If a number be a part of a number, and another be the same part of another, alternately also, whatever part or parts the first is of the third, the same part, or the same parts, will the second also be of the fourth.
若一数是另一数的一部分，且另一数是另一数的相同部分，则交替地，第一数是第三数的无论几分之一或几分之几，第二数也是第四数的相同几分之一或几分之几。
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For let the number A be a part of the number BC, and another, D, the same part of another, EF, that A is of BC; I say that, alternately also, whatever part or parts A is of D, the same part or parts is BC of EF also.
设数A是数BC的一部分，且数D是数EF的相同部分，即A是BC的相同部分。
For since, whatever part A is of BC, the same part also is D of EF, therefore, as many numbers as there are in BC equal to A, so many also are there in EF equal to D.
因A是BC的一部分，D是EF的相同部分，故BC中有多少个等于A的数，EF中也有多少个等于D的数。
Let BC be divided into the numbers equal to A, namely BG, GC, and EF into those equal to D, namely EH, HF; thus the multitude of BG, GC will be equal to the multitude of EH, HF.
将BC分为等于A的数BG、GC，将EF分为等于D的数EH、HF，则BG、GC的个数等于EH、HF的个数。
Now, since the numbers BG, GC are equal to one another, and the numbers EH, HF are also equal to one another, while the multitude of BG, GC is equal to the multitude of EH, HF, therefore, whatever part or parts BG is of EH, the same part or the same parts is GC of HF also; so that, in addition, whatever part or parts BG is of EH, the same part also, or the same parts, is the sum BC of the sum EF.
由于BG、GC相等，EH、HF相等，且个数相等，故BG是EH的几分之一或几分之几，GC也是HF的相同几分之一或几分之几；因此，BG是EH的几分之一或几分之几，总和BC也是总和EF的相同几分之一或几分之几。