Euclid's Elements

is contained, it will be given meaning "defined" or "known". 149.b
If more is multiplied by less, or less by more, the product is less.
A space composed of means is irrational. 155.b
To find the square side, or root, of a binomial space. 160
If a straight line is cut into unequal parts, the squares of those parts are greater than the rectangle which is contained by the two aforementioned parts. 162.b
Let there be four magnitudes A, B, C, E, F, G, and let AB exceed C by the same amount by which EF exceeds G. I say that also by permuting, AB exceeds EF by the same amount, or is exceeded by it, by which C exceeds G, or is exceeded by it. 171

IN THE ELEVENTH BOOK.

Converse 10. If there are two equal angles contained by straight lines not existing in the same plane, and one of them is parallel to one of the lines containing the equal angle, the remaining one will also be parallel to the remaining one. 195
Converse 14. If two planes are parallel, a straight line which is perpendicular to one of them will also be perpendicular to the other. 196
Converse 18. If all planes produced through a certain straight line are at right angles to a certain plane, the straight line itself will be at right angles to that same plane. 197.b
Converse 19. Of planes intersecting each other, if their common section is at right angles to a certain plane, the intersecting planes will also be at right angles to the same plane.
If there are any number of plane angles, of which one is greater than the others, however they are taken; and if they are contained by equal straight lines. I say that also of the straight lines subtending the angles, one is greater than the others, however they are taken; that is, it is possible for a figure of many sides to be constructed from those which connect the straight lines. 200
If equal straight lines fall into some plane from a certain elevated point, they will be on the circumference of a circle: and that which is drawn from the said point to the center of the circle will be perpendicular to the circle. 201
Every solid angle, which is contained by isosceles plane figures, has its base described within a circle. 201.b
From any given plane angles of which one is greater than the others, however they are taken, to construct a solid angle; it is necessary, however, that the given angles be less than four right angles.
If a solid is contained by parallel planes, its opposite planes are both equal and similar. 202.b
If a parallelepiped solid is cut by a plane parallel to the bases, the solid will be to the solid as the height is to the height. 203
Parallelepiped solids constructed on the same base, or on equal bases, have the same ratio between them as their heights. 205
Prisms having triangular bases, which are constructed either on the same or equal bases, and have the same height, are equal to each other. And moreover, those which have the same height are to each other as their bases. And those which are constructed on the same or equal bases are to each other as their heights.
Of equal prisms having triangular bases, the bases correspond to the heights on the opposite side. And those prisms having triangular bases whose bases correspond to the heights on the opposite side are equal to each other. 207.b
Proposition 38 is demonstrated otherwise. 209.b

IN THE TWELFTH BOOK.

In a given circle, to describe a polygon similar to the polygon described in the circle. 212.b
All prisms which have the same height are to each other as their bases. 215.b
All prisms and pyramids which are constructed on the same or equal bases have the same ratio between them as their heights.
All prisms and pyramids have a ratio between them compounded from the ratio of their bases and the ratio of their heights. 216
Similar pyramids, which have multi-angled bases, are divided into triangular-based pyramids that are similar, equal in number, and homologous to the wholes. 216.b