transverie parallell to the length to be measured, as in the index the difference of the greater segment is unto the lesser; so is the difference of the second station unto the length.
This kinde of Geodæsy is somewhat more subtile than the former were. The figure is thus; in which let the first ayming be from a, the beginning of the transome, and out of ai the length sought by o, the end of the Index, unto e, the toppe of the heighth: And let the segment of the Index be ou: The second ayming let it be from y, the beginning of the transome, out of a greater distance by s, the end of the Index, unto e, the same note of the heighth: And let the segment of the Index be sr.
Here the measuring performed, is the taking of the difference betweene ou and sr. The rest are faigned onely for demonstrations sake. Therefore in the first station let aml, be from the beginning of the transome, be parallell to ye. Here first mu, is equall to sr. For the triangles
mua, and sry, are equall in their shankes ua, and ry, by the grant (Because the transome standeth still in his owne place:) And the angles at mua, uam, are equall to the angles: And all right angles are equall, by the [14 e iij]. These are the outter and inner opposite one to another: And such are equall by the 1 e v. Therefore they are equilaters, by the [2 e vij]; And om, is the difference of the segments of the Index. Then as om is to mu, so is el, to li; as the equation of three degrees doth shew. For, by the [12 e vij], as om is to ma: so is el to la: And as ma is to mu; so is la, to li. Therefore by right, as om, is to mu: so is el, to li: And by the [12 e vj], so is ya, to ai: As if the difference of the first segment be 36 parts: The second segment be 72 parts: The difference of the second station 40 foote. The length sought shall be 80 foote. And here indeed is no heighth definitely given, that may make any bound of the principall proportion. Notwithstanding the Heighth, although it be of an unknowne measure, is the bound of the length sought: And therefore it is an helpe and meanes to argue the question. Because it is conceived to stand plumbe upon the outmost end of the length.
Therefore that third kinde of measuring of length is oftentimes necessary, when by neither of the former wayes the length may possibly be taken, by reason of some impediment in the way, to wit of a wall, or tree, or house, or mountaine, whereby the end of the length may not be seene, which was the first way: Nor an height next adjoyning to the end of the length is given, which is the second way.
Hitherto we have spoken of the threefold measure of longitude, the first and second out of an heighth given the third cut of a double distance: The measuring of heighth followeth next, and that is also threefold. Now heighth is a perpendicular line falling from the toppe of the magnitude, unto the ground or plaine whereon the measurer doth stand, after which manner Altitude or
heighth was defined at the [9 e iiij]. The first geodesy or manner of measuring of heighths is thus.
10. If the sight be from the beginning of the transome perpendicular unto the height to be measured, as the segment of the transome, is unto the segment of the Index, so shall the length given be to the height.
Let the segment of the transome be 60 parts: the segment of the Index 36: the Length given 120 foote: the height sought shall be, by the golden rule, 72 foote.