Theorem 204 (Landau 111). A/1 > B/1, or ~, or <, if and only if A > B, or A = B, or A < B, respectively.
A•1 > B•1, or =, or <, says the same as A > B, or =, or <, by Theorem 30.
Ratios and Whole Ratios: Whole Ratios
Theorems 204–208 · Definitions 40–42
Definition 40 (Landau 25). A ratio is a whole ratio when it is the ratio of some fraction A/1.
By Theorem 204 this A is only one. Conversely each A has exactly one whole ratio.
Theorem 205 (Landau 112). A/1 + B/1 ~ (A + B)/1, and A/1 • B/1 ~ (A•B)/1.
The first is Theorem 150. The second: (A•B)/(1•1) is (A•B)/1 by Theorem 30. So sums and products of whole ratios are whole ratios.
Theorem 206 (Landau 113). The whole ratios answer one to one to the unit and the numbers: the ratio of 1/1 to the unit, and the ratio of A′/1 to the successor of what answers to the ratio of A/1. Order, sum and product agree. So Axioms 1 to 3 hold of whole ratios, and with them Theorems 1 to 38.
One to one: Theorem 204. Order: Theorem 204. Sum and product: Theorem 205. Axiom 2 and its corollaries: the ratio of A′/1 is never that of 1/1, since A′ ≠ 1; and if A′/1 ~ B′/1 then A′ = B′ and A = B. Axiom 3: what is true of the ratio of 1/1, and of the ratio of A′/1 whenever of A/1, is true of every A by Axiom 3 itself, and so of every whole ratio.
Definition 41 (Landau 26). Where it causes no confusion, A also denotes the whole ratio of A/1.
Then X•1 = X, since A1/A2 • 1/1 is (A1•1)/(A2•1), which is A1/A2.
Theorem 207 (Landau 114). If Z is the ratio of A/B, then B•Z = A.
B/1 • A/B is (B•A)/(1•B), which is equivalent to A/1 by Theorems 31 and 133.
This is the sense in which a ratio is a specific operation of division: Z is that which, taken B times, gives A.
Definition 42 (Landau 27). The U of Theorem 203 is the quotient of X by Y, written X/Y.
If X and Y are the whole ratios of A and B, then X/Y is the ratio of the fraction A/B, by Theorem 207. So the one sign A/B serves for both without ambiguity, and every ratio can be written so.
Theorem 208 (Landau 115). For any X and Y there is a number C with C•X > Y.
Proof: Y/X is a ratio. By Theorem 182 there are numbers C and E with C/E > Y/X. Since E ≧ 1 (Theorem 19), C•X = X•((C/E)•E) = (X•(C/E))•E ≧ X•(C/E) > X•(Y/X) = Y, by Theorems 196, 198 and 202.
Remark. This is Euclid V, definition 4: ratios are such that one, multiplied, can exceed another. It is what later lets the difference between a lesser and a greater bound be made smaller than any given ratio.