DIE NATURWISSENSCHAFTEN

63. Jahrgang, 1976 Heft 2 Februar

History of Classical Mechanics

Part I, to 1800

C. Truesdell Professor of Rational Mechanics, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A.

This article sketches the main discoveries in the quantitative, mathematical science of mechanics. Part I, after referring to the origins in Greek and Mediaeval times, traces the growth of the science in the seventeenth and eighteenth centuries. The savants of those centuries, Euler above all, gave to mechanics most of its major concepts, and they formulated several parts of it as clear, succinct, and fairly definitive disciplines, which have retained their identies to this day. The second part of the article will carry the history of mechanics down to the present.

I know that it will bee said by many, That I might have of "pure" mathematics. At the other, it is invention beene more pleasing to the Reader, if I had written the of machines or practice with them, described sugges- Story of mine owne times; having been permitted to tively if loosely in terms which some will think to have draw water as neare the Well-head as another. To this been borrowed from mathematical science, while I answer, that who-so-ever in writing a moderne Historie, others will regard them as providing the basis in experi- shall follow truth too neare the heeles, it may happily ence for the corresponding concepts of mathematics. strike out his teeth. There is no Mistresse or Guide, that Between these two extremes lies what physicists call hath led her followers and servants into greater miseries. mechanics. ... It is enough for me (being in that state I am) to write of the eldest times: wherein also why may it not be said, The Folklorish History of Mechanics that in speaking of the past, I point at the present, and taxe the vices of those that are yet lyving, in their persons However "objective" may be science itself, it is the that are long since dead; and have it laid to my charge ? certified preserve of a jealous guild or cult and thus has its folklore. This folklore is usually inculcated Sir Walter Ralegh, imprisoned on a capital charge in the Tower of through notices labelled "history" in textbooks. London, 1614 Because most of the natural sciences pursued today are inventions of the last century, in order to produce a history of science having a respectable antiquity it Background of this Sketch is necessary to fix attention on mechanics, and, so as to fit preconceptions as to what natural science Mechanics is or at least ought to be, this "history" of mechanics Next to mathematics itself, mechanics is the 01dest is largely invented. Although purveyed originally in of the logical sciences. While algebra and analysis refer textbooks of physics, it is now disseminated even only to sets and functions, irrespective of how they more intensely in introductory courses on the biologi- may be suggested by experience, and while geometry cal sciences and psychology. reflects experience only through relations among posi- This folklorish history may be summarized in a few tions, irrespective of what occupies them or how it lines. Despite the temporary advance toward godfree may come to do so, mechanics enriches the discourse and hence "scientific" thought made by the Greeks, by providing logical models for time, mass, force, mankind fell back into piety. The Middle Ages lay energy, and, finally, heat and temperature. Thus inert beneath a pall of scholastic repetitions, appealing mechanics has many aspects. At the one extreme, it to Aristotle instead of to experiment. Copernicus said is mathematics-indeed, to the layman indistinguish- Aristotle was wrong, and Galileo dropped weights able from algebra, analysis, geometry, and other parts from the tower of Pisa so as to shatter scholasticism

Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976 53 54 Naturwissenschaften 63, 53--62 (1976) O by Springer-Verlag 1976 by an irrefutable, simultaneous thud. From then on, It is full of gaps, gaps which reflect my ignorance only reactionaries (non-scientists) set up theories with- of some parts of mechanics as well as of many parts out first collecting heaps of data. Kepler proved Coper- of its history, but I prefer to reveal that ignorance nicus right to within eccentricity and set up empirical by silence rather than by parroting the fruits of others' laws of planetary motion. Descartes and Leibniz were labor or carelessness. The reader will perceive my bias. put to shame for their failure to adhere to the nine- I hope he will perceive also that that bias is the result teenth century program, but Newton showed Kepler's not of prejudice but of judgment, judgment induced laws to be consequences of the fact that each body from a lifetime of study in which the past and present in the universe attracts every other one by a force of creative mechanics have illuminated each other inversely proportional to the square of the intervening mutually. distance. To do so, he had to lay down (on the basis of experiment) three Laws of Motion. In the words Ernst Mach (1838-1916) wrote in his famous history Old Cultures of mechanics-the acknowledged classic of this folk- Archimedes (A.C. 287-212) lore- "Newton discovered universal gravitation and completed the formal enunciation of the mechanical Archimedes' amazing insight into mathematical pro- principles now generally accepted. Since his time no cesses and superb standard of mathematical rigor in essentially new principle has been stated. All that has reasoning about problems of mechanics have inspired been accomplished in mechanics since his day has been everyone who could read his works. He showed how a deductive, formal, and mathematical development to use the law of the lever, and ever since his day of mechanics on the basis of Newton's laws." Mach it has been a keystone of mechanics. He set up axioms refers, of course, to "classical" mechanics, the history for the equilibrium of floating bodies. His theory did of which makes up all but the last eighty years of not reveal much about fluids but was specially designed the history of all kinds of mechanics. for the particular problem he studied: to find all posi- Nearly a century has passed since Mach's book was tions in which a given body, such as a segment of published. More serious students, willing to take the a paraboloid of revolution, may float, and to determine trouble to learn before generalizing, have scrutinized their stability or instability. He regarded a position and analyzed, line by line, the works of discovery, as stable if the body when displaced from it and then have brought to light and published dozens of volumes constrained to remain at rest suffered a torque tending of sources previously unknown. Mach's picture of the to restore it to that position. development of mechanics, though it remains Although ancient Greece produced many other fine accepted folklore, as history has been shown to be mathematicians, I can find no evidence that any of so largely false as to mislead even in regard to such them turned his attention to mathematical theories details as it does recount more or less justly. of mechanics, or that Archimedes' discoveries were enlarged. The only culture to develop a mathematical science of mechanics - rational mechanics - is the Eur- Present State of the True History of Mechanics opean West. Although other cultures, both older and younger ones, had expert and sometimes creative No competent man today would dare to write a general schools of arithmetic, algebra, number theory, geo- history of mechanics. Too much remains dead on metry, and celestial kinematics, in their remains I can library shelves, buried in print as yet unread by any find no evidence of even the most special or primitive of the inframinority who know enough science and original mathematical thinking about mechanics. To enough history to follow and comprehend them word see that achievements of the utmost skill and success by word and equation by equation-for science with- in mechanical construction and contrivance need not out accurate detail, "general" science, is mere sociolo- lead to any scientific discipline or quantitative theory gy or propaganda, neo-Machism. The now numerous of mechanics, we need only consider that notoriously competent studies of various special topics remain un- unmathematical people, the Ancient Romans. The digested and unconnected. The gaps between them central idea of scientific mechanics is force as a mathe- are as long as the stretches they cover. The student matical quantity, and force is a characteristic concept who searches for historical truth is blocked by the of the European West. vast middens of words on which those eager to exalt their own incompetence in the science of all periods into professional expertise in the history of science Origins in the West raise their hogans ever higher, words which, while rarely trustworthy, nevertheless cannot be flatly ignored. The Middle Ages The nature and properties of motion were studied again and again in the Middle Ages. Jordanus de The Nature of this Sketch Nemore (before 1260) asserted that the weight of a The text I present here expresses my view of some of body was diminished when it fell along a direction the major stages in the creation of classical mechanics. oblique to the vertical. In his reasoning he appealed

Naturwissenschaften63, 53-62 (1976) by Springer-Verlag 1976 55 to a special case of what came much later to be called in steady flow and distinguished accurately between the principle of virtual work. He set up for study and the whirling of a vortex and the rotation of a wheel. in some cases claimed to solve a number of problems He formulated numerous scientific questions, to which that were to become typical of Western mechanics: he usually demanded an answer in terms of quantities, the resistance offered by water to the prow of a ship not merely qualities. Such few answers as he asserted or by air to a projectile, the form assumed by a beam are generally quantitative but unsupported by reason- loaded in various ways, the load required so as to ing or fact. He projected many experiments, but there break a beam, etc. For his ideas no ancient antecedent is no firm evidence he ever did any of them. Almost has ever been found. all of his writings must be regarded as literature or William Heytesbury (ft. 1330-1348), Richard Swines- engineering or pure observation of nature or clair- head (ft. 1344-1355), and John of Dumbleton (ft. 1338- voyance, not as the organized inquiry which alone 1348) distinguished kinematics, the geometry of can produce science. The value of his numerous pro- motion, from dynamics, the theory of the agents of nouncements, his originality in projection of the motion. They succeeded in formulating a fairly clear machines he drew, and the influence his work may concept of instantaneous speed, which means that they have exerted upon subsequent students of natural foreshadowed the central concepts of function and science and engineering, will always remain in dispute. derivative, and they proved that the space traversed by a body in uniformly accelerated motion in a given time is the same as that traversed by a body in uniform The Late Renaissance and Early Baroque motion at the speed which is the mean of the greatest Simon Stevin (1548-1620) and least speeds of the accelerated motion. The folklor- ish history attributes this theorem and its main conse- Stevin seems to have been a disciple of the Greek quences to Galileo, who worked three hundred years mathematicians, unconcerned with philosophical later (see below). In principle, Mediaeval studies re- aspects and uninfluenced either by the mediaeval tradi- placed the qualities of Greek physics by the numerical tion or by then current questions in astronomy. His quantities that have ruled Western science ever since. writings on mechanics, short and easy to read, reveal Almost immediately Giovanni di Casale (ft. 1346- consummate skill with the triangle of forces and the 1375) and Nicole Oresme (13307-1382) found how law of the lever. A reader who mastered their contents to represent the results by geometrical graphs, introduc- could solve any statically determinate problem regard- ing the connection between geometry and the physical ing a discrete set of forces. Stevin took up hydrostatics world that became a second characteristic habit of from a standpoint more practical than Archimedes'. Western thought. Jean Buridan (c. 1330-1357 ...) pro- On the basis of explicit axioms he concluded that the posed a concept of "impetus" that comes close to pressure of quiet water upon the bottom of a container what is now called "momentum", but he did not was independent of the shape of the sides. A further achieve a formal science of dynamics even in the sim- assumption enabled him to calculate rigorously the plest cases. resultant force of water upon a plane of any inclination The Mediaeval work was entirely mathematical. It was as the limit of the sum of pressures upon thin strips. inferred or at least described by means of common Thus he was the first to consider successfully a system experience, but, if we may judge by the documents of infinitely many forces. Nowadays we express his that have been preserved, it did not lead to experi- reasoning concisely in terms of integrals. ments. In much later and largely derivative work Blaise Pascal Mediaeval mechanics was quickly made a part of (1623-1662) asserted that at a given point in a fluid advanced university instruction on the Continent of at rest, the pressure upon a plane was independent Europe; the main works were printed in the late fif- of its inclination. This conclusion, which was inherent teenth century, but by then their mode of thought in Stevin's theory, is one of the earliest regarding the as well as their language was becoming unacceptable, forces that parts of one and the same body exert on for the literary humanists of the Renaissance insisted each other. upon a Latin classical in style as well as form and upon reverence for the works of the Ancients. Galileo Galilei (1564.1642) Galileo's work in mechanics is contained in his Dis- Leonardo da Vinci (1452-1519) courses Concerning Two New Sciences (1638). The first The notebooks of Leonardo suggest indirect contact new science is the theory of strength of materials, in with the dying Mediaeval tradition, but he certainly which Galileo gave a rule for comparing the load that did not master it. He was an accurate and discerning will break a beam transversely with the load that will observer of all that could be seen, and what he saw break it in pulling straight on. His rule was seriously he recorded in splendid drawings. Machines and in error but was accepted later by practising engineers mechanical phenomena were among his favorite sub- and was judged responsible for a railway disaster two jects. From his observations of flows in channels and hundred years afterward. On the contrary, theorists rivers he inferred the principle of constant discharge rejected the rule as soon as it was published, and an

56 Naturwissenschaften 63, 53--62 (1976) by Springer-Verlag 1976 important branch of mechanics grew from attempts ment Galileo concluded that a heavy cord would itself to correct it. hang in a parabola, From this false statement began The second new science is that of local motion. How the celebrated problem of the "catenary curve". much Galileo may have known of the Mediaeval tradi- Beeckman claimed to show that the frequency of tion regarding uniformly accelerated motions is ob- vibration of a taut string would be inversely as the scure, but he obtained in his way all the old theorems length. Although correct only for the first instant of and many subsidiary ones. He employed a good deal motion, his reasoning provides the earliest mathemati- of geometry in rigorous proofs of minor details but cal proof in acoustics. concealed his central assumptions by rhetorical per- suasion and sarcasm directed against those who did Ren6 Descartes (1596-1650) not value his discoveries sufficiently. Whether or not he performed the experiments he described on bodies Descartes, who seems to have learned mechanics and sliding down inclined planes, is also unknown, but physics from Beeckman, made few if any specific con- it has been demonstrated that means and concepts tributions to those fields, but he left a powerful impres- available to him were sufficient for him to have done sion upon them and upon natural science in general. so, and that the failure of Matin Mersenne (1588-1648) First, he pictured nature as a great machine, the to confirm his results arose from use of rolling balls outward workings of which were systematic and un- rather than sliding ones. changing and hence could be described by reason Galileo asserted that motions of projectiles were com- alone: that is, by mathematics. Natural science in this posed of independent horizontal and vertical motions. sense was thus neither impious nor, within its avowed Thus he concluded that the path of a projectile would limitations, futile. Second, he asserted that nature be a parabola. This extremely inaccurate rule, which operated in such a way as to keep constant the sum neglects friction and wind and spin, formed a basis of the products of the masses and the speeds of all of ballistic tables for centuries thereafter. bodies. Had he taken the pains to work out specific Galileo was fascinated by the concept of vibration. consequences of this idea, he would certainly have The idea that sound is a vibratory motion of definite seen that the particular sum he claimed was conserved frequency was already current; it had been emphasized generally could not be, but his assertion gave rise to and published by Mersenne. Galileo made more pre- the search for principles of conservation, which con- cise one of the three laws of vibrating strings Mersenne tinues on into the present time. had inferred from experiments; he may well have dis- covered those laws independently, before Mersenne's book appeared. He asserted that the motion of the The Age of Reason and High Baroque circular pendulum was absolutely isochrone, and he Christiaan Huygens (1629-1695) regarded sound as being somehow similar to the swing- ing of such a pendulum as well as to the motion of In his demand for mathematical rigor and elegance waves on the surface of water. Of his brilliantly written Huygens seems like a new Archimedes, but he brought treatise on local motion the only part that is surely to mechanics also a balance between pure reason and original is that in which he attempts to show that experience of nature achieved never before and seldom descent along a circular arc with horizontal bottom afterward. Adopting, or perhaps re-creating, Galileo's is the fastest possible between any two given points. idea that the tangential acceleration of a descending Galileo's conclusion is false. He supports it only by constrained body should be the same as that of the trying to prove that descent along the circular arc same body if sliding downward along the tangent line, is more rapid than along any inscribed polygon. His he proved that the period of oscillation of a pendulum argument breaks down, but the principle upon which was independent of the amplitude of its excursions it rests is prophetic, for he assumes in effect that a if and only if the constraining curve were a cycloid. body sliding upon a circular arc has at each instant Huygens solved also the problem of the center of oscil- the tangential acceleration it would have if it were lation: If several bodies in the same plane are joined sliding along the tangent line there. together rigidly, through what axis perpendicular to that plane can the rigid assembly rotate freely ? Equiva- lently, if the assembly is allowed to oscillate about Isaac Beeckman (1570-1637) an arbitrary axis perpendicular to the plane, what is Continuing the tradition of Stevin, by use of the princi- the length of the simple pendulum whose period of ples of statics Beeckman analyzed a suspended cord vibration is the same function of its amplitude of to which weights were attached. He indicated that if vibration? To solve this problem Huygens introduced the horizontal distances between the weights were a principle of conservation of energy: In oscillation equal, the points of attachment would lie upon a para- subject to gravity, the center of gravity rises to the bola. Thus he solved the problem of the suspension height from which it has fallen. His result is expressed bridge. This work, done before Galileo's book in terms of what later came to be called "the moment appeared, remained unpublished though not undiscov- of inertia". Thus a measure of inertia altogether dif- ered to others. Through an altogether fallacious argu- ferent from weight entered mechanics.

Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976 57 Huygens' solutions of these problems have stood as of bodies subject to forces that are functions of dis- pinnacles of mechanics to this day. It was recognized tances alone. It contains also a great deal of infinitesi- at once, however, that they rested upon separate princi- mal geometry and analysis, explained in geometrical ples, and that to determine the orbits of the planets notations and terms which have given centuries of or the tidal motion would require still other principles. uncritical readers the idea that Newton concealed his Mechanics in becoming more precise and mathemati- use of what came to be called "differential and integral cal was at the same time splitting into a group of calculus". The treatment of mechanics is fairly syste- sciences only loosely related to each other. matic and remains roughly within Newton's program In his youth Huygens had discerned Galileo's error of rational deduction from the axioms. To a great regarding the catenary and had discovered for himself extent Book I is a retrospective work in that it unifies Beeckman's solution for the suspension bridge. In his and generalizes many previously known but more or last years he attacked the problem of the catenary less separated results. Its most celebrated achievement curve in the same spirit, regarding the hanging body is the proof that the rules of planetary motion painfully as being a number of discrete weights joined by incor- induced from celestial data by Johann Kepler (1571- poreal cords. His methods and concepts were strained 1630) follow by a few lines of mathematics from New- to the limit to solve this problem. He was able to ton's Second Law and the assumption that the universe treat any given particular case but could not synthesize contains only two bodies, which are points or homo- a general description of the curve assumed. geneously layered spheres attracting each other by Huygens may stand as the perfect scientist. His exam- equal and oppositely directed central forces inversely ple shows that one and the same man may create major proportional to the square of the distance between the results in pure mathematics, may develop and apply bodies. physical theory with the highest requirements of logical Book II concerns the motion of bodies in resisting rigor, may introduce new concepts and propositions media. The media are idealized in various ways: as of physics and correlate major physical phenomena agents of gross friction, as assemblies of numerous previously ill comprehended, may project and carry static particles, as incompressible fluids devoid of in- out important experiments, may design and patent ternal friction, as stacks of dilatating and condensing useful devices and cause them to be manufactured. elastic material, as layers sliding upon one another. Mechanics was only one of his fields of activity. Newton claimed to determine the resistance opposed by rare or dense media to the passage of spheres or cylinders; the oscillation of water in a U-tube; the Isaac Newton (1643-1727) efflux of water from a vessel with a hole in its bottom; the progress of surface waves; the speed of sound in Newton's Mathematical Principles of Natural Philoso- air; the resistance arising from want of slipperiness phy, published in 1687, was the first general treatise in a fluid; and several other things. on rational mechanics, It attracted great attention Although many who merely glanced at Newton's work because of its ambitious plan of deduction. The reader were persuaded that its contents really did follow gained the impression that everything should follow mathematically from his Laws of Motion, and many from three "Axioms, or Laws of Motion". These persons today are so persuaded, in Book II his program axioms were phrased in terms of "forces". Force was of rational deduction breaks down altogether. New a common word, loosely used in scientific writing, hypotheses are introduced, either avowedly or tacitly, but never before considered fundamental. Readers of every few pages; the "proofs" are often rhetorical the time were puzzled to encounter it as what we should or even circular; and some passages are little more today call a "primitive variable" in the mathematics, than bluffirig. Many of the problems Newton takes neither a philosophical quality nor a quantity defined up in this book had never before been subjected to mathematically in terms of motions. Newton showed any kind of mathematical treatment. For ingenuity how to use forces, but he never explained them or and insight, and, above all, for sense of problem, Book set forth their properties in general. II is the most brilliant and fertile work ever written His operative axiom is the Second Law, which states in mechanics; it was so received at once by the dozen in effect that for any body in any circumstances the men who could understand it; and so it remains. For acceleration is equal to the applied force per unit mass. over one hundred years the finest geometers devoted Later students were to see that such a statement could themselves to criticism, development, and correction make sense only if the concept of "body" were refined: of the ideas and arguments sketched in it, to solution Typically, but not exclusively, the "body" occupies of the problems it attacked unsuccessfully, to explo- only a point in space. The law itself seems nowadays ration and conquest of the ranges of mechanics it to be a natural extension of principles used by Galileo opened. and Huygens, but unlike those, it refers not only to Book III concerns the "System of the World", that uniform gravity but also to all kinds of forces, and is, the sun, the planets and their moons, and certain not to only one component of acceleration but to the comets. In it Newton attempted to show that his law acceleration vector itself. of universal gravitation sufficed to explain in detail Book I of Newton's great treatise concerns the motion the phenomena of celestial mechanics. Since the math-

58 Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976 ematical structure presented in Book I was insufficient by Jakob Bernoulli gives the first example of the law to treat a universe containing more than two bodies, of rotational momentum, which in time became and Newton had to resort here to approximations of var- has remained the second fundamental principle of all ious sorts. Here, too, he showed himself a consummate mechanics. master of mathematics and of ingenious guessing. Jakob Bernoulli's second masterpiece is his theory of Newton published only a small fraction of what he the bent elastic band. Considering the cross-section wrote. At last it is becoming possible to follow the as if it spanned a spring, he found the differential course of his thought, for a definitive edition of all equation of equilibrium the curve must satisfy, and his mathematical works, superbly edited by D.T. he solved it in one major case. Probing deeply the Whiteside, began to appear in 1967. Most of Newton's nature of elasticity, he saw that response of a material effort was put out on algebra, geometry, and analysis, should be described as a relation between tensile force partly but by no means entirely in response to problems per unit area and elongation per unit length, what arising in his study of mechanics. is called today a "stress-strain relation". His attempt to determine the unstretched fibre in a bent beam was a failure. Gottfried Wilhelm Leibniz (1646-1716) So as to solve various particular problems of mechanics concerning flexible lines or sliding bodies, Leibniz's mechanics was developed at about the same he created the calculus of variations, which has time as Newton's and was entirely independent of it. remained ever afterward an important tool in all parts After refuting Descartes' principle of conservation, he of mathematical physics. In particular, correcting defined the "live force" of a body as being the product Galileo, he proved that the curve of quickest descent of its quantity by the square of its speed and claimed between two given points is a cycloid. that any loss of live force was compensated by an equal gain of "dead force", and vice versa. The ideas of kinetic and potential energy, the sum of which is The Enlightenment and Rococo conserved in many important cases, descend from his principle. The Age of Euler (c. 1730-c. 1780) Leibniz proposed also a"law of continuity", in conse- No other man has ever dominated mechanics to such quence of which "everything goes by degrees in nature a degree or for so long a time as did Leonhard Euler and nothing by jumps ". Thus he easily dispensed with (1707-1783). Mechanics as it is taught today to discrete models and considered continuous bodies dir- engineers and mathematicians is largely his creation. ectly. Gaston Pardies (1636/8-1673) had already He took up every aspect of it and every important attempted to do so but had lacked the mathematics special problem then under study. Everything he to express and solve the conditions he obtained. Leib- touched he transformed, clarified, refounded, and niz's differential and integral calculus provided logical enriched. His work on any given subject always sub- tools perfectly fit for analysis of a universe conceived sumed and rendered obsolete all previous studies of as being one of perpetual and infinitesimal change. it. With them, Leibniz himself easily solved the problem Euler was a student of Johann Bernoulli (1667-1748), of the catenary curve on the basis of Pardies' mechani- who had been trained by his brother Jakob. Thus cal principle. Also, correcting Galileo's rule of Euler inherited not only the mathematics of Leibniz strength, he was the first person to calculate the effect and the Bernoullis but also the tradition of earthly of tensions distributed unequally over the cross-section statics as developed by Stevin, Huygens, and Pierre of a loaded beam. Varignon (1654-1722). At the same time, he mastered Elegant and definitive as are Leibniz's solutions of Newton's methods and concepts, with their emphasis a few specific problems, his importance for mechanics on dynamics and on celestial phenomena, and in him comes more from the fact that the kind of differential the two main streams of thought in mechanics came and integral calculus that lent itself best to use in together. They have remained so forever afterward. natural science developed on the basis of his ideas In the second quarter of the eighteenth century great rather than Newton's. attention was given to systems undergoing small oscil- lations. Both discrete systems, such as many linked Jakob Bernoulli (1655-1705) masses or weights or rigid rods, and continuous ones, such as taut or heavy strings and elastic bars, were Jakob Bernoulli's mechanics was strongest in the considered. Differential equations of motion for such aspects where Newton's was weakest: problems con- bodies were still unknown, for the principles of cerning rigid or elastic solids. He united Huygens' mechanics had not yet been formulated in generality theory of the center of oscillation with static principles sufficient to obtain them. Johann Bernoulli, his son by showing that it followed from the law of the lever Daniel (1700-1784), and Euler developed great skill if the reversed accelerations were treated as being in using special assumptions sufficient to determine forces. This is one of the several ideas called "d'Alem- the forms of the principal modeS and to calculate the bert's Principle" in textbooks. The application of it corresponding proper frequencies. They saw that these

Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976 59 modes could be excited simultaneously, and Daniel in the strict form enunciated by Leibniz and upheld Bernoulli finally claimed that every vibratory motion by d'Alembert. Euler proposed no substitute for it, could be obtained by superimposing a sufficient but his example in the treatment of the vibrating string number of simple vibrations. He regarded this asser- has been followed in all of mathematical physics ever tion as a new law of physics, not as a provable theorem since, for to this day smooth solutions are always of mechanics. sought, and continuity is relinquished in a particular Newton and others had formulated through intrinsic problem only when smooth functions fail to solve it. differential relations the problem of determining the A great controversy about the vibrating string arose motion of certain systems that could be reduced in between d'Alembert, Euler, and Daniel Bernoulli, but effect to a single mass-point subject to a central force. it concerned analysis rather than mechanics for the Newton's famous solution of the problem of two gravi- most part. Euler obtained also the equation governing tating bodies is of this kind. Up to 1743 all who had small oscillations of taut membrane and solved it for treated more complicated systems, such as three gravi- circular and for rectangular drums. tating bodies or two linked bodies, had resorted to Daniel Bernoulli had used special devices so as to guesswork or confined themselves to particularly sim- obtain a relation between the speed of water in a tube ple, special motions, such as the simple vibrations just and the pressure it exerts on the wall of that tube. mentioned. In 1743 the first typical differential equa- Johann Bernoulli showed that his son's results could tions of motion were obtained by Johann Bernoulli be obtained in simpler but more general form by direct and Jean leRond d'Alembert (1717-1783): The latter appeal to the" ordinary principles" of mechanics, that was the first to state a partial differential equation is, the principle of linear momentum. In order to do as the law of motion of a mechanical system, namely, so he had to introduce the concept of internal pres- a heavy cord in small horizontal oscillation. In order sure, the pressure exerted by any one part of the fluid to do so he formulated a general law concerning con- upon its neighboring part, and to consider the effect strained systems, which he stated very obscurely. It of the difference of the pressures upon two sides of amounts to an assertion that the accelerations effected a thin slice of fluid. Perhaps he was influenced here by the constraints constitute by themselves an equili- by an incomplete early study of the vibrating string brated system of forces per unit mass. As Lagrange by Brook Taylor (1685-1731). Certainly he made use was to remark later, d'Alembert's principle is difficult of ideas and methods which his older brother and to use. Euler immediately extended Johann Bernoulli's he had used in their work on the statics of flexible method so as to obtain the equations governing the lines in the 1690's. motion of linked systems, and then, as if by an after- Euler saw at once that this was the way to unify much thought, he saw that the general motion of n mass- of existing mechanical theory. He was led to propose points subject toany given forces could likewise be what he called a new principle of mechanics, namely, formulated in terms of a system of differential equa- that the acceleration of each infinitesimal part of any tions referred to a single Cartesian co-ordinate system. body was equal to the resultant force per unit mass These equations, on which nearly all subsequent re- acting upon it. This is the general principle of balance searches in celestial mechanics have been based, are of linear momentum, a broad extension of Newton's called" Newtonian" in the folklore, although nothing Second Law. like them appears anywhere in Newton's work. Once On the basis of this principle Euler was able to derive he had found them, Euler used them again and again. at once the differential equations of motion of a rigid For example, he determined the effect of a harmonic body. He proved then that every such body has an driving force upon a harmonic oscillator; thus he axis of free rotation; later Johann Andreas v. Segner discovered the mathematical theory of resonance, a (1704-1777) proved that every body has at least three phenomenon which had been understood before then mutually orthogonal axes of this kind. Euler was led only pictorially. to introduce the tensor of inertia and thus to separate From this time onward Euler always formulated prob- rotational inertia from translational inertia and to dis- lems of mechanics in terms of definite ordinary or tinguish both from weight. He showed that Newton's partial differential equations. D'Alembert in his way Second Law must refer strictly to the center of mass, obtained the equation governing the small transverse which is not necessarily the center of gravity. Euler vibrations of a taut string; although he perceived the determined various special motions of a spinning body. solution in arbitrary functions, representing waves By another application of his principle of linear propagated in both directions at a determined speed, momentum Euler obtained differential equations to he.imposed other restrictions so strong as almost to determine the motion of ideal fluids. Earlier, on the forbid use of his own results. Euler, denying the restric- basis of less complete principles, he had extended and tions, exhibited explicit solutions for all cases of inter- applied the hydraulics of the Bernoullis. For example, est: strings plucked to arbitrary form, the propagation introducing what was later called the "Coriolis force", and reflection of pulses, etc. Most of these solutions he determined the pressures within a whirling turbine had discontinuous derivatives, and some were them- and established criteria of cavitation. In an even earlier selves discontinuous. By exhibiting propagating dis- essay on ballistics he had shown that a body submerged continuities Euler overturned the" Law of Continuity" in steady uniform flow of an ideal fluid would suffer

60 Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976 no resistance, a fact now usually called "the d'Alem- direction is parellel to the surface on which it acts bert paradox ". later came to be called a "shear force". Euler, repre- Euler considered also the propagation of sound in senting the beam by a deformable line, replaced the air. He imbedded Daniel Bernoulli's theory of the action of one part upon its neighbor by a force acting modes and frequencies of organ pipes within the at the junction. The components of this stress resultant, general theory of small motions of compressible fluids. the longitudinal one being the tension and the trans- By applying his general equations he was able to deter- verse one the shear force, are those that enter the mine the sequence of overtones of a conical horn. principle of linear momentum; supplementing them He obtained also the partial differential equations, by reversed accelerations thus yields equations of now called "wave equations ", which govern the pro- motion, In this way Euler finally succeeded in separat- pagation of cylindrical and spherical waves and, ing the generic principles of mechanics from the consti- finally, waves of any form, and he determined some tutive relations defining particular bodies or materials. particular solutions of them. He attempted many times The program of Jakob Bernoulli was thus achieved, to correct Newton's value for the speed of sound but though only for plane one-dimensional bodies. succeeded only in showing that the fault in it could Jakob Bernoulli had determined the variables with not be laid to mathematical mistakes, to inexact which to state a constitutive relation for an elastic mechanical principles, to effects of large amplitude, bar, but he had not applied his idea. In very early or to Newton's having considered only plane waves. work Euler had successfully derived Jakob Bernoulli's Euler came to see that he had been mistaken in claim- law of bending by assuming the bar to be composed ing that the principle of linear momentum was the of stretched fibres obeying Hooke's "law" of exten- one and only fundamental law of mechanics. From sion. In order to do so, he had had to integrate the his earliest years he had studied Jakob Bernoulli's moments of the tensions over the cross-section of the theory of bent elastic bands. He had succeeded in clas- beam, and integration requires a stress-strain relation sifying and detmanining explicitly all forms that a that is independent of the size of the element consi- naturally straight band might assume if loaded at its dered. If the relation is linear, as Euler assumed, the ends only. One product of this research was the" Euler constant of proportionality is the modulus of elasticity. buckling formula", which gives the minimum longitu- Euler had thus been forced to introduce this material dinal thrust sufficient to make a column bend. Extend- descriptor, which the folklore attributes to Young. ing work of Daniel Bernoulli, Euler had determined In his final work on elasticity Euler came to see the all the modes and proper frequencies of straight bars central importance of this modulus, so he defined it in small transverse oscillations subject to various end clearly and expressly and gave numerical estimates conditions. This mass of fine theory rested in part for it in common materials. upon a special assumption which was not a conse- Euler deserves to be regarded as the perfect theorist. quence of any general law of mechanics, and in part He mastered all existing theories, solved with finality upon balance of moments, not balance of forces, so long outstanding central problems, and found new the principle of linear momentum could not be applied applications. He improved, simplified, consolidated, directly. How could equations of motion for elastic and united what was known; he forged new concepts bars be obtained? and new definitions so as to render fruitful the general Two elements were lacking. The first of these was principles he had induced by elimination of specializ- a properly general concept of the force that one part ing hypotheses. In subject matter his papers range from of a bar exerts upon another. The second was the collections of simple rules for engineers, with tables extension of the principle of static balance of torques prepared for their blind use, through the projection so as to include the effects of motion. To obtain the and even design of machines and the systematic exploi- second, Euler had recourse to Jakob Bernoulli's treat- tation of ideas he acknowledged as being largely due ment of the center of oscillation. In this way he arrived, to others, to presentations of pure and abstract general late in life, at the second of the two fundamental princi- principles little likely to be understood by any of his ples that serve as the foundation of all mechanics: contemporaries. He was not ashamed to publish a the principle of balance of rotational momentum. Euler paper in which the mathematics was no more than expressed the two principles as integral relations: The elementary arithmetic if the results were beautiful or rate of change of linear momentum of a body equals promised to be useful; at the other extreme, he created the total force acting upon it, and the rate of change new mathematical tools and even whole disciplines of rotational momentum of a body equals the total of mathematics so as to solve problems of physics, torque acting upon it. These laws are now coming and he did not shrink from writing papers in which to be called "Euler's Laws of Motion". In contrast he treated problems of mechanics by use of analysis with Newton's Laws, they are unambiguous, explicit, that could be followed then by at most a handful of and generic. men. He.wrote with perfect candor; he was the first The first missing element had been seen by Antoine scientific author to cite the works of others in acknow- Parent (1666-1716), but his work had attracted no ledgment of their deserts rather than to point out their notice. Namely, within a loaded beam there must be mistakes or deficiencies; when he could go only part transverse as well as longitudinal forces. A force whose way on a problem, he disclosed his imperfect results

Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976 61 in the hope that others could use them; and in at idea. In justly claiming this principle for himself, least one case he published a paper which he knew Lagrange used such tactful expressions that it has come to be partly wrong and then caused to be printed to be named after d'Alembert instead. Lagrange's on the immediately following pages an explanation book is a monograph on howto derive differential of just where the error lay, followed by correction equations of equilibrium and motion by his method. of it through adjustment of the parts of his foregoing Unlike Newton's Principia, it contains scarcely any work that remained valid. Finally, he left behind him examples, applications, interpretations in physical hints toward new concepts and the beginnings of new contexts, or new results. It omits all problems not solutions he did not live to complete. In addition to easily amenable to Lagrange's methods. In the early the indirect influence through the tradition of nineteenth century it was read as the bible of mechanics, Euler's papers and books have themselves mechanics, and its use of a scalar potential inspired continued to inspire occasional researches on much good work by those who studied it. Its effect mechanics down to the present day. upon subsequent conceptions of the history of If we survey Euler's writings, we find in them all the mechanics was largely unfortunate, first because, con- elements necessary to construct a general system of centrating upon facile algebra, it failed to mention mechanics, including all kinds of materials and all the deepest work done in the eighteenth century, and forms of body, but to do so was left for his successors second because Lagrange included historical sketches to achieve. which are so capriciously lacunary as to lie in effect even if not in fact. The influence of Lagrange's history can be traced easily by later writer's uncritical repeti- Domination by Mathematical Formalism tions of his errors, some of which continue to appear in catalogues issued by merchants of antiquarian Joseph-Louis Lagrange (1736-1813) books today. Lagrange made his name in mechanics by analysis of the vibrations of a taut string loaded by many Experimentists of the Late Eighteenth Century masses. He determined the general motion and then, by passage to the limit as the masses became more The concepts developed by the mathematicians of the numerous and their separation approached zero, he eighteenth century were insufficient to construct claimed to establish Euler's solution in arbitrary func- theories for two centrally important but still special tions for the motion of the continuous string. Thus phenomena of elasticity: the response of beams to his program was akin to Huygens'. His analysis of torsion, and the oscillations of plates. the discrete string rested heavily on earlier but by him In reconsidering the equilibrium of a cantilever beam unacknowledged work of Euler; as d'Alembert Charles Augustin Coulomb (1736-1806) recognized pointed out at once, Lagrange's passage to the limit the importance of shear forces, as had Parent. Cou- is irremediably fallacious. Most of the rest of lomb developed the idea further, for he regarded the Lagrange's work in mechanics presents simple exten- rupture of a masonry pier as due to one part's sliding sions or alternative forms of Euler's results. Of that along another. He saw that different planes at one which is original, much is erroneous. It appears to and the same point suffer different stresses, and he have been given but little critical study by historians. showed that the shear stress is greatest on the planes An exception is the famous "Lagrangean equations", inclined at 45 ~ to the direction of the compressive which convert Euler's general equations for discrete load. Studying torsion by experiment, he found that systems in Cartesian co-ordinates into an invariant the twist was proportional to the torque. To explain expression, valid for- all descriptions of the system. this fact from a general theory and to calculate the Another exception is Lagrange's general formulation constant of proportionality corresponding to a given and analysis of the principle of least action, which shape of the cross section, stood as a challenge to had been proposed inaccurately by Pierre-Louis-Mor- theorists thereafter. eau de Maupertuis (1698 1759); Euler had corrected Ernst Florens Friedrich Chladni (1756-1827) deter- Maupertuis' bungling, but only for a single body. mined experimentally the nodal curves and the corres- These two achievements suffice to justify Lagrange's ponding proper frequencies for circular and square great name. plates. To explain these by theory remained an open In 1788 Lagrange published his celebrated treatise, problem for half a century or more thereafter. As MOchanique Analitique, in which he claimed to reduce Chladni himselfremarked, his plan rested upon a thor- all of mechanics to a few algebraic formulae. His statics ough knowledge of the theory of simple vibrating sys- is based on the principle of virtual work. While this tems invented by Daniel Bernoulli and Euler. principle may be traced from antiquity through the Chladni's work provides a rare example of the use work of Jordanus, and while it had been revised and of good theory for simple cases so as to guide experi- extended by Johann Bernoulli, Lagrange was the first ment in exploring phenomena which lie beyond the to formulate it in generality. Lagrange's dynamics rests range of theoretical principles as yet known. on this very same principle, extended by adding reversed accelerations according to Jakob Bernoulli's Received January 3, 1975

62 Naturwissenschaften 63, 53-62 (1976) by Springer-Verlag 1976