which have their definite signification flowing from the rules of the notation.
But Professor Miller, who has treated the subject of Crystallography in the most general and symmetrical manner, adopts the plan of marking each crystalline plane by three numerical indices. Thus in the Octahedral System, the cube is {100}; the octahedron is {111}; the rhombic dodecahedron is {011}; the pentagonal dodecahedron is π {012}; where π indicates that the form is not holohedral but hemihedral, only half the number of faces being taken which the law of derivation would give. This system is the most mathematically consistent, and affords the best means of calculation, as Professor Miller has shown; but there appears to be in it this defect, that though an essential part of the scheme is the division of crystalline forms into Systems,—the Octahedral, Pyramidal, Rhombohedral and Prismatic,—this division does not at all appear in the notation.
But whatever be the notation which the crystallographer adopts, it is evident that he must employ some notation; and that, without it, he will be unable to express the forms and relations of forms with which he has to deal.
2. Chemistry.
The same has long been the case in Chemistry. As I have stated elsewhere[67], the chemical nomenclature of the oxygen theory was for a time very useful and effective. But yet it had defects which could not be overlooked, as I have already stated under [Aphorism II.] The relations of elements were too numerous, and their numerical properties too important, to be expressed by terminations and other modifications of words. Thus the compounds of Nitrogen and Oxygen are the Protoxide, the Deutoxide, Nitrous Acid, Peroxide of Nitrogen, Nitric Acid. The systematic nomenclature here, even thus loosely extended, does not express our knowledge. And the Atomic Theory, when established, brought to view numerical 360 relations which it was very important to keep in sight. If N represents Nitrogen and O Oxygen, the compounds of the two elements just mentioned might be denoted by N + O, N + 2O, N + 3O, N + 4O, N + 5O. And by adopting a letter for each of the elementary substances, all the combinations of them might be expressed in this manner.
[67] Hist. Ind. Sc. b. xiv. c. 6.
But in chemistry there are different orders of combination. A salt, for instance, is a compound of a base and an acid, each of which is already compound. If Fe be iron and C be carbon, Fe + O will be the protoxide of iron, and C + 2O will be carbonic acid; and the carbonate of iron (more properly carbonate of protoxide of iron), may be represented by
(Fe + O) + (C + 2O)
where the brackets indicate the first stage of composition.
But these brackets and signs of addition, in complex cases, would cumber the page in an inconvenient degree; and oxygen is of such very wide occurrence, that it seems desirable to abridge the notation so far as it is concerned. Hence Berzelius proposed[68] that in the first stage of composition the oxygen should be expressed by dots over the letter; and thus the carbonate of iron would be Ḟe + C̈. But Berzelius further introduced into his notation indexes such as in algebra denote involution to the square, cube, &c. Thus Cu being copper, the sulphate of copper is represented by S⃛²C̈u. This notation, when first proposed, was strongly condemned by English chemists, and Berzelius’s reply to them may be taken as stating the reasons in favour of such notation. He says[69], ‘We answer to the opponents, that undoubtedly the matter may be looked at in various lights. The use of Formulæ has always, for a person who has not accustomed himself to them, something repulsive; but this is easy to overcome. I agree with my opponent, 361 who says that nothing can be understood in a Formula which cannot be expressed in words; and that if the words express it as easily as the Formula, the use of the latter would be a folly. But there are cases in which this is not so; in which the Formula says in a glance what it would take many lines to express in words; and in which the expression of the Formula is clearer and more easily apprehended by the reader than the longer description in words. Let us examine such a Formula, and compare it with the equivalent description in words. Take, for example, crystallized sulphate of copper, of which the Formula is