If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.
若一数是另一数的一部分，且另一数是另一数的相同部分，则两数之和也是两数之和的相同部分。
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For let the number A be a part of BC, and another, D, the same part of another EF that A is of BC; I say that the sum of A, D is also the same part of the sum of BC, EF that A is of BC. For since, whatever part A is of BC, D is also the same part of EF, therefore, as many numbers as there are in BC equal to A, so many numbers are there also in EF equal to D.
设数A是BC的一部分，数D是EF的相同部分，即A是BC的几分之一，D也是EF的几分之一。
Let BC be divided into the numbers equal to A, namely BG, GC, and EF into the numbers equal to D, namely EH, HF; then the multitude of BG, GC will be equal to the multitude of EH, HF. And, since BG is equal to A, and EH to D, therefore BG, EH are also equal to A, D.
则BC中有多少个等于A的数，EF中也有同样多个等于D的数。将BC分为等于A的数BG、GC，EF分为等于D的数EH、HF，则BG、GC的个数等于EH、HF的个数。
For the same reason GC, HF are also equal to A, D. Therefore, as many numbers as there are in BC equal to A, so many are there also in BC, EF equal to A, D.
由于BG等于A，EH等于D，故BG与EH之和等于A与D之和。同理，GC与HF之和也等于A与D之和。
Therefore, whatever multiple BC is of A, the same multiple also is the sum of BC, EF of the sum of A, D.
因此，BC中有多少个等于A的数，BC与EF之和中就有同样多个等于A与D之和的数。故BC是A的多少倍，BC与EF之和就是A与D之和的相同倍数。