Archimedes.
by Thomas Little Heath.
CHAPTER I.
ARCHIMEDES.
If the ordinary person were asked to say off-hand what he knew of Archimedes, he would probably, at the most, be able to quote one or other of the well-known stories about him: how, after discovering the solution of some problem in the bath, he was so overjoyed that he ran naked to his house, shouting [Greek: eureka, eureka] (or, as we might say, "I've got it, I've got it"); or how he said "Give me a place to stand on and I will move the earth"; or again how he was killed, at the capture of Syracuse in the Second Punic War, by a Roman soldier who resented being told to get away from a diagram drawn on the ground which he was studying.
And it is to be feared that few who are not experts in the history of mathematics have any acquaintance with the details of the original discoveries in mathematics of the greatest mathematician of antiquity, perhaps the greatest mathematical genius that the world has ever seen.
History and tradition know Archimedes almost exclusively as the inventor of a number of ingenious mechanical appliances, things which naturally appeal more to the popular imagination than the subtleties of pure mathematics.
Almost all that is told of Archimedes reaches us through the accounts by Polybius and Plutarch of the siege of Syracuse by Marcellus. He perished in the sack of that city in 212 B.C., and, as he was then an old man (perhaps 75 years old), he must have been born about 287 B.C. He was the son of Phidias, an astronomer, and was a friend and kinsman of King Hieron of Syracuse and his son Gelon. He spent some time at Alexandria studying with the successors of Euclid (Euclid who flourished about 300 B.C. was then no longer living). It was doubtless at Alexandria that he made the acquaintance of Conon of Samos, whom he admired as a mathematician and cherished as a friend, as well as of Eratosthenes; to the former, and to the latter during his early period he was in the habit of communicating his discoveries before their publication. It was also probably in Egypt that he invented the water-screw known by his name, the immediate purpose being the drawing of water for irrigating fields.
After his return to Syracuse he lived a life entirely devoted to mathematical research. Incidentally he became famous through his clever mechanical inventions. These things were, however, in his case the "diversions of geometry at play," and he attached no importance to them.
In the words of Plutarch, "he possessed so lofty a spirit, so profound a soul, and such a wealth of scientific knowledge that, although these inventions had won for him the renown of more than human sagacity, yet he would not consent to leave behind him any written work on such subjects, but, regarding as ignoble and sordid the business of mechanics and every sort of art which is directed to practical utility, he placed his whole ambition in those speculations in the beauty and subtlety of which there is no admixture of the common needs of life".
During the siege of Syracuse Archimedes contrived all sorts of engines against the Roman besiegers. There were catapults so ingeniously constructed as to be equally serviceable at long or short range, and machines for discharging showers of missiles through holes made in the walls. Other machines consisted of long movable poles projecting beyond the walls; some of these dropped heavy weights upon the enemy's ships and on the constructions which they called _sambuca_, from their resemblance to a musical instrument of that name, and which consisted of a protected ladder with one end resting on two quinqueremes lashed together side by side as base, and capable of being raised by a windlass; others were fitted with an iron hand or a beak like that of a crane, which grappled the prows of ships, then lifted them into the air and let them fall again. Marcellus is said to have derided his own engineers and artificers with the words, "Shall we not make an end of fighting with this geometrical Briareus who uses our ships like cups to ladle water from the sea, drives our _sambuca_ off ignominiously with cudgel-blows, and, by the multitude of missiles that he hurls at us all at once, outdoes the hundred-handed giants of mythology?" But the exhortation had no effect, the Romans being in such abject terror that, "if they did but see a piece of rope or wood projecting above the wall they would cry 'there it is,' declaring that Archimedes was setting some engine in motion against them, and would turn their backs and run away, insomuch that Marcellus desisted from all fighting and assault, putting all his hope in a long siege".
Archimedes died, as he had lived, absorbed in mathematical contemplation. The accounts of the circumstances of his death differ in some details. Plutarch gives more than one version in the following passage: "Marcellus was most of all afflicted at the death of Archimedes, for, as fate would have it, he was intent on working out some problem with a diagram, and, his mind and his eyes being alike fixed on his investigation, he never noticed the incursion of the Romans nor the capture of the city. And when a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration; whereat the soldier was so enraged that he drew his sword and slew him. Others say that the Roman ran up to him with a drawn sword, threatening to kill him; and, when Archimedes saw him, he begged him earnestly to wait a little while in order that he might not leave his problem incomplete and unsolved, but the other took no notice and killed him. Again, there is a third account to the effect that, as he was carrying to Marcellus some of his mathematical instruments, sundials, spheres, and angles adjusted to the apparent size of the sun to the sight, some soldiers met him and, being under the impression that he carried gold in the vessel, killed him."
The most picturesque version of the story is that which represents him as saying to a Roman soldier who came too close, "Stand away, fellow, from my diagram," whereat the man was so enraged that he killed him.
Archimedes is said to have requested his friends and relatives to place upon his tomb a representation of a cylinder circumscribing a sphere within it, together with an inscription giving the ratio (3/2) which the cylinder bears to the sphere; from which we may infer that he himself regarded the discovery of this ratio as his greatest achievement.
Cicero, when quaestor in Sicily, found the tomb in a neglected state and restored it. In modern times not the slightest trace of it has been found.
Beyond the above particulars of the life of Archimedes, we have nothing but a number of stories which, if perhaps not literally accurate, yet help us to a conception of the personality of the man which we would not willingly have altered. Thus, in illustration of his entire preoccupation by his abstract studies, we are told that he would forget all about his food and such necessities of life, and would be drawing geometrical figures in the ashes of the fire, or, when anointing himself, in the oil on his body. Of the same kind is the story mentioned above, that, having discovered while in a bath the solution of the question referred to him by Hieron as to whether a certain crown supposed to have been made of gold did not in fact contain a certain proportion of silver, he ran naked through the street to his home shouting [Greek: eureka, eureka].
It was in connexion with his discovery of the solution of the problem _To move a given weight by a given force_ that Archimedes uttered the famous saying, "Give me a place to stand on, and I can move the earth"
([Greek: dos moi pou sto kai kino ten gen], or in his broad Doric, as one version has it, [Greek: pa bo kai kino tan gan]). Plutarch represents him as declaring to Hieron that any given weight could be moved by a given force, and boasting, in reliance on the cogency of his demonstration, that, if he were given another earth, he would cross over to it and move this one. "And when Hieron was struck with amazement and asked him to reduce the problem to practice and to show him some great weight moved by a small force, he fixed on a ship of burden with three masts from the king's arsenal which had only been drawn up by the great labour of many men; and loading her with many passengers and a full freight, sitting himself the while afar off, with no great effort but quietly setting in motion with his hand a compound pulley, he drew the ship towards him smoothly and safely as if she were moving through the sea." Hieron, we are told elsewhere, was so much astonished that he declared that, from that day forth, Archimedes's word was to be accepted on every subject! Another version of the story describes the machine used as a _helix_; this term must be supposed to refer to a screw in the shape of a cylindrical helix turned by a handle and acting on a cog-wheel with oblique teeth fitting on the screw.
Another invention was that of a sphere constructed so as to imitate the motions of the sun, the moon, and the five planets in the heavens.
Cicero actually saw this contrivance, and he gives a description of it, stating that it represented the periods of the moon and the apparent motion of the sun with such accuracy that it would even (over a short period) show the eclipses of the sun and moon. It may have been moved by water, for Pappus speaks in one place of "those who understand the making of spheres and produce a model of the heavens by means of the regular circular motion of water". In any case it is certain that Archimedes was much occupied with astronomy. Livy calls him "unicus spectator caeli siderumque". Hipparchus says, "From these observations it is clear that the differences in the years are altogether small, but, as to the solstices, I almost think that both I and Archimedes have erred to the extent of a quarter of a day both in observation and in the deduction therefrom." It appears, therefore, that Archimedes had considered the question of the length of the year. Macrobius says that he discovered the distances of the planets. Archimedes himself describes in the _Sandreckoner_ the apparatus by which he measured the apparent diameter of the sun, i.e. the angle subtended by it at the eye.
The story that he set the Roman ships on fire by an arrangement of burning-glasses or concave mirrors is not found in any authority earlier than Lucian (second century A.D.); but there is no improbability in the idea that he discovered some form of burning-mirror, e.g. a paraboloid of revolution, which would reflect to one point all rays falling on its concave surface in a direction parallel to its axis.
CHAPTER II.
GREEK GEOMETRY TO ARCHIMEDES.
In order to enable the reader to arrive at a correct understanding of the place of Archimedes and of the significance of his work it is necessary to pass in review the course of development of Greek geometry from its first beginnings down to the time of Euclid and Archimedes.
Greek authors from Herodotus downwards agree in saying that geometry was invented by the Egyptians and that it came into Greece from Egypt. One account says:--
"Geometry is said by many to have been invented among the Egyptians, its origin being due to the measurement of plots of land. This was necessary there because of the rising of the Nile, which obliterated the boundaries appertaining to separate owners. Nor is it marvellous that the discovery of this and the other sciences should have arisen from such an occasion, since everything which moves in the sense of development will advance from the imperfect to the perfect. From sense-perception to reasoning, and from reasoning to understanding, is a natural transition. Just as among the Phoenicians, through commerce and exchange, an accurate knowledge of numbers was originated, so also among the Egyptians geometry was invented for the reason above stated.
"Thales first went to Egypt and thence introduced this study into Greece."
But it is clear that the geometry of the Egyptians was almost entirely practical and did not go beyond the requirements of the land-surveyor, farmer or merchant. They did indeed know, as far back as 2000 B.C., that in a triangle which has its sides proportional to 3, 4, 5 the angle contained by the two smaller sides is a right angle, and they used such a triangle as a practical means of drawing right angles. They had formulae, more or less inaccurate, for certain measurements, e.g. for the areas of certain triangles, parallel-trapezia, and circles. They had, further, in their construction of pyramids, to use the notion of similar right-angled triangles; they even had a name, _se-qet_, for the ratio of the half of the side of the base to the height, that is, for what we should call the _co-tangent_ of the angle of slope. But not a single general theorem in geometry can be traced to the Egyptians. Their knowledge that the triangle (3, 4, 5) is right angled is far from implying any knowledge of the general proposition (Eucl. I., 47) known by the name of Pythagoras. The science of geometry, in fact, remained to be discovered; and this required the genius for pure speculation which the Greeks possessed in the largest measure among all the nations of the world.
Thales, who had travelled in Egypt and there learnt what the priests could teach him on the subject, introduced geometry into Greece. Almost the whole of Greek science and philosophy begins with Thales. His date was about 624-547 B.C. First of the Ionian philosophers, and declared one of the Seven Wise Men in 582-581, he shone in all fields, as astronomer, mathematician, engineer, statesman and man of business. In astronomy he predicted the solar eclipse of 28 May, 585, discovered the inequality of the four astronomical seasons, and counselled the use of the Little Bear instead of the Great Bear as a means of finding the pole. In geometry the following theorems are attributed to him--and their character shows how the Greeks had to begin at the very beginning of the theory--(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground (this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids).
After Thales come the Pythagoreans. We are told that the Pythagoreans were the first to use the term [Greek: mathemata] (literally "subjects of instruction") in the specialised sense of "mathematics"; they, too, first advanced mathematics as a study pursued for its own sake and made it a part of a liberal education. Pythagoras, son of Mnesarchus, was born in Samos about 572 B.C., and died at a great age (75 or 80) at Metapontum. His interests were as various as those of Thales; his travels, all undertaken in pursuit of knowledge, were probably even more extended. Like Thales, and perhaps at his suggestion, he visited Egypt and studied there for a long period (22 years, some say).
It is difficult to disentangle from the body of Pythagorean doctrines the portions which are due to Pythagoras himself because of the habit which the members of the school had of attributing everything to the Master ([Greek: autos epha], _ipse dixit_). In astronomy two things at least may safely be attributed to him; he held that the earth is spherical in shape, and he recognised that the sun, moon and planets have an independent motion of their own in a direction contrary to that of the daily rotation; he seems, however, to have adhered to the geocentric view of the universe, and it was his successors who evolved the theory that the earth does not remain at the centre but revolves, like the other planets and the sun and moon, about the "central fire".
Perhaps his most remarkable discovery was the dependence of the musical intervals on the lengths of vibrating strings, the proportion for the octave being 2 : 1, for the fifth 3 : 2 and for the fourth 4 : 3. In arithmetic he was the first to expound the theory of _means_ and of proportion as applied to commensurable quantities. He laid the foundation of the theory of numbers by considering the properties of numbers as such, namely, prime numbers, odd and even numbers, etc. By means of _figured_ numbers, square, oblong, triangular, etc.
(represented by dots arranged in the form of the various figures) he showed the connexion between numbers and geometry. In view of all these properties of numbers, we can easily understand how the Pythagoreans came to "liken all things to numbers" and to find in the principles of numbers the principles of all things ("all things are numbers").
We come now to Pythagoras's achievements in geometry. There is a story that, when he came home from Egypt and tried to found a school at Samos, he found the Samians indifferent, so that he had to take special measures to ensure that his geometry might not perish with him. Going to the gymnasium, he sought out a well-favoured youth who seemed likely to suit his purpose, and was withal poor, and bribed him to learn geometry by promising him sixpence for every proposition that he mastered. Very soon the youth got fascinated by the subject for its own sake, and Pythagoras rightly judged that he would gladly go on without the sixpence. He hinted, therefore, that he himself was poor and must try to earn his living instead of doing mathematics; whereupon the youth, rather than give up the study, volunteered to pay sixpence to Pythagoras for each proposition.
In geometry Pythagoras set himself to lay the foundations of the subject, beginning with certain important definitions and investigating the fundamental principles. Of propositions attributed to him the most famous is, of course, the theorem that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the sides about the right angle (Eucl. I., 47); and, seeing that Greek tradition universally credits him with the proof of this theorem, we prefer to believe that tradition is right. This is to some extent confirmed by another tradition that Pythagoras discovered a general formula for finding two numbers such that the sum of their squares is a square number. This depends on the theory of the _gnomon_, which at first had an arithmetical signification corresponding to the geometrical use of it in Euclid, Book II. A figure in the shape of a _gnomon_ put round two sides of a square makes it into a larger square. Now consider the number 1 represented by a dot. Round this place three other dots so that the four dots form a square (1 + 3 = 2). Round the four dots (on two adjacent sides of the square) place five dots at regular and equal distances, and we have another square (1 + 3 + 5 = 3); and so on. The successive odd numbers 1, 3, 5 ... were called _gnomons_, and the general formula is
1 + 3 + 5 + ... + (2n - 1) = n.
Add the next odd number, i.e. 2n + 1, and we have n + (2n + 1) = (n + 1). In order, then, to get two square numbers such that their sum is a square we have only to see that 2n + 1 is a square. Suppose that 2n + 1 = m; then n = (m - 1), and we have {(m - 1)} + m = {(m + 1)}, where m is any odd number; and this is the general formula attributed to Pythagoras.
Proclus also attributes to Pythagoras the theory of proportionals and the construction of the five "cosmic figures," the five regular solids.
One of the said solids, the dodecahedron, has twelve pentagonal faces, and the construction of a regular pentagon involves the cutting of a straight line "in extreme and mean ratio" (Eucl. II., 11, and VI., 30), which is a particular case of the method known as the _application of areas_. How much of this was due to Pythagoras himself we do not know; but the whole method was at all events fully worked out by the Pythagoreans and proved one of the most powerful of geometrical methods.
The most elementary case appears in Euclid, I., 44, 45, where it is shown how to apply to a given straight line as base a parallelogram having a given angle (say a rectangle) and equal in area to any rectilineal figure; this construction is the geometrical equivalent of arithmetical _division_. The general case is that in which the parallelogram, though _applied_ to the straight line, overlaps it or falls short of it in such a way that the part of the parallelogram which extends beyond, or falls short of, the parallelogram of the same angle and breadth on the given straight line itself (exactly) as base is similar to another given parallelogram (Eucl. VI., 28, 29). This is the geometrical equivalent of the most general form of quadratic equation ax mx = C, so far as it has real roots; while the condition that the roots may be real was also worked out (= Eucl. VI., 27). It is important to note that this method of _application of areas_ was directly used by Apollonius of Perga in formulating the fundamental properties of the three conic sections, which properties correspond to the equations of the conics in Cartesian co-ordinates; and the names given by Apollonius (for the first time) to the respective conics are taken from the theory, _parabola_ ([Greek: parabole]) meaning "application" (i.e. in this case the parallelogram is applied to the straight line exactly), _hyperbola_ ([Greek: hyperbole]), "exceeding" (i.e. in this case the parallelogram exceeds or overlaps the straight line), _ellipse_ ([Greek: elleipsis]), "falling short" (i.e. the parallelogram falls short of the straight line).
Another problem solved by the Pythagoreans is that of drawing a rectilineal figure equal in area to one given rectilineal figure and similar to another. Plutarch mentions a doubt as to whether it was this problem or the proposition of Euclid I., 47, on the strength of which Pythagoras was said to have sacrificed an ox.
The main particular applications of the theorem of the square on the hypotenuse (e.g. those in Euclid, Book II.) were also Pythagorean; the construction of a square equal to a given rectangle (Eucl. II., 14) is one of them and corresponds to the solution of the pure quadratic equation x = ab.
The Pythagoreans proved the theorem that the sum of the angles of any triangle is equal to two right angles (Eucl. I., 32).
Speaking generally, we may say that the Pythagorean geometry covered the bulk of the subject-matter of Books I., II., IV., and VI. of Euclid (with the qualification, as regards Book VI., that the Pythagorean theory of proportion applied only to commensurable magnitudes). Our information about the origin of the propositions of Euclid, Book III., is not so complete; but it is certain that the most important of them were well known to Hippocrates of Chios (who flourished in the second half of the fifth century, and lived perhaps from about 470 to 400 B.C.), whence we conclude that the main propositions of Book III. were also included in the Pythagorean geometry.
Lastly, the Pythagoreans discovered the existence of incommensurable lines, or of _irrationals_. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of [root]2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a _reductio ad absurdum_ proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable was bound to cause geometers a great shock, because it showed that the theory of proportion invented by Pythagoras was not of universal application, and therefore that propositions proved by means of it were not really established. Hence the stories that the discovery of the irrational was for a time kept secret, and that the first person who divulged it perished by shipwreck. The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus (408-355 B.C.) discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.
By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded _pari passu_ with the completion of the body of the Elements.
Hippocrates himself is an example of the concurrent study of the two departments. On the one hand, he was the first of the Greeks who is known to have compiled a book of Elements. This book, we may be sure, contained in particular the most important propositions about the circle included in Euclid, Book III. But a much more important proposition is attributed to Hippocrates; he is said to have been the first to prove that circles are to one another as the squares on their diameters (= Eucl. XII., 2), with the deduction that similar segments of circles are to one another as the squares on their bases. These propositions were used by him in his tract on the squaring of _lunes_, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial.
Anaxagoras for instance (about 500-428 B.C.) is said to have worked at the problem while in prison. The essential portions of Hippocrates's tract are preserved in a passage of Simplicius (on Aristotle's _Physics_), which contains substantial fragments from Eudemus's _History of Geometry_. Hippocrates showed how to square three particular lunes of different forms, and then, lastly, he squared the sum of a certain circle and a certain lune. Unfortunately, however, the last-mentioned lune was not one of those which can be squared, and so the attempt to square the circle in this way failed after all.
Hippocrates also attacked the problem of doubling the cube. There are two versions of the origin of this famous problem. According to one of them, an old tragic poet represented Minos as having been dissatisfied with the size of a tomb erected for his son Glaucus, and having told the architect to make it double the size, retaining, however, the cubical form. According to the other, the Delians, suffering from a pestilence, were told by the oracle to double a certain cubical altar as a means of staying the plague. Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i.e. finding x, y such that a : x = x : y = y : b, where a, b are the two given straight lines. It is easy to see that, if a : x = x : y = y : b, then b/a = (x/a), and, as a particular case, if b = 2a, x = 2a, so that the side of the cube which is double of the cube of side a is found.
The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals. Two significant early solutions are on record.
(1) Archytas of Tarentum (who flourished in first half of fourth century B.C.) found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: (a) a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0.
(2) Menaechmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, ([alpha]) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x = ay, y = bx, and ([beta]) by the intersection of a parabola and a rectangular hyperbola, the corresponding equations being x = ay, and xy = ab respectively. It would appear that it was in the effort to solve this problem that Menaechmus discovered the conic sections, which are called, in an epigram by Eratosthenes, "the triads of Menaechmus".