292 REV. T. p. KIBKMAN ON LINEAR CONSTRUCTIONS. 



or, taking for our axes of x and y, the lines (34) and (32), 

 putting thus X3 =: 1/3 =z Xi =2 yiZZx^z=i a:J = ye = y? = Oj 



(y . x-x — yx) (ti . x^)(f^ , x) (58) — (y .x^—yx) (y . x^) (pg . x) (57) =r. « =- o, 

 o a a ae 2 C 8 8 1 4 c/ e; a a e/ 2 S 7 7 1 4 



is the line which is to be drawn ; where 

 a is the point 12 . 45, c is the point 23 .58, e is 34 . 81 

 ci 23 . 57, ^1 is 34 . 71 



Let the co-ordinates of the eight points be drawn, which 

 amount to twelve lines besides the axes, and on these let 

 any two positive lengths from the origin, as y^ rz. k^ x^zn 

 m, be assumed. 



Let that diagonal of any parallelogram made by the co- 

 ordinates produced, which produced cuts from the axes seg- 

 ments of like sign, be called positive, and negative when 

 those signs are unlike. 



The rhomboid k . x^ — x^ is a determined portion of the 

 figure ; and has a side w^ — x^ in the axis of a;. Draw that dia- 

 gonal d of it which has the sign oixe-x^, and parallel to d from 

 c, draw cAto A in the axis of a;; then is 3/0. (a?, — x^):=zk, OA. 

 Draw the ordinate AAi, meeting a?, in Ai; draw the 

 diagonal di of the rhomboid k x„ which differs in sign firom 

 x^ and parallel to di draw A,X to X in the axis of x : then 

 is X'. AX zz. — y^Xc and ^.OX zz y^ . {x^-x^ — y^ x^. Draw 

 next J2 the diagonal of ^ . (x^ — x^) like-signed with x^ — Xs, and 

 parallel to d.^ draw from 2 to Xi in the axis of x the line 

 2 X, : y, .{x,'-'X,)=.K.OXi. Parallel to 24 draw a line Oi B, 

 meeting on the axis of y at O, one of the abscissoe x^ and Xi 

 so as to cut the other in B on the positive or negative side 

 of that axis, according as y^ — yi is positive or negative. The 

 ordinate BX, will cut off OX2 such that K, OX2 zz y^ — y, 

 . ^4. Thus by drawing the ten lines d^ c Ay AAxy di, AiX, d„ 

 2X1, 24, OiB, BXif we have effected the reduction 

 (y . x — X — yx) (y . x — x) y — y . x zz ^ OX OX OX. 



ceaae3588Il 13 



Join now 2 X„ and draw the lines Xk^^ k,X', X'k^ k„X" ; 

 'Xk, and X'k' both parallel to 24, and hX' and k^^X" in ordex 



