172 On ^n Improvement in tlie 



and s p are all too long by the small qtjantity s, and thsi 

 the arches nf, fe., e d, and d b are all too short hy the 

 same quantity, which is the supposition the mostuntavour- 

 able of any lo the exactness of the operation; (hen the 

 ^rror in the position oF ,6 =± g, and the point /» errs 4s in the 

 game direction, and therefore the point assumed as the true 



point ot quinquesection, 'will' be JTt th^ distance of -4- from 



/3, and the error in the position of this point = s x If. 



By the saine way of reasoning, the error in the position 

 of the point taken between d and ^ = g x 2^-. 



In trisecting the error of each point = e X 1-^j and in 

 bisecting, the error = s; and in quadrisecting, the error of 

 Vhe middle point = 2 s. 



It appears therefore that in trisecting, the greatest error 

 we are liable to does not exceed that of bisection in a greater 

 proportion than that of 4 to 3 ; but in quinquesecting the 

 error of the two middle points is 2f times greater than m 

 bisecting. It must be considered, however, that in the 

 method of continued bisection, the two opposite points 

 must be found by quadrisection ; and the error of quinque- 

 section exceeds that of quadrisection in no greater propor- 

 tion than that of six to five; so that we may fairly say, 

 that if we begin with quinquesection, this method of di- 

 viding is not greatly inferior, in point of accuracy, to that 

 by continued bisection. 



Second Method, 



This differs from the foregoing, i n placing dots or scratches 

 in the true points of quinquesection and trisection, before 

 we begin to subdivide. For this purpose, we must have a 

 niicroscoi|ie placed as in page 170, first par. at the same di- 

 stance frfMT) the centre of motion as the point is ; and 

 this microscope must be furnished with a moveable wire 

 and micrometer, as in page 171 ; ^nd then having first 

 made the fixed wire of this microscope correspond exactly 

 with the point, we must draw the scratcht^s b and /3, d and 

 ^, &c. as before, and bring the fixed wire lo the true, point 

 of quinquesection between b and /3, in the manner directed 

 in page 226, and with the point strike the scratch or dot : 

 and if we please, we may, for further security, as soon as 

 this is done, examine, by means of the moveable wire, 

 whether this intermediate scratch or dot is well placed. 



The advantage of this method is, that when this is done, 

 we may subdivide and cut the true divisions, by making 

 the wire of the microscope bisect the intermediate scratches, 

 instead of being obliged to^ use the more troublesome ope- 

 ration 



