A DISSERTATION ON THE DEVELOPMENT 
 
 OF THE 
 
 SCIENCE OF MECHANICS 
 
 BEING A STUDY OF THE CHIEF CONTRIBUTIONS OF ITS 
 EMINENT MASTERS, WITH A CRITIQUE OF THE FUN- 
 DAMENTAL MECHANICAL CONCEPTS, AND A 
 BIBLIOGRAPHY OF THE SCIENCE 
 
 EMBODYING RESEARCH SUBMITTED IN PARTIAL FULFILLMENT OF 
 
 THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF 
 
 SCIENCE IN NEW YORK UNIVERSITY, 1908 
 
 BY 
 
 DAVID HEYDORN RAY 
 
 BACHELOR OF ARTS, COLLEGE OF CITY OF NEW YORK; BACHELOR OF SCIENCE AND MASTER OF 
 
 ARTS, COLUMBIA UNIVERSITY; CIVIL ENGINEER, NEW YORK UNIVERSITY; INSTRUCTOR 
 
 IN THE COLLEGE OF THE CITY OF NEW YORK; CONSULTING ENGINEER 
 
 », 1 » 
 
 LANCASTER, PA. 
 
\\^ 
 
CONTENTS. 
 
 INTRODUCTION. 
 
 NATURAL SCIENCE. 
 
 PART I. 
 
 BEGINNINGS IN MECHANICS. 
 
 The Period of Antiquity, iogoo B. C. to 500 A. D. 
 
 PAGE. 
 
 1. The Science of Mechanics 8 
 
 2. Science in Antiquity 12 
 
 3. Archimedes 19 
 
 PART II. 
 The Medieval Period, 500 A. D. to 1500 A. D. 
 
 1. The Medieval Attitude toward Science 33 
 
 2. The Influence of Arabian Culture 39 
 
 3. The Period of the Renaissance 42 
 
 4. The Contribution of Stevinus 45 
 
 5. The Contribution of Galileo 52 
 
 PART III. 
 
 MODERN MECHANICS. 
 
 The Modern Period, 1500 to 1900. 
 
 1. Characteristics of the Modern Period 60 
 
 Huygens 63 
 
 2. Newton 69 
 
 3. The Contributions of Varignon, Leibnitz, the 
 
 Bernoullis, Euler and D'Alembert 77 
 
 4. The Contributions OF Lagrange AND Laplace. . . 106 
 
 iii 
 
iv contents. 
 
 5. Recent Contributions. The Law of Conserva- 
 
 tion 117 
 
 6. The Ether. Energy. Dissociation of Matter. 125 
 
 PART IV. 
 
 CONCLUSION. 
 
 1. Conclusions and Critique of the Fundamental 
 
 Concepts of the Science 133 
 
 2. Tabular View of the Development of Me- 
 
 chanics 134 
 
 3. Bibliography 146 
 
INTRODUCTORY CHAPTER. 
 NATURAL SCIENCE. 
 
 The word mechanics, though it indicated of old the study 
 of machines, has long since outgrown this limited meaning 
 and now embraces the entire study of moving bodies, both 
 large and small, suns and satellites, as well as atoms and mole- 
 cules. The phenomena of nature present to us a world of 
 change through ceaseless motion. Mechanics is the "Science 
 of Motion" as the physicist Kirchhoff has defined it, and has 
 all natural phenomena for its field of investigation. Why 
 things happen and how they happen are the questions that 
 here present themselves. 
 
 It was a long time before the distinction between "why" 
 and "how" was drawn, but when once the question "why" 
 was turned over to the metaphysician and the theologian, 
 and attention was concentrated on "Aow," then mechanics 
 made progress. Men then began to discover "how things 
 go," and to try their hand at invention. 
 
 It is not the purpose here to touch upon either the meta- 
 physical or the psychological aspect of phenomena, nor the 
 mystery of vegetable or animal activities, but to trace the 
 development of Mechanics as a science from the earliest records 
 to the present time, first analyzing the contributions made to 
 it, step by step, and then touching upon their use and value. 
 
 As the French philosopher Comte first noted, three stages 
 are apparent in the growth of human knowledge. In the 
 first stage, man ascribed every act to the direct interposition 
 of the Deity, in the second he tried to analyze the Deity's 
 motives and so tried to learn "why," while in the third, men 
 came to regard the inquiry "why" as profitless and ask "how." 
 In this last stage, they accept the universe and are content 
 with learning all they can of how it goes. With this last 
 attitude, called positivism, science flourishes. Out of it grew 
 the notion of utilitarianism, — the devotion of all energies 
 
2 THE SCIENCE OF MECHANICS. 
 
 toward the improvement of the conditions of life on earth. 
 Though this later philosophy cannot entirely justify itself, 
 it is commonly identified with the scientific attitude of mind. 
 
 By the long road of experience, by blunder, trial and experi- 
 ment, men first gathered, it seems, ideas of things that appear 
 always to happen together as by a necessary sequence of 
 "cause and effect." Of the stream of appearances continu- 
 ously presenting themselves, some are invariably bound to- 
 gether, being either simultaneous or successive, the presence 
 or absence of the others apparently making no difference. 
 Those having no influence may reasonably be ignored and 
 eliminated as of no consequence. In this way, the method 
 of abstracting from the great multitude of phenomena those 
 that are mutually dependent seems to have been evolved. 
 
 Barbarous peoples do not possess a clear notion of sequence 
 or of the interdependence of things. They are prone to regard 
 the consequence of an action as accessory, as something done 
 by an invisible being or a god. An action is performed by 
 them, and what is commonly called by us the result is con- 
 ceived by them as the simultaneous act of their god. Their 
 medicineman is thought of, as one proficient in the art of 
 appealing to the moods and whims of their gods propitiously. 
 Even the Greeks and Romans, the founders of our European 
 civilization, were accustomed to be guided in affairs of state 
 and of the home by omens, by the flight of birds, and the 
 inspection of the entrails of animals, — most naive examples 
 of traditional error in the interdependence of simultaneous 
 phenomena. 
 
 Things which we now understand to have not the slightest 
 relation with each other were systematically confounded by 
 the ancients. For thousands of years belief in astrology was 
 general in Europe and the universahty of the belief is at- 
 tested by such words as ill-starred, disastrous, consider and 
 saturnine, all of which are manifestly of astrological ety- 
 mology. It was only very slowly and gradually, step by step, 
 that men came to think of phenomena quantitatively rather 
 than qualitatively, and to arrive at a more rational concep- 
 tion of nature through experience and reflection. 
 
NATURAL SCIENCE. 3 
 
 As the interrelation of things came to be more clearly per- 
 ceived, people began to say they could "explain things," 
 meaning that they had arrived at a familiarity with, and had 
 begun to recognize certain permanent elements and sequences 
 in the variety of phenomena. By joining these elements, they 
 constructed a chain and attained to a more or less extensive 
 and consistent comprehension of the relations of phenomena 
 by a co-ordination of their permanent elements. 
 
 If these elements are linked together logically, the satis- 
 factoriness of "the explanation" depends upon the length of 
 the chain. The longer the chain, the further it reaches, and 
 the more satisfied one is, the more one "understands" the 
 matter. This is the general method of "learning things," and 
 the information so collected may be called, as Prof. Karl 
 Pearson has called it, an "intellectual resume of experience." 
 But it should be noted that it is rarely the simple correlation of 
 things that will stand the test of experiment. 
 
 There is in this method abundant chance to go wrong. It is 
 difficult, and especially troublesome for a beginner, untrained 
 in this process, to decide what things really do not have effect 
 and hence may be excluded from consideration. And if it is 
 difficult for the beginner in science to-day, surely it was im- 
 mensely more so for primitive men. Students are wont to 
 complain of the artificiality of geometry and mechanics. Fac- 
 tors which they feel do make a difference in reality do not 
 seem to them to be fully allowed for, or they are troubled by a 
 feeling of uncertainty as to the equity of the allowance. The 
 peculiar value of mathematical studies lies just here in the 
 rigorous training in reasoning. Whatever a student's success 
 with his mathematics, few make its acquaintance without 
 receiving wholesome lessons of patient application of the in- 
 tellectual method by which mankind has won its mastery 
 over natural forces. 
 
 We may quote here to advantage Prof. Faraday.^ "There 
 are multitudes who think themselves competent to decide, 
 after the most cursory observation, upon the cause of this or 
 
 1 Lecture delivered before Royal Institution of Great Britain, — "On Edu- 
 cation of the Judgment." 
 
4 THE SCIENCE OF MECHANICS. 
 
 that event, (and they may be really very acute and correct 
 in things familiar to them) : — a not unusual phrase with them 
 is, that 'it stands to reason,' that the effect they expect should 
 result from the cause they assign to it, and yet it is very dif- 
 ficult, in numerous cases that appear plain, to show this reason, 
 or to deduce the true and only rational relation, of cause and 
 effect. 
 
 "If we are subject to mistake in the interpretation of our 
 mere sense impressions, we are much more liable to error 
 when we proceed to deduce from these impressions (as sup- 
 plied to us by our ordinary experience), the relation of cause 
 and effect; and the accuracy of our judgment, consequently, 
 is more endangered. Then our dependence should be upon 
 carefully observed facts, and the laws of nature; and I shall 
 proceed to a further illustration of the mental deficiency I 
 speak of, by a brief reference to one of these. 
 
 "The laws of nature, as we understand them, are the founda- 
 tion of our knowledge in natural things. So much as we 
 know of them has been developed by the successive energies 
 of the highest intellects, exerted through many ages. After 
 a most rigid and scrutinizing examination upon principle and 
 trial, a definite expression has been given to them; they have 
 become, as it were, our belief or trust. From day to day we 
 still examine and test our expression of them. We have no 
 interest in their retention if erroneous; on the contrary, the 
 greatest discovery a man could make would be to prove that 
 one of these accepted laws was erroneous, and his greatest 
 honour would be the discovery. . . . 
 
 "These laws are numerous, and are more or less compre- 
 hensive. 'They are also precise; for a law may present an 
 apparent exception, and yet not be less a law to us, when 
 the exception is included in the expression. Thus, that eleva- 
 tion of temperature expands all bodies is a well-defined law, 
 though there be an exception in water for a limited tempera- 
 ture; we are careful, whilst stating the law to state the excep- 
 tion and its limits. Pre-erriinent among these laws, because 
 of its simplicity, its universality, and its undeviating truth, 
 stands that enunciated by Newton (commonly called the law 
 
NATURAL SCIENCE. 5 
 
 of gravitation), that matter attracts matter with a force in- 
 versely as the square of the distance. Newton showed that, 
 by this law, the general condition of things on the surface of 
 the earth is governed; and the globe itself, with all upon it 
 kept together as a whole. He demonstrated that the motions 
 of the planets round the sun, and of the satellites about the 
 planets, were subject to it. During and since his time, certain 
 variations in the movements of the planets, which were called 
 irregularities, and might, for aught that was then known, be 
 due to some cause other than the attraction of gravitation, 
 were found to be its necessary consequences. By the close 
 and scrutinizing attention of minds the most persevering and 
 careful, it was ascertained that even the distant stars were 
 subject to this law; and, at last, to place as it were the seal 
 of assurance to its never-failing truth, it became, in the minds 
 of Leverrier and Adams (1845), the foreteller and the dis- 
 coverer of an orb rolling in the depths of space, so large as 
 to equal nearly sixty earths, yet so far away as to be invisible 
 to the unassisted eye. What truth, beneath that of revelation, 
 can have an assurance stronger than this!" 
 
 Such is the process of scientific induction. It was by linking 
 ideas together in an orderly way, by forming and verifying 
 hypotheses, that men finally came to the "principles," and 
 "formulae," which embody these general "truths" or "laws of 
 nature." In this way knowledge has been built up, chain by 
 chain, into a more or less complete system of the relations of 
 things. Without asking the "why" of it all one can see 
 "how" it goes together by running along the chains from link 
 to link. In a word this knowledge is relative, and therefore 
 quantitative,- and that is why numbers and mathematics play 
 so large a part in the exact sciences, and in mechanics. 
 
 The guiding principle in all this is the belief in the con- 
 stancy of the order of nature founded on the experience of 
 the human race. On this belief are based all scientific calcu- 
 lations and deductions. This is sometimes formulated as a 
 "Law of Causality," affirming that every effect has a sufficient 
 cause and that the relation of cause and effect is one of in- 
 variable sequence, if not interfered with by conditions or 
 circumstances that make the cases dissimilar. 
 
6 THE SCIENCE OF MECHANICS. 
 
 Information thus systematized, verified and formulated into 
 truths or general principles is called Natural Philosophy or 
 Natural Science. The Science of Mechanics is the oldest and 
 one of the most important divisions of Natural Philosophy. 
 This knowledge of the interdependence and inter-relation of 
 phenomena makes it possible to "predict" and "control" them, 
 and keeps us from making hasty and erroneous inferences. 
 When developed with this view, applied science or applied 
 mechanics is the usual designation, and that such information 
 is power to one who has the skill to apply it, need not be dwelt 
 upon. As Herbert Spencer says in his volume on Education:^ 
 "On the application of rational mechanics depends the success 
 of nearly all modern manufacture. The properties of the 
 lever, the wheel and axle, etc., are involved in every machine 
 — every machine is a solidified mechanical theorem; and to 
 machinery in these times we owe nearly all production.'' 
 Elsewhere he says : "All Science is prevision ; and all prevision 
 ultimately helps us in greater or less degree to achieve the 
 good and to avoid the bad."^ 
 
 It is not the intention here to discuss or even to enumerate 
 the triumphs in the practical applications of mechanics. The 
 utilization of power, of the strength of animals, the power 
 of the wind, of waterfalls, of steam and of electromagnetic 
 attraction, constitutes the art of machine contrivance 
 rather than the science of mechanics. Progress in theoretical 
 mechanics has always brought in its train an advance in 
 machinery. 
 
 The innumerable engines for enlightenment and destruction, 
 the cylinder-printing-press and the machine-gun which have 
 changed and are altering the economic, social and religious 
 prospect of nations and tribes are the direct result of the 
 application of the principles of the science of mechanics. With 
 further advance in theory and systematic experimentation even 
 more revolutionizing contrivances will inevitably follow. 
 When invention has realized the theoretical surmise that the 
 "molecular energy" in a cup of tea is sufficient to tumble down 
 
 ip. 30. 
 
 2"First Principles," p. 15. 
 
NATURAL SCIENCE. 7 
 
 a town, we may expect an Age of Power ushering in wonders 
 untold.^ 
 
 With the philosophy that denies the existence of reaUties 
 outside of the mind we shall not trouble ourselves here. 
 Mechanics regards a "truth" or a "law" not as subjective but 
 as objective, holding that an external world exists and that 
 truth is a relation of conformity between the mental world 
 of perceptions and inferences, and really existing objects and 
 their relations. Unless this and the validity of the principle 
 of logical inference be conceded, our science is futile. The 
 mental processes by which the victories of Science are won are 
 in no wise different from those used by all in daily affairs. As 
 Huxley says : "Science is nothing but organized common sense. 
 The man of Science simply uses with scrupulous exactness 
 the methods which we all habitually and at every moment, 
 use carelessly. Nor does that process of induction and de- 
 duction by which a lady, finding a stain of a peculiar kind 
 on her dress, concludes that somebody has upset the inkstand 
 thereon, differ in any way, in kind from that by which Adams 
 and Leverrier discovered a new planet." 
 
 Nevertheless there will always remain certain ultimate truths 
 which cannot be proved and which must be consideredas axiom- 
 atic and intuitive. This should not invalidate our conclusions 
 and we will not enter upon a discussion of these questions here. 
 
 The science of mechanics has then, for its subject matter, 
 the motion-phenomena of the universe. Its growth is co- 
 extensive with that of the race, and one of its functions is the 
 widening of its perceptions. It is obviously a subject of 
 primary importance, for from apparent chaos, it evolves rules 
 and principles of practical utility, and so increases knowledge 
 and efhciency, and consequently happiness, through power 
 and dominion over nature. 
 
 ^Suppose that a cup of tea (about lOO cubic centimeters) could be 
 suddenly and completely dissociated, after the manner of the radio-active 
 emissions of radium, into a cloud of particles with a velocity similar to 
 radium emanations of say 100,000 kilometers a second (about one-third 
 the velocity of light), then a simple calcu lation by th e theoretical formula 
 for energy, J^twi)^, gives 3^ X.1/9.8X 100, 000,000^ = 50,000,000,000,000 
 kilogramme-meters, equal to the energy of explosion of about 500,000 tons 
 of rifle powder, or enough energy to drive an express train around the globe 
 a hundred times. 
 
PART I. 
 
 I. THE SCIENCE OF MECHANICS. 
 
 The most common of all our experiences is the motion of 
 solid bodies. No idea is more frequently with us than the 
 idea of such movements. It seems to be the first experience 
 of the dawning intellect and it is soon fully developed by 
 boyhood's games of marbles and tops. Indeed, there is 
 nothing that our imagination pictures with greater ease and 
 readiness, than a moving speck or particle. There is there- 
 fore considerable satisfaction, and an appealing reasonableness 
 and inevitableness in the idea of classifying phenomena on 
 the basis of this familiar experience. 
 
 This idea and another, quite as familiar, namely, that com- 
 mon objects can be crushed and broken into many small par- 
 ticles and ground to dust so small as to seem indivisible, are 
 fundamental, and upon them the science of mechanics, as a 
 scheme of motions and equilibrium of particles has been built 
 up. Masses either change their relative position or they do 
 not. How they move, rather than why they move, is the 
 question of Mechanics. It is especially the circumstances of 
 motion or of rest that are the subject of investigation of the 
 science. 
 
 In its formal presentation in textbooks, Mechanics is now 
 defined by an American Professor, Wright, as "the science 
 of matter, motion, and force"; by an English Professor, Ran- 
 kine, as the "science of rest, motion and force"; by a German 
 Professor, Mach, as that branch of Science which is "concerned 
 with the motions and equilibrium of masses." These defini- 
 tions do not differ essentially. 
 
 The questions at once present themselves what is force, 
 what is matter, what is mass? Etymology does not help us. 
 The further back one goes, the more indistinctive and general 
 is the idea corresponding to a scientific term. The terms, 
 matter, mass, force and weight lose precision as we trace them 
 
THE SCIENCE OF MECHANICS. 9 
 
 back. Matter leads us back to the Latin, materia, i. e., 
 substance for construction or building. Mass appears to be 
 derived from the Greek root (Mdaaetv), to knead. So by 
 derivation, matter means the substance or pith of a body, 
 and mass means anything kneaded together like a lump of 
 dough. The fundamental idea of mass is then an agglutinated 
 lump. Weight is of Saxon derivation from a root meaning to 
 bear, to carry, to lift. Force appears to come from the Latin 
 root, fortia, meaning muscular vigor and strength for violence. 
 It is an anthropomorphic concept, and is suggestive of myth- 
 ology in its application to inanimate things. 
 
 All these terms are derived from words expressing distinct 
 muscular sensations. Here in the last analysis we come back 
 to sense-impressions. A mass is an agglutinated lump as of 
 kneaded dough, weight is resistance to lifting, and force is some- 
 thing that produces results analogous to those produced by 
 muscular exertion. We cannot analyze these simple, immediate 
 perceptions, nor can we analyze motion. Motion is a sense of 
 free, unrestricted muscular action. Muscular action impeded 
 gives us our sense of force. Perhaps our primitive perception 
 of force was muscular action under restraint or not accom- 
 panied by motion. From these sense-impressions we attain, 
 by inference, the idea of space, i. e., room to move in, and the 
 notion of time or uniformity of sequence. Mechanics might ^; 
 then be crudely defined as a scheme of the relations of lumps 
 of matter acted upon by muscular exertion or by anything ; 
 that produces like effects. ) 
 
 Observe that we are conscious of these sense-impressions, 
 comparatively only. We are aware of them only through 
 change in their intensity. Here in our endeavors to com- 
 prehend and to define the ultimate elements of mechanics we 
 have borne in upon us the relativity of knowledge. The con- 
 viction that the human intelligence is incapable of absolute 
 knowledge is the one idea upon which philosophers, scientists, 
 and theologians are in accord. It is a characteristic of con- 
 sciousness that it is only possible in the form of a relation. 
 
 "Thinking is relationing and no thought can express more 
 than relations," says Herbert Spencer in his Chapter on the 
 
10 THE SCIENCE OF MECHANICS. 
 
 Relativity of Knowledge. And he concludes: "Deep down in 
 the very nature of Life, the relativity of our knowledge is dis- 
 cernible. The analysis of vital actions in general, leads not 
 only to the conclusion that things in themselves cannot be 
 known to us, but also to the conclusion that knowledge of 
 them, were it possible, would be useless."^ 
 
 But though we are limited in this way we have a large field 
 in the building of a scheme of inter-relations of the relations 
 which comprise our conscious perceptions. This is the purpose 
 of our science of mechanics. In general it endeavors to inter- 
 pret for us the complex relativity of phenomena in terms of 
 the most common and simplest of our experiences, namely 
 the relativity of motion of a particle and the relativity of 
 the divided parts of bodies. 
 
 As science progresses the ideas, mental pictures, and terms 
 found serviceable in the earlier stages are bound to prove 
 inadequate later. The process of reorganizing these ideas, 
 and perfecting terminology is slow, but in it there is unmistak- 
 able evolutionary progress. 
 
 As the philologist Nietzsche says, "Wherever primitive man 
 put up a word, he believed he had made a discovery. How 
 utterly mistaken he really was! He had touched a problem, 
 and while supposing he had solved it, he had created an 
 obstacle to its solution. Now, with every new knowledge, we 
 stumble over flint-like petrified words. "^ 
 
 The prehistoric races probably explained phenomena by 
 associating with everything that produces motion, some in- 
 visible god whose muscular strength was the force of wind, 
 wave or waterfall. We find in all languages, survivals of this 
 in the genders ascribed to things inanimate. Indeed, one can 
 dig out of philology and mythology a petrified primitive natu- 
 ral philosophy. 
 
 To-day we sometimes hear that all phenomena of the material 
 world are explainable, in terms of matter, motion, and force, 
 or by the whirl of molecules. One may endeavor to make 
 this a truism by defining matter as anything that occupies 
 
 ^Spencer, "First Principles," Chapter IV. 
 ''Nietzsche, "Morgenrote," vol. i, 47. 
 
THE SCIENCE OF MECHANICS. II 
 
 space, and by defining force as any agent which changes the 
 relative condition as to rest or motion between two bodies, 
 or which tends to change any physical relation between them, 
 whether mechanical, thermal, chemical, electrical, magnetic, 
 or of any other kind. But here one does not say what force 
 is, nor what matter is. The chain hangs in the air; it does 
 not begin or end anywhere, but the relation of the links is 
 apparent and serviceable. Indeed, the idea of force is still 
 fundamentally the same, it is still an agent, as was the ancient 
 nature-god, though much less definite, nor does it help matters 
 to subdivide force and mass. 
 
 The idea of force as a latent unknown cause is a historical "-^ 
 survival of our primitive conceptions and undergoes trans- 
 formation with the idea of force as a "circumstance of motion," { 
 which was developed about the year 1700. It is now held \ 
 by some that force is a purely subjective conception. For 
 example, Tait says in his "Newton's Laws of Motion": "We 
 have absolutely no reason for looking upon force as a term 
 for anything objective; we can, if we choose, entirely dispense 
 with the use of it. But we continue to employ it; partly 
 because of its undoubted convenience, mainly because it is 
 essentially involved in the terminology of Newton's Laws of 
 Motion, which still form the simplest foundation of our subject. 
 It must be remembered that even in strict science we use such 
 obvious anthropomorphisms as the 'sun rises,' 'the wind blows,' 
 etc." 
 
 Yet though there may be no such reality as force, mechanics ^ 
 will probably long continue to be known as the dictionary 
 defines it, as "the science which treats of the action of forces 
 on bodies, whether solid, liquid or gaseous." We do not 
 disparage the use of the idea and term force; we shall have 
 occasion to use them often. But it should be noted that an 
 evolution in terminology is involved in the evolution of 
 science. 
 
 Such changes in conception and in terminology are inevi- 
 table. They are essential characteristics of progressive science 
 which seeks continually to improve the definiteness of relation 
 between phenomena by making clearer vague connections, or 
 
12 THE SCIENCE OF MECHANICS. 
 
 by discovering new relations. The relations formerly classed 
 as acoustic, luminous, thermal, electric, magnetic and chem- 
 ical expressing certain constant connections of antecedents 
 and consequents are now generally expressible with exactness 
 in the terms of the science of mechanics which is built on the 
 familiar notions of motion and divisibility. 
 
 As a matter of convenience, the science has come to be 
 divided into Phoronomics or Kinematics, the study of pure 
 motion without reference to the nature of the body moved, 
 or how the motion is produced, and Dynamics, the "science 
 of force," "the study of the push or pull of bodies," or "the 
 science of the properties of matter in motion." It is evident 
 that in some cases the "forces balance," giving the condition 
 of rest; this branch of the study is called Statics. The study 
 of unbalanced forces producing motions of various kinds is 
 called Kinematics. These divisions are purely arbitrary and 
 were made late in the development of the subject. His- 
 torically, the study of Statics, or of bodies relatively at rest, 
 was the first to be undertaken for obvious reasons. 
 
 2. THE SCIENCE OF MECHANICS IN ANTIQUITY. 
 
 It is the verdict of conservative geologists and physicists 
 that the earth's crust is at least 25,000,000 years old, that 
 period of time being required for the deposit of the depth of 
 about 50,000 feet^ of sedimentary rocks that research discloses; 
 and it is the opinion of conservative authorities that rude com- 
 munities of men were dwelling in the broad alluvial valleys 
 of the Nile, Euphrates, Ganges, Hoang-Ho, (perhaps also on 
 the ancient Thames-Rhine system), as early as 25,000 years 
 ago. The subsidence of these broad rivers into narrower 
 channels left exposed fertile plains in their old bottoms and 
 islands in the estuaries, which favored the development of 
 progressive communities. 
 
 This was particularly true of the Euphrates valley and along 
 
 the Nile, where the wild wheat and barley offered food and 
 
 made life a less severe struggle for existence. Here perhaps 
 
 the first rude camps and villages were developed. But even 
 
 ^130,000 feet is the average figure suggested by Dr. E. Haeckel — p. 9 
 "Evolution of Man," Vol. 2. 
 
THE SCIENCE OF MECHANICS. 1 3 
 
 these early communities were probably in possession of rude 
 tools and weapons. Darwin^ cites instances of tools used by 
 animals and we must imagine that even the very earliest com- 
 munities of men were acquainted with such crude mechanical 
 appliances as the lever and cord. 
 
 The researches of geologists and archaeologists present in- 
 numerable stone wedges, flint axes, bone and horn implements, 
 and primitive tools found in graves of the stone age, or on the 
 site of ancient cave and lake dwellings, indicating extensive 
 mechanical experience in prehistoric times.^ An instinctive 
 familiarity through long experience, with some of the com- 
 mon natural processes, and a knowledge of crude cutting and 
 grinding tools must then be accepted as very ancient, at least 
 twelve or fifteen thousand years old. 
 
 This must be distinguished, however, from a mechanical 
 theory of science, which is the product of reflection. The 
 latter was a very slow and gradual evolution. From a 
 long experience of measuring and bartering, a knowledge 
 of numbers probably arose, and then a more definite knowl- 
 edge of the simple mechanical devices was developed. From 
 these, by reflection and generalization, rules and principles were 
 evolved. In the ancient Sanskrit language the word from 
 which "man" comes, appears to mean to estimate, to measure. 
 Man first became conscious of himself, it appears, therefore, as 
 the being who measures and weighs, compares and reflects. 
 
 Wedges, pulleys, windlasses, oars and the lever in various 
 forms were used before any rule for them was conceived of; 
 and then the rules for centuries remained but disjointed 
 unrelated statements of experience. Only very, very slowly 
 were they mastered and made into a body of mechanical 
 knowledge. As this process proceeded, the fetishism and 
 mythology invented to explain natural phenomena declined 
 before a more rational and logical group of mechanical prin- 
 ciples. But traces of it long survived. For example, the 
 idea that "nature abhors a vacuum" is a late survival of such 
 
 iThe Descent of Man, Chapter III, "Tools and Weapons used by 
 Animals." 
 
 -Prehistoric Times, Sir J. Lubbock; Ancient Stone Implements, Evans; 
 Man and the Glacial Period, D. F. Wright; Man's Place in Nature, T. H. 
 Huxley; Origin of Species, etc., C. Darwin. 
 
14 THE SCIENCE OF MECHANICS. 
 
 fanciful conceptions, and was cited as late as 1600 A.D. But 
 for Science, as Spencer says, we should be still worshipping 
 fetishes; or with hecatombs of victims be propitiating dia- 
 bolical deities. 
 
 It seems that it was only among the people of the Eastern 
 Mediterranean coast that a true science of mechanics was 
 developed. There is no evidence to show that among any 
 of the peoples of the Far East any true science of mechanics 
 was even begun. Indeed some of the people of the yellow 
 and darker races still live in the stone or bronze age. Cer- 
 tainly the whole development of the science as. we have it is 
 European. Of the world's population of 1,500,000,000, the 
 200,000,000 of Europe and the 100,000,000 of America who 
 have a grasp on mechanical science are in control. Half of 
 Asia's 700,000,000 are held subject by Europe's Science, and 
 the destiny of the other half is the topic of the hour. 
 
 To the Babylonians and Phoenicians, skilled in measuring, 
 in plane surveying, in keeping accounts, and in seafaring, 
 the science of Europe is traced back. Centuries before the 
 era of Greece, the Phoenicians had developed a crude astron- 
 omy and were practicing and slowly improving the common 
 mechanic arts and trades. They are not to be credited with 
 originating them however, for scholars have traced these people 
 back to a mingling of tribes of primitive Semetic and Aryan 
 stock which took place in the Tigris-Euphrates region of Asia, 
 about 8000-10000 B.C. 
 
 Here a remarkable civilization of teeming cities had devel- 
 oped by 5000 B.C. The trials and troubles, the institutions, 
 arts, literature, and the wail of the prophets, the complete 
 life history of growth and decay of these cities may be read 
 in the cuneiform inscriptions on the clay tablets in the British 
 Museum. With the shifting of the trade routes to the north 
 and west, through the Dardanelles, their prosperity declined 
 and they passed out of existence. 
 
 Perhaps the oldest relic of their mechanical arts is the 
 splendid tablet or "stele" set up in the -temple of Lagash by 
 Eannatum (c. 4200 B.C.). One side shows the king in his 
 chariot leading his army to victory, the other shows the wreck 
 
THE SCIENCE OF MECHANICS. 1 5 
 
 and ruin of the vanquished whose mangled corpses are left 
 to the vultures. The great king of these people, Sargon I 
 (c. 3800 B.C.), is said to have extended his conquests west- 
 ward as far as the Island of Cyprus, the land of copper. 
 Bartering expeditions then as now spread information and 
 developed the arts and trades. As early as 3000 B.C. the 
 Eg^'ptians seem to have become a power. 
 
 It seems, then, that the European development of mechanics 
 as a science is founded on at least 3,000 or 4,000 years^ develop- 
 ment of the recognized mechanic arts and trades,^ and it is 
 probable that it began with the systematizing of craft experi- 
 ence and the formulation of this experience in connection with 
 the instruction of apprentices. 
 
 Reflection on methods, and endeavors to train novices by 
 the experience and mistakes of older craftsmen, formed a sort 
 of groundwork of experience, and tended to develop a nomen- 
 
 iThe Egyptian pyramid of Cochrome is referred by archaeologists to 
 the first dynasty of Manetho, 3600 B.C., making it fifty-five centuries 
 old. It exhibits well developed skill in the trades, "dating from a time 
 nearly coincident, according to Biblical authority, with the creation of the 
 world itself (3761 B.C.)" — Reber, History of Ancient Art, p. 3. See also, 
 Petrie; Maspero; Perrot and Chipiez. 
 
 ^The Egyptians' sculptured wall reliefs and wall paintings exhibit con- 
 siderable specialization in the trades several thousand years B.C. As for 
 the Greeks, the picture of Vulcan's smithy in Iliad XVIII is that of a 
 most busineslike and efficient shop. There is no mention of iron or steel, 
 but it indicates the tools employed 1000 B.C. 
 
 So speaking he withdrew, and went where they lay 589 
 
 The bellows, turned them toward the fire, and bade 
 
 The work begin. From twenty iellows came 
 
 Their breath into the furnaces, — a blast 
 
 Varied in strength as need might he; . . . 
 
 And as the work required. Upon the fire 
 
 He laid impenetrable brass, and tin 595 
 
 And precious gold and silver; and on its block 
 
 Placed the huge anvil and took the ponderous sledge 
 
 And held the pincers in the other hand. 
 
 When the great artist Vulcan saw his task 757 
 
 Complete, he lifted all that armor up 
 
 And laid it at the feet of her who bore 
 
 Achilles. Like a falcon in her flight, 
 
 Down plunging from Olympus capped with snow, 
 
 She bore the shining armor Vulcan gave. 
 
 William Cullen Bryant's Translation. 
 
1 6 THE SCIENCE OF MECHANICS. 
 
 clature, and a set of rules. This indicates in its very genesis 
 the practical and economical character of mechanical science. 
 It generalizes experience. It is not only a mental labor-sav- 
 ing device, but also a guide to the fashioning of physical 
 labor-saving apparatus. 
 
 Mechanics began, then, with the theory and rules of the 
 trades. The very common origin of its twin-brother geometry, 
 is seen on translating this Greek word into English : Teco/xeTpia, 
 rj, — the science of measuring the earth.^ Herodotus attributes 
 the origin of this science to the necessity of resurveying the 
 Egyptian fields after each inundation of the Nile and refers 
 to the system of taxation of Rameses II (c. 1340-1273 B.C.), 
 which required such survey. Early geometry was therefore a 
 crude theory of land surveying. Its abstractions and rules 
 were brought to bear upon mechanical problems and there 
 followed that intimate connection in the development of these 
 sciences which has been so useful. Formal mechanics has in- 
 deed been called by one of the masters,^ a geometry of four 
 dimensions, i. e., the three spatial dimensions and time. 
 
 The Ahmes papyrus of the British Museum, "Directions for 
 Obtaining Knowledge of all Dark Things" (about 2000 B.C.), 
 is perhaps the oldest treatise on arithmetic in existence. 
 The Egyptians appear to have had manuscripts on arithmetic 
 as early as 2500 B.C. But what every school boy is now 
 taught was then a dark mystery known to but a few priests 
 and scribes. The hieroglyphic numerals are a vertical line 
 for I, a kind of horse shoe for 10, a spiral for 100, a pointing 
 finger for 10,000, a frog for 100,000 and the figure of a man 
 in the attitude of wonder for 1,000,000; a rather hopeless 
 notation for mechanical calculations from the modern point 
 of view. 
 
 Building on the accumulations of Egyptian and Phoenician 
 civilization, the Greeks began the Science of Mechanics by 
 applying in the trades the rules of geometry and the inductive 
 and deductive methods of thought. They labored under the 
 
 1 Pickering's Greek Lexicon; Aristoph. Nub. 202; Th. yia and tiirpov; also 
 Herodt. 
 
 ^Joseph Louis Lagrange (1736-1813). 
 
THE SCIENCE OF MECHANICS. I7 
 
 erroneous conceptions of nature taught in their mythological 
 religion, and they were further handicapped by the notion 
 that it was not necessary to investigate nature at first hand, 
 but that the scheme of things could be evolved by ratiocina- 
 tion. 
 
 Mechanics as a science may be said to begin with the Greeks, 
 as they formulated the fl^=st principle of mechanics. But their 
 speculations were limited to problems of equilibrium, that is, 
 to Statics. They never evolved any rational theory of moving 
 bodies. Dynamics was unknown to them and did not take 
 form as a branch of mechanics until about 1600 A.D. The 
 great bulk of the correct theory of mechanics known in an- 
 tiquity is commonly attributed to Archimedes. Before con- 
 sidering his work, it will be profitable to glance at the work 
 of several of his predecessors. 
 
 Thales, probably of Greek and Phoenician ancestry, tradi- 
 tion declares, brought the art of geometry from Egypt into 
 Greece about 600 B.C. He taught half a dozen theorems by 
 the inductive method. Proclus a Greek teacher of about 450 
 A.D., speaks of him as the father of geometry in Greece, and 
 declares that he learned it in Egypt. 
 
 His method was later extended by Pythagoras who, about 
 500 B.C., prepared two books of geometry on the deductive 
 plan. He appears to have been the first to separate clearly 
 the studies of geometry and of numbers. By pointing out 
 that quantity is incommensurable, but that measure of quan- 
 tity or a unit may be enumerated or counted, he drew the 
 distinction between geometry and arithmetic, and set apart 
 the study of numbers or arithmetic as a branch of mathematics. 
 
 One finds it difficult to realize the mysticism and magic with 
 which so commonplace an idea as a number was then mingled. 
 Pythagoras regarded numbers as having celestial natures, the 
 even numbers as feminine and the odd as masculine!^ 
 
 Hippocrates (420 B.C.) invented the method of reducing 
 one theorem to another for proof instead of going back to 
 the axioms with each proposition; while Eudoxus (355 B.C.) 
 invented proportion and devised the method of exhaustions, 
 
 ^"The Philosophy of Arithmetic," Dr. Edw. Brooks. 
 
1 8 THE SCIENCE OF MECHANICS. 
 
 one form of the idea of limits which he applied in geometry. 
 
 About 300 B.C., Euclid collected and systemized the geom- 
 etry and number-work of his time, invented some new propo- 
 sitions and made a volume on the "Elements of Geometry." 
 This work of fifteen books remained the standard text-book 
 of geometry the "Euclid," of the following twenty centuries. 
 The work gives rules for the geometrical construction of various 
 figures, as well as the proof of numerous theorems. He also 
 wrote a volume on Conies and Geometrical Optics. 
 
 Aristotle (384-322 B.C.), the famous Greek teacher, often 
 mentioned as one of the founders of Science, is notable for 
 his voluminous writings on philosophy, on natural history and 
 on geometry, which in part directed attention to the study of 
 nature by direct observation. But there is no doubt that his 
 teaching on the theory of motion and some of his notions on 
 equilibrium were erroneous. His great reputation as a natural 
 philosopher gained acceptance for some of his opinions for 
 eighteen centuries after his time, and as they were wrong, this 
 was a great impediment to the development of the science of 
 mechanics. Even as late as 1590, Galileo felt the strength of 
 the partisans of the erroneous Aristotelian philosophy who 
 forced him from the University of Pisa. 
 
 By 200 B.C., four centuries after Thales, the Greeks had 
 brought their geometry to a high stage of perfection. Apol- 
 lonius, of Perga (d. 205 B.C.), published about this time a 
 treatise on conic sections and geometry containing over four 
 hundred problems which left little for his successors to improve. 
 His problem, "to draw a circle tangent to three given circles 
 in a plane," found in his tretaise on "Tangency," has baffled 
 many later mathematicians. 
 
 His studies on astronomy were the basis of Ptolemy's expo- 
 sition of planetary motions and his goemetry has been dis- 
 covered in two distinct Arabic editions, indicating its influence 
 on Moorish mathematics of the ninth and tenth centuries. He 
 also wrote on methods of arithmetical calculation and on statics, 
 but this work is overshadowed by that of his contemporary 
 Archimedes, who appears to have co-ordinated the scattered 
 information on statics and to have contributed largely to it. 
 
THE SCIENCE OF MECHANICS. I9 
 
 What Euclid did for geometry Archimedes tried to do for Sta- 
 tics. In this he was in part at least successful. For he de- 
 veloped a body of correct mechanical doctrine which still finds 
 place to-day in our elementary text books of this science. 
 
 3. THE CONTRIBUTIONS OF ARCHIMEDES. 
 (287-212 B.C.) 
 
 Archimedes, the greatest mathematician of antiquity, the 
 son of a Greek astronomer, had the advantage of a good train- 
 ing in the schools of Alexandria, and then retired to Syracuse 
 in Sicily, where he devoted himself to the study of mathematics 
 and mechanics. 
 
 We know his work through the manuscripts and the books 
 which have come down to us, and by references to him in the 
 classics which give us some slight additional data. Some of 
 his writings we have in the original Greek, while others exist 
 only in the Latin or Arabic translation. They may be briefly 
 summarized as follows: 
 
 Extant Works. 
 
 1 . On the Sphere and Cylinder. 
 
 Two books containing sixty propositions relative to the 
 dimensions of cones and cylinders, all demonstrated by 
 rigorous geometric proof. 
 
 2. The Measure of the Circle. 
 
 A book of three propositions. Prop. I proves that the area 
 of a circle is equal to a triangle whose base is equal to 
 the circumference and whose altitude is equal to the 
 radius. Prop. II shows that the circumference exceeds 
 three times the diameter by a fraction greater than 10/70 
 and less than 10/71. Prop. Ill proves that a circle is to 
 its circumscribing square nearly as 11 to 14. 
 
 3. Conoids and Spheroids. 
 
 A treatise of 40 propositions on the superficial areas and 
 volume of solids generated by the revolution or conic 
 sections about their axis. 
 
 4. On Spirals. 
 
 A book of 28 propositions upon the curve known as the 
 
20 THE SCIENCE OF MECHANICS. 
 
 spiral of Archimedes which is traced by a radius vector 
 whose length varies as the angle through which it turns. 
 
 5. On Equiponderants and Centers of Gravity. 
 
 Two volumes which are the foundation of Archimedes' 
 theory of Mechanics. They deal with statics. The first 
 book contains fifteen propositions and eight postulates. 
 The methods of demonstration are those often given 
 to-day for finding the center of gravity of — 
 
 (a) any two weights, 
 
 (b) any triangle, 
 
 (c) any parallelogram, 
 
 (d) any trapezium. 
 
 The second volume is devoted to finding the center of 
 gravity of parabolic segments. 
 
 6. The Quadrature of the Parabola. 
 
 A book of 24 propositions demonstrating the quadrature 
 of the parabola by a process of summation — a kind of 
 crude integration. 
 
 7. On Floating Bodies. 
 
 A treatise of two volumes on the principles of buoyancy 
 and equilibrium of floating bodies and of floating para- 
 bolic conoids. 
 
 8. The Sand Reckoner, or Arenarius. 
 
 A book of arithmetical numeration which indicates a method 
 of representing very large numbers. He indicates that 
 the number of grains of sand required to fill the universe, 
 is less than 10^^ It contains an idea which might have 
 been developed into a system of logarithms. 
 
 9. A collection of Lemmas, — fifteen propositions in plane 
 
 geometry. 
 
 Archimedes is also credited with these lost books, though 
 some authorities dispute the fact that he ever wrote such 
 volumes; that he worked upon the subjects there is little 
 doubt. 
 
the science of mechanics. 21 
 
 Lost Works.^ 
 
 1. On Polyhedra. 
 
 2. On the Principles of Numbers. 
 
 3. On Balances and Levers. 
 
 4. On Center of Gravity. 
 
 5. On Optics. 
 
 6. On Sphere Making. 
 
 7. On Method. 
 
 8. On a Calendar or Astronomical Work. 
 
 9. A Combination of Wheels and Axles. 
 
 10. On the Endless Screw or Screw of Archimedes. 
 Archimedes is to be credited with the development of a 
 theory of the lever, the principle of buoyancy, the theory of 
 numbers and numeration. He was the first to apply correctly 
 geometry and arithmetic to mechanical problems of equi- 
 librium, and he thus founded the science of applied or mixed 
 mathematics. He founded and developed the theory of statics 
 in reference both to rigid solids and fluids, but he by no means 
 completed it. He developed no correct theory of dynamics. 
 The following quotations from his book on Equilibrium, or 
 the "Center of Gravity of Plane Figures," give an insight 
 to his mental attitude and an idea of his method of approaching 
 problems in mechanics. 
 
 Book L 
 
 "I postulate the following: 
 
 1. "Equal weights at equal distances are in equilibrium, and 
 equal weights at unequal distances are not in equilibrium but 
 incline toward the weight which is at the greater distance. 
 
 2. "If, when weights at certain distances are in equilibrium, 
 something be added to one of the weights, they are not in 
 equilibrium but incline toward that weight to which addition 
 is made. 
 
 3. "Similarly, if anything be taken away from one of the 
 weights, they are not in equilibrium but incline toward the 
 weight from which nothing was taken. 
 
 •Accounts of the recently discovered "lost works" of Archimedes will 
 be found in the following periodicals: Hermes, vol. 42; Bulletin of the Amer- 
 ican Mathematical Society, May, 1908; Bibliotheca mathematica, vol. 7, 
 p. 321. 
 
22 THE SCIENCE OF MECHANICS. 
 
 4. "When equal and similar plane figures coincide if applied 
 to one another, their centers of gravity similarly coincide." 
 
 5. "In figures which are unequal but similar the centers 
 of gravity will be similarly situated. By points similarly 
 situated in relation to similar figures, I mean points such that, 
 if straight lines be drawn from them to the equal angles, they 
 make equal angles with the corresponding sides." 
 
 6. "If magnitudes at certain distances be in equilibrium 
 (other) magnitudes equal to them will also be in equilibrium 
 at the same distances." 
 
 7. "In any figure whose perimeter is concave in (one and) 
 the same direction the center of gravity must be within the 
 figure." This is the way he proves the equilibrium of the 
 
 lever. 
 
 ^^Proposition i." 
 
 "Weights which balance at equal distances are equal." 
 "For, if they are unequal, take away from the greater the 
 
 difference between the two. The remainders will then not 
 
 balance — (Postulate 3); which is absurd." 
 "Therefore the weights cannot be unequal." 
 
 ^^Proposition 2." 
 
 "Unequal weights at equal distances will not balance but 
 will incline toward the greater weight." 
 
 "For take away from the greater the difference between the 
 two. The equal remainders will therefore balance (Postulate 
 i). Hence if we add the difference again the weights will not 
 balance but will incUne toward the greater (Postulate 2)." 
 
 Proposition j . 
 
 Proves that weights will balance at unequal distances, the 
 greater weight being at the lesser distance, by a similar kind 
 of reasoning. 
 
 Proposition 4. 
 
 Shows similarly that two equal weights have the center of 
 gravity of both at the middle point of the line joining their 
 centers of gravity. 
 
THE SCIENCE OF MECHANICS. 
 
 23 
 
 Proposition 5. 
 Proves, if three equal magnitudes have their centers of 
 gravity on a straight Hne at equal distances, the center of 
 gravity of the system will coincide with that of the middle 
 magnitude. He then proves, 
 
 Propositions 6-7. 
 
 Two magnitudes, whether commensurable (Prop. 6) or in- 
 commensurable (Prop. 7) balance at distances reciprocally 
 proportional to the magnitudes. 
 
 I. Suppose the magnitudes .4, 5 to be commensurable and 
 the points A, B to he their centers of gravity. 
 
 Let DE be a straight line so divided that at C 
 
 A : B = DC : CE 
 
 We have then to prove that, if A be placed at E and 
 B at D, C is the center of gravity of the two taken together. 
 
 
 A 
 
 N 
 
 
 
 B 
 
 
 
 
 
 
 c 
 
 Fig. I. 
 
 H 
 
 K 
 
 Since A and B are commensurable, so are DC, CE. Let N 
 be a common measure of DC, CE. Make DH, DK each equal 
 to CE and EL (on CE produced) equal to CD. Then EH = 
 CD. Since DH = CE therefore LH is bisected at E, as HK 
 is bisected at D. 
 
 Thus LH, HK must each contain N an even number of 
 times. 
 
 Take a magnitude such that is contained as many 
 times in ^ as iV is contained in LH whence 
 
 A :0 = LH :N 
 
24 THE SCIENCE OF MECHANICS. 
 
 But 
 
 B :A = CE:DC 
 = HK :LH 
 
 "Hence B : = HK : N, or is contained in B as many- 
 times as N is contained in HK." 
 
 "Thus is a common measure of A, B. Divide LH, HK 
 into parts each equal to N, and A, B, into parts each equal to 
 0. The parts A will therefore be equal in number to those of 
 LH, and the parts of B equal in number to those of HK. 
 Place one of the parts of A at the middle point of each of the 
 parts N of LH, and one of the parts of B at the middle point 
 of each of the parts N of HK. 
 
 "Then the center of gravity of the parts of A placed at 
 equal distances on LH will be at E, the middle point of LH 
 (Proposition 5, Cor. 2), and the center of gravity of the parts 
 of B placed at equal distances along HK will be at D the middle 
 point of HK. 
 
 "Thus we may suppose A itself applied at E, and B itself 
 applied at D." 
 
 "But the system formed by the parts oi A and B to- 
 gether is a system of equal magnitudes even in number and 
 placed at equal distances along LK, and, since LE = CD and 
 EC = DK, LC = CK so that C is the middle point of LK. 
 Therefore C is the center of gravity of the system ranged 
 along LK. 
 
 "Therefore A acting at E and B acting at D balance about 
 the point C" 
 
 The incommensurable case. 
 
 "Suppose the magnitudes to be incommensurable and let 
 them be {A ± a) and B respectively. Let DE be a line divided 
 at C so that 
 
 (A + a) : B = DC : CE 
 
 "Then, if {A + a) placed at E and B placed at D do not 
 balance about C, (A + a) is either too great to balance B 
 or not great enough." 
 
 "Suppose, if possible that (A + o-) is too great to balance B. 
 
THE SCIENCE OF MECHANICS. 
 
 25 
 
 Take from (A + a) a. magnitude smaller than the deduction 
 which would make the remainder balance B, but such that 
 the remainder A and the magnitude B are commensurable. 
 
 a : A 
 
 B 
 
 Fig. 2. 
 
 "Then, since A, B are commensurable and 
 A : B < DC : CR 
 
 A and B will not balance (Prop. 6) but D will be depressed. 
 
 "But this is impossible since the deduction A was an in- 
 sufificient deduction from (A + a) to produce equilibrium, so 
 that E was still depressed. 
 
 "Therefore (A + a) is not too great to balance B; and 
 similarly it may be proved that B is not too great to balance 
 {A + a). 
 
 "Hence (A -\- a), B taken together have their center of 
 gravity at C." 
 
 Thus it is seen that the demonstration rests upon the axiom 
 that equal bodies at the ends of equal arms of a rod supported 
 at its middle point will balance each other. From this he 
 proves that the bodies will be in equilibrmm when their dis- 
 tances from the fulcrum are inversely as their weight, and all 
 his determinations are based on these propositions. All his 
 investigations are limited to -the case of forces perpendicular 
 to straight lever arms, he does not appear to have grasped 
 the idea of "moments" or of "equal work" up and down. 
 These conceptions were not fully attained until eighteen 
 centuries later. 
 
 To Archimedes also belongs the fame of establishing the 
 principle of buoyancy commonly known as Archimedes' prin- 
 
26 THE SCIENCE OF MECHANICS. 
 
 ciple. The account of this discovery given by Vitruvius in 
 De Architectura, Liber IX, is as follows: "Although Archi- 
 medes discovered many curious things proving his great intelli- 
 gence, that which I now narrate is the most remarkable. 
 Hiero, when he obtained the regal power in Syracuse, having 
 on the happy turn of his fortunes decreed a votive crown of 
 gold to be placed in a certain temple, commanded it to be made 
 of great value, and assigned for the purpose an appropriate 
 weight of metal to the goldsmith. The latter in good time 
 presented the crown to the king beautifully wrought and of 
 correct weight. 
 
 But a report having been circulated, that some of the gold 
 had been supplanted with silver of equal weight Hiero was 
 indignant at the fraud, and appealed to Archimedes for a 
 method by means of which the theft might be detected. 
 Charged with this commission he by chance went to a bath, 
 and on getting into the tub perceived that just in proportion 
 that his body became immersed, in the same proportion the 
 water ran out of the vessel. Whence catching at the method 
 to be used in solving the king's difficulty he leapt out of the 
 vessel in joy, and ran naked shouting in a loud voice, evprjKa, 
 I have found it!" 
 
 It seems that Archimedes' conception was, that a body 
 immersed in water must raise an equivalent quantity of water 
 just as though the body lay on one arm of a balance and the 
 water on the other arm. Buoyancy he conceived as a case 
 of equilibrium or equipoise by a balance of weights. If the 
 object overbalances the water displaced it sinks. These ideas 
 he elaborated in his book on Floating Bodies. 
 
 One of his fundamental assumptions in this work is that 
 it is an essential property of a liquid that the portion that 
 suffers less pressure is forced upward by that which suffers 
 greater pressure and that each part of the liquid suffers 
 pressure from the portions directly above it, if the latter be 
 sinking or suffer from another portion. From this he elabo- 
 rates the ideas, 
 
 (i) That when a heavy body is entirely surrounded by liquid 
 it is buoyed up or balanced in part, by a force equal to the 
 weight of the liquid it displaces; 
 
THE SCIENCE OF MECHANICS. 27 
 
 (2) That when bodies lighter than a fluid are wholly im- 
 mersed in it, they displace an amount of liquid greater than 
 their own weight and so if left free to adjust themselves they 
 rise to the surface and float so that only so much of their bulk 
 is submerged as will displace sufficient liquid to balance them- 
 selves ; 
 
 (3) When a submerged body displaces a magnitude of liquid 
 which just balances itself it is in equilibrium anywhere below 
 the surface of the liquids. 
 
 It follows from the story of the gold and silver crown that 
 Archimedes must have arrived at the idea of relative density 
 or specific gravity but he could not distinguish mass and weight. 
 The favorite word in his discussions is the abstract mathemati- 
 cal term magnitude by which he often seems to mean mass. 
 But how mass gets that drag downwards, or how the force or 
 weight is related to mass he did not attempt to explain. Cer- 
 tainly he presents no theory on the subject in any of his extant 
 works. 
 
 Some convenient mechanical appliances are by tradition 
 commonly attributed to Archimedes, notably the Archimedean 
 screw or pump, a device said to have been invented by him 
 while in Egypt for use in the irrigation works. His practical 
 inventions indicate that he, in common with all the eminent 
 masters, was not so lost in his theoretical studies as to be out 
 of touch with practical affairs. 
 
 It should be noted that the geometry of Euclid was a geom- 
 etry of forms and positions whereas that of Archimedes was 
 a geometry of measurement. This new trend is seen in the 
 attention that Archimedes gives to problems of the quadrature 
 of curvilinear plane figures (such as the parabola), and to the 
 cubature of curved surfaces. This development of geometry 
 placed it in most intimate connection with mechanics, for 
 progress in the latter depended upon accuracy of measure- 
 ment. 
 
 Therefore we are indebted to Archimedes not only for the 
 mechanical devices and rules commonly associated with his 
 name, but also for having given to geometry that trend of de- 
 velopment into a science of measurements which made it of 
 
28 THE SCIENCE OF MECHANICS. 
 
 such assistance in developing mechanics. Archimedes himself 
 well illustrated this in the field of statics. Later when the 
 labors of Galileo and Stevinus had developed the method of 
 representing forces, velocities and acceleration by lines, it was 
 by similar geometrical methods that Newton in his Principia 
 presented the proofs of theorems in dynamics. 
 
 Ctesibius and Hero (cir. 150 B.C.) are sometimes mentioned 
 as the successors of Archimedes. Following Archimedes' 
 method they formulated a table of mechanical appliances set- 
 ting forth the five simple principles or "simple machines," 
 about as they are listed in our elementary textbooks of physics 
 to-day. But though they made several practical inventions 
 such as the forcing pump, the clepsydra and air-gun and con- 
 trived curious fountains and syphons, they do not appear to 
 have added anything to the principles of mechanics, nor do 
 they appear to have comprehended the theory of their mechan- 
 ical appliances, except in so far as the principles of Archimedes 
 could explain them. 
 
 There is no evidence to show that the principle of work 
 was understood or appreciated in ancient times in spite of 
 
 Fig. 3. 
 
 the fact that we feel almost instinctively now, that in a lever 
 such as Fig. ;t, the force times the distance it moves {i. e., 
 the work applied), is equal to the resistance times the distance 
 it moves, {i. e., the work done) if we ignore friction, and that 
 the algebraic sum of the positive and negative work is zero. 
 The simple equation of work, F X S = R X S' was Chinese to 
 Archimedes, for algebraic symbolic notation was not known 
 in mechanics in his time. Archimedes does not appear to have 
 
THE SCIENCE OF MECHANICS. 29 
 
 attained to the conception of "moment," nor "principle of 
 work," nor "conservation of center of gravity." He made 
 equilibrium in the lever depend on the length of the lever-arm 
 and the "magnitude" of the bodies hung on the ends of the 
 lever-arms without understanding the terms moment, mass, 
 work or weight in the modern sense. 
 
 The physical science of "the Greeks was limited to calcula- 
 tions based upon: 
 
 (i) The law of the lever, 
 
 (2) Center of gravity, 
 
 (3) Density, 
 
 (4) Hydrostatic pressure, 
 
 (5) Arithmetical relations of tones, 
 
 (6) The law of the reflection of light. 
 Ancient Greece was a slave country. At the height of its 
 
 glory Athens contained twenty slaves to one free citizen. The 
 slaves were permitted no initiative and there was no incentive 
 to mechanical invention. Indeed, the application of natural 
 forces and the substitution of machines for slave-labor would 
 have been viewed with alarm by all classes of the Greek state 
 as ushering in an industrial and social revolution. Inventors 
 and innovators therefore met scant encouragement in ancient 
 Greece. 
 
 The government was a close corporation of capitalist citizens 
 whose profits depended upon the slaves. Furthermore it was 
 to the interest of the government to keep the slaves steadily 
 employed yet not oppressively burdened. Conditions of life 
 were favorable in the Greek peninsula, and history records 
 very few slave insurrections. There was no urgent demand for 
 mechanical invention and no reward for it. Only free men 
 have an interest in the improvement of their tools and only 
 under the laws of property and of patents is there encourage- 
 ment and incentive to mechanical invention. 
 
 With the extension of the Roman power on the fall of 
 Syracuse, in which Archimedes lost his life, conditions were 
 not favorable to the advance of science. The Romans, a 
 practical, commerical, military people did not advance the 
 theory of mechanics. Their talents lay in administration 
 
30 THE SCIENCE OF MECHANICS. 
 
 rather than in science. They used the simple machines prac- 
 tically and successfully on land and on sea, in war and in 
 peace, and by trireme and catapult extended their dominon 
 over the known world. In their marvellous public works 
 aqueducts, baths, fountains, sewers, roads, public buildings, 
 and monuments, we find examples of the art of construction 
 rather than of the science of engineering. 
 
 They built by "rule of thumb" and experience based on 
 trial, using a large surplus of materials. No delicate appre- 
 ciation of stresses is apparent in their architecture. It is 
 massive, heavy and monumental, without subtlety of artistic 
 conception or of scientific design. It is true the Romans in- 
 troduced as common features in their buildings, arches, vaults 
 and domes which were used by the Greeks, Persians and Egyp- 
 tians but rarely, but they used them without theoretical 
 calculation. Some of their buildings were supplied with run- 
 ning water carried in lead pipes, and were heated by hot air 
 in tile flues but they had not grasped even the elements of 
 hydraulics or thermodynamics. It was not till after 1600 A.D. / 
 that the principle of moments and the law of action and re- ^ 
 action upon which the common engineering calculations are^ 
 based, were fully apprehended. 
 
 Nor did Roman philosophers and writers busy themselves 
 with mechanical science. The works of Lucretius (95-52 
 B.C.), Vitruvius (85-26 B.C.), Seneca (2-66 A.D.) and Pliny 
 (23-79 A.D.) contain no new idea in mechanics. For prac- 
 tically twenty centuries no advance was made in the theory 
 of mechanics after the time of Archimedes. "Vir stupendae 
 sagacitatis, qui prima fundamenta posuit inventionum fere 
 omnium in quibus promovendis setas nostra gloriatur" is the 
 tribute Wallis penned two thousand years later, when Latin 
 was still the language of scholars and engineers. 
 
 The Romans left their mark on civilization as the annals of 
 government, law and language testify, but there does not 
 appear to their credit the discovery of a single scientific prin- 
 ciple or the invention of an important mechanical appliance 
 for mitigating the drudgery and toil of mankind. Their slaves 
 and captives labored long and hard, tilling the fields, in the 
 galleys, or with brick, tiles and concrete on the aqueducts. 
 
THE SCIENCE OF MECHANICS. 3I 
 
 Every conquest delivered to the Imperial City a new supply 
 of labor. Besides, in tranquil times the legions, kept from mis- 
 chief by employment on roads, bridges and wall building, 
 supplied abundant labor. In a word the Romans had neither 
 interest in the theory of mechanics, nor the pressing necessity 
 for improved mechanical appliances, as they commanded an 
 abundance of cheap labor. For these reasons this intensely 
 practical people appears to have made no contribution to the 
 science of mechanics. 
 
32 THE SCIENCE OF MECHANICS. 
 
 REFERENCES. 
 
 Ball, J. J. The History of Mathematics. 
 
 Burr, W. H. Ancient and Modern Engineering. 
 
 Darwin, C. Descent, Origin of Species, etc. 
 
 Diihring. Geschichte der Principien der Mechanik. 
 
 Faraday, M. Proc. Royal Ins. G. B. 
 
 Fletcher, B. History of Architecture. 
 
 Grote. History of Greece. 
 
 Hamlin, A. D. F. History of Architecture. 
 
 Heiburg, Leipsic (1881); Heath, Cambridge (1891). Opera Archimedis 
 
 Huxley, T. H. Science and Educ, Man's Place in Nature. 
 
 Lubbock, J. J. Prehistoric Times. 
 
 Mach, E. The Science of Mechanics. 
 
 Maspero and Sayce. The Dawn of Civilization. 
 
 Perrot and Chipiez. History of Ancient Art. 
 
 Petrie, W. M. F. A History of Egypt; Tales; Royal Tombs. 
 
 Reber, F. History of Ancient Art. 
 
 Renan, E. Mission de Phenicie. 
 
 Robinson, J. H. A History of Western Europe. 
 
 Rogers, New York (1900). History of Babylonia and Assyria. 
 
 Spencer, H. First Principles, Education. 
 
 Stark. Archaologie der Kunst. 
 
 Tylor. Primitive Culture. 
 
 Tylor. The Early History of Mankind. 
 
 Tyndall, J. Essays, Notes and Papers. 
 
 Winckelmann. Geschichte der Kunst des Alterthums. 
 
 Winckler (1900). Die politische Entwicklung Babyloniens und Assyriens. 
 
 Wright, D. F. Man and the Glacial Period, etc. 
 
PART II. 
 I. THE MEDIAEVAL PERIOD, 500-1500 A.D. 
 
 I. The Medieval Attitude toward Science. 
 
 The period of societal reconstruction which followed the 
 decay of the Roman Empire was not a time of scientific re- 
 search or achievement. It was an age of semi-barbarism, 
 tumult and superstititon. Those of gentle and scholarly dis- 
 position who sought the quiet asylum of the Church found 
 there a faith in an established cosmography, which did not 
 encourage independent research and investigation of natural 
 phenomena. 
 
 Dr. Andrew D, White, of Cornell, says:^ "The establishment 
 of Christianity, beginning a new evolution of theology, arrested 
 the normal development of physical sciences for fifteen hundred 
 years." This is in part true and it was due, during the first 
 thousand years at least, to a widespread belief, based on the 
 New Testament, that the end of the world was soon at hand. 
 St. Paul had preached: "For ye know perfectly that the day 
 of the Lord so cometh as a thief in the night," and St. Peter 
 had reiterated: "The day of the Lord will come as a thief in 
 the night in the which the heavens shall pass away with a 
 great noise and the elements shall melt with fervent heat and 
 the earth also and the works that are therein shall be burned 
 up." 
 
 It was widely proclaimed that the world was in its last days, 
 that just as the antediluvian world was destroyed in the flood, 
 so now the coming of the Lord in a cataclysm of fire was to be 
 awaited from day to day. With such a stupendous supernatural 
 event impending and the termination of the world imminent, 
 devotion to mechanical science was sheer folly. Even such 
 science as had been developed was now become vain and trivial, 
 and was neglected in the face of the duty to watch and to pray. 
 
 The end of the world was announced for various specific 
 
 ^"Warfare of Science and Theology," vol. i, p. 375. 
 3 33 
 
34 THE SCIENCE OF MECHANICS. 
 
 dates notably looo A.D., and all endeavor except "saving 
 souls" was pronounced folly and the inspiration of the evil 
 one. And when, after centuries of waiting, the existing order 
 was found going along just as ever, and curious men began 
 to turn again to worldly affairs, they found theology had woven 
 a magic circle and defied any one to find truth outside of it. 
 In place of verified experience, a literal belief in the Old 
 and New Testament offered a precarious theophany and 
 created a frenzied terror of supernatural agencies. Demons, 
 ( imps and devils rode the wind and disported themselves to 
 the fevered imagination of the time as the cause of the most 
 common occurrences.^ Any prying into the secrets of nature 
 was held to be dangerous to body and soul. Physics and 
 chemistry, such as there was, were tabooed as the devil's 
 own arts, and experimental research was anathema. 
 I Stories of interference with the law of gravitation by the 
 \ devil and the saints are common among the legends of this 
 \period. A story published in the Dialogues of St. Gregory 
 the Great, Vol. II, illustrates this belief. During the con- 
 struction of Monte Cassino about 530, one day the builders 
 found a stone which their united efforts could not move. They 
 reported this to St. Benedict, "who instantly knew the devil 
 was hanging on to it." He exorcised the devil and the stone 
 which before was too heavy for six men became so light that St. 
 Benedict lifted it with ease and put it into the wall. A similar 
 account of the devil increasing the gravity of two marble 
 columns at the Cathedral of St. Virgile, Bishop of Aries, about 
 600, is given in "Les Petits Bollandistes," Vol. Ill, p. 162. 
 
 Even after the year 1000 A.D., ideas, which to us appear 
 most fantastic, were handed down for generations apparently 
 without anyone doubting their verity or making any endeavor 
 
 ipor the spirit of the time refer to Longfellow's "Christus; a mystery." 
 
 Safe in this Wartburg tower I stand 
 
 Where God hath led me by the hand, . . . 
 Safe from the overwhelming blast 
 Of the mouths of Hell, that followed me fast, 
 
 And the howling demons of despair 
 
 That hunted me as a beast to his lair. 
 (Second interlude.) 
 
THE MEDIEVAL PERIOD. 35 
 
 to verify them by experiment. When Albertus Magnus (cir. 
 1250), a famous philosopher of the thirteenth century, beHeved 
 that the diamond could be softened in the blood of a stag fed 
 on parsley and that a sapphire would drive away boils, it is 
 hardly to be expected that even the learned of this period 
 would have any conception or appreciation of a science of 
 mechanics. 
 
 Though St. Paul had advised, "Prove all things, hold fast 
 to what is good," St. Augustine commanded in vigorous Latin 
 — "Major est Scripturse auctoritas quamomnis humani ingenii 
 capacitas," i. e., accept nothing except on authority of Scrip- 
 ture for that is greater than all the powers of the human mind. 
 When asked, might there not be inhabitants on the other side 
 of the earth, he answered, it is impossible that there should 
 be inhabitants on the other side of the earth, for on judgment 
 day such men could not see the Lord descending through the 
 air. Discussion was closed by authority and debate came 
 to be restricted to such questions as, whether an angel in 
 passing from one spot to another, had to pass through the 
 intervening space. 
 
 It came to be considered blasphemous to wish for or to 
 attempt to better earthly conditions, and presumptuous to 
 attempt to explain phenomena except in terms of mystic 
 theology. So, in the course of the centuries, an unfortunate 1 
 conviction was developed that science was dangerous and evil. \ 
 This persisted beyond the Reformation. Martin Luther (1483- 
 1 546) complained : ' 'The people give ear to an upstart astrologer 
 (Copernicus) who strives to show that the earth revolves, — 
 but Sacred Scripture tells us that Joshua commanded the 
 sun to stand still, not the earth." 
 
 In much the same spirit, Melanchthon (1497-1560) declared: 
 "It is the want of honesty and decency to say that the earth 
 revolves and the example is pernicious. It is the part of a good 
 mind to accept the truth as revealed by God and to acquiesce 
 in it." Indeed, theologians of all persuasions, have, at some 
 time, denounced the Copernican idea, for Scripture declares the 
 "sun Cometh forth as a bridegroom" — and "the earth standeth 
 fast forever." When a theologian did deign to debate such a 
 
36 THE SCIENCE OF MECHANICS. 
 
 topic, his argument was likely to be like that of Fromundus of 
 Antwerp who, in refuting the revolution of the earth, declared 
 that "the buildings would fly off with such rapid motion, and 
 that men would have to be provided with claws like cats to 
 enable them to hold onto the earth's surface." 
 
 The theologian who declared in Galileo's time (1600) that 
 "geometry is of the devil," and "that mathematicians should 
 be banished as the authors of all heresies," was but fanatically 
 defending his traditions. The matter is summed up by Huxley 
 in his Essay on Science and Culture (p. 145), where he says: 
 "The business of the philosopher of the middle ages was to 
 deduce from the data furnished by theologians, conclusions in 
 accordance with ecclesiastical decrees. They were allowed the 
 high privilege of showing by logical process how and why that 
 which the Church said was true and must be true and if their 
 demonstrations fell short of or exceeded this limit, the church 
 was maternally ready to check their aberrations; if need be by 
 the secular arm. 
 
 Between the two, our ancestors were furnished with a com- 
 pact and complete criticism of life. They were told how the 
 world began and how it would end ; they learned that material 
 existence was a base and insignificant blot on the fair face of 
 the spiritual world and that nature was to all intents and 
 purposes, the playground of the devil; they learned that the 
 earth is the center of the visible universe, and that man is the 
 cynosure of things terrestrial, and more especially was it incul- 
 cated that the course of nature had no fixed order but that it 
 could be and constantly was, altered by the agency of innumer- 
 able spiritual beings, good and bad, according as they were 
 moved by the deeds and prayers of man. The sum and sub- 
 stance of the whole doctrine was to produce the conviction that 
 the only thing really worth knowing in this world was how to 
 secure that place in a better, which under certain conditions 
 the church promised." There was no place in such a scheme 
 for a science of mechanics. 
 
 To the unbiased student there is a measure of truth in the 
 remarks of Dr. White and Dr. Huxley and yet in justice it 
 must be said that they also carry a sting and a reproach which 
 
THE MEDIEVAL PERIOD. 37 
 
 is somewhat unfair. The attitude of the Middle Ages was a 
 growth; it was developed in, and was not imposed on Europe. 
 Whatever reproach there is, should be placed where it belongs 
 on the general ignorance and stupidity of the inhabitants of 
 Europe during this period. There was no conscious conspiracy 
 to retard progress if we except the bigotry, fanaticism and 
 perversion which inevitably .accompany ignorance anywhere 
 and in any time. As the general average of intelligence rose, 
 the situation improved. It cannot be denied that the Church 
 was the great conserving and civilizing agency of this era. 
 
 All the learning of the ancients was not lost or destroyed 
 outright, but continued to filter through Europe until it gained 
 force under the favorable circumstances of the so-called 
 Renaissance about 1600. But it is most unfortunate that 
 early Christian fanaticism burned and destroyed so much that 
 was good, though "Pagan." In the East, the Greek School 
 at Alexandria preserved, for some centuries A.D., the learning 
 of the ancients, but, about the fifth century, there developed 
 within the church a pronounced spirit of hostility toward the 
 scientific spirit. 
 
 Persecutions became common and culminated in 415 A.D. 
 with the murder of Hypatia and the breaking up of the Alex- 
 andrian University. In a burst of religious fervor, the schools 
 and a portion of the library were destroyed. The scholars 
 were forced to flee to Byzantium, where schools grew up. 
 
 In the west, after 500 A.D., Roman civilization finally went 
 down under the successive inroads of barbarians from the 
 North and culture and refinement were eclipsed in Italy. 
 
 Under Constantine (306-337), Christianity was recognized 
 (313) and Byzantium rebuilt as the Eastern capital. Until 
 476, there were two capitals but the center of wealth and 
 population shifted steadily to this new "City of Constantine." 
 Constantinople soon became the wealthiest and most enlight- 
 ened city of the world, a quiet retreat for scholarly pursuits. 
 Here many ancient manuscripts on mathematics and mechanics 
 were read, copied, and preserved for posterity. 
 
 On the capture of Constantinople, a thousand years later 
 in 1453, this Greek and Byzantine learning was spread to 
 
( 
 
 38 THE SCIENCE OF MECHANICS. 
 
 various cities of Western Europe by traveling scholars, and 
 stimulated scholarship in the West. As the mediaeval uni- 
 versities grew up from the church schools, their enthusiasm 
 was naturally not in the direction of scientific or mechanical 
 investigation. An age of faith is not inclined to be an age 
 of investigation, and the mediaeval period developed little 
 mechanical progress. 
 
 The one notable indirect contribution to mechanics in this 
 period was the introduction about 1200, of Hindu arithmetic 
 and Arabic algebra into Europe through the Moors of Spain. 
 Among the ancients, primitive number pictures such as the 
 Egyptian hieroglyphics and the Babylonian cuneiform sym- 
 bols were used for the digits. 
 
 The Greeks used the letters of their alphabet a, /3, 7, 8, etc., 
 to represent numbers. The Roman system was little better 
 and no extensive calculation could be performed without the 
 aid of a registering instrument of colored beads called an 
 abacus. Our present powerful system of ten symbols, the 
 "ten digits," and the "method of position," whereby their 
 value depends on their place, seems to have originated with 
 the Hindus, and was carried into Europe by the Arabs. 
 
 Leonardo Fibonacci of Pisa (1175) among others is credited 
 with introducing it into Italy by his book Liber Abaci (1202). 
 His introduction reads — "The nine figures of the Hindoos are 
 9, 8, 7, 6, 5, 4, 3, 2, I. With these nine and with the sign o 
 which in Arabic is called sifr, any number may be written." 
 It is likely that convenience and serviceableness in commerce 
 brought the system into vogue through the trade of Genoa 
 and Venice with the Orient. From these ports the merchants 
 probably spread it by the great overland trade routes through 
 Nuremberg and the Rhine to Antwerp, Bruges and the 
 towns of the Hanseatic League. The money exchanges and 
 the channels of trade probably had more to do with spreading 
 it over Europe than the philosophers. 
 
 The college accounts in the English Universities are found to 
 have been kept in Roman numerals up to about the year 1550 
 and even later. After this date the Arabic system generally 
 displaced the Roman method. 
 
the medieval period. 39 
 
 2. The Influence of Moorish Culture. 
 
 Before the time of Mohammed (570-632), the Arabs had 
 played an inactive part in history; but, when the wandering 
 tribes of the desert had been welded into a nation by the 
 fiery enthusiasm of the prophet and his fanatical followers, 
 they began to be a factor in the civilization of both East and 
 West. 
 
 Within a decade after Mohammed's death, the faith had 
 conquered Arabia, Palestine, Syria and Persia, and within a 
 few years more, the Moslems threatened Europe from the 
 northern coast of Africa. By 711 all Spain, except Asturia, 
 was subject to their sway, and their dominion began to ap- 
 proach in extent the glorious Empire of Rome. In their 
 opinion, the Koran, the new revelation, was destined to sup- 
 plant the Bible. 
 
 The Koran is evidently based on the Hebrew and Christian 
 scriptures. Islam is an offshoot of Christianity, modified to 
 suit the Arabic temperament, by a coloring of Oriental imagery 
 and fatalism. The theological structure of both is the same. 
 There is much the same scheme of rewards and punishments 
 with a tinge of predestination. The prohibition in the Koran 
 against "graven images" was held to forbid the representation 
 of any human or animal form. 
 
 This had a marked effect upon their arts, and no doubt encour- 
 aged the study of geometry and mathematics in general. On 
 the whole the Moslems seem to have been rather favorably dis- 
 posed toward pagan culture, regarding it with placid superi- 
 ority rather than enmity, and they never were hostile to the 
 scientific spirit. When in Europe the practice of medicine 
 was looked at askance, the Arabs were adept in medicine, and 
 their surgeons were in demand in the courts of Europe. 
 
 The nomadic Arabs had neither need nor desire for a science 
 of mechanics and the earlier caliphs were too busy establishing 
 their empire, to develop any of the arts and sciences. But 
 toward the end of the eighth century, when their religious 
 fervor was no longer at a white heat, the caliphs became 
 patrons of learning. 
 
40 THE SCIENCE OF MECHANICS. 
 
 Through the Greeks of the conquered provinces, the Moham- 
 medans became acquainted with classical learning and blended 
 it with the wisdom of the Orient, which they had from India 
 and Persia. From the tenth to the thirteenth centuries, the 
 Arabs were the teachers of Europe. They brought about, 
 within their dominions, a renaissance of the Greek culture. 
 
 As early as 800, under the caliphate of Haroun-al-Raschid, 
 Bagdad was a famous center of culture. Later, the Western 
 caliphs developed, at Cordova and Seville, schools and libraries 
 which equalled those of Constantinople and Bagdad, and made 
 these Spanish cities famous seats of learning. 
 
 In the tenth century, Cordova was one of the greatest 
 centers of commerce of the world and supported eighty schools. 
 The University of Cordova, with its library of 500,000 volumes, 
 became famous throughout Christendom. Philosophy, mathe- 
 matics, medicine, geography, astronomy and mechanics were 
 taught from Arabian translations of the masters of ancient 
 Greece, Persia, and India. 
 
 In working over this material, the Moorish scholars, as was 
 to be expected, developed new ideas and methods, especially 
 in mathematics, astronomy and alchemy. In mechanics and 
 geometry they studied and preserved for posterity the writings 
 of Archimedes, Euclid, and Aristotle. They developed known 
 principles and perfected methods, but it does not appear that 
 they made any very important advance. Extensive fortifica- 
 tions and irrigation works were developed by their engineers, 
 who were well versed in algebra and statics, but had but little 
 grasp of dynamics. 
 
 The English champion of Science, Professor Huxley, may 
 be again quoted to advantage on this topic. He says, "Even 
 earlier than the thirteenth century, the development of 
 Moorish civilization in Spain and the great movement of the 
 Crusades had introduced the leaven which, from that day to 
 this, has never ceased to work. At first, through the inter- 
 mediation of Arabic translations, afterwards by the study of 
 the originals, the western nations of Europe became acquainted 
 with the writings of the ancient philosophers and poets, and 
 in time with the whole of the vast literature of antiquity. 
 
THE MEDIiEVAL PERIOD. 4 1 
 
 Whatever there was of high intellectual aspiration or dominant 
 capacity in Italy, France, Germany, and England, spent itself 
 for centuries in taking possession of the rich inheritance left 
 by the dead civilizations of Greece and Rome. . . . There 
 was no physical science but that which Greece had created." 
 This found its way into Europe in part through Arabic trans- 
 lations of Greek and Latin texts. 
 
 The celebrated Moslem scholar, Al-Khuwarizmi or Mo- 
 hammed Ibn Musa (cir. 900 A.D.) wrote voluminously on 
 mathematics, on Hindu arithmetic, the sun-dial, and the 
 astrolabe. His "al-jabr-w'al-muqabalah," that is the "red- 
 integration and the comparison," a treatise on algebra, gave 
 the name to this science. 
 
 The Arabic numbers and the algebraic method were an 
 immense advance over the clumsy Roman numbers. Without 
 these, it is hardly possible to apply mathematics extensively 
 to mechanical problems. This indicates one reason why the 
 ancients did not advance further in the practical applications 
 of mechanical science. Much advance in mechanics was 
 simply impossible with the old Roman arithmetic which 
 possessed a most awkward duodecimal system of fractions. 
 Decimal fractions date from 1600 when Stevinus, the Flemish 
 engineer, recommended them in his writings. 
 
 Of the numerous Christian scholars who attended the Moor- 
 ish Universities of Cordova and Seville, the most famous was 
 Gerbert, who later became Archbishop of Rheims, and who as 
 Pope Sylvester H (999-1003), exercised a wide influence in 
 Christendom. He is credited with the introduction in Europe 
 of the Arabic mathematics. 
 
 The Moors made small original contributions to the science 
 of mechanics, but they are to be credited with the preservation 
 and development of the Greek and Indian knowledge of arith- 
 metic, geometry, and mechanics and the diffusion of it through- 
 out Europe. The Arabic words in our language indicate the 
 breadth of their influence: — algebra, alcohol, Aldebaran, 
 almanac, amalgam, alkali, borax, cipher, carat, minaret, 
 nadir, Vega, zenith, zero. Their invention of algebra and 
 development of the Arabic numerals and notation, while not 
 
42 THE SCIENCE OF MECHANICS. 
 
 a contribution to mechanics proper, had a most direct bearing 
 upon the future progress of the science, for without it, the 
 development of our analytical mechanics would probably have 
 been long delayed. 
 
 With the coming of the ignorant and fanatical Turks under 
 Genghis Khan in the middle of the thirteenth century, Arabic 
 civilization rapidly declined and the development of mathe- 
 matics and mechanics was arrested in the Moslem domain. 
 Four hundred years of the Turks has made the once world- 
 renowned Byzantium, one of the most backward cities on the 
 globe. It was not till about 1890 that the Sultan would permit 
 a railway to run into Constantinople. 
 
 3. The Period of the Renaissance. 
 
 The new order which slowly overcame and displaced the 
 conceptions of mediaeval times was the expression of a revo- 
 lution in the realm of thought. The Renaissance was a period 
 of breaking away from the ideas and ideals of the Middle Ages. 
 It was in part the result of the recognition of certain provinces 
 of thought and endeavor, which the mediaeval spirit either 
 ignored or condemned and in part the victory of certain 
 superior features of the civilization of Athens and Rome. The 
 inventions of printing, of gunpowder, of the mariner's compass 
 and the discovery of America, accelerated this tendency, and 
 the religious, political and social changes followed. 
 
 With the weakening of the dictates of established authority, 
 men credited personal experience more. They slowly became 
 less biased and more open-minded in their opinions. Pagan 
 writings, which, in mediaeval times, were regarded with aver- 
 sion, if not fear and distrust, came to be studied with interest. 
 Good was found in the manuscripts of the infidel Moslems; 
 their writings were read with interest and appreciation, and 
 their arithmetic was adopted throughout Europe. All this 
 prepared the way for a new start in Science. 
 
 Even the theologians began to be dissatisfied with barren 
 dogmas. One of the first to break with the prevailing scholas- 
 ticism was Cardinal Nicholas of Cusa (d. 1464), who possessed 
 the independence to say that man was prone to err, that it 
 
THE MEDIEVAL PERIOD. 43 
 
 was good to hold one's opinions lightly, and to reject them 
 when they began to appear erroneous. He cultivated mathe- 
 matics and is said to have taught an imperfect heliocentric 
 theory. 
 
 The dawn of the period of the Renaissance may be set 
 at about 1450. Then humanity's native curiosity overcame 
 the terrors of narrow theology. Confidence in the persistence 
 of the order of the universe gained ground and a general 
 interest in the things of the world resulted. The spirit of 
 inquiry soon became rife and with it came a healthy scepticism. 
 In the words of Machiavelli, men began to follow the real 
 truth of things rather than an imaginary view of them. 
 
 The mute evidence of cathedral churches left half completed, 
 or with one spire, or none, after the year 1400 or 1500, testifies 
 to the flow of human enthusiasm and energy toward other 
 channels. That so many mighty cathedrals could be con- 
 structed in Europe from 900 to 1400 A.D., without advance 
 in the science of mechanics, seems remarkable. But their 
 excellence is in the field of art and not in that of engineering. 
 
 Close acquaintance with them reveals to the engineer, poor 
 foundations, cracked arches,^ crooked walls and leaning towers^ 
 and settled piers,' quite in accord with the annals* of failure 
 and collapse which is the history of their construction. In 
 what constituted the spirit of their time, in imagination, in 
 fancy, in inversion of idea, in naivete of conception, they are 
 
 »In 1284 the central tower and the apse vaulting of Beauvais Cathedral 
 collapsed utterly. The dome of St. Peter's at Rome would have fallen 
 long since but for the iron bandage of chains placed about the dome in 
 1742 by Vanvitelli under the direction of Poleni. 
 
 ^The Campanile at Bologna is a well-known example. 
 
 sAnnales de Sevilla, 1677, "On Dec. 28, 151 1. a split pillar (of Seville 
 Cathedral) brought down all the central tower and three great arches with 
 a noise that stunned the city. . . . By a miracle of Our Lady of the Sea it 
 did not fall at once. . . . The Archbishop granted indulgences to all who 
 would assist in clearing away the debris." In 1890 it collapsed again. — The 
 utter and complete collapse of the Campanile of San Marco at Venice in 
 1902 is recent history. 
 
 *See Hamlin, p. 197, and Feree, Chronology of Cathedral Churches in 
 France. — 'The unscientific Romanesque vaulting, etc., resulted in the entire 
 reconstruction of the cathedrals of Bayeux Bayonne, Cambray, Evreux, 
 Laon, Lisieux, Le Mans, Noyon, Poitiers, Senlis, Soissons and Troyes about 
 1200," etc. 
 
44 THE SCIENCE OF MECHANICS. 
 
 wonderful, but to the trained eye of the engineer, the method 
 of trial and blunder through which they were achieved is 
 apparent. 
 
 They are works of art par excellence, there is little science 
 here, except the experienced skill of the master masons, 
 whose closely guarded guild secrets seem to have been trade 
 
 / tricks rather than a science of statics. No evidence has been 
 discovered tending to prove that the cathedral builders had 
 
 \ any clear conception of the law of action and reaction, or of 
 I the general principle of moments, but they may have used a 
 crude method of determining the ratio of stresses by the funi- 
 cular method of using weighted strings passing over pulleys. 
 The first formal exposition of this method seems to be in the 
 works of the Flemish engineer Stevinus who was not born till 
 
 v_ 1548- 
 
 The opening of the Renaissance found the science of me- 
 chanics not very much further advanced than where Archi- 
 medes had left it. Now men began to study and speculate 
 on the subject. Most eminent and successful among those 
 who so occupied themselves are the following : 
 
 1. Copernicus (1473-1543), of Thorn in Prussia, who set 
 forth the system of astronomy since identified with his 
 name in "De Revolutionibus Orbium Coelestium." He main- 
 tained that the sun is at rest and that the planets revolve 
 about it, and hinted that theology and mechanics are two 
 distinct branches of knowledge. This quotation dimly presag- 
 ing the law of gravitation is interesting: "I am of the opinion 
 that gravity is nothing more than a natural tendency im- 
 planted in particles by the Divine Master by virtue of which, 
 they collecting together in the shape of a sphere do form their 
 own proper unity and integrity. And it is also to be assumed 
 that this propensity is inherent in the sun, the moon and the 
 other planets." 
 
 2. Leonardo da Vinci (1452-1519), the Italian painter, 
 whose manuscripts give a crude idea of the statical moment. 
 
 3. Peter Ramus (1515-1572), who contended in his thesis 
 for the Master's degree at the College de Navarre that all 
 that Aristotle taught was false. In his "Animadversiones in 
 
4- The Contribution of Simon Stevinus 
 ( 1 548-1 620). 
 
 Simon Stevinus of Bruges, a military engineer of Prince 
 Maurice of Orange seems to have been a man of genius in 
 experimental research as well as in practical engineering. His 
 earliest extant work is the "Beghinseln der Weegkonst" pub- 
 lished in Dutch at Leyden in 1586. The full account of his 
 researches is given in "Hypomnemata Mathematica" (Mathe- 
 matical Memoranda) a large volume in Latin published at 
 Leyden 1608. 
 
 This volume covers in six books, the topics, arithmetic, 
 geometry, cosmography, practical geometry, statics, optics and 
 fortifications. The division on Statics treats of, 
 
 1. The elements of statics. 
 
 2. The theory of center of gravity. 
 
 3. Practical statics. 
 
 4. First principles of hydrostatics. 
 
 5. Practical hydrostatics. , 
 
 6. Miscellaneous topics. 
 
 This curious medley of theory and practical hints was no doubt 
 the encyclopedia of mathematics and mechanics of the period. 
 A revised edition in French was published by Albert Gerard in 
 1634. Both editions are very fully illustrated with wood cuts. 
 We do not find in it any mention of dynamics. Statics is 
 defined as the interpretation of the computations, proportions 
 and conditions of equilibrium (pondus) and of weight (gravi- 
 
 THE MEDIEVAL PERIOD. 45 
 
 Dialecticam Aristotelis," 1543, he strenuously opposed the 
 scholastic dogmas. ^ 
 
 4. Guido Ubaldi, an Italian, who published in his "Mechani- / 
 corum Liber" (1577), an imperfect idea of the statical moment. I 
 
 Of this period is the work of two of the great contributors 
 to the science of mechanics, one in the field of statics, and the 
 other in the field of dynamics, of which he was the founder — 
 Simon Stevinus, an engineer of Bruges (1548-1620), and 
 Galileo, a professor of Florence (1564-1642). 
 
46 THE SCIENCE OF MECHANICS. 
 
 tas). The weight of a body is defined as its (potentia descen- 
 sus in dato loco) force of descent in a given place. Center of 
 gravity is clearly conceived and defined. 
 
 Whereas Archimedes considered only the action of parallel 
 forces at right angles to the lever, Stevinus considers the 
 action of forces in any direction and at any angle. He was 
 the first to give a solution of the problem of stability or in- 
 stability on an inclined plane. His presentation of the simple 
 machines differs from that of Archimedes in that he uses the 
 graphic method of the triangle of forces in the solution of them. 
 
 His principal contribution to the science is this idea of the 
 parallelogram or triangle of forces which he gives by many 
 graphical examples without definitely proving it as a general 
 principle at the beginning. It was not completely stated and 
 generally admitted as a principle until about ninety years 
 later when Varignon proved it geometrically and set it forth 
 in a paper before the Paris Academy (1687). In the same year 
 Newton and Lami also published a proof. 
 
 It is worthy of note that the first practical exposition of 
 the solution of engineering problems by graphical representa- 
 tion of forces or funicular polygons, now so commonly in use 
 to-day under the name of "graphic statics" was published by 
 engineer Stevinus, about three hundred years ago. 
 
 He arrived at the conception of the triangle of forces and 
 the conditions of stability on an inclined plane by his famous 
 "chain of balls on prism" experiment. This is given in the 
 Hypomnemata Mathematica as follows: 
 
 /^I Theorem. Proposition 19. If a plane triangle is placed 
 vertically with the base parallel to the horizon, and upon the 
 other two sides are placed single globes in equilibrium then 
 according as the right side of the triangle is to the left so is the 
 balancing effect of the left globe to the counterbalancing effect 
 of the right globe. -;,5 
 
 Given: Let ABC be the vertical triangle (Fig.^t) with base 
 parallel to the horizon with side AB double BC, ar^d let the 
 globe Aon AB be of equal size and weight to that £ on" 5 C. 
 
 Question: Demonstrate to us that as AB (2) is to BC (i) 
 so the balancing effect of globe E is to the counterbalancing 
 globe D. 
 
THE MEDI/EVAL PERIOD. 47 
 
 Construction. Let us arrange a crown of fourteen balls of 
 equal size and weight strung together at equal intervals, and 
 let there be three fixed points STV which are touched by the 
 string so to admit of motion of ascent or descent of the string 
 of balls. 
 
 Demonstration: If the balancing effect of the globes DRQP 
 is not equal to that of EF they must be heavier. Suppose 
 they are, then ONML being equal to GHIK, the eight globes 
 
 Fig. 5. 
 (From Liber Statica, Vol. IV, p. 34.) 
 
 D, R, Q, P, 0, N, M and L must overbalance the six E, F, G, 
 H, I and K and the eight will go down and the six will rise up. 
 D will go down to and I and K will take the place of E and 
 F. But, if this is so, the string of globes will now be situated 
 as before and by the same cause the eight globes on the left 
 will go down and the six on the right will go up, which is 
 saying that the globes of themselves produce continual and 
 eternal motion. This is false. Therefore the part of the 
 string DRQPNML holds the part EFGHIK in equilibrium. 
 If from equal things equal are taken, equals remain, therefore 
 subtracting ONML and GHIK, DRPQ balances EF. But 
 four being held in equilibrium by two, E must be doubly as 
 effective as D. Therefore as the side BA (2) is to the side 
 BC (i) so the balancing effect of globe E is the counter- 
 balancing effect of globe D. 
 
 As a corollary it follows that the four balls and the two balls 
 
48 
 
 THE SCIENCE OF MECHANICS. 
 
 may be concentrated in globes of corresponding magnitudes 
 as indicated in Fig. ^. Or a device like Fig. ^ may be em- 
 ployed. 3p'] 3;o 
 (..^o-- 'From this Stevinus comes by corollary to the study of the 
 condition of equilibrium on an inclined plane, which he proves 
 
 Fig. 6. 
 
 (From Liber Statica, Vol. IV, p. 36.) 
 
 by the diagram, Fig. 4. This diagram is the earliest exposition 
 of the triangle of forces. 
 
 He then generalizes the principle for practical use, in Figs;, 
 5 and 6, where CE is to EO as the weight of the body is to the/ 
 pull P. From this principle the theory of the funicular polygon 
 is then developed as indicated in Figs. 1^ and ^. 
 
 jji- 31:5 
 
 Fig. 8. 
 
 Fig. 
 
 From the funicular polygon he advanced to the considera- 
 tion of the conditions of statical equilibrium in each of the 
 simple machines, referring back to his proof of the inclined 
 plane. Nowhere does he state the principle of the parallelo- 
 gram of forces explicitly as a general rule from which all cases 
 of equilibrium in machines may be deduced. In the chapter 
 on practical statics the simple machines are fully expounded 
 
THE MEDIEVAL PERIOD. 
 
 49 
 
 and their applications indicated in illustrations showing cask 
 being moved into warehouses, etc. 
 
 The chapter on hydrostatics is also very practical. The 
 weight of a cubic foot of water at Leyden is noted (62 pounds) , 
 
 Fig. 10. 
 
 Fig. II. 
 
 (From Liber Statica, Vol. IV, p. 162.) 
 
 and suggestions on ship design are given. It is a question as 
 to how much of Archimedes' Hydrostatics was known to 
 Stevinus, but the probability is strong that Stevinus dis- 
 covered, or at least proved the principle of Archimedes by his 
 
 ^ 
 
 WW 
 
 B 
 
 Fig. 12. Fig. 13. Fig. 14. 
 
 (From de Hydrostatices Elementis, p. 119.) 
 
 own method. He clearly set forth for the first time the fact 
 that the pressure of a liquid is independent of the shape of the 
 containing vessel and depends upon the height and area of the 
 base. His method of reasoning is simple and convincing and 
 worthy of quotation. 
 
50 THE SCIENCE OF MECHANICS. 
 
 Suppose a mass of water A in a jar of still water. This cube 
 is in equilibrium. For, if not, let us suppose it descends, then 
 the water which comes into its place must also descend when 
 it comes into that same place and under similar conditions; 
 but, this leads to perpetual motion which is absurd and con- 
 trary to our experiences. Therefore the cube A does not move 
 down nor up. It is in equilibrium. If now we suppose the 
 surface of the cube A to become solidified, this surface or 
 "vas superficiarium" will be subjected to the same circum- 
 stances of pressure. 
 
 When it is empty it will suffer an upward pressure equal to 
 the weight of the absent water which balanced the upward 
 pressure. If we fill it with any other substance of any specific 
 gravity it is plain that the loss in weight of that substance in 
 water is equal to this same upward pressure which is equal 
 to the weight of the water displaced. Figs. 9 and 10 illustrate 
 experimental proofs with cubes of specific gravities 1-5 and 
 4 times that of the fluid. 
 
 Granted that the pressure on the base of a cube or vertical 
 parallelopiped of liquid is equal to its weight, by following 
 a similar method of imagining portions of the liquid to become 
 solidified or to be cut out, Stevinus shows that the pressure 
 on the base of a vessel is independent of its form, and proves 
 the laws of pressure of communicating vessels and tubes. 
 
 Perusal of Stevinus' notes indicates that he had a hazy 
 idea of the principle of virtual displacements. He had ob- 
 served that what a simple machine gains in force it loses in 
 distance. In his discussion of pulleys he notes that, "Ut 
 spatium agentis ad spatium patientis, sic potentia patientis 
 ad potentiam agentis" (Vol. IV, L. 3) as the space passed 
 over by the force is to the space passed over by the resistance 
 so is the resisting force to applied force. Here he strikes close 
 to the principle of work, namely that in a perfect machine the 
 product of the force and distance traversed is equal to the 
 resistance times the distance through which it is overcome. 
 
 We have here in Stevinus' book the germ of the idea of 
 virtual displacements. That is, if in a simple machine we 
 consider any virtual or possible displacement of the agent, 
 
THE MEDIEVAL PERIOD. 5I 
 
 the resistance moves over a corresponding displacement so 
 that in every case the product of the acting force and its dis- 
 placement is equal to the resisting force times its displace- 
 ment. 
 
 Stevinus utilized this idea in a narrow limited way, applying 
 it in the calculations on the simple machines but not attaining 
 to the idea of work, as the measure of force acting through 
 distance, nor to the idea of the balance of positive and negative 
 work in a machine. His chief contributions are the statical 
 principle of the triangle of forces, the founding of Graphic 
 Statics, and the exposition of the conditions of buoyancy and 
 liquid pressure. While he was developing Statics in these 
 directions, his young contemporary Galileo had been experi- 
 menting with moving bodies and was laying the foundations 
 of Dynamics. 
 
 REFERENCES. 
 
 Ennemoser. The History of Magic. 
 
 Rydberg. Magic in the Middle Ages. 
 
 Dollinger. Studies in European History. 
 
 Adams, B. The Law of Civilization and Decay. 
 
 Whewell. History of the Inductive Sciences. 
 
 Melanchton. Initia Doctrinae Physicse. 
 
 Bacon. Novum Organum. 
 
 Duhring. Geschichte der Mechanik. 
 
 Heller. Geschichte der Physik. 
 
 Eichen. Mittelalterische Weltanschauung. 
 
 Schneider. Geschichte der Alchemic. 
 
 Figuier. L'alchimie et les Alchimistes. 
 
 Cuvier. Histoire de Sciences Naturelles. 
 
 Maury. L'Antiquite et au Moyen Age. 
 
 Fahie, J. J. Galileo, his life and work. 
 
 Vivian. Life of Galileo. 
 
 Boundry, F. Galilee, sa vie. 
 
 Lord. Beacon Lights of History. 
 
 Mach. The Science of Mechanics. 
 
 Ball, J. W. The History of Mathematics. 
 
 Cajori, F. A History of Physics. 
 
 Alberi. Opere — Geo lettore Galilei, 16 vols. 
 
 Mahafy. Des Cartes. 
 
 Stevinus. Hyponemata Mathematica. 
 
 Nasmith. Pascal. 
 
 Gerland & Traumiiller. Geschichte der Physical Experimentier Kunst. 
 
 Brewster. Martyrs of Science. 
 
 Lodge, Sir. O. Pioneers of Science. 
 
52 the science of mechanics. 
 
 5. The Contribution of Galileo Galilei 
 (f^o/ (1564-1642). 
 
 Taking up now the work of Galileo we find that he caused 
 a revolution in mental attitude toward the study of natural 
 phenomena. The Aristotelian Natural Philosophy had for 
 centuries been regarded as an infallible authority in the schools. 
 In 1543, Petrus Ramus (1515-1572), a scholar of the University 
 of Paris, was forbidden by an edict of Francis I, under pain 
 of punishment, to teach or write against it. To Galileo, a 
 young medical student of noble Florentine family, who had 
 come to disbelieve in the dogmas of the old philosophy belongs 
 in part the glory of emancipating men's mind from this author- 
 ity of antiquity. 
 
 Galileo appealed from apriori axioms, presuppositions and 
 syllogistic deductions to an investigation of the actual facts. 
 The teachings of Aristotle had been received, "ipse dixit," 
 up to this time, in spite of the fact that some of them were 
 contradicted by daily experience, and in spite of the fact that 
 easy, simple experiments proved them wrong. 
 
 To quote but a few of these Aristotelian notions which 
 were blindly accepted and believed — 
 
 1. Substances were divided into "corruptible" and "in- 
 corruptible," chief among the latter were the heavenly bodies. 
 
 2. Bodies were classified as absolute heavy bodies and ab- 
 solute light bodies and "sought their places"; the light bodies 
 belonging up and the heavy bodies down. 
 
 3. Motions were classified as "natural motions" and "violent 
 motions." 
 
 4. Large bodies were believed to fall quicker than small 
 ones, or the velocity of falling bodies was believed to be in 
 proportion to their weight. 
 
 Galileo vigorously attacked this Aristotelian philosophy; 
 he appealed from authority to experiment, to nature. He 
 boldly contradicted the teachings of Aristotle, which had been 
 accepted and believed for over a thousand years. By direct 
 experiment, as for example, by dropping weights from the 
 leaning tower of Pisa he proved that Aristotle was wrong. 
 
THE MEDIEVAL PERIOD. 53 
 
 The schoolmen of the time did not readily relinquish their 
 errors and carried on long and bitter controversies with him 
 so that he was obliged to leave Pisa for Padua. 
 
 His keenness of perception is well illustrated by the story 
 of the discovery of the isochronism of the pendulum through 
 observations on the gradually decreasing swing of a hanging 
 lamp in Pisa Cathedral. He counted his pulse as the lamp 
 oscillated over a smaller and smaller arc and found that the 
 number remained constant, thus verifying his suspicion of 
 isochronism. He is also credited with the first determination 
 of the relation between the time and length of a pendulum 
 and the application of it in a metronome for the use of physi- 
 cians. A scheme for a pendulum clock, which he never realized, 
 is found among his manuscripts. 
 
 Galileo seems to have been the first to set forth clearly : 
 
 1 . The idea of force as a mechanical agent. 
 
 2. The conception of mechanical invariability of cause and 
 effect. 
 
 3. The principle of the independence of action of simul- 
 taneous forces. 
 
 The rigorous mechanical explanation of motion dates from 
 Galileo. He studied carefully the motions of falling bodies 
 and projectiles and found their laws setting forth: 
 
 1. All bodies fall from the same height in equal times. 
 
 2. In falling the final velocities are proportional to the times. 
 
 3. The spaces fallen through are proportional to the squares 
 of the times. 
 
 He came to these laws experimentally by collecting data 
 on the time of descent, the final velocities and the distances 
 traversed, as in the following table, g being a constant. 
 
 Time. Velocity. Space. 
 
 1. Ig lXlg/2 
 
 2. 2g 2X2g/2 
 
 3. 32 3X3g/2 
 
 4. 42 4X4g/2 
 t tg tX tgl2 
 
 The experiment on grooved planes by which these results 
 were obtained are now well known. An inspection of the 
 table shows at once that the numbers follow the simple law, 
 
54 THE SCIENCE OF MECHANICS. 
 
 V varies as gi, 
 
 which expresses the relation between the first and second 
 columns, 
 
 5 varies as gt^/2, 
 
 which expresses the relation between column one and column 
 three, while 
 
 5 varies as v^/2g, 
 
 is the relation of the second and third columns. 
 
 The first two of these expressions, vccgt and socgt^/2 were 
 used by Galileo in his development of dynamics to the neglect 
 of S(xv^/2g. Later, Huygens took up the expression s<xv^l2g, 
 and made important advances based upon it. 
 
 It was observed in the discussions of moving bodies after 
 Galileo's time that a moving body had a certain "efficacy" — 
 that there was inherent in a moving body, something that cor- 
 responds to force. Later philosophers debated strenuously as to 
 whether this efficacy was propottional to the velocity or to the 
 velocity squared. But it will be perceived from an inspection of 
 the above expressions that a body with double the velocity can 
 overcome a given force through double the time, but through 
 four times the distance. With respect to time, therefore, its 
 efficacy is proportional to velocity; but with respect to dis- 
 tance, or space traversed, its efficacy is proportional to the 
 velocity squared. 
 
 Before the time of Galileo, force was treated in mechanics 
 only as pressure; after his time the ideals of force, velocity 
 and acceleration as we know them to-day came into use. That 
 either acceleration of motion or change of shape is the imme- 
 diate effect of force is the fact that Galileo perceived and set 
 down as a fundamental and invariable rule of dynamics. 
 
 He determines force by the change of velocity, or the ac- 
 celeration it produces, and he may be said to have discovered 
 the law of inertia indirectly. At all events, his conception of 
 dynamics might be expressed by the formula F = m.a, though 
 he did not so express it because his conception of mass was 
 not clear. He made no use of the expression .S = v^/2g, which 
 led Huygens to his conception of energy, later formulated as 
 
THE MEDIEVAL PERIOD. 55 
 
 F.S = mv^/2. His failure to do so appears to be due to the 
 same cause — he did not fully grasp the modern conception of 
 mass. - / 
 
 In Galileo's "Delia Scienza Mecanica" (1655), Tom. i, p. 
 265, appears the first clear presentation of the principle of 
 virtual velocities. This idea is now common property in sev- 
 eral forms, one of which is the familiar dictum, "what is gained 
 in speed is lost in power." It developed slowly into the law 
 of conservation. 
 
 In completer form it is now stated as follows: If a material 
 system, acted on by any forces whatever, be in equilibrium; 
 and we conceive the system to experience, consistently with 
 its geometrical relations, any indefinitely small arbitrary dis- 
 placement; the sum of the forces multiplied each of them by 
 the resolved part parallel to its direction, of the space described 
 by its point of application, will be equal to zero; this resolved 
 part being considered positive when it lies in the direction of 
 its corresponding force, and negative when in an opposite 
 direction. 
 
 Though Guido Ubaldi in his "Mechanicorum Liber" called 
 attention to the idea of virtual displacement and moments in 
 connection with the lever, and though Stevinus makes mention 
 of it, Galileo appears to have been the first to apply these 
 ideas to all the simple machines. The term "moment" of a 
 force seems to have meant to Galileo, the effort that tended 
 to set a machine in motion. Therefore, in order that a 
 machine should remain at rest or in equilibrium under the 
 action of two forces it is necessary that their moments balance. 
 He showed that the moments of a force are always proportional 
 to the force times its virtual velocity. 
 
 In his "Mechanica, sive de Motu," Wallis uses the term 
 moment in this same sense and bases his statics on the equality 
 of moments as a fundamental principle. 
 
 This idea is most prolific and many later writers used it in 
 varied form as the basis of their formal presentation of me- 
 chanics Descartes for example in Lettre 73, Tom. i, "de 
 Mechanica Tractatus" (1657), bases his whole treatment of 
 Statics on a single principle which is essentially Galileo's idea 
 
56 THE SCIENCE OF MECHANICS. 
 
 of virtual velocities. His conception of it is, that it requires 
 exactly the same energy to raise a weight P through an altitude 
 .4, as a weight Q through an altitude B, provided that P is to 
 ^ as 5 is to ^. It follows from this that any two weights 
 attached to a machine will be in equilibrium when they are 
 disposed in such a way that the small paths they can simul- 
 taneously describe are reciprocally as their weights. 
 
 The same idea was presented in another aspect by Torricelli 
 in "De Motu Gravium Naturaliter descendentium" (1644). 
 His conception was, that when any two weights rigidly con- 
 nected together, are so placed that the center of gravity is in 
 the lowest position which it can assume consistently with the 
 geometrical conditions, they will be in equilibrium. Torri- 
 celli's principle was finally presented in the form, — any system 
 of heavy bodies will be in equilibrium when their center of 
 gravity is in its lowest or highest position. His presentation 
 was based on Galileo's conception of virtual velocities. 
 
 Finally a century later, it was stated about as we have it 
 to-day in general terms by John Bernoulli, in his letter to Varig- 
 non dated. Bale, Jan. 26, 1717, published in "Nouvelle Mecan- 
 ique," Tom. H, sect. 9. 
 
 These ideas gave a new trend to the development of mechan- 
 ics. In 1743, D'Alembert built the first Treatise on Dynamics 
 on this principle. Had this idea of virtual displacements been 
 clearly perceived and appreciated by Archimedes, mechanics as 
 a science would have developed much more rapidly. This 
 principle is of such universal application that a separate rule 
 is no longer necessary for each of the simple machines; it 
 suffices for them all. 
 
 To Bernoulli belongs the credit of showing that the prin- 
 ciple of virtual displacements may be made the basis of a 
 whole theory of equilibrium, but the idea originated with 
 Galileo. He also applied the principle in his "Discourse on 
 Floating Bodies" demonstrating by it the theory of buoyancy. 
 In spite of these expositions some of Galileo's opponents still 
 held blindly to the Aristotelian theory that the breadth or 
 form of a body was the factor that determined whether it 
 sank or floated. 
 
THE MEDIAEVAL PERIOD. 57 
 
 In general, Galileo prepared and familiarized men's minds 
 with the correct notion of interdependence of force and motion 
 thus clearing the way for the generalizations of Newton's Laws 
 of Motion. Nowhere does he state these laws explicitly, but 
 their perception is involved in the solution of some of the 
 dynamical problems in his books. He even gives two of the 
 laws of motion in an incomplete way. The first law of Newton 
 is a generalization of Galileo's theory of uniform motion. The 
 second law, that change of motion is due to force and is pro- 
 portional to the force that makes the change, and takes place 
 in the direction of the force, is a generalization of Galileo's 
 theory of projectile motion. 
 
 Before this time, it was commonly believed that a body 
 could not be affected by more than one force at a time, and 
 it was even held that a ball shot horizontally, moved in a 
 straight line until the force was spent and then fell vertically 
 to the earth. Galileo demonstrated in his fourth Dialogue 
 that the path of a projectile must be a parabola, the resultant 
 of a uniform transverse motion and a uniformly accelerated 
 vertical motion. 
 
 He, however, did not attain to a clear discrimination between 
 mass and weight, and he failed to see that acceleration might 
 be made a means of measuring the magnitude of the force of 
 gravity. There is no statement of the third law of motion, 
 in reference to action and reaction, anywhere in Galileo's work, 
 though there is a suggestion of the idea of it in some of his 
 statements in the "Delia Scienza Mecanica." 
 
 Galileo not only founded Dynamics but he made perfectly 
 clear the fact that force may produce two effects upon bodies, 
 change their motion, that is give them acceleration, or it may 
 change their form or shape, that is deform them. In his study 
 of the first effect he developed the dynamical laws of falling 
 bodies, of projectiles and of the pendulum, in the later he 
 founded the study of the resistance of materials. His crude 
 investigations as to the internal structure of matter and his 
 theory of its deformation and resistance in the form of col- 
 umns, posts, beams and cantilevers is set forth in his book 
 "Discorsi e dimostrazioni matematiche intorno a due nuove 
 
58 THE SCIENCE OF MECHANICS. 
 
 scienze," published in Leyden, 1638. This work attracted 
 little or no attention at the time but it is one of Galileo's most 
 substantial contributions. Although he wrote some sixteen 
 volumes in all, this work and his "Discorso interno alle cose 
 che stanno in sur I'acqua" in which he proves the static law 
 of fluid pressure, contain nearly all his research in mechanics. 
 
 After Galileo's time we no longer find such naive and obscure 
 phraseology as, "motions are of two orders, natural and vio- 
 lent." Henceforth the notions of the Aristotelians became 
 untenable. Men soon came to recognize that all bodies, even 
 the heavenly bodies, were probably of one kind. Force came 
 to be understood as that which causes acceleration in a body, 
 or deformation in a body. It became apparent that in- 
 stantaneous and continuous forces produce unlike effects, and 
 that weight is a continuous force drawing bodies toward the 
 earth. A little later the great Newton explained how it was 
 that all bodies fall with equal velocities from the same height, 
 barring the unequal resistance of the air. 
 
 When these things had been pointed out and verified by 
 experiment, the foundati^on of the study of moving bodies was 
 laid and the progress ^f Dynamics was sure and steady. 
 Newton's generalizations followed logically upon Galileo's dis- 
 cussions of motion, and D'Alembert's Treatise on Dynamics 
 came as an expansion of these ideas. It is no exaggeration 
 to say that to Galileo we owe modern mechanics. Lagrange 
 in his "Mecanique Analytique" testifies to Galileo's greatness 
 in these words: 
 
 "Dynamics is the science of forces accelerating or retarding, 
 and of the various movements which these forces can produce. 
 This science is entirely due to moderns, and Galileo is the one 
 who laid its foundations. Before him philosophers considered 
 the forces which act on bodies in a state of equilibrium only; 
 and although they could only attribute in a vague way the 
 acceleration of heavy bodies, and the curvilinear movements 
 of projectiles, to the constant action of gravity, nobody had 
 yet succeeded in determining the laws of these daily phe- 
 nomena on the basis of a cause so simple. Galileo made the 
 first important steps, and thereby opened a way, new and 
 immense, to the advance of mechanics as a Science. 
 
THE MEDIEVAL PERIOD. 59 
 
 "These discoveries did not bring to him while living as 
 much celebrity as those which he had made in the heavens; 
 but to-day his work in mechanics forms the most soHd and 
 most real part of the glory of this great man. The discovery 
 of Jupiter's satellites, of the phases of Venus, of the Sun-spots, 
 etc., required only a telescope and assiduity; but it required 
 an extraordinary genius to unravel the laws of nature in 
 phenomena which one has always under the eye, but the 
 explanation of which, nevertheless, had always baffled the 
 researches of philosophers." 
 
PART III. 
 
 THE MODERN PERIOD, 1500 TO 1900. 
 
 We no longer believe with the cave-men that thunder is 
 the roar of an angry god, nor with Luther that a stone thrown 
 into a pond will cause a dreadful storm because of the wrath 
 of devils kept in prison there; but we still believe with them 
 that wood floats, and we have clear ideas of the conditions 
 of its buoyancy. The characteristics of the modern period 
 are its empiricism, the great and increasing part played by 
 natural knowledge, and a strong conviction of the importance 
 of sense impressions as a source of knowledge. Added to this 
 we observe an enthusiasm for research and a determination 
 to expose error regardless of controversy or consequences. 
 
 Since 1700 the whole outlook upon the universe has changed. 
 Science has routed the old theology, and altered the habits 
 of life of millions by its influence in the trades and industries. 
 Though there is still nothing more mysterious than force, the 
 imps of that weird ante-world of Science that lurked in every 
 zephyr and grinned from every tree and dark nook are now 
 no more. Quite apart from the comforts of life that we owe 
 to mechanics, we are indebted to the science for the peace 
 of mind which a rational Natural Philosophy has brought us. 
 
 Since Galileo's time mechanics has been characterized by 
 an attitude of direct experimental inquiry which has sought 
 to test and extend the conceptions already formed. The 
 science has grown by slow expansion and accretion, and it has 
 often been some time before new conceptions have become 
 susceptible of precise statement. It seems as though an idea 
 must be considered and turned over by many minds before it 
 can be clearly set forth. Even the great masters have not 
 always presented the principles which they have contributed, 
 in the form in which we now state them. 
 
 Thus it is, that a clear statement of the ideas developed 
 in this later period is not attained to, until a number of 
 
 60 
 
THE MODERN PERIOD. 6l 
 
 workers have cultivated the field, and until each has developed 
 separately facts, which when correlated and worked over, by 
 a master mind, give us an illuminating view of the whole 
 subject. 
 
 So, in passing from the work of Galileo to that of the next 
 great master Christian Huygens (i 629-1 696), we pass over 
 a number of men whose work was one of preparation, ex- 
 tension and amplification, rather than of new contribution. 
 Such were these workers whose activities can be but briefly 
 referred to in this outline of. the history of mechanics. 
 
 Johann Kepler, 1 571-1630, of Wiirtemburg, Germany, who 
 in "Astronomia Nova," 1609, and "Harmonice Mundi," 1619, 
 set forth the three laws of planetary motion, viz: 
 
 (i) Each planet revolves in an elliptic orbit having the 
 sun as its focus; (2) the straight line joining the sun and 
 planet passes over equal areas in equal times; (3) the square 
 of the time of revolution of each planet is proportional to 
 the cube of its mean distance from the sun. Here we have 
 the statement that the solar system is disposed according to 
 mathematical and mechanical law. 
 
 Francis Bacon (i 561-1626), author of Novum Organum 
 (1620), who declared for science on the ground that "knowledge 
 is power" and who advocated an experimental study of the 
 world with a view to improving human conditions. 
 
 Marcus Marci, 1595-1667, published at Prague in 1639 
 "De Proportione Motus" in which he gives correct elementary 
 notions of impact. 
 
 Rene Descartes, 1596-1650, published at Amsterdam in 
 1644 his "Principia Philosophia;" in which we have the first 
 notable modern endeavor to formulate a system of mechanics 
 from the universal point of view. His scheme is objectionable, 
 but he called attention to the problem of a universal mechanical 
 philosophy. 
 
 Gilles Personne de Roberval, 1 602-1 675, published in the 
 Memoirs of the French academy, 1668, a notable paper "Sur 
 la Composition de Movements." 
 
62 THE SCIENCE OF MECHANICS. 
 
 Otto Von Guericke, 1602-1686, of Magdeburg, Germany, an 
 engineer in the army of Gustavus Adolphus published "De 
 Vacuo Spatio," 1663, and "Experimenta Nova," 1672, giving 
 an account of his invention of the air-pump, 1650, and various 
 experiments performed with it. 
 
 Pierre de Fermat, 1601-1665, of Montauban, France, pub- 
 lished between 1670 and 1680 a series of monographs on 
 maxima and minima, tangents, curves, centers of gravity 
 and copies of his correspondence with Descartes, Huygens 
 and Pascal on mechanical problems, under the title "Opera 
 Mathematica," which cleared the way for later advance. 
 
 Evangelista Torricelli, 1608-1647, constructed the first 
 mercurial barometer about 1643 and applied it to the measure- 
 ment of variations in atmospheric pressure. 
 
 Edme Mariotte, c. 1620-1684, who in his "Trait^ du Mouve- 
 ment des Eaux," 1686, published the first treatise on hydraulics 
 and advocated and developed experimental research on gravi- 
 tation, hydraulics and pneumatics. 
 
 Robert Boyle, 1627-1691, who first formulated what is now 
 known as'^Buyles Law," viz : that when a gas is at a constant 
 temperature, the product of the pressure and volume remains 
 constant, howsoever one of these be varied. 
 
 Blaise Pascal, 1 623-1 662, who published his studies on the 
 question of fluid pressure in "Recit de la grande experience de 
 I'equilibre des liquers" (1648), and "Traite de I'equilibre des 
 liquers et de la presanteur de la masse de I'air" (1662). One 
 of his conclusions, commonly known as Pascal's principle is, 
 that "external pressure is transmitted by fluids in all directions 
 without change in the intensity." 
 
 All of these investigators either set forth an idea of some 
 importance or simplified and extended the presentation of 
 accepted ideas. Later investigators were familiar with their 
 work and mounting upon it attained to the higher conceptions 
 of the science. The work of Kepler in astronomy was partic- 
 ularly useful to Newton in attaining to his grand generalization 
 of the law of universal gravitation. The work of Descartes 
 besides presenting a highly useful method of combined alge- 
 
THE MODERN PERIOD. 63 
 
 braic and geometric analysis, suggested the idea of Conserva- 
 tion, while the branches of hydrostatics and pneumatics 
 could not have progressed far without the researches of 
 Guericke, Torricelli, Mariotte, Pascal and Boyle. Their 
 work was one of investigation, experiment and study in rather 
 narrow fields. It was a work of preparation upon which their 
 successors built grandly. 
 
 I. The Contribution of Christian Huygens 
 (1629- 1 696). 
 
 Most of the great masters of mechanics have left a record 
 of practical invention as well as of theoretical advance in the 
 science. Huygens is remembered as the man who first made 
 a good clock. Born at The Hague, and educated at Leyden 
 and Breda, where he studied law and mathematics, he won 
 fame in astronomy as well as in mechanics. In 1665 he dis- 
 covered the rings of Saturn with a telescope which he had con- 
 structed. We are concerned however only with his contri- 
 bution to mechanics. He carried forward Dynamics by de- 
 veloping precise statements of accelerated motion and solving 
 the first problems in the dynamics of several masses. Galileo 
 had always restricted his speculations to a single body. 
 
 Huygens' contributions are set forth in his publications; — 
 "A summary account of the laws of motion," Philosophical 
 Transactions, 1669, "Horologium Oscillatorium," Paris, 1673, 
 and "Opuscula Posthuma," Leyden, 1703. The complete 
 mathematical theory of the pendulum and his invention of 
 the escapement is completely set forth in the "Horologium" 
 (1673), which is a work worthy to rank with Newton's Prin- 
 cipia. He was the first to determine the acceleration of gravity 
 by the pendulum, and also the first to enunciate the formula 
 of centrifugal force, F=mv^/r; his discovery of the laws of 
 collision of elastic bodies was announced simultaneously 
 with that of Wallis and Wren. 
 
 Huygens' great work was a complete exhaustive theory of 
 the pendulum and the solution of the problem of center of 
 oscillation. He considered the pendulum to be made of 
 particles, and originated the idea of the dynamics of more 
 
64 THE SCIENCE OF MECHANICS. 
 
 than one mass or particle. He employed the method of ag- 
 gregating the motions of particles, but though he used symbols 
 for "moment of inertia" and "statical moment," he did not 
 define and assign these names. Euler appears to be responsible 
 for the term moment of inertia (Sw.r^). This method, now 
 so general in its application in the mechanics of solids and 
 liquids and in daily use by engineers and physicists, is one of 
 the great inventions of Huygens. It gave to the science of 
 mechanics a new trend, and when the invention of the calculus 
 made the process of summation easy, this method brought a 
 wealth of progress in dynamics and hydrodynamics. 
 
 In the dispute as to whether the so-called "efficacy" of a 
 moving body is proportional to the first or second power of 
 the velocity, Huygens, who had originated the later idea, 
 maintained it strenuously. The dispute was really one of 
 terms. The "efficacy" of a moving body varies as its velocity 
 in reference to the time and as the square of the velocity in 
 reference to the space passed over. Reference to the time 
 leads to what Descartes called the "quantity of motion" 
 (momentum), w.z?. This makes the notion of force the primary 
 concept. Reference to the distance passed over gives the 
 expression, m.v^, which makes work or energy the primary 
 concept. The first view is expressed by F.t = m.v as the fun- 
 damental equation of mechanics, the second gives: F.s = m.v^ 
 as the fundamental equation. 
 
 In 1847, Belanger proposed the name impulse for the ex- 
 pression F.t, which was later adopted and popularized by 
 Clerk Maxwell in his writing on matter and motion. 
 
 Leibnitz gave the name "vis viva" to the expression m-v"^ 
 in a memoir published in the "Acta Eruditorum," 1695, en- 
 titled "Specimen dynamicum pro admirandis naturae legibus 
 circa corporum vires et mutuas actiones detengendis et ad 
 suas causas revocandis," or "A dynamic illustration of the 
 astonishing laws of the power of bodies in their reciprocal 
 action revealed and traced back to their causes." He in- 
 tended, as the name visa viva suggest, to indicate a measure 
 of the force of a body in actual motion. The term "vis 
 motrix" was also used interchangeably with vis viva to dis- 
 
THE MODERN PERIOD. 65 
 
 tinguish the moving force from statical pressure which was 
 called "vis mortua." 
 
 The difference of opinion really hinged on whether force or 
 energy is to be considered the fundamental notion. Huygens 
 and his party maintained the latter position, while Descartes 
 and later Newton accepted force, mass and momentum as 
 fundamental notions. This dispute went on for fifty years 
 until D'Alembert in 1740 in his "Dynamique" showed it to 
 be a misunderstanding as to terms, not facts. 
 
 Later Coriolis (i 792-1 843) introduced the more common 
 notation of ^w.t;^ for vis viva or kinetic energy, and Poncelet 
 adopted the same plan, but the conception of energy we owe 
 then to Huygens. 
 
 Another of his important achievements was the solution 
 of the problem of center of oscillation. This cannot be done 
 without recourse to the new method which he used so suc- 
 cessfully, namely the method of the dynamics of particles. 
 It is a matter of every-day observation that a long pendulum 
 oscillates more slowly than a shorter one. Therefore if we 
 consider the component particles of a compound pendulum 
 as so many simple pendulums, it is manifest that owing to 
 their connections they all vibrate with only one determinate 
 period of oscillation. There must exist a simple pendulum 
 that has the same time of oscillation as the compound pendulum. 
 Its length measured off on the compound pendulum gives us 
 the particle or point that preserves the same period of oscil- 
 lation as if it were detached and vibrating as a simple pendulum. 
 This point is the center of oscillation. 
 
 The idea which Huygens applied in the solution of this prob- 
 lem is, that in whatever manner the particles of the pendulum 
 may by their mutual interaction modify each other's motions, 
 in every case the velocities acquired in the descent of the pen- 
 dulum will be such only that the center gravity of the particles, 
 whether still in connection or with their connections dissolved, 
 is able to rise to the same height as that from which it fell. 
 
 Huygens' proof in brief is: — Let OK be a linear pendulum 
 made up of a large number of masses set in a line OK. If it 
 be set free it will swing through B to OK' where KX = XK'. 
 s 
 
66 
 
 THE SCIENCE OF MECHANICS. 
 
 The center of gravity will ascend just as high on the second 
 side as it fell on the first. Now if at OX we should suddenly 
 release the individual masses from their connections the masses 
 could by virtue of the velocities impressed upon them by their 
 connections, only attain the same height with respect to center 
 
 ,-^*K' 
 
 Fig. 15. 
 
 Fig. 16. 
 
 of gravity. If the free outward-swinging masses be arrested 
 at the greatest heights they attain, the shorter pendulums 
 will be found below the line 0K\ the longer ones will have 
 passed beyond it, but the center of gravity of the system will 
 be found on OK' in its former position. 
 
 The enforced velocities are proportional to the distances 
 from the axis; therefore, one being given all are determined, 
 and the height of ascent of the center of gravity may be found. 
 Conversely the velocity of any particle is determined by the 
 known height of the center of gravity. So if we know in a 
 pendulum the velocity corresponding to a given distance of 
 descent we know its motion is defined.' 
 
 If now on a compound linear pendulum we cut off the portion 
 L equal to /, and if the pendulum move from its position of 
 greatest displacement to the position of equilibrium, the point 
 
THE MODERN PERIOD. 67 
 
 at the distance / from the axis falls through the distance K. 
 The masses m, m' , m" at the distances r, r', r", will fall the 
 distances rk, r'k, r"k . . . and the distance of descent of the 
 center of gravity will be: 
 
 mrk-hin'r'k-\-ni"r"k-\- ' • • _ 2wr 
 
 If now, the point at the distance / acquires on passing through 
 the point of Equilibrium a velocity v, the height of ascent 
 assuming the dissolution of connections will be v^l2g and the 
 heights of the other particles will be (rvy/2g, (r'vyi2g, {r"vYl2g 
 . . . and the height of ascent of the center of gravity of the 
 liberated masses will be: 
 
 {rvY Ar'vY , m"(r"vy 
 m + m' 1 + • • • 2^ 2 
 
 2g 2g 2g _ v^Zmr^ 
 
 m+m'+m" 2gLm' 
 
 and 
 
 2m 2gSw 
 
 {a) 
 
 But, to find the length of the simple pendulum that has the 
 same period of oscillation as the compound pendulum, it is 
 necessary that the same relation must exist between the dis- 
 tance of its descent and its velocity as in the case of unimpeded 
 fall. If y is the length of this pendulum, ky is the distance 
 of its descent and vy its velocity. Therefore, 
 
 vy'^ 
 
 — = ky 
 2g 
 
 or, 
 
 y-=k. (b) 
 
 2g 
 
 Multiplying equation (a) by equation (&) we get 
 
 y = 
 
 Xmr' 
 
 Here we note Huygens' recognition of work as the deter- 
 minative of velocity, and we see that he measures it in terms 
 
68 THE SCIENCE OF MECHANICS. 
 
 of the second power of the velocity, i. e., V^. This work he 
 called the vis viva of any system of masses such as m, m' , m" , 
 having the velocities v, v', v" as expressed in the formula. 
 
 2 2 2 
 
 Huygens thus clearly indicated that the center of gravity is 
 conserved or cannot rise higher than it falls and in establishing 
 it he sets forth what he calls the principle of vis viva, or the 
 rule that the work or energy is proportional to the mass times 
 the velocity squared. The equation for work and energy 
 F.s = m.v^ is really an algebraic statement of Newton's second 
 law of motion and is fundamental in dynamics. Lagrange 
 based his work on it. 
 
 The writers before Huygens did not have a clear conception 
 of mass as distinguished from weight nor was he perfectly 
 clear on this point, but his endeavors to explain the error of 
 the pendulum clock of the Jean Richer expedition to Cayenne 
 (167 1), by the greater centrifugal force at the equator, show 
 that he had the idea of mass as somehow different from 
 weight. 
 
 Huygens may then be said to have contributed the mechan- 
 ical principles symbolized by those type-expressions and their 
 simple derivatives, 
 (i) 2mr2, 
 
 (2) F = mv'^/r, 
 
 (3) F.s'^m.v^. 
 
 Thus Huygens originated the mathematical method by 
 which the ideas of Galileo were applied to a variety of problems. 
 The development and amplification of these contributions 
 by their successors brought a wealth of progress. Huygens' 
 general way of attacking problems of masses under the action 
 of forces by the method of the dynamics of a particle is in 
 almost daily use by the physicist and the engineer. Like 
 some of the contributions of Archimedes, the contributions 
 of this great master Huygens have an eternal value. 
 
THE MODERN PERIOD. 69 
 
 REFERENCES— HuYGENS. 
 
 (Euvres completes de Christian Huygens, 6 vols. (1888 La Haye). 
 
 Christiani Huygenii de circuli magnitude inventa. 
 
 Christiani Huygenii Zuilichemii Opera Reliqua (Amsterdam, 1728). 
 
 Christiani Huygenii a Zulichem Opera Varia, 1724. 
 
 Christiani Huygenii Zulichemii Opera Mechanica; Geometrica, Astro- 
 
 nomica et miscellanea, 1751. 
 Christiani Huygenii Kosmotheros, 1698. 
 
 2. The Contribution of Sir Isaac Newton 
 
 (1642-1727). 
 
 The year of Galileo's death was the year of Newton's birth. 
 With the passing of the great Italian scientist, a worthy 
 successor, the greatest of all experimental philosophers, was 
 born in England. He established order in the domain of 
 Science and set forth the great laws by and through which 
 mechanics has been able to grow and prosper. 
 
 Newton's contribution may be considered under two head- 
 ings: the development of dynamics, and the applications 
 of dynamics to the great problem of planetary motions. His 
 work is a logical sequence to that of Stevinus, Galileo, Kepler 
 and Huygens. The "Philosophiae Naturalis Principia Mathe- 
 matica" of Newton (1687), commonly called the Principia, 
 is one of the most extraordinary products of human genius, 
 not only in itself but in the revolution which it effected in 
 theoretical and practical mechanics. In it we find much more 
 than a re-statement of the general principles of equilibrium, 
 center of gravity and mechanical powers which were common 
 property at this time. It is a body of doctrine based upon the 
 contributions of all preceding inquirers reduced to the lowest 
 terms. 
 
 The whole body of doctrine on motion of projectiles which 
 had been developed by Galileo, Huygens and others, is re- 
 duced to the concise, comprehensive "axioms" or laws of 
 motion. We have then in the very opening pages of the 
 Principia, a clarification, precipitation and crystallization 
 of all previous contributions. This of itself was a great gain, 
 but it was done by Newton as preliminary to further advance, 
 preliminary to dynamical discussions which in their grand 
 
70 THE SCIENCE OF MECHANICS. 
 
 scope, sweep from the earth to the planets and beyond to 
 the utmost limits of the universe. 
 
 The "composition of forces" he indicates as a corollary to 
 
 I the laws of motion. This was a new idea. In Newton's 
 
 point of view if a body in space is acted upon by an impulsive 
 
 force it moves in a straight line. If at the same time, another 
 
 force acts upon it at an angle inclined to the first force, the 
 
 body takes an intermediate course called a "resultant" path 
 
 / determined by drawing the diagonal of the parallelogram, the 
 
 (_ two sides of which represent the magnitude of the two forces. 
 
 The truth of this principle is made to depend upon the laws 
 of motion. From being a mere statement of experience, as 
 it was with Galileo and Stevinus the parallelogram of forces 
 is correlated to the fundamental laws of motion and deduced 
 as following at once from them. What Galileo and Stevinus 
 said in pages, Newton said in a paragraph. The principle 
 of the parallelogram of forces is not merely stated, it is deduced 
 from three fundamental laws of motion. 
 
 On the same axioms Newton based the whole theory of 
 central forces. He supposes a body to be acted upon by 
 two forces as above, but supposes that the second force acts 
 on it in a new direction in succeeding instants. Then the 
 successive diagonals of the parallelograms of the forces will 
 be successive sides of a polygonal figure and the lines of the 
 deflecting forces will cut one another within the figure. If 
 they meet in a point forming a series of triangles of equal 
 area it is easy to see that the path of the body is the same as 
 though a single force acted upon the body to produce motion 
 forever in a straight line and a second force acted upon it to 
 deflect it continually to a point within the polygon. The limit 
 of such a path, as the polygonal sides become smaller, is a 
 curve. 
 
 In this manner centrifugal forces and curvilinear motion 
 are demonstrated and their laws set forth. Thus a whole 
 system of dynamics is developed from the geometrical and 
 mathematical relations of diagrams of the parallelogram of 
 forces. Newton founded the correct theory of motion about a 
 center and the whole system of dynamics involved with it. 
 
THE MODERN PERIOD. 7 1 
 
 He set forth the principle of equal areas described in equal 
 times as the test of a central force and gave the mathematical 
 proof. 
 
 A body exposed to the action of a central force and given 
 an impulse in a straight line will neither fall toward the central 
 force nor proceed in a straight line, but will take an inter- 
 mediate diagonal or curvilinear path. Newton brought to 
 bear on this problem all the geometrical and mathematical 
 knowledge of his day, as well as his own fluxional calculus, 
 and established the theorem that a body projected in a straight 
 line and subjected to the action of a central force will revolve 
 in some one of the conic sections if the force vary inversely 
 as the square of the distance from the focus, — which of the 
 conic sections, he shows, depends on the ratio of the forces. 
 This dynamical theorem is the starting point of Newton's 
 system of celestial mechanics. 
 
 A variety of consequences follows mathematically from this 
 theorem. The dynamics of elliptic orbits is established at 
 once upon a sound basis and the interrelation of the functions 
 of motion follow inevitably. 
 
 He also took up the abstract theory of the attractions which 
 portions of matter may be conceived to exert upon each other, 
 showing that if the particles be attracted according to the 
 law of inverse square of the distance and if they be aggregated 
 into spherical masses these spheres will themselves attract 
 accordingly to the same law, and that the attraction would be 
 directed to the centers of the spheres and be proportional to 
 the matter contained in them, divided by the square of the 
 distance between the centers. 
 
 From this, to the law of universal gravitation seems but a 
 step. But though Newton is said to have entertained this 
 theory as early as 1666, it was not till 1672, when the data of 
 Picard on the figure of the earth were obtained, that Newton 
 justified it and became convinced of its truth. He first gave 
 it out in his lectures in 1684. It was published latter in his 
 treatise "De Motu" and in the "Principia." In Book III, 
 Proposition 4 of the latter, he calculates the acceleration of 
 the moon toward the earth and shows that starting from 
 
72 THE SCIENCE OF MECHANICS. 
 
 rest with this acceleration, it would fall towards the earth 
 i6 feet in the first minute and that at the earth's surface 60 times 
 nearer, the same distance would here be fallen through in 
 one second which was almost exactly the value obtained by 
 Huygens in his experiments. 
 
 The action of force without a medium to transmit it, appears 
 to have troubled Newton, and this is not surprising when one 
 considers that the pull of the sun on the earth is equal to a 
 force sufficient to break a million million round steel rods 
 each twenty-five feet in diameter. 
 
 The law of universal gravitation is the basic principle of 
 Newton's applications of his dynamics to planetary motions; 
 when he had achieved it, all the mechanism of the universe 
 lay like an open book before him. It was now possible to 
 apply mathematical analysis with absolute precision to the 
 problems of astronomy. 
 
 At once many new conceptions came into view. Neither 
 Galileo nor Huygens had clearly distinguished mass from 
 weight, but now it followed at once, that the same body must 
 have a different weight at different places on the surface of 
 the earth, and might even be conceived imder certain condi- 
 tions to have no weight. We arrive now for the first time at 
 a clear idea of mass. The idea of force ^s first propounded by 
 Galileo was now seen to be of universal application. Finally 
 the law of action and reaction was clearly stated and set forth. 
 These are most illuminating conceptions. 
 
 It began to be evident after this, that gravity was a force 
 measurable like any force in terms of mass and acceleration, 
 though it was some time before the principle was stated in 
 the concise algebraic form, 
 
 F=m.a, 
 W=m.g, 
 
 Newton's concept of mass is in fact the corner-stone of his 
 dynamics. 
 
 At the beginning of the Principia we find a series of funda- 
 mental conceptions given in a series of definitions. The final 
 one refers to mass, as follows: 
 
THE MODERN PERIOD. 73 
 
 Definition I. (As to mass.) "The quantity of any matter 
 is the measure of it by its density and volume conjointly. 
 This quantity is what I shall understand by mass of a body in 
 the discussion below. It is ascertainable from the weight of 
 the body for I have found by pendulum experiments of high 
 precision, that the mass of a body is proportional to its weight, 
 as will hereafter be shown. 
 
 Definition II. "Quantity of Motion" is the measure of 
 it by its velocity and quantity of matter conjointly. 
 
 Definition III. The resident force "vis insita," i. e., 
 inertia of matter is a power of resisting, by which every body, 
 so far as in it lies, persists in its state of rest or of motion in a 
 straight line. 
 
 Definition IV. "An impressed force is any action which 
 changes or tends to change the state of rest or of uniform 
 motion in a straight line." This defines force as the cause 
 of acceleration or tendency to acceleration of a body. 
 
 The laws of motion as Newton enunciates them are: 
 
 Law I. Every body persists in its state of rest or of uni- 
 form motion in a straight line, except in so far as it is com- 
 pelled to change that state by Impressed forces. 
 
 Law II. Change of motion {i. e., of momentum) is pro- 
 portional to the moving force impressed, and takes place in 
 the direction of the straight line in which such is impressed. 
 
 Law III. "Reaction is always equal and opposite to 
 action, that is to say the actions of two bodies upon each 
 other are always equal and directly opposite." 
 
 To these are added a number of corollaries. The first 
 and second relate to the principle of the parallelogram of 
 forces, the others are logical consequences of the laws. Then 
 follow the propositions in two books, the first treating of the 
 motion of bodies in non-resisting media and the second in 
 resisting media. 
 
 The work of Newton may be summed up in his definitions 
 and laws. The great result of his work was the clear concept 
 of mass and the conception that bodies mutually cause ac- 
 celeration in each other dependent upon space and material 
 circumstances. 
 
74 THE SCIENCE OF MECHANICS. 
 
 Mach^ sums the matter up by saying that in reality only 
 one great fact was established, viz: that "different pairs of 
 bodies determine independently of each other, and mutually, 
 in themselves, pairs of accelerations whose terms exhibit a 
 constant ratio, the criterion and characteristic of each pair." 
 
 Newton did not state his results in algebraic terms. He 
 developed the deductions in the "Principia" geometrically 
 though there seems to be no doubt but that many of his con- 
 clusions were arrived at, and perhaps first proven by his 
 method of fluxions or fluxional calculus. 
 
 Had Newton stated his principles algebraically, the formula 
 for force, F=m.a would probably have been expressed by 
 him as F=in.v/t or F.t = m.v. Acceleration he would have 
 expressed as the time rate of change of velocity vjt, and forces 
 he would have measured by their momentum, mXv. Up to 
 Newton's time such formal presentation of the Science of 
 Mechanics as there was, had been made on the geometrical 
 method and Newton following in the steps of Archimedes, 
 Stevinus and Galileo, used this method in his "Principia," 
 representing forces by lines by the graphical method. 
 
 The fact that acceleration may also be expressed as a space 
 rate of velocity squared, or analytically by the formula 
 a = v^l2s is not apparent geometrically. Substitution in the 
 algebraic formula F=m.a gives F.s — m.v'^l2, or the expression 
 for work and energy which is not considered in Newton's 
 geometrical analysis, and which was the point of view from 
 which Descartes, Huygens and Leibnitz approached the 
 subject. 
 
 This difference in point of view gave rise to the long con- 
 troversy already mentioned between the English disciples of 
 Newton and the continental school or the adherents of Huygens 
 and Leibnitz. The so-called Galileo-Newtonian school main- 
 taining that momentum {F.t = m.v) was the only correct 
 measure of force and the Leibnitzian-Huygenian school main- 
 tained with equal vigor that force was a function of the "vis- 
 
 viva or energy I F- 5 = — — I • 
 
 iDr. E. Mach, Science of Mechanics. 
 
THE MODERN PERIOD. 75 
 
 For fifty years mechanics developed along these two sep- 
 arate paths. The English investigators long followed the 
 formal geometrical presentation of Newton, and the French, 
 Germans and Swiss developed their mechanics on the work of 
 Huygens, using the calculus of Leibnitz. It was not till 
 mathematical analysis came to be applied to mechanics and an 
 analytical scheme was developed by D'Alembert in his "Traite 
 de Dynamique" (1743) that they were reconciled, and brought 
 into accord. 
 
 It was then seen that both can be derived from Newton's 
 fundamental equation F=m.a. If the acceleration is meas- 
 ured as a time rate of velocity, we get a — v/t and if the accel- 
 eration is measured as a space rate of the velocity squared 
 we get a = v^/s, which are of the same dimensions. 
 
 Newton made very clear the conception that the effect 
 of a force is to change the size or shape of a body or to change 
 its velocity, that is to give it acceleration. For studying the 
 flux or flowing relations of quantities he devised his "Method 
 of Fluxions" now commonly known as the calculus. The 
 basic idea of his Fluxions is this. He considers a "fluent" as 
 a quantity that gradually and indefinitely increases or flows. 
 The velocities at which such fluents move he defines as fluxions 
 ("Quas Velocitates appello Fluxiones, aut simpliciter Veloci- 
 tates vel Celeritates"). With the development of this method 
 we have at hand an instrument for tracing changing phenomena 
 by the relation or ratio of elements. This is an invention 
 of inexpressible value to Mechanics.^ The method appears 
 to have been first used by Newton as early as 1666 and is 
 found in his MS. "De Analysi per Equationes Numero 
 Terminorum Infinitas," which was given to his students in 
 1669. 
 
 Sir Isaac Newton is to be credited then with a general 
 clarification and formulation of the investigations of all his 
 predecessors, and these specific contributions: 
 
 1. The concept of mass. 
 
 2. The generalization of the idea of force. 
 
 'Prof. John Perry's "Calculus for Engineers" exemplifies the practical 
 value of the Calculus of Newton. 
 
76 THE SCIENCE OF MECHANICS. 
 
 3. The laws of motion. 
 
 4. The theory of central forces. 
 
 5. The theory of attraction. 
 
 6. The system of dynamics based on the conception 
 
 F.t = M.v. 
 
 7. The method of fluxions or the fluxional calculus. 
 
 8. The law of universal gravitation, and the application 
 
 of his abstract dynamics to planetary motions. 
 
 Before Newton the science consisted of the more apparent 
 rules of statics as developed by Archimedes and Stevinus 
 and the uncorrelated principles of dynamics as worked out by 
 Galileo and Huygens. Newton reduced these cumbersome 
 unconnected rules of statics and dynamics to three formal laws 
 of motion and founded a system of dynamics of universal 
 application which has been found all-sufficient to co-ordinate 
 the mechanical phenomena of the universe. 
 
 In cqnclusion we may say that the principles formulated by 
 Newton cover all statical and dynamical problems. Much of 
 the work of later masters has been a verification and an exten- 
 sion of the work begun by him. The scienc of mechanics, as 
 now generally taught, is founded upon them. 
 
 Playfair says^ in his dissertation on Newton, "No one ever 
 left knowledge in a state so different from that in which he 
 found it. Men were instructed not only in new truths, but in 
 new methods of discovering truth ; they were made acquainted 
 with the great principle which connects together the most 
 distant regions of space as well as the most remote periods of 
 duration and which was to lead to future discoveries, far beyond 
 what the wisest or most sanguine could anticipate." 
 
 REFERENCES— Newton. 
 
 Brewster, D. Memoirs of Sir Isaac Newton. (2 vols.) 
 
 Ball, W. R. An essay on Newton's Principia. 
 
 Newton. Principia. 
 
 Mach, E. The Science of Mechanics. 
 
 Ball, W. R. A Short History of Mathematics. 
 
 Cittenden. Life of Newton. 
 
 iPage 133, Dissertation 11, Playf air's "Progress of Math, and Phys. 
 Sciences," 1820. 
 
THE MODERN PERIOD. 77 
 
 Hosley, Opera Omnia, 1779. 
 
 Maxwell. Matter and Motion. 
 
 Thomson & Trait. Natural Philosophy. 
 
 Cajori, F. A History of Physics. 
 
 Hersley's Newton. London, 1785. 
 
 Pearson, C. The Grammar of Science. 
 
 Pemberton. View of Newton's Philosophy. 
 
 Playfair. Progress of Mathematical and Physical Sciences, 1820. 
 
 Perry, J. Calculus for Engineers. 
 
 3. The Contributions of Varignon, Leibnitz, the 
 Bernoullis, Euler and D'Alembert. 
 
 Pierre Varignon (1654-1722). S 
 
 In the same year that Newton pubHshed his Principia (1687), 
 there was presented before the Paris Academy a work on 
 Statics by Pierre Varignon based on the principle of moments 
 which he developed geometrically from the parallelogram of 
 forces. 
 
 The book was published after his death under the title, 
 "Project d'une Nouvelle Mecanique" with the dedication, 
 "Illustrissimo clarrissimoque viro D. D. Isaaco Newton." It 
 begins, "La Mecanique en general est la Science du Mouve- 
 ment, de la cause de ses effects; en un mot de tout ce qui y a 
 rapport. Par consequent elle est aussi la science de proprietez 
 et des usuages de Machines ou Instruments propres a faciliter 
 le mouvement;" i. e., "Mechanics is in general the science of 
 motion, of its cause and of its effects; in a word of all that 
 pertains to motion. Consequently it is also the science of 
 machines." We meet here in Varignon's book a system of 
 mechanics which is essentially dynamical, including statics 
 as the special case where forces counterbalance. 
 
 After defining mechanics thus, as the science of motion and 
 the theory of machines, he says this treatise will be divided 
 into ten sections: 
 
 (i) Axioms, postulates and propositions; 
 
 (2) Weights suspended or supported by strings; 
 
 (3) Pulleys; 
 
 (4) Wheel and axle; 
 
 (5) The lever; 
 
78 THE SCIENCE OF MECHANICS. 
 
 (6) The inclined plane; 
 
 (7) The screw; 
 
 (8) The wedge; 
 
 (9) The general principle of the simple machines; 
 
 (10) The equilibrium of fluids. 
 
 The first section treats of the definitions, axioms and hy- 
 potheses upon which the work is based. The idea that forces 
 may act each upon other and maintain a body at rest is 
 emphasized. Then the suppositions are made that in the 
 geometrical treatment of machines the parts are to be con- 
 sidered as without weight and friction, perfectly mobile upon 
 their axes, cords are to be considered as perfectly flexible 
 without weight, without elasticity and without stretch or 
 elongation. 
 
 The principle of the parallelogram of velocities is now stated 
 geometrically as, 
 
 Lemma I. 
 
 In order to help the mind to conceive compounded motions 
 let us conceive the point A without weight to move toward B 
 along the line AB, and at the same time suppose that the 
 line itself moves uniformly towards CD along AC remaining 
 always parallel to itself, that is to say maintaining always 
 the same angle with AC. Of these two movements com- 
 mencing at the same time let the velocity of the first and of 
 the second be as the sides AB and ^C of the parallelogram 
 A BCD. Then in the parallelogram, I say that by the action 
 of the two forces upon A, this point will travel along the di- 
 agonal AD oi the parallelogram during the time that AB and 
 AC are being traversed. 
 
 Lemma II. 
 
 If the point A without weight is pushed in the same time 
 and uniformly by two forces E and F acting upon it, along 
 the lines AC and AB acting at the angle CAB. The united 
 action of these two forces will move A along the diagonal 
 of the parallelogram AD in the same time that A would move 
 to C or to B and as though a force in the proportion ol AD to 
 CA or AB had acted upon it. 
 
THE MODERN PERIOD. 
 
 79 
 
 The parallelogram of velocities and of forces is here con- 
 ceived as axiomatic. On these conceptions Varignon builds 
 up a logical geometrical development of mechanics. He dem- 
 onstrates the principle of statical moments by a geometrical 
 theorem in which he shows that the product of a force F 
 (represented graphically by a line), times its lever arm, a 
 
 X /\ 
 
 Fig. 17. Fig. 18. 
 
 (Diagram from "Nouvelle Mecanique.") 
 
 (another line), the product of which is a certain area, is equal 
 to the complementary moment also represented by an equal 
 area in the diagram. For example, in the diagram. Fig. 12, 
 from the Nouvelle Mecanique we have then F X a -{- F' X c 
 = R X b or the moment of the diagonal of a parallelogram of 
 forces is equal to the sum of the moments of the other two 
 sides. The point O may be chosen either without the paral- 
 lelogram within it or on one of the sides. Varignon demon- 
 strates all three cases by proving that the areas are equal by 
 geometry. 
 
 If the point be taken within the parallelogram and the 
 perpendiculars be then drawn we have FXa— F'Xc = 
 R X b. Finally if be taken on the diagonal the moment of 
 the diagonal is zero and we have Fa = F'c. In every case the 
 
8o 
 
 THE SCIENCE OF MECHANICS. 
 
 proof consists in a geometrical proof of equality of areas, the 
 area being the representation of the product of a line, the lever 
 arm, by another line representing the force. The principle 
 thus proven is often called Varignon's principle. It is hardly 
 credible that this principle of moments was not established 
 until 1687, but such appears to be the fact. The proof of 
 
 Fig. 19. 
 
 Fig. 20. 
 
 this principle was quite within the reach of Archimedes, but 
 was not established until nearly twenty centuries after his 
 time. 
 
 As an inevitable corollary of Varignon's proof of the prin- 
 ciple of moments we arrive at the mechanical rule that in all 
 cases of the parallelogram of forces and in all cases of statical 
 equilibrium of forces in a plane the algebraic sum of the 
 moments of the forces must be zero. 
 
 Having established the principle of moments, Varignon then 
 applies it to many examples of equilibrium in rigid bodies and 
 in machines. He used it for the solution of all problems of all 
 the simple machines and founded a whole system of statics 
 on this idea of balanced moments. 
 
 For the exposition of the simple machines it is perhaps easier 
 for the student to grasp than the principle of virtual velocities 
 which was established earlier. 
 
THE MODERN PERIOD. 8 1 
 
 Varignon in his Mecanique refers all cases of equilibrium 
 back to his proof of moments as a criterion and presents 
 therefore a harmonious theory of mechanics. His exposition 
 is essentially geometrical and graphical, and is based upon 
 Stevinus' triangle of forces the proof of which Varignon states 
 as axiomatic in Lemma II of his book. His method of pre- 
 sentation and his proofs are a great advance over those of 
 Archimedes and Stevinus. 
 
 Many of the modern text-book methods in statics are 
 copied directly from this Mecanique and used verbatim to 
 this day in class-rooms. The method of the parallelogram 
 of velocities and of forces and the method of moments as 
 applied in the simple machines are in daily use among engineers. 
 
 Algebra and geometry deal with fixed quantities, but with 
 the development of dynamics, mathematics was called upon 
 to investigate and express quantities whose value is continually 
 changing. In the latter half of the seventeenth century, this 
 need was met by the invention of that branch of mathematics, 
 called the calculus. As has been noted Newton invented one 
 method of studying the relative changes in dependent quanti- 
 ties by considering the ratio of change of their elements, which 
 method of studying "flowing quantities" he called fluxions. 
 
 At about the same time Leibnitz, feeling the need of some such 
 method, developed his system of studying change by infinitely 
 small differences or by the "method of infinitesimals." The fun- 
 damental idea and the purpose of the two systems is much the 
 same. Each calculus consists of two branches: (i) differential 
 calculus which comprises methods of deducing the relations 
 between infinitely small differences of quantities from the 
 relations of the quantities themselves; (2) the integral calculus 
 which treats of the inverse process of determining the relations 
 of the quantities themselves when the relations of their in- 
 finitely small differences is known. 
 
 Newton's theory of flux or flow was better suited to Me- 
 chanics than Leibnitz's concept of instantaneous changes but 
 the latter's notation was found more serviceable and has 
 generally displaced Newton's symbols. It was perhaps a 
 hundred years before this method was generally accepted, 
 6 
 
82 THE SCIENCE OF MECHANICS. 
 
 recognized and developed. When we consider that all nature 
 varies continually, the importance of this mathematical method 
 of treating variables is obvious. With this instrument once 
 mastered by the investigators, advance in Mechanics became 
 rapid. 
 
 The transactions of the learned societies of this period are 
 filled with discussions as to the integrity of the calculus 
 methods, and with numerous isolated memoirs on its utility in 
 one problem or another in mechanics. Among the earliest and 
 most noteworthy of these is perhaps that on page 22, "Memoirs 
 de I'Academie des Sciences de Paris, 1700," in which Varignon 
 clearly presents for the first time, the differential equations of 
 motion : 
 
 dx dv 
 
 dt ^'"' Jt^^' 
 
 d^x dv . 
 
 df "•'' ^dx^''' 
 
 These equations express completely the circumstances of 
 rectilinear motion for every condition of acceleration or retar- 
 dation. Newton had stated these laws geometrically (Principia, 
 Lib. I, sect. 7; Lib. II, sect, i) but Varignon seems to have 
 been the first to advocate their expression in the notation of 
 
 , the calculus. He aspired to free Dynamics from the encum- 
 brance of purely geometrical proofs by using this lately invented 
 
 ) method of the calculus, and showed how acceleration might 
 
 J be expressed by the calculus, thus helping to clear the way for 
 
 I an analytical mechanics. 
 
 '^ Like his predecessors he was greatly interested in hydro- 
 statics and hydraulics and is to be credited with the earliest 
 clear proof of the important principle that the velocity of 
 efflux of a liquid is equal to ^2gh. 
 
 Starting with the relation between force and the momentum 
 Ft = mv and denoting the area of the orifice by a, the head of 
 liquid by h, specific gravity by S, acceleration due to gravity 
 by g, the velocity of effiux by v, and by 7" a small interval of 
 time we have, 
 
THE MODERN PERIOD. 
 
 83 
 
 avTS 
 ahS.T= V, 
 
 gh = v^. 
 
 In the formula ahS represents the pressure acting during 
 the time T on the mass of Hquid avTS/g. But since z; is a 
 final velocity we get more exactly, 
 
 V 
 
 a-TS 
 ahS.T = ■ -v 
 
 or 
 
 v^ = 2gh or, V = ^2gh. 
 
 Varignon's contributions may be summed up then as: 
 
 1 . Proof of the principle of moments. 
 
 2. A complete system of statics based on moments. 
 
 3. The differential equations 
 
 dx 
 dt 
 
 dv 
 dt 
 
 d^x 
 df 
 
 = V, 
 
 = a, 
 
 = a. 
 
 4. The equation for velocity of efflux in hydraulics, v^ = 2gh. 
 That these are masterly contributions to the science of 
 mechanics is self-evident. 
 
 REFERENCES. 
 
 Nouvelle Mecanique, ouvrage posthume de M. Varignon, 2 vols., 1725. 
 Memoirs d'l Academie des Sciences de Paris. 
 Geschichte der Principien der Mechanik, Diihring. 
 Mach's Science of Mechanics. 
 
 Leibnitz, the Bernoullis and Euler. 
 Two hundred years ago the facilities for the spread of 
 scientific progress and invention were meagre, and in general 
 
84 THE SCIENCE OF MECHANICS. 
 
 unless the studies of the philosophers developed something 
 that bore directly upon some immediate practical problem of 
 the time, no general notice was taken of their work. However 
 after the principles of Statics and Dynamics which the re- 
 searches of Stevinus, Galileo, Huygens, Newton and Varignon 
 had developed, came to be understood among scholars, the 
 custom of sending out challenge problems arose. 
 
 These competitions developed many small points which in 
 the aggregate amounted to a very considerable contribution. 
 Slowly a universal method of mechanical reasoning and nota- 
 tion came into use, which was understood in Florence and Paris 
 ^as well as in Berlin, London and St. Petersburg. It consisted 
 
 \ 
 
 in the reduction of questions concerning force and motion to 
 problems in pure geometry and calculus. 
 
 This method which began with the crude picture diagrams 
 of Stevinus grew into the formal abstract geometrical me- 
 chanics of Newton's "Principia" and by the genius of Varignon 
 and others was then expressed analytically. Once the analyt- 
 ical method was developed it became the custom for the 
 Philosophical Societies of Paris, London, Berlin and St. Peters- 
 burg to offer prizes for solutions to various problems in me- 
 chanics, and there resulted a period of great activity in the 
 application and extension of the fundamental contributions of 
 Stevinus, Galileo, Huygens, Varignon and Newton. 
 
 Thus the calculus of Newton and Leibnitz came to be applied 
 in a great variety of problems in mechanics, and ultimately, 
 this method displaced entirely the geometrical method. Dur- 
 ing this period both methods were often used. Numerous 
 mechanical problems were proven by both methods separately 
 and much was achieved in a disjointed unconnected way. 
 
 Most active among those who took part in this development 
 were Gottfried Wilhelm von Leibnitz (1646-1716), Leonhard 
 Euler (1707-1783), and the Bernoullis — James (1654-1705), 
 John (1667-1748) and Daniel (1700-1782). 
 
 Leibnitz, through his papers in the "Acta Eruditorum," 
 which he founded, familiarized continental writers with his 
 powerful method of analysis. Though Newton probably de- 
 veloped the calculus earlier by the method of fluxions, he had 
 
THE MODERN PERIOD. 85 
 
 presented his Principia on the geometrical method and the 
 English for a long time followed the geometrical method. It 
 was not till 181 7 that the differential calculus was introduced 
 into the curriculum at Cambridge and came into general use 
 in England. On the continent however, the method of Leib- 
 nitz came into use almost immediately and generally. Leib- 
 nitz and James Bernoulli, his brother John, the latter's sons 
 Nicolas and Daniel, and their friend Euler were most active 
 in this work of applying the Leibnitzian analysis to various 
 problems in mechanics. 
 
 This was a period of development during which there was 
 often acrimonious controversy. Two men sometimes arrived 
 at similar or very similar results by different routes and then 
 entered into a wordy conflict over their methods, both of 
 which often proved to be correct. But great and lasting good 
 came of all these discussions for they served to clear up and 
 define the fundamental concepts of the science and develop 
 methods and forms of proof which later masters correlated 
 into formal treatises. 
 
 Leibnitz does not appear to have had a perfectly clear con- 
 ception of mass. He speaks of a body as "corpus" and of a 
 load or weight as "moles" and only once does "massa" occur. 
 He makes the distinction however between "vis mortua" (pres- 
 sure) and "vis viva" (moving force). His ideas of the measure 
 of force also were not clear. He noticed that in machines in 
 equilibrium the loads are inversely proportional to the veloci- 
 ties of displacement, so he measured the force by the product 
 of the body ("corpus" or "moles"), into velocity. So far he 
 was in accord with the notion of Newton and Descartes who 
 regarded momentum as the measure of force, but Leibnitz 
 held that such measure of force is only accidental and that the 
 true measure of force is determined by the method of Galileo 
 and Huygens, viz: by the mass times the velocity squared. 
 
 In 1686 in the "Acta Eruditorum," Leibnitz attacked Des- 
 cartes' conception under the title "A short Demonstration of 
 a Remarkable Error of Descartes and Others concerning the 
 Natural Law by which they think the Creator always preserves 
 the same Quantity of Motion ; by which however, the science 
 
86 THE SCIENCE OF MECHANICS. 
 
 of mechanics is totally perverted." This dispute continued 
 to agitate philosophers, until the constancy of Smy and 2mz^^ 
 was realized. Certainly Leibnitz was not perfectly clear upon 
 this. His discussion, however, helped toward a solution of 
 the difficulty. 
 
 James Bernoulli, a friend and admirer of Leibnitz, applied 
 the calculus to various problems in mechanics with marked 
 success. He was the first to work out a formula for the 
 isochronous curve and for the catenary curve.^ 
 
 Galileo had assumed that a uniform flexible string supported 
 at its two extremities and acted upon by gravity would hang 
 in a parabolic curve. James Bernoulli successfully applied 
 analysis to this problem of the "chainette" as he called it, 
 and to derive the formula for the catenary curve. Having 
 solved it himself he proposed it as a challenge problem in 1691. 
 The four mathematicians who solved it successfully were Leib- 
 nitz, Huygens and James and John Bernouilli. Their solu- 
 tions appear in the Acta Eruditorum for 1691, pages 273-282, 
 and also in the Philosophical Transactions of the Royal Society 
 at London, 1697. 
 
 From the physical point of view it is easily seen that equi- 
 librium exists when all the links of the chain have sunk as 
 low as possible, so that no link can sink lower without raising 
 part of the chain of equal mass higher, in consequence of the 
 connections. This state of equilibrium exists when the center 
 of gravity has sunk as low as it can sink. The mathematical 
 problem then resolves itself into the problem of determining 
 the curve that has the lowest center of gravity for a given 
 length between A and B. The equation of such a curve 
 Bernoulli determined with the aid of the calculus. 
 
 James Bernoulli also extended the method of analysis to the 
 study of the curve of an elastic rod with a weight fixed at the 
 end. This he put forth under the title of "Elastica." He also 
 discussed the problem of the flexible sheet or impervious sail 
 filled with a liquid which he presented under the titles "line- 
 taria" and "volaria," and investigated the theory of cycloidal 
 lines and various spiral and logarithmic curves. His writings 
 
 ^"Opera," Tom. I, p. 449. 
 
THE MODERN PERIOD. 87 
 
 include an edition of Algebraic Geometry and the "Ars Con- 
 jectandi," in which he established the fundamental notions 
 of the calculus of probabilities. 
 
 Johann Bernoulli, brother of James, was the most prominent 
 and successful professor of mathematics of his time, holding 
 the chair at Basel and Groningen. His lectures contain the 
 earliest use of the term "integral" and show the first effort to 
 construct an integral calculus as a set of general rules or a 
 body of mathematics. Before this, investigators had treated 
 each problem of integration by itself. He made himself a 
 master of the calculus and applied it with marked success to 
 many problems. 
 
 He was the author of the famous challenge problem of the 
 "brachistochrone" which he propounded, 1697. 
 
 In abbreviated form it is as follows: 
 
 "Acutissimis qui toto orbe florent Mathematicis" 
 
 Johannes Bernoulli, Math. P. P. 
 
 "Problema Mechanico-Geometricum de linea celerrimi de- 
 scensus 
 
 "Determinare lineam curvam data duo puncta, in diversis 
 ab horizonte distantiis, et non in eadem recta verticali posita, 
 connectentem, super qua mobile, propria gravitate decurrens 
 et a superiori puncto moveri incipiens, citissime descendat ad 
 punctum inferius 
 
 Groningse ipsis Gal. Jan. 1697." 
 Which may be freely translated : 
 A Challenge to the Keenest Mathematicians of the World, 
 
 From John Bernoulli, Prof, of Mathematics. 
 A Mechanico-Geometrical Problem of the Gurve of Swiftest 
 Descent. 
 
 Find the curve of quickest descent between any two given 
 points at different distances from a horizontal line and not in 
 the same vertical straight line. 
 
 Groningen, January i, 1697. 
 
88 THE SCIENCE OF MECHANICS. 
 
 The correct solutions were given by Leibnitz (Acta Erudit., 
 1697, p. 203), by Newton (Phil. Trans., 1697, No. 224, p. 389), 
 by L'Hopital (Acta Erudit., 1697, p. 217) and John Bernoulli 
 (Acta Erudit., 1697, p. 207). 
 
 Bernoulli's methods are the methods of the present time as 
 the following quotations from his "Lectiones Mathematice, 
 38, 39, 40, Opera, Tom. Ill," will show: 
 
 A flexible string fixed at any two points A and B is acted 
 upon by gravity. If we suppose the mass of the string to vary 
 according to any assigned law as we pass from one point to 
 another, to find the equation of the catenary of rest. Con- 
 versely the curve being known, to determine the law of mass 
 of the string. 
 
 Let the axis of y extend vertically upwards, and the axis of 
 X be horizontal, the plane xOy coinciding with the plane which 
 contains the catenary. Then since, 
 
 X = o, y = - g, 
 
 we have, by previous equations, section I, 
 
 d I dx\ , ^ 
 
 d / dy\ 
 
 JsVTs) = '^^' (^) 
 
 Integrating the equation (a) we get 
 
 dx 
 ds 
 
 Where C is a constant quantity. 
 
 Let T denote the tension at the lowest point, then evidently 
 T = C, and therefore 
 
 dx 
 
 '5-.= ^- W 
 
 From {h) and (c), we have 
 
 d dy 
 dsdx 
 
THE MODERN PERIOD. 89 
 
 And therefore, 
 
 dy 
 dx 
 
 = I mgds, (d) 
 
 but at the lowest point of the catenary dy/dx = o and there- 
 fore, supposing a to be the value of S, at the lowest point, 
 
 dy 
 dx 
 
 I mds. (e) 
 
 If m be given in terms of the variable x, y, s the form of the 
 catenary may be determined from (d). 
 Again, differentiating (d) we obtain 
 
 a formula by which m may be computed for every point of 
 the string when the form of the catenary is given. Also from 
 (c) we get 
 
 which gives the tension at any point of the catenary when its 
 form is known. 
 
 Another example is that on page 497, Tom. Ill, Lectiones 
 Mathematice; Opera. 
 
 A flexible string AOB fixed at two points A and B is acted 
 upon by gravity, the mass at any P varies inversely as the 
 square root of the length OP measured from the lowest point 
 0; to find the equation of the catenary. 
 
 Let the origin of co-ordinates be taken at 0, x being hori- 
 zontal, and y vertical, and the plane of xy coinciding with the 
 plane of the catenary, also let be the origin of 5. 
 
 Then, if /x be the mass at end of a length C from the lowest 
 point, 
 
90 THE SCIENCE OF MECHANICS. 
 
 and therefore i, d, a being in the present case zero, we have 
 
 Jo ^'^ 
 
 hence putting for sake of brevity 
 
 dy _ 
 
 2gnC^ I 
 
 we get 
 
 dy _ {S\^dy'^ S 
 dx ~ [jsJ ^2 = "^ » 
 
 ^d^df ^ds ^/ df\i^ 
 
 dxdx^ dx \ dx^/ 
 
 d_dy^ 
 dxdx^ 
 
 7 = 1; 
 
 {'+%) 
 
 integrating with respect to x we obtain, 
 
 ^^(■+f)' = '= + ^' 
 
 but X =o, dy/dx = o simultaneously; hence C = 2B and 
 therefore 
 
 squaring and transposing 
 df 
 
 dy' I \2 
 
 2^dy = {{x + 2^y - 4/32}^(fx; 
 integrating we have 
 
 C + 2^y = l{x-\- 2/3)(3c2 +4^^)1-2/32 log{x + 2/3 + (x2 + ^^xf] . 
 But X = o, 3' = o, simultaneously; hence 
 
 C = 2^2 log (2^), 
 
THE MODERN PERIOD. 9I 
 
 hence, eliminating C, 
 
 2Py = i{x + 2^){x- + 4^:.)^ - 2^ log " + ^^ + f f + ^^^^^ , 
 
 which is the required equation of the catenary. 
 Cor. From (a) we get 
 
 ds X + 2/3 
 dx 2/3 
 
 and therefore, by (i,/) 
 
 ^ds T ^ 
 '=^dx=T^^'+'^^^ 
 
 which gives the tension at any point of the curve. 
 
 On page 502, of Tom. Ill, Opera, we find the interesting 
 problem : 
 
 To find the law of variation of the mass of a catenary acted 
 upon by gravity so that it may hang in the form of a semi- 
 circle with its diameter horizontal. 
 
 The notation remains the same as in the preceding problem, 
 and the equation of the catenary is 
 
 x^ = 2ay — y^j 
 
 where a denotes the radius of the semi-circle; hence 
 
 a^ — x'^ = {a — yY, 
 
 y = a — (a^ — x-)^, 
 
 dy X d'^y a^ 
 
 also 
 
 dx {a^ — x^)^ dx^ (a^ — x"^)^ * 
 
 ds^ dy"^ a^ ds a 
 
 dx"^ dx"^ ~ a^ — x^ ' dx (a^ — x^)^* 
 
 and therefore by (i, e) 
 
 .d^y 
 
 T 
 dx^ Ta Ta 
 
 m = 
 
 ds ga^ — x"^ g(a — yY 
 dx 
 
92 THE SCIENCE OF MECHANICS. 
 
 or the mass at any point varies inversely as the square of the 
 depth below the horizontal diameter of the circle. Cor. By 
 (i,/) we have for tension at any point 
 
 ds Ta Ta 
 
 dx {a^ — x^)* a— y^ 
 
 These proofs show great facility in handling the calculus 
 but they are an extension of known ideas rather than a new 
 contribution. Nevertheless the methods of Johann Bernoulli 
 exerted a great influence upon the development of the science 
 in extending the mathematical or analytical method of treat- 
 ment. 
 
 As to his new contributions, he set forth the principle of 
 virtual velocities in a letter to Varignon, introduced the symbol 
 g, and assisted him in arriving at the formula v^ = 2gh, which 
 had been previously stated, 
 
 v^ : vi^ :: hi : h. 
 
 This Bernoulli was a profound scholar and wrote on a great 
 
 variety of topics as will be seen from the following selection 
 
 of titles from his Opera Omnia published at Lausanne in 1742. 
 
 I. Dissertatio de Effervescentia and Fermenta- 
 
 tione. 
 II. Novum Theorema pro Doctrina Conicarum. 
 
 III. Inventio curvse geometricae quae spirali aequa- 
 
 tione. 
 
 IV. Solutio Problematis Funicularii. 
 
 V. Curvae sui evolutione se ipsas describentes. 
 XVIII. Dissertatio physico anatomica de motu mus- 
 culorum. 
 LIII. Disputatio medico physica de nutritione. 
 XC. De motu pendulorum et projectilium. 
 XCIX. Demonstratio principii Hydraulici de veloci- 
 tate per foramen et vase erumpentis. 
 CXL. Meditationes de Chordis vibrantibus. 
 XXIV. Cycloidis evoluta ipsa est cyclois. 
 XXXIII. Varia Problemata Physico-Mechanica. 
 
THE MODERN PERIOD. 93 
 
 The most interesting of these is the first part of the third 
 volume, ;"Discours sur les Loix de la communication du mouve- 
 ment, contenant la solution de la premiere Question proposee 
 par Messieurs de Tacad^mie Royale des Sciences pour I'annee 
 1724." 
 
 As a preface to it Bernoulli says: "The author of this dis- 
 course has the honor to present it to the Academy. It was 
 composed on the occasion of the first of the questions pro- 
 pounded by the Society to the savants of Europe. Messrs. 
 Huygens, Mariotte, Wren, Wallis and various other mathe- 
 maticians have written worthily on this subject and given 
 us rules for impulse. But not satisfied with taking a general 
 rule for the most simple cases, by a kind of induction, the 
 author has followed a method different from theirs and more 
 natural. 
 
 "He goes back to the sources and taking up all that is 
 known of the subject, it is on principles of mechanics that 
 he deduces like corollaries particular rules for each case. Up 
 to this time we have had only a confused idea of the force of 
 bodies in motion to which M. Liebnitz has given the name 
 vis viva. The author has not only attempted to bring the 
 discussion down to date and to explain the difficulty between 
 Leibnitz and those of the opposite party, he has attempted to 
 prove by demonstrations direct and entirely new, a truth which 
 M. Leibnitz never proved except indirectly. 
 
 "He proposes to show that the vis viva of a body is not 
 proportional to its simple velocity as commonly believed but 
 to the square of the velocity and he hopes to prove what he 
 shall say, so that no one shall any longer doubt the truth of this 
 proposition: moreover, he proposes to determine that which 
 results from the shock of a body which encounters two or 
 several others following different directions, a problem so diffi- 
 cult that no one has yet solved it. And indeed, how could 
 any one, since its solution presupposes an exact comprehension 
 of the theory of vis viva? 
 
 "This theory opens an easy way to several important 
 truths. It has given the author a solution of the preced- 
 ing problem which seems somewhat peculiar and a method 
 
94 THE SCIENCE OF MECHANICS. 
 
 of determining the actual loss of velocity in a resisting 
 medium and an easy way of finding the center of oscillation 
 in compound pendulums." He then goes on to expound the 
 principle of virtual velocities and the notion of energy as 
 measured by the mass and the velocity squared. 
 
 Talent for mathematics seems to have run in the Bernoulli 
 family. Several of the younger generations were famed for 
 their writings and teaching, among them, the three sons of 
 John, viz: Nicholas, Daniel and John, Jr., and the two sons of 
 John, Jr., John, 3rd, and Jacob. 
 
 Of these Daniel Bernoulli (i 700-1 782) was the most promi- 
 nent. He was professor at St. Petersburg and at Basel, a 
 famous mathematician, and winner of the French Academy 
 prize ten times. His chief work in mechanics was on hydro- 
 dynamics and the solution of the problems of vibrating cords, 
 in which we find ingenious extensions of known principles of 
 mechanics by the aid of the calculus. 
 
 Euler ( 1 707-1 783), the pupil and friend of Johann Bernoulli 
 and friend of his sons, carried the integral calculus to a high 
 degree of perfection and invented numerous solutions of me- 
 chanical problems. His strength lay rather in pure than in 
 applied mathematics. Euler's principal contributions are set 
 forth in his "Methodus inveniendi lineas curvas maximi mini- 
 mive proprietate gaudentes" (1744) in which he presents the 
 method of co-ordinate analysis and shows the properties of 
 maximum and minimum of various curves. 
 
 He also published at St. Petersburg in 1736 his "Mechanica 
 sive Motus Scientia Analytice Exposita" which is sometimes 
 referred to as the first book of Analytical Mechanics. In 
 this, he still adheres in part to the old method of geometrical 
 presentation of mechanics, but his general method is to resolve 
 all forces in three fixed directions, X, Y, Z. This makes his 
 presentations and computations lucid and symmetrical. 
 
 As an example, note the method of the following discussion 
 from page 237, Tom. I of "Mechanica," on tangential and 
 normal resolution in curvilinear motion. 
 
 A particle is projected with a given velocity in a given 
 direction, and is acted upon by a constant force in parallel lines, 
 to determine the path of the particle. 
 
THE MODERN PERIOD. 95 
 
 Let the axis of X be taken so as to pass through the initial 
 
 place of the particle, and let the axis of Y be taken parallel 
 
 to the constant force which acts toward the axis of X. Let / 
 
 denote the constant force. 
 
 dy 
 Then, the tangential resolved part being ~ f IT^ ^^d the 
 
 dx 
 normal one being / -7- we have for the motion of the particle 
 
 dv dy , 
 
 'ds=-fds^ ^'^ 
 
 Integrating (i) 
 
 v''- = C — 2jy. 
 
 Let V be the initial velocity; then y being zero initially 
 V^ = C\ therefore 
 
 j;2 ^ 72 _ 2/3,. 
 
 Hence substituting this expression for v^ in (2) 
 
 but 
 
 I dx 
 
 dx^ 
 d^y 
 dx^ 
 
 hence 
 
 
 Integrating, we have 
 
 Where C is an arbitrary constant. 
 
96 THE SCIENCE OF MECHANICS. 
 
 Let /3 be the angle which the direction of projection makes 
 with the axis of x; then 
 
 hence 
 
 Cii + tan2 ^) = 72^ 
 
 V'{^ +%) = sec^ ^ (^' - 2/3'). 
 V^~ = F2 tan ;8 - 2fy sec^ jS, 
 
 7^/3; = (72 tan2 ,3 - 2fy sec^ |S)' c^x; 
 
 whence by integration, 
 
 C - V{V^ tan2 ^ - 2fy sec^ /3)* = /x sec^ j8. 
 
 But X = o, 3' = o simultaneously; hence 
 
 C - F2 tan 18 = o, 
 and therefore 
 
 F2 tan jS - F (72 tan^ ^ - 2fy sec^ j8)^ = /x sec^ fi. 
 
 Clearing the equation of radicals and simplifying, we obtain, 
 
 /sec^jS 
 
 y = tan B.x — 77^ x^. 
 
 ^ f- 2F2 
 
 He also solves in a similar manner various problems such as: 
 
 "A particle always acted on by a force in parallel lines, 
 describes a given curve; to determine the nature of the force, 
 the velocity and the direction of projection being given." 
 
 And, "A particle describes a given curve about a center of 
 force; to determine the motion of the particle and the law 
 of force." 
 
 As has been stated the name Moment of Inertia of a body 
 was given by Euler to the sum of all the products result- 
 ing from the multiplication of each element of the mass by 
 the square of the distance from the axis. In his "Theoria 
 Motus Corporum Solidorum," page 167, Euler says: "Ratio 
 hujus denominationis ex similitudine motus progressivi est 
 desumpta: quemadmodum enim in motu progressivo, si a vi 
 
THE MODERN PERIOD. 97 
 
 secundum suam directionem sollicitante acceleretur, est in- 
 crementum celeritatis ut vis sollicitans divisa per massam seu 
 inertiam; ita in motu gyratorio, quoniam loco ipsius vissollici- 
 tantis ejus momentum considerari oportet, eam expressionem 
 J rHM quai loco inertise in calculum ingreditur, momentum 
 inerticB appelemus, ut incrementum celeritatis angularis simili 
 modo proportionale fiat momento vis sollicitattis diviso per 
 momentum inertise."^ This very useful expression used so com- 
 monly by engineers was developed by Euler for various plane 
 figures and for solids of revolution. His method for finding the 
 moment of inertia for the sphere, right cone, cylinder and 
 other figures is given on page 198 and the following pages of 
 "Theoria Motus Corporum Solidorum," as follows: 
 
 To find the radius of gyration of a homogeneous sphere about 
 a diameter. Let x, x -\- dx be the distances of the circular 
 faces of a thin circular slice of a sphere at right angles to the 
 diameter, from the center and let y be the radius of the section; 
 then p denoting the density of the sphere, the moment of 
 inertia of this slice about the diameter will be equal to 
 
 lirpy^dx, 
 
 and therefore the moment of inertia of the whole sphere, a 
 being its radius, will be equal to 
 
 ^TTp I y^dx = Ittp / (a^ — x'^ydx = -{^ irpa^. 
 
 %} —a k) —a 
 
 As the mass of the sphere is f Trpa' the radius of gyration 
 
 Similarly on page 203 the radius of gyration of a hollow 
 
 ^Translation. — The scheme of this notation is derived by analogy with 
 rectilinear motion; for as in rectilinear motion if it be increased by a dis- 
 turbing force in its own direction the increase of velocity (acceleration) is 
 equal to the disturbing force divided by the mass or inertia, thus in rotary 
 motion, since in place of the disturbing force itself we must consider its 
 moment we call that expression fr-dM which comes into calculation in 
 place of inertia — the moment of inertia — so that the increase of angular 
 velocity in a similar way is made proportional to the moment of the 
 disturbing force, divided by the moment of inertia. 
 
98 THE SCIENCE OF MECHANICS. 
 
 sphere with external and internal diameters a, b, is proven to be 
 
 2 a^ — b^ 
 
 k^ = 
 
 5 a^ — b^ 
 
 Euler systematized and perfected the mathematical knowl- 
 edge of the time. Among his publications are, 
 
 Introductio in analysin infinctorum 1748 
 
 Institutiones Calculi differential 1755 
 
 Institutiones Calculi Integral 1768 
 
 also a development of the Calculus of Variations. 
 
 He set forth the principle of least action, — though Mau- 
 pertuis is usually given the credit of having originated the 
 notion, — expressing it in that curious blending of theology 
 and science common in this period, in this fashion: the all- 
 wise Maker would not make anything in which some maximal 
 and minimal property is not shown. 
 
 The original Latin form is "Quum enim universi fabrica sit 
 perfectissima, ataque a creatore sapientissimo absoluta, nihil 
 omnino is mundo contingit in quo non maximi minimive ratio 
 qusepiam eluceat; quam ob rem dubium prorsus est nullum, 
 quin omnes mundi effectus ex causis finalibus ope methodi 
 maximorum et minimorum, aeque feliciter determinari quseant, 
 atque ex ipsis causis eflficientibus," or "For since the fabric of 
 the universe is most perfect and finished by a most wise crea- 
 tor, nothing occurs in the world in which some plan of maxima 
 and minima does not show forth ; therefore there is no doubt at 
 all (!) but that all phenomena of the world are equally well to 
 be determined from final causes by the method of maxima 
 and minima and from the same effecting causes." 
 
 This idea was taken up by Euler (Proc. Berlin Acad., 1751) 
 and developed into a theory of equilibrium of utility by the 
 application of the method of maxima and minima. If in any 
 system we cause infinitely small displacements we produce a 
 sum of virtual moments 
 
 Pp + P'p' +P"p" +••• 
 
 which only reduces to zero in the case of equilibrium. The 
 sum is the work corresponding to the displacements, or since 
 
THE MODERN PERIOD. 
 
 99 
 
 for minute displacements it is itself infinitely small, the cor- 
 responding element of work. If the displacements are con- 
 tinuously increased till a finite displacement results, their 
 summation is a finite amount of work. 
 
 Therefore if we start with any initial configuration of the 
 system and pass to any given final configuration, a certain 
 amount of work will have to be done. This work done when 
 a final configuration or a configuration of equilibrium or equi- 
 librium is reached is a maximum or a minimum. That is if 
 any system is carried through the configuration of equilibrium 
 the work done is previously and subsequently less or greater 
 than at the configuration of equilibrium itself. For equilib- 
 rium, therefore 
 
 Pp' + P'p' + P"p" + • • • =0. 
 
 From this Euler deduced the principle that the element of 
 work or the differential of work is equal to zero in equilibrium; 
 and if the differential of a function can be put equal to zero, 
 the function has generally a maximum or minimum value. 
 
 FH- 21. 
 
 This highly ingenious method of determining the equilibrium 
 of a system was later developed by others. In 1749 Courtiv- 
 ron, in a paper before the Paris Academy gave it the form: 
 "For the configuration of stable or unstable equilibrium at 
 
100 THE SCIENCE OF MECHANICS. 
 
 which work done is a maximum or a minimum, the vis viva 
 of the system in motion, is also a maximum or minimum in its 
 transit through these configurations." 
 
 Euler also assisted in the development of the so-called prin- 
 ciple of vis viva. He showed that if a body M is attracted 
 to a fixed center C according to a certain law the increase in 
 vis viva in the case of rectilinear approach is calculable from 
 the initial and terminal distances ro, n. But the increase is 
 the same if M passes at all from the position ro to ri inde- 
 pendently of the form of the path MN. The elements of the 
 work done are to be calculated from the projections on the radius 
 of the actual displacements and are thus ultimately the same. 
 
 Euler is also to be credited with the first general use of r 
 for 3.1416 + and the application of his methods of analysis 
 to hydrodynamics. 
 
 We may sum up his contributions then as follows: 
 
 1. A perfect systematizing of the calculus. 
 
 2. The foundations of analytical mechanics. 
 
 3. The analytical method of resolving tangential and 
 
 normal components of curvilinear forces. 
 
 4. The development of moment of inertia. 
 
 5. The principle of least action or maxima and minima 
 
 in equilibrium. 
 
 6. The principle that the increase of vis viva is inde- 
 
 pendent of the path. 
 
 Much of the work of D'Alembert and Lagrange is based 
 on the contributions or methods of Euler, and perhaps would 
 not have been possible without Euler's work. 
 
 REFERENCES. 
 
 Jacobi Bernoulli Basilensis Opera. Geneva, 1744. 
 
 Johanni Bernoulli Basilensis Opera Omnia. 1742. 
 
 Opuscules et Fragments inedits de Leibnitz. Paris, 1903. 
 
 Euler, Mechanica Sive Motus, 1736. 
 
 Euler, Institutiones Calculi Differentialis. 
 
 Euler, Introductio in Analysin Infinitorum. 
 
 Correspondenz von Nicolaus Bernoulli. Basel Library. 
 
 Nouvelle Mecanique. Varignon. Paris, 1725. 
 
 Euler, Methodus, 1744. 
 
 Harnack, Leibnitz's Bedeutung in der Geschichte der Mathematik. 
 
 D. Bernoulli, Hydrodynamica. 
 
 Cantor, Geschichte der Mathematik. 
 
the modern period. loi 
 
 Jean-le-Rond D'Alembert (1717-1783). ; 
 
 As a result of the labors of a host of contributors much had 
 now been evolved in mechanics in a disjointed way and from 
 diverse points of view. The prize and challenge problems 
 were usually very special and did not tend to develop a 
 formal presentation of the science. It was now in order for 
 some one to verify, consolidate and formulate all these con- 
 tributions. 
 
 This D'Alembert did in his "Traite de Dynamique" (i743-) 
 While his work rests upon the work of all his predecessors, 
 and while he is particularly indebted to Euler, yet his Treatise 
 possesses distinctly original features. He shows that all 
 problems in dynamics may be regarded as problems in statics 
 and he applies in their solution one single unifying principle 
 known by his name as D'Alembert's principle. It is to the 
 effect that in any system of bodies the impressed forces are 
 equivalent to the effective force. 
 
 This formal presentation of mechanics in a treatise is a 
 memorable event. It typifies the coming of age of the science. 
 Henceforth it has a character and unity which it did not pre- 
 viously possess. This is due to the fact that now there is one 
 general guiding principle, D'Alembert's Principle, to which 
 all problems in mechanics can be referred for solution. 
 Namely: — 
 
 If a material system connected together in any way, and 
 subject to any constraints, be in motion under the influ- 
 ence of any forces, each point of the system has at 
 any instant a certain acceleration. If now to each point 
 an acceleration were imparted equal and opposite to its 
 actual acceleration, the velocities of all points of the system 
 would become constant, that is, each particle would move 
 as if free and unacted on by any force whatever. The applied 
 accelerations, the external forces, and the constraints and 
 mutual or internal forces of the system, would equilibrate 
 one another. 
 
 In the "Traite de Dynamique" this idea of which the 
 above is a condensed translation is expressed as follows: — 
 
 J 
 
102 THE SCIENCE OF MECHANICS. 
 
 "Probleme General." 
 
 "Soit donne un systeme de corps disposes les uns par rapport 
 aux autres d'une maniere quelconque; et supposons qu'on 
 imprime a chacun de ces corps un mouvement particulier, 
 qu'il ne puisse suivre a cause de raction des autres corps; 
 trouver le mouvement que chaque corps doit prendre." 
 
 "Solution." 
 
 "Soient A, B, C, etc., les corps qui composent le systeme et 
 supposons qu'on leur ait imprime les mouvemens a, b, c, etc., 
 qu'ils soient forces, a cause de leur action mutuelle, de changer 
 dans les mouvements a, b, c, etc. II est clair qu'on pent 
 regarder les mouvemens b, c, etc., comme composes des mouve- 
 mens b, /3; c, 7; etc.; d'ou il s'ensuit que le mouvement des 
 corps A,B, C, etc.; entr' eux auroit ete le meme, si au lieu de 
 leur donner les impulsions a, b, c, etc., on leur eut donn6 
 a-la-fois les doubles impulsions a, a; 6, j9; C, 7, etc. Or par la 
 supposition, les corps A, B, C, etc., out pris d'eux-memes 
 les mouvemens a, b, c, etc., done les mouvemens a, j8, 7, etc., 
 doivent etre tels qu'ils ne derangent rien dans les mouvemens 
 a, b, c, etc., c'est a-dire que, si les corps n'avoient recu que les 
 mouvemens, a, j8, 7, etc., ces mouvemens auroient dil se 
 detruire mutuellement et le systeme demeurer en repos. 
 
 "De la resulte le principe suivant, pour trouver le mouve- 
 ment de plusieurs corps qui agissent les uns sur les autres. 
 De Composez les mouvemens a, b, c, etc., imprimes a chaque 
 corps, chacun en deux autres, a, a; 6, /3; c, 7; etc.; qui soient 
 tels, que si Ton n'etlt imprime aux corps que les mouvemens, 
 a, b, c, etc., ils eussent pu conserver ces mouvemens sans se 
 nuire reciproquement ; et que si on ne leur eilt imprime que 
 les mouvemens a, /3, 7, etc., le systeme fut demeure en repos; 
 il est clair que a, b, c, etc., seront les mouvemens que ces corps 
 prendront en vertu de leur action. Ce qu'il falloit trouver." 
 
 The idea was not entirely new. James Bernoulli in a 
 memoir published in Acta Eruditorum, 1686, p. 356, "Nar- 
 ratio Controversise inter Dn. Hugenuim et Abbatem Catela- 
 num agitatae de Centro oscillationis," set forth the idea of 
 reducing the determination of the motions of material systems 
 
THE MODERN PERIOD. IO3 
 
 to the solution of statical problems. It is a direct conse- 
 quence of Newton's laws rather than a new principle. How- 
 ever, to D'Alembert belongs the credit of clearly setting forth 
 this idea and of founding a formal mechanics upon it. 
 
 In algebraic language the principle is: If the co-ordinates 
 of any particle w of a material system be x, y, z and the ex- 
 ternal forces there applied be X, Y, Z the system of forces 
 
 d^-x d?y d'z 
 
 ^^-^^^7' ^^-^^7/^' ^^-^^^2- 
 
 d'^x d^y d^x^ 
 
 etc., acting at the points x, y, z and X2, y^, 22, etc., will be in 
 equilibrium in virtue of the constraints and mutual reactions 
 of the system. 
 
 The force whose components are 
 
 d^x d^y dH 
 
 is called the force of inertia of the mass m. D'Alembert 's 
 principle states that. The applied forces and the forces of 
 inertia in any system are in equilibrium. 
 
 If in any problem the work be o, the particular case of the 
 principle of virtual displacement results. This principle 
 follows therefore as a special case of D'Alembert's principle. 
 
 The equation of vis viva also follows from D'Alembert's prin- 
 ciple. The integral of the equations of motion can usually be 
 obtained from D'Alembert's principle, viz: 
 
 = o. 
 
 Here bx, by, 8z are arbitrary displacements consistent with 
 the conditions of the problem. When the equations of con- 
 dition do not contain the time explicitly, dx (the actual move- 
 ment along the axis of x during an infinitely short time) is 
 always a value which can be assigned to 8x. In most problems 
 dx is a possible value of 8x and the same holds for dy and dz 
 similarly. Therefore if this be admitted as a legitimate sub- 
 
104 THE SCIENCE OF MECHANICS. 
 
 stitution as is usually the case, if we write dx, dy, dz for 5rc, by, 
 bz, D'Alembert's equation becomes 
 
 (d^x d^v d^z \ 
 
 -^idx^^,dy + -^,dz)= MXdx + Ydy + Zdz). 
 
 Integrating we have 
 
 This is the equation of vis viva. 
 
 If the vis viva at any particular /' is Xmv^ we have 
 
 2wz;2 - Smy'^ = 2Xf{Xdx + Ydy + Zdz). 
 
 If there be no forces acting on the system its vis viva remains 
 constant. The equations of vis viva are among the most 
 important in dynamics. They are the foundation of the 
 theory of energy. 
 
 By means of D'Alembert's principle the equation of motion 
 of a rigid body can be written at once. We have only to 
 write the six equations of equilibrium, taking into account 
 applied forces and the forces of inertia and we have at once 
 
 d^x ,^ d^y „ d^z 
 
 (Pz d^y ' 
 
 If ~^~d^ 
 
 d^x d^z 
 
 I d'z d:'y\ , ^ 
 
 / dH dH\ 
 
 These equations express the moments about the axes. 
 
 The comprehensive character and broad application of 
 D'Alembert's principle are apparent; other principles follow 
 from it as corollaries. It supplies a routine-form of solution 
 for problems, in a masterly fashion, with great economy of 
 thought. 
 
THE MODERN PERIOD. IO5 
 
 In his study of equilibrium and motion in fluids, and in the 
 theory of vibrating strings D'Alembert encountered a partial 
 differential equation of the forms, 
 
 8f ~ 8x^ ' 
 
 which he finally solved in 1747. The solution is given in a 
 paper before the Berlin Academy as follows: 
 
 If 7— be denoted by p, and — by q, then du = pdx-{-pdL But 
 ox ot 
 
 8q 8p 
 8t 8x 
 
 by the given equation, therefore pdt-{-qdx is also an exact 
 differential, denote it by dv. 
 Therefore 
 
 dv = pdt-\-qdx. 
 Hence 
 
 du-\-dv = {pdx -\-qdt)-\- (pdt + qdx) = {p-\-q) {dx -\-dt) 
 and 
 
 du — dv = (pdx + qdt) — (pdt + qdx) = (p — q) (dx — dt) . 
 
 Thus u -{-V must be a function of x + / and ti — v must be a 
 function of x — /. We may therefore put 
 
 tl-\-V = 2<j)(x-\-t), 
 U — V = 2yp(x — t). 
 
 Hence 
 
 u = (l>(x-\-t)-{-\p(x — t) 
 
 in which <}> and 1/' are arbitrary functions. 
 
 In 1749 D'Alembert published the first analytical solution 
 of the precession of the Equinoxes and of the rotation of the 
 earth's axis. He also published a work entitled "Reflexions 
 sur la Cause Generale des Vents," 1744, and three volumes on 
 the "Systemedu Monde" in which his calculations and theories 
 in astronomy are set forth. 
 
 One of D'Alembert's chief claims to distinction, in addition 
 
I06 THE SCIENCE OF MECHANICS. 
 
 to his special contributions, is that he put Newton's results 
 into the form of the Calculus and made possible their study 
 and extension. He presented in his "Traite de Dynamique," 
 the first treatise on Analytical Mechanics. When this had 
 been done the way was prepared for a complete exhaustive 
 treatment of the entire domain of Mechanics by the An- 
 alytical method. This was done by Lagrange within five 
 years of D'Alembert's death. 
 
 REFERENCES. 
 
 Traite de Dynamique, 1743. 
 
 Traite de I'Equibre du Mouvement de Fluides, 1744. 
 
 Reflexions sur la Cause Generale des Vents, 1747. 
 
 Recherches sur la Precession des Equinoxes, 1749. 
 
 Reserches sur difi^erent Points Importans du Systeme du Monde, 1756. 
 
 Systeme du Monde, 3 vols., 1754. 
 
 W. W. R. Ball, History of Mathematics. 
 
 Mach, The Science of Mechanics. 
 
 Williamson, Treatise on Dynamics. 
 
 Bertrand, D'Alembert. 
 
 Condorcet, Eloge (French Acad., 1784). 
 
 4. The Contribution of Lagrange and Laplace. 
 
 Although it is probable that Newton used his method of 
 fluxions or calculus in arriving at the ideas set forth in the 
 Principia, still he presented them in geometrical form. Even 
 so, it was some fifty years before they were accepted and as- 
 similated. The next big step in advance was to be the full 
 and complete development of mechanics by the analytical 
 method based on Newton's laws. 
 
 It was necessary first that the calculus and its notation 
 should be perfected and that its use and value in problems of 
 mechanics should come to be recognized. The labors of Leib- 
 nitz, the Bernoullis and Euler brought this to pass. Secondly 
 it was necessary that the co-ordinate method should be devel- 
 oped. This was done by Descartes, Euler and Maclaurin and 
 D'Alembert. When this had been done it was possible to ex- 
 press the results of Newton in the language of the calculus and 
 have them generally received and accepted. 
 
the modern period. io7 
 
 Joseph Louis Lagrange ( 1736-18 13). 
 
 Comte Lagrange, one of the greatest masters of pure and 
 mixed mathematics that ever lived, was born at Turin though 
 of French extraction. A Senator of France, a Count with 
 the Grand Cross of the Legion of Honor, professor in the 
 Artillery School of Turin and in the Polytechnic School of 
 France, Director of the Berlin Academy under Frederick the 
 Great, his life is one glorious record of achievement. 
 
 His great work the "Mecanique Analytique" is analytical as 
 opposed to geometrical. There is not a geometric diagram in 
 it, whereas the Principia is full of them, page on page. Writ- 
 ten 100 years after Newton's great work, it is a grand com- 
 prehensive treatise gathering up the scattered methods and 
 principles of the preceding century, harmonizing them and 
 setting them forth in concise harmonious algebraic form. 
 He gives a general method by which every mechanical ques- 
 tion of solids, liquids or gases may be stated in a single alge- 
 braic equation. The entire mechanics of any system, even 
 the solar system, can be summed up in a few equations by this 
 method. This is a wonderful labor-saving and thought- 
 saving device. 
 
 It was his boast that he had transformed Mechanics, (de- 
 fined by him as "a geometry of four dimensions") into a branch 
 of analysis. He exhibited the mechanical principles of his 
 predecessors as simple results of the calculus, and introduced 
 the method of regarding a fluid as a material system charac- 
 terized by free mobility of its molecules. With this the sep- 
 aration between the mechanics of solids, liquids and gases 
 disappeared, for the fundamental equations of forces could 
 now be extended to hydraulics and pneumatics. He formu- 
 lated a universal science of matter and motion, deduced from 
 the principle of virtual velocities by the method of generalized 
 co-ordinates. 
 
 Departing from the method of D'Alembert and Euler, in- 
 stead of considering the motion of each individual part of a 
 material system, Lagrange shows how to determine its config- 
 uration by a number of variables corresponding to the degrees 
 of freedom of the system. The kinetic and potential energies 
 
I08 THE SCIENCE OF MECHANICS. 
 
 of the system can be expressed in terms of these variables and 
 the equations of motion obtained by differentiation. 
 
 He gave to analytical mechanics a complete logical per- 
 fection, reducing the science to differential equations and 
 developing the calculus of variations. The introduction of 
 the "Mecanique Analytique" (1788) is so simple and direct 
 a statement of the author's purpose that it is worthy of literal 
 quotation. 
 
 "There are already several treatises on Mechanics but the 
 plan of this one is entirely new. I have attempted to reduce 
 the theory of this science and the art of solving the problems 
 connected with it to general formulas, whose simple develop- 
 ment will have all the necessary equations for the solutions 
 of each problem. I hope that the manner in which I have 
 tried to accomplish my object will leave nothing to be desired, 
 
 "This work will have in addition another advantage: it will 
 collect and present under the same point of view the different 
 principles, so far found, to facilitate the solution of mechanical 
 questions. It will show their connection, their mutual de- 
 pendence, leaving one to judge of their accuracy and value." 
 
 "iVo diagram will be found in the work. The method which 
 I follow requires neither figures nor arguments geometrical or 
 mechanical, but merely algebraic operations arranged in a 
 regular and uniform order. Those who are fond of analysis 
 will anticipate this mechanics with pleasure, and be pleased 
 that I have set it forth in this way." 
 
 Concerning the fundamental principle of the work, he says 
 after stating D'Alembert's principle: 
 
 "But there is another manner of treatment more general and 
 more severe which merits the attention of geometers. M. 
 Euler gave the first hint of it at the end of his treatise on 
 isomerism printed at Lausanne, 1744, showing that in the paths 
 described by central forces, the integral of velocity by the 
 element of the curve always is a maximum or a minimum. 
 This property M. Euler had not noticed except in the motion 
 of isolated bodies. 
 
 Since that time I have considered the motion of bodies 
 acting upon each other in any fashion whatsoever, and there 
 
THE MODERN PERIOD. IO9 
 
 has resulted this new general principle that the sum of the 
 products of the masses by the integral of velocities multiplied 
 by the elements of the spaces covered is constantly a maximum 
 or a minimum. Such is the principle to which I give here, 
 although improperly, the name of "least action," and which 
 I consider not as a metaphysical principle, but as a simple and 
 general result of the laws of mechanics. One may see in 
 volume 2, "Memoirs of Jarin," the use I have made of it for 
 solving several difficult problems of dynamics." 
 
 This principle combined with that of the conservation of 
 energy, and developed according to the rules of the calculus of 
 variations, gives directly all the necessary equations for the 
 solution of any problem. 
 
 He then proceeds to develop his "general dynamic formula 
 for the motion of a system of bodies acted upon by any forces 
 whatsoever," after the manner briefly indicated here. He says 
 if the forces acting upon a body do not mutually destroy or 
 equilibrate themselves as in statics, then the forces produce 
 accelerations. When these forces act freely and uniformly 
 they necessarily produce velocities which increase with the 
 time. One may regard these velocities as measures of the 
 forces. Let us suppose now that of every accelerating force 
 we know the velocity that it is capable of impressing upon a 
 free body during a unit time. We measure accelerating force 
 by the velocity it produces in a unit time supposing the body 
 to move uniformly for that time, and we know by the theorems 
 of Galileo that this space that the body would pass over is 
 twice the distance that the body moves under a constant 
 accelerating force, such as gravity; therefore we have as the 
 velocity by which to measure a constant force twice the 
 distance that the body passes over in a unit time. We must 
 choose our units accordingly. 
 
 After a careful development of these notions Lagrange says 
 let us now consider a system of bodies disposed as you will and 
 acted upon by any accelerating forces you please. Let m be 
 the mass of any of the bodies regarded as point and let it be 
 referred to three co-ordinate axes by the co-ordinates x, y, z 
 at any instant t, then dx/dt, dy/dt, dz/dt, will represent the 
 
no THE SCIENCE OF MECHANICS. 
 
 velocities in the directions of the axes, if the body is abandoned 
 to itself and moves uniformly. But if by reason of the action 
 of accelerating forces the velocities take on during the instant 
 t, the increments 
 
 dx dy dz 
 
 dt ^ dt ' dt 
 
 one may regard these increments as new velocities and dividing 
 them by dt, one will have a measure of the accelerating forces 
 that produce them. Taking the element of time dt, as constant, 
 the accelerating forces will be proportional to d^x/df^, dy'^/dt'^, 
 dz^jdf, and multiplying these forces by the mass of the body 
 upon which it acts we have 
 
 dH d^y d'^z 
 
 ^^' ^J/^' ^^' 
 
 for the forces moving the body during the time dt. We may 
 regard each body m of the system as acted upon by parallel 
 forces, then the total force will be equal the sum of these parallel 
 forces. Employing now the sign d, to represent differentials 
 relative to the time, and representing the variations which 
 express the virtual velocity by 5, we have 
 
 d^x d^y dH 
 
 ^^^^' ^^^^' ^^^^ 
 
 for the momenta of the forces 
 
 dH d^y d^z 
 
 ^^' ^^' ^^' 
 
 and for the sum of the momenta 
 
 ' d^x d'^x dH 
 
 (d^x d^x d^z , \ 
 
 ^8x+^,8y+j^dzjm. 
 
 Now let P, Q, R, etc., be accelerating forces acting upon the 
 system and p, q, r their distances, then the differentials bp, 
 8q, 8r, etc., represent the variations of the lines p, q, r during 
 the variations 8x, 8y, 8z; but these forces P, Q, R, tend to 
 shorten the lines, therefore their virtual velocities should be 
 
m 
 
 THE MODERN PERIOD. Ill 
 
 written — Sp, — dq, — 8r, and their moments — mP8p, 
 — mQ8p, — mRbr and the sum of all these forces will be 
 
 - -LiPbp + Qbg + Rhr + etc.)m. 
 
 Therefore the sum of all the forces acting upon the body will be 
 
 / d^x d^v d^z \ 
 
 ^ (^^^ + ^2 53' + ^52)m = - X(P8p + Q8q = RSr,etc.) 
 
 or 
 
 (d^x d^y d^z \ 
 
 dt^ ^^+^^3'+ ^^2 jw+2(P5^+<252+i?5r+etc.)w = o. 
 
 "C'est la formule generate de la Dynamique pour le mouve- 
 ment d'un systeme quelconque de corps." This formula does 
 not differ from the formula given in his Statics says Lagrange, 
 except in the terms 
 
 d'^x d^y d^z 
 
 ^^' ^^' ^J^ 
 
 which express the accelerating forces. In statics where the 
 acceleration is o, these terms drop out. Therefore this is a 
 general formula applying to statics and dynamics and to solids 
 and fluids. In fact the distinction between statics and dy- 
 namics and solids and fluids vanishes except for the difference 
 in substitution in the formulas. 
 
 Lagrange then applied this formula to many problems such 
 as, "Sur le mouvement d'un systems de corps libres regardes 
 comme des points et animes par de forces d'attraction." He 
 was the first to make extensive use of the calculus of variations. 
 The idea of this is present in Euler's work in an undeveloped 
 form, but Lagrange was the first to recognize the supreme 
 importance of these ideas and to develop the method of varying 
 arbitrary constants in analysis. He successfully applied this 
 method to the investigation of periodical and secular inequali- 
 ties of any system of interacting bodies. These methods gave 
 beautiful solutions of such intricate problems as the effect of 
 the disturbance produced in the rotation of the planets by 
 external action on their equatorial protuberances. He also 
 determined the first maximum and minimum values for the 
 slowly varying planetary eccentricities, and contributed 
 
112 THE SCIENCE OF MECHANICS. 
 
 memoirs on the "Propagation of Sound" on the "Motion of 
 Fluids," on the "Calculus of Variations," and a "Treatise 
 on Functions and Equations." His notes on the Problem of 
 the Three Bodies, on Variations of the Element of Planetary- 
 Orbits, on Attractions of Ellipsoids, and on the Moon's Secular 
 Inequality are noteworthy. 
 
 Lagrange verified Newton's theory and developed his sug- 
 gestions much as Newton did those of Galileo. He reduced 
 the whole theory of mechanics to one fundamental formula, 
 and drew clearly the line between physics and metaphysics. 
 After his time we hear no more such fantastic speculations as 
 were set forth by Descartes and Leibnitz. 
 
 Duhring in his "Geschicte der Principien der Mechanik," 
 page 305, sums Lagrange's contribution in these words: 
 
 "Die Anwendung eines Fundamentalprincips, welches sich 
 fiir den Calciil eignet, und die grunsatzliche Durchfiihring 
 der analytischen Entwicklungen als der Haupt eitfadens fiir 
 die Verbindung aller Wahrheiten der rationellen Mechanik 
 zu einem einheitlichen System, — das sind die beiden Hauptei- 
 genschaften, durch welche sich die Behandlungsart Lagranges 
 auszeichnet." /. e., The application of a fundamental principle 
 adapted to the calculus and the consistent utilization of an- 
 alysis as his main guide for the combination of all the truths 
 of rational mechanics into a unified system, these are the two 
 points which distinguish Lagrange's method. 
 
 Laplace, Simon Pierre, Marquis De (1749-1827). 
 
 The genius of Lagrange was at its best in generalization 
 and abstraction and he brought his mind to practical physical 
 problems with difficulty. It was not so with his contemporary 
 Laplace, who was gifted with shrewd practical sagacity in 
 addition to the wonderful mathematical power which won 
 for him the title of the "Newton of France." 
 
 He applied himself especially to the great problems of de- 
 veloping an analytical exposition of celestial motions and per- 
 turbations, based upon the law of gravitation, and he spent 
 his life in tracing the consequences of the law of gravitation 
 as applied to the solar system. 
 
 The solar system does not consist of several bodies, but of 
 
THE MODERN PERIOD. II3 
 
 a crowd of them traveling about the sun, many of them at- 
 tended by satelHtes; thus the compHcation of attractions 
 is evident. Again the motion of a planet at any time de- 
 pends not merely upon its relative position with reference 
 to the sun, but also upon the position of the other planets ^ 
 and of its own satellites. Added to this is the difficulty that 
 no planet is where it seems to be, owing to the effects of 
 atmospheric refraction and of the finite velocity of light. The 
 magnitude of the task that Laplace set himself is appalling. 
 
 Yet he produced in his "Mecanique Celeste" a work in 
 which the whole theory of planetary motions is investigated, 
 and which offers a complete solution of the great mechanical 
 problem presented by the solar system. It was his constant 
 endeavor to "bring theory to coincide so closely with observa- 
 tion that empirical equations should no longer find a place in 
 astronomical tables." His work is based on the Principia of 
 Newton, which he translates into the language of the calculus, 
 and carries forward and completes so as to produce a mechan- 
 ical theory of celestial motions. 
 
 The "Mecanique Celeste," in five volumes, gives a full an- 
 alytical discussion of the solar system. The first two give 
 methods for calculating the motions of translation and rotation 
 of the planets, determining their figures and solving tidal 
 problems. The third and fourth volumes contain applications 
 of these formulae and astronomical tables. The fifth volume 
 is historical. The work is a complete treatise on physical 
 astronomy. The "Exposition du systeme du Monde" is the 
 "Mecanique Celeste" in popular form without the analysis. 
 The results only are given and the nebular theory is pro- 
 pounded. 
 
 Laplace's special contributions to the notation of mechanics 
 are the Laplace Coefficient and the Potential Function. In 
 the course of his work of investigating the figure of a rotating 
 fluid mass, the stability of Saturn's rings, etc., he came upon 
 expressions for the attraction of an ellipsoid involving an in- 
 tegration, which he could not solve. He discovered however 
 that the attracting force in any direction could be obtained 
 by the direct process of differentiating a single function. He 
 8 
 
114 THE SCIENCE OF MECHANICS. 
 
 was then able to translate the forces of nature into the 
 language of analysis so that he could consider also by mathe- 
 matical analysis the phenomena of heat, electricity and 
 magnetism. 
 
 The function V which was named the Potential Function 
 by Green and Gauss about 1840 is defined as the sum of the 
 masses of the molecules of the attracting bodies divided by 
 their respective distance from the attracting point. 
 
 In general terms m being the mass, and r the distance from 
 
 the attracting point, we have 
 
 ^^ _ . 2Aw 
 
 V = Limit , 
 
 r 
 
 or 
 
 Am = o, 
 
 if p is the density of the body at the point x, y, z and a, j8, 7 
 the co-ordinates of the attracted point 
 
 v= fff 
 
 J J J [{x — a 
 
 pdxdydz 
 
 )2 + (3, _ ^)2 + (2 _ y)]' ' 
 
 the limits of the integration being determined by the form of 
 the attracting mass. Therefore F is a function of a, /3, 7, 
 that is, it depends on the position of the point, and its several 
 differentials furnish the components of the attractive force. 
 As the integrations did not usually give V in finite terms, 
 Laplace introduced (1785) the partial differential equation 
 
 Since known as Laplace's equation. Here V^ is called the 
 operator. This equation forms the basis of all Laplace's re- 
 search in attractions and opened up the whole field of potential. 
 This equation is now used in every branch of physical science. 
 The quantity V^F may be viewed as the measure of the con- 
 centration of F. Its value at any point indicates the excess 
 of the value of F at that point over its mean value in the neigh- 
 borhood of the point. This potential function laid the foun- 
 dation of the mathematical development of heat, electricity 
 and magnetism. 
 
THE MODERN PERIOD. 115 
 
 The form in which Laplace first gave his equation, "Re- 
 cherches sur I'attraction des Spheroides homogenes" in Divers 
 Savans, v. lo, 1873, is, in the polar co-ordinate form, 
 
 where ix is substituted for the cos 9. 
 
 If two points in space are determined by their polar co- 
 ordinates r, 6, CO and /, 6', w', and T be the reciprocal of the 
 distance between them expressed in these co-ordinates, then 
 
 T= {r^- 2rr' W + Vi-,j?Vi- ^'^ ^^g (^ _ ^.)] _,_ ^>^y, 
 
 where n and n' represent the cos B and cos 6' . 
 
 If this expression be expanded into a series of the form 
 
 -, (Po + Pi-, + P2-2 + • • • P„-,i • • •). 
 
 where Po, Pi, P^ are known as Laplace's coefficients of the orders 
 o, I, ... a, these are found to be rational integral func- 
 tions of yi and /, of -v/l — /i^ cos w and '^ i — /t'^ cos w and 
 -v^i — /i^ sin CO and Vi — ti'^ sin co or of the rectangular co- 
 ordinates of the two points divided by their distances from the 
 origin. The general coefficient P„ is of a dimensions and its 
 maximum value Laplace shows to be unity so that the above 
 series will converge if r' is greater than r. He proves that T 
 satisfies the differential equation 
 
 ^^' ~^^^lin I d'T dKrT) 
 
 dix "^ I - m' * <^co2 + ^ dr'' ~ °' 
 
 and if for T the expanded form is substituted we obtain the 
 general differential equation of which Laplace's coefficients 
 are particular integrals 
 
 . ^dPa 
 
 ^ + -o • -TT + «(«^ + l)^^ = O- 
 
 dix I — )u aco'' 
 
 Laplace's theorem of these functions is to the effect that if 
 
Il6 THE SCIENCE OF MECHANICS. 
 
 Expressions that satisfy this are called Laplace functions. 
 Y and Z be two such functions, i and i' being whole numbers 
 and not identical then 
 
 J— I Jo 
 
 YiZidfidco = o. 
 
 The great value of these functions in physical research de- 
 pends on the fact that every function of the co-ordinates of a 
 point on a sphere can be expanded in a series by Laplace's 
 functions. They are therefore useful in mechanics in researches 
 in which spheres figure, as in the problem of the figure of the 
 earth, the general theory of attraction, and in electricity and 
 magnetism. 
 
 Laplace also published in 1812 his "Theorie analytique des 
 Probabilities," an exhaustive treatment of the subject of 
 probability. 
 
 It cannot be said of Laplace that he created a new branch 
 of science like Galileo or Archimedes, new principles or a 
 radically new method like Newton, Leibnitz, or Descartes. His 
 work was one of verification and formulation of known ideas 
 into grand generalizations. He possessed a genius for tracing 
 out the remote consequences of the great principles already 
 developed, and he brought within the range of analysis a great 
 number of physical truths which it did not appear probable 
 could ever be brought subject to laws of mechanics. His great 
 contribution was the invention of the potential function in 
 analysis, which, as developed by him and later by Green, Gauss 
 and Lord Kelvin, brought fluid motion, heat, electricity, and 
 magnetism under the dominion of analytical mechanics. 
 
 REFERENCES. 
 
 Mecanique Analytique. Paris, 1788. 
 
 Mecanique Analytique. Paris, i8ii. 
 
 Exposition du Systeme du Monde. Paris, 1873. 
 
 Mecanique Celeste. Translated by Bowditch. 
 
 Kelvin, General Integration of Laplace's Differential Equations of Tides. 
 
 Diihring, Geschichte der Principien der Mechanik. 
 
 Todhunter, Treatise on Laplace's Functions. 
 
 Mach, The Science of Mechanics. 
 
 Williamson, Treatise on Dynamics. 
 
 Todhunter, History of the Mathematical Theory of Attraction. 
 
 Thomson and Tait, Treatise on Natural Philosophy. 
 
the modern period. ii7 
 
 5. Recent Contributions. 
 The Contribution of Louis Poinsot (i 777-1 859). 
 
 The contribution of Poinsot to the science of mechanics is 
 one of method rather than of principle. In fact, since the 
 time of Lagrange and Laplace no radically new principle in the 
 science of mechanics has been brought forth, with the excep- 
 tion of the principle of conservation of matter and of energy. 
 
 Poinsot's work is set forth in two volumes: "Les Elemens 
 de Statique" and "Theorie Nouvelle de la Rotation des Corps." 
 He follows Newton's method, and builds the science on force, T 
 mass, and acceleration as fundamental concepts, but in his 
 exposition the notion of couples, i. e., pairs of parallel forces 
 acting on the same body in opposite directions has a prominent 
 part. This idea of a couple was now new; Poinsot did not 
 originate it. It follows from the principle of moments as set 
 forth by Varignon in 1687, but nothing worth mentioning 
 had been made of the idea till Poinsot based a system of 
 mechanics on it, in his Elemens de Statique in 1803. Perhaps 
 no idea in mechanics is so easily comprehended, so useful and 
 so fruitful in the presentation of equilibrium of rigid bodies. 
 But it does not express the historical development of the 
 science. Once mechanics had been developed, it was easy to 
 formulate a system of mechanics by the idea of the couple, 
 but as a rational primitive conception, the idea of equilibrium 
 established in this way does not appeal to the mind. 
 
 Poinsot says, in the preface of the "Elemens": "Dans la 
 solution mathematique des problemes, on doit regarder un 
 corps en equilibre comme s'il etait en repos; et reciproquement, 
 si un corps est en repos, on sollicite par des forces quelconques, 
 on peut lui supposer appliquees telles nouvelles forces qu'on 
 voudra, qui soient en equilibre d'elles-memes, et I'etat du 
 corps ne sera point change. On verra bientot de nombreuses 
 applications de cette remarque." One may regard a body in 
 equilibrium as if at rest, and one may regard a body at rest 
 as being so, because the forces applied to it balance each other. 
 One may assume various other pairs of forces applied to the 
 body and it will still remain at rest. This idea has many 
 useful applications. 
 
 ) 
 
Il8 THE SCIENCE OF MECHANICS. 
 
 He then develops the idea of a couple and sets forth a 
 number of theorems on couples from which he evolves the 
 theory of the simple machines. He says: "Nous reduirons 
 les machines simples a trois principales que Ton peut considerer 
 si Ton dans I'ordre suivant en regard a la nature de I'obstacle 
 qui gene le mouvement du corps: le levier le tour et le plan 
 incline." The simple machines may be reduced to three prin- 
 ciples according to the nature of points considered as fixed, 
 viz: the lever, the screw and the inclined plane. 
 
 In the first, the obstacle or impediment is a fixed point; in 
 the second, it is a straight line; in the third, it is a fixed plane. 
 From these he develops geometrical theorems on the simple 
 machines. 
 
 In general, Poinsot's method is distinctly his own develop- 
 ment of a synthetic mechanics, based on Newton's ideas. 
 He does not use the calculus, but develops the whole system 
 by a judicious choice of fixed points and by the action of 
 couples. He gives a self-contained exposition of the science 
 which is useful rather as a practical text-book than as a system 
 for advancing the science. The Theorie Nouvelle de la Rota- 
 tion des Corps treats of the motion of a rigid body by geometry 
 and shows that the most general motion of such a body can 
 be represented at any instant by a rotation about an axis 
 combined with a motion of translation parallel to the axis, 
 and that any motion of a body, of which one point is fixed, 
 may be produced by the rolling of a cone fixed in a body on a 
 cone fixed in space. This enables one to picture the motion 
 of a rigid body as clearly as the motion of a point. The 
 previous treatment of the motion of such a body had been 
 analytical, and gave no mental picture of the moving body. 
 
 Poinsot's exposition of statics and of rotation by the action 
 of couples about arbitrarily chosen fixed points, lines, or planes, 
 is valuable as offering ready practical conceptions of mechan- 
 ical action for every-day use. It is just such a system as 
 one would expect a professor in a technical school to develop 
 for the use of students who were preparing for professional 
 work rather than for research. The diagrams demonstrate 
 the theorems so as to make the proof almost axiomatic and 
 
THE MODERN PERIOD. II9 
 
 intuitive. His theorems are to be found to-day in modern 
 text-books and are of service to the mechanical and civil 
 engineer. 
 
 Among his memoirs are contributions on: "Sur la composi- 
 tion des moments et des aires." "Sur la geometric de I'equi- 
 libre et du mouvement des Systemes." "Sur la plan invariable 
 du systeme du monde." His Mechanics is valuable for its 
 ready practical methods, rather than for new contributions 
 to the science. 
 
 The Contributions of Simeon Denis Poisson 
 (1781-1840). 
 
 Poisson, the distinguished young contemporary of Laplace 
 and Lagrange, was their equal in mathematical analysis and 
 their superior in grasp of physical principles. A large number 
 of memoirs, on a wide range of scientific subjects, testify to 
 his ability. In some of these he corrected errors in the work 
 of Laplace and Lagrange. 
 
 Poisson applied himself particularly to mathematical 
 physics. He explored heat, light, electricity and magnetism 
 by analysis and originated the method of investigation by 
 "potential." He evolved the correct equation for potential 
 
 V^V = — 47rp 
 
 in place of Laplace's equation 
 
 V^F = o. 
 
 This equation now appears in all branches of mathematical 
 physics, and, according to some writers, it follows that it 
 must so appear from the fact that the operator V^ is a scalar 
 operator. Indeed it may be that this equation represents 
 analytically some law of nature not yet reduced to words. 
 
 Poisson's work, "Traite de Mecanique" (1853), is an excel- 
 lent exposition of rational mechanics by the method of the 
 calculus It proceeds logically from the definitions of "corps," 
 "masse" and "force," and a definition of Mechanics "la science 
 qui traite de I'equilibrium et du mouvement des corps" through 
 
120 . THE SCIENCE OF MECHANICS. 
 
 statics and dynamics, section by section. Though it contains 
 some variations in mathematical presentation, it contains no 
 new principle. 
 
 His work on the theory of Electricity and Magnetism and 
 his "Theorie Mathematique de la Chaleur," 1835, present 
 methods by which nearly all physical phenomena may be 
 explained in terms of mathematical mechanics. With this 
 the science of mechanics approaches its highest development. 
 From the time of Poisson up to the present, a number of 
 investigators have worked over the field and developed the 
 applications of known principles and methods. Among them 
 must be mentioned : 
 
 Fourier, Theorie analytique de la chaleur, 1822. 
 
 Gauss, De figura fluidorum in statu sequilibrie, 1828. 
 
 Poncelet, Cours de mecanique, 1828. 
 
 Belanger, Cours de mecanique, 1847. 
 
 Mobius, Statik, 1837. 
 
 Coriolis, Traite de Mecanique, 1829. 
 
 Grausmann, Ausdehnungslehre, 1844. 
 
 Hamilton, Lectures on Quaternions, 1853. 
 
 Jacobi, Vorlesungen iiber Dynamik, 1866. 
 
 Joule, J. P., Scientific Papers, 1887. 
 
 As a result of the earnest labors of these and others, and more 
 particularly by the patient research of those mentioned below, 
 the nineteenth century saw the establishment of the great 
 mechanical principle of conservation, the most unifying and 
 fruitful of all scientific dogmas. It is the result of the accu- 
 mulated experience of many inquirers rather than the achieve- 
 ment of any individual. 
 
 The Law of Conservation. 
 
 In 1775, the French Academy declined to consider any 
 further devices for obtaining "perpetual motion," but it was 
 not till one hundred years later, about 1875, that the generali- 
 zations known as the Conservation of Matter and the Con- 
 servation of Energy, or the Law of Conservation came to be 
 generally admitted after long experiment and careful study. 
 
THE MODERN PERIOD. 121 
 
 The principle of the Conservation of Matter was established 
 about 1780 by Lavoisier, (1743-94), as a result of a series of 
 experiments with the chemist's balance which indicated that 
 the mass of a given quantity of matter remains constant 
 regardless of change of state or of chemical combination. 
 
 The principle of conservation of energy was of slow growth. 
 The idea of conservation in nature seems to have been dimly 
 felt as far back as the time of Descartes (1596-1650). New- 
 ton, also, seems to have had an idea of it, though his de- 
 velopment of mechanics by the concepts of work, force and 
 distance, blinded him to the appreciation of the measure of 
 activity by energy. Still in the scholium to his third law, we 
 read : "If the action of an agent be measured by the product of 
 the force into its velocity, and if similarly the reaction of the 
 resistance be measured by the velocities of its several parts 
 multiplied into their several forces, whether they arise from 
 friction, cohesion, weight or acceleration, action and reaction 
 in all combination of machines will be equal and opposite." 
 It is probable that the popularity of the Newtonian exposition 
 of mechanics from the point of view of force and work, had a 
 tendency to delay the establishment of this principle of con- 
 servation. The concept of Energy was foreign to Newton's 
 mechanics. 
 
 The principle was rather a slow development of the Huy- 
 genian idea of energy and it came to the fore, with the recog- 
 nition of a relation between mechanical energy and heat. The 
 idea that heat is a form of energy for which there is an exact 
 mechanical equivalent was first suggested about 1798, by the 
 experiments of Count Rumford on the heat resulting from 
 the boring of cannon and by the experiments the following 
 year, of Sir Humphrey Davy on melting ice by friction. This 
 conception was at variance with the generally held hypothesis 
 that heat was of the nature of a material fluid. 
 
 The idea languished till 1842, when Julius Robert Mayer 
 began experimental research on the subject. Choosing as the 
 unit of heat, the quantity necessary to raise one gram of water 
 ato° C, one degree centigrade, commonly called a "calorie," 
 and for the unit of work, one gram lifted one meter or a 
 
122 THE SCIENCE OF MECHANICS. 
 
 "gram-meter," the determination of the number of gram- 
 meters that are equivalent to a calorie in energy was stated 
 by Mayer as 365 from his experiments on the heat evolved in 
 compressing air. 
 
 In 1843 J. P. Joule (1818-89) undertook the investigation of 
 the subject and invented a variety of apparatus for determin- 
 ing the dynamical equivalent of heat and among other forms 
 the common laboratory method of descending weights turning 
 paddle wheels in a vessel of water, the temperature of which 
 is determined by thermometers. The subject now came up 
 for thorough investigation and discussion by scientists. Helm- 
 holtz maintained the principle in "Ueber die Erhaltung der 
 Kraft," 1847, and Rankine, Kelvin, Clausius and Maxwell 
 contributed either experimentally or theoretically to its estab- 
 lishment. 
 
 It is worded in various ways, one form being: In any system 
 of bodies the energy remains constant during any reaction or 
 transformation between its part. It is also stated as: "The 
 energy of the universe is constant." 
 
 In 1850 Joule obtained his value 423.5 gram-meters for the 
 dynamical equivalent of heat which for two decades was the 
 accepted value. By i860 research had verified this figure by 
 transformations of energy through mechanical, electric, mag- 
 netic and chemical transformations in sufficient number to 
 warrant the acceptance of the principle of conservation of 
 energy. Prof. Rowland in 1879 made a series of very careful 
 determinations of the dynamical equivalent of heat using 
 Joule's stirring or paddle apparatus, and finally gave the value 
 425.9 for water at 10° C. 
 
 This principle is, as Maxwell says, "the one generalized 
 statement which is found to be consistent with fact, not in 
 one physical science only but in all. When once apprehended, 
 it furnishes to the physical inquirer a principle on which he 
 may hang every known law relating to physical actions, and 
 by which he may be put in the way to discover the relations 
 of such actions in new branches of science." He states the 
 principle as follows: "The energy of a system is a quantity 
 which can neither be increased nor diminished by any action 
 
THE MODERN PERIOD. 1 23 
 
 between the parts of the system, though it may be transformed 
 into any of the forms of which energy is susceptible." The 
 total energy of a closed system is invariable quantity. 
 
 Whether the energy of a system is partially in the kinetic 
 and partially in the potential form, whether the energy exists 
 as potential energy of arrangement of the gross parts of a 
 system, or as molecular energy, or electrical energy, or as 
 kinetic energy of moving masses, or of moving molecules, or 
 of vibrations of the ether or of electrical currents, the total 
 quantity of energy in an isolated system is constant. 
 
 We have no acquaintance w^ith "absolute energy" or of 
 energy apart from matter. Our knowledge is limited to energy 
 changes in matter. Work done upon a body or a system 
 increases its energy, or work done by it upon another body 
 confers energy upon it. If we do work upon a body weighing 
 100 lbs. so as to raise it vertically 5 ft. we store 500 ft. lbs. of 
 energy in it, which is said to be in the "potential" form. The 
 mathematical expression of energy always requires two factors. 
 For instance, in doing mechanical work we may measure the 
 energy by the product of the force times the distance, F XS, 
 or if the work has produced kinetic energy we measure it by 
 the mass of the body multiplied by the square of the velocity, 
 i. e., mv^J2. In case the mechanical work is transformed into 
 heat the factors become the specific heat and the rise in tem- 
 perature. If the heating is produced by a transformation of 
 electrical energy, the electrical energy is measured by the 
 quantity of electricity and the electromotive force. 
 
 From the principle of conservation have been evolved the 
 three principles of thermodynamics or of energetics which are 
 commonly listed as: 
 
 (i) the conservation of energy; 
 
 (2) the distribution of energy or the principle of Carnot; 
 
 (3) the law of least action. 
 
 The second principle is given by Clausius in the form: 
 "Heat cannot of itself pass from a colder body to a warmer 
 one." Lord Kelvin put it thus: "It is impossible, by means 
 of inanimate material agencies to derive mechanical effect 
 from any portion of matter by cooling it below the tem- 
 perature of the coldest surrounding objects." 
 
124 THE SCIENCE OF MECHANICS. 
 
 This was later generalized and put into the form : The trans- 
 fer of energy can only be effected by a fall in tension. This 
 is the principle of Carnot and signifies that energy always 
 goes from the point where the tension is high to the point 
 where it is low. This applies not only to heat but to all 
 known forms of energy. 
 
 If we imagine a system of bodies taken at random in various 
 conditions of temperature, electrification, etc., they will not 
 remain as thrown together, but a readjustment, with trans- 
 ferences and transformations of energy will begin, until one 
 of the factors of the energy of all the bodies has the same 
 value or intensity in all parts of the system. 
 
 That is, if the electromotive force or the temperature is 
 the same in all parts of the system, no transference takes 
 place; or, if for the kinetic energy, the velocity is the same, 
 there is no change; but whenever there is a difference there 
 will follow a change within the system. The third principle 
 of thermodynamics says that these changes always follow a 
 path which requires the least effort. This is sometimes named 
 Hamilton's principle. With these theories of readjustment 
 and flux of energy the occasion and character of the various 
 changes or phenomena of the material world may be 
 schematized. 
 
 It is worthy to note that no one has succeeded in exactly 
 and completely reversing a series of natural processes. There 
 is always a loss of energy usually as heat, in any series of 
 transferences or transformations of energy. The researches 
 of Clausius and Planck seem to prove that there is a constant 
 "degradation" of energy or a reduction to the condition of a 
 dead level. Without tension or difference in potential there 
 is no transmission of energy, nor can there be any work done. 
 
 Having attained then, the mechanical conception of energy 
 and the principles of conservation, we come into possession of 
 a unified theory and a workable scheme of antecedents and 
 sequences of the gross phenomena of nature, which now become 
 a subject of calculation by mathematical analysis as formulated 
 in Analytical and Celestial Mechanics. 
 
 Granted a certain quantity of energy in a material system, 
 
THE MODERN PERIOD. 125 
 
 the conditions of its transfer and transformation are now be- 
 come a matter of mathematical calculation, and the concomi- 
 tant gross phenomena may be predicted with certainty and 
 precision. The great principle of conservation of energy is a 
 wider generalization than the Newtonian mechanics. It has 
 enabled us to advance our explanation of the motion-phenom- 
 ena of the universe, but we are still far from explaining all 
 phenomena by Mechanics. The result of recent efforts to ex- 
 tend the science so as to explain the minuter and more subtle 
 phenomena of the universe will now be briefly commented 
 upon. 
 
 6. The Ether. Energy. Dissociation of Matter. 
 
 The nineteenth century saw the general acceptance of 
 Lavoisier's adage, "Nothing is created, nothing is lost." With 
 the gradual establishment of the idea of conservation came an 
 enthusiastic endeavor to unite the various separate sciences 
 into Science by means of the concept of energy. Energy being 
 conceived as a measure of activity and the quantity of energy 
 being considered invariable, it is logical to expect that all the 
 phenomena of the universe might be co-ordinated by this idea. 
 
 Mechanics which had developed the concept of energy and 
 a series of mathematical equations expressing its relations, 
 from a study of the gross motion phenomena of the world, 
 had arrived at what appeared to be a universal law. And 
 now the various separate chains of phenomena which had 
 been linked together by the chemist, the physicist, the botanist 
 and the biologist were to be welded into one Science by the 
 principles of mechanics. The chemists had been working 
 toward the idea of conservation for nearly a century and 
 when chemistry and mechanics came into accord upon the 
 idea of conservation, it was felt that it must fit the other 
 sciences too, and that it was the key to nature's secrets. 
 
 A review of the scientific beliefs of twenty-five years ago 
 reveals a faith in the duality of natural phenomena. They 
 were conceived as the result of the action of indestructible 
 energy through indestructible matter which was conceived as 
 floating in an all pervading medium called the ether of space. 
 
126 THE SCIENCE OF MECHANICS. 
 
 This medium was conceived as penetrating and pervading all 
 matter. 
 
 The idea of an ether of space appears to be very old. The 
 term is derived from the Greek word aether, meaning the 
 brilliant upper air. The hypothesis in later times was the 
 result of the logic that demanded a medium to transmit light 
 and heat through interplanetary space and through a vacuum. 
 Hence it was at first called the light-bearing or luminiferous 
 ether. 
 
 Fresnel (i 788-1 827), the French physicist, in his undulatory 
 theory of light first gave this hypothesis definition. Later 
 Faraday (1791-1867) likewise postulated a medium in connec- 
 tion with his researches in electricity and magnetism and 
 suggested that perhaps one and the same medium would serve 
 for both light and electricity. The researches and calculations 
 of numerous investigators among whom Maxwell was promi- 
 nent finally gave decision in favor of one medium or ether, 
 possessing certain characteristics. 
 
 Being a purely arbitrary hypothesis the ether could and 
 soon came to be endowed with such properties as were called 
 for by the logic of the situation, and these properties were 
 altered from time to time as seemed necessary. The ether 
 was declared to possess inertia, because time was required 
 for the propagation of light through it. It was conceived as 
 having density and elasticity by analogy with matter, and it 
 was pictured as an "elastic jelly." In this medium, waves 
 varying in length from miles to less than two millionths of a 
 millimeter were conceived as explaining various phenomena of 
 light, heat, electricity and magnetism. Though nothing is 
 positively known of the existence or structure of the ether, 
 this convenient assumption has been developed with great 
 definiteness. 
 
 Once this hypothesis was established Mechanics entered 
 upon a new phase of development. It was called upon to 
 deal with molecular and atomic energy and invited to explain 
 by its principles the minute phenomena of light, electricity and 
 biology. In this it relied upon the unifying power of the law 
 of conservation and the license to warp and model the sup- 
 posititious ether to the exigencies of the occasion. 
 
THE MODERN PERIOD. 127 
 
 How far this has been successful can be but briefly considered 
 here. It soon became apparent that the molecule or smallest 
 portion of physical matter, sometimes pictured as bearing to 
 a drop of water the ratio that a golf ball bears to the earth, 
 must give up its simplicity as a dense hard sphere and become 
 constituted of at least several atoms of various densities to 
 comply with the chemist's notions of elementary and com- 
 pound substances. 
 
 Before long, these atoms had assumed the complexity of 
 solar systems and were conceived as composed of thousands 
 of particles or electrons in rapid motion, and as being of many 
 varieties. Here we see at work the familiar old primitive no- 
 tions of division, moving particles and pictorial representation. 
 In the hands of such investigators as Fizeau, Crookes, Kelvin, 
 Lodge, Le Bon, Michelson, Morley, Rayleigh, Ramsey, Roent- 
 gen, J. J. Thomson, Rutherford and others, the method has 
 been applied in linking up, by the principles of gross mechanics 
 a variety of minute phenomena. It has led experimental re- 
 search through numerous novel and remarkable investigations 
 in light, heat and electricity from which much is expected. 
 
 With the discovery of the X-Rays by Roentgen in 1895, and 
 of radioactivity by Becquerel and the Curies in 1898, and with 
 the discovery by J. J. Thomson that the passage of these 
 activities through the air makes it a conductor of electricity, 
 new conceptions arise. The air as we commonly know it, is 
 a non-conductor of electricity but "ionized" air produced by 
 radioactivity, or by the emanations from such substances as 
 radium, thorium, and polonium, is a conductor. It soon be- 
 came evident that a great many bodies in nature are spon- 
 taneously active and are constantly giving out emanations. 
 
 Investigation showed that these emanations have the power 
 of dissociating a gas, or of breaking it up into particles, com- 
 parable with hydrogen atoms, and particles approximately one 
 thousandth as large, called electrons. The velocity of these 
 particles approximates that of light and their total mass or 
 inertia appears to be due to an electric charge in motion. In 
 other words the one characteristic invariable property of mat- 
 ter, viz: mass, is explained as an electric charge in motion. 
 
128 THE SCIENCE OF MECHANICS. 
 
 Larmor, in his "Ether and Matter," says the atom of matter is 
 composed of electrons and of nothing else. This conception 
 builds matter of electricity in motion, though it is a question as 
 to whether this is a simplification or a complication of theory. 
 
 The question as to where these electrons get their motion, 
 or what is the origin of the energy which expels these emana- 
 tions with such terrific velocity, has been met by a mechanical 
 hypothesis of the atoms as whirling "solar systems" of thou- 
 sands of electron-satellites, some of which, when equilibrium is 
 disturbed, fly off tangentially with great velocities. This is 
 practically saying that molecules and atoms of matter on their 
 disruption or dissociation set free energy. Experiments on 
 radioactivity show that a gram of radium will raise the tem- 
 perature of 100 grams of water i° C. an hour without per- 
 ceptible loss of weight on the chemist's balance. But the re- 
 searches of Prof. Crookes and Dr. Heydweiller,^ estimate the 
 duration of a gram of radium at about lOO years after which 
 there is no longer any radium, therefore a quantity of highly 
 heated water may be left as a result of its emanations if we 
 conceive it to act upon water. Here matter disappears and 
 energy in the form of steam pressure appears in exact ratio. 
 
 This brings us face to face with a contradiction of the law 
 of conservation as we have stated it. We have matter fading 
 into the ghost of matter losing its one distinguishing unalter- 
 able characteristic, namely, mass, and liberating an enormous 
 quantity of energy in the process. From a mechanical point 
 of view this is a contradiction in terms but the advance guard 
 on the skirmish line of science necessarily uses the terms that 
 are at hand with various mental reservations and modifications 
 until nomenclature can be revised and remodeled. With every 
 advance in Science there is inevitably a period of temporary 
 anarchy in theory and terminology. The concepts of energy 
 and electricity appear to be about to go through some such 
 period of transformation as has happened with the term force. 
 
 We find ourselves now on the threshold of the realization 
 of the dream of the alchemist. These X-rays, emanations, 
 
 ^P. 237. 
 
 'Phys. Zeitschrift, October 15, 1903. 
 
THE MODERN PERIOD. 1 29 
 
 ions, electrons and electricity appear to be phases of the 
 dematerialization of matter, stages in the breaking down of 
 matter into intra-atomic energy. As Professor de Heen of 
 Liege says, "it seems we find ourselves confronted by condi- 
 tions which remove themselves from matter by successive 
 stages of cathode and X-ray emissions and approach the sub- 
 stance designated as the ether." 
 
 Further researches indicate that electricity is one of the 
 forms of energy that result from the breaking up of atoms, 
 that it is composed of these imponderable electrons, the ghostly 
 emanations of fading matter which themselves have been pic- 
 tured as but minute whirls in the all pervasive ether. We come 
 here to a new conception, matter is conceived as built up of 
 electrons, pictured as little whirl-pools in a fundamental ether 
 of which the universe is composed. However this may be, we 
 are made acquainted with stores of energy and activities as 
 little known as electricity was before Volta's day. The estab- 
 lishment of the fact of the dissociation of matter opens up 
 unsuspected and inconceivable sources of energy. The energy 
 liberated from the partial dissociation of a tub of water would 
 probably equal that of all the anthracite coal fields of America. 
 
 This theory hints at an explanation of some of the mysterious 
 activities of vegetable and animal life. The researches of bio- 
 logical chemistry are just beginning to reveal some of the 
 secrets of the flux and reflux of intra-atomic energy in highly 
 complicated and unstable compounds and the incidental liber- 
 ation of (electrical) energy. The theory also offers suggestions 
 as to the character of allotropy, catalytic action, diastases, 
 toxins and protoplasmic action. These minute phenomena of 
 nature are motion-phenomena and as such come within the 
 purview of mechanics, but in the development of a theory of 
 the grosser phenomena they have had scant attention. It 
 may be that the laws of gross mechanics do not apply here 
 exactly, at any rate it seems that there is enough suspicion of 
 mutation of matter and flow of energy to put the law of con- 
 servation on the defensive. 
 
 The most radical contradiction of the now commonly ac- 
 cepted doctrine of conservation is that given by Dr. Gustave 
 
130 THE SCIENCE OF MECHANICS. 
 
 Le Bon in his "Evolution of Matter," 1905, from which the 
 following summary is taken. 
 
 "i. Matter, hitherto deemed indestructible, vanishes slowly 
 by the continuous dissociation of its component atoms. 
 
 "2. The products of the dematerialization of matter con- 
 stitute substances placed by their properties between ponder- 
 able bodies and the unponderable ether — that is to say between 
 two worlds hitherto considered as widely separate. 
 
 "3. Matter, formerly regarded as inert and only able to 
 give back the energy originally applied to it, is on the other 
 hand, a colossal reservoir of energy — of intra-atomic energy 
 — which it can expend without borrowing anything from 
 without. 
 
 "4. It is from the intra-atomic energy, manifested during 
 the dissociation of matter that most of the forces in the uni- 
 verse are derived, notably electricity and solar heat. 
 
 "5. Force and matter are two different forms of one and 
 the same thing. Matter represents a stable form of intra- 
 atomic energy; heat, light, electricity, etc., represent unstable 
 forms of it. 
 
 "6. By the dissociation of atoms, — that is to say, by the 
 dematerialization of matter, the stable form of energy termed 
 matter is simply changed into those unstable forms known by 
 the names electricity, light, heat, etc. 
 
 "7. The law of evolution applicable to living beings is also 
 applicable to simple bodies; chemical species are no more in- 
 variable than are living species." 
 
 These are bold generalizations made from comparatively 
 scanty experimental data on very minute and delicate phe- 
 nomena, and they are not unchallenged. But they suggest 
 a new departure and a new phase of development in mechanics 
 and hint at marvels until now undreamt of. 
 
 As to the possibility of producing energy for industrial 
 purposes by breaking down or using up matter and thus turn- 
 ing it into energy, the expectation is certainly as bright as 
 was the prospect, that Volta's early electrical experiments 
 with frogs' legs and a copper wire would ever lead to the 
 operation of heavy railroad trains by electricity or to the 
 
THE MODERN PERIOD. I3I 
 
 transmission of the voice from city to city, by wire, or of 
 "wireless messages" from mid-ocean to shore. 
 
 The wonders of aerial telegraphy and telephony are the 
 result of careful investigation and study in this new field of 
 what might be called the mechanics of the ether. When the 
 "activities in the ether" are more thoroughly understood we 
 may expect greater wonders. It is to be noted that it is not 
 always the most intense action that will produce a desired 
 result. A thunder clap will not move a tuning fork to vibra- 
 tion, whereas the vibration of a violin string will do so if of 
 the proper key. A spark is ridicuously inadequate as com- 
 pared with the explosion of energy it may cause. The simple 
 striking of a phosphorus-match by moving it with a velocity 
 of about ten feet a second, serves to set up disturbances 
 which have a velocity of 186,000 miles a second. 
 
 Atomic energy, of the existence of which there seems to be 
 no doubt, is practically inexhaustible in amount, as simple 
 calculations show. The energy that would flow from the dis- 
 sociation of a one cent copper coin is equal to the energy of 
 1,000 tons of coal applied in the production of steam. Me- 
 chanics has brought us from the dim gropings of the Stone 
 Age, for "more power to the arm," to an outlook upon an 
 immense universe of ceaseless energy. When mechanical con- 
 trivance shall have caught up with, and exploited this vision 
 we may expect a conquest of power that will accomplish incon- 
 ceivable wonders. 
 
 This then is the fruit of fifty centuries of patient endeavor 
 in mechanics, of 2,000 years of geometrical mechanics and 200 
 years of analytical mechanics. It is the heritage of the patient 
 fidelity and stern integrity of the great inquirers and their nu- 
 merous minor coadjutors, and it presages a greater and more 
 marvellous harvest of enlightenment and benefaction for the 
 future. The indomitable courage and patience of these 
 searchers for ultimate and invariable truth have emancipated 
 the race from much of the incubus of the superstitious fetish- 
 ism, and from some of the drudgery of daily life, and they point 
 prophetically to greater conquests to come. But in the words 
 
132 THE SCIENCE OF MECHANICS. 
 
 of one of the eminent sages^ of the science, all have thus far 
 been as little children picking pebbles on the shore, while the 
 great ocean of the unknown glooms beyond. The words of 
 Laplace are still all too true, "What we know is little, what we 
 do not know, immense." 
 
 ^Sir Isaac Newton. 
 
PART IV. 
 CONCLUSION. 
 
 The history of the science of mechanics has now been traced 
 in outHne. We have noted its aspirations; we must now note 
 its limitations. Science is human experience tested and ar- 
 ranged in order. It is not its purpose to offer a philosophy 
 of the universe, nor is it essentially in conflict with religion. 
 It seeks, rather to co-ordinate experiences into a systematic 
 theory of relations, of causes and effects. The discovery of 
 natural truths and the extension of the field of knowledge by 
 a process of correlation, rejection, revision and verification is 
 its province. 
 
 We note that the science is a mental resume of the growing 
 experience of the race, a development founded on many cen- 
 turies of endeavor in the arts and trades. It had its origin 
 in the dim past with geometry which evolved from land- 
 surveying as mechanics did from the trades. The science is 
 essentially the product of European thought. In the nature 
 of things its development consisted in abstracting from the 
 numerous phenomena of nature the constant elements, this 
 method obviously indicating itself as the path of progress. 
 Once the abstractions of form and position were realized, study 
 of forms and positions led to the development of a geometry 
 of measurement and an arithmetic. Until this point is reached 
 not much can be expected in physical science, for the spur of 
 progress is the question "how," and no satisfactory answer 
 can be given to it until a system of measurements is developed. 
 
 When once the abstract conceptions of form and position 
 are firmly established and a method of measurements devised, 
 then the conditions and circumstances of change of position 
 and of change of form and size present themselves as questions 
 of possible investigation. 
 
 Even after the Greeks had developed geometry, their ideas 
 
 133 
 
134 
 
 THE SCIENCE OF MECHANICS. 
 
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CONCLUSION. 135 
 
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136 THE SCIENCE OF MECHANICS. 
 
 on natural phenomena are found expressed in such naive state- 
 ments as those of Aristotle to the effect that all bodies have a 
 place and seek their place; bodies float because of their form; 
 if they move, their motion is either natural or violent; heavy 
 bodies go down because they belong down under the lighter 
 ones; the heavier they are, the farther down they belong and 
 the faster they move to get there! It was not till the abstract 
 concept of force was completely attained to, eighteen hundred 
 years later, and the concomitant circumstances of it studied 
 and measured that these ideas gave place to our modern gen- 
 eralizations. 
 
 The measurement of the constant elements in natural phe- 
 nomena naturally began with bodies near at hand and at 
 rest, with statics and hydrostatics. We find that the great 
 contribution of Archimedes is not so much the rules and 
 methods commonly associated with his name, as the develop- 
 ment of a science of measurement and its application in the 
 study of natural phenomena. With this the science of me- 
 chanics began. 
 
 Here at its very basis we find the abstract concept and the 
 mathematical notion of relativity as fundamental to the 
 science. Those who ask a physics or mechanics without ab- 
 stractions and mathematics are seeking science without its 
 most essential and useful features. The further review of 
 the science indicates that an evolution of abstract concepts 
 and a development of the application of analysis in connection 
 with them is an inevitable necessity of its progress. 
 
 It is not the purpose here to consider metaphysical and 
 psychological questions, therefore we will not speculate upon 
 the probability or possibility of developing a science of me- 
 chanics on another basis than the abstract concept linked up 
 mathematically. That we cannot know matter or the phe- 
 nomena of nature except as mental percepts or concepts is a 
 trite saying; it seems to follow logically therefore that our 
 mechanics must be built up of these elements, however much 
 experience and study may change and elaborate them. 
 
 We have seen how the notion of force developed and 
 changed. How at first it was probably anthropomorphic, con- 
 
CONCLUSION. 137 
 
 ceived as the muscular vigor of an invisible demi-god, how it 
 underwent transformation, served a very useful purpose in 
 the development of mechanics and is now discarded in some 
 text-books as an outworn subjective idea to which there is 
 nothing in nature to correspond. It is so with other ideas, 
 and perhaps many of them will continue to go through some 
 such development and transformation. They will be changed, 
 modified or discarded. Mechanics was developed by geo- 
 metric methods of measurement up to the year 1700; then, 
 the method of fluxions and limits made possible the investi- 
 gation of quantities whose value is continually changing, and 
 the science made a wonderful advance. 
 
 In all mechanical experience there are two conceptions, how- 
 ever, which are constantly present, and form, as it were, the 
 background — Time and Space. They appear to be funda- 
 mental and ultimate; being irreducible, they cannot be com- 
 pared with anything, and are indefinable. With them a third 
 conception variously pictured and called, matter, energy, ether 
 or electron, suffices to form the rational resume of phenomena 
 which is called mechanics. The riddle of time and space is a 
 question of metaphysics, not of mechanics. Considering it 
 briefly we note there are several points of view. One may 
 hold after the manner of Trendelenberg in his "Logische Un- 
 tersuchungen," Chap. V, that there exists the conception of 
 motion quite apart from the ideas of space and time, which 
 he derives from it. His endeavors to prove a knowledge of 
 motion prior to any idea of position or of sequence are not 
 convincing. Though the theory is plausible, it is not proven. 
 A second theory due largely to the philosophy of Kant and 
 Hume would make time and space merely modes of perception, 
 ways in which the perceptive faculty distinguishes objects. 
 Though this is not admitted generally as expressing the full 
 truth, it indicates clearly the intimate relation between time 
 and space. 
 
 They are bound together in a way which may be pictured 
 by supposing space to represent the breadth of our field of 
 perception, then time would represent its length. Space 
 marks the co-existence of perceptions at a point in time, so 
 
138 THE SCIENCE OF MECHANICS. 
 
 time marks the progression of perceptions at a position in 
 space. The two modes combined give us motion as a funda- 
 mental way in which we conceive phenomena. 
 
 If we admit this, the mode of perceiving things in this way 
 would seem an essential feature of our conscious life. With 
 only the one space mode of perception, we could know only 
 of co-existing things, of number, position and measurement; 
 our science would be limited to arithmetic, algebra and geo- 
 metry, and the phenomena of motion would not exist for us. 
 We could not conceive of warmth, weight, hardness, etc., for 
 these depend upon sequence, on time. On this theory the 
 perceptive faculty sorts sense impressions by these two modes, 
 as upon a rack or frame-work; and if the simile be permitted 
 one may say both co-ordinates are necessary. Neither infinity 
 of space or of time, or empty time find place on the frame- 
 work, nor have they meaning in the field of perception. 
 
 Space and time in this view are modes of perceiving things. 
 They are not necessarily, per se, infinitely large or infinitely 
 divisible, but are essentially relative. The reality of time 
 and space is not a point of discussion in this paper, but this 
 philosophical theory is often involved with the philosophy 
 that denies or is agnostic as to the reality of matter, — re- 
 garding sense perceptions under the modes of time and space 
 as the only realitites. 
 
 A more rational point of view and one that will appeal to a 
 greater number is the theory that neither denies the existence 
 of matter and motion apart from perception nor affirms that 
 time and space are but modes of thought, and that motion 
 is the concomitant of their relation to each other. According 
 to this theory the reality of moving bodies is not denied, but 
 it recognizes that our knowledge of them is limited to what 
 we may perceive and conceive of under the modes or limita- 
 tions of time and space. It recognizes our hypotheses and 
 principles as approximations, and questions whether we can 
 ever fully conceive of or comprehend the realities of nature 
 in all their completeness. 
 
 The concepts developed with and within these modes of time 
 and space, such as geometrical surface, atom, molecule, force, 
 
CONCLUSION. 139 
 
 etc., are not necessarily the realities. Often they are obviously 
 not so. But these terms are very useful in picturing the correla- 
 tion and sequence of phenomena. The continual shifting and 
 the evolution which we see going on in our scientific concepts 
 indicate that they are approximations, and points the necessity 
 for caution in speaking of them as realities, or of projecting 
 ideal dancing or whirling molecules into the world of the actual. 
 As to this third conception which, together with time and 
 space, suffices to illustrate the phenomena of nature, it is 
 variously conceived and described as matter, as ether, and 
 as electrons. The first and oldest of these was the idea out 
 of which the others have arisen as later and broader knowledge 
 imposed more precise requirements. The older books on 
 mechanics went along with "corpus" and "moles" until the 
 distinction between weight and mass was made. Then 
 "masses" sufficed until more recent times when matter came 
 to be commonly defined in the text-book by its properties 
 of extension and inertia. The modern view is given in the 
 little book "Matter and Motion," by Clerk Maxwell thus, 
 (page 163): "All that we know about matter relates to the 
 series of phenomena in which energy is transferred from one 
 portion of matter to another till in some part of the series our 
 bodies are affected, and we become conscious of a sensation. 
 We are acquainted with matter only as that which may have 
 energy communicated to it from other matter. Energy, on 
 the other hand, we know only as that which in all natural 
 phenomena is continually passing from one portion of matter 
 to another. It cannot exist except in connection with matter." 
 
 In effect, this paragraph defines this third thing as a medium 
 for the storage and communication of energy, without telling 
 what it is or what energy is. Having arrived at "energy" as 
 a convenient conception and being under the necessity of con- 
 ceiving of it as stored and transmitted, the idea of matter is 
 made to assist to that end. 
 
 In the well known Treatise on Natural Philosophy by Thom- 
 son and Tait we read (p. 207): "We cannot of course, give 
 a definition of matter which will satisfy the metaphysician, 
 but the naturalist may be content to know matter as that 
 
I40 THE SCIENCE OF MECHANICS. 
 
 which can be perceived by the senses or as that which can be 
 acted upon, or can exert force." This definition, Hke the 
 first, is of a dual character indicating matter as the medium of 
 the action of force instead of energy. Either of these defini- 
 tions will serve for a development of mechanics from the view 
 of energy or of force. With either of these premises granted 
 a logical mechanics is possible. 
 
 If we consult Tait's "Properties of Matter" (pp. 12-13 and 
 pp. 287-91), we read: "We do not know, and are probably 
 incapable of discovering what matter is," and, "The discovery 
 of the ultimate nature of matter is probably beyond the range 
 of human intelligence.'" This is at least decisive, but some 
 would probably say it is unnecessarily blunt and discouraging. 
 
 The idea of matter, proving inadequate with the advance 
 of the science, and the application of molecular mechanics 
 in the study of light, electricity and heat, the theory of the 
 ether as a perfect fluid medium and a perfect-jelly medium 
 was introduced. The jelly-theory of the ether has undoubt- 
 edly been of value in simplifying many of our views of physical 
 phenomena, but not being entirely satisfactory, the "vortex 
 atom" and "vortex ring" in the ether were invented. This 
 was followed by Kelvin's "ether-squirt." From periodic varia- 
 tions of the rate of squirting as influenced by the mutual 
 action of groups of squirts, he was able to picture and deduce 
 many of the phenomena of chemical action, cohesion, light 
 and electro-magnetism. 
 
 A more recent theory is the corpuscular or electron theory 
 as expounded by J. J. Thomson in "The Corpuscular Theory 
 of Matter," 1907, which supposes that "the various properties 
 of matter may be regarded as arising from electrical effects." 
 On page 2 of this volume we read: "This theory supposes 
 that the atom is made up of positive and negative electricity. 
 A distinctive feature of the theory — the one from which it 
 derives its name — is the peculiar way in which the negative 
 electricity occurs both in the atom and free from the atom." 
 He supposes that the negative electricity always occurs as 
 exceedingly fine particles called corpuscles, and that all these 
 corpuscles, whenever they occur, are always of the same size 
 
CONCLUSION. 141 
 
 and always "carry the same quantity of electricity." "What- 
 ever may prove to be the constitution of the atom we have 
 direct experimental proof of the existence of these corpuscles." 
 This theory has the advantage of being able to explain elec- 
 trical metallic conduction. It explains mechanical inertia as 
 the self induction of an electric current, and mass on the basis 
 of the velocity of the corpuscles. On this theory matter is 
 conceived of, as in part at least identical with electricity, and 
 the properties of matter are explained as electrical effects. 
 But this still leaves a third concept, electricity in addition 
 to time and space. It is interesting to note that in the exposi- 
 tion of this theory, analysis and the elementary mechanical 
 concepts of velocity, mass and energy, etc., are used to attain 
 to this new idea. This illustrates the evolution of new con- 
 cepts from the old, as the path of advance in mechanics, and 
 in all science. 
 
 Yet even these closer approximations cannot be regarded 
 as realities. They indicate mechanical actions which may be 
 close approximations of the reality but there is no ground for 
 calling them identities. 
 
 Dr. Ernst Mach says,^ "purely mechanical phenomena do 
 not exist. They are abstractions made either intentionally or 
 from necessity for facilitating our comprehension of things." 
 Though this is not conclusively established, the mechanical 
 theory of nature does seem artificial. There is no reason for 
 believing that an actual mechanism of atoms and molecules 
 as some scientists present, is at the bottom of nature. In fact, 
 recent researches in the electron idea tend to indicate that this 
 is unlikely. But this pictorial mechanical method developed 
 by European thought is a highly serviceable and valuable ex- 
 pedient for generalizing experiences, teaching them, and apply- 
 ing them. It serves a most useful purpose in investigation. 
 
 As J. J. Thomson says in his "Corpuscular Theory of Mat- 
 ter, "^ "From the point of view of the physicist, a theory of 
 matter is a policy rather than a creed; its object is to connect 
 or co-ordinate apparently diverse phenomena, and above all to 
 
 '"Mechanics," p. 404. 
 *Page I. 
 
142 THE SCIENCE OF MECHANICS. 
 
 suggest, Stimulate and direct experiment." Mechanics will no 
 doubt continue to develop in this way in the future as in the 
 past. As Professor Ziwet says:^ "It is now pretty generally 
 recognized that Newton's laws of motion including his defini- 
 tion of force are not unalterable laws of thought but merely 
 arbitrary postulates, assumed for the purpose of interpreting 
 natural phenomena in the most simple and adequate manner. 
 . . . It is now coming to be recognized, as researches are made 
 in the electron theory, that the abandonment or generalization 
 of the older mechanics must lead to a more general mechanics. 
 It will probably be non-Newtonian, based on the development of 
 the electron theory including Newton's laws as a special case." 
 Prof. Pearson says:^ "We must hope to ultimately conceptual- 
 ize an ether, from the simple structure of which several of 
 the laws of motion postulated for particles of gross matter 
 may directly flow. . . . The customary definitions of mass 
 and force, as well as Newton's statements of the laws of 
 motion, abound in metaphysical obscurities. It is also ques- 
 tionable whether the principles involved in the current state- 
 ments as to superposition and combination of forces are scien- 
 tifically correct when applied to atoms and molecules. The 
 hope for future progress lies in clearer conceptions of the nature 
 of ether and of the structure of gross matter." 
 
 The history of mechanics in the past indicates that before 
 each step in advance, there is a period of readjustment and 
 of assimilation of previous ideas. We appear to be passing 
 through such a period now. The great generalizations of the 
 law of gravitation and of the principles of conservation of 
 matter and of energy have about done their work of readjust- 
 ment, and have been assimilated to the previous ideas, forming 
 a body of doctrine as a basis for further progress. 
 
 As indications of where this advance may be expected, one 
 may look toward the points at which investigators are dis- 
 satisfied with the science, or are not in accord with each other. 
 These are in the direction of the ultimate character of this 
 third concept variously termed matter-energy, ether-squirt, 
 
 ^Science, Vol. XXIII, p. 50. 
 
 *" Grammar of Science," p. 321. 
 
CONCLUSION. 143 
 
 vortex-ring, electron, etc., and in the direction of the law of 
 gravitation. This principle never has fitted in well with the 
 other principles of mechanics. It is found unsatisfactory in 
 that its action seems to be different in kind. It is a convenient 
 generalization, but there is no explanation of how the pull of 
 one body is conducted across space or what conducts it. The 
 principle of action at a distance is not satisfactory. It was 
 not satisfactory to Newton. Progress is to be expected in this 
 direction, and a beginning has already been made with the 
 Electron Theory. It is possible that a more general law may 
 be evolved from the study of potential as expressed in the 
 equation of Poisson. 
 
 V^F = — 47rp. 
 
 This formula appears to express a mode of conception of 
 natural phenomena which is ai^Qst_asultimate as the time 
 and space modes. It may be that a law or mode of conception 
 may be evolved that will include all three modes in one. 
 But though this is foreshadowed in analysis it is not possible 
 yet, to state such a law in words. 
 
 As some of the fundamental concepts of formal mechanics, 
 such as matter and the law of gravitation, are not beyond criti- 
 cism, later advances will very probably reformulate the science. 
 It may indeed be necessary partially to tear down the present 
 system and build it anew on different lines, when the funda- 
 mentals are more correctly perceived and comprehended. The 
 science originated with, and developed from a study of gross 
 bodies with motions of considerable amplitude, and the notions 
 thus obtained have been refined and applied in picturing the 
 unseen. The minute operations which produce the large ap- 
 pearances may in the future be pictured as of a different kind 
 and order from the gross things. While the endeavors of 
 the German professors Hertz and Boltzmann in this direction 
 cannot be called successful, they indicate the tendency. We 
 should be careful not to let prejudice in favor of present ideas 
 and methods hamper progress as prejudices have done in the 
 past. 
 
 The curious disintegrating effects of radium and uranium 
 
144 THE SCIENCE OF MECHANICS. 
 
 and their derivative products, each with a characteristic "rate 
 of decay," tend to weaken the notions of immutability and 
 conservation, and to reinforce the idea that formal mechanics 
 at best gives but a hazy picture of the realities of the world. 
 But it is a model or a picture that can be improved and brought 
 more in accord with a wider and more varied number of 
 phenomena. Though imperfect, its value and utility in ap- 
 plied science and engineering is marvellous. Its economic 
 value is beyond question, and is indeed the reason of its exist- 
 ence, and one of the strong incentives to its improvement. 
 
 The "conceptual shorthand," by which the resume of phe- 
 nomena is made, will no doubt be improved, but it seems im- 
 possible that the fundamental concepts of time and space 
 shall give place. And the third idea which is at present neces- 
 sary for a formal presentation of the science appears to contain 
 an ultimate element not resolvable into these other two. The 
 future may evolve from electricity or energy a more precise 
 idea of this third fundamental concept and make clearer its 
 connections. 
 
 At present, in spite of the fact that some of the generaliza- 
 tions recorded under the head of mechanics are widely appli- 
 cable in the world of phenomena, we cannot claim that the 
 science comprises a knowledge of the foundations of the world 
 of phenomena, nor indeed a true picture of any reality of the 
 world. The most that may be legitimately claimed is that it 
 gives a tentative mental resume, as Dr. Mach says, an "as- 
 pect" of the world of phenomena which is fairly satisfactory 
 and prodigiously useful and valuable. 
 
 Withal we should be on our guard lest our science be too 
 much with us, late and soon, lest we come to reverence these 
 apparent constancies of relation and these serviceable fancies 
 too highly. When in the glories of sunset and rainbow one 
 sees and thinks of nothing but molecules and refractions, then 
 truly there is "little we see in nature that is ours." 
 
 The historical review of the development of the Science indi- 
 cates that it is not essentially in conflict with Philosophy or 
 Religion. It speculates not why, but asks how, and it is only 
 the tyro who finds it incompatible with piety. 
 
CONCLUSION. 145 
 
 While, with some, a mechanical explanation of all nature 
 is an avowed ideal, the scientist who ponders the world of 
 phenomena with an open-mind, cannot but be impressed with 
 a causal activity immanent therein, which is more than blind 
 chance. The universe is more than a fortuitous concourse of 
 molecules. The more we study it, the more need we have to 
 predicate as a cause of the cosmos as a whole, and of its cease- 
 lessly varying infinity of phenomena, an Immanence of Con- 
 trol, incomprehensible to our finite mind. 
 
146 THE SCIENCE OF MECHANICS. 
 
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BIBLIOGRAPHY. 1 47 
 
 Larmor, Jos. Ether and Matter, 1901. 
 
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 Maspero. The Dawn of Civilization. 
 
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 Geschichte der exacten Wissenschaften. Leipzig, 1863. 
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 Todhunter, L A History of the Calculus of Variation during Nineteenth 
 
 Century. 
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