QUA 



C A ; but P G = G H, 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 plumb-line 

 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 I, Z I S : therefore the 

 angle ZIS = QAC;butZSI =Q C A, 

 as being both right; hence, in the equi- 

 angular triangles AC Q, S Z I, we 'nave 

 (by 4. 6. Eucl.) Z S : S I : : C Q : CA; 

 that is, 100 : 40 : : 300 : C A, or C A = 



4_XJ2P.:=:120 ; 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 R : 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 : S B : : B 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, which 

 will be 21, the height of the tower in 

 poles. 



QUADRAT, in printing, a piece of me- 

 tal cast like the letters, to fill up the void 

 spaces between words, &c. There are 

 quadrats of different sizes, as m quad- 

 rats, n quadrats &c. which are, respect- 

 ively, of the dimensions of these letters. 



QUADRATIC equation, in algebra, that 

 wherein the unknown equality is of two 

 dimensions, or raised to the second power. 

 See ALGEBRA. 



QUADRATURE, in geometry, denotes 

 the squaring, or reducing a figure to a 

 square. Thus, the finding of a square., 

 which shall contain just as much surface, 

 or area, as a circle, an ellipsis, a triangle, 

 &c. is the quadrature of a circle, ellipsis, 

 &,c. The quadrature of rectilinear figures, 

 or method of finding their areas, has beea 

 already delivered. See MENSURATION. 



But the quadrature of curvilinear 

 spaces, as the circle ellipsis, parabola, 

 &c. is a matter of much deeper specula- 

 tion, making a part of the higher geo- 

 metry ; wherein the doctrine of fluxions 

 is of singular use. We shall give an ex- 

 ample or two. 



Let A R C (Plate XIII. Miscell. fig. 5) 

 be a curve of any kind, whose ordinates 

 R b, C B, are perpendicular to the axis 

 A B. Imagine a right line, 6R-, perpen. 

 dicular to A B, to move parallel t 

 itself from A towards B ; and let the ve- 

 locity thereof, or the fluxion of the ab- 

 sciss, A&, in any proposed position of 

 that line, be denoted by b d, then will b n t 

 the rectangle under b d and the ordinate, 

 b R, express the corresponding fluxion 

 of the generating area, A b R ; which 

 fluxion, if A b = x t and b R = t/, will be 

 yx. From whence, by substituting for 

 y or x, according to the equation of the 

 curve, and taking the fluent, the area 

 itself, A b R, will become known. 



But in order to render this still more 

 plain, we shall give some examples, where- 

 in x, y, z, and u are all along put to de- 

 note the absciss, ordinate, curve-line, and 

 the area, respectively, unless where the 

 contrary is expressly specified. Thus, if 

 the area of a right angled triangle be re- 

 quired ; put the base A H (fig 6) = a % 

 the perpendicular H M = b, and let A B 

 = x, be any portion of the base, con- 

 sidered as a flowing quantity ; and lei B R 

 = y be the ordinate, or perpendicular 

 corresponding. Then because of the 

 similar triangles, A H M and A B R, we 



shall have a : b : : x : y=. . Whencf 





