, and there existed, in consequence, between the two great men, "a feeling more than coldness." The potential function, , is much used by Laplace, and is shown by him to satisfy the partial differential equation . This is known as Laplace's equation, and was first given by him in the more complicated form which it assumes in polar co-ordinates. The notion of potential was, however, not introduced into analysis by Laplace. The honour of that achievement belongs to Lagrange.[49]

Among the minor discoveries of Laplace are his method of solving equations of the second, third, and fourth degrees, his memoir on singular solutions of differential equations, his researches in finite differences and in determinants, the establishment of the expansion theorem in determinants which had been previously given by Vandermonde for a special case, the determination of the complete integral of the linear differential equation of the second order. In the Mécanique Céleste he made a generalisation of Lagrange's theorem on the development of functions in series known as Laplace's theorem.

Laplace's investigations in physics were quite extensive. We mention here his correction of Newton's formula on the velocity of sound in gases by taking into account the changes of elasticity due to the heat of compression and cold of rarefaction; his researches on the theory of tides; his mathematical theory of capillarity; his explanation of astronomical refraction; his formulæ for measuring heights by the barometer.

Laplace's writings stand out in bold contrast to those of Lagrange in their lack of elegance and symmetry. Laplace looked upon mathematics as the tool for the solution of physical problems. The true result being once reached, he spent little time in explaining the various steps of his analysis, or in polishing his work. The last years of his life were spent mostly at Arcueil in peaceful retirement on a country-place, where he pursued his studies with his usual vigour until his death. He was a great admirer of Euler, and would often say, "Lisez Euler, lisez Euler, c'est notre maître à tous."

Abnit-Théophile Vandermonde (1735–1796) studied music during his youth in Paris and advocated the theory that all art rested upon one general law, through which any one could become a composer with the aid of mathematics. He was the first to give a connected and logical exposition of the theory of determinants, and may, therefore, almost be regarded as the founder of that theory. He and Lagrange originated the method of combinations in solving equations.[20]

Adrien Marie Legendre (1752–1833) was educated at the Collège Mazarin in Paris, where he began the study of mathematics under Abbé Marie. His mathematical genius secured for him the position of professor of mathematics at the military school of Paris. While there he prepared an essay on the curve described by projectiles thrown into resisting media (ballistic curve), which captured a prize offered by the Royal Academy of Berlin. In 1780 he resigned his position in order to reserve more time for the study of higher mathematics. He was then made member of several public commissions. In 1795 he was elected professor at the Normal School and later was appointed to some minor government positions. Owing to his timidity and to Laplace's unfriendliness toward him, but few important public offices commensurate with his ability were tendered to him.

As an analyst, second only to Laplace and Lagrange, Legendre enriched mathematics by important contributions, mainly on elliptic integrals, theory of numbers, attraction of ellipsoids, and least squares. The most important of Legendre's works is his Fonctions elliptiques, issued in two volumes in 1825 and 1826. He took up the subject where Euler, Landen, and Lagrange had left it, and for forty years was the only one to cultivate this new branch of analysis, until at last Jacobi and Abel stepped in with admirable new discoveries.[52] Legendre imparted to the subject that connection and arrangement which belongs to an independent science. Starting with an integral depending upon the square root of a polynomial of the fourth degree in , he showed that such integrals can be brought back to three canonical forms, designated by , , and , the radical being expressed in the form . He also undertook the prodigious task of calculating tables of arcs of the ellipse for different degrees of amplitude and eccentricity, which supply the means of integrating a large number of differentials.

An earlier publication which contained part of his researches on elliptic functions was his Calcul intégral in three volumes (1811, 1816, 1817), in which he treats also at length of the two classes of definite integrals named by him Eulerian. He tabulated the values of for values of between 1 and 2.

One of the earliest subjects of research was the attraction of spheroids, which suggested to Legendre the function , named after him. His memoir was presented to the Academy of Sciences in 1783. The researches of Maclaurin and Lagrange suppose the point attracted by a spheroid to be at the surface or within the spheroid, but Legendre showed that in order to determine the attraction of a spheroid on any external point it suffices to cause the surface of another spheroid described upon the same foci to pass through that point. Other memoirs on ellipsoids appeared later.

The two household gods to which Legendre sacrificed with ever-renewed pleasure in the silence of his closet were the elliptic functions and the theory of numbers. His researches on the latter subject, together with the numerous scattered fragments on the theory of numbers due to his predecessors in this line, were arranged as far as possible into a systematic whole, and published in two large quarto volumes, entitled Théorie des nombres, 1830. Before the publication of this work Legendre had issued at divers times preliminary articles. Its crowning pinnacle is the theorem of quadratic reciprocity, previously indistinctly given by Euler without proof, but for the first time clearly enunciated and partly proved by Legendre.[48]

While acting as one of the commissioners to connect Greenwich and Paris geodetically, Legendre calculated all the triangles in France. This furnished the occasion of establishing formulæ and theorems on geodesics, on the treatment of the spherical triangle as if it were a plane triangle, by applying certain corrections to the angles, and on the method of least squares, published for the first time by him without demonstration in 1806.

Legendre wrote an Élements de Géométrie, 1794, which enjoyed great popularity, being generally adopted on the Continent and in the United States as a substitute for Euclid. This great modern rival of Euclid passed through numerous editions; the later ones containing the elements of trigonometry and a proof of the irrationality of and . Much attention was given by Legendre to the subject of parallel lines. In the earlier editions of the Élements, he made direct appeal to the senses for the correctness of the "parallel-axiom." He then attempted to demonstrate that "axiom," but his proofs did not satisfy even himself. In Vol. XII. of the Memoirs of the Institute is a paper by Legendre, containing his last attempt at a solution of the problem. Assuming space to be infinite, he proved satisfactorily that it is impossible for the sum of the three angles of a triangle to exceed two right angles; and that if there be any triangle the sum of whose angles is two right angles, then the same must be true of all triangles. But in the next step, to show that this sum cannot be less than two right angles, his demonstration necessarily failed. If it could be granted that the sum of the three angles is always equal to two right angles, then the theory of parallels could be strictly deduced.

Joseph Fourier (1768–1830) was born at Auxerre, in central France. He became an orphan in his eighth year. Through the influence of friends he was admitted into the military school in his native place, then conducted by the Benedictines of the Convent of St. Mark. He there prosecuted his studies, particularly mathematics, with surprising success. He wished to enter the artillery, but, being of low birth (the son of a tailor), his application was answered thus: "Fourier, not being noble, could not enter the artillery, although he were a second Newton."[53] He was soon appointed to the mathematical chair in the military school. At the age of twenty-one he went to Paris to read before the Academy of Sciences a memoir on the resolution of numerical equations, which was an improvement on Newton's method of approximation. This investigation of his early youth he never lost sight of. He lectured upon it in the Polytechnic School; he developed it on the banks of the Nile; it constituted a part of a work entitled Analyse des equationes determines (1831), which was in press when death overtook him. This work contained "Fourier's theorem" on the number of real roots between two chosen limits. Budan had published this result as early as 1807, but there is evidence to show that Fourier had established it before Budan's publication. These brilliant results were eclipsed by the theorem of Sturm, published in 1835.

Fourier took a prominent part at his home in promoting the Revolution. Under the French Revolution the arts and sciences seemed for a time to flourish. The reformation of the weights and measures was planned with grandeur of conception. The Normal School was created in 1795, of which Fourier became at first pupil, then lecturer. His brilliant success secured him a chair in the Polytechnic School, the duties of which he afterwards quitted, along with Monge and Berthollet, to accompany Napoleon on his campaign to Egypt. Napoleon founded the Institute of Egypt, of which Fourier became secretary. In Egypt he engaged not only in scientific work, but discharged important political functions. After his return to France he held for fourteen years the prefecture of Grenoble. During this period he carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy. The trigonometric series represents the function for every value of , if the coefficients , and be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function. In 1827 Fourier succeeded Laplace as president of the council of the Polytechnic School.

Before proceeding to the origin of modern geometry we shall speak briefly of the introduction of higher analysis into Great Britain. This took place during the first quarter of this century. The British began to deplore the very small progress that science was making in England as compared with its racing progress on the Continent. In 1813 the "Analytical Society" was formed at Cambridge. This was a small club established by George Peacock, John Herschel, Charles Babbage, and a few other Cambridge students, to promote, as it was humorously expressed, the principles of pure "-ism," that is, the Leibnizian notation in the calculus against those of "dot-age," or of the Newtonian notation. This struggle ended in the introduction into Cambridge of the notation , to the exclusion of the fluxional notation . This was a great step in advance, not on account of any great superiority of the Leibnizian over the Newtonian notation, but because the adoption of the former opened up to English students the vast storehouses of continental discoveries. Sir William Thomson, Tait, and some other modern writers find it frequently convenient to use both notations. Herschel, Peacock, and Babbage translated, in 1816, from the French, Lacroix's treatise on the differential and integral calculus, and added in 1820 two volumes of examples. Lacroix's was one of the best and most extensive works on the calculus of that time. Of the three founders of the "Analytical Society," Peacock afterwards did most work in pure mathematics. Babbage became famous for his invention of a calculating engine superior to Pascal's. It was never finished, owing to a misunderstanding with the government, and a consequent failure to secure funds. John Herschel, the eminent astronomer, displayed his mastery over higher analysis in memoirs communicated to the Royal Society on new applications of mathematical analysis, and in articles contributed to cyclopædias on light, on meteorology, and on the history of mathematics.

George Peacock (1791–1858) was educated at Trinity College, Cambridge, became Lowndean professor there, and later, dean of Ely. His chief publications are his Algebra, 1830 and 1842, and his Report on Recent Progress in Analysis, which was the first of several valuable summaries of scientific progress printed in the volumes of the British Association. He was one of the first to study seriously the fundamental principles of algebra, and to fully recognise its purely symbolic character. He advances, though somewhat imperfectly, the "principle of the permanence of equivalent forms." It assumes that the rules applying to the symbols of arithmetical algebra apply also in symbolical algebra. About this time D. F. Gregory wrote a paper "on the real nature of symbolical algebra," which brought out clearly the commutative and distributive laws. These laws had been noticed years before by the inventors of symbolic methods in the calculus. It was Servois who introduced the names commutative and distributive in 1813. Peacock's investigations on the foundation of algebra were considerably advanced by De Morgan and Hankel.

James Ivory (1765–1845) was a Scotch mathematician who for twelve years, beginning in 1804, held the mathematical chair in the Royal Military College at Marlow (now at Sandhurst). He was essentially a self-trained mathematician, and almost the only one in Great Britain previous to the organisation of the Analytical Society who was well versed in continental mathematics. Of importance is his memoir (Phil. Trans., 1809) in which the problem of the attraction of a homogeneous ellipsoid upon an external point is reduced to the simpler problem of the attraction of a related ellipsoid upon a corresponding point interior to it. This is known as "Ivory's theorem." He criticised with undue severity Laplace's solution of the method of least squares, and gave three proofs of the principle without recourse to probability; but they are far from being satisfactory.

 

The Origin of Modern Geometry.

 

By the researches of Descartes and the invention of the calculus, the analytical treatment of geometry was brought into great prominence for over a century. Notwithstanding the efforts to revive synthetic methods made by Desargues, Pascal, De Lahire, Newton, and Maclaurin, the analytical method retained almost undisputed supremacy. It was reserved for the genius of Monge to bring synthetic geometry in the foreground, and to open up new avenues of progress. His Géométrie descriptive marks the beginning of a wonderful development of modern geometry.

Of the two leading problems of descriptive geometry, the one—to represent by drawings geometrical magnitudes—was brought to a high degree of perfection before the time of Monge; the other—to solve problems on figures in space by constructions in a plane—had received considerable attention before his time. His most noteworthy predecessor in descriptive geometry was the Frenchman Frézier (1682–1773). But it remained for Monge to create descriptive geometry as a distinct branch of science by imparting to it geometric generality and elegance. All problems previously treated in a special and uncertain manner were referred back to a few general principles. He introduced the line of intersection of the horizontal and the vertical plane as the axis of projection. By revolving one plane into the other around this axis or ground-line, many advantages were gained.[54]

Gaspard Monge (1746–1818) was born at Beaune. The construction of a plan of his native town brought the boy under the notice of a colonel of engineers, who procured for him an appointment in the college of engineers at Mézières. Being of low birth, he could not receive a commission in the army, but he was permitted to enter the annex of the school, where surveying and drawing were taught. Observing that all the operations connected with the construction of plans of fortification were conducted by long arithmetical processes, he substituted a geometrical method, which the commandant at first refused even to look at, so short was the time in which it could be practised; when once examined, it was received with avidity. Monge developed these methods further and thus created his descriptive geometry. Owing to the rivalry between the French military schools of that time, he was not permitted to divulge his new methods to any one outside of this institution. In 1768 he was made professor of mathematics at Mézières. In 1780, when conversing with two of his pupils, S. F. Lacroix and Gayvernon in Paris, he was obliged to say, "All that I have here done by calculation, I could have done with the ruler and compass, but I am not allowed to reveal these secrets to you." But Lacroix set himself to examine what the secret could be, discovered the processes, and published them in 1795. The method was published by Monge himself in the same year, first in the form in which the shorthand writers took down his lessons given at the Normal School, where he had been elected professor, and then again, in revised form, in the Journal des écoles normales. The next edition occurred in 1798–1799. After an ephemeral existence of only four months the Normal School was closed in 1795. In the same year the Polytechnic School was opened, in the establishing of which Monge took active part. He taught there descriptive geometry until his departure from France to accompany Napoleon on the Egyptian campaign. He was the first president of the Institute of Egypt. Monge was a zealous partisan of Napoleon and was, for that reason, deprived of all his honours by Louis XVIII. This and the destruction of the Polytechnic School preyed heavily upon his mind. He did not long survive this insult.

Monge's numerous papers were by no means confined to descriptive geometry. His analytical discoveries are hardly less remarkable. He introduced into analytic geometry the methodic use of the equation of a line. He made important contributions to surfaces of the second degree (previously studied by Wren and Euler) and discovered between the theory of surfaces and the integration of partial differential equations, a hidden relation which threw new light upon both subjects. He gave the differential of curves of curvature, established a general theory of curvature, and applied it to the ellipsoid. He found that the validity of solutions was not impaired when imaginaries are involved among subsidiary quantities. Monge published the following books: Statics, 1786; Applications de l'algèbre à la géométrie, 1805; Application de l'analyse à la géométrie, The last two contain most of his miscellaneous papers.

Monge was an inspiring teacher, and he gathered around him a large circle of pupils, among which were Dupin, Servois, Brianchion, Hachette, Biot, and Poncelet.

Charles Dupin (1784–1873), for many years professor of mechanics in the Conservatoire des Arts et Métiers in Paris, published in 1813 an important work on Développements de géométrie, in which is introduced the conception of conjugate tangents of a point of a surface, and of the indicatrix.[55] It contains also the theorem known as "Dupin's theorem." Surfaces of the second degree and descriptive geometry were successfully studied by Jean Nicolas Pierre Hachette (1769–1834), who became professor of descriptive geometry at the Polytechnic School after the departure of Monge for Rome and Egypt. In 1822 he published his Traité de géométrie descriptive.

Descriptive geometry, which arose, as we have seen, in technical schools in France, was transferred to Germany at the foundation of technical schools there. G. Schreiber, professor in Karlsruhe, was the first to spread Monge's geometry in Germany by the publication of a work thereon in 1828–1829.[54] In the United States descriptive geometry was introduced in 1816 at the Military Academy in West Point by Claude Crozet, once a pupil at the Polytechnic School in Paris. Crozet wrote the first English work on the subject.[2]

Lazare Nicholas Marguerite Carnot (1753–1823) was born at Nolay in Burgundy, and educated in his native province. He entered the army, but continued his mathematical studies, and wrote in 1784 a work on machines, containing the earliest proof that kinetic energy is lost in collisions of bodies. With the advent of the Revolution he threw himself into politics, and when coalesced Europe, in 1793, launched against France a million soldiers, the gigantic task of organising fourteen armies to meet the enemy was achieved by him. He was banished in 1796 for opposing Napoleon's coup d'état. The refugee went to Geneva, where he issued, in 1797, a work still frequently quoted, entitled, Réflexions sur la Métaphysique du Calcul Infinitésimal. He declared himself as an "irreconcilable enemy of kings." After the Russian campaign he offered to fight for France, though not for the empire. On the restoration he was exiled. He died in Magdeburg. His Géométrie de position, 1803, and his Essay on Transversals, 1806, are important contributions to modern geometry. While Monge revelled mainly in three-dimensional geometry, Carnot confined himself to that of two. By his effort to explain the meaning of the negative sign in geometry he established a "geometry of position," which, however, is different from the "Geometrie der Lage" of to-day. He invented a class of general theorems on projective properties of figures, which have since been pushed to great extent by Poncelet, Chasles, and others.

Jean Victor Poncelet (1788–1867), a native of Metz, took part in the Russian campaign, was abandoned as dead on the bloody field of Krasnoi, and taken prisoner to Saratoff. Deprived there of all books, and reduced to the remembrance of what he had learned at the Lyceum at Metz and the Polytechnic School, where he had studied with predilection the works of Monge, Carnot, and Brianchion, he began to study mathematics from its elements. He entered upon original researches which afterwards made him illustrious. While in prison he did for mathematics what Bunyan did for literature,—produced a much-read work, which has remained of great value down to the present time. He returned to France in 1814, and in 1822 published the work in question, entitled, Traité des Propriétés projectives des figures. In it he investigated the properties of figures which remain unaltered by projection of the figures. The projection is not effected here by parallel rays of prescribed direction, as with Monge, but by central projection. Thus perspective projection, used before him by Desargues, Pascal, Newton, and Lambert, was elevated by him into a fruitful geometric method. In the same way he elaborated some ideas of De Lahire, Servois, and Gergonne into a regular method—the method of "reciprocal polars." To him we owe the Law of Duality as a consequence of reciprocal polars. As an independent principle it is due to Gergonne. Poncelet wrote much on applied mechanics. In 1838 the Faculty of Sciences was enlarged by his election to the chair of mechanics.

While in France the school of Monge was creating modern geometry, efforts were made in England to revive Greek geometry by Robert Simson (1687–1768) and Matthew Stewart (1717–1785). Stewart was a pupil of Simson and Maclaurin, and succeeded the latter in the chair at Edinburgh. During the eighteenth century he and Maclaurin were the only prominent mathematicians in Great Britain. His genius was ill-directed by the fashion then prevalent in England to ignore higher analysis. In his Four Tracts, Physical and Mathematical, 1761, he applied geometry to the solution of difficult astronomical problems, which on the Continent were approached analytically with greater success. He published, in 1746, General Theorems, and in 1763, his Propositiones geometricœ more veterum demonstratœ. The former work contains sixty-nine theorems, of which only five are accompanied by demonstrations. It gives many interesting new results on the circle and the straight line. Stewart extended some theorems on transversals due to Giovanni Ceva (1648–1737), an Italian, who published in 1678 at Mediolani a work containing the theorem now known by his name.