Euclid's Elements

IN THE SIXTH BOOK.

Triangles and parallelograms constructed on equal bases have the same ratio to one another as their heights. 72.b
Proposition VI is demonstrated differently. 74.b
To cut a given straight line in a given ratio. 75.b
To describe a square in a given triangle. 76
Given three straight lines AB, BC, and D, to find such that as AB is to BC, so another certain one is to the line D. 76.b
If rectilinear figures are equal and similar, their corresponding sides will be equal to one another. 81
Triangles which have one angle equal to one angle have a ratio composed of their sides. 81.b
How a ratio is composed from two given ratios, or even more.
How a ratio is taken away from a given greater ratio.
How ratios are both composed and taken away in numbers.
Triangles, of which one angle is equal to one angle, have the same ratio to one another as the rectangles which are contained by the sides containing the equal angle.
Equiangular parallelograms have the same ratio to one another as the rectangles which are contained by their sides. 82
Triangles and parallelograms have to one another a ratio composed of the ratio of their bases and the ratio of their heights.
Proposition XXVII is explained differently. 84
To find the excess of two unequal rectilinear figures by which the greater exceeds the smaller. 84.b
The theorem of Pappus, which is much more universal than the XXXI of Euclid.

IN THE SEVENTH BOOK.

Given two numbers prime to one another, if the smaller is always taken from the greater, this kind of subtraction will not cease until unity is reached. 90
Given two numbers composite to one another, if the smaller is always taken from the greater, the subtraction will not reach unity.
Given two numbers, to discover whether they are prime to one another or composite.
If a number measures many numbers, to measure their common measure. 91
If a number is a multiple of a number, and another is the same multiple of another, then both together will be the same multiple of both together as one is of one. 91.b
If there are any number of numbers, each being the same multiple of as many numbers of equal multitude, then as one is of one, so all will be of all.
If any number of smaller numbers are referred to as many larger numbers, and each is the same part or same parts, then as one is of one, so all will be of all. 92
If a number is the same multiple of a number as the taken-away is of the taken-away, then the remainder will be the same multiple of the remainder as the whole is of the whole.
Ratios of numbers which are the same to the same are also the same to one another. 93.b
If four numbers are proportional, they will also be proportional by conversion. 94.b
If four numbers are proportional, they will also be proportional by composition.
If four numbers are proportional, they will also be proportional by division.
If four numbers are proportional, they will also be proportional by conversion of ratio.
If the first has the same ratio to the second as the third has to the fourth, but the fifth also has the same ratio to the second as the sixth has to the fourth, then the composite first and fifth will have the same ratio to the second as the third and sixth have to the fourth.
If any number multiplying many numbers produces as many others, the produced numbers will have the same ratio as the multiplied numbers. 95.b