PKOBLEM xxxiii xxxv.] MATHEMATICS. PR ACTIOAL GEOMETRY. 



607 



METHOD. Bisect AB in C (Fig. 54), and from B 

 draw BD perpendicular to AB, 

 and equal to A C or B C ; join 

 A D aud produce it ; with the 

 centre D, and radius D B, de- 

 scribe a circle cutting A D pro- 

 duced, in E ; join B E, and 

 with the centres A and B and 

 radii, each equal to BE, describe 

 arcs intersecting in O, with 

 which as a centre and radius 

 O A or O B, describe a circle. 

 With the centres A and B and 

 radii, each equal to AB, describe 

 arcs cutting the circle in G and 

 H ; lastly, with the centre G or H, and the same radius, 

 describe an arc cutting the circle 

 in F, and join A G, G F, F H, 

 H B ; then A G F H B is the 

 pentagon required. 



SRD METHOD. With the cen- 

 tres A and B (Fig. 55), and radii, 

 each equal to AB, describe circles 

 intersecting in C and D ; join 

 C D, cutting A B in E, and pro- 

 duce it. Make E K equal to 

 A B ; join A K, and produce it, 

 till K L is equal to A E or E B ; 

 with the centre A, and radius 

 A L, describe an arc cutting D C 

 produced in F ; and with the 

 centre F and radius A B describe a circle cutting the 

 circles D A C and D B C in H and G respectively ; join 

 A G, F G, F H, and B H ; then A B H F G is tie pen- 

 tagon required. 



PROBLEM XXXIII. 



To describe a regular hexagon upon a given straight line. 

 (Euclid, Book IV., Prop. XV.) 



With the centiei A and B (Fig. 56), and radii, each 

 equal to A B, describe circles 

 intersecting in O ; with the 

 centre O, and the same radius, 

 describe another circle, cutting 

 the two former in F and C re- 

 spectively ; join A O and B O, 

 and produce them till they 

 meet the latter circle in D and 

 E respectively ; join BC, CD, 

 D E, E F, and FA; then 

 ABC DBF is the hexagon 

 required. From this it will be seen that the side of a 

 hexagon is equal to the radius of the circumscribed circle; 

 and by merely finding the centre O as above, and setting 

 off the length of the given line six times round the cir- 

 cumference, the hexagon will be constructed. 



PROBLEM XXXIV. 



To describe a regular octagon on a given straight line (AB). 

 With the centres A and B (Fig. 57), and radii equal 



Fig. 57. 



to AB, describe circles inter- 

 secting in K : with the centre 

 K, and the same radius, describe 

 a circle cutting the two former 

 in L and M ; with the centres 

 L and M, and still the same 

 radius, describe arcs cutting the 

 , last circle in N and O ; join AN 

 \ and B O, by lines cutting the 

 two first circles in P and Q, and 

 produce them ; join A Q and 

 B P, and produce them till P G 

 and Q D are each equal to A B ; and lastly, with the 

 centres G and D, and the same radii, equal to AB, 

 describe circles cutting the two first circles in H and C, 

 and the lines A N and B O produced, in F and E. 

 Join B C, C D, D E, E F, F G, G H and H A ; then 

 ABCDEFGHis the octagon required. 



2ND METHOD. With the centres A and B (Fig. 58), 

 and radii, each equal to A B, 

 describe circles cutting each 

 other in K and L ; join K L, 

 and through A and B draw 

 A F and B E parallel to K L, 

 cutting the circles in M and 

 N ; join A N and B M, and 

 parallel to them through A 

 and B draw AH and BC, 

 meeting the circles in H and 

 C ; through H and C draw 

 H G and C D parallel to KL, 

 and meeting B M and A N 

 produced in G and D ; and 

 lastly, through G and D draw 

 G F and D E parallel to A D and B G, cutting A M and 

 B N produced in F and E ; join FE ; then ABCDEFGH 

 is the octagon required. 



PROBLEM XXXV. 



To describe a regular polygon (containing any number of 

 given tides) upon a given straight line (A B). 



With the centre B (Fig. 69), and radius A B, describe 

 a semicircle, meeting A B 

 produced in H ; divide the 

 semicircle into as many equal 

 parts as in the proposed 

 polygon ; join B and the 

 second point of division C ; 

 bisect each of the sides A B 

 and B C by perpendiculars 

 intersecting in O ; then with 

 the centre O, and radius 

 O A, O B, or O C, describe 

 a circle, which will pass 

 through the points A, B, 

 and C ; this circle will be 

 that described about the re- 

 quired polygon ; and therefore, by setting off A B as 

 many times as required round the circle thus formed, 

 the polygon will be described. The example given in 

 the figure is a nonagon, or nine-sided figure. 



Fig. 59. 



