Criticism
Euclid's list of axioms in the Elements was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms.
For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.1 – I.3 can be proved trivially by using superposition.
Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."
Read more about this topic: Euclid's Elements
Famous quotes containing the word criticism:
“The critic lives at second hand. He writes about. The poem, the novel, or the play must be given to him; criticism exists by the grace of other men’s genius. By virtue of style, criticism can itself become literature. But usually this occurs only when the writer is acting as critic of his own work or as outrider to his own poetics, when the criticism of Coleridge is work in progress or that of T.S. Eliot propaganda.”
—George Steiner (b. 1929)
“Unless criticism refuses to take itself quite so seriously or at least to permit its readers not to, it will inevitably continue to reflect the finicky canons of the genteel tradition and the depressing pieties of the Culture Religion of Modernism.”
—Leslie Fiedler (b. 1917)
“Of all the cants which are canted in this canting world—though the cant of hypocrites may be the worst—the cant of criticism is the most tormenting!”
—Laurence Sterne (1713–1768)