One may againe also out of the toppe of a Turret measure the distance of two turrets one from another: For it is the first manner of measuring of longitudes, neither doth it here differ any whit from it, more than the yard is hang'd without the heighth given. The figure is thus: And the Demonstration is by the [12. e vij]. For as ae, the segment of the yard, is unto ei the segment of the transome: so is the assigned altitude ao, unto the length ou.
The geodesy or measuring of altitude is thus, where either the length, or some part of the length is given, as in the first and second way: Or where the distance is double, as in the third.
17 If the sight be from the beginning of the yard being right or perpendicular, by the vanes of the transome, unto the ends of the breadth; as in the yard the difference of the segment is unto the differēce of the distance, so is the distance of the vanes unto the breadth.
The measuring of breadth, that is of a thwart or crosse
line, remaineth. The Figure and Demonstration is thus: The first ayming, let it be aei, by o, and u, the vanes of the transome ou. The second, let it be yei, by s, and r, the vanes of the transome sr. Then by the point s, let the parallell lsm, be drawne against aoe. Here first, the triangles oua, and sil, are equilaters, by the [2 e vij]. Because the angles at n and j, are right angles: And uao, and jls, the outter and inner, are equall in their bases ou, and sj, by the grant: Because here the segment of the transome remaineth the same: Therefore ua, is equall to jl. These grounds thus laid, the demonstration of the third altitude here taken place. For as yl, is unto ya: so is sj, unto er: And, because parts are proportionall unto their multiplicants, so is sr, unto ei: for the rest doe agree.
The same shall be the geodesy or manner of measuring, if thou wouldest from some higher place, measure the breadth that is beneath thee, as in the last example. But from the distance of two places, that is, from latitude or breadth, as of Trees, Mountaines, Cities, Geographers and Chorographers do gaine great advantages and helpes.
