identical with the larger one either by magnification all round (as in a), or simply by an increment at one end (as in b); indeed, in the case of the cone, we have yet a third possibility, for the same result is attained when it increases all round, save only at the base, that is to say when the triangular section increases {509} on two of its sides, as in c. All this is closely associated with the fact, which we have already noted, that the Nautilus shell is but a cone rolled up; in other words, the cone is but a particular variety, or ãlimiting case,ã of the spiral shell.
This property, which we so easily recognise in the cone, would seem to have engaged the particular attention of the most ancient mathematicians even from the days of Pythagoras, and so, with little doubt, from the more ancient days of that Egyptian school whence he derived the foundations of his learning5; and its bearing on our biological problem of the shell, though apparently indirect, is yet so close that it deserves our further consideration.
Fig. 249. Fig. 250.
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 ãcarpenterãs square,ã the resulting figure is still a square; and the portion which we have added is called, by Aristotle (Phys. III, 4), a ãgnomon.ã Euclid extends the term to include the case of any parallelogram6, whether rectangular or not (Fig. 250); and Hero of Alexandria specifically defines a ãgnomonã (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 öçÃ¥¯öÇöñüö¿ö¤ö¢Ã§Ñ Ã¥üö¿ö¡ö¥ö¢ö₤, which can be translated