A Short Account of the History of Mathematics · W. W. Rouse Ball

Chapter 3: The Schools of Athens and Cyzicus

5 of 28 · 28 min of reading

The history of these schools is discussed at length in G. Loria's Le Scienze Esatte nell' Antica Grecia, Modena, 1893--1900; in G. J. Allman's Greek Geometry from Thales to Euclid, Dublin, 1889; and in J. Gow's Greek Mathematics, Cambridge, 1884; it is also treated by Cantor, chaps. ix, x, and xi; by Hankel, pp. 111--156; and by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; a critical account of the original authorities is given by S. P. Tannery in his Géométrie Grecque, Paris, 1887, and other papers.

It was towards the close of the fifth century before Christ that Athens first became the chief centre of mathematical studies. Several causes conspired to bring this about. During that century she had become, partly by commerce, partly by appropriating for her own purposes the contributions of her allies, the most wealthy city in Greece; and the genius of her statesmen had made her the centre on which the politics of the peninsula turned. Moreover, whatever states disputed her claim to political supremacy her intellectual pre-eminence was admitted by all. There was no school of thought which had not at some time in that century been represented at Athens by one or more of its leading thinkers; and the ideas of the new science, which was being so eagerly studied in Asia Minor and Graecia Magna, had been brought before the Athenians on various occasions.

Anaxagoras. Amongst the most important of the philosophers who resided at Athens and prepared the way for the Athenian school I may mention Anaxagoras of Clazomenae, who was almost the last philosopher of the Ionian school. He was born in 500 b.c., and died in 428 b.c. He seems to have settled at Athens about 440 b.c., and there taught the results of the Ionian philosophy. Like all members of that school he was much interested in astronomy. He asserted that the sun was larger than the Peloponnesus: this opinion, together with some attempts he had made to explain various physical phenomena which had been previously supposed to be due to the direct action of the gods, led to a prosecution for impiety, and he was convicted. While in prison he is said to have written a treatise on the quadrature of the circle.

The Sophists. The sophists can hardly be considered as belonging to the Athenian school, any more than Anaxagoras can; but like him they immediately preceded and prepared the way for it, so that it is desirable to devote a few words to them. One condition for success in public life at Athens was the power of speaking well, and as the wealth and power of the city increased a considerable number of "sophists" settled there who undertook amongst other things to teach the art of oratory. Many of them also directed the general education of their pupils, of which geometry usually formed a part. We are told that two of those who are usually termed sophists made a special study of geometry--these were Hippias of Elis and Antipho, and one made a special study of astronomy--this was Meton, after whom the metonic cycle is named.

Hippias. The first of these geometricians, Hippias of Elis (circ. 420 b.c.), is described as an expert arithmetician, but he is best known to us through his invention of a curve called the quadratrix, by means of which an angle can be trisected, or indeed divided in any given ratio. If the radius of a circle rotate uniformly round the centre O from the position OA through a right angle to OB , and in the same time a straight line drawn perpendicular to OB move uniformly parallel to itself from the position OA to BC , the locus of their intersection will be the quadratrix.

Let OR and MQ be the position of these lines at any time; and let them cut in P , a point on the curve. Then

angle AOP: angle AOB = OM:OB.

Similarly, if OR' be another position of the radius,

& angle AOP': angle AOB = OM':OB & angle AOP: angle AOP' = OM:OM'; & angle AOP': angle P'OP = OM':MM.

Hence, if the angle AOP be given, and it be required to divide it in any given ratio, it is sufficient to divide OM in that ratio at M' and draw the line M'P' ; then OP' will divide AOP in the required ratio.

If OA be taken as the initial line, OP = r , the angle AOP = theta , and OA = a , we have theta :1/2pi = r sin theta :a , and the equation of the curve is pi r = 2atheta cosec theta .

Hippias devised an instrument to construct the curve mechanically; but constructions which involved the use of any mathematical instruments except a ruler and a pair of compasses were objected to by Plato, and rejected by most geometricians of a subsequent date.

Antipho. The second sophist whom I mentioned was Antipho (circ. 420 b.c.). He is one of the very few writers among the ancients who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. He began by inscribing an equilateral triangle (or, according to some accounts, a square); on each side he inscribed in the smaller segment an isosceles triangle, and so on ad infinitum. This method of attacking the quadrature problem is similar to that described above as used by Bryso of Heraclea.

No doubt there were other cities in Greece besides Athens where similar and equally meritorious work was being done, though the record of it has now been lost; I have mentioned here the investigations of these three writers, chiefly because they were the immediate predecessors of those who created the Athenian school.

The Schools of Athens and Cyzicus. The history of the Athenian school begins with the teaching of Hippocrates about 420 b.c.; the school was established on a permanent basis by the labours of Plato and Eudoxus; and, together with the neighbouring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 b.c.) of the university at Alexandria drew thither most of the talent of Greece.

Eudoxus, who was amongst the most distinguished of the Athenian mathematicians, is also reckoned as the founder of the school at Cyzicus. The connection between this school and that of Athens was very close, and it is now impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, Menaechmus, and Aristaeus belonged to that of Cyzicus. There was always a constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil Theodorus of Cyrene, and there was no difference in their treatment of the subject, so that they may be conveniently treated together.

Before discussing the work of the geometricians of these schools in detail I may note that they were especially interested in three problems: namely (i), the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube; (ii) the trisection of an angle; and (iii) the squaring of a circle, that is, the determination of a square whose area is equal to that of a given circle.

Now the first two of these problems (considered analytically) require the solution of a cubic equation; and, since a construction by means of circles (whose equations are of the form x^2 + y^2 + ax + by + c = 0 ) and straight lines (whose equations are of the form x + beta y + gamma = 0 ) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of the conic sections be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have been long known to be incommensurable, but it is only recently that it has been shewn by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence this problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three problems, but the attempts to solve them led to the discovery of many new theorems and processes.

Besides attacking these problems the later Platonic school collected all the geometrical theorems then known and arranged them systematically. These collections comprised the bulk of the propositions in Euclid's Elements, books i--ix, xi, and xii, together with some of the more elementary theorems in conic sections.

Hippocrates. Hippocrates of Chios (who must be carefully distinguished from his contemporary, Hippocrates of Cos, the celebrated physician) was one of the greatest of the Greek geometricians. He was born about 470 b.c. at Chios, and began life as a merchant. The accounts differ as to whether he was swindled by the Athenian custom-house officials who were stationed at the Chersonese, or whether one of his vessels was captured by an Athenian pirate near Byzantium; but at any rate somewhere about 430 b.c. he came to Athens to try to recover his property in the law courts. A foreigner was not likely to succeed in such a case, and the Athenians seem only to have laughed at him for his simplicity, first in allowing himself to be cheated, and then in hoping to recover his money. While prosecuting his cause he attended the lectures of various philosophers, and finally (in all probability to earn a livelihood) opened a school of geometry himself. He seems to have been well acquainted with the Pythagorean philosophy, though there is no sufficient authority for the statement that he was ever initiated as a Pythagorean.

He wrote the first elementary text-book of geometry, a text-book on which probably Euclid's Elements was founded; and therefore he may be said to have sketched out the lines on which geometry is still taught in English schools. It is supposed that the use of letters in diagrams to describe a figure was made by him or introduced about this time, as he employs expressions such as "the point on which the letter A stands" and "the line on which AB is marked." Cantor, however, thinks that the Pythagoreans had previously been accustomed to represent the five vertices of the pentagram-star by the letters upsilon gamma iota theta alpha ; and though this was a single instance, perhaps they may have used the method generally. The Indian geometers never employed letters to aid them in the description of their figures. Hippocrates also denoted the square on a line by the word d'unamis, and thus gave the technical meaning to the word power which it still retains in algebra: there is reason to think that this use of the word was derived from the Pythagoreans, who are said to have enunciated the result of the proposition Euc. i, 47, in the form that "the total power of the sides of a right-angled triangle is the same as that of the hypotenuse."

In this text-book Hippocrates introduced the method of "reducing" one theorem to another, which being proved, the thing proposed necessarily follows; of this method the reductio ad absurdum is an illustration. No doubt the principle had been used occasionally before, but he drew attention to it as a legitimate mode of proof which was capable of numerous applications. He elaborated the geometry of the circle: proving, among other propositions, that similar segments of a circle contain equal angles; that the angle subtended by the chord of a circle is greater than, equal to, or less than a right angle as the segment of the circle containing it is less than, equal to, or greater than a semicircle (Euc. iii, 31); and probably several other of the propositions in the third book of Euclid. It is most likely that he also established the propositions that [similar] circles are to one another as the squares of their diameters (Euc. xii, 2), and that similar segments are as the squares of their chords. The proof given in Euclid of the first of these theorems is believed to be due to Hippocrates.

The most celebrated discoveries of Hippocrates were, however, in connection with the quadrature of the circle and the duplication of the cube, and owing to his influence these problems played a prominent part in the history of the Athenian school.

The following propositions will sufficiently illustrate the method by which he attacked the quadrature problem.

( alpha ) He commenced by finding the area of a lune contained between a semicircle and a quadrantal arc standing on the same chord. This he did as follows. Let ABC be an isosceles right-angled triangle inscribed in the semicircle ABOC , whose centre is O . On AB and AC as diameters describe semicircles as in the figure. Then, since by Euc. i, 47, sq. on BC = sq. on AC + sq. on AB, therefore, by Euc. xii, 2, area 12odot on BC = area 12odot on AC + area 12odot on AB. Take away the common parts area ABC = sum of areas of lunes AECD and AFBG. Hence the area of the lune AECD is equal to half that of the triangle ABC .

( beta ) He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O , and on OA , AB , BC , and CD as diameters described semicircles of which those on OA and AB are drawn in the figure. Then AD is double any of the lines OA , AB , BC , and CD ,

sq. on AD = sum of sqs. on OA, AB, BC, and CD, area 12odot ABCD = sum of areas of 12odot s on OA, AB, BC, and CD.

Take away the common parts area trapezium ABCD = 3 lune AEBF + 12odot on OA. If therefore the area of this latter lune be known, so is that of the semicircle described on OA as diameter. According to Simplicius, Hippocrates assumed that the area of this lune was the same as the area of the lune found in proposition ( alpha ); if this be so, he was of course mistaken, as in this case he is dealing with a lune contained between a semicircle and a sextantal arc standing on the same chord; but it seems more probable that Simplicius misunderstood Hippocrates.

Hippocrates also enunciated various other theorems connected with lunes (which have been collected by Bretschneider and by Allman) of which the theorem last given is a typical example. I believe that they are the earliest instances in which areas bounded by curves were determined by geometry.

The other problem to which Hippocrates turned his attention was the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube.

This problem was known in ancient times as the Delian problem, in consequence of a legend that the Delians had consulted Plato on the subject. In one form of the story, which is related by Philoponus, it is asserted that the Athenians in 430 b.c., when suffering from the plague of eruptive typhoid fever, consulted the oracle at Delos as to how they could stop it. Apollo replied that they must double the size of his altar which was in the form of a cube. To the unlearned suppliants nothing seemed more easy, and a new altar was constructed either having each of its edges double that of the old one (from which it followed that the volume was increased eightfold) or by placing a similar cubic altar next to the old one. Whereupon, according to the legend, the indignant god made the pestilence worse than before, and informed a fresh deputation that it was useless to trifle with him, as his new altar must be a cube and have a volume exactly double that of his old one. Suspecting a mystery the Athenians applied to Plato, who referred them to the geometricians, and especially to Euclid, who had made a special study of the problem. The introduction of the names of Plato and Euclid is an obvious anachronism. Eratosthenes gives a somewhat similar account of its origin, but with king Minos as the propounder of the problem.

Hippocrates reduced the problem of duplicating the cube to that of finding two means between one straight line ( a ), and another twice as long ( 2a ). If these means be x and y , we have a : x = x : y = y : 2a , from which it follows that x^3 = 2a^3 . It is in this form that the problem is usually presented now. Hippocrates did not succeed in finding a construction for these means.

Plato. The next philosopher of the Athenian school who requires mention here was Plato. He was born at Athens in 429 b.c., and was, as is well known, a pupil for eight years of Socrates; much of the teaching of the latter is inferred from Plato's dialogues. After the execution of his master in 399 b.c. Plato left Athens, and being possessed of considerable wealth he spent some years in travelling; it was during this time that he studied mathematics. He visited Egypt with Eudoxus, and Strabo says that in his time the apartments they occupied at Heliopolis were still shewn. Thence Plato went to Cyrene, where he studied under Theodorus. Next he moved to Italy, where he became intimate with Archytas the then head of the Pythagorean school, Eurytas of Metapontum, and Timaeus of Locri. He returned to Athens about the year 380 b.c., and formed a school of students in a suburban gymnasium called the "Academy." He died in 348 b.c.

Plato, like Pythagoras, was primarily a philosopher, and perhaps his philosophy should be regarded as founded on the Pythagorean rather than on the Socratic teaching. At any rate it, like that of the Pythagoreans, was coloured with the idea that the secret of the universe is to be found in number and in form; hence, as Eudemus says, "he exhibited on every occasion the remarkable connection between mathematics and philosophy." All the authorities agree that, unlike many later philosophers, he made a study of geometry or some exact science an indispensable preliminary to that of philosophy. The inscription over the entrance to his school ran "Let none ignorant of geometry enter my door," and on one occasion an applicant who knew no geometry is said to have been refused admission as a student.

Plato's position as one of the masters of the Athenian mathematical school rests not so much on his individual discoveries and writings as on the extraordinary influence he exerted on his contemporaries and successors. Thus the objection that he expressed to the use in the construction of curves of any instruments other than rulers and compasses was at once accepted as a canon which must be observed in such problems. It is probably due to Plato that subsequent geometricians began the subject with a carefully compiled series of definitions, postulates, and axioms. He also systematized the methods which could be used in attacking mathematical questions, and in particular directed attention to the value of analysis. The analytical method of proof begins by assuming that the theorem or problem is solved, and thence deducing some result: if the result be false, the theorem is not true or the problem is incapable of solution: if the result be true, and if the steps be reversible, we get (by reversing them) a synthetic proof; but if the steps be not reversible, no conclusion can be drawn. Numerous illustrations of the method will be found in any modern text-book on geometry. If the classification of the methods of legitimate induction given by Mill in his work on logic had been universally accepted and every new discovery in science had been justified by a reference to the rules there laid down, he would, I imagine, have occupied a position in reference to modern science somewhat analogous to that which Plato occupied in regard to the mathematics of his time.

The following is the only extant theorem traditionally attributed to Plato. If CAB and DAB be two right-angled triangles, having one side, AB , common, their other sides, AD and BC , parallel, and their hypotenuses, AC and BD , at right angles, then, if these hypotenuses cut in P , we have PC : PB = PB : PA = PA : PD . This theorem was used in duplicating the cube, for, if such triangles can be constructed having PD = 2PC , the problem will be solved. It is easy to make an instrument by which the triangles can be constructed.

Eudoxus. Of Eudoxus, the third great mathematician of the Athenian school and the founder of that at Cyzicus, we know very little. He was born in Cnidus in 408 b.c. Like Plato, he went to Tarentum and studied under Archytas the then head of the Pythagoreans. Subsequently he travelled with Plato to Egypt, and then settled at Cyzicus, where he founded the school of that name. Finally he and his pupils moved to Athens. There he seems to have taken some part in public affairs, and to have practised medicine; but the hostility of Plato and his own unpopularity as a foreigner made his position uncomfortable, and he returned to Cyzicus or Cnidus shortly before his death. He died while on a journey to Egypt in 355 b.c.

His mathematical work seems to have been of a high order of excellence. He discovered most of what we now know as the fifth book of Euclid, and proved it in much the same form as that in which it is there given.

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He discovered some theorems on what was called "the golden section." The problem to cut a line AB in the golden section, that is, to divide it, say at H , in extreme and mean ratio (that is, so that AB : AH = AH : HB ) is solved in Euc. ii, 11, and probably was known to the Pythagoreans at an early date. If we denote AB by l , AH by a , and HB by b , the theorems that Eudoxus proved are equivalent to the following algebraical identities. (i) (a + 1/2l)^2 = 5(1/2l)^2 . (ii) Conversely, if (i) be true, and AH be taken equal to a , then AB will be divided at H in a golden section. (iii) (b + 1/2a)^2 = 5(1/2a^2) . (iv) l^2 + b^2 = 3a^2 . (v) l + a : l = l : a , which gives another golden section. These propositions were subsequently put by Euclid as the first five propositions of his thirteenth book, but they might have been equally well placed towards the end of the second book. All of them are obvious algebraically, since l = a + b and a^2 = bl .

Eudoxus further established the "method of exhaustions"; which depends on the proposition that "if from the greater of two unequal magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there will at length remain a magnitude less than the least of the proposed magnitudes." This proposition was placed by Euclid as the first proposition of the tenth book of his Elements, but in most modern school editions it is printed at the beginning of the twelfth book. By the aid of this theorem the ancient geometers were able to avoid the use of infinitesimals: the method is rigorous, but awkward of application. A good illustration of its use is to be found in the demonstration of Euc. xii, 2, namely, that the square of the radius of one circle is to the square of the radius of another circle as the area of the first circle is to an area which is neither less nor greater than the area of the second circle, and which therefore must be exactly equal to it: the proof given by Euclid is (as was usual) completed by a reductio ad absurdum. Eudoxus applied the principle to shew that the volume of a pyramid or a cone is one-third that of the prism or the cylinder on the same base and of the same altitude (Euc. xii, 7 and 10). It is believed that he proved that the volumes of two spheres were to one another as the cubes of their radii; some writers attribute the proposition Euc. xii, 2 to him, and not to Hippocrates.

Eudoxus also considered certain curves other than the circle. There is no authority for the statement made in some old books that these were conic sections, and recent investigations have shewn that the assertion (which I repeated in the earlier editions of this book) that they were plane sections of the anchor-ring is also improbable. It seems most likely that they were tortuous curves; whatever they were, he applied them in explaining the apparent motions of the planets as seen from the earth.

Eudoxus constructed an orrery, and wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the sun, moon, and stars were attached, and which by their rotation produced the effects observed. In all he required twenty-seven spheres. As observations became more accurate, subsequent astronomers who accepted the theory had continually to introduce fresh spheres to make the theory agree with the facts. The work of Aratus on astronomy, which was written about 300 b.c. and is still extant, is founded on that of Eudoxus.

Plato and Eudoxus were contemporaries. Among Plato's pupils were the mathematicians Leodamas, Neocleides, Amyclas, and to their school also belonged Leon, Theudius (both of whom wrote text-books on plane geometry), Cyzicenus, Thasus, Hermotimus, Philippus, and Theaetetus. Among the pupils of Eudoxus are reckoned Menaechmus, his brother Dinostratus (who applied the quadratrix to the duplication and trisection problems), and Aristaeus.

Menaechmus. Of the above-mentioned mathematicians Menaechmus requires special mention. He was born about 375 b.c., and died about 325 b.c. Probably he succeeded Eudoxus as head of the school at Cyzicus, where he acquired great reputation as a teacher of geometry, and was for that reason appointed one of the tutors of Alexander the Great. In answer to his pupil's request to make his proofs shorter, Menaechmus made the well-known reply that though in the country there are private and even royal roads, yet in geometry there is only one road for all.

Menaechmus was the first to discuss the conic sections, which were long called the Menaechmian triads. He divided them into three classes, and investigated their properties, not by taking different plane sections of a fixed cone, but by keeping his plane fixed and cutting it by different cones. He shewed that the section of a right cone by a plane perpendicular to a generator is an ellipse, if the cone be acute-angled; a parabola, if it be right-angled; and a hyperbola, if it be obtuse-angled; and he gave a mechanical construction for curves of each class. It seems almost certain that he was acquainted with the fundamental properties of these curves; but some writers think that he failed to connect them with the sections of the cone which he had discovered, and there is no doubt that he regarded the latter not as plane loci but as curves drawn on the surface of a cone.

He also shewed how these curves could be used in either of the two following ways to give a solution of the problem to duplicate a cube. In the first of these, he pointed out that two parabolas having a common vertex, axes at right angles, and such that the latus rectum of the one is double that of the other will intersect in another point whose abscissa (or ordinate) will give a solution; for (using analysis) if the equations of the parabolas be y^2 = 2ax and x^2 = ay , they intersect in a point whose abscissa is given by x^3 = 2a^3 . It is probable that this method was suggested by the form in which Hippocrates had cast the problem; namely, to find x and y so that a : x = x : y = y : 2a , whence we have x^2 = ay and y^2 = 2ax .

The second solution given by Menaechmus was as follows. Describe a parabola of latus rectum l . Next describe a rectangular hyperbola, the length of whose real axis is 4l , and having for its asymptotes the tangent at the vertex of the parabola and the axis of the parabola. Then the ordinate and the abscissa of the point of intersection of these curves are the mean proportionals between l and 2l . This is at once obvious by analysis. The curves are x^2 = ly and xy = 2l^2 . These cut in a point determined by x^3 = 2l^3 and y^3 = 4l^3 . Hence l : x = x : y = y : 2l .

and Theaetetus. Of the other members of these schools, Aristaeus and Theaetetus, whose works are entirely lost, were mathematicians of repute. We know that Aristaeus wrote on the five regular solids and on conic sections, and that Theaetetus developed the theory of incommensurable magnitudes. The only theorem we can now definitely ascribe to the latter is that given by Euclid in the ninth proposition of the tenth book of the Elements, namely, that the squares on two commensurable right lines have one to the other a ratio which a square number has to a square number (and conversely); but the squares on two incommensurable right lines have one to the other a ratio which cannot be expressed as that of a square number to a square number (and conversely). This theorem includes the results given by Theodorus.

The contemporaries or successors of these mathematicians wrote some fresh text-books on the elements of geometry and the conic sections, introduced problems concerned with finding loci, and systematized the knowledge already acquired, but they originated no new methods of research.

Aristotle. An account of the Athenian school would be incomplete if there were no mention of Aristotle, who was born at Stagira in Macedonia in 384 b.c. and died at Chalcis in Euboea in 322 b.c. Aristotle, however, deeply interested though he was in natural philosophy, was chiefly concerned with mathematics and mathematical physics as supplying illustrations of correct reasoning. A small book containing a few questions on mechanics which is sometimes attributed to him is of doubtful authority; but, though in all probability it is not his work, it is interesting, partly as shewing that the principles of mechanics were beginning to excite attention, and partly as containing the earliest known employment of letters to indicate magnitudes.

The most instructive parts of the book are the dynamical proof of the parallelogram of forces for the direction of the resultant, and the statement, in effect, that if alpha be a force, beta the mass to which it is applied, gamma the distance through which it is moved, and delta the time of the motion, then alpha will move 1/2beta through 2gamma in the time delta , or through gamma in the time 1/2delta : but the author goes on to say that it does not follow that 1/2alpha will move beta through 1/2gamma in the time delta , because 1/2alpha may not be able to move beta at all; for 100 men may drag a ship 100 yards, but it does not follow that one man can drag it one yard. The first part of this statement is correct and is equivalent to the statement that an impulse is proportional to the momentum produced, but the second part is wrong.

The author also states the fact that what is gained in power is lost in speed, and therefore that two weights which keep a [weightless] lever in equilibrium are inversely proportional to the arms of the lever; this, he says, is the explanation why it is easier to extract teeth with a pair of pincers than with the fingers. Among other questions raised, but not answered, are why a projectile should ever stop, and why carriages with large wheels are easier to move than those with small.

I ought to add that the book contains some gross blunders, and as a whole is not as able or suggestive as might be inferred from the above extracts. In fact, here as elsewhere, the Greeks did not sufficiently realise that the fundamental facts on which the mathematical treatment of mechanics must be based can be established only by carefully devised observations and experiments.

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End of part 5 of 28