XXXIII.—On Superposition

Extract

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 be capable 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 common, 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 geometry, 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 modification 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 converted 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.