MATHEMATICS. PRACTICAL GEOMETRY. [PROBLEM xxv. 



Fig. 47. 



strai-jht line (A B) in a girtn point (E), and ihtill also 

 touch a given arc (D U 1 1 . 



Draw E C (Fig. 46) per- 

 pendicular to A B ; and 

 through the centre K of 

 H the given arc DG H, draw 

 K L parallel to E C, and 

 meeting the are D G H 

 continued in the point L ; 

 join L K. and produce it 

 to meet the given arc in 

 the point D, and join 1)K, 

 cutting EC in O ; then 

 with the centre O, and 

 radius O E, describe a part of a circle which will touch 

 the given line A B in the given point E, and also the 

 given arc D G U in the point D. 



PROBLEM XXV. 



To describe two arcs that shall meet each other in the line 

 of their centres, and shall touch two given straight lines 

 (A B, C D) at the given points (E, F) in those lines, the 

 radius (E G) of the lesser arc being also given. 



From the point E (Fig. 47) 

 draw KG perpendicular to 

 A B, and equal to the given 

 radius, and from the point F 

 draw F H perpendicular to 

 C D, aud make it equal to 

 E G ; join GH ; bisect G H 

 by a perpendicular K. O, cut- 

 ting F H produced, in the 

 point O ; join O G, and pro- 

 duce it, then with O as a 

 centre, and radius O F, de- 

 scribe an arc meeting O G pro- 

 duced in L. With the centre 

 G, and radius GE or GL, 

 describe an arc meeting the 

 former arc in L ; then the arcs F L and E L are those 

 required, meeting each other at the point L in the line 

 L O, passing through both their centres G and O, and 

 also touching A I) and C D in E and F. 



PROBLEM XXVI. 

 Tu find a straight line nearly equal to the length of a given 



arc (A B C). 



Fig. 4g. Bisect the given arc in 



B B (Fig. 48), and draw the 



lines AB, AC; with the 

 centre A, and radius A B, 

 describe an arc cutting 

 A C in D ; make D E in 

 A C produced, equal to 

 A D, divide C E into three equal parts, and make E F 

 equal to one of the parts ; then A F will be nearly equal 

 to the arc A B C. 



PROBLEM XXVIL 



To draw a straight line 

 nearly equal to the semi- 

 circumference of a given 

 eircfe(ADBC). 

 Draw the diameters A B 

 and C D (Fig. 49) at right 

 angles to each other ; pro- 

 duce CD until DE is 

 equal to three-fourths of 

 D O ; through C draw F G 

 parallel to A B ; join E A, 

 E B, and produce them 

 until tljey meet F G in F 

 and G ; then F G will be nearly equal to the semi-cir- 

 cumference of the circle A D B C. 



PROBLEM XXVIII. 



To intfrihf a circle in a given triangle (A B C) that shall 

 touch all its iide. (Euclid, Book IV., Prop. IV.) 

 Produce A C (Fig. 60) both ways ; with the centre A, 



and any radius, describe an arc cutting A C produced in 



D, and A B in E ; 

 also with the centre 

 C, and the same 

 radius, describe an 

 arc cutting AC pro- 

 duced in F, and 

 BCinG; joinDE 

 and F G, and draw 

 through A and C 

 lines parallel to DE 

 and F G respec- 

 tively, and cutting each other in the point O ; then O is 

 the centre of the required circle (the lines A and O C 

 will bisect the angles BAG and A C B, which might be 

 done by Problem XII.) From Odraw O H perpendicular 

 to A C ; then with the centre O, aud radius O H, describe 

 a circle which will touch all the sides of the given tri- 

 angle ABC. 



PROBLEM XXIX. 



To describe a circle about a given triangle (A B C). 



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



This Problem is simply to describe a circle passing 

 through three given points, A B C, as given before in 

 Problem XVIII, 



PROBLEM 



To draw a square within or about a given circle (A C B D). 

 (Euclid, Book IV., Props. VI. and VII.) 



Draw through the centre O (Fig. 

 61) any two diameters, A B and C D, 

 at right angles to each other, meeting 

 the circle in the points A C B D. To 

 inscribe a square in the circle, join 

 those four points ; then A C B 1) is 

 the square required ; and to describe 

 a square about the circle, draw tan- 

 gents or lines parallel to the diame- 

 ters at those four points, then EFHG 

 is the square required. 



PROBLEM XXXI. 

 circumscribe and inscribe circles to a given 



To 

 (ABC D). 



.. square 



(Euclid, Book IV., Props. VIIL and IX.) 



Fig. 52. Draw the diagonals, A C, B D 



(Fig. 62), intersecting in the 

 point O, the centre of the re- 

 quired circles. With the centre 

 O, aud radius O A, describe a 

 circle which will pass through 

 A, B, C and D, and be circum- 

 scribed about the given square 

 A B C D. From O, draw O E 

 perpendicular to DC; and 

 again with tlie centre O, and 

 radius O E, describe a circle which will touch all the four 

 sides of the square, and bo inscribed in the given square 

 A BCD. 



PROBLEM XXXIL 



To describe a pentagon on a given straight line (A B) 

 (Euclid, Book IV., Props. XI. and XII.) 



IST METHOD . Bisect A B in 

 C (Fig. 63), and from B draw 

 B D perpendicular to A B, and 

 equal to A C or B C ; join A D 

 and produce it ; with the centre 

 D and radius D B describe a 

 circle cutting A D produced, in 

 E ; with the centres A and B 

 and radii, each equal to A E, 

 describe arcs intersecting in F ; 

 lastly, with the centres A and 

 F, and radii, each equal to A B, 

 describe arcs intersecting in G, 

 and with B and Fas cciitr.s, 

 and the same radii, describe arcs 



intersecting in H ; join A G, F G, F U, aud B H ; then 

 A B H F G is the pentagon required. 



Fig. 5S. 



