The History of Mathematics: A Brief Course

282 10. EUCLIDEAN GEOMETRY FIGURE 10. Trisection of an arbitrary angle by neusis construction. A mechanical (curvilinear) soluti ...

THE EARLIEST GREEK GEOMETRY 283 FIGURE 11. A mechanical device for drawing the conchoid of Nicomedes. the late nineteenth and ...

284 10. EUCLIDEAN GEOMETRY point ahead of the faster. Hence the race can be thought of as beginning again at that instant, with ...

THE EARLIEST GREEK GEOMETRY 285 to their bases, or the theorem (Book 12, Proposition 2) that the areas of two circles are prop ...

286 10. EUCLIDEAN GEOMETRY Laertius quotes a letter allegedly from Archytas to Dionysus urging that Plato be released. Plato ret ...

THE EARLIEST GREEK GEOMETRY^287 Chapter 3, where he laments the lack of public support for research into solid ge- ometry.) Th ...

288 10. EUCLIDEAN GEOMETRY A G II Β C D Ê FIGURE 12. A fundamental theorem in the theory of proportion. Proposition 1 of Book 6 ...

THE EARLIEST GREEK GEOMETRY 289 "arranged many of the theorems of Eudoxus"; (2) an anonymous scholium (com- mentary) on Euclid ...

290 10. EUCLIDEAN GEOMETRY The method of exhaustion. Eudoxus' second contribution is of equal importance with the first; it is t ...

THE EARLIEST GREEK GEOMETRY 291 Ε Ç FIGURE 14. Proof that circles are proportional to the squares on their diameters. Let ΑΒÃΑ ...

292 10. EUCLIDEAN GEOMETRY be added (first to first and third to third), that is, if á : b :: c : d and e :&::/: d, then (a+ ...

THE EARLIEST GREEK GEOMETRY 293 and less philosophical. It is likely that even Plato realized that his attempt to un- derstand ...

294 10. EUCLIDEAN GEOMETRY FIGURE 15. How do we exclude the possibility that two lines per- pendicular to the same line may inte ...

THE EARLIEST GREEK GEOMETRY 295 FIGURE 16. The exterior angle theorem. Euclid deals with this issue in the Elements by stating ...

296 10. EUCLIDEAN GEOMETRY As Aristotle's discussion of begging the question continues, further evidence comes to light that thi ...

EUCLID 297 Β A C D A FIGURE 17. Expression of a rectangle as the difference of two squares. by Euclid, we give just a summary ...

298 10. EUCLIDEAN GEOMETRY part. After it is established that four lines are proportional when the rectangle on the means equals ...

ARCHIMEDES 299 cones and cylinders are proportional to the cubes of their linear dimensions, end- ing with the proof that sphe ...

300 10. EUCLIDEAN GEOMETRY Marcellus took the city of Syracuse after a long siege, and Archimedes was killed by a Roman soldier ...

ARCHIMEDES 301 References by Archimedes himself and other mathematicians tell of the exis- tence of other works by Archimedes ...