. Memoirs and proceedings of the Manchester Literary & Philosophical Society. imilarlyobtain the points Q\ Q and Q. Draw BPy BQ. BPand BQ are then the approximate trisectors of arcs HP, PQ and QK are obviously exactlyequal, and to each of these is added an approximation toone-third of the sum of the arcs PP\ QQ. Method F.—Although the construction of method Egives a very close approximation to trisection, a muchcloser approximation is obtained by the following varia-tion devised by one of the writers. Let ABC {Fig. 25) be the angle to be trisected. WithB as centre and any radius descri
. Memoirs and proceedings of the Manchester Literary & Philosophical Society. imilarlyobtain the points Q\ Q and Q. Draw BPy BQ. BPand BQ are then the approximate trisectors of arcs HP, PQ and QK are obviously exactlyequal, and to each of these is added an approximation toone-third of the sum of the arcs PP\ QQ. Method F.—Although the construction of method Egives a very close approximation to trisection, a muchcloser approximation is obtained by the following varia-tion devised by one of the writers. Let ABC {Fig. 25) be the angle to be trisected. WithB as centre and any radius describe a circle ERG cuttingBA at E and BC at G. With B as centre and radiusequal to three times BE describe a circle H VK cuttingBA at H and BC at K. Join EG, HK. Now with H ascentre and radius equal to EG describe an arc cuttingHK at M and the circle HVK at P. Join E to M and Manchester Memoirs, Vol. lix. (1915), No. 13- H produce EM to meet the circle HVK at P. Draw BFand BF cutting the circle ERG at R and S the chord RS. With F as centre and radius equal. Fig. 25. Modification of Durers Method. to RS describe an arc cutting the arc FF at a point B to P, then BP is one of the approximate tri-sectors. It will now be proved that the angle ABP only exceeds one-third of the angle ABC by JL (*-) 648 \3/ radians, where the angle ABC is /3 radians and the termsneglected involve (^ J to at least the 13th power. DrawBV perpendicular to HK and P W parallel to the circular measure of angles ABC, ABP, PBP,PBP and ABP be /3, a, y, <S and 0 , since angle PBW=QJ2, 3 sin 0/2 = sin/3/2 ;hence by expanding, But •-£-!(£) neglecting.^)7. 3 3^3 ^3 34 Gee AND ADAMSON, Trisecting an Also hence But 3 = -, -), neglecting (-i 2V3/ * * v3/ sin - = sin 0 neglecting / - j 7 = ■ 3 3^3^ \3 X -r /S\3 a=0+d-y= + d -_ + -(_) 3 3 3W - i7 .13 = - + -(-) neglecting(- = - + (-) neglecting (-) 3 648 W s ^3 In addition to the preceding metho
Size: 2202px × 1134px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1880, booksubjectscience, bookyear1888
