G.313 there many numbers

VIII.13

If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products is proportional; and, if the original numbers by multiplying the products make certain numbers, the latter is proportional.

there_many_numbers_viii_13(A, B, C)

given there be as many numbers as we please, A, B, C, in continued proportion, so that, as A is to B, so is B to C

prove D, E, F and G, H, K are in continued proportion

given there be as many numbers as we please, A, B, C, in continued proportion, so that, as A is to B, so is B to C

assert Let A, B, C by multiplying themselves make D, E, F, and by multiplying D, E, F let them make G, H, K

assert Let A by multiplying B make L, and let the numbers A, B by multiplying L make M, N

assert Again let B by multiplying C make O, and let the numbers B, C by multiplying O make P, Q

assert In manner similar to the foregoing, we can prove that D, L, E and G, M, N, H are continuously proportional in the ratio of A to B, and further E, O, F and H, P, Q, K are continuously proportional in the ratio of B to C

assert As A is to B, so is B to C

assert D, L, E are also in the same ratio with E, O, F, and further G, M, N, H in the same ratio with H, P, Q, K

assert The multitude of D, L, E is equal to the multitude of E, O, F, and that of G, M, N, H to that of H, P, Q, K

∵ there_many_numbers_vii_14assert :term:`ex aequali <G.174>`, as D is to E, so is E to F, and, as G is to H, so is H to K

assert If $a, b, c \dots$ be a series in geometrical progression, then

assert $$\left. \begin{array}{l} a^2,\ .. step:: assert B^2,\ .. step:: assert C^2 \dots \\ a^3,\ .. step:: assert B^3,\ .. step:: assert C^3 \dots \end{array} \right\} \text{are also in geometrical progression.}$$

assert Heiberg brackets the words added to the enunciation which extend the theorem to any powers. The words are “and this always occurs with the extremes” (καὶ ἀεὶ περὶ τοὺς ἄκρους τοῦτο συμβαίνει). They seem to be rightly suspected on the same grounds as the same words added to the enunciation of VII. 27. There is no allusion to them in the proof, much less any proof of the extension

assert Euclid forms, besides the squares and cubes of the given numbers, the products $ab$, $a^2b$, $ab^2$, $bc$, $b^2c$, $bc^2$. When he says that “we prove in manner similar to the foregoing,” he indicates successive uses of VII. 17, 18 as in VIII. 12

assert With our notation the proof is as easy to see for any powers as for squares and cubes

assert To prove that $a^n, b^n, c^n \dots$ are in geometrical progression

assert Form all the means between $a^n, b^n$, and set out the series

assert $$a^n,\quad a^{n-1}b,\quad a^{n-2}b^2\ .. step:: assert \dots\ .. step:: assert Ab^{n-1},\quad b^n.$$

assert The common ratio of one term to the next is $a : b$

assert Next take the geometrical progression

assert $$b^n,\quad b^{n-1}c,\quad b^{n-2}c^2\ .. step:: assert \dots\ .. step:: assert Bc^{n-1},\quad c^n,$$

assert The common ratio of which is $b : c$

assert Proceed thus for all pairs of consecutive terms

assert Now

assert $$a : b = b : c = \dots$$

assert any pair of succeeding terms in one series are in the same ratio as any pair of succeeding terms in any other of the series

assert The number of terms in each is the same, namely $(n + 1)$

assert ex aequali,

∴ $$a^n : b^n = b^n : c^n = c^n : d^n = \dots$$

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