Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when equal multiples of the first and the third either both exceed, are both equal to, or are both less than, equal multiples of the second and the fourth, respectively, being taken in corresponding order, according to any kind of multiplication whatever.
If for positive real numbers $\alpha,\beta,\gamma,\delta$ and for any natural numbers $n,m,k$ we have
$$\begin{array}{rcll} n\alpha > \beta&\Rightarrow&n\gamma > \delta&\wedge\\ m\alpha = \beta&\Rightarrow&m\gamma =\delta&\wedge\\ k\alpha < \beta&\Rightarrow&k\gamma < \delta&\wedge\\ \end{array}$$
then $$\frac\alpha\beta=\frac\gamma\delta.$$
This definition is the kernel of Eudoxus's theory of proportion, and is valid even if $\alpha$, $\beta$, etc., are irrational.
Corollaries: 1
Definitions: 2 3 4 5 6
Proofs: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
Propositions: 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103