Book 7
20
Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
21
21
Similar plane and solid numbers are those which have their sides proportional.
22
22
A perfect number is that which is equal to its own parts.
BOOK VII. PROPOSITIONS.
PROPOSITION I.
1
Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers will be prime to one another.
1
For, the less of two unequal numbers AB, CD being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left; I say that AB, CD are prime to one another, that is, that an unit alone measures AB, CD.
1
For, if AB, CD are not prime to one another, some number will measure them.
1
Let a number measure them, and let it be E; let CD, measuring BF, leave FA less than itself, let AF, measuring DG, leave GC less than itself, and let GC, measuring FH, leave an unit HA.
1
Since, then, E measures CD, and CD measures BF, therefore E also measures BF.
1
But it also measures the whole BA; therefore it will also measure the remainder AF.
1
But AF measures DG; therefore E also measures DG.
1
But it also measures the whole DC therefore it will also measure the remainder CG.
1
But CG measures FH; therefore E also measures FH.
