Euclid's Elements

The space which is at the center is double the angle which is at the circumference, when they have the same circumference as a base. 44

Proposition XXI is demonstrated differently.

In the same straight line, on neither side, can similar and unequal portions of circles be constructed. 44.b

In the same straight line, or in equal straight lines, equal portions of circles are similar. 45

If equal straight lines cut off equal and similar circumferences, the circles of which those are the circumferences will be equal. 46.b.47

In unequal circles, equal straight lines cut off dissimilar circumferences. 47

In unequal circles, unequal straight lines subtend similar circumferences.

Unequal straight lines subtend similar and unequal circumferences.

If from a point taken outside a circle, as many straight lines as one wishes are drawn cutting the circle, the rectangles which are contained by the wholes and their outer segments are equal to one another. 50

Two straight lines drawn from a point outside a circle touching the circle are equal to one another.

IN THE FOURTH BOOK.

In a given circle, to fit a straight line equal to a given straight line, which is not greater than the diameter, and parallel to another given one.

IN THE FIFTH BOOK.

If the first has the same ratio to the second as the third has to the fourth, but the third has a lesser ratio to the fourth than the fifth has to the sixth, then the first will also have a lesser ratio to the second than the fifth has to the sixth. 64.b

If the first has a greater ratio to the second than the third has to the fourth, but the third has a greater ratio to the fourth than the fifth has to the sixth, then the first will also have a greater ratio to the second than the fifth has to the sixth. 64.b

If the first has the same ratio to the second as the third has to the fourth, and the first is greater than the second, then the third will also be greater than the fourth, and if equal, equal, and if lesser, lesser. 65.b

If there are three proportional magnitudes, the greatest and the least of them will be greater than double the remaining one. 68.b

If the first has a greater ratio to the second than the third has to the fourth, then by conversion, the second will have a lesser ratio to the first than the fourth has to the third. 69

If the first has a greater ratio to the second than the third has to the fourth, then by permutation, the first will have a greater ratio to the third than the second has to the fourth. 69

If the first has a greater ratio to the second than the third has to the fourth, then also by composition, the first and second will have a greater ratio to the second than the third and fourth have to the fourth. 69.b

If the first and second have a greater ratio to the second than the third and fourth have to the fourth, then by division, the first will have a greater ratio to the second than the third has to the fourth.

If the first and second have a greater ratio to the second than the third and fourth have to the fourth, then by conversion of ratio, the first and second will have a lesser ratio to the first than the third and fourth have to the third. 70

If the first has a greater ratio to the third than the second has to the fourth, then also the first will have a greater ratio to the third than the first and second have to the third and fourth.

If the whole has a greater ratio to the whole than the part taken away has to the part taken away, then the remainder will also have a greater ratio to the remainder than the whole has to the whole.

If there are three magnitudes, and others equal to them in number, and the first of the former has a greater ratio to the second than the first of the latter has to the second, and the second of the former has a greater ratio to the third than the second of the latter has to the third, then by equality, the first of the former will have a greater ratio to the third than the first of the latter has to the third. 70.b