15 Out of the Geodesy of heights, the difference of two heights is manifest.
Or thus: By the measure of one altitude, we may know the difference of two altitudes: H.
For when thou hast taken or found both of them, by some one of the former wayes, take the lesser out of the greater; and the remaine shall be the heighth desired. From hence therefore by one of the towers of unequall heighth, you may measure the heighth of the other. First out of the lesser, let the length be taken by the first way: Because the height of the lesser, wherein thou art, is easie to be taken, either by a plumbe-line, let fall from the toppe to the bottom, or by some one of the former waies. Then measure
the heighth, which is above the lesser: And adde that to the lesser, and thou shalt have the whole heighth, by the first or second way. The figure is thus, and the demonstration is out of the [12. e vij]. For as ae, is to ei, so is ao, to ou. Contrariwise out of an higher Tower, one may measure a lesser.
16 If the sight be first from the toppe, then againe from the base or middle place of the greater, by the vanes of the transome unto the toppe of the lesser heighth; as the said parts of the yards are unto the part of the first yard; so the heighth betweene the stations shall be unto his excesse above the heighth desired.
Let the unequall heights be these, as, the lesser, and uy, the greater: And out of the assigned greater uy, let the lesser, as, be sought. And let the sight be first from u, the toppe of the greater, unto a, the toppe of the lesser,
making at the shankes of the staffe the triangle urm. Then againe let the same sight be from the base, or from the lower end of uy, the heighth given, unto a, the same toppe of the lesser, making by the shankes of the staffe the triangle yln, so that the segments of the yard be, the upper one, I meane, ur, the neather one ul: I say the whole of ur, and nl, is unto ur: so is the uy, greater heighth assigned, unto as, the lesser sought.
The Demonstration, by drawing of ao, a perpendicular unto uy, is a proportion out of two triangles of equall heighth. For the forth of the totall equally heighted triangles uao, and yas, although they be reciprocall in situation, they have their bases uo, and as, as if their were oy. Then they have the same with the whole triangles; as also the subducted triangles urm, and ynl, of equal heighth; to wit whose common heighth is the segment of the transome remained still in the same place, there rm, here yl. And therefore the bases of these, namely, the segments of the yards ur, and nl, have the same rate with uo, unto oy.