( 471 ) 



XXXIII. — On Superposition. By the Rev. Philip Kelland, M.A., F.R.S., Pro- 

 fessor of Mathematics in the University of Edinburgh. Part II. (Continued 

 from Vol. XXI. p. 273.) (Plate XX.) 



(Bead 7tli March 1864.) 



In my former paper on the subject, I selected the following problem : — 



From a given square, one quarter is cut off, to divide the remaining gnomon 

 into four such parts that they shall he caipable of forming a square. 



The gnomon is, I assume, incapable of being formed into a square by being cut 

 into three parts, and consequently the number of different ways in which it can 

 be so formed, by cutting it into four parts, must be very limited. But, to show the 

 fertility of the method of superposition, I exhibited the solution of the problem in 

 twelve different manners. Many of these, no doubt, have much that is in com- 

 mon, whilst, on the other hand, some (such as the 12th) differ in every feature from 

 the rest. I had thoughts of following up my plea for the study of the old geo- 

 metry, by exhibiting the solutions of the 47th proposition of Euclid's first book 

 in their beautiful variety. I have indeed temptation to do so. The modifica- 

 tion which I gave of the demonstration of this proposition in the notes to my 

 edition of Playfair's Geometry (edition 1846, p. 273), has had the honour of being 

 exhibited in two different mechanical forms. The first by two rotations without 

 sliding, whereby the two squares on the sides, when placed together, are con- 

 verted into the square on the hypothenuse ; the second, by two transpositions 

 (slidings) without rotation, whereby the same change is effected The former is 

 obvious enough, and could have escaped nobody. The latter is described by 

 Professor De-Morgan in the " Quarterly Journal of Mathematics," vol. i. p. 236. 



I venture, however, to think that the problem before us is still more curious, 

 as an exemplification of the method of superposition, than the 47th of Euclid's 

 first book. With this feeling I have overcome the hesitation I long experienced 

 in presenting the following twelve additional solutions to the Society. The solu- 

 tions are numbered in continuation of my former paper, and the values of x and 

 a are the same as in that paper. 



Constructions and Demonstrations. 



XIII. — BY = 57, EG = «— «. Place (3) and (4) on (1) as in the second figure, 

 they will just fill the diagonal of (1). And since the remaining portion of the first 

 figure is a rectangle, whose sides are a and x—a, it exactly completes the second 

 figure ; hence the conclusion. 



VOL. XXIII. PART II. 6 M 



