If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.
若一数是另一数的若干部分，且减去的数是减去的数的相同部分，则余数也是余数的相同部分，如同整体是整体的相同部分。
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For let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted; I say that the remainder EB is also the same parts of the remainder FD that the whole AB is of the whole CD. For let GH be made equal to AB. Therefore, whatever parts GH is of CD, the same parts also is AE of CF.
设数AB是数CD的若干部分，且减去的AE是减去的CF的相同部分。
Let GH be divided into the parts of CD, namely GK, KH, and AE into the parts of CF, namely AL, LE; thus the multitude of GK, KH will be equal to the multitude of AL, LE. Now since, whatever part GK is of CD, the same part also is AL of CF, while. CD is greater than CF, therefore GK is also greater than AL.
取GH等于AB，则GH是CD的部分与AE是CF的部分相同。将GH分为CD的部分GK、KH，AE分为CF的部分AL、LE，则GK、KH的个数等于AL、LE的个数。
Let GM be made equal to AL. Therefore, whatever part GK is of CD, the same part also is GM of CF; therefore also the remainder MK is the same part of the remainder FD that the whole GK is of the whole CD. [VII. 7] Again, since, whatever part KH is of CD, the same part also is EL of CF, while CD is greater than CF, therefore HK is also greater than EL.
由于GK是CD的部分与AL是CF的部分相同，且CD大于CF，故GK大于AL。取GM等于AL，则GK是CD的部分与GM是CF的部分相同，因此余数MK是余数FD的部分与整体GK是整体CD的部分相同。
Let KN be made equal to EL. Therefore, whatever part KH is of CD, the same part also is KN of CF; therefore also the remainder NH is the same part of the remainder FD that the whole KH is of the whole CD. [VII. 7] But the remainder MK was also proved to be the same part of the remainder FD that the whole GK is of the whole CD; therefore also the sum of MK, NH is the same parts of DF that the whole HG is of the whole CD.
同理，KH是CD的部分与EL是CF的部分相同，取KN等于EL，则余数NH是余数FD的部分与整体KH是整体CD的部分相同。因此MK与NH之和是DF的部分与整体HG是整体CD的部分相同。