460 Notices respecting New Books. 



example, are grossly erroneous, viz. the propositions relating 

 to the constructions of figures 34, 37, 41, 43, 45, 48, 50, 52, 

 and 53. 



The propositions relating to figures 34 and 53 profess to give a 

 general method of constructing regular polygons. Now the method 

 fails in evo'y case excejDting for the triangle, square, and hexagon ; 

 it even fails for the polygon given by the author as an example. 

 The following is Mr. Burchett's proposition, given at page 22 of his 

 work : — 



"To construct any regular polygon, the circumscribing circle 

 being given. 



" Fig. 34. The same — another method. 



"Draw a diameter, AB, of the given circle. Divide AB into as 

 many equal parts as the polygon is required to have sides. From A 

 and B, with the line AB as radius, describe arcs, cutting in C. Draw 

 a line from C through the second division (E) of the diameter, and 

 produce it, cutting the circle in D. BD will be the side of the 

 required polygon. Set off BD round the circumference of the circle, 

 and points for the angles of the polygon will be obtained. Note. — 

 Whatever number of sides the polygon may have, the line from C 

 must always be drawn through the second division of the dia- 

 meter." 



In order to show the error of this proposition, let O be the centre 

 of the circle, DK a perpendicular let fall on AB, n = the number of 

 sides of the polygon, = ZBOD, the angle which the chord BD 

 subtends from the centre O, and radius circle =1. Then we get 

 from the similar triangles OCE and EKD, 



OCxEK=OExDK. 

 But 



OC=tan60°=A/3; OE=I--; DK=sine; EK=cos0-A \ 



n \ ny 



.■.V3{c„,e-(i-i)}=(i-l)si„», 



Whence we find 



i-i 



fi^A'W'-'Hy} 



Now, if the construction be correct, we should always have m0=36O°; 

 and if it be erroneous, the error, or the number of degrees which the 

 last side overlaps or falls short of the first, will be expressed by 

 «e-360°. When m=12, we find 0=30° 28' nearly, and nd-860'' 

 = 5° 36', which is the overlap or error in this case. And so on to 

 other cases, the error increasing with the number of sides. 



