If, as in Fig. [249], we add to two sides of a square a symmetrical L-shaped portion, similar in shape to what we call a “carpenter’s square,” the resulting figure is still a square; and the portion which we have added is called, by Aristotle (Phys. III, 4), a “gnomon.” Euclid extends the term to include the case of any parallelogram[500], whether rectangular or not (Fig. [250]); and Hero of Alexandria specifically defines a “gnomon” (as indeed Aristotle implicitly defines it), as any figure which, being added to any figure whatsoever, leaves the resultant figure similar to the original. Included in this important definition is the case of numbers, considered geometrically; that is to say, the εἰδητικοὶ ἀριθμοί, which can be translated into form, by means of rows of dots or other signs (cf. Arist. Metaph. 1092 b 12), or in the pattern of a tiled floor: all according to “the mystical way of {510} Pythagoras, and the secret magick of numbers.” Thus for example, the odd numbers are “gnomonic numbers,” because
0 + 1 = 12 ,
12 + 3 = 22 ,
22 + 5 = 32 ,
32 + 7 = 42 etc.,
which relation we may illustrate graphically σχηματογραφεῖν by the successive numbers of dots which keep the annexed figure a perfect square[501]: as follows:
There are other gnomonic figures more curious still. For instance, if we make a rectangle (Fig. [251]) such that the two sides
| Fig. 251. | Fig. 252. |
are in the ratio of 1 : √2, it is obvious that, on doubling it, we obtain a precisely similar figure; for 1 : √2 :: √2 : 2; and {511} each half of the figure, accordingly, is now a gnomon to the other. Another elegant example is when we start with a rectangle (A) whose sides are in the proportion of 1 : ½(√5 − 1), or, approximately, 1 : 0·618. The gnomon to this figure is a square (B) erected on its longer side, and so on successively (Fig. [252]).
| Fig. 253. | Fig. 254. |