If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.
若一数是另一数的若干部分，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 AB be parts of the number C, and another, DE, the same parts of another, F, that AB is of C; I say that the sum of AB, DE is also the same parts of the sum of C, F that AB is of C. For since, whatever parts AB is of C, DE is also the same parts of F, therefore, as many parts of C as there are in AB, so many parts of F are there also in DE.
设数AB是数C的若干部分，另一数DE是另一数F的相同部分，即AB是C的部分与DE是F的部分相同。
Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of F, namely DH, HE; thus the multitude of AG, GB will be equal to the multitude of DH, HE.
由于AB是C的部分与DE是F的部分相同，故AB中所含C的部分数等于DE中所含F的部分数。
And since, whatever part AG is of C, the same part is DH of F also, therefore, whatever part AG is of C, the same part also is the sum of AG, DH of the sum of C, F.
将AB分为C的部分AG、GB，DE分为F的部分DH、HE，则AG、GB的个数等于DH、HE的个数。
[VII. 5] For the same reason, whatever part GB is of C, the same part also is the sum of GB, HE of the sum of C, F.
根据第七卷第五命题，AG是C的部分与DH是F的部分相同，故AG与DH之和是C与F之和的相同部分；同理，GB与HE之和也是C与F之和的相同部分，因此AB与DE之和是C与F之和的相同部分。