MR ROBERT BRODIE ON 



IV. 



To cut a parallelogram, ABCD, (fig. 6), into pieces so as to recompose into a parallelo- 

 gram, OCFH, similar to a given parallelogram. Find by usual geometrical construction, 

 Euclid, vi. 25, the length of the sides OC, CF. From B or C insert BE = CF, one of 

 the above-found sides, produce AE to F, draw CF II BE, and the parallelogram is obtained 

 having CF = one of the required sides. If E comes on one of the sides AD or DC, then 

 this first step is done with one cut, BE. Next, taking CF as the base of the parallelogram 

 EBCF, make i FCO = one of the given angles, or insert CO = the other required side, 

 and complete the parallelogram OCFH, which is = ABCD, and has the required sides. 

 As the areas are equal, the angles at and H must be = those of the given parallelo- 

 gram. Hence a parallelogram can be cut into a square. 



V. 



To cut a triangle ABC, (fig. 7), into a parallelogram, or vice versa. Bisect AB, AC, in 

 D and E, cut in line DE, and place the triangle to right or left as in the figure, and FBCE 

 and DBCG are obtained. There are evidently six solutions — two for each side of the 

 triangle. The cutting a parallelogram into a triangle is manifest. 



VI. 



To cut a A ABC, (fig. 8), into a rectangle. Bisect AB, AC in D and E, and proceed 

 as in the figure, and FBCG, FHKG are obtained. There are six solutions for an acute- 

 angled triangle, but only two in an obtuse-angled triangle, and four for a right-angled 

 triangle. 



VII. 



To cut any rectilineal figure into a parallelogram with given side and given angle. 

 Divide the figure into triangles, cut each triangle into a parallelogram, each parallelogram 

 into another with given angle (Case 3), and this parallelogram into another with the 

 given side (Case 1) ; then place all the parallelograms together, and it is done. N.B. — 

 This is nearly the same as Euclid, i. 45. 



VIII. 



To cut a rectilineal figure into a square. Ascertain the length of the side of the 

 square, cut the figure into triangles, each triangle into a rectangle, and each rectangle 

 into another whose length = side of the square, place them together and the square 

 must be produced. 



