A remarkable feature of Euclid's, and of all Greek geometry before Archimedes is that it eschews mensuration . Thus the theorem that the area of a triangle equals half the product of its base and its altitude is foreign to Euclid.
To avoid any assertion about the infinitude of the straight line, Euclid says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in Euclid's statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point.
On an attentive examination of the methods adopted by modern elementary writers, in laying down the first principles of ratios and proportion, and especially in commenting upon Euclid, I long since experienced a conviction of the extremely unsatisfactory nature of most of their views; and this chiefly, as appearing to me to involve inadequate ideas of Euclid's real principle in treating of proportionals in his 5th book, and of the nature of the quantities which form the subject of investigation.
Euclid's Elements is certainly one of the greatest books ever written, and one of the most perfect monuments of the Greek intellect.
Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.
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