a rudimentary history of dynamics - semantic scholar

Modeling, Identification and Control, Vol. 30, No. 4, 2009, pp. 223–235, ISSN 1890–1328

A Rudimentary History of Dynamics

Andrew Ross

Norwegian Marine Technology Research Institute (MARINTEK), Marine Technology Centre, NO-7450 Trondheim,Norway. E-mail: [email protected]

Abstract

This article presents a condensed overview of the development of dynamics from Ancient Greece throughto the late 18th century. Its purpose is to bring an often neglected topic to the control community in asinteresting a fashion as the author can achieve.

Keywords: History of science, dynamics, mechanics, kinematics, Newtonian equations, Newton’s Laws

1 Introduction

This article describes a few key points in the develop-ment of the field of dynamics. It begins with a briefoverview of dynamics’ origins in Ancient Greece, andprogresses through the Middle–Ages and the Renais-sance, stopping at the times, places, and people thatthe author considers most relevant. Following that, itsummarises the appearance of the Principia, and pro-ceeds to explain its development into what we now callClassical Mechanics.

It should be clear that the contents are not new: theyare well known in many fields, and have been for manyyears. They are presented to the control communityin the hope that knowledge of, and interest in, the his-tory of science might be improved. Any contributionis solely in bringing the subject to an audience that isperhaps generally unacquainted with it. Not only arethe contents not new, they are also by no means com-plete or comprehensive. With those caveats, the authorhopes that some of the intended audience might findvalue in what is presented.

Take what the pre–eminent physicist and philoso-pher of science, Ernst Mach, wrote of Newton’s contri-butions (Mach, 1886):

Newton discovered universal gravitationand completed the formal enunciation of themechanical principles now generally accepted.Since his time no essentially new principle has

been stated. All that has been accomplishedin mechanics since his day has been a deduc-tive, formal, and mathematical developmentof mechanics on the basis of Newton’s Laws.1

The sentiments expressed by Mach are broadly inline with the opinions of many in academia. The pop-ularity of a belief does not allow us to conclude whetherit is factual or not.

The motivation for writing this paper stems fromthe sentiments of the great physicist, mathematician,historian of science, and polemicist, Clifford Trues-dell2. In his essay A Program Toward Rediscoveringthe Rational Mechanics of the Age of Reason(Truesdell,1968), he wrote:

The scientists, in so far as they take anynote of history at all, not only have shared thehistorians’ neglect of the later [i.e., after New-ton] mathematical development of mechanicsbut also, in the main, have ignored what thehistorians have learned about the earlier pe-riods and have rested content with Mach’swhole view or a rudimentary abstract of it.

1That general relativity has since superseded classical mechan-ics is of no importance here.

2The term “natural philosopher” ought to suffice, but unfor-tunately has not enjoyed popular or correct usage in recenttimes (Noll, 1996).

doi:10.4173/mic.2009.4.3 c© 2009 Norwegian Society of Automatic Control

http://dx.doi.org/10.4173/mic.2009.4.3

Modeling, Identification and Control

Whether reading a textbook on robotics, marine en-gineering, aerospace engineering or, indeed, on me-chanics itself, the statement above is too often applica-ble. Discussions of dynamics almost invariably beginby citing the work of Newton (1687) in his Principia,and seldom proceed further than this opus. It is asif classical mechanics arose from the genius of Newtonalone.

This hagiography does a great disservice to at leastfour parties: firstly, to the scholars whose works pre-date Newton; secondly, to Newton’s contemporariesand successors, who actually synthesised the dynamicsthat we know and apply today; thirdly, to the present–day students who desire an accurate history of dynam-ics; and fourthly to Newton himself, whose memory issullied by the misrepresentation of his work.

This neglect leads us to question why this historyis being contemplated here. The problem is that thehistory of science, a large and growing field, seems tofilter little of its knowledge to the practitioners of sci-ence. Scientific careers can be built on advanced topicswith absolutely no concept of what lies in the founda-tions. The history of a science is vital to a humbleunderstanding of that science.

Awkwardly, the history of a science can only reallybe grasped and analysed after the subject itself is suf-ficiently well understood. However, once that scienceis understood and a university degree comes to its end,there is no drive to put the learning into a historicalperspective. A modern engineering degree is then morelike an advanced tradeschool diploma than the higherform of learning and understanding that it ought to be,and that it used to be.

A cursory and often inaccurate glance at historyseems sufficient nowadays, but should it be? The de-velopment of dynamics is a much longer, halting andlaborious story that neither began nor ended with IsaacNewton.

2 Dynamics in Antiquity

An analysis of motion ought to begin with the AncientGreeks3. Since the influence of the Greeks lasted twomillennia, it is inconceivable to describe the growth ofdynamics without mentioning them.

The dominant figure in the ancient development ofdynamics was Aristotle (384 BC– 322 BC). His writ-ings (Aristotle, 330 B.C.) on this and on many othersubjects held sway over much of science for the nexttwo thousand years. Much of his reasoning on motion

3Greek philosophy is generally taken to begin with the work ofThales of Miletus (ca. 624 BC–ca. 546 BC), in modern–dayTurkey.

Figure 1: Replica of Lysippos’ Aristotle.

stemmed from the faulty concept of the classical el-ements (fire, air, water and earth). Each of these isgiven its own natural place in the world: fire at thetop; air underneath fire; water below air; and finallyearth resting beneath them all.

Whenever an element was taken from its naturalplace it would endeavour to return. This reasoning ex-plained why an air bubble breathed underwater floatsto the surface, and why a rock thrown upwards fallsback to the Earth. Each object was then a combina-tion of all of these. A feather, lighter than a rock, musthave more air than the rock, but less than the air it-self. From this line of thinking arose “natural motion”:motion that occurs due to the nature of the object. Allother motion was violent; it had a separate cause. Abrick falling to the ground would be natural, but abrick thrown through the air would be violent.

Aristotle concluded that heavier objects fall fasterthan light objects, and that this fall–rate is propor-tional to their weights: an object twice as heavy fallstwice as fast. He also reasoned that the speed of pro-gression through a medium was inversely proportionalto the density of that medium. This reasoning impliedthat the speed of progression in the void would be in-finite; thus he concluded that the very existence of avoid was impossible (Aristotle (330 B.C.), Book IV:8).

...between any two movements there is aratio (for they occupy time, and there is aratio between any two times, so long as both

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are finite), but there is no ratio of void to full.

In the same section, he wrote that if a void were toexist, heavy objects would fall at the same rate as lightones (“Therefore all will possess equal velocities. Butthis is impossible.”). He used this supposed equalityof fall rates to then say by modus tollens that a voidcannot exist. He further wrote that, in a void, therewould be no reason for a body to stay in one place ormove to another, and so motion would continue for-ever. It is often said, based on this statement, that heenunciated or foresaw a principle of inertia, but this isonly possible by a selective reading of his works.

Among the various physical questions pondered byancient philosophers, the question of why an arrow con-tinues to fly after it has left its bowstring was partic-ularly perplexing. Aristotle reasoned that the arrowdisplaced the air in front of it, which rushed behindand then pushed the arrow forwards. The idea of athing moving violently without some other thing push-ing it along the way; moving without a mover, wasentirely alien to Aristotelians. This fallacious separa-tion of natural and violent motion would haunt physicsfor two thousand more years.

The progress towards a truer representation was slowand halting. Aristotle’s worldview became ingrainedupon both Western and Arabic science and theology.His prevalence in the latter of these fields impacted theprogression of the former. Much of it became Churchdogma. By raising his theology above and beyond crit-icism, it raised a protective wall, by proxy, around hisphysics.

The lengthy dominance of Aristotle is now difficultto imagine. Even into the early Renaissance entire con-tributions on physics from philosophers would consistsolely of commentaries on Aristotle’s works: two mil-lennia after they were composed.

The 6th century Alexandrian philosopher, JohnPhiloponus (ca. 490–ca. 570), wrote extensive cri-tiques of Aristotelian physics (Philoponus, 2006), andit is here that the inklings of a modern approach todynamics can be seen.

Philoponus found little satisfaction in Aristotle’s ap-proach to motion, indeed he also found little satisfac-tion in his other approaches. In his commentaries hedemolished Aristotle’s work on both natural and vio-lent motion. For natural motion, Philoponus positedthat an object has a natural rate of fall. Fallingthrough a medium would hinder this natural rate:

But a certain additional time is requiredbecause of the interference of the medium.

He introduced a natural fall rate in the void, andsubtracted from this the effect of the resistance of the

medium. This concept allowed him to reject the Aris-totelian concept that the speeds at which objects fallat are in proportion to their weights. He did this withappeal to the same kind of experiment carried out inRenaissance Italy around a millennium later4. Philo-ponus did not believe in the equality of fall–rates inthe void. In fact he concluded that this concept waswrong. His belief was that heavier objects do fall fasterthan light ones in a void.

For violent motion, he asserted that when an objectis moved, it is given a finite supply of forcing impetus5:a supply of force that, while it lasted, would explain theobject’s continuing motion:

Rather it is necessary to assume that someincorporeal motive energeia is imparted bythe projector to the projectile...

This incorporeal motive energeia is exhausted overthe course of an object’s motion, which rests once thisexhaustion is complete. This property was internalto the body. He struck fairly close to some kind ofrudimentary concept of kinetic energy. At the veryleast, he struck close to some concept which we cannow relate to kinetic energy.

The conclusion of the sentence quoted above is:

...and that the air set in motion contributeseither nothing at all or else very little to thismotion of the projectile.

The strongest and most groundbreaking insight thatPhiloponus made was that a medium does not playa role in maintaining motion. It acts as a retardingforce. This notion was in direct opposition to Aristotle,who required that the medium should cause the con-tinuing motion. This paradigm shift that John Philo-ponus introduced allowed him to explain that motionin a void was possible. His lasting contribution is withthese qualitative analyses. His quantitative explana-tions are without merit, although these analyses res-onate through Galileo’s dynamics.

In the centuries that followed Philoponus, otherphilosophers followed in a staggering and haphazardprogression towards Newton. It would be another mil-lennium before Aristotelian motion would be disre-garded. The reasons are various, but much of them aretheological in nature. Philoponus’ writings on Trithe-ism were declared anathema by the Church, which ledto the neglect, condemnation, and ridicule of his writ-ings. Zimmerman had the following to say (Zimmer-man, 1987):

4The comparison of the speeds of falling objects, carried out byBenedetti around 1553 (commonly attributed to Galileo).

5Impetus theory, the precursor to the modern principle of iner-tia, can properly be attributed to Philoponus.

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His writings, then and later, enjoyed noto-riety rather than authority.

The inferior works on mechanics from his contem-poraries, such as Simplicius, were treated in a morefavourable light.

3 The Middle Ages

In the following centuries, the development of dynamicswas very slight. There is a pernicious popular beliefthat science stood still from the fall of the WesternRoman Empire (476 A.D.) until the Renaissance: theso called Dark Ages. While the remark may hold waterfor certain periods of the Early Middle Ages, it has nostanding whatsoever with the High and Late MiddleAges. The idea that the world of understanding stoodstill for a millennium is a hopelessly incorrect one.

Aristotle’s views, or variations on these, were anal-ysed further by the likes of the Andalusian–ArabsAvempace and Averroes6 in the mid–13th century. Thegratitude owed to these philosophers should not beunderstated. It is through their works that Philo-ponus’ thoughts were preserved: his books were notpublished in Western Europe until the early 16th cen-tury. Averroes wrote such extensive treatises on Aris-totelian physics and theology that he was nicknamedThe Commentator by Thomas Aquinas. The intellec-tual stupor existed in the West because an Aristoteliantheological worldview was dogma. Those studying me-chanics were reticent to go further than simple rein-terpretation of Aristotle, even when so much of it wasclearly wrong.

The stimulus that reinvigorated the field can betraced to the Condemnations of 1277. In this year,Tempier, the Bishop of Paris, condemned various doc-trines enveloping much of radical Aristoelianism andAverroeism, among others. This event is importantbecause the condemnation of Aristotle’s theology ledphilosophers to question the truth of the rest of hisworldview. Deviating from dogma was then, and re-mained for centuries more, very dangerous for philoso-phers, but now Aristotle’s physics were no longer pro-tected. The importance of the Condemnations led towhat Duhem (1917) called:

...a large movement that liberated Chris-tian thought from the shackles of Peripateticand Neoplatonic philosophy and producedwhat the Renaissance archaically called thescience of the ‘Moderns.’

6Ibn Bajjah and Ibn Rushd in Anglicised Arabic respectively.

Soon after, in the early 14th century, the Oxford Cal-culators7 explained, in a kinematic sense, the motion ofobjects under uniform acceleration. Importantly, thesemen did not concentrate solely on the qualitative de-scription of motion. What was previously a murky de-scription of motion became a quantitative derivation.They answered kinematic questions numerically. Whatis fantastic is that the notion of instantaneous speedwas within their grasp, even without the strong grip af-forded us by calculus. The mean–speed theorem datesfrom this period, and is attributed to William Heytes-bury8. That theorem sprung from the investigationsinto how two bodies moving along a path at differentspeeds might arrive at an endpoint at the same time(see the essay “Laws of Motion in Medieval Physics” inMoody (1975)). They were additionally responsible forseparating motion itself from its causes: the separationof kinematics and kinetics. Bradwardine9 also noted:

All mixed bodies10 of similar compositionwill move at equal speeds in a vacuum.

The statement above shows that the Mertonianswere well aware of the principle that objects of thesame composition fall at the same rate, regardless oftheir mass. The fall rates were still explained in termsof the nonsense classical elements of Ancient Greece,but they were explained. Within their work can befound thorough analyses of uniform and acceleratedmotion. Their analytical approaches to motion werewell received Europe–wide.

French priest Jean Buridan (1300–1358) was by mostaccounts the giant of fourteenth century philosopy. Heexpounded a theory that can properly be described asan early and rudimentary concept of what we now callinertia.

He posited in a similar manner to Philoponus thatthe motion of an object was internal to it, and impor-tantly recognised that this impetus does not dissipatethrough its own motion: that something else must actupon the object to slow its motion. His insights intothe implications of this were more advanced than any-thing prior. In discussing a thrown projectile, he saidthat it would:

...continue to be moved as long as the im-petus remained stronger than the resistance,

7The Oxford Calculators were a group of 14th cen-tury academics based at Merton College, Oxford, andincluded William Heytesbury, Thomas Bradwardine, RichardSwineshead and John Dumbleton.

8Bizarrely attributed to Galileo by many.9The selfsame Bradwardine spoken of in Chaucer’s Canterbury

Tales.10A mixed body is one that consists of two or more of the clas-

sical elements: fire, air, water and earth.

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0 5 10

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−6

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Horizontal Position (m)

Ver

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Buridan’s Projectile Motion

ViolentViolent and NaturalNatural

Figure 2: Albert of Saxony’s motions.

and would be of infinite duration were it notdiminished and corrupted by a contrary forceresisting it or by something inclining it to acontrary motion.

His statement is an early and rudimentary notionthat is qualitatively similar to Newton’s First Law. Heentertained this notion of infinite motion, a full threecenturies before. His talent in descriptions of the qual-itative properties was not matched by his talent in thequantitative.

Buridan’s student, Nicolo Oresme (ca. 1323–1382),developed geometrical descriptions of motion. Morethan that, he used geometry as a method of explainingthe variations of any physical quantity. As great as thiswas, he had a poorer understanding of dynamics thanhis tutor, and treated impetus as something which de-cays with motion (Wallace, 1981). Oresme’s work isa prime example of the stumbling advancement of dy-namics: it was rare that any one person could advancein all areas at once.

Albert of Saxony (ca. 1316–1390), another studentof Buridan, took impetus theory forwards in projectilemotion. For an object propelled horizontally, he rea-soned that the motion had three distinct periods. Thefirst of these was purely horizontal, where the bodymoved by its own impetus. The second was a curvetowards the ground, as gravity began to take effect.The third was a vertical drop, as gravity took over andimpetus died. Although maintaining the distinctionbetween natural and violent motion, Albert at leastcame closer to the true shape of projectile motion.

It is quite difficult to conceive the true effect that thephilosophers from the Oxford and Parisian schools hadon mechanics, and on science in general. Mechanics

had moved from indistinct qualities into defined quan-tities: if an object moves at this speed, how far doesit go in this amount of time? If an object acceleratesin this manner, what will its speed be after a givenperiod? These questions were asked and answered.

Shortly after Giovannia di Casale (d. ca. 1375) re-turned to Genoa from studying at Oxbridge, he devel-oped a geometric approach in his book “On the ve-locity of the motion of alteration” similar to that ofOresme. This work influenced the Venetian, Giambat-tista Benedetti, in his 1553 demonstration of the equal-ity of fall–rates. The influence that Casali’s geomet-ric approach wielded is evident while reading Galileo’sworks on kinematics.

An important point is then evident: the field of kine-matics had leapt ahead of dynamics. Truesdell (1968)speaks of the impact of the Calculators in the followingglowing terms:

In principle, the qualities of Greek physicswere replaced, at least for motions, by thenumerical quantities that have ruled Westernscience ever since.

While kinematics was becoming more and more ca-pable of describing both uniform and accelerated mo-tion, and was able to quantify these analytically, nu-merically and geometrically, philosophers remained un-able to explain the why behind them. The causes ofmotion, now separate and distinct from kinematics,were not very much closer to being discovered. Thissituation changed very little until the late 16th cen-tury.

4 Galileo

Galileo Galilei (1564–1642), shown in Figure 3, soughtthese causative descriptions of motion: he was the firstof the modern dynamicists. The Italian was well–readin the workings of both the Parisian and Mertonianschools. From these, he set out into the still poorlyunderstood field of kinetics.

To move forward, he examined the most successfulof the ancient sciences: Archimedes’ hydrostatics. Hetook those principles as inspiration to examine the mo-tion of a falling object. He utilised no mixed–bodytheory of matter. Instead, he treated bodies, and themedia through which they travel, in terms of their den-sities11.

Archimedes’ propositions explain the forces of buoy-ancy in equilibrium: they detail where an object willrest in a body of water. Galileo extended these prin-ciples from static into dynamic concepts. Archimedes

11or, rather, their specific gravity.

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Figure 3: One of Sustermans’ portaits of Galileo.

explained the behaviour of bodies and their natural po-sitions of rest. Galileo took this notion and applied itto bodies in motion. His monumental postulation wasthat buoyancy, in addition to determining a body’s po-sition of rest, furthermore determines how fast a bodywill reach that position of rest. He used this force ofbuoyancy to try to explain why objects fell at the speedthey did.

It is wrong to say that he devised a dynamical lawbased on static principles. His theorems are a general-isation of Archimedes’ static principles, which are thenderivable from Galileo’s: the converse is untrue.

These notions were not wholly new. Instead of us-ing a ratio of weight to resistance in order to explainmotion, Galileo described it as a natural motion fromwhich was subtracted the effect of the medium. In-stead of having velocity determined by the ratio of abody’s weight to the medium’s resistance, it was to bedetermined by a natural value minus some part due tothe resistance of the medium. The approach, ingeniousthough it was, led to no hoped for grand principle.

The comparison between Galileo and Avempace iscommonly drawn, as Avempace had postulated thesame kind of thing: discarding the Aristotelian ra-tio. Galileo was certainly aware of Avempace’s work,through what Averroes wrote of it12. It is unfair tosay that Avempace was the originator of this sort ofanalysis, as it predates him by hundreds of years. Thistheory again goes back to John Philoponus, who wasalso well known to Galileo. Additionally, Avempacedid not postulate Galileo’s explanations for the causesof motion (i.e., a dynamic buoyancy law).

A note should be made on his supposed discovery

12In his early notes, Galileo cites Averroes ahead of all,save Aquinas. See the essay “Galileo and the DoctoresParisienses” in (Wallace, 1981).

of the equality of fall–rates. Galileo did not make thisdiscovery. The story is that in 1589 he dropped variouscannon balls from the Leaning Tower of Pisa, and thusthe world came to know that all objects fall at the samerate. The story is wrong on several counts. Firstly, thisexperiment does not even demonstrate equal fall rates:it only shows that objects of the same composition fallat the same rate, independent of their weight. Sec-ondly, in 1589, Galileo did not believe in the equalityof fall–rates. His notes from this period (Galileo, 1590)state that objects of equal density fall at the same rate,but that denser objects fall faster than less dense ones:

I say therefore that in a vacuum, heav-ier bodies would descend more rapidly thanlighter ones, because the excess of the heav-ier bodies over the medium would be greaterthan the excess of the lighter ones.

That bodies of the same density fell at the same ratehad been stated already by Bradwardine two hundredyears before. Thirdly, the selfsame experiment had al-ready been performed by Giambattista Benedetti yearsbefore, and his work was known to Galileo.

It is difficult to ascertain when Galileo concludedthat all objects fall at the same rate. He withheldpublication on this subject for many years. The initialcause of his withholding was his own desire to bringthe subject to a completion before revealing it. Thelater cause was the restrictions placed on him by theInquisition. He knew of it by 1604, as he revealed itin correspondence with a confidant. He felt betrayedwhen a friend of his mistakenly revealed it to the worldin the early 1630s, and only published anything on thematter towards the end of his life.

For all Galileo’s effort, he never satisfied himselfwith his explanations of the causes of motion. In Dia-logues Concerning Two New Sciences (Galileo, 1638),towards the close of his life, he sadly confessed to thisfailure. After listening to the thoughts on dynamicsvoiced by Simplicio and Sagredo, Salviati, proponentof Galileo’s philosophy, makes the comment:

Now, all these fantasies, and others too,ought to be examined; but it is really notworth while. At present it is the purposeof our Author merely to investigate and todemonstrate some of the properties of accel-erated motion (whatever the cause of this ac-celeration may be)...

Here Galileo resigns himself to never finding whathe sought. His characters instead progress throughthorough discussions of the kinematics of motion alone:

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Salviati and Sagredo dragging Simplicio13 by the coat-tails into modern science:

...we have decided to consider the phenom-ena of bodies falling with an acceleration suchas actually occurs in nature and to make thisdefinition of accelerated motion exhibit theessential features of observed accelerated mo-tions.

Over the course of the rest of the book, Galileo setsforth his definitions of uniform and accelerated mo-tion in lightning fast demonstrations. The topics ofdiscussion then go through motions of various things,especially that of projectiles. This work sounded thedeath knell of Aristotle’s physics. In discussing a bodythrown upwards, then falling back downwards, Simp-licio voices the two thousand year old distinction be-tween natural and violent motion. Sagredo replies:

...this distinction between cases which youmake [i.e., violent and natural] is superfluousor rather non–existent.

Much is said about Newton unifying motion in theheavens and motion on Earth: that is, recognising thatthe laws apply equally so to the orbit of a massiveplanet as they do to the fall of a tiny apple. Very littleis said about unifying natural and violent terrestrialmotion, and yet it took two millennia of thought be-fore the two were recognised as one and the same. Theseparation of the two had permeated Western sciencefor two thousand years, and it was Galileo who demon-strated the insight to finally and permanently demolishthis concept.

His dynamics were noteworthy and had a very largeinfluence over his successors, but the true contributionof Galileo is in his kinematics, not in his dynamics. Hecould explain what he observed: uniformly acceleratedmotion, but he could not explain the causes behind it.While he failed, he set a great example for his suc-cessors: one that was well learnt and will never beforgotten. His dynamics were laden with no mysticalindistinct properties. They were laden with definitionsand analyses.

Aristotelian dynamics had been staggered by manydeserved blows, but had kept standing, in variousposes, for two thousand years. Galileo delivered thecoup de grace, putting it to its long overdue rest.

By the mid 17th century, dynamics was understoodin the murkiest of ways. A myriad of problems, each re-quiring its own ingenious solution, was solvable almostsolely by special cases. Much of this problem solving

13The name is a portmanteau of Simplicius, the classical Aris-totelian, and the word for simpleton.

was needed before any general laws of dynamics couldfinally be grasped. Today we learn the principles, andthen how to apply them. The developers of the fieldsolved extraordinarily difficult problems, and lots ofthem, before any true governing principles were found.We learn the principles and then tackle problems, butthe physicists had to tackle the problems before theycould see the principles.

The giant of mechanics in the years leading upto Newton was the Dutchman, Christiaan Huygens(1629–1695). He was the first to explain oscillations ofa finite pendulum, which he did so for a special case.His writings on solid body collisions had a monumentaleffect on the world. He observed that after two solidbodies collide, their collective speed may well be in-creased or decreased, but their collective momentumremains the same: perhaps the first true expressionof the conservation of momentum14. He furthermorerecognised that in rigid body collisions, the centre ofgravity of the system remains in uniform motion: ahugely penetrating notion.

5 Sir Isaac Newton

Sir Isaac Newton (1643–1728), depicted in Figure 4,made contributions to virtually every area of naturalphilosophy, mathematics, optics and astronomy. Hismonumental publication, Philosophæ Naturalis Prin-cipia Mathematica, usually called The Principia inshort, was published in 1687. It is likely the most in-fluential book in the field of classical mechanics, yet islittle read. Its purpose was set forth in its preface:

...mechanics will be the science of motionresulting from any forces whatsoever, and ofthe forces required to produce any motion...

Newton set out to explain phenomena throughoutthe universe. What lay within was to apply every-where, and to every process. The trajectory taken bya cannon ball was to be governed by the same lawswhich governed the orbits of the planets.

As the start of his work, he states his definitionsof mass, momentum, inertia and forces, both throughcontact and at a distance. He then states his laws:

First Law Every body perseveres in its state of rest,or of uniform motion in a right line, unless it becompelled to change that state by forces impressedupon it thereon.

14Descartes is often credited with this concept, but his notionwas the bulk of an object times its scalar speed. His rationalewas entirely different too.

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Figure 4: One of Kneller’s portaits of Newton.

Second Law The alteration of motion is proportionalto the motive force impressed; and is made in thedirection of the right line in which that force isimpressed.

Third Law To every action there is always opposed anequal reaction: or the mutual actions of two bodiesupon each other are always equal, and directed tocontrary parts.

It is broadly divided into three books, each of whichalone would eclipse almost any other. Books One andTwo are titled Of the Motion of Bodies, being splitinto two exhaustive analyses. The third is titled TheSystem of the World.

The first book analyses motions in a void. From hislaws, he analyses a multitude of motions, such as ellip-tic, parabolic and hyperbolic orbits around some focus.He investigates the forces that maintain these, i.e. thecentripetal forces. Universal gravitation is introduced.After showing how point masses behave in the void un-der gravitation, he demonstrates that finite bodies canbe treated as such. Kepler’s Laws follow directly.

The first book organised and systematised principles,some of which were at least dimly understood before,but these principles had never been organised togetherinto a system of analysis for application everywhere.

The second book sets out to explain motion onEarth, where motion does not occur in a void: hesought the details of motion in resisting media. It ishere that Newton departs from his program of deduc-ing physical behaviour based on his laws: he finds but

little use for them. For example, in all his treatmentsof fluidic motion he finds no room to apply his princi-ple of momentum. In contrast, he conjures ingenioushypotheses to explain a myriad of things ranging fromprojectile motions to the speed of sound in air. Thisbook is a testament to Newton’s towering stature asa mathematician and dynamicist. The second book ofthe Principia is almost entirely new. The scholium ofthe first section of it reads:

But, yet, that the resistance of bodies is inthe ratio of the velocity, is more a mathemat-ical hypothesis than a physical one.

This sentiment is applicable to much of the hypothe-ses in the book. Today it is mostly forgotten. The bookis dominated by hypothesis after hypothesis, with New-ton displaying his flair for creative solutions: often pre-cise, often an excellent approximation, but also oftenwrong and today of only historical value.

There are veins of gold hidden within. His observa-tion that fluidic resistance is proportional to the squareof velocity can be found, as can the first description ofinternal fluidic friction:

The resistance arising from the want oflubricity in the parts of a fluid is, cæterisparibus, proportional to the velocity withwhich the parts of the fluid are separated fromeach other.

That most of the results were incorrect cannot be acriticism of Newton either as a physicist or mathemati-cian. The contribution of this book is immeasurable.For instance, it constitutes the beginning of fluid dy-namics, and studied many of its problems for the firsttime. From his efforts, his contemporaries and suc-cessors were gifted with a bridgehead from which toattack these subjects in earnest. A myriad of poten-tial motions through fluids is contemplated. The bookis the staging point for hydrodynamics. Newton con-templated which hullform might pass through the wa-ter with least resistance, introducing an optimisationproblem that found application throughout the 19thcentury.

The third book set forth his solutions to problems incelestial dynamics, with great success. Kepler’s Lawsof Planetary Motion had resulted from Newton’s own,and he performed exhaustive analyses of the Solar Sys-tem.

The deficiencies in the Principia are little discussed.To the modern scholar, it is often impenetrable andconfusing; the language of mathematics having evolvedso much since then. A common remark made about thePrincipia is that Newton strangely resorts to geometri-cal methods instead of his own calculus. Newton does

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not use his notation of fluxions, but even as soon as wearrive at Lemma II of Book I, the notion of calculus ispresent, if in an unfamiliar form.

For rigid body mechanics, there is no treatment ofrotation. Although Newton says that the spinning top“does not stop spinning except insofar as it is slowedby air.” there is no justification given. His statementappears directly after his statement of the First Law,but this law cannot tell us anything of the spinning top.Newton might have perceived that the top continues tospin, just as it would continue in linear motion if so im-pelled, but it is not possible to explain the spinning topusing what is within the Principia. There is certainlyno treatment of angular momentum. The motion of arigid body cannot be described by the methods givenin the Principia.

There is no treatment of flexible bodies, such as thecatenary curve or the vibrating string, nor is there anyanalysis of the finite body pendulum. No equationsof motion appear for systems of more than two freemasses, or one constrained. A prime example of thefield’s infancy is the three–body problem. Newton at-tempted to solve this problem, but the contents of thePrincipia are insufficient to do so. He devised insightfulapproximations and valid inequalities, but the three–body problem was insoluble from his principles. Histalent in this area is evident, as his work would not besurpassed until the mid–18th century by the efforts ofEuler and Lagrange.

That Newton did not solve all of mechanics’ prob-lems is not a criticism at all, but only part of a clear–headed appraisal of what he did do. His achievementswere monumental. He ought not to be credited withthe completion of classical mechanics, but rather itsbeginnings.

In the century following The Principia’s publication,the field of mechanics swelled immensely. For all thecredit given to Newton, the world ought to be equallygrateful to his contemporaries and successors, espe-cially Leonhard Euler, the Bernoullis Jakob and John,and Joseph Lagrange. These are the men who synthe-sised what we now apply today.

6 Newton’s Contemporaries andSuccessors

If The Principia contains no treatment of angular mo-mentum, contrary to popular belief, then where andwhen did this law arise and who discovered it? Theanswer is difficult to ascertain, as the principle was ap-plied for many years before it was recognised for whatit was. For a comprehensive analysis, see Whence thelaw of moment of momentum? in (Truesdell, 1968).

Figure 5: Jakob Bernoulli: towering mathematicianand physicist of the 1700s and early 1800s.

The law of angular momentum is commonly treatedas a consequence of linear momentum, but that is nomore true than the common statement that Newton’sFirst Law can be ascertained from his Second. Thisapproach works for special cases only: it is not true ingeneral. The Newtonian equations cannot contemplatedeformable bodies or motion of a continuum withoutsevere restriction. Angular momentum is a law of me-chanics independent from any other. It took most ofthe 18th century for this to be realised.

In the Acta Eruditorum of 1686, Jakob Bernoulli(1654–1705), shown in Figure 5, analysed the motionof a pendulum using the ancient law of the lever: i.e.,by balancing the moments. By applying this staticproblem to the dynamic problem of the pendulum, hesought a new methodology for mechanics. His attemptat this stage was flawed, but was published in correctform in 1703 (Bernoulli, 1703). His efforts led him tointroduce many concepts which are now elementary, oreven obvious, today.

By balancing the moments, the law of the lever isfound. This static equilibrium condition was gener-alised into a dynamic equilibrium. Jakob Bernoulliwrote that the force of the lever can be regarded asequivalent to the acceleration per unit mass reversedin sign, thereby restoring a static condition. His wasthe earliest proper explanation of inertial force, andis a progenitor of what is now by convention calledD’Alembert’s Principle.

Jakob Bernoulli’s statement of the moment of mo-mentum can not be considered a consequence of apply-

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ing Newton’s Second Law. Its first appearance, flawedthough it was, predates the publication of the Prin-cipia by a year. To emphasise: the appearance of thelaw governing angular motion predates Newton’s 2ndlaw, which governs only linear motion. That said, it isonly to be found and understood with great difficulty.The impenetrability of Jakob Bernoulli’s work is evi-dent, since it took many years before the genius withinwas recognised and developed by Euler.

Through analysing the catenary curve15, it wasJakob Bernoulli who first recognised that solutionscould be derived by balancing forces applied to in-finitesimal portions of the cable, and furthermore thatthe selfsame solutions could be derived by balancingthe moments acting on those infinitesimal portions:two essential principles of mechanics are equivalent forcertain systems. This apparent equivalency caused thegreat physicists of the 18th century to seek, in vain, thesingle unifying principle through which all of mechanicscould be analysed.

Using this work on the catenary as a springboard,Jakob Bernoulli vaulted into an analysis of elasticbeams: the bending of finite bodies, which he pub-lished in the Acta Eruditorum of 1694. Here he recog-nised that the balancing of forces or moments alonewas insufficient to the task. Between each infinitesimalstretch, there must be a contact force and moment.

In addition to these mammoth contributions, JakobBernoulli was the first to state how the motion of aconstrained system can be analysed. Given the con-straints, propose the forces which maintain these con-straints. The motion of a system of constrained massescan then be analysed. Seemingly obvious today, theidea finds no ancestor before Jakob Bernoulli.

If it is typical to elevate some beyond their trueachievements, it is equally typical to undermine thosewith achievements beyond measure. The laws, equa-tions and principles named after Leonhard Euler(1707–1783) devastate Stigler’s Law of Eponymy16,and yet he is the very reason that the law applies vir-tually everywhere else. It is through Euler that muchof dynamics was delivered to the modern world.

Between 1747 and 1750, Euler took his own works onconstrained systems, and applied them to the three–body problem. In this work, he wrote:

The foundation... is nothing else than theknown principle of mechanics, du = pdt, ...wecan see that this principle holds equally foreach partial motion into which the true mo-tion is thought of as reduced.

15The shape that a thin hanging cable assumes under its ownweight.

16“No scientific discovery is named after its original discoverer.”

Euler is here saying that what had been found wasan approach that applied to any dynamic process, andthat it additionally applied to every part of that pro-cess. It was simply not understood prior to Euler’spaper. In retrospect it seems too obvious to even men-tion. This retrospection emphasises the difficulty ofanalysing the history of science: today, it is thoroughlydifficult not to see this principle as self–evident in New-ton’s writings. What is obvious to us today is supposedto have been obvious then, but that is wrong.

In Euler’s paper, Discovery of a New Principle ofMechanics (Euler, 1750), he wrote down the following:

Fx = Max, Fy = May, Fz = Maz.

Following his statement of the new principle, hederived the tensor of inertia by taking the momentsabout the centre of gravity. By these equations, Eu-ler claimed, all mechanical problems could be solved.That we now call these equations Newton’s Second Lawis immaterial. In the words of Truesdell:

...they occur nowhere in the work of New-ton or of anyone else prior to 1747. It istrue that we, today, can easily read them intoNewton’s words, but we do so by hindsight.

Although Euler initially believed that the issue wasresolved, he shortly came to realise his mistake. Theprinciple of angular momentum lay hidden. The fullclassical equations would not be written down for an-other two decades.

The rotations of even a rigid body were problem-atic, let alone of a system of particles or continuum.That a rigid body could rotate ad infinitum was dimlyperceived for almost a century before it was properlyexplained. As mentioned, Newton’s spinning top is akey example, but we simply cannot admit stabs–in–the–dark. Throughout the early to mid 18th century,physicists had been unable to explain rotational motionin more than a single axis.

Euler contemplated the problem in the early 1730s,but did not approach it again for another decade. Thedriving force was his work on naval architecture in Sci-entia Navalis (Euler, 1749). Here he was forced to dealwith oscillations very different from the simple planartype. He hypothesised that each body has three or-thogonal axes about which it may rotate.

The first rigorous treatment of these axes was by theHungarian physicist, Jan Andrej Segner (1704–1777).He proved that free rotation is possible through a min-imum of three individual axes, there being more thanthree for special cases of symmetry (spheres, etc.). Eu-ler recognised the strength of Segner’s reasoning, andwas the first to reason that these axes all had to passthrough the centre of gravity.

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1776 saw the birth of a foundation of classical me-chanics. It was in this year that Euler published hisFirst and Second Axioms (Euler, 1776):

F = P, P = Mv

L = H, H = Iω.

At last, the road devised by Newton, and hewn bymany, had been paved by Euler. The laws of vectorialmechanics were understood and formulated then justas they are today.

7 The Indirect Approach

Newton’s approach takes force and momentum as itsbasis. It is often called the direct approach, or vectorialdynamics. D’Alembert wrote what is anything but ashining endorsement of the direct approach in his bookTreatise on Dynamics in 1743:

Why should we appeal to that principleused by everybody nowadays, that the ac-celerating or retarding force is proportionalto the element [i.e. differential] of velocity, aprinciple resting only on that vague and ob-scure axiom that the effect is proportional tothe cause? ...we shall be content to remarkthat the principle, be it true or be it dubi-ous, be it clear or be it obscure, is useless tomechanics and ought therefore to be banishedfrom it.

The lionisation of the Newtonian approach by theBritish certainly was not quite mirrored everywhereon the continent. In contrast to the direct approach,and with equal validity, the scalar quantities of energyand work can serve as a basis for an approach calledthe indirect approach, or analytical dynamics (WilliamsJr., 1996).

The history of analytical dynamics is just as cloudyand obscure as the development of Newton’s Laws.Gottfried Wilhelm Leibniz (1647–1716) is the earli-est true standard bearer for using energy and work asthe bases for mechanical principles. He posited thatthrough any process, a vis viva (living force) is pre-served. This term he equated to mv2 and so Leibniz’sliving force is just twice the kinetic energy. He believed,and so did his contemporaries, that conservation of visviva contradicted the Cartesian and Newtonian notionof conservation of momentum. Leibniz wished to usethis vis viva, along with his dead force (potential en-ergy), as the basic principles of mechanics. This “liv-ing” force irritated the delicate sensibilities of many,and with good reason. A living force seemed to invite

the teleological and theological qualities of the ancientsciences to return from the dead.

One must question why Leibniz objected so muchto the Cartesians’ and Newtonians’ use of momentum.For them, landmark results were already inbound usingthe principle of momentum and its conservation. Whatwas special about this vis viva, and why did velocityappear twice? These notions stem from Galileo’s TwoNew Sciences.

In his final Dialogue (Galileo, 1638), Galileo madekey observations related to the indirect approach.Falling from a given height, a body acquires a veloc-ity that is precisely the same velocity required to raisethe body back to the given height. Since the squareof the velocity acquired is proportional to the height,it seemed reasonable to surmise that v2 has a link tosome fundamental property of motion. Running alongthe same vein of gold, he noted that velocity acquiredby a body rolling down an incline is only influenced bythe height of the fall and not by the inclination itself:he recognised that the velocity was independent of thepath.

Leibniz considered many cases using his live anddead forces. Indeed, many problems can be solvedby examining the energy quantities at key points. In-stead of the Newtonian momentum, and its alterationthrough impressed forces, Leibniz considered kineticenergy, and its alteration through the work done byimpressed forces. The mathematics behind what layahead was undeveloped in his day, and the true leapforward for the indirect approach would have to waitmany years from Leibniz, until it found its home withEuler and Lagrange. Before the indirect method couldcome to fruition, the calculus of variations was re-quired.

Along those lines, an interesting period in this devel-opment is the brachistochrone challenge of the late 17thcentury. Brachistochrone is a portmanteau of brachis-tos and chronos: Greek for shortest and time respec-tively. The problem is to take a bead on a frictionlesswire, acted upon only by gravity, and to then deter-mine the quickest possible route between two points Aand B. It can be seen in Figure 6. The first versionof the problem was posed by Galileo in Two New Sci-ences (Galileo, 1638), but his solution was incorrect,mistakenly thinking that the solution was the arc of acircle. He at least recognised that it was not a straightline.

The first person to solve the problem was JohnBernoulli. He posed this problem to the mathemati-cians of Europe as a challenge in the Acta Eruditorumof 1696.

In choosing the wording of his challenge, Bernoulligave an unmistakable hint at how he had solved the

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Figure 6: John Bernoulli’s Brachistochrone.

problem. He invoked both Pascal and Fermat. Pas-cal had offered prizes for challenges on the cycloid fourdecades prior, while Fermat showed that light alwaystakes the path of least time (Fermat’s Principle). Thesolution to John Bernoulli’s challenge was the former’scurve, and his method was by applying Fermat’s prin-ciple.

Solutions soon arrived from his elder brother Jakob,Leibniz, de L’Hopital and an anonymous one fromNewton17; all showing the solution to be the cycloid.

This challenge led to a clash between the fragile egosof the two Bernoullis. Following publication of the so-lutions to the brachistochrone problem in 1697, JakobBernoulli posed a more difficult version, again in theActa Eruditorum. The first version sought the mini-mum time to a certain point. Jakob Bernoulli insteadposed a problem to minimise the travel time to a ver-tical line. That is, to find out which of all the possiblecycloids reaches the line first. The reposed problem isdepicted in Figure 7.

This problem was quickly tackled by both Bernoullis,Leibniz and Euler. The answer to the problem is thecycloid which passes through the line at a right angleto it. It is not, however, the answer to the problemwhich makes this challenge especially noteworthy.

The second version of the brachistochrone was obvi-ously another minimum time problem, but now it wasa problem to be solved by contemplating all possiblepaths. It was hardly the first optimisation problem to

17Though anonymous, John Bernoulli realised its author, andremarked in a letter that “we know indubitably that theauthor is the celebrated Mr Newton; and, besides, it wereenough to understand so by this sample, as ex ungueLeonem.”(Whiteside, 2008). The phrase is usually given in-correctly as “tanquam ex ungue leonem.”

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Figure 7: Jakob Bernoulli’s Brachistochrone.

be contemplated, but it was the particular one which,in being solved, led to the development of the calculusof variations. In the years of the early 18th century,the calculus of variations was formalised and organisedby the likes of John Bernoulli, D’Alembert and Euler.

It was during this timeframe that D’Alembert andEuler finally generalised Jakob Bernoulli’s inertialforce. They formalised the pre–existing principle andshowed that the principle of virtual work appliedequally to bodies in motion. Mechanics was then giftedwith a single variational principle.

The furtherance of the attempts to use energy andwork to ascertain physical principles is due to the workof the Frenchman, Pierre de Maupertuis (1698–1759).He posited that in many processes, an action is min-imised, with this action being defined according tothe process. His perceptive qualities were remarkable.Most of his definitions of action found no use, but oth-ers did. For light, he posited in a 1744 paper, Agree-ment of several natural laws that had hitherto seemedto be incompatible, that this action was the integral ofthe speed over the path taken. With this principle, hestruck gold, deriving Snell’s Law from an indirect ap-proach. His genius here mirrors what John Bernoullidid in his brachistochrone solution.

This principle of least action became ubiquitousthroughout much of mechanics. In the same year, Eulerpublished his own results on the matter. He posited itin a far more general and accurate way, and applied theintegral of the momentum of a body over its path trav-elled, giving the reduced action, i.e., the action withkinetic energy alone. In the following years, he appliedthe same to static problems, taking variations of thepotential energy. This method lead him to the classi-cal result that a body, or a system of bodies, at rest

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always lie at a minimum of potential energy.The thrust in this direction was taken up by

the French–Italian mathematician, Joseph Lagrange(1736–1813). A contemporary of Euler in his lateryears and common collaborator, the two formulated theEuler–Lagrange equation, bringing the formulation ofthe indirect method on by leaps and bounds. Lagrangewas the true champion of this method throughout hisMecanique Analytique (Lagrange, 1788). In addition,he recognised the usefulness of generalised coordinates,giving the method a towering strength: the invarianceof the method to coordinate changes.

The future development of this field through Hamil-ton and Jacobi is rather well understood, and so hereis a good point to close this history.

8 Conclusion

This article summed up, as concisely as the authorcould achieve, the development of the broad field ofdynamics, progressing speedily from Aristotle, throughPhiloponus, on to Galileo, and Newton, and finishing inthe details of how the great minds of the 18th centurysynthesised many of the modern principles of mechan-ics that still serve humanity well today.

The deficiencies of this article are numerous. It isperhaps placed in too smooth a narrative, implyingthe development of dynamics occurred along a singletimeline, but that is emphatically not the case. Forexample, although the article transitions from Galileoto Newton, it is wholly unclear if Newton benefitedfrom Galileo’s work. Were the essay rather to havelooked to Newton’s inspiration, it would more likelyhave looked back to Apollonius of Perga, and how hisworks were used by Newton. Such an article wouldbe of a wholly different nature to the one presentedhere, which attempted to bring a short and reasonablyconcise history to a new audience.

References

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Duhem, P. Research on the history of physi-cal theories. Synthese, 1917. 83(2):189—200.doi:10.1007/BF00413756. Reprinted May 1990.

Euler, L. Scientia Navalis. Birkhauser Basel, 1749.Written ca. 1738, edited by C. Truesdell.

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Galileo. De Motu. University of Wisconsin Press, 1590.Trans. S. Drake., 1960.

Galileo. Dialogue Concerning Two New Sciences. El-sevier, 1638.

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Mach, E. Die Mechanik in ihrer Entwicklung:Historisch–kritisch dargestellt. 1886.

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Newton, I. Philosophæ Naturalis Principia Mathemat-ica. Prometheus Books, 3rd ed. 1995 edition, 1687.Translated by Andrew Motte.

Noll, W. On the past and future of natural phi-losophy. Journal of Elasticity, 1996. 84(1):1–11.doi:10.1007/s10659-006-9068-y.

Philoponus, J. Ancient Commentators on Aristotle,chapter On Aristotle’s Physics. Gerald Duckworth& Co. Ltd., 2006.

Truesdell, C. Essays in the history of mechanics.Springer–Verlag, Berlin, Germany, 1968.

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Zimmerman, F. Philoponus and the Rejection of Aris-totelian Science, chapter Philoponus’ Impetus The-ory in the Arabic Tradition, pages 121–129. CornellUniversity Press, 1987.

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