By A.N. Parshin (editor), I.R. Shafarevich (editor), V.L. Popov, T.A. Springer, E.B. Vinberg

Contributions on heavily similar matters: the idea of linear algebraic teams and invariant idea, by way of recognized specialists within the fields. The e-book could be very precious as a reference and learn advisor to graduate scholars and researchers in arithmetic and theoretical physics.

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**Extra resources for Algebraic geometry 04 Linear algebraic groups, invariant theory**

**Example text**

Eudoxus observed that his axiom on the existence of ratios yields at once the following so-called Exhaustion theorem: Exhaustion theorem If from a given magnitude, one subtracts a part at least equal to half of it, if from the remaining magnitude one subtracts a part at least equal to half of it, and if this process is repeated, one ends up eventually with a magnitude which is smaller than any prescribed magnitude of the same nature. In contemporary terms, ∀n an+1 ≤ an 2 =⇒ lim an = 0. n→∞ The proof is essentially the one we would give today.

This supposes implicitly (in modern language) that the ratio of the measures of two given segments is always a rational number: a fact which we know today to be false. It was perhaps the Pythagorean Hippasus, around 420 BC, who discovered the existence of incommensurable magnitudes: two magnitudes that you cannot possibly measure with the same unit segment. This discovery destroyed a large part of Greek geometry. In any case all results depending on Thales’ theorem, and in particular on the widely used theory of similar triangles, were to be called into question again.

X y 2a The intersection point of the two curves (see Fig. 18) thus yields the expected solution to the “double proportional mean problem”, namely √ √ 3 3 (x, y) = a 2, a 4 . The magnitude x is thus the side of the cube with volume 2a 3 . Menaechmus’ approach to the problem reduces at once the problem of the double proportional mean to the intersection of two parabolas in the plane. This is a major progress compared with the techniques of Archytas. 6 Incommensurable Magnitudes 29 these operations are indeed “constructible” with ruler and compass (see Sect.