Kitap 7
15
A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
16
16
And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.
17
17
And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.
18
18
A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
19
19
And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
20
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.
