L K N, A C K, are similar triangles, and (by prop. 4. lib. 6. of Euclid) NK:KL : : K C (i. e. B D) : C A ; that is, 80 : 100 : : 96 : CA : therefore, by the rule of three, , 96 x 100 C ---= ucv feet, and C B 6 feet being added, the whole height B A is 126 feet.
If the observer's distance, as D E, be such, that, when the instrument is direct ed as formerly towards the summit A, the perpendicular fall on the angle P, and the distance, B E or C G, be 120 feet, C A will also be 120 feet: forPG:GII:: GC: C A; but P G = G If, therefore G C C A; that is, C A will be 120 feet, and the whole height B A = 126 feet, as before.
But let the distance B F (ibid.) be 300 feet, and the perpendicular or plumbline cut off 40 equal parts from the reclining side. Now, in this case, the angles Q A C, Q Z I, are equal (29. 1. Eucl.) as are also the angles Q Z f, Z I S: therethre the angleZIS=QAC; but ZSI=4CA, as being both right ; hence, in the equi angular triangles A C Q, S Z 1, we have (by 4. 6. Encl.) ZS:S1::CQ: CA; that is, 100: 40 : : 300 : C A, or C A 40 ; and by adding 6 feet, 100 the observer's height, the whole height B A will be 126 feet.
To measure any distance at land or sea, by the quadrat. In this operation the index, A H, is to be applied to the instrument, as was shown in the descrip tion; and, by the help of a support, the instrument is to be placed horizontally at the point A (fig. 4,) then let it be turned till the remote point F, whose distance is to be measured, be seen through the fixed sights : and bringing the index to be parallel with the other side of the in strument, observe through its sights any accessible mark, B, at a distance ; then carrying the instrument to the point B, let the immoveable sights be directed to the first station A, and the sights of the index to the point F. If the index cut
the right side of the square, as in K, the proportion will be (by 4. 6.) B RK : B A (the distance of the stations to be measured with a chain) : A F, the dis tance sought. But if the index cut the reclined side of the square in the point L ; then the proportion is L S: SB:: 13 A: A G, the distance sought; which, accord ingly, may be found by the rule of three.
The quadrat may be used without calculation, where the divisions of the square are produced both ways so as to form the area into little squares. Ex. Suppose the thread to fall on 40 in the side of right shadows, and the distance to be measured 20 poles ; seek among the little squares for that perpendicular, to the side of which is 20 parts from the thread, this perpendicular will cut the side of the square next the centre, in the point 50, which is the height of the required poles. If the thread cut the side of the versed shadows in the point 60, and the distance be 35 poles, count 35 parts on the side of the quadrat from the centre, count also the divisions of the perpendicular from the point 35 to the thread, olliCh will be 21, the height of the tower in poles.