Geometry.

If two surfaces equal in quantity and different in figure are placed one upon the other, three effects will follow, of which one is that they will never integrally cover one another; the second, that whatever of them touches will be equal in quantity and similar in figure; the third, that whatever of them does not touch will be equal in quantity and of different figure.

Those parts will remake their whole which are restored to their remainder.

The thing that moves acquires as much space as it loses.

a - b d - c e; a - b d - c e

a n m - b d g - c e f

If the quantity of the square b d c e is extended to the irrational length b a, by how much will it diminish its width? Produce the side of the square b c as far as the given distance b a, and from a to d produce the straight line orthogonally in a continuous straight line as far as f, and from side to side of the square c e produce as far as f; and thus you will have made the right-angled figure a c f, to which add another similar right-angled figure, but upside down, which will be a m f; and thus you will have made the quadrilateral a m e f, in which draw the lines n e and b g orthogonally according to the sides of the square b g and d e, and you will have made the quadrilateral n m b g, equal to the given square b d c e. It is proved by removing, from the two equal right-angled figures, the two greater right-angled figures which are similar, and similar are the two lesser right-angled figures, which being removed, there remains b d c e similar to n m d g.

It is asked, when such a cube has been elongated, what will be its thickness.

This is on the reverse of this sheet.