Posted by Evilbunny on November 28, 19101 at 20:16:56:
Archimedes’ mathematics:
The most notable characteristic of Archimedes’ mathematical work is its freedom from the trammels of traditional Greek mathematics. It is true that in the proofs of those theorems for which the integral calculus would now be used (e.g. those determining the surface-area and volume of a sphere, the area of a parabola, and the volume of conoids) he uses the traditional Greek method of byping infinitesimals (invented by Eudoxus, and deployed in Euclid, bk. 10; it has been misnamed 'method of exhaustion' in modern works). But the Method reveals that for the discovery of these theorems he used a technique which consists essentially of dividing two figures into infinitely thin strips, weighing these strips against each other, and then summing them to get the ratio of the two whole figures. This is ogous to the procedure of the first practitioners of the integral calculus in the seventeenth century, but unlike them Archimedes recognized its lack of logical rigour, and used it only as a heuristic method. The same freedom of thought appears in the arithmetical field in the Sand-reckoner, which shows an understanding of the nature of a numerical system immeasurably superior to anything else from antiquity. It is this breadth and freedom of vision, rather than the amazing ingenuity which Archimedes everywhere displays in the solution of particular problems, which justifies his title not only as the greatest mathematician of antiquity, but as one of the greatest ever. His work in hydrostatics (see no. (9) above) was epoch-making (though the effect in antiquity was negligible). The same is true of statics, though here he probably had predecessors.
Archimedes’ astronomy:
All his work in astronomy is lost except for an ingenious method of finding the sun’s apparent diameter described in the Sand-reckoner, and a page giving the distances of the heavenly bodies preserved in Hippolytus (Haer. 41, 18 ff. Wendland). This (highly corrupt) page suggests that he had no mathematical theory of astronomy. However, his construction of a planetarium suggests the reserve. On this he wrote a work ("Peri sphairopoiias", Pappus 8. 3), now lost.
READ THE GREAT BOOKS
TERM PAPERS, RESEARCH PAPERS, ESSAYS
DR. ELLIOT'S NORTH AMERICAN GREAT BOOKS TOUR--COMING TO A BOOK
STORE NEAR YOU
[Shakespeare Forums]
[Bible Forums]