BIATHEMATICS. PRACTICAL GEOMETRY. [PROBLEM xm.xix. 



I :- ML 



PROBLEM XIIL 



To trisect or divide into thrtt equnl angles a given right 

 angle (A Bf). 



With the centre B (Fig. 29) and 

 any radius, describe an arc, cutting 

 11 A in 1), and B C in E, and from 

 the |K>iuU D and E at centres, and 

 he same radius as before, describe 

 cutting the former arc in F 

 ; join B and the points of 

 Intersection F and G ; then the an- 

 gle A B C is divided into three equal 

 angles, ABG, DBF, FBC. 



PROBLEM XIV. 



To tritcci or divide into three equal angles any given angle 



(A B C). 



f 'f- *> With the centre B (Fig. 30) 



and any radius, describe a 

 circle cutting B A in D, and 

 B C in E ; bisect the angle 

 A B C by the line B L (Pro- 

 blem XII.), and produce A B 

 till it meets the circle in G ; 

 on the edge of the ruler mark 

 off a distance (bd) equal to 

 the length of the radius B D 

 or B E, and lay the ruler at 

 the point G, moving it until 

 the point 6 cuts the circle, 

 and the point d intersects 

 the line B L, in H and F 

 respectively : join B H, and 

 set off HK equal to EH, 

 leaving KD also equal to 

 E H, and join B K ; then the 



angle ABC will be divided into three equal angles, 



A B K, K B H, and H B C. 



PROBLEM XV. 



To describe an equilateral triangle 

 worn a given straight line ( AB). 

 (Euclid, Book I., Prop. I.) 

 With the centres A and B 

 (Fig. 31) and radii equal to 

 A B, describe arcs intersecting 

 in the point C ; join A C and 

 B C ; the triangle ABC will 

 be equilateral. 



"L 



PROBLEM XVI. 



To construct a triangle, whose side shall be respectively 

 equal to three given straiyht lines (1, 2, 3), any two of 

 which are greater than the third. (Euclid, Book I.. 

 Prop. XXII ) 



Draw any straight line, Fig. 32. 



AB, (Fig. 32) equal to 

 the given line 1, with the 

 centre A and a radius 

 equal to the given line 2 

 describe an arc, and with 

 the centre B, and a radius 

 equal to the given line 3, 

 describe another arc, cut- 

 ting the former at the ' ".'' 



nil ' JO '"J^ C aud B : then ABC will be the 

 triangle required. 



PROBLEM XVII. 

 To find Oif ffntre mul rarfins n f a q h tn circle.* 



(Euclid, Book III., Prop. I.) 

 : METHOD. In the circumference of the given circle 



ABC (Fig. 33), take any point D, and from it as a 

 c, -litre, with any radius, describe 

 a circle cutting the circmu: 

 uf the givi-n circle in the points 

 E and F, with which as centres, 

 and the same radius, describe 

 arcs intersecting the circle K F 

 in the points G H and K Ij. 

 Draw lines pasting through the 

 points H and G, L and K, and 

 produce them until they intersect 

 each other in the point O, which 

 is the centre of the circle, aud 



O M or O N the radius. 



By this method the arcs D E and D F are bisected, and 

 the problem may be performed by merely drawing any two 

 chords, such at D E and D F, bisecting those chords by per- 

 pendiculars, which being produced, until they intersect each 

 otlier, witt give the centre. 



2.VD METHOD. Take any five points, B, E, D, G, C 



Fig. 3. 



(Fig. 34), in the circumference of 

 the given circle ABC, equidistant 

 from each other ; draw lines pass- 

 ing through B E and G D, meet- 

 ing in the point K ; also through 

 C G and E D, meeting in the 

 point L ; join E G, and draw B D 

 and C D, cutting the line E G in 

 M and N respectively ; and 

 through the points of intersection 

 K M, and L N, draw lines, and 

 produce them until they intersect 

 each other in the point 0, which is the centre of the 

 given circle ABC. 



PROBLEM XVIII. 



To describe a circle that shall pass through any three given 

 points (ABC), which are not in a straight line. 



From the point A (Fig. 35), as a centre, with any 



Fig. 35. 



convenient radius greater 

 than half the distance AB 

 or A C, describe an arc ; 

 with B and C as centres, 

 and the same radii, de- 

 scribe arcs cutting the 

 former in the points D, F 

 and E G ; draw lines pass- 

 ing through those points 

 intersecting each other in 

 the point O, and with the 

 centre O, and a radius equal to the distance O A, OB, or 

 O C, describe a circle which will necessarily pass through 

 the three given points A, B, and C. 



This problem will be found most applicable in the 



Fig. 30. 



case of a circular arch, 

 whoso span (B C) and 

 rise (AH) (Fig. 36) 

 are given, and it is 

 rei|iiired to describe it; 

 join AB and AC, 

 bisect them by the per- 

 pendiculars D F and 

 E G ; then, as before, 

 the point of intersec- 

 tion O is the centre 



with which to describe the circle, t 



The joints of the stones or vmtssoirs forming the arch 

 are drawn from the centre, and form continuations of 

 the radii. 



PROBLEM XIX. 



To find the point in a, given straight line (AB), <lrnn-n 

 from the one extremity of a given arc (A EC), ihrmigh 

 ii-hirh the other extremity (C) of the given air, if 

 Untied, would pass, and without using the centre. 



From A (Fig. 37) draw any chord A D, and also any 



+ Sw Euclid, Book III., Prop. XXV. 



