economy of the animal, but because such variations are proved by empirical observation to coincide with most important differences in the general affinities. It is found that the minor classes and genera of mammals can be discriminated accurately by their teeth, especially by the foremost molars and the hindmost pre-molars. Some teeth, indeed, are occasionally missing, so that zoologists prefer to trust to those characteristic teeth which are most constant,1 and to infer from them not only the arrangement of the other teeth, but the whole conformation of the animal.

It is

very difficult matter to mark out a boundary-line between the animal and vegetable kingdoms, and it may even be doubted whether a rigorous boundary can be established. The most fundamental and important difference of a vegetable as compared with an animal substance probably consists in the absence of nitrogen from the constituent membranes. Supposing this to be the case, the difficulty arises that in examining minute organisms we cannot ascertain directly whether they contain nitrogen or not. Some minor but easily detected circumstance is therefore needed to discriminate between animals and vegetables, and this is furnished to some extent by the fact that the production of starch granules is restricted to the vegetable kingdom. Thus the Desmidiaceæ may be safely assigned to the vege table kingdom, because they contain starch. But we must not employ this characteristic negatively; the Diatomaceæ are probably vegetables, though they do not produce starch.

Diagnostic Systems of Classification.

We have seen that diagnosis is the process of discovering the place in any system of classes, to which an object has been referred by some previous investigation, the object being to avail ourselves of the information relating to such an object which has been accumulated and recorded. It is obvious that this is a matter of great importance, for, unless we can recognise, from time to time, objects or substances which have been investigated, recorded discoveries would lose their value. Even a single investi

1 Owen, Essay on the Classification and Geographical Distribution of the Mammalia, p. 20.

gator must have means of recording and systematising his observations of any large groups of objects like the vegetable and animal kingdoms.

Now whenever a class has been properly formed, a definition must have been laid down, stating the qualities and circumstances possessed by all the objects which are intended to be included in the class, and not possessed completely by any other objects. Diagnosis, therefore, consists in comparing the qualities of a certain object with the definitions of a series of classes; the absence in the object of any one quality stated in the definition. excludes it from the class thus defined; whereas, if we find every point of a definition exactly fulfilled in the specimen, we may at once assign it to the class in question. It is of course by no means certain that everything which has been affirmed of a class is true of all objects afterwards referred to the class; for this would be a case of imperfect inference, which is never more than matter of probability. A definition can only make known a finite number of the qualities of an object, and it always remains possible that objects agreeing in those assigned qualities will differ in others. An individual cannot be defined, and can only be made known by the exhibition of the individual itself, or by a material specimen exactly representing it. But this and other questions relating to definition must be treated when I am able to take up the subject of language in another work.

Diagnostic systems of classification should, as a general rule, be arranged on the bifurcate method explicitly. Any quality may be chosen which divides the whole group of objects into two distinct parts, and each part may be subdivided successively by any prominent and well-marked circumstance which is present in a large part of the genus and not in the other. To refer an object to its proper place in such an arrangement we have only to note whether it does or does not possess the successive critical differentiæ. Dana devised a classification of this kind1 by which to refer a crystal to its place in the series of six or seven classes already described. If a crystal has all its edges modified alike or the angles replaced by three or six similar planes,

1 Dana's Mineralogy, vol. i. p. 123 ; quoted in Watts' Dictionary of Chemistry, vol. ii. p. 166.

it belongs to the monometric system; if not, we observe whether the number of similar planes at the extremity of the crystal is three or some multiple of three, in which case it is a crystal of the hexagonal system; and so we proceed with further successive discriminations. To ascertain the name of a mineral by examination with the blowpipe, an arrangement more or less evidently on the bifurcate plan, has been laid down by Von Kobell. Minerals are divided according as they possess or do not possess metallic lustre; as they are fusible or not fusible, according as they do or do not on charcoal give a metallic bead, and so on.

Perhaps the best example to be found of an arrangement devised simply for the purpose of diagnosis, is Mr. George Bentham's Analytical Key to the Natural Orders and Anomalous Genera of the British Flora, given in his Handbook of the British Flora. In this scheme, the great composite family of plants, together with the closely approximate genus Jasione, are first separated from all other flowering plants by the compound character of their flowers. The remaining plants are sub-divided according as the perianth is double or single. Since no plants are yet known in which the perianth can be said to have three or more distinct rings, this division becomes practically the same as one into double and not-double. Flowers with a double perianth are next discriminated according as the corolla does or does not consist of one piece; according as the ovary is free or not free; as it is simple or not simple; as the corolla is regular or irregular; and so on. On looking over this arrangement, it will be found that numerical discriminations often occur, the numbers of petals, stamens, capsules, or other parts being the criteria, in which cases, as already explained (p. 697), the actual exhibition of the bifid division would be tedious.

Linnæus appears to have been perfectly acquainted with the nature and uses of diagnostic classification, which he describes under the name of Synopsis, saying:

1 Instructions for the Discrimination of Minerals by Simple Chemical Experiments, by Franz von Kobell, translated from the German by R. C. Campbell. Glasgow, 1841.

2 Edition of 1866, p. lxiii.

3 Philosophia Botanica (1770), § 154, p. 98.

Synopsis tradit Divisiones arbitrarias, longiores aut breviores, plures aut pauciores: a Botanicis in genere non agnoscenda. Synopsis est dichotomia arbitraria, quæ instar viæ ad Botanicem ducit. Limites autem non determinat."

The rules and tables drawn out by chemists to facilitate the discovery of the nature of a substance in qualitative analysis are usually arranged on the bifurcate method, and form excellent examples of diagnostic classification, the qualities of the substances produced in testing being in most cases merely characteristic properties of little importance in other respects. The chemist does not detect potassium by reducing it to the state of metallic potassium, and then observing whether it has all the principal qualities belonging to potassium. He selects from among the whole number of compounds of potassium that salt, namely the compound of platinum tetra-chloride, and potassium chloride, which has the most distinctive appearance, as it is comparatively insoluble and produces a peculiar yellow and highly crystalline precipitate. Accordingly, potassium is present whenever this precipitate can be produced by adding platinum chloride to a solution. The fine purple or violet colour which potassium salts communicate to the blowpipe flame, had long been used as a characteristic mark. Some other elements were readily detected by the colouring of the blowpipe flame, barium giving a pale yellowish green, and salts of strontium a bright red. By the use of the spectroscope the coloured light given off by an incandescent vapour is made to give perfectly characteristic marks of the elements contained in the vapour.

Diagnosis seems to be identical with the process termed by the ancient logicians abscissio infiniti, the cutting off of the infinite or negative part of a genus when we discover by observation that an object possesses a particular difference. At every step in a bifurcate division, some objects possessing the difference will fall into the affirmative part or species; all the remaining objects in the world fall into the negative part, which will be infinite in extent. Diagnosis consists in the successive rejection from further notice of those infinite classes with which the specimen in question does not agree.

Index Classifications.

Under classification we may include all arrangements of objects or names, which we make for saving labour in the discovery of an object. Even alphabetical indices are real classifications. No such arrangement can be of use unless it involves some correlation of circumstances, so that knowing one thing we learn another. If we merely arrange letters in the pigeon-holes of a secretaire we establish a correlation, for all letters in the first hole will be written by persons, for instance, whose names begin with A, and so on. Knowing then the initial letter of the writer's name, we know also the place of the letter, and the labour of search is thus reduced to one twenty-sixth part of what it would be without arrangement.

Now the purpose of a catalogue is to discover the place in which an object is to be found; but the art of cataloguing involves logical considerations of some importance. We want to establish a correlation between the place of an object and some circumstance about the object which shall enable us readily to refer to it; this circumstance therefore should be that which will most readily dwell in the memory of the searcher. A piece of poetry will be best remembered by the first line of the piece, and the name of the author will be the next most definite circumstance; a catalogue of poetry should therefore be arranged alphabetically according to the first word of the piece, or the name of the author, or, still better, in both ways. It would be impossible to arrange poems according to their subjects, so vague and mixed are these found to be when the attempt is made.

It is a matter of considerable literary importance to decide upon the best mode of cataloguing books, so that any required book in a library shall be most readily found. Books may be classified in a great number of ways, according to subject, language, date, or place of publication, size, the initial words of the text or title-page, or colophon, the author's name, the publisher's name, the printer's name, the character of the type, and so on. Every one of these modes of arrangement may be useful, for we may happen to remember one circumstance about a book