Algebraic Geometry 3 Curves Jaobians by Parshin Shafarevich

By Parshin Shafarevich

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The choice of undefined concepts and axioms is free, apart from the constraint of consistency. Mathematicians following Pasch’s path introduced various elements and axioms and developed their geometries with greater or lesser elegance and trouble. The most successful of these systematizers was the Göttingen professor David Hilbert (1862–1943), whose The Foundations of Geometry (1899) greatly influenced efforts to axiomatize all of mathematics. The Real World Euclid’s Elements had claimed the excellence of being a true account of space.

Archimedes also came West in the 12th century, in Latin translations from Greek and Arabic sources. Apollonius arrived only by bits and pieces. Ptolemy’s Almagest appeared in Latin manuscript in 1175. Not until the humanists of the Renaissance turned their classical learning to mathematics, however, did the Greeks come out in standard printed editions in both Latin and Greek. These texts affected their Latin readers with the strength of revelation. Europeans discovered the notion of proof, the power of generalization, and the superhuman cleverness of the Greeks; they hurried to master techniques that would enable them to improve their calendars and horoscopes, fashion better instruments, and raise Christian mathematicians to the level of the infidels.

58 CHAPTER 2 BRAnCHes oF GeoMetRY euclideAn geometry A s discussed in a general way, in chapter 1, several ancient cultures had developed a form of geometry suited to the relationships among lengths, areas, and volumes of physical objects. This geometry was codified in Euclid’s Elements about 300 BCE on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic. The Elements epitomized the axiomatic-deductive method for many centuries. In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.

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