720 



PRACTICAL DIAGRAMS. 



Fig. 1016. 



the cross-staff at b, with one of the sights 

 pointing to a, make b r perpendicular to 

 a b, and measure its length. Then, re- 

 moving the cross-staff to c, make c r per- 

 pendicular to b c, and equal to b r ; and 

 make the line b r d equal to a r c. Then 

 dis a point in the curve; and in the same 

 manner other points may be found suc- 

 cessively. 



Fig. 1016 differs from the above only 

 in this, that the angles are taken outside. 



Set up three pegs, say 

 fifty links apart, as be- 

 fore, and fix the cross- 

 staff in r, with one sight 

 on the line r b a, and 

 the other pointing to c. 

 Then measure r b and 

 r c, and remove to the 

 line e c b ; draw e c 

 equal to r b, and e d 

 equal to r c, and so on. 

 The same end may be 

 obtained by a theodo- 

 lite, or by any other 

 instrument for taking 

 angles ; or even with 

 three needles stuck in 

 a board, forming the 

 requisite oblique angle. 

 Setting the instrument in b, fifty links 

 from a, with one leg of the angle on 

 the line b a, and by the other peg direct- 

 ing an assistant to place the peg c at the 

 distance of fifty links ; then remove to c, 

 and so on. 



To find the centre of a circle, whose cir- 

 cumference will 

 pass through 

 three given 

 points (not in a 

 straight line,) 

 connect the 

 three points a 

 b c together ; 

 from the middle 

 of each, erect 

 lines perpendi- 

 cular to them, 

 and where these 

 perpendiculars cut each other is the 

 centre required. 



To find the centre of a circle. — Con- 

 nect three points in the circumfer- 

 ence, and from the middle of the two 

 lines erect perpendiculars ; where these 



intersect each other is the centre re- 

 quired. 



To construct a hexagon. — Divide the 

 circle into three equal parts; from the 

 middle of each line erect a perpendicular ; 

 and where these cut the circumference 

 of the circle are the points where the 

 sides of the hexagon meet. 



To construct an octagon. — Divide the 

 circle into four equal parts, by describing 

 a square within it ; erect perpendiculars 

 from the middle of each side of the 

 square ; and where they intersect the 

 circle are the points where the sides of 

 the octagon meet. 



To construct a pentagon. — Draw a line 

 through the centre of the circle, from the 

 centre of which erect a perpendicular, 

 c d ; divide the 

 Fig. 1018. straight line 



^ from c to b 



into two equal 

 parts; take e d 

 as a radius, and 

 describe a cir- 

 cle, making e 

 the centre, and 

 when that cir- 

 cle cuts the 

 straight line at / the distance from / 

 to d is the length of the side of the 

 pentagon. 



To describe a circle the centre of which 

 is occupied with a square, say the base 

 of the pedestal of a statue, fountain, &c. — 

 Tie a cord round the square, not over 

 tight; to that attach a line, in length 

 equal to the radius, minus half the size of 

 the square base ; with that line describe 

 the circle. — This is a plain working plan, 

 and near enough for all practical pur- 

 poses in laying out grounds. The same 

 rule may be applied when the base is 

 circular, or of any equal-sided figure, a 

 pentagon, hexagon, &c. 



To describe a circle when the base of 

 the fountain, statue, &c. is oblong. — Lay 

 the oblong correctly down on paper; find 

 its centre, by drawing two lines diagonally 

 through it ; from that describe a circle 

 of any size; draw two lines across the 

 circle parallel to the longest sides of the 

 oblong figure ; from these erect perpen- 

 diculars, at equal distances, and note their 

 respective lengths ; on the ground draw 

 two lines parallel to the longest sides of 



