the simple hypothesis that the elasticity and approximation of the particles vary in the directions of the crystalline axes allows of the application of deductive reasoning. The whole of the phenomena are gradually being proved to be consistent with this hypothesis, so that we have in this subject of crystallography a beautiful instance of successful classification, connected with a nearly perfect physical hypothesis. Moreover this hypothesis was verified experimentally as regards the mechanical vibrations of sound by Savart, who found that the vibrations in a plate of biaxial crystal indicated the existence of varying elasticity in varying directions.

Classification an Inverse and Tentative Operation.

If all attempts at so-called natural classification be really attempts at perfect induction, it follows that they are all subject to the remarks which were made upon the inverse character of the inductive process, and upon the difficulty of every inverse operation (vol. i. pp. 14, 15, 140, &c.). There will of necessity be no royal road to the discovery of the best system, and it will even be impossible to lay down any series of rules of procedure to assist those who are in search of a good arrangement. The only invariable logical rule which could be stated would be as follows:--Having given certain objects, group them in every way in which they can be grouped, and then observe in which method of grouping the coincidence. of properties is most conspicuously manifested. But this method of exhaustive classification will in almost every case be impracticable, owing to the immensely great number of modes in which a comparatively small number of objects may be grouped together. About sixty-three elements have been classified by chemists in six principal groups as Monad, Dyad, Triad, &c. elements, the numbers

in the classes varying from three to twenty elements. Now if we were to calculate the whole number of ways in which sixty-three objects can be arranged in six groups, we should find the number to be so great that the life of the longest lived man would be wholly inadequate to enable him to go through these possible groupings. The rule of exhaustive arrangement, then, is absolutely impracticable. It follows also that mere haphazard trial cannot as a general rule give any useful result. If we were to write the names of the elements in succession upon sixty-three cards, throw them into a ballot-box, and draw them out haphazard in six handfuls time after time, the probability is excessively small that we take them out at any one trial in a specified order, for instance that at present adopted by chemists.

The usual mode in which an investigator proceeds to form a classification of any new group of objects, seems to consist in tentatively arranging them according to their most obvious similarities. Any two objects which present a close resemblance to each other will be joined and formed into the rudiment of a class, the definition of which will at first include all the apparent points of resemblance. Other objects as they come to our notice will be gradually assigned to those groups with which they present the greatest number of points of resemblance, and the definition of a class will often have to be altered in order to admit them. The early chemists, for instance, could hardly avoid classing together the common metals, gold, silver, copper, lead, and iron, which present such conspicuous points of similarity as regards density, metallic lustre, malleability, &c. With the gress of discovery, however, difficulties begin to present themselves in such a grouping. Antimony, bismuth, and arsenic are distinctly metallic as regards lustre, density, and some chemical properties, but are wanting in malle

pro

ability. The more recently discovered and rare tellurium presents greater difficulties, for it has many of the physical Įroperties of metal, and yet all its chemical properties are analogous to those of sulphur and selenium which have never been regarded as metals. Great chemical differences again are by degrees discovered between the five metals just mentioned; and the class, if it is to have any chemical validity, must be made to include other elements, having none of the original properties on which the class was founded. Hydrogen is a transparent colourless gas and the least dense of all substances, yet in its chemical analogies it is a metal, as suggested by Faraday m in 1838, and almost proved by the late Professor Graham; it must be placed in the same class as silver. In this way it comes to pass that almost every classification which is proposed in the early stages of a science will be found to break down as the deeper similarities of the objects come to be detected. The most obvious points of difference will have to be neglected. Chlorine is a gas, bromine a liquid, and iodine a solid, and at first sight these might have seemed formidable circumstances to overlook; but in chemical analogy the substances are closely united. The progress of organic chemistry, too, has yielded wholly new ideas of the similarities of compounds. Who, for instance, would recognise without extensive research a close similarity between glycerine and alcohol, or between fatty substances and ether. The class of paraffins contains three substances gaseous at ordinary temperatures, several liquids, and some crystalline solids. It required much insight to detect the perfect affinity which exists between such apparently different substances.

The science of chemistry now depends to a great extent on a correct classification of the elements, as will be learnt by consulting the able article on Classification by Prom Life of Faraday,' vol. ii. p. 87.

fessor G. C. Foster in Watts's Dictionary of Chemistry.' But the present theory of classification was not reached until at least three previous false systems had been long entertained. And though there is much reason to believe that the present system of classification according to atomicity is substantially correct, many errors may yet be discovered in the details of the grouping.

Symbolic Statement of the Theory of Classification.

The whole theory of classification can be explained in the most complete and general manner, by reverting for a time to the use of the Logical Abecedarium, which was found to be of supreme importance in Formal Logic (vol. i. p. 109). That form expresses in fact the necessary classification of all objects and ideas as depending on the laws of thought, and there is no point concerning the purpose and methods of classification which may not be explained most precisely by the use of letter combinations, the only inconvenience being the somewhat abstract and repulsive form in which the subject is thus represented.

If we pay regard only to three qualities or circumstances in which things may resemble each other, namely the qualities A, B, C, then there are according to the laws of thought eight possible classes of objects. If there exist objects belonging to all these eight classes, thus indicated,

it follows that the qualities A, B, C are subject to no conditions except the primary laws of thought and nature (vol. i. p. 6). There is then no special law of nature to

discover, and, if we arrange the classes in any one order rather than another, it must be for the purpose of showing that the combinations are logically complete. It will be obvious that there are three different possible arrangements which may be of some use; firstly, that employed above in which all the combinations containing A stand first, and those devoid of it follow; secondly, and thirdly, the similar arrangements in which the combinations containing B, and C, respectively stand first.

Suppose now that there are but four kinds of objects possessing the qualities A, B, C, and that these kinds are represented by the combinations ABC, AbC, aBc, abc. The order of arrangement will now be of importance; for if we place them in the order

ABC
La Bc
[AbC
Labe

placing the B's first and those which are b's last, we shall perhaps overlook the law of correlation of properties involved. But if we arrange the combinations as follows

ABC
АЪС

[aВc
Labc

it becomes apparent at once that where A is, and only where A is, the property C is to be found, B being indifferently present and absent. The second arrangement then would be called a natural one, as rendering manifest the conditions under which the combinations exist.

As a further instance, let us suppose that eight objects are presented to us for classification, which exhibit combinations of the five properties, A, B, C, D, E, in the following manner :