Euclid

If two numbers by multiplying two numbers make two numbers which are prime to one another, the original numbers will also be prime to one another.

If two numbers by multiplying any number make two numbers which are prime to one another, the original numbers will also be prime to one another.

Numbers prime to one another are also prime to all their products.

If two numbers be prime to one another, and one of them be a measure of any number, the remaining one will not measure this number.

If one be a measure of any number, and the remainder be prime to the number, the remainder will be a measure of the number.

If two numbers be prime to one another, and one of them be a measure of a number, then the second will measure any number measured by the first, and any number which it measures the first will measure.

If two numbers be commensurable in square only, that is, their squares have to one another the ratio which a square number has to a square number, and if one of them measure any number, the square on the second will measure the square on that number.

The square on the greater number measures the square on the less, and conversely, the square on the less measures the square on the greater.

If two magnitudes have to one another the ratio which a square number has to a square number, and the first be square, the second will also be square.

If a medial straight line be bisected and a straight line be added to it in a straight line which is equal to the greater segment, the whole straight line has been rationally applied to the whole, and the square on the whole is equal to the squares on the whole and on the added straight line.

The square on the lesser of two unequal numbers measures the square on the greater, and the square on the greater measures twice the square on the lesser plus the square on the difference between the numbers.

If a medial area be applied to a rational straight line, it is possible to find a rational area such that the excess is a square.

The square on a bimedial straight line applied to a rational straight line produces as breadth a straight line which is rational and incommensurable in length with it.

The square on the side of the sum of two medial straight lines commensurable in square only and containing a rational rectangle produces as breadth two rational straight lines commensurable in length.

If two rational straight lines commensurable in square only contain a rectangle, and one of them be added to the other, the square on the sum is triple the rectangle contained by the rational straight lines plus the square on the greater.

The square on the sum of two unequal medial straight lines commensurable in square only and containing a rational rectangle is equal to the sum of the squares on the medial straight lines plus twice the rectangle contained by the medial straight lines.

The square on the difference of two medial straight lines commensurable in square only and containing a rational rectangle is equal to the difference of the squares on the medial straight lines minus twice the rectangle contained by the medial straight lines.

To two given unequal straight lines to find a third proportional.

To a given straight line in a given rectilineal angle, to apply a parallelogram equal to a given triangle.

To a given straight line to apply a parallelogram equal to a given rectilineal figure in a given rectilineal angle.