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:
“I hold with the old-fashioned criticism that Browning is not really a poet, that he has all the gifts but the one needful and the pearls without the string; rather one should say raw nuggets and rough diamonds.”
—Gerard Manley Hopkins (1844–1889)
“Cubism had been an analysis of the object and an attempt to put it before us in its totality; both as analysis and as synthesis, it was a criticism of appearance. Surrealism transmuted the object, and suddenly a canvas became an apparition: a new figuration, a real transfiguration.”
—Octavio Paz (b. 1914)
“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)