Euclid's Elements

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If many numbers multiplying a number produce as many others, the produced numbers will have the same ratio as the multiplying numbers. 95.b
Given any number of numbers proportional in succession, to find two minimum numbers which have the same ratio as they do. 99.b

IN THE EIGHTH BOOK.

Plane numbers which have the same ratio as a square number to a square number are similar to one another. 109
Solid numbers which have the same ratio as a cube number to a cube number are similar to one another. 109.b

IN THE NINTH BOOK.

If a cube number multiplying a number that is not a cube produces a number, the product will not be a cube. 111
If a cube number multiplying a certain number produces a number that is not a cube, then the multiplied number will not be a cube.
If two numbers are proposed, and one of them is divided into any number of parts, the plane number which is made from the two numbers originally proposed will be equal to the plane numbers which are made from the undivided number and the individual parts of the divided number. 114
If a number is divided into two numbers, the two plane numbers which are made from the whole and each part added together are equal to the square number which is made from the whole.
If a number is divided into two numbers, the plane number which is made from the whole and one part is equal to the plane number which is made from the parts together with that square which is made from the aforementioned part. 115
If a number is divided into two numbers, the square which is made from the whole is equal to the squares which are made from the parts, and to that plane number which is made twice from said parts.
If an even number is divided into two equal parts, but also divided into unequal numbers, the plane number made from the unequal parts together with the square of the intermediate number is equal to the square made from the half.
If an even number is divided into two equal parts, and a certain number is added to it, the plane number made from the whole with the addition and the addition together with the square of the half is equal to the square of that which consists of the half and the addition. 115.b
If a number is divided into two numbers, the square which is made from the whole together with the square of one part is equal to the plane number which is made twice from the whole and the said part together with the square of the remaining part. 116
If a number is divided into two numbers, the plane number which is made four times from the whole and one part together with the square of the remaining part is equal to the square which is made from the whole and the said part as if from one. 116*
If an even number is divided into two equal parts, but also divided into unequal numbers, the squares which are made from the unequal parts are double the square which is made from the half together with the square of the number lying between them.
If an even number is divided into two equal parts, and another number is added to it, the square made from the whole with the addition and the square made from the addition to both are double the square from the half and the square which is made from the half and the addition as if from one. 116.b
However, that which corresponds to the eleventh of the second book, namely to divide a number such that the plane number made from the whole and one part is equal to the square made from the remaining part, can in no way be done.

IN THE TENTH BOOK.

Given two commensurable magnitudes, to find what ratio they have to one another in numbers. 126.b
Given two numbers and a straight line, to make it so that the ratio of number to number is as the square of the straight line to the square of another straight line. 130.b
To find two dissimilar plane numbers. 131
Magnitudes which are commensurable to incommensurable ones will also be incommensurable to one another. 132