The main results obtained are (1) that the diameter of the sun is between 18 and 20 times the diameter of the moon, (2) that the diameter of the moon is between 2/45ths and 1/30th of the distance of the centre of the moon from our eye, and (3) that the diameter of the sun is between 19/3rds and 43/6ths of the diameter of the earth. The book contains a good deal of arithmetical calculation.
Archimedes was born about 287 B. C. and was killed at the sack of Syracuse by Marcellus’s army in 212 B. C. The stories about him are well known, how he said ‘Give me a place to stand on, and I will move the earth’ (πα βω και κινω ταν γαν; how, having thought of the solution of the problem of the crown when in the bath, he ran home naked shouting ἑυρηκα, ἑυρηκα; and how, the capture of Syracuse having found him intent on a figure drawn on the ground, he said to a Roman soldier who came up, ‘Stand away, fellow, from my diagram.’ Of his work few people know more than that he invented a tubular screw which is still used for pumping water, and that for a long time he foiled the attacks of the Romans on Syracuse by the mechanical devices and engines which he used against them. But he thought meanly of these things, and his real interest was in pure mathematical speculation; he caused to be engraved on his tomb a representation of a cylinder circumscribing a sphere, with the ratio 3/2 which the cylinder bears to the sphere: from which we infer that he regarded this as his greatest discovery.
Archimedes’s works are all original, and are perfect models of mathematical exposition; their wide range will be seen from the list of those which survive: On the Sphere and Cylinder I, II, Measurement of a Circle, On Conoids and Spheroids, On Spirals, On Plane Equilibriums I, II, the Sandreckoner, Quadrature of the Parabola, On Floating Bodies I, II, and lastly the Method (only discovered in 1906). The difficult Cattle-Problem is also attributed to him, and a Liber Assumptorum which has reached us through the Arabic, but which cannot be his in its present form, although some of the propositions in it (notably that about the ‘Salinon’, salt-cellar, and others about circles inscribed in the αρβηλος, shoemaker’s knife) are quite likely to be of Archimedean origin. Among lost works were the Catoptrica, On Sphere-making, and investigations into polyhedra, including thirteen semi-regular solids, the discovery of which is attributed by Pappus to Archimedes.
Speaking generally, the geometrical works are directed to the measurement of curvilinear areas and volumes; and Archimedes employs a method which is a development of Eudoxus’s method of exhaustion. Eudoxus apparently approached the figure to be measured from below only, i. e. by means of figures successively inscribed to it. Archimedes approaches it from both sides by successively inscribing figures and circumscribing others also, thereby compressing them, as it were, until they coincide as nearly as we please with the figure to be measured. In many cases his procedure is, when the analytical equivalents are set down, seen to amount to real integration; this is so with his investigation of the areas of a parabolic segment and a spiral, the surface and volume of a sphere, and the volume of any segments of the conoids and spheroids.
The newly-discovered Method is especially interesting as showing how Archimedes originally obtained his results; this was by a clever mechanical method of (theoretically) weighing infinitesimal elements of the figure to be measured against elements of another figure the area or content of which (as the case may be) is known; it amounts to an avoidance of integration. Archimedes, however, would only admit that the mechanical method is useful for finding results; he did not consider them proved until they were established geometrically.
In the Measurement of a Circle, after proving by exhaustion that the area of a circle is equal to a right-angled triangle with the perpendicular sides equal respectively to the radius and the circumference of the circle, Archimedes finds, by sheer calculation, upper and lower limits to the ratio of the circumference of a circle to its diameter (what we call π). This he does by inscribing and circumscribing regular polygons of 96 sides and calculating approximately their respective perimeters. He begins by assuming as known certain approximate values for √3, namely 1351/780 > √3 > 265/153, and his calculations involve approximating to the square roots of several large numbers (up to seven digits). The text only gives the results, but it is evident that the extraction of square roots presented no difficulty, notwithstanding the comparative inconvenience of the alphabetic system of numerals. The result obtained is well known, namely 3-1/7 > π > 3-10/71.
The Plane Equilibriums is the first scientific treatise on the first principles of mechanics, which are established by pure geometry. The most important result established in Book I is the principle of the lever. This was known to Plato and Aristotle, but they had no real proof. The Aristotelian Mechanics merely ‘refers’ the lever ‘to the circle’, asserting that the force which acts at the greater distance from the fulcrum moves the system more easily because it describes a greater circle. Archimedes also finds the centre of gravity of a parallelogram, a triangle, a trapezium and finally (in Book II) of a parabolic segment and of a portion of it cut off by a straight line parallel to the base.
The Sandreckoner is remarkable for the development in it of a system for expressing very large numbers by orders and periods based on powers of myriad-myriads (10,000²). It also contains the important reference to the heliocentric theory of the universe put forward by Aristarchus of Samos in a book of ‘hypotheses’, as well as historical details of previous attempts to measure the size of the earth and to give the sizes and distances of the sun and moon.
Lastly, Archimedes invented the whole science of hydrostatics. Beginning the treatise On Floating Bodies with an assumption about uniform pressure in a fluid, he first proves that the surface of a fluid at rest is a sphere with its centre at the centre of the earth. Other propositions show that, if a solid floats in a fluid, the weight of the solid is equal to that of the fluid displaced, and, if a solid heavier than a fluid is weighed in it, it will be lighter than its true weight by the weight of the fluid displaced. Then, after a second assumption that bodies which are forced upwards in a fluid are forced upwards along the perpendiculars to the surface which pass through their centres of gravity, Archimedes deals with the position of rest and stability of a segment of a sphere floating in a fluid with its base entirely above or entirely below the surface. Book II is an extraordinary tour de force, investigating fully all the positions of rest and stability of a right segment of a paraboloid floating in a fluid according (1) to the relation between the axis of the solid and the parameter of the generating parabola, and (2) to the specific gravity of the solid in relation to the fluid; the term ‘specific gravity’ is not used, but the idea is fully expressed in other words.
Almost contemporary with Archimedes was Eratosthenes of Cyrene, to whom Archimedes dedicated the Method; the preface to this work shows that Archimedes thought highly of his mathematical ability. He was indeed recognized by his contemporaries as a man of great distinction in all branches, though the names Beta and Pentathlos[4] applied to him indicate that he just fell below the first rank in each subject. Ptolemy Euergetes appointed him to be tutor to his son (Philopator), and he became librarian at Alexandria; he recognized his obligation to Ptolemy by erecting a column with a graceful epigram. In this epigram he referred to the earlier solutions of the problem of duplicating the cube or finding the two mean proportionals, and advocated his own in preference, because it would give any number of means; on the column was fixed a bronze representation of his appliance, a frame with right-angled triangles (or rectangles) movable along two parallel grooves and over one another, together with a condensed proof. The Platonicus of Eratosthenes evidently dealt with the fundamental notions of mathematics in connexion with Plato’s philosophy, and seems to have begun with the story of the origin of the duplication problem.