‘The rigid test of mathematical calculation’ is then applied to the case of mimetic butterflies, with the view of showing that they could not have been produced simply according to the laws of variation, inheritance, and natural selection. In the application of this rigid test, the very first step is a perfectly gratuitous assumption, ‘that it would require, at the very lowest calculation, one thousand steps to enable the normal _Leptalis_ to put on its protective form.’ Who is to prove that fifty differences would be insufficient? An interval of a thousand years might be granted for establishing each one of these variations. Suppose even fifty thousand, instead of only fifty steps, to be necessary, it is another gratuitous assumption that ‘the smallest change in the direction of the _Ithomia_, which we can conceive on any hypothesis to be beneficial to the _Leptalis_, is at the very lowest one-fiftieth of the change required to produce perfect resemblance.’ How small a difference must decide the choice made by a donkey placed equidistant between two bundles of hay! Certainly, then, a bird on the wing, having to choose amidst myriads of butterflies, may be determined by an almost infinitesimal distinction. Further, though the whole change may be produced by an immense number of small changes, it is not necessary to suppose that all the changes will be equally small. It is merely begging the question to assume that the first change could not possibly be large enough to be of any use. And if it may be of use, the whole mathematical calculation, based on its being useless, breaks down from the beginning. Again, since the _Leptalis_ may have spent one million years in arriving at its present likeness to the present _Ithomia_, it is impossible to assert that the normal forms of the two butterflies were as wide apart at the beginning of that period as they are at present. The mimicry having once set in, might be retained by parallel variations. This, indeed, cannot fail to be the case, if the protection is to be a lasting one; for when the _Ithomia_ varies in outward appearance, unless the _Leptalis_ varies in the same direction, the resemblance will be lost. This progressive mimicry would be more valuable than an imitation in which no changes occurred, since the enemies of a mimetic species would in time become aware of a fraud which had no variations at its command, as birds are said now-a-days to pounce without hesitation upon caterpillars which very much resemble twigs[67]. Even ‘a rough imitation’ may be useful in the first instance, and yet when hostile eyes have long been exercised, and have acquired greater and greater sharpness, finally nothing less than _absolute identity_ of appearance may be thoroughly effective. Thus the perfecting of the resemblance will be no ‘mere freak of Nature,’ nor shall we be ‘landed in the dilemma that the _last_ stages are comparatively useless’ in this procedure.