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ARCHIMEDES was a native of Syracuse, one of the greatest cities of the West Grecian world. His letters to Dositheus of Alexandria show him to have been in constant communication with the students of geometry in that city. Plutarch, in his life of Marcellus, speaks of his intimate friendship with King Hiero of Syracuse, who induced him to apply his mechanical principles to the construction of military engines; though the time thus withdrawn from his theoretical researches was most unwillingly given.
During the second Punic War Hiero had been in close alliance with Rome. But after his death, Hippocrates, an ambitious general, enlisted the city on the side of Carthage, and a Roman force, under the command of Marcellus, besieged it by sea and land. The fleet, equipped with the usual engines of war, especially the sambuca, a vast scaling-ladder erected on a platform fixed on the decks of two ships lashed together broadside, found themselves in face of a strange foe. Balistæ of surpassing power suddenly arose above the battlements, and hurled vast masses of stone or lead against the approaching ships, then were as suddenly withdrawn. When the ships reached the walls, huge beams were let fall, end downward, on the decks; or iron hooks, attached to a chain, grappling the prows, lifted the ships endwise from the water, then let them go and sank them. On the land side the besiegers were met in the same way. These engines were the work of Archimedes. To prosecute the siege actively was impossible. The city was reduced by famine, and at last stormed. Archimedes, meanwhile, having done as a citizen all that in him lay, had returned to his diagrams; and was still bending over the sand on which these were traced when a soldier slew him. Marcellus erected a monument to his memory, which was discovered more than a hundred years afterwards by Cicero. It bore the image of a cylinder circumscribing a sphere, with a verse indicating, what Archimedes had held to be his greatest achievement, the measurement and mutual proportion of these two bodies. Dramatic, surely, was the contrast offered by the siege of Syracuse between the scientific intellect of Greece and the disciplined force of Rome; and not less remarkable is the admiration of the conqueror for the conquered, which, in a few generations, would weld the Greco-Roman world into one.
Geometry, when Archimedes began his career, had made more progress than is shown by the thirteen books of Euclid's Elements. For a century at least thinkers had been exercised by the Delian problem, suggested by an oracular command to build a new altar in Apollo's shrine at Delos similar in shape to the old altar, but double the size. The duplication of the cube was not possible by Euclid's Elements. Involving as it did the finding of two mean proportionals between two given magnitudes, it led to the study of various curved lines resulting from mechanical movements more complicated than those of the compasses. Euclid himself investigated Conic Sections. Dinostratus discovered the quadratrix resulting from the intersection of a rotating line with another moving parallel to itself. An immense field of research was thus opened. Archimedes made an elaborate study of the spiral formed by a straight line rotating with uniform velocity round one of its extremities, while, at the same time, a point in the line moved forward from the fixed point with similar velocity. This curve, usually handled by the transcendental calculus of modern mathematics, was attacked by Archimedes with extraordinary skill and success. Tangents were drawn to any point in the curve, and its quadrature was found.
Great thinkers, while heedless of immediate utility, have always been guided by a secret purpose of increasing man's power to control his destiny: they have served Humanity, though not always their own generation. In dealing with curved figures, Archimedes fixed always on the most important and difficult of the many problems connected with them--their comparison with spaces bounded by straight lines--in other words, their quadrature. Of all curved lines and surfaces, none could be so important as the circle and the sphere; not because of the imaginary perfection attributed to them by Plato, but because the heavenly bodies moved in circles, and the earth's figure was apparently a sphere fixed within the vaster sphere of the heavens. To measure the area of a circle, the surface, and the solid content of a sphere, were obviously problems of vast moment.
Inserting a polygon in a circle, it was clear that the polygon was the sum of the triangles of which the side was the base, and the perpendicular from the centre the altitude. Increasing the number of sides indefinitely, the sum of the sides became equivalent to the circumference of the circle. Thus the area of the circle was equal to that of a triangle of which the circumference was the base and the radius the altitude.
The next step was to compare the radius, or the diameter, with the circumference. Artificers had always known the proportion to be less than one to three; and, since Pythagoras, it had been suspected to belong to the class of incommensurable quantities. Describing round a circle a succession of equilateral figures, of six, twenty-four, forty-eight, and, finally, of ninety-six sides, he found by an extremely laborious calculation that the circumference was three times the diameter plus a quantity which was less than a seventh but more than ten seventy-oneths.
There remained the problem of measuring the surface of the sphere. If a polygon, described round a circle, were made to rotate round the diameter, the resulting solid would consist of a series of truncated cones. There was no difficulty about measuring the surface of any of these. It was equal to the rectangle of which one side was that of the truncated cone, and the other the length of the circle drawn midway between the greater and lesser base and parallel to them. When the sides of the polygon were numerous, ultimately merging into a circle, it became impossible to find the sum of these infinitely small areas. But by a happy transformation, substituting for the cone-side its projection on the diameter of the solid, the integration was effected. The surface of the sphere was found equal to the length of the great circle multiplied by its diameter: in other words, to four great circles.
The next problem was to effect the cubature of the sphere; i.e. to determine its solid content. By regarding it as an assemblage of pyramids with their summits at the centre and bases at the surface, the sum of these bases, when very numerous, became equivalent to the sphere. The content of the sphere was therefore the area of four great circles mutliplied by two-thirds of the diameter. And as the cylinder of equal height and diameter was equal to the great circle multiplied by the diameter, it followed that the sphere was equal of two-thirds of the circumscribing cylinder. Henceforth the simple measurement of the diameter of one of the heavenly bodies would suffice to estimate it's cubical content.
Archimedes did not confine himself to geometry. He enlarged the Greek numeration-scale, and showed its capacity for expressing the number of grains of sand that would fill a radius equal to the distance between the earth and the fixed stars. Of far more importance were his investigations in Statics. He proved mathematically that the weights attached to the unequal arms of a lever were in equilibrium when either weight was inversely proportional to the length of the arms. He followed up this truth by a long series of investigations into the centre of gravity and different geometrical forms.
The incident which is said to have directed his attention to the measurement of the density or specific gravity of bodies is well known. A body immersed in water will displace an ammount of water equal to its bulk. A crown made of gold and silver will therefore displace more water than one made of pure gold, the weight of the two crowns being equal. Starting from this point he composed his remarkable treatise upon floating bodies, the first attempt made to estimate the pressure exercised by the elements of a fluid. Lagrange (in his Mécanique Analytique, part I) speaks of it as "one of the most striking monuments of the genius of Archimedes, containing a theory of the stability of floating bodies to which little has been added in later times."
We have, then, in the work of this great thinker the whole field of inquiryaccessible to mathematical research opened up and cultivated: the measurement of space, of number, and of force. He was perhaps the most perfect type of scientific intellect that has appeared in the world. Comte, in his mathematical treatise, always refers to him as the Great Geometer.
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| This biography is
reprinted from The New Calendar of Great Men. Ed. Frederic
Harrison. London: Macmillan and Co., 1920. |
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