### Ball: A Short Account of the History of Mathematics

#### by Carson Reynolds

For brevity, I will focus only on the mathematicians Ball details who are directly related to the development of dynamics. Dynamics is intimately linked to the development of mechanics, so I will work through the mathematicians who contributed to the development of mechanics but focus on those who contributed to dynamics in particular.

Archytas (c. 400 BCE) “is said to have treated it [mechanics] with the aid of geometry. He is alleged to have invented and worked out the theory of the pulley . . .” “Archytas was also interested in astronomy he taught the earth was a sphere rotating round its axis in twenty-four hours, and around which the heavenly bodies moved.”

Aristotle (384-322 BCE) “A small book containing a few questions on mechanics which is sometimes attributed to him is of doubtful authority; but though in all probability it is not his work, it is interesting, party as shewing that the principles of mechanics were beginning to excite attention, and partly as containing the earliest known employment of letters to indicate magnitudes.”

Archimedes (287-212 BCE) “The *Mechanics* is a work on statics with special reference to the equilibrium of plane laminas and to properties of their centres of gravity.” “the science of statics rested on his theory of the lever until 1586, when Stevinus publishes his treatise on statics.” “It will be noticed that the mechanical investigations of Archimedes were concerned with statics. It may be added that thought the Greeks attacked a few problems in dynamics, they did it with but indifferent success: some of their remarks were acute, but they did not sufficiently realise that the fundamental facts on which the theory must be based can be established only by carefully devised observations and experiments. It was not until the time of Galileo and Newton that this was done.” “We know both from occasional references in his works and from remarks by other writers, that Archimedes was largely occupied in *astronomical observations*.”

Pappus (end of 3rd century CE) “In mechanics Pappus shewed that the centre of mass of a triangular lamina is the same as that of an inscribed triangular lamina whose vertices divide each of the sides of the original triangle in the same ratio. He also discovered the two theorems on the surface and volume of a solid of revolution . . .”

Stevinus (1548-1620) “It is, however, on his *Statics and Hydrostatics*, published (in Flemish) at Leyden in 1586, that his fame rests. In this work he enunciates the triangle of forces–a theorem which some think was first propounded by Leonardo da Vinci.” “He further distinguishes between stable and unstable equilibrium.”

Galileo (1564-1642) “Just as the modern treatment of statics originates with Stevinus, so the foundation of the science of dynamics is due to Galileo.” “It was there that he noticed that the great bronze lamp hanging from the roof of the cathedral, performed its oscillations in equal times, and independently of whether the oscillations were large or small–a fact which he verified by counting his pulse.” “During the next three years he carried on, from the leaning tower, that series of experiments on falling bodies which established the first principles of dynamics.” “His lectures there seem to have been chiefly on mechanics and hydrostatics, and the substance of them in contained in his treatise on mechanics, which was published in 1612. In these lectures he repeated his Pisan experiments, and demonstrated that falling bodies did not (as was then commonly believed) descend with velocities proportional, amongst other things, to their weights. “The laws of motion are, however, nowhere enunciated in a precise and definite form, and Galileo must be regarded rather as preparing the way for Newton than as being himself the creator of the science of dynamics.” “It is, however, as an astronomer that most people regard Galileo . . .” “In 1637 he lost his sight, but with the aid of pupils he continued his experiments on mechanics and hydrostatics, and in particular on the possibility of using a pendulum to regulate a clock, and on the theory of impact.”

Huygens (1629-1695) “His astronomical observations required some exact means of measuring time, and he was thus led in 1656 to invent the pendulum clock, as descried in his tract *Holologium*. “. . . he proved by experiment that the momentum in a certain direction before the collision of two bodies is equal to the momentum in that direction after the collision. This was one of the points in mechanics on which Descartes had been mistaken.” “The most important of Huygens’s work was his *Horologium Oscillatorium* published at Paris in 1673. The first chapter is devoted to pendulum clocks. The second chapter contains a complete account of the descent of heavy bodies under their own weights in a vacuum, either vertically down or on smooth curves. Amongst other propositions he shews that the cycloid is tautochronous. In the third chapter he defines evolutes and involutes, proves some of their more elementary properties, illustrates his methods by finding the evolutes of the cycloid and the parabola . . . This work contains the first attempt to apply dynamics to bodies of finite size and not merely to particles.”

Newton (1642-1727) “No sooner had Newton proved this superb theorem … than all the mechanism of the universe at once lay spread before him.” “In it [the first book of the *Principia*] also Newton generalizes the law of attraction into a statement that every particle of matter in the universe attracts every other particle with a force which varies directly as the distance between them; and he thence deduces the law of attraction for spherical shells of constant density. The book is prefaced by an introduction on the science of dynamics, which defines the limits of mathematical investigation. His object, he says, is to apply mathematics to the phenomena of natural among these phenomena motion is one of the most important; now motion is the effect of force, and though he does not know what is the nature or origin of force, still many of its effects can be measured; and it is these that form the subject-matter of the work.” “This book [the second book] treats of motion in a resisting medium, and of hydrostatics and hydrodynamics, with special applications to waves, tides, and acoustics.” “He proceeds [in the third book] to apply the theorems obtained in the first book to the chief phenomena of the solar systems, and to determine the masses and distances of the planets and (whenever sufficient data existed) of their satellites.” “The adoption of geometrical methods in the *Principia* for purposes of demonstration does not indicate a preference on Newton’s part for geometry over analysis as an instrument of research, for it is known that Newton used the fluxional calculus in the first instance in finding some of the theorems” “within ten years of its publication it was generally accepted in Britain as giving a correct account of the laws of the universe” “The invention of the infinitesimal calculus was one of the great intellectual achievements of the seventeenth century.” “In mechanics also, by integration, velocities could be deduced from known accelerations, and distances traveled from known velocities.” “Newton, further, was the first to place dynamics on a satisfactory basis, and from dynamics he deduced the theory of statics: this was in the introduction to the *Principia* published in 1687. The theory of attractions, the application of the principles of mechanics to the solar system, the creation of physical astronomy, and the establishment of the law of universal gravitation are due to him . . .” “Not less remarkable is the homage rendered by Gauss; for other great mathematicians or philosophers he used the epithets magnus, or clarus, or clarissimus: for Newton alone he kept the prefix summus.”

Leibnitz (1646-1716) “As to Leibnitz’s system of philosophy it will be enough to say that he regarded the ultimate elements of the universe as individual percipient beings whom he called monads. According to him the monads are centres of force, and substance is force, while space, matter, and motion are merely phenomenal” “The ideas of infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials.” “There is no question that the differential notation is due to Leibnitz”

John Bernoulli (1667-1748) “. . . the chief discoveries of John Bernoulli were exponential calculus . . . the solution of the brachistochrone” “I believe that he was the first to denote the accelerating effect of gravity by an algebraical sign *g*, and thus arrived at the formula *v^2=2gh*.

L’Hospital (1661-1704) “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. ”

Varigon (1654-1722) “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.”

Cramer (1704-1752) “wrote on the physical cause of the spheroidal shape of the planets and the motion of their apses”

Clairaut (1713-1765) “shewn that a mass of homogenous fluid set in rotation about a line through its center mass would, under the mutual attraction of its particles, take the form of a spheroid.” “explanation of the motion of the apse which had previously puzzled astronomers” “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.”

D’Alembert (1717-1783) “The first of these was his *Traite de dynamique*, published in 1743, in which he enunciates the principle known by his name, namely, that of 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 studying “fluids . . . the motion of the air . . . a vibrating string . . . [he] arrived at a partial differential equation.” “The chief remaining contributions of D’Alembert to mathematics were on physical astronomy, especially on the procession of the equinoxes, and on variations in the obliquity of the ecliptic.”

Daniel Bernoulli (1700-1782) “His chief work is his *Hyrdrodynamica*, published in 1738; it resembles Lagrange’s *Mecanique analytique* in being arranged so that all the results are consequences of a single principle, namely, in this case, the conservation of energy.” “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 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.”

Taylor (1685-1731) “he discussed the motion of projectiles, the centre of oscillation, and the forms taken by liquids when raised by capillarity.” The most important of them is the theory of the transverse vibrations of strings, a problem which had baffled previous investigators.” “The form of the catenary and the determination of the centres of oscillation and percussion are also discussed.”

Cotes (1682-1716) “It contains essays on Newton’s *Methodus Differentialis*, on the construction of tables by the method of differences, on the decent of a body under gravity, on the cycloid pendulum, and on projectiles.”

Maclaurin (1698-1746) “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.” “This treatise is, however, especially valuable for the solutions it contains of numerous problems in geometry, statics, the theory of attractions, and astronomy.” “No further advance in the theory of attractions was made until Lagrange in 1773 introduced the idea of the potential.”

Stewart (1717-1785) “a mathematician of considerable power, to whom I allude in passing, for his theorems on the problem of three bodies”

Simpson (1710-1761) “In his 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.”

Euler (1707-1783) “The classic problem on isoperimetrical curves, the barchistochrone in a resisting medium, and the theory of geodesics (all of which had been suggested by his master; John Bernoulli) had engaged Euler’s attention at an early date; and in solving them he was led to the calculus of variations. The idea of this was given in his *Curvarum Maximi Minimive Propreitate Gaudentium Inventio*, published in 1741 and extended in 1744, but the complete development of the new calculus was first effected by Lagrange in 1759.” “In the mechanics of a rigid system he determined the general equations of motion of a body about a fixed point . . . and he gave the general equations of motion of a free body . . .” “He also defended and elaborated the theory of ‘least action’ which had been propounded by Maupertuis in 1751 in his *Essai de cosmologie* [p.70]” “In hydrodynamics Euler established the general equations of motion . . .” “At the time of his death he was engaged in writing a treatise on hydromechanicas . . .” “In these he attacked the problem of three bodies: he supposed the body considered (*ex. gr.* the moon) to carry three rectangular axes with it in its motion, the axes moving parallel to themselves, and to these axes all the motions were referred.”

Lambert (1728-1777) “for the first time he expressed Newton’s second law of motion in the notation of the differential calculus.”

Lagrange (1736-1813) “he enunciated the principles of the calculus of variations. Euler recognized the generality of the method adopted . . .” “The first volume [of *Miscellanea Traurinensia*] contains a memoir on the theory of the propagation of sound in this he indicates a mistake by Newton, obtains the general differential equations for the motion and integrates it for motion in a straight line. This volume also contains the complete solution of the problem of a string vibrating transversely . . .” “The second volume contains . . . the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics. The third volume includes the solution of several dynamical problems by means of the calculus of variations . . . and the general differential equations of motion for three bodies moving under their mutual attractions.” “The next work he produced was in 1764 on the libration of the moon, and an explanation as to why the same face was always turned to the earth . . . containing the germ of the idea of generalized equations of motion, equations which he first formally proved in 1780.” “. . . he contributed a paper on the pressure exerted by fluids in motion . . .” “During the years from 1772 to 1785 he contributed a long series of memoirs which created the science of *differential equations*, at any rate as far as partial differential equations are concerned.” “Lastly, there are numerous memoirs on problems in *astronomy*” “Over and above these various papers he composed his great treatise, the *Mecanique analytique*. In it he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids.” “The method of generalized co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis.” “Lagrange held that mechanics was really a branch of pure mathematics analogous to a geometry of four dimensions, namely, the time and the three co-ordinates of the point in space . . .” “The theory of planetary motions had formed the subject of some of the most remarkable of Lagrange’s Berlin papers.” “But above all he impressed on mechanics . . . that generality and completeness towards which his labours invariably tended.”

Laplace (1749-1827) “. . . he shewed that the planetary motions were stable . . .” “. . . he completely determined the attraction of a spheroid on a particle outside it.” “Laplace now set himself the task to write a work which should ‘offer a complete solution of the great mechanical problem presented by the solar system, and bring theory to coincide so closely with observation that empirical equations should no longer find a place in astronomical tables.’ The result is embodied in the *Exposition du system du monde* and *Mecanique celeste*.”

Legendre (1752-1833) “The earliest of these memoirs, presented in 1783, was on the attraction of spheroids.”

Fourier (1768-1830) “In 1822 he published his *Theorie analytique de la chaleur*, in which he bases his reasoning on Newton’s law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the infinitely small difference of their temperatures.”

Sadi Carnot (1796-1832) “He made the mistake of assuming heat was material, but his essay may be taken as initiating the modern theory of thermodynamics.”

Poisson (1781-1840) “The chief treatise which he wrote were his *Traite de mecanique* . . .” “The most important of the those on physical astronomy are the two read in 1806 (printed in 1809) on the secular inequalities of the mean motions of the planets, and on the variation of arbitrary constants introduced into the solutions of questions on mechanics; in these Poisson discusses the question of the stability of the planetary orbits (which Lagrange had already proved to the first degree of approximation for the disturbing forces), and shews that the result can be extended to the third order of small quantities . . .” “Poisson also published a paper in 1821 on the libration of the moon; and another in 1827 on the motion of the earth about its center of gravity.”

Gauss (1777-1855) “In electrodynamics Gauss arrived (in 1835) at a result . . . that the attraction between two electrified particles *e* and *e’*, whose distance apart is *r*, depends on their relative motion and position . . .”

Jacobi (1804-1851) “his development of the calculus of variational and his contributions to the problem of three bodies, and other particular dynamical problems.”

Hamilton (1805-1865) “This was followed by a paper in 1827 on the principle of *Varying Action*, and in 1834 and 1835 by memoirs on a *General Method in Dynamics*–the subject of theoretical dynamics being properly treated as a branch of pure mathematics.”

Ball ends his book with a brief survey of then-recent work on Kinematics, Analytical Mechanics, Theoretical Dynamics, and Theoretical Astronomy. Here he mentions Poincare, and the work of Hamilton, Mobius, Laplace and Largrange in slightly more depth.

HAY QUE AÑADIR QUE EULER FUE EL PRIMERO EN ESCRIBIR LA SEGUNDA LEY DE NEWTON EN LA FORMA, FUERZA IGUAL A MASA POR LA ACELERACIÓN.