Any number is either a part or parts of any number, the less of the greater.
任意一个数要么是另一个数的部分，要么是几个部分，即较小数是较大数的部分或几个部分。
本页以“较小数对较大数的部分关系”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let A, BC be two numbers, and let BC be the less; I say that BC is either a part, or parts, of A. For A, BC are either prime to one another or not. First, let A, BC be prime to one another.
设A和BC是两个数，BC较小。则BC与A要么互素，要么不互素。
Then, if BC be divided into the units in it, each unit of those in BC will be some part of A; so that BC is parts of A. Next let A, BC not be prime to one another; then BC either measures, or does not measure, A.
若互素，将BC分解为单位，每个单位是A的一部分，故BC是A的几个部分。
If now BC measures A, BC is a part of A. But, if not, let the greatest common measure D of A, BC be taken; [VII. 2] and let BC be divided into the numbers equal to D, namely BE, EF, FC.
若不互素，则BC要么量尽A，要么不量尽。若量尽，BC是A的一个部分。
Now, since D measures A, D is a part of A. But D is equal to each of the numbers BE, EF, FC; therefore each of the numbers BE, EF, FC is also a part of A; so that BC is parts of A.
若不量尽，取A和BC的最大公度量D，将BC分为等于D的BE、EF、FC。因D是A的一部分，且等于BE、EF、FC，故每个都是A的一部分，所以BC是A的几个部分。