The first and second kinde of measuring of heights is thus: The third followeth.
14 If the sight be from the beginning of the Index perpendicular to the heighth, as in the Index the difference of the segment, is unto the difference of the distance or station; so is the segment of the transome unto the heighth.
Hitherto you must recall that subtilty, which was used in the third manner of measuring of lengths.
Let the first aime be taken from a, the beginning of the Index perpendicular unto the height to be measured: And from an unknowne length ai, by o, the end of the transome, unto e, the toppe of the height ei: And let the segment of the Index be ua. The second ayme, let it be taken from y, the beginning of the same Index; and out of a
greater distance, by s, the end of the transome, unto the same toppe e. And the segment of the Index let it be ry.
Here, as afore, the measuring is performed and done, by the taking of the difference of the said yr, above au: Now the demonstration is concluded, as in the former was taught. Let the parallell lsm, be erected against aoe.
Here first the triangles oua, & srl, are equilaters, by the [2 e vij].; (seeing that the angles at a, and l, the externall and internall, are equall in bases ou, and sr, for the segment in each distance is the same still:) Therefore ua, is equall to rl. Now the rest is concluded by a sorites of foure degrees: As yr, is unto yi: so by the [12. e vij]. is sr, that is, ou, unto ei: And as ou, is unto ei, so is au, that is, lr, unto ai. Therefore the remainder yl, unto the remainder ya; shall be as yr, is unto the whole yi, and therefore from the first unto the last, as sr, is to ei.
Therefore let the difference of the Index be 23. parts: The difference of the distance 30. foote: The segment of the transome 44. parts: The height shall be 57.9/23. or foote.
Therefore