rich in intellectual treasures. Chasles, in his History, says that the
cycloid interweaves itself with all the great discoveries of the
seventeenth century.

A curve which fulfils more perfectly the demands of our _dictum_ is
that of an elastic thread, to which we have already alluded. If the two
ends of a straight steel hair be brought towards each other by simple
pressure, the intervening spring may be put into a series of various
forms,--simple undulations, and those more complicated, a figure 8,
loops turning alternately opposite ways, loops turning all one way, and
finally a circle. Now the whole of this variety is the result of
subjecting each part of the curve to a law more simple than that of the
cycloid. The elastic curve is a curve which bends or curves exactly in
proportion to its distance from a given straight line. According to the
canon, therefore, this curve should be beautiful; and it is
acknowledged to be so in the examples given by the bending osier and
the waving grain,--also by the few who have seen full drawings of all
the forms. And the mathematician finds in it a new beauty, from its
marvellous correspondence with the motions of a pendulum,--the
algebraic expression of the two being identical.

The forms of organic life afford, however, the best examples of the
dominion of our fundamental canon. The infinite variety of vegetable
forms, all beautiful, and each one different in its beauty, is all the
result of simple laws. It is true that these simple laws are not as yet
all discovered; but the one great discovery of Phyllotaxis, which shows
that all plants follow one law in the arrangement of their leaves upon
the stem, thereby intimates in unmistakable language the simplicity and
unity of all organic vegetable laws; and a similar assurance is given
by the morphological reduction of all parts to a metamorphosed leaf.