Euclid's Elements - Criticism

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:

    It is from the womb of art that criticism was born.
    Charles Baudelaire (1821–1867)

    I consider criticism merely a preliminary excitement, a statement of things a writer has to clear up in his own head sometime or other, probably antecedent to writing; of no value unless it come to fruit in the created work later.
    Ezra Pound (1885–1972)

    I am opposed to writing about the private lives of living authors and psychoanalyzing them while they are alive. Criticism is getting all mixed up with a combination of the Junior F.B.I.- men, discards from Freud and Jung and a sort of Columnist peep- hole and missing laundry list school.... Every young English professor sees gold in them dirty sheets now. Imagine what they can do with the soiled sheets of four legal beds by the same writer and you can see why their tongues are slavering.
    Ernest Hemingway (1899–1961)