great mathematician, in a lecture delivered at the Lowell Institute last winter, showed that in his science, also, similarity of outline does not always indicate identity of character.  Compare the different circles,—­the perfect circle, in which every point of the periphery is at the same distance from the centre, with an ellipse in which the variation from the true circle is so slight as to be almost imperceptible to the eye; yet the latter, like all ellipses, has its two foci by which it differs from a circle, and to refer it to the family of circles instead of the family of ellipses would be overlooking its true character on account of its external appearance; and yet ellipses may be so elongated, that, far from resembling a circle, they make the impression of parallel lines linked at their extremities.  Or we may have an elastic curve in which the appearance of a circle is produced by the meeting of the two ends; nevertheless it belongs to the family of elastic curves, in which may even be included a line actually straight, and is formed by a process entirely different from that which produces the circle or the ellipse.

But it is sometimes exceedingly difficult to find the relation between structure and form in Families, and I remember a case which I had taken as a test of the accuracy of the views I entertained upon this subject, and which perplexed and baffled me for years.  It was that of our fresh-water Mussels, the Family of Unios.  There is a great variety of outline among them,—­some being oblong and very slender, others broad with seemingly square outlines, others having a nearly triangular form, while others again are almost circular; and I could not detect among them all any feature of form that was connected with any essential element of their structure.  At last, however, I found this test-character, and since that time I have had no doubt left in my mind that form, determined by structure, is the true criterion of Families.  In the Unios it consists of the rounded outline of the anterior end of the body reflected in a more or less open curve of the shell, bending more abruptly along the lower side with an inflection followed by a bulging, corresponding to the most prominent part of the gills, to which alone, in a large number of American Species of this Family, the eggs are transferred, giving to this part of the shell a prominence which it has not in any of the European Species.  At the posterior end of the body this curve then bends upwards and backwards again, the outline meeting the side occupied by the hinge and ligament, which, when very short, may determine a triangular form of the whole shell, or, when equal to the lower side and connected with a great height of the body, gives it a quadrangular form, or, if the height is reduced, produces an elongated form, or, finally, a rounded form, if the passage from one side to the other is gradual.  A comparison of the position of the internal organs of different Species of Unios with the outlines of their shells will leave no doubt that their form is determined by the structure of the animal.