Definition 18 (Euclid VII, Def. 20). Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
The same multiple, part or parts comes to this: A is to B as C is to D when, for some m and n (numbers or the unit), n times A is equal to m times B, and n times C is equal to m times D. Multiple, part and parts are the three cases. In this sense sameness of ratio is reflexive, symmetric and transitive, and numbers to which one number has the same ratio are equal, by Theorems 31 to 35. A : B = C : D is read "as A is to B, so is C to D." A • B is the number A makes by multiplying B.
Theorem 51 (Euclid VII.11). If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.
Proof: Let AB : CD = AE : CF. Then whatever part or parts AB is of CD, the same is AE of CF (Definition 18). So the remainder EB is the same part or parts of the remainder FD (Theorem 47, Theorem 48). So EB : FD = AB : CD (Definition 18).
Theorem 52 (Euclid VII.12). If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.
Proof: Let A : B = C : D. Whatever part or parts A is of B, the same is C of D (Definition 18). So A, C together are the same part or parts of B, D together (Theorem 45, Theorem 46). So A : B = (A, C together) : (B, D together).
Theorem 53 (Euclid VII.13). If four numbers be proportional, they will also be proportional alternately.
Proof: Let A : B = C : D. Whatever part or parts A is of B, the same is C of D (Definition 18). So, alternately, whatever part or parts A is of C, the same is B of D (Theorem 49, Theorem 50). So A : C = B : D.
Theorem 54 (Euclid VII.14). If there be as many numbers as we please, and others equal to them in multitude, which taken two and two are in the same ratio, they will also be in the same ratio ex aequali.
Proof: Let A : B = D : E and B : C = E : F. Alternately, A : D = B : E and B : E = C : F (Theorem 53). So A : D = C : F. Alternately, A : C = D : F.
Note. Ratios that are the same with the same ratio are the same with one another, by the sense of proportion fixed at Definition 18.
Theorem 55 (Euclid VII.15). If a unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.
Proof: Let the unit A measure BC, and D measure EF the same number of times. Divide BC into its units BG, GH, HC, and EF into EK, KL, LF equal to D. The multitudes are equal. The units are equal to one another and the numbers EK, KL, LF are equal to one another, so BG : EK = GH : KL = HC : LF. So as one antecedent to one consequent, so all to all (Theorem 52): BG : EK = BC : EF. BG is equal to A and EK to D. So the unit A measures D the same number of times that BC measures EF.
This is Theorem 49 for the unit. Euclid proves it apart because the unit is not a number. Axiom 3 lets the unit be taken as a number by analogy; Euclid's separate proof shows that the analogy holds here.
Theorem 56 (Euclid VII.17). If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.
Proof: Let A by multiplying B, C make D, E. B measures D according to the units in A, and the unit F measures A according to the units in it. So F : A = B : D (Definition 18). For the same reason F : A = C : E. So B : D = C : E, and alternately B : C = D : E (Theorem 53).
Theorem 57 (Euclid VII.18). If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.
Proof: Let A, B by multiplying C make D, E. Then C by multiplying A makes D, and C by multiplying B makes E (Theorem 31). So A : B = D : E (Theorem 56).
Theorem 58 (Euclid VII.19). If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.
Proof: Let A : B = C : D. Let A by multiplying D make E, B by multiplying C make F, and A by multiplying C make G. A by multiplying C, D makes G, E, so C : D = G : E (Theorem 56). But C : D = A : B, so A : B = G : E. Again, A, B by multiplying C make G, F, so A : B = G : F (Theorem 57). So G : E = G : F, and E is equal to F.
Conversely, let E be equal to F. Then G : E = G : F. But G : E = C : D and G : F = A : B. So A : B = C : D.
Note. Numbers to which the same number has the same ratio are equal, and conversely, by the sense of proportion fixed at Definition 18.
Remark. This theorem is the ground of Definition 29, where two fractions are called equivalent when the product of the antecedent of the first and the consequent of the second is equal to the product of the antecedent of the second and the consequent of the first. Equivalence of fractions and sameness of ratio are one thing reached from two sides.
Theorem 59 (Euclid VII.20). The least numbers of those which have the same ratio with them measure those which have the same ratio the same number of times, the greater the greater and the less the less.
Proof: Let CD, EF be the least numbers of those which have the same ratio with A, B. CD is not parts of A. For, if possible, let it be so. Then EF is the same parts of B that CD is of A (Theorem 53, Definition 18). Divide CD into the parts of A, namely CG, GD, and EF into the parts of B, namely EH, HF. The multitudes are equal. CG is equal to GD and EH to HF, so CG : EH = GD : HF, and so, as one to one, so all to all: CG : EH = CD : EF (Theorem 52). So CG, EH are in the same ratio with CD, EF, being less than they: which is impossible, since CD, EF are the least. So CD is a part of A, and EF is the same part of B (Theorem 53, Definition 18). So CD measures A the same number of times that EF measures B.
Theorem 60 (Euclid VII.21). Numbers prime to one another are the least of those which have the same ratio with them.
Proof: Let A, B be prime to one another. If they are not the least, let C, D be the least in the same ratio. Then C measures A the same number of times that D measures B (Theorem 59). As many times as C measures A, so many units let there be in E. Then D also measures B according to the units in E. So E measures A according to the units in C, and E measures B according to the units in D (Theorem 31). So E measures A, B, which are prime to one another: which is impossible.
Theorem 61 (Euclid VII.22). The least numbers of those which have the same ratio with them are prime to one another.
Proof: Let A, B be the least of those which have the same ratio with them. If they are not prime to one another, some number C measures them. As many times as C measures A, so many units let there be in D; and as many times as C measures B, so many units let there be in E. Then C by multiplying D makes A, and by multiplying E makes B. So D : E = A : B (Theorem 56). So D, E are in the same ratio with A, B, being less than they: which is impossible.
Remark. For any ratio of numbers there is exactly one pair of least numbers in it; that pair is prime to one another; and every other pair in the ratio is that pair taken the same number of times. So there is something determinate that all equivalent fractions share (Definition 37), and it is a pair of numbers. No class of fractions is needed.