History of scientific ideas

by William Whewell

[1] Hist. Ind. Sc. b. xvii. c. vi.

It will of course be understood that by the term Symmetry I here intend, not that more indefinite attribute of form which belongs to the domain of the fine arts, as when we speak of the ‘symmetry’ of an edifice [68] or of a sculptured figure, but a certain definite relation or property, no less rigorous and precise than other relations of number and position, which is thus one of the sure guides of the scientific faculty, and one of the bases of our exact science.

2. In order to explain what Symmetry is in this sense, let the reader recollect that the bodies of animals consist of two equal and similar sets of members, the right and the left side;—that some flowers consist of three or of five equal sets of organs, similarly and regularly disposed, as the iris has three straight petals, and three reflexed ones, alternately disposed, the rose has five equal and similar sepals of the calyx, and alternate with these, as many petals of the corolla. This orderly and exactly similar distribution of two, or three, or five, or any other number of parts, is Symmetry; and according to its various modifications, the forms thus determined are said to be symmetrical with various numbers of members. The classification of these different kinds of symmetry has been most attended to in Crystallography, in which science it is the highest and most general principle by which the classes of forms are governed. Without entering far into the technicalities of the subject, we may point out some of the features of such classes.

The first of the figures (1) in the margin may represent the summit of a crystal as it appears to an eye looking directly down upon it; the center of the figure represents the summit of a pyramid, and the spaces of various forms which diverge from this point represent sloping sides of the pyramid. Now it will be observed that the figure consists of three portions exactly similar to one another, and that each part or member is repeated in each of these portions. The faces, or pairs of faces, are repeated in threes, with exactly similar forms and angles. This figure is said to be three-membered, or to have triangular symmetry. The same kind of [69] symmetry may exist in a flower, as presented in the accompanying figure, and does, in fact, occur in a large class of flowers, as for example, all the lily tribe.

The next pair of figures (2) have four equal and similar portions, and have their members or pairs of members four times repeated. Such figures are termed four-membered, and are said to have square or tetragonal symmetry. The pentagonal symmetry, formed by five similar members, is represented in the next figures (3). It occurs abundantly in the vegetable world, but never among crystals; for the pentagonal figures which crystals sometimes assume, are never exactly regular. But there is still another kind of symmetry (4) in which the opposite ends are exactly similar to each other and also the opposite sides; this is oblong, or two-and-two-membered symmetry. And finally, we have the case of simple symmetry (5) in which the two sides of the object are exactly alike (in opposite positions) without any further repetition.

3. These different kinds of symmetry occur in various ways in the animal, vegetable, and mineral kingdom. Vertebrate animals have a right and a [70] left side exactly alike, and thus possess simple symmetry. The same kind of symmetry (simple symmetry) occurs very largely in the forms of vegetables, as in most leaves, in papilionaceous, personate, and labiate flowers. Among minerals, crystals which possess this symmetry are called oblique-prismatic, and are of very frequent occurrence. The oblong, or two-and-two-membered symmetry belongs to right-prismatic crystals; and may be seen in cruciferous flowers, for though these are cross-shaped, the cross has two longer and two shorter arms, or pairs of arms. The square or tetragonal symmetry occurs in crystals abundantly; to the vegetable world it appears to be less congenial; for though there are flowers with four exactly similar and regularly-disposed petals, as the herb Paris (Paris quadrifolia), these flowers appear, from various circumstances, to be deviations from the usual type of vegetable forms. The trigonal, or three-membered symmetry is found abundantly both in plants and in crystals, while the pentagonal symmetry, on the other hand, though by far the most common among flowers, nowhere occurs in minerals, and does not appear to be a possible form of crystals. This pentagonal form further occurs in the animal kingdom, which the oblong, triangular, and square forms do not. Many of Cuvier’s radiate animals appear in this pentagonal form, as echini and pentacrinites, which latter have hence their name.

4. The regular, or as they may be called, the normal types of the vegetable world appear to be the forms which possess triangular and pentagonal symmetry; from these the others may be conceived to be derived, by transformations resulting from the expansion of one or more parts. Thus it is manifest that if in a three-membered or five-membered flower, one of the petals be expanded more than the other, it is immediately reduced from pentagonal or trigonal, to simple symmetry. And the oblong or two-and-two-membered symmetry of the flowers of cruciferous plants, (in which the stamens are four large and two small ones, arranged in regular opposition,) is held by botanists to result [71] from a normal form with ten stamens; Meinecke explaining this by adhesion, and Sprengel by the metamorphosis of the stamens into petals[2].

[2] Sprengel, Gesch. d. Bot. ii. 304.