His book on the Measure of the Circle, is a kind of sup plement to those on the sphere and cylinder. In this, he demonstrates that any circle is equal to a triangle having its base equal to the circumference, and its height equal to the radius ; and he proves, that if the diameter of a cir cle be reckoned unity, the circumference will be between 344 and 344. The principles laid down by Archimedes were sufficient to carry the approximation to any degree of nearness ; but he appears to have aimed at nothing more than a simple rule, sufficiently accurate for the common concerns or life.
His treatise on Conoids and Spheroids relates to the so lids generated by the conic sections revolving about their axes: those produced by the rotation of the parabola and hyperbola he called Conoids; and such as are generated by the revolution of the ellipse about either axis are his Spheroids. Here he compares the area of an ellipse with that of a circle ; he also proves that the sections of conoids and spheroids are conic sections, and he treats of their tan gent planes. Ile proves, for the first time, that a parabo lic conoid is equal to three times the half of a cone of the same base and altitude, and he also shews what is the ra tio of any segment of a hyperbolic conoid, or of a sphe roid to a cone of the same base and altitude. His reason ing is a model of accuracy ; and it exhibits the true spirit of the ancient synthetic method ; it is however exceeding ly prolix and difficult, insomuch that few will have patience to follow the steps of the venerable mathematician, more especially as the same conclusions may be found with equal certainty by the modern analysis, at an infinitely less ex pence of thought.
His treatise on Spirals treats of a curve which was the invention of his friend Conon, who it seems had found its properties, but died before he had time to investigate their demonstrations : these Archimedes has supplied. The whole subject is, however, so much his own, that what is properly the spiral of Conon, is usually called the spiral of Archimedes. He has also treated Of the Equilibrium of Planes, or of their centres of gravity, in two books ; and next Of the Quadrature of the Parabola. This is the first complete quadrature of a curve that was ever found. He here shews, that the area of any segment of a parabola cut of by a chord, is two-thirds of the circumscribing paral lelogram, and this he proves by two different methods. His ?lrenarius was written to evince the possibility of ex pressing, by numbers, the grains of sand that might fill the whole space of the universe. Here he introduces a
property of a geometrical progression, that has since been made the foundation of the theory of logarithms ; but it would be going too far to suppose that Archimedes had made any approach to that valuable invention. This tract is valuble, not on account of the subject on which he treats, but because of. the information it contains respecting the ancient astronomy, and the application which it gives of the Greek arithmetic. In addition to the works we have enu merated, there is a treatise On bodies which are carried on a fluid, in two books, and a book of Lemmas, which is a collection of theorems and problems, curious in them selves, and useful to the geometrical analysis. These are all the writings of Archimedes now extant, but many have been lost.
The writings of Archimedes are the most precious re lict of the ancient geometry : they chew to what an extent such a genius as his could carry its method of demonstra tion ; but they likewise prove, that there were certain limits beyond which it became inapplicable, on account of the un wieldiness or the machinery. In general, the progress of discovery is slow ; but Archimedes took up the subject where men of ordinary capacities were at a stand, and, by the vigour of his mind, anticipated the labour of ages : he was undoubtedly the Newton of antiquity.
Eutocius has written a commentary on a part of the works of Archimedes, viz. on the books De Sphcera et cy lindro, de dimen.lione circuli et de tequiponderantibus. In the year 1543, Nicolas Tartalea translated from Greek into Latin, and published at Venice, the treatises, I. De Centris Gravium, f5'c. 2. Quadratura Parabola. 3. De Insidentibus aqu.r, fiber primus ; and, in 1555, the two books De Thai dentibus ague appeared at Venice. In 1543, an edition of the works of Archimedes was published at Basle, with the Latin translation of John of Cremona, and revised by Re gimontanus. In this, the two books De Insidentibus in Fluido, and the Lemmata, were wanting, but it contained the commentary of Eutocius in Greek and Latin. Other editions of his works, or parts of them, have been given by Commandinus, Renault, Greaves and Foster, Borelli, Bar row, Maurolicus, Wallis, some with commentaries ; but these are in a manner superseded by the Oxford edition of Torelli in Greek and Latin, printed in 1792,and the French translation of Peyrard in 4to and 8vo, the latter printed in 1808. For farther information respecting this geometer, see ARCHIMEDES.