PROFESSOR KELLAND'S PROBLEM ON SUPERPOSITION. 311 



IX. 



To cut a rectilineal figure, A, so as to be similar to another rectilineal figure, B. 



It is manifest that all the preceding processes are reversible. By Euclid, vi. 25, con- 

 struct C = A and similar to B, cut A and C into squares, and from the square obtained 

 from A, which is = that obtained from C, work backwards to C from the square derived 

 from C. 



Examples. 



To cut a A ABC (fig. 9) into another, IKL, with same base and height, and with 

 the L K = a given angle or side IK = a given line not less than the height of ABC. 



Bisect the sides in D,E, cut DE, and complete parallelogram DC. Draw BF, making 

 L FBC = K, or the line BF = ^IK, complete the parallelogram HB. Bisect FH in G, cut 

 from C to G, turn over AGCH, and IKL is obtained. Or bisect the sides, (fig. 10), 

 in D and E. Through D draw MDN, making the required angle at N, or the line 

 DN = ^ given side, draw AM II BC, draw MEF through E, and proceed as in the 

 figure, and IKL is obtained. This illustrates Euclid, i. 38, by superposition. 



To cut a A ABC, (fig. 11), into pieces, so as to recompose another Aabc, of equal 

 area. Cut off ADE, ade, and form the parallelograms DC, dc. Cut parallelogram DC 

 into dc, as in Case 4, proceed as indicated in the figures, and the pieces obtained in ABC 

 compose abc. 



To cut a triangle into a square. Cut into a rectangle, (fig. 8), and then into a 

 square (figs. 1 and 2). 



To cut a regular pentagon into a square (fig. 12). Cut off ECD and place it at BCF, 

 bisect BF in H, draw KHL II AE, and the parallelogram EK is obtained = pentagon. 

 Insert EM = side of required square, complete the parallelogram LMNE and the square 

 EP, &c. The six pieces of the pentagon compose the square EP. 



To cut a regular hexagon, ABCDEF, (fig. 13), into a square. Cut off AFED and 

 place it at CDHG. Insert AI = side of required square. Draw HK || AI, and com- 

 plete the square HL, &c. The five pieces of the hexagon compose the square. 



VOL. XXXVI. PART II. (NO. 12). 3 B 



