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Let as many Numbers, as you please, be proposed to be Combined: Suppose Five, which we will call a b c d e. Put, in so many Lines, Numbers, in duple proportion, beginning with 1. The Sum is the Number of Sumptions, or Elections; wherein, one or more of them, may several ways be taken. Hence subduct the Number of the Numbers proposed; because each of them may once be taken singly. And the Remainder shews how many ways they may be taken in Combination; And, consequently, how many Products may be had by the Multiplication of any two or more of them so taken. But the same Sum without such Subduction, shews how many Aliquot Parts there are in the greatest of those Products, a b c d e. For every one of those Sumptions, are Aliquot Parts of a b c d e, except the last, and instead thereof, 1 is also an Aliquot Part; which makes the number of Aliquot Parts, the same with the Number of Sumptions. Only here is to be understood, that, all the Numbers proposed, are to be Prime Numbers, and each distinct from the other. For if any of them be Compound Numbers, or any Two of them be the same, the Rule for Aliquot Parts will not hold. (en) |
