elem.5.4
若第一量对第二量之比等于第三量对第四量之比，则第一量与第三量的任意等倍分别与第二量与第四量的任意等倍之比相等（对应顺序）。
A:B = C:D；E、F 是 A、C 的等倍量，G、H 是 B、D 的等倍量；K、L 是 E、F 的等倍量，M、N 是 G、H 的等倍量。各量按行竖直排列，水平方向按其相对大小估算长度。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For let a first magnitude A have to a second B the same ratio as a third C to a fourth D; and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D; I say that, as E is to G, so is F to H. For let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.
设第一量A对第二量B之比等于第三量C对第四量D之比。
Since E is the same multiple of A that F is of C, and equimultiples K, L of E, F have been taken, therefore K is the same multiple of A that L is of C.
取A、C的等倍E、F，以及B、D的任意等倍G、H。
[V. 3] For the same reason M is the same multiple of B that N is of D. And, since, as A is to B, so is C to D, and of A, C equimultiples K, L have been taken, and of B, D other, chance, equimultiples M, N, therefore, if K is in excess of M, L also is in excess of N, if it is equal, equal, and if less, less.
再取E、F的等倍K、L，以及G、H的任意等倍M、N。由V.3，K是A的倍与L是C的倍相同；同理，M是B的倍与N是D的倍相同。
[V. Def. 5] And K, L are equimultiples of E, F, and M, N other, chance, equimultiples of G, H; therefore, as E is to G, so is F to H.
因A:B=C:D，由V.定义5，若K大于M则L大于N，相等则相等，小于则小于；故由定义5，E:G=F:H。