CHAPTER XXX.

CLASSIFICATION.

The extensive subject of Classification has been deferred to a late part of this treatise, because it involves questions of difficulty, and did not seem naturally to fall into an earlier place. But it must not be supposed that, in now formally taking up the subject, we are for the first time entertaining the notion of classification. All logical inference involves classification, which is indeed the necessary accompaniment of the action of judgment. It is impossible to detect similarity between objects without thereby joining them together in thought, and forming an incipient class. Nor can we bestow a common name upon objects without implying the existence of a class. Every common name is the name of a class, and every name of a class is a common name. It is evident also that to speak of a general notion or concept is but another way of speaking of a class. Usage leads us to employ the word classification in some cases and not in others. We are said to form the *general notion* parallelogram when we regard an infinite number of possible four-sided rectilinear figures as resembling each other in the common property of possessing parallel sides. We should be said to form a *class*, Trilobite, when we place together in a museum a number of specimens resembling each other in certain defined characters. But the logical nature of the operation is the same in both cases. We form a *class* of figures called parallelograms and we form a *general notion* of trilobites.

Science, it was said at the outset, is the detection of identify, and classification is the placing together, either in thought or in actual proximity of space, those objects between which identity has been detected. Accordingly, the value of classification is co-extensive with the value of science and general reasoning. Whenever we form a class we reduce multiplicity to unity, and detect, as Plato said, the one in the many. The result of such classification is to yield generalised knowledge, as distinguished from the direct and sensuous knowledge of particular facts. Of every class, so far as it is correctly formed, the principle of substitution is true, and whatever we know of one object in a class we know of the other objects, so far as identity has been detected between them. The facilitation and abbreviation of mental labour is at the bottom of all mental progress. The reasoning faculties of Newton were not different in nature from those of a ploughman; the difference lay in the extent to which they were exerted, and the number of facts which could be treated. Every thinking being generalises more or less, but it is the depth and extent of his generalisations which distinguish the philosopher. Now it is the exertion of the classifying and generalising powers which enables the intellect of man to cope in some degree with the infinite number of natural phenomena. In the chapters upon combinations and permutations it was made evident, that from a few elementary differences immense numbers of combinations can be produced. The process of classification enables us to resolve these combinations, and refer each one to its place according to one or other of the elementary circumstances out of which it was produced. We restore nature to the simple conditions out of which its endless variety was developed. As Professor Bowen has said,[560] “The first necessity which is imposed upon us by the constitution of the mind itself, is to break up the infinite wealth of Nature into groups and classes of things, with reference to their resemblances and affinities, and thus to enlarge the grasp of our mental faculties, even at the expense of sacrificing the minuteness of information which can be acquired only by studying objects in detail. The first efforts in the pursuit of knowledge, then, must be directed to the business of classification. Perhaps it will be found in the sequel, that classification is not only the beginning, but the culmination and the end, of human knowledge.”

[560] *A Treatise on Logic, or, the Laws of Pure Thought*, by Francis Bowen, Professor of Moral Philosophy in Harvard College, Cambridge, United States, 1866, p. 315.

*Classification Involving Induction.*

The purpose of classification is the detection of the laws of nature. However much the process may in some cases be disguised, classification is not really distinct from the process of perfect induction, whereby we endeavour to ascertain the connexions existing between properties of the objects under treatment. There can be no use in placing an object in a class unless something more than the fact of being in the class is implied. If we arbitrarily formed a class of metals and placed therein a selection from the list of known metals made by ballot, we should have no reason to expect that the metals in question would resemble each other in any points except that they are metals, and have been selected by the ballot. But when chemists select from the list the five metals, potassium, sodium, cæsium, rubidium, and lithium and call them the Alkaline metals, a great deal is implied in this classification. On comparing the qualities of these substances they are all found to combine very energetically with oxygen, to decompose water at all temperatures, and to form strongly basic oxides, which are highly soluble in water, yielding powerfully caustic and alkaline hydrates from which water cannot be expelled by heat. Their carbonates are also soluble in water, and each metal forms only one chloride. It may also be expected that each salt of one of the metals will correspond to a salt of each other metal, there being a general analogy between the compounds of these metals and their properties.

Now in forming this class of alkaline metals, we have done more than merely select a convenient order of statement. We have arrived at a discovery of certain empirical laws of nature, the probability being very considerable that a metal which exhibits some of the properties of alkaline metals will also possess the others. If we discovered another metal whose carbonate was soluble in water, and which energetically combined with water at all temperatures, producing a strongly basic oxide, we should infer that it would form only a single chloride, and that generally speaking, it would enter into a series of compounds corresponding to the salts of the other alkaline metals. The formation of this class of alkaline metals then, is no mere matter of convenience; it is an important and successful act of inductive discovery, enabling us to register many undoubted propositions as results of perfect induction, and to make a great number of inferences depending upon the principles of imperfect induction.

An excellent instance as to what classification can do, is found in Mr. Lockyer’s researches on the sun.[561] Wanting some guide as to what more elements to look for in the sun’s photosphere, he prepared a classification of the elements according as they had or had not been traced in the sun, together with a detailed statement of the chief chemical characters of each element. He was then able to observe that the elements found in the sun were for the most part those forming stable compounds with oxygen. He then inferred that other elements forming stable oxides would probably exist in the sun, and he was rewarded by the discovery of five such metals. Here we have empirical and tentative classification leading to the detection of the correlation between existence in the sun, and the power of forming stable oxides and then leading by imperfect induction to the discovery of more coincidences between these properties.

[561] *Proceedings of the Royal Society*, November, 1873, vol. xxi. p. 512.

Professor Huxley has defined the process of classification in the following terms.[562] “By the classification of any series of objects, is meant the actual or ideal arrangement together of those which are like and the separation of those which are unlike; the purpose of this arrangement being to facilitate the operations of the mind in clearly conceiving and retaining in the memory the characters of the objects in question.”

[562] *Lectures on the Elements of Comparative Anatomy*, 1864, p. 1.

This statement is doubtless correct, so far as it goes, but it does not include all that Professor Huxley himself implicitly treats under classification. He is fully aware that deep correlations, or in other terms deep uniformities or laws of nature, will be disclosed by any well chosen and profound system of classification. I should therefore propose to modify the above statement, as follows:--“By the classification of any series of objects, is meant the actual or ideal arrangement together of those which are like and the separation of those which are unlike, the purpose of this arrangement being, primarily, to disclose the correlations or laws of union of properties and circumstances, and, secondarily, to facilitate the operations of the mind in clearly conceiving and retaining in the memory the characters of the objects in question.”

*Multiplicity of Modes of Classification.*

In approaching the question how any given group of objects may be best classified, let it be remarked that there must generally be an unlimited number of modes of classifying a group of objects. Misled, as we shall see, by the problem of classification in the natural sciences, philosophers seem to think that in each subject there must be one essentially natural system of classification which is to be selected, to the exclusion of all others. This erroneous notion probably arises also in part from the limited powers of thought and the inconvenient mechanical conditions under which we labour. If we arrange the books in a library catalogue, we must arrange them in some one order; if we compose a treatise on mineralogy, the minerals must be successively described in some one arrangement; if we treat such simple things as geometrical figures, they must be taken in some fixed order. We shall naturally select that arrangement which appears to be most convenient and instructive for our principal purpose. But it does not follow that this method of arrangement possesses any exclusive excellence, and there will be usually many other possible arrangements, each valuable in its own way. A perfect intellect would not confine itself to one order of thought, but would simultaneously regard a group of objects as classified in all the ways of which they are capable. Thus the elements may be classified according to their atomicity into the groups of monads, dyads, triads, tetrads, pentads, and hexads, and this is probably the most instructive classification; but it does not prevent us from also classifying them according as they are metallic or non-metallic, solid, liquid or gaseous at ordinary temperatures, useful or useless, abundant or scarce, ferro-magnetic or diamagnetic, and so on.

Mineralogists have spent a great deal of labour in trying to discover the supposed natural system of classification for minerals. They have constantly encountered the difficulty that the chemical composition does not run together with the crystallographic form, and the various physical properties of the mineral. Substances identical in the forms of their crystals, especially those belonging to the first or cubical system of crystals, are often found to have no resemblance in chemical composition. The same substance, again, is occasionally found crystallised in two essentially different crystallographic forms; calcium carbonate, for instance, appearing as calc-spar and arragonite. The simple truth is that if we are unable to discover any correspondence, or, as we may call it, any *correlation* between the properties of minerals, we cannot make any one arrangement which will enable us to treat all these properties in a single system of classification. We must classify minerals in as many different ways as there are different groups of unrelated properties of sufficient importance. Even if, for the purpose of describing minerals successively in a treatise, we select one chief system, that, for instance, having regard to chemical composition, we ought mentally to regard the minerals as classified in all other useful modes.

Exactly the same may be said of the classification of plants. An immense number of different modes of classifying plants have been proposed at one time or other, an exhaustive account of which will be found in the article on classification in Rees’s “Cyclopædia,” or in the introduction to Lindley’s “Vegetable Kingdom.” There have been the Fructists, such as Cæsalpinus, Morison, Hermann, Boerhaave or Gaertner, who arranged plants according to the form of the fruit. The Corollists, Rivinus, Ludwig, and Tournefort, paid attention chiefly to the number and arrangement of the parts of the corolla. Magnol selected the calyx as the critical part, while Sauvage arranged plants according to their leaves; nor are these instances more than a small selection from the actual variety of modes of classification which have been tried. Of such attempts it may be said that every system will probably yield some information concerning the relations of plants, and it is only after trying many modes that it is possible to approximate to the best.

*Natural and Artificial Systems of Classification.*

It has been usual to distinguish systems of classification as natural and artificial, those being called natural which seemed to express the order of existing things as determined by nature. Artificial methods of classification, on the other hand, included those formed for the mere convenience of men in remembering or treating natural objects.

The difference, as it is commonly regarded, has been well described by Ampére,[563] as follows: “We can distinguish two kinds of classifications, the natural and the artificial. In the latter kind, some characters, arbitrarily chosen, serve to determine the place of each object; we abstract all other characters, and the objects are thus found to be brought near to or to be separated from each other, often in the most bizarre manner. In natural systems of classification, on the contrary, we employ concurrently all the characters essential to the objects with which we are occupied, discussing the importance of each of them; and the results of this labour are not adopted unless the objects which present the closest analogy are brought most near together, and the groups of the several orders which are formed from them are also approximated in proportion as they offer more similar characters. In this way it arises that there is always a kind of connexion, more or less marked, between each group and the group which follows it.”

[563] *Essai sur la Philosophie des Sciences*, p. 9.

There is much, however, that is vague and logically false in this and other definitions which have been proposed by naturalists to express their notion of a natural system. We are not informed how the *importance* of a resemblance is to be determined, nor what is the measure of the *closeness* of analogy. Until all the words employed in a definition are made clear in meaning, the definition itself is worse than useless. Now if the views concerning classification here upheld are true, there can be no sharp and precise distinction between natural and artificial systems. All arrangements which serve any purpose at all must be more or less natural, because, if closely enough scrutinised, they will involve more resemblances than those whereby the class was defined.

It is true that in the biological sciences there would be one arrangement of plants or animals which would be conspicuously instructive, and in a certain sense natural, if it could be attained, and it is that after which naturalists have been in reality striving for nearly two centuries, namely, that *arrangement which would display the genealogical descent of every form from the original life germ*. Those morphological resemblances upon which the classification of living beings is almost always based are inherited resemblances, and it is evident that descendants will usually resemble their parents and each other in a great many points.

I have said that a natural is distinguished from an arbitrary or artificial system only in degree. It will be found almost impossible to arrange objects according to any circumstance without finding that some correlation of other circumstances is thus made apparent. No arrangement could seem more arbitrary than the common alphabetical arrangement according to the initial letter of the name. But we cannot scrutinise a list of names of persons without noticing a predominance of Evans’s and Jones’s, under the letters E and J, and of names beginning with Mac under the letter M. The predominance is so great that we could not attribute it to chance, and inquiry would of course show that it arose from important facts concerning the nationality of the persons. It would appear that the Evans’s and Jones’s were of Welsh descent, and those whose names bear the prefix Mac of Keltic descent. With the nationality would be more or less strictly correlated many peculiarities of physical constitution, language, habits, or mental character. In other cases I have been interested in noticing the empirical inferences which are displayed in the most arbitrary arrangements. If a large register of the names of ships be examined it will often be found that a number of ships bearing the same name were built about the same time, a correlation due to the occurrence of some striking incident shortly previous to the building of the ships. The age of ships or other structures is usually correlated with their general form, nature of materials, &c., so that ships of the same name will often resemble each other in many points.

It is impossible to examine the details of some of the so-called artificial systems of classification of plants, without finding that many of the classes are natural in character. Thus in Tournefort’s arrangement, depending almost entirely on the formation of the corolla, we find the natural orders of the Labiatæ, Cruciferæ, Rosaceæ, Umbelliferæ, Liliaceæ, and Papilionaceæ, recognised in his 4th, 5th, 6th, 7th, 9th, and 10th classes. Many of the classes in Linnæus’ celebrated sexual system also approximate to natural classes.

*Correlation of Properties.*

Habits and usages of language are apt to lead us into the error of imagining that when we employ different words we always mean different things. In introducing the subject of classification nominally I was careful to draw the reader’s attention to the fact that all reasoning and all operations of scientific method really involve classification, though we are accustomed to use the name in some cases and not in others. The name *correlation* requires to be used with the same qualification. Things are correlated (*con*, *relata*) when they are so related or bound to each other that *where one is the other is, and where one is not the other is not*. Throughout this work we have then been dealing with correlations. In geometry the occurrence of three equal angles in a triangle is correlated with the existence of three equal sides; in physics gravity is correlated with inertia; in botany exogenous growth is correlated with the possession of two cotyledons, or the production of flowers with that of spiral vessels. Wherever a proposition of the form A = B is true there correlation exists. But it is in the classificatory sciences especially that the word correlation has been employed.

We find it stated that in the class Mammalia the possession of two occipital condyles, with a well-ossified basi-occipital, is correlated with the possession of mandibles, each ramus of which is composed of a single piece of bone, articulated with the squamosal element of the skull, and also with the possession of mammæ and non-nucleated red blood-corpuscles. Professor Huxley remarks[564] that this statement of the character of the class mammalia is something more than an arbitrary definition; it is a statement of a law of correlation or co-existence of animal structures, from which most important conclusions are deducible. It involves a generalisation to the effect that in nature the structures mentioned are always found associated together. This amounts to saying that the formation of the class mammalia involves an act of inductive discovery, and results in the establishment of certain empirical laws of nature. Professor Huxley has excellently expressed the mode in which discoveries of this kind enable naturalists to make deductions or predictions with considerable confidence, but he has also pointed out that such inferences are likely from time to time to prove mistaken. I will quote his own words:

[564] *Lectures on the Elements of Comparative Anatomy, and on the Classification of Animals*, 1864, p. 3.

“If a fragmentary fossil be discovered, consisting of no more than a ramus of a mandible, and that part of the skull with which it articulated, a knowledge of this law may enable the palæontologist to affirm, with great confidence, that the animal of which it formed a part suckled its young, and had non-nucleated red blood-corpuscles; and to predict that should the back part of that skull be discovered, it will exhibit two occipital condyles and a well-ossified basi-occipital bone.

“Deductions of this kind, such as that made by Cuvier in the famous case of the fossil opossum of Montmartre, have often been verified, and are well calculated to impress the vulgar imagination; so that they have taken rank as the triumphs of the anatomist. But it should carefully be borne in mind, that, like all merely empirical laws, which rest upon a comparatively narrow observational basis, the reasoning from them may at any time break down. If Cuvier, for example, had had to do with a fossil Thylacinus instead of a fossil Opossum, he would not have found the marsupial bones, though the inflected angle of the jaw would have been obvious enough. And so, though, practically, any one who met with a characteristically mammalian jaw would be justified in expecting to find the characteristically mammalian occiput associated with it; yet, he would be a bold man indeed, who should strictly assert the belief which is implied in this expectation, viz., that at no period of the world’s history did animals exist which combined a mammalian occiput with a reptilian jaw, or *vice versâ*.”

One of the most distinct and remarkable instances of correlation in the animal world is that which occurs in ruminating animals, and which could not be better stated than in the following extract from the classical work of Cuvier:[565]

[565] *Ossemens Fossiles*, 4th edit. vol. i. p. 164. Quoted by Huxley, *Lectures*, &c., p. 5.

“I doubt if any one would have divined, if untaught by observation, that all ruminants have the foot cleft, and that they alone have it. I doubt if any one would have divined that there are frontal horns only in this class: that those among them which have sharp canines for the most part lack horns.

“However, since these relations are constant, they must have some sufficient cause; but since we are ignorant of it, we must make good the defect of the theory by means of observation: it enables us to establish empirical laws which become almost as certain as rational laws when they rest on sufficiently repeated observations; so that now whoso sees merely the print of a cleft foot may conclude that the animal which left this impression ruminated, and this conclusion is as certain as any other in physics or morals. This footprint alone then, yields, to him who observes it, the form of the teeth, the form of the jaws, the form of the vertebræ, the form of all the bones of the legs, of the thighs, of the shoulders, and of the pelvis of the animal which has passed by: it is a surer mark than all those of Zadig.”

We meet with a good instance of the purely empirical correlation of circumstances when we classify the planets according to their densities and periods of axial rotation.[566] If we examine a table specifying the usual astronomical elements of the solar system, we find that four planets resemble each other very closely in the period of axial rotation, and the same four planets are all found to have high densities, thus:--

[566] Chambers, *Descriptive Astronomy*, 1st edit. p. 23.

Name of Period of Axial Planet. Rotation. Density.

Mercury 24 hours 5 minutes 7·94 Venus 23 " 21 " 5·33 Earth 23 " 56 " 5·67 Mars 24 " 37 " 5·84

A similar table for the other larger planets, is as follows:--

Jupiter 9 hours 55 minutes 1·36 Saturn 10 " 29 " ·74 Uranus 9 " 30 " ·97 Neptune -- " -- 1·02

It will be observed that in neither group is the equality of the rotational period or the density more than rudely approximate; nevertheless the difference of the numbers in the first and second group is so very well marked, the periods of the first being at least double and the densities four or five times those of the second, that the coincidence cannot be attributed to accident. The reader will also notice that the first group consists of the planets nearest to the sun; that with the exception of the earth none of them possess satellites; and that they are all comparatively small. The second group are furthest from the sun, and all of them possess several satellites, and are comparatively great. Therefore, with but slight exceptions, the following correlations hold true:--

Interior planets. Long period. Small size. High Density. No satellites. Exterior " Short " Great " Low " Many "

These coincidences point with much probability to a difference in the origin of the two groups, but no further explanation of the matter is yet possible.

The classification of comets according to their periods by Mr. Hind and Mr. A. S. Davies, tends to establish the conclusion that distinct groups of comets have been brought into the solar system by the attractive powers of Jupiter, Uranus, or other planets.[567] The classification of nebulæ as commenced by the two Herschels, and continued by Lord Rosse, Mr. Huggins, and others, will probably lead at some future time to the discovery of important empirical laws concerning the constitution of the universe. The minute examination and classification of meteorites, as carried on by Mr. Sorby and others, seems likely to afford us an insight into the formation of the heavenly bodies.

[567] *Philosophical Magazine*, 4th Series, vol. xxxix. p. 396; vol. xl. p. 183; vol. xli. p. 44. See also Proctor, *Popular Science Review*, October 1874, p. 350.

We should never fail to remember the slightest and most inexplicable correlations, for they may prove of importance in the future. Discoveries begin when we are least expecting them. It is a significant fact, for instance, that the greater number of variable stars are of a reddish colour. Not all variable stars are red, nor all red stars variable; but considering that only a small fraction of the observed stars are known to be variable, and only a small fraction are red, the number which fall into both classes is too great to be accidental.[568] It is also remarkable that the greater number of stars possessing great proper motion are double stars, the star 61 Cygni being especially noticeable in this respect.[569] The correlation in these cases is not without exception, but the preponderance is so great as to point to some natural connexion, the exact nature of which must be a matter for future investigation. Herschel remarked that the two double stars 61 Cygni and α Centauri of which the orbits were well ascertained, evidently belonged to the same family or genus.[570]

[568] Humboldt, *Cosmos* (Bohn), vol. iii. p. 224.

[569] Baily, British *Association Catalogue*, p. 48.

[570] *Outlines of Astronomy*, § 850, 4th edit. p. 578.

*Classification in Crystallography.*

Perhaps the most perfect and instructive instance of classification which we can find is furnished by the science of crystallography (p. 133). The system of arrangement now generally adopted is conspicuously natural, and is even mathematically perfect. A crystal consists in every part of similar molecules similarly related to the adjoining molecules, and connected with them by forces the nature of which we can only learn by their apparent effects. But these forces are exerted in space of three dimensions, so that there is a limited number of suppositions which can be entertained as to the relations of these forces. In one case each molecule will be similarly related to all those which are next to it; in a second case, it will be similarly related to those in a certain plane, but differently related to those not in that plane. In the simpler cases the arrangement of molecules is rectangular; in the remaining cases oblique either in one or two planes.

In order to simplify the explanation and conception of the complicated phenomena which crystals exhibit, an hypothesis has been invented which is an excellent instance of the Descriptive Hypotheses before mentioned (p. 522). Crystallographers imagine that there are within each crystal certain axes, or lines of direction, by the comparative length and the mutual inclination of which the nature of the crystal is determined. In one class of crystals there are three such axes lying in one plane, and a fourth perpendicular to that plane; but in all the other classes there are imagined to be only three axes. Now these axes can be varied in three ways as regards length: they may be (1) all equal, or (2) two equal and one unequal, or (3) all unequal. They may also be varied in four ways as regards direction: (1) they may be all at right angles to each other; (2) two axes may be oblique to each other and at right angles to the third; (3) two axes may be at right angles to each other and the third oblique to both; (4) the three axes may be all oblique. Now, if all the variations as regards length were combined with those regarding direction, it would seem to be possible to have twelve classes of crystals in all, the enumeration being then logically and geometrically complete. But as a matter of empirical observation, many of these classes are not found to occur, oblique axes being seldom or never equal. There remain seven recognised classes of crystals, but even of these one class is not positively known to be represented in nature.

The first class of crystals is defined by possessing three equal rectangular axes, and equal elasticity in all directions. The primary or simple form of the crystals is the cube, but by the removal of the corners of the cube by planes variously inclined to the axes, we have the regular octohedron, the dodecahedron, and various combinations of these forms. Now it is a law of this class of crystals that as each axis is exactly like each other axis, every modification of any corner of a crystal must be repeated symmetrically with regard to the other axes; thus the forms produced are symmetrical or regular, and the class is called the *Regular System* of crystals. It includes a great variety of substances, some of them being elements, such as carbon in the form of diamond, others more or less complex compounds, such as rock-salt, potassium iodide and bromide, the several kinds of alum, fluor-spar, iron bisulphide, garnet, spinelle, &c. No correlation then is apparent between the form of crystallisation and the chemical composition. But what we have to notice is that the physical properties of the crystallised substances with regard to light, heat, electricity, &c., are closely similar. Light and heat undulations, wherever they enter a crystal of the regular system, spread with equal rapidity in all directions, just as they would in a uniform fluid. Crystals of the regular system accordingly do not in any case exhibit the phenomena of double refraction, unless by mechanical compression we alter the conditions of elasticity. These crystals, again, expand equally in all directions when heated, and if we could cut a sufficiently large plate from a cubical crystal, and examine the sound vibrations of which it is capable, we should find that they indicated an equal elasticity in every direction. Thus we see that a great number of important properties are correlated with that of crystallisation in the regular system, and as soon as we know that the primary form of a substance is the cube, we are able to infer with approximate certainty that it possesses all these properties. The class of regular crystals is then an eminently natural class, one disclosing many general laws connecting together the physical and mechanical properties of the substances classified.

In the second class of crystals, called the dimetric, square prismatic, or pyramidal system, there are also three axes at right angles to each other; two of the axes are equal, but the third or principal axis is unequal, being either greater or less than either of the other two. In such crystals accordingly the elasticity and other properties are alike in all directions perpendicular to the principal axis, but vary in all other directions. If a point within a crystal of this system be heated, the heat spreads with equal rapidity in planes perpendicular to the principal axis, but more or less rapidly in the direction of this axis, so that the isothermal surface is an ellipsoid of revolution round that axis.

Nearly the same statement may be made concerning the third or hexagonal or rhombohedral system of crystals, in which there are three axes lying in one plane and meeting at angles of 60°, while the fourth axis is perpendicular to the other three. The hexagonal prism and rhombohedron are the commonest forms assumed by crystals of this system, and in ice, quartz, and calc-spar, we have abundance of beautiful specimens of the various shapes produced by the modification of the primitive form. Calc-spar alone is said to crystallise in at least 700 varieties of form. Now of all the crystals belonging both to this and the dimetric class, we know that a ray of light passing in the direction of the principal axis will be refracted singly as in a crystal of the regular system; but in every other direction the light will suffer double refraction being separated into two rays, one of which obeys the ordinary law of refraction, but the other a much more complicated law. The other physical properties vary in an analogous manner. Thus calc-spar expands by heat in the direction of the principal axis, but contracts a little in directions perpendicular to it. So closely are the physical properties correlated that Mitscherlich, having observed the law of expansion in calc-spar, was enabled to predict that the double refracting power of the substance would be decreased by a rise of temperature, as was proved by experiment to be the case.

In the fourth system, called the trimetric, rhombic, or right prismatic system, there are three axes, at right angles, but all unequal in length. It may be asserted in general terms that the mechanical properties vary in such crystals in every direction, and heat spreads so that the isothermal surface is an ellipsoid with three unequal axes.

In the remaining three classes, called the monoclinic, diclinic, and triclinic, the axes are more or less oblique, and at the same time unequal. The complication of phenomena is therefore greatly increased, and it need only be stated that there are always two directions in which a ray is singly refracted, but that in all other directions double refraction takes place. The conduction of heat is unequal in all directions, the isothermal surface being an ellipsoid of three unequal axes. The relations of such crystals to other phenomena are often very complicated, and hardly yet reduced to law. Some crystals, called pyro-electric, manifest vitreous electricity at some points of their surface, and resinous electricity at other points when rising in temperature, the character of the electricity being changed when the temperature sinks again. This production of electricity is believed to be connected with the hemihedral character of the crystals exhibiting it. The crystalline structure of a substance again influences its magnetic behaviour, the general law being that the direction in which the molecules of a crystal are most approximated tends to place itself axially or equatorially between the poles of a magnet, respectively as the body is magnetic or diamagnetic. Further questions arise if we apply pressure to crystals. Thus doubly refracting crystals with one principal axis acquire two axes when the pressure is perpendicular in direction to the principal axis.

All the phenomena peculiar to crystalline bodies are thus closely correlated with the formation of the crystal, or will almost certainly be found to be so as investigation proceeds. It is upon empirical observation indeed that the laws of connexion are in the first place founded, but 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 attempts at so-called natural classification are really attempts at perfect induction, it follows that they are subject to the remarks which were made upon the inverse character of the inductive process, and upon the difficulty of every inverse operation (pp. 11, 12, 122, &c.). There will be no royal road to the discovery of the best system, and it will even be impossible to lay down rules of procedure to assist those who are in search of a good arrangement. The only logical rule 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 correlation 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 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 should take them out in a specified order, that for instance at present adopted by chemists.

The usual mode in which an investigator proceeds to form a classification of a 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 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 progress of discovery, however, difficulties began 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 malleability. The recently discovered tellurium presents greater difficulties, for it has many of the physical properties 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 discovered by degrees between the five metals 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[571] in 1838, and almost proved by Graham;[572] 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, again, 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 analogy which exists between such apparently different substances.

[571] *Life of Faraday*, vol. ii. p. 87.

[572] *Proceedings of the Royal Society*, vol. xvii. p. 212. *Chemical and Physical Researches*, reprint, by Young and Angus Smith, p. 290.

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 Professor G. C. Foster in Watts’ *Dictionary of Chemistry*. But the present system of chemical 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 mode of classification according to atomicity is substantially correct, errors may yet be discovered in the details of the grouping.

*Symbolic Statement of the Theory of Classification.*

The theory of classification can be explained in the most complete and general manner, by reverting for a time to the use of the Logical Alphabet, which was found to be of supreme importance in Formal Logic. That form expresses 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 stated precisely by the use of letter combinations, the only inconvenience being the abstract form in which the subject is thus represented.

If we pay regard only to three qualities in which things may resemble each other, namely, the qualities A, B, C, there are according to the laws of thought eight possible classes of objects, shown in the fourth column of the Logical Alphabet (p. 94). If there exist objects belonging to all these eight classes, it follows that the qualities A, B, C, are subject to no conditions except the primary laws of thought and things (p. 5). There is then no special law of nature to discover, and, if we arrange the objects in any one order rather than another, it must be for the purpose of showing that the combinations are logically complete.

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

{ ABC { A*b*C { *a*B*c* { *abc*

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 { *a*B*c* { A*b*C { *abc*

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:--

ABC*d*E *a*BC*d*E AB*cde* *a*B*cde* A*b*CDE *ab*CDE A*bc*D*e* *abc*D*e*

They are now classified, so that those containing A stand first, and those devoid of A second, but no other property seems to be correlated with A. Let us alter this arrangement and group the combinations thus:--

ABC*d*E A*b*CDE AB*cde* A*bc*D*e* *a*BC*d*E *ab*CDE *a*B*cde* *abc*D*e*

It requires little examination to discover that in the first group B is always present and D absent, whereas in the second group, B is always absent and D present. This is the result which follows from a law of the form B = d (p. 136), so that in this mode of arrangement we readily discover correlation between two letters. Altering the groups again as follows:--

ABC*d*E AB*cde* *a*BC*d*E *a*B*cde* A*b*CDE A*bc*D*e* *ab*CDE *abc*D*e*,

we discover another evident correlation between C and E. Between A and the other letters, or between the two pairs of letters B, D and C, E, there is no logical connexion.

This example may seem tedious, but it will be found instructive in this way. We are classifying only eight objects or combinations, in each of which only five qualities are considered. There are only two laws of correlation between four of those five qualities, and those laws are of the simplest logical character. Yet the reader would hardly discover what those laws are, and confidently assign them by rapid contemplation of the combinations, as given in the first group. Several tentative classifications must probably be made before we can resolve the question. Let us now suppose that instead of eight objects and five qualities, we have, say, five hundred objects and fifty qualities. If we were to attempt the same method of exhaustive grouping which we before employed, we should have to arrange the five hundred objects in fifty different ways, before we could be sure that we had discovered even the simpler laws of correlation. But even the successive grouping of all those possessing each of the fifty properties would not necessarily give us all the laws. There might exist complicated relations between several properties simultaneously, for the detection of which no rule of procedure whatever can be given.

*Bifurcate Classification.*

Every system of classification ought to be formed on the principles of the Logical Alphabet. Each superior class should be divided into two inferior classes, distinguished by the possession and non-possession of a single specified difference. Each of these minor classes, again, is divisible by any other quality whatever which can be suggested, and thus every classification logically consists of an infinitely extended series of subaltern genera and species. The classifications which we form are in reality very small fragments of those which would correctly and fully represent the relations of existing things. But if we take more than four or five qualities into account, the number of subdivisions grows impracticably large. Our finite minds are unable to treat any complex group exhaustively, and we are obliged to simplify and generalise scientific problems, often at the risk of overlooking particular conditions and exceptions.

Every system of classes displayed in the manner of the Logical Alphabet may be called *bifurcate*, because every class branches out at each step into two minor classes, existent or imaginary. It would be a great mistake to regard this arrangement as in any way a peculiar or special method; it is not only a natural and important one, but it is the inevitable and only system which is logically perfect, according to the fundamental laws of thought. All other arrangements of classes correspond to the bifurcate arrangement, with the implication that some of the minor classes are not represented among existing things. If we take the genus A and divide it into the species AB and AC, we imply two propositions, namely that in the class A, the properties of B and C never occur together, and that they are never both absent; these propositions are logically equivalent to one, namely AB = A*c*. Our classification is then identical with the following bifurcate one:--

A | +----------+----------+ | | AB A*b* | | +------+------+ +------+------+ | | | | ABC = 0 AB*c* A*b*C A*bc* = 0

If, again, we divide the genus A into three species, AB, AC, AD, we are either logically in error, or else we must be understood to imply that, as regards the other letters, there exist only three combinations containing A, namely AB*cd*, A*b*C*d*, and A*bc*D.

The logical necessity of bifurcate classification has been clearly and correctly stated in the *Outline of a New System of Logic* by George Bentham, the eminent botanist, a work of which the logical value has been quite overlooked until lately. Mr. Bentham points out, in p. 113, that every classification must be essentially bifurcate, and takes, as an example, the division of vertebrate animals into four sub-classes, as follows:--

Mammifera--endowed with mammæ and lungs. Birds without mammæ but with lungs and wings. Fish deprived of lungs. Reptiles deprived of mammæ and wings but with lungs.

We have, then, as Mr. Bentham says, three bifid divisions, thus represented:--

Vertebrata | +-----------+-----------+ | | Endowed with lungs deprived of lungs | = Fish. +--------+----------------+ | | Endowed with deprived of mammæ mammæ = Mammifera. | +------+------+ | | with wings without wings = Birds. = Reptiles.

It is quite evident that according to the laws of thought even this arrangement is incomplete. The sub-class mammifera must either have wings or be deprived of them; we must either subdivide this class, or assume that none of the mammifera have wings, which is, as a matter of fact, the case, the wings of bats not being true wings in the meaning of the term as applied to birds. Fish, again, ought to be considered with regard to the possession of mammæ and wings; and in leaving them undivided we really imply that they never have mammæ nor wings, the wings of the flying-fish, again, being no exception. If we resort to the use of our letters and define them as follows--

A = vertebrata, B = having lungs, C = having mammæ, D = having wings,

then there are four existent classes of vertebrata which appear to be thus described--

ABC AB*c*D AB*cd* A*b*.

But in reality the combinations are implied to be

ABC*d* = Mammifera, AB*c*D = Birds, AB*cd* = Reptiles, A*bcd* = Fish,

and we imply at the same time that the other four conceivable combinations containing B, C, or D, namely ABCD, A*b*CD, A*b*C*d*, and A*bc*D, do not exist in nature.

Mr. Bentham points out[573] that it is really this method of classification which was employed by Lamarck and De Candolle in their so-called analytical arrangement of the French Flora. He gives as an example a table of the principal classes of De Candolle’s system, as also a bifurcate arrangement of animals after the method proposed by Duméril in his *Zoologie Analytique*, this naturalist being distinguished by his clear perception of the logical importance of the method. A bifurcate classification of the animal kingdom may also be found in Professor Reay Greene’s *Manual of the Cœlenterata*, p. 18.

[573] *Essai sur la Nomenclature et la Classification*, Paris, 1823, pp. 107, 108.

The bifurcate form of classification seems to be needless when the quality according to which we classify any group of things admits of numerical discrimination. It would seem absurd to arrange things according as they have one degree of the quality or not one degree, two degrees or not two degrees, and so on. The elements are classified according as the atom of each saturates one, two, three, or more atoms of a monad element, such as chlorine, and they are called accordingly monad, dyad, triad, tetrad elements, and so on. It would be useless to apply the bifid arrangement, thus:--

Element | +-----+-------+ | | Monad not-Monad | +---------+---------+ | | Dyad not-Dyad | +---------+---------+ | | Triad not-Triad | +---------+--------+ | | Tetrad not-Tetrad.

The reason of this is that, by the nature of number (p. 157) every number is logically discriminated from every other number. There can thus be no logical confusion in a numerical arrangement, and the series of numbers indefinitely extended is also exhaustive. Every thing admitting of a quality expressible in numbers must find its place somewhere in the series of numbers. The chords in music correspond to the simpler numerical ratios and must admit of complete exhaustive classification in respect to the complexity of the ratios forming them. Plane rectilinear figures may be classified according to the numbers of their sides, as triangles, quadrilateral figures, pentagons, hexagons, heptagons, &c. The bifurcate arrangement is not false when applied to such series of objects; it is even necessarily involved in the arrangement which we do apply, so that its formal statement is needless and tedious. The same may be said of the division of portions of space. Reid and Kames endeavoured to cast ridicule on the bifurcate arrangement[574] by proposing to classify the parts of England into Middlesex and what is not Middlesex, dividing the latter again into Kent and what is not Kent, Sussex and what is not Sussex; and so on. This is so far, however, from being an absurd proceeding that it is requisite to assure us that we have made an exhaustive enumeration of the parts of England.

[574] George Bentham, *Outline of a New System of Logic*, p. 115.

*The Five Predicables.*

As a rule it is highly desirable to consign to oblivion the ancient logical names and expressions, which have infested the science for many centuries past. If logic is ever to be a useful and progressive science, logicians must distinguish between logic and the history of logic. As in the case of any other science it may be desirable to examine the course of thought by which logic has, before or since the time of Aristotle, been brought to its present state; the history of a science is always instructive as giving instances of the mode in which discoveries take place. But at the same time we ought carefully to disencumber the statement of the science itself of all names and other vestiges of antiquity which are not actually useful at the present day.

Among the ancient expressions which may well be excepted from such considerations and retained in use, are the “Five Words” or “Five Predicables” which were described by Porphyry in his introduction to Aristotle’s Organum. Two of them, *Genus* and *Species*, are the most venerable names in philosophy, having probably been first employed in their present logical meanings by Socrates. In the present day it requires some mental effort, as remarked by Grote, to see anything important in the invention of notions now so familiar as those of Genus and Species. But in reality the introduction of such terms showed the rise of the first germs of logic and scientific method; it showed that men were beginning to analyse their processes of thought.

The Five Predicables are Genus, Species, Difference, Property, and Accident, or in the original Greek, γένος, εἶδος, διαφορά, ἴδιον, συμβεβηκός. Of these, Genus may be taken to mean any class of objects which is regarded as broken up into two minor classes, which form Species of it. The genus is defined by a certain number of qualities or circumstances which belong to all objects included in the class, and which are sufficient to mark out these objects from all others which we do not intend to include. Interpreted as regards intension, then, the genus is a group of qualities; interpreted as regards extension, it is a group of objects possessing those qualities. If another quality be taken into account which is possessed by some of the objects and not by the others, this quality becomes a difference which divides the genus into two species. We may interpret the species either in intension or extension; in the former respect it is more than the genus as containing one more quality, the difference: in the latter respect it is less than the genus as containing only a portion of the group constituting the genus. We may say, then, with Aristotle, that in one sense the genus is in the species, namely in intension, and in another sense the species is in the genus, namely in extension. The difference, it is evident, can be interpreted in intension only.

A Property is a quality which belongs to the whole of a class, but does not enter into the definition of that class. A generic property belongs to every individual object contained in the genus. It is a property of the genus parallelogram that the opposite angles are equal. If we regard a rectangle as a species of parallelogram, the difference being that *one* angle is a right angle, it follows as a specific property that all the angles are right angles. Though a property in the strict logical sense must belong to each of the objects included in the class of which it is a property, it may or may not belong to other objects. The property of having the opposite angles equal may belong to many figures besides parallelograms, for instance, regular hexagons. It is a property of the circle that all triangles constructed upon the diameter with the apex upon the circumference are right-angled triangles, and *vice versâ*, all curves of which this is true must be circles. A property which thus belongs to the whole of a class and only to that class, corresponds to the ἴδιον of Aristotle and Porphyry; we might conveniently call it *a peculiar property*. Every such property enables us to make a statement in the form of a simple identity (p. 37). Thus we know it to be a peculiar property of the circle that for a given length of perimeter it encloses a greater area than any other possible curve; hence we may say--

Curve of equal curvature = curve of greatest area.

It is a peculiar property of equilateral triangles that they are equiangular, and *vice versâ*, it is a peculiar property of equiangular triangles that they are equilateral. It is a property of crystals of the regular system that they are devoid of the power of double refraction, but this is not a property peculiar to them, because liquids and gases are devoid of the same property.

An Accident, the fifth and last of the Predicables, is any quality which may or may not belong to certain objects, and which has no connexion with the classification adopted. The particular size of a crystal does not in the slightest degree affect the form of the crystal, nor does the manner in which it is grouped with other crystals; these, then, are accidents as regards a crystallographic classification. With respect to the chemical composition of a substance, again, it is an accident whether the substance be crystallised or not, or whether it be organised or not. As regards botanical classification the absolute size of a plant is an accident. Thus we see that a logical accident is any quality or circumstance which is not known to be correlated with those qualities or circumstances forming the definition of the species.

The meanings of the Predicables can be clearly explained by our symbols. Let A be any definite group of qualities and B another quality or group of qualities; then A will constitute a genus, and AB, A*b* will be species of it, B being the difference. Let C, D and E be other qualities or groups of qualities, and on examining the combinations in which A, B, C, D, E occur let them be as follows:--

ABCDE A*b*C*d*E ABCD*e* A*b*C*de*.

Here we see that wherever A is we also find C, so that C is a generic property; D occurs always with B, so that it constitutes a specific property, while E is indifferently present and absent, so as not to be related to any other letter; it represents, therefore, an accident. It will now be seen that the Logical Alphabet represents an interminable series of subordinate genera and species; it is but a concise symbolic statement of what was involved in the ancient doctrine of the Predicables.

*Summum Genus and Infima Species.*

As a genus means any class whatever which is regarded as composed of minor classes or species, it follows that the same class will be a genus in one point of view and a species in another. Metal is a genus as regards alkaline metal, a species as regards element, and any extensive system of classes consists of a series of subordinate, or as they are technically called, *subaltern* genera and species. The question, however, arises, whether such a chain of classes has a definite termination at either end. The doctrine of the old logicians was to the effect that it terminated upwards in a *genus generalissimum* or *summum genus*, which was not a species of any wider class. Some very general notion, such as substance, object, or thing, was supposed to be so comprehensive as to include all thinkable objects, and for all practical purposes this might be so. But as I have already explained (p. 74), we cannot really think of any object or class without thereby separating it from what is not that object or class. All thinking is relative, and implies discrimination, so that every class and every logical notion must have its negative. If so, there is no such thing as a *summum genus*; for we cannot frame the requisite notion of a class forming it without implying the existence of another class discriminated from it; add this new negative class to the supposed *summum genus*, and we form a still higher genus, which is absurd.

Although there is no absolute summum genus, nevertheless relatively to any branch of knowledge or any particular argument, there is always some class or notion which bounds our horizon as it were. The chemist restricts his view to material substances and the forces manifested in them; the mathematician extends his view so as to comprehend all notions capable of numerical discrimination. The biologist, on the other hand, has a narrower sphere containing only organised bodies, and of these the botanist and the zoologist take parts. In other subjects there may be a still narrower summum genus, as when the lawyer regards only reasoning beings of his own country together with their property.

In the description of the Logical Alphabet it was pointed out (p. 93) that every series of combinations is really the development of a single class, denoted by X, which letter was accordingly placed in the first column of the table on p. 94. This is the formal acknowledgment of the principle clearly stated by De Morgan, that all reasoning proceeds within an assumed summum genus. But at the same time the fact that X as a logical term must have its negative *x*, shows that it cannot be an absolute summum genus.

There arises, again, the question whether there be any such thing as an *infima species*, which cannot be divided into minor species. The ancient logicians were of opinion that there always was some assignable class which could only be divided into individuals, but this doctrine appears to be theoretically incorrect, as Mr. George Bentham long ago stated.[575] We may put an arbitrary limit to the subdivision of our classes at any point convenient to our purpose. The crystallographer would not generally treat as different species crystalline forms which differ only in the degree of development of the faces. The naturalist overlooks innumerable slight differences between animals which he refers to the same species. But in a strictly logical point of view classification might be carried on as long as there is a difference, however minute, between two objects, and we might thus go on until we arrive at individual objects which are numerically distinct in the logical sense attributed to that expression in the chapter upon Number. Either, then, we must call the individual the *infima species* or allow that there is no such thing at all.

[575] *Outline of a New System of Logic*, 1827, p. 117.

*The Tree of Porphyry.*

Both Aristotle and Plato were acquainted with the value of bifurcate classification, which they occasionally employed in an explicit manner. It is impossible too that Aristotle should state the laws of thought, and employ the predicables without implicitly recognising the logical necessity of that method. It is, however, in Porphyry’s remarkable and in many respects excellent *Introduction to the Categories of Aristotle* that we find the most distinct account of it. Porphyry not only fully and accurately describes the Predicables, but incidentally introduces an example for illustrating those predicables, which constitutes a good specimen of bifurcate classification. Translating his words[576] freely we may say that he takes Substance as the genus to be divided, under which are successively placed as Species--Body, Animated Body, Animal, Rational Animal, and Man. Under Man, again, come Socrates, Plato, and other particular men. Now of these notions Substance is the genus generalissimum, and is a genus only, not a species. Man, on the other hand, is the species specialissima (infima species), and is a species only, not a genus. Body is a species of substance, but a genus of animated body, which, again, is a species of body but a genus of animal. Animal is a species of animated body, but a genus of rational animal, which, again, is a species of animal, but a genus of man. Finally, man is a species of rational animal, but is a species merely and not a genus, being divisible only into particular men.

[576] *Porphyrii Isagoge*, Caput ii. 24.

Porphyry proceeds at some length to employ his example in further illustration of the predicables. We do not find in Porphyry’s own work any scheme or diagram exhibiting this curious specimen of classification, but some of the earlier commentators and epitome writers drew what has long been called the Tree of Porphyry. This diagram, which may be found in most elementary works on Logic,[577] is also called the Ramean Tree, because Ramus insisted much upon the value of Dichotomy. With the exception of Jeremy Bentham[578] and George Bentham, hardly any modern logicians have shown an appreciation of the value of bifurcate classification. The latter author has treated the subject, both in his *Outline of a New System of Logic* (pp. 105–118), and in his earlier work entitled *Essai sur la Nomenclature et la Classification des Principales Branches d’Art-et-Science* (Paris, 1823), which consists of a free translation or improved version of his uncle’s Essay on Classification in the *Chrestomathia*. Some interest attaches to the history of the Tree of Porphyry and Ramus, because it is the prototype of the Logical Alphabet which lies at the basis of logical method. Jeremy Bentham speaks truly of “the matchless beauty of the Ramean Tree.” After fully showing its logical value as an exhaustive method of classification, and refuting the objections of Reid and Kames, on a wrong ground, as I think, he proceeds to inquire to what length it may be carried. He correctly points out two objections to the extensive use of bifid arrangements, (1) that they soon become impracticably extensive and unwieldy, and (2) that they are uneconomical. In his day the recorded number of different species of plants was 40,000, and he leaves the reader to estimate the immense number of branches and the enormous area of a bifurcate table which should exhibit all these species in one scheme. He also points out the apparent loss of labour in making any large bifurcate classification; but this he considers to be fully recompensed by the logical value of the result, and the logical training acquired in its execution. Jeremy Bentham, then, fully recognises the value of the Logical Alphabet under another name, though he apprehends also the limit to its use placed by the finiteness of our mental and manual powers.

[577] Jevons, *Elementary Lessons in Logic*, p. 104.

[578] *Chrestomathia; being a Collection of Papers, &c.* London, 1816, Appendix V.

*Does Abstraction imply Generalisation?*

Before we can acquire a sound comprehension of the subject of classification we must answer the very difficult question whether logical abstraction does or does not imply generalisation. It comes to exactly the same thing if we ask whether a species may be coextensive with its genus, or whether, on the other hand, the genus must contain more than the species. To abstract logically is (p. 27), to overlook or withdraw our notice from some point of difference. Whenever we form a class we abstract, for the time being, the differences of the objects so united in respect of some common quality. If we class together a great number of objects as dwelling-houses, we overlook the fact that some dwelling-houses are constructed of stone, others of brick, wood, iron, &c. Often at least the abstraction of a circumstance increases the number of objects included under a class according to the law of the inverse relation of the quantities of extension and intension (p. 26). Dwelling-house is a wider term than brick-dwelling-house. House is more general than dwelling-house. But the question before us is, whether abstraction *always* increases the number of objects included in a class, which amounts to asking whether the law of the inverse relation of logical quantities is *always* true. The interest of the question partly arises from the fact, that so high a philosophical authority as Mr. Herbert Spencer has denied that generalisation is implied in abstraction,[579] making this doctrine the ground for rejecting previous methods of classifying the sciences, and for forming an ingenious but peculiar method of his own. The question is also a fundamental one of the highest logical importance, and involves subtle difficulties which have made me long hesitate in forming a decisive opinion.

[579] *The Classification of the Sciences*, &c., 3rd edit. p. 7. *Essays: Scientific, Political, and Speculative*, vol. iii. p. 13.

Let us attempt to answer the question by examination of a few examples. Compare the two classes *gun* and *iron gun*. It is certain that there are many guns which are not made of iron, so that abstraction of the circumstance “made of iron” increases the extent of the notion. Next compare *gun* and *metallic gun*. All guns made at the present day consist of metal, so that the two notions seem to be coextensive; but guns were at first made of pieces of wood bound together like a tub, and as the logical term gun takes no account of time, it must include all guns that have ever existed. Here again extension increases as intension decreases. Compare once more “steam-locomotive engine” and “locomotive engine.” In the present day, as far as I am aware, all locomotives are worked by steam, so that the omission of that qualification might seem not to widen the term; but it is quite possible that in some future age a different motive power may be used in locomotives; and as there is no limitation of time in the use of logical terms, we must certainly assume that there is a class of locomotives not worked by steam, as well as a class that is worked by steam. When the natural class of Euphorbiaceæ was originally formed, all the plants known to belong to it were devoid of corollas; it would have seemed therefore that the two classes “Euphorbiaceæ,” and “Euphorbiaceæ devoid of Corollas,” were of equal extent. Subsequently a number of plants plainly belonging to the same class were found in tropical countries, and they possessed bright coloured corollas. Naturalists believe with the utmost confidence that “Ruminants” and “Ruminants with cleft feet” are identical terms, because no ruminant has yet been discovered without cleft feet. But we can see no impossibility in the conjunction of rumination with uncleft feet, and it would be too great an assumption to say that we are certain that an example of it will never be met with. Instances can be quoted, without end, of objects being ultimately discovered combining properties which had never before been seen together. In the animal kingdom the Black Swan, the Ornithorhynchus Paradoxus, and more recently the singular fish called Ceratodus Forsteri, all discovered in Australia, have united characters never previously known to coexist. At the present time deep-sea dredging is bringing to light many animals of an unprecedented nature. Singular exceptional discoveries may certainly occur in other branches of science. When Davy first discovered metallic potassium, it was a well established empirical law that all metallic substances possessed a high specific gravity, the least dense of the metals then known being zinc, of which the specific gravity is 7·1. Yet to the surprise of chemists, potassium was found to be an undoubted metal of less density than water, its specific gravity being 0·865.

It is hardly requisite to prove by further examples that our knowledge of nature is incomplete, so that we cannot safely assume the non-existence of new combinations. Logically speaking, we ought to leave a place open for animals which ruminate but are without cleft feet, and for every possible intermediate form of animal, plant, or mineral. A purely logical classification must take account not only of what certainly does exist, but of what may in after ages be found to exist.

I will go a step further, and say that we must have places in our scientific classifications for purely imaginary existences. A large proportion of the mathematical functions which are conceivable have no application to the circumstances of this world. Physicists certainly do investigate the nature and consequences of forces which nowhere exist. Newton’s *Principia* is full of such investigations. In one chapter of his *Mécanique Céleste* Laplace indulges in a remarkable speculation as to what the laws of motion would have been if momentum, instead of varying simply as the velocity, had been a more complicated function of it. I have already mentioned (p. 223) that Airy contemplated the existence of a world in which the laws of force should be such that a perpetual motion would be possible, and the Law of Conservation of Energy would not hold true.

Thought is not bound down to the limits of what is materially existent, but is circumscribed only by those Fundamental Laws of Identity, Contradiction and Duality, which were laid down at the outset. This is the point at which I should differ from Mr. Spencer. He appears to suppose that a classification is complete if it has a place for every existing object, and this may perhaps seem to be practically sufficient; but it is subject to two profound objections. Firstly, we do not know all that exists, and therefore in limiting our classes we are erroneously omitting multitudes of objects of unknown form and nature which may exist either on this earth or in other parts of space. Secondly, as I have explained, the powers of thought are not limited by material existences, and we may, or, for some purposes, must imagine objects which probably do not exist, and if we imagine them we ought to find places for them in the classifications of science.

The chief difficulty of this subject, however, consists in the fact that mathematical or other certain laws may entirely forbid the existence of some combinations. The circle may be defined as a plane curve of equal curvature, and it is a property of the circle that it contains the greatest area within the least possible perimeter. May we then contemplate mentally a circle not a figure of greatest possible area? Or, to take a still simpler example, a parallelogram possesses the property of having the opposite angles equal. May we then mentally divide parallelograms into two classes according as they do or do not have their opposite angles equal? It might seem absurd to do so, because we know that one of the two species of parallelogram would be non-existent. But, then, unless the student had previously contemplated the existence of both species as possible, what is the meaning of the thirty-fourth proposition of Euclid’s first book? We cannot deny or disprove the existence of a certain combination without thereby in a certain way recognising that combination as an object of thought.

The conclusion at which I arrive is in opposition to that of Mr. Spencer. I think that whenever we abstract a quality or circumstance we do generalise or widen the notion from which we abstract. Whatever the terms A, B, and C may be, I hold that in strict logic AB is mentally a wider term than ABC, because AB includes the two species ABC and AB*c*. The term A is wider still, for it includes the four species ABC, AB*c*, A*b*C, A*bc*. The Logical Alphabet, in short, is the only limit of the classes of objects which we must contemplate in a purely logical point of view. Whatever notions be brought before us, we must mentally combine them in all the ways sanctioned by the laws of thought and exhibited in the Logical Alphabet, and it is a matter for after consideration to determine how many of these combinations exist in outward nature, or how many are actually forbidden by the conditions of space. A classification is essentially a mental, not a material thing.

*Discovery of Marks or Characteristics.*

Although the chief purpose of classification is to disclose the deepest and most general resemblances of the objects classified, yet the practical value of a system will depend partly upon the ease with which we can refer an object to its proper class, and thus infer concerning it all that is known generally of that class. This operation of discovering to which class of a system a certain specimen or case belongs, is generally called *Diagnosis*, a technical term familiarly used by physicians, who constantly require to diagnose or determine the nature of the disease from which a patient is suffering. Now every class is defined by certain specified qualities or circumstances, the whole of which are present in every object contained in the class, and *not all present* in any object excluded from it. These defining circumstances ought to consist of the deepest and most important circumstances, by which we vaguely mean those probably forming the conditions with which the minor circumstances are correlated. But it will often happen that the so-called important points of an object are not those which can most readily be observed. Thus the two great classes of phanerogamous plants are defined respectively by the possession of two cotyledons or seed-leaves, and one cotyledon. But when a plant comes to our notice and we want to refer it to the right class, it will often happen that we have no seed at all to examine, in order to discover whether there be one seed-leaf or two in the germ. Even if we have a seed it will often be small, and a careful dissection under the microscope will be requisite to ascertain the number of cotyledons. Occasionally the examination of the germ would mislead us, for the cotyledons may be obsolete, as in Cuscuta, or united together, as in Clintonia. Botanists therefore seldom actually refer to the seed for such information. Certain other characters of a plant are correlated with the number of seed-leaves; thus monocotyledonous plants almost always possess leaves with parallel veins like those of grass, while dicotyledonous plants have leaves with reticulated veins like those of an oak leaf. In monocotyledonous plants, too, the parts of the flower are most often three or some multiple of three in number, while in dicotyledonous plants the numbers four and five and their multiples prevail. Botanists, therefore, by a glance at the leaves and flowers can almost certainly refer a plant to its right class, and can infer not only the number of cotyledons which would be found in the seed or young plant, but also the structure of the stem and other general characters.

Any conspicuous and easily discriminated property which we thus select for the purpose of deciding to which class an object belongs, may be called a *characteristic*. The logical conditions of a good characteristic mark are very simple, namely, that it should be possessed by all objects entering into a certain class, and by none others. Every characteristic should enable us to assert a simple identity; if A is a characteristic, and B, viewed intensively, the class of objects of which it is the mark, then A = B ought to be true. The characteristic may consist either of a single quality or circumstance, or of a group of such, provided that they all be constant and easily detected. Thus in the classification of mammals the teeth are of the greatest assistance, not because a slight variation in the number and form of the teeth is of importance in the general 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,[580] and to infer from them not only the arrangement of the other teeth, but the whole conformation of the animal.

[580] Owen, *Essay on the Classification and Geographical Distribution of the Mammalia*, p. 20.

It is a 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 vegetable 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 investigator 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 sub-divided 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 kind[581] 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, 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 blow-pipe, an arrangement more or less evidently on the bifurcate plan, has been laid down by Von Kobell.[582] 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.

[581] Dana’s *Mineralogy*, vol. i. p. 123; quoted in Watts’ *Dictionary of Chemistry*, vol. ii. p. 166.

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

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*.[583] 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.

[583] Edition of 1866, p. lxiii.

Linnæus appears to have been perfectly acquainted with the nature and uses of diagnostic classification, which he describes under the name of Synopsis, saying:[584]--“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.”

[584] *Philosophia Botanica* (1770), § 154, p. 98.

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 when we have forgotten all others; but as we cannot usually go to the expense of forming more than two or three indices, we must select those circumstances which will lead to the discovery of a book most frequently. Many of the criteria mentioned are evidently inapplicable.

The language in which a book is written is definite enough, provided that the whole book is written in the same language; but it is obvious that language gives no means for the subdivision and arrangement of the literature of any one people. Classification by subjects would be an exceedingly useful method if it were practicable, but experience shows it to be a logical absurdity. It is a very difficult matter to classify the sciences, so complicated are the relations between them. But with books the complication is vastly greater, since the same book may treat of different sciences, or it may discuss a problem involving many branches of knowledge. A good account of the steam-engine will be antiquarian, so far as it traces out the earliest efforts at discovery; purely scientific, as regards the principles of thermodynamics involved; technical, as regards the mechanical means of applying those principles; economical, as regards the industrial results of the invention; biographical, as regards the lives of the inventors. A history of Westminster Abbey might belong either to the history of architecture, the history of the Church, or the history of England. If we abandon the attempt to carry out an arrangement according to the natural classification of the sciences, and form comprehensive practical groups, we shall be continually perplexed by the occurrence of intermediate cases, and opinions will differ *ad infinitum* as to the details. If, to avoid the difficulty about Westminster Abbey, we form a class of books devoted to the History of Buildings, the question will then arise whether Stonehenge is a building, and if so, whether cromlechs, mounds, and monoliths are so. We shall be uncertain whether to include lighthouses, monuments, bridges, &c. In regard to literary works, rigorous classification is still less possible. The same work may partake of the nature of poetry, biography, history, philosophy, or if we form a comprehensive class of Belles-lettres, nobody can say exactly what does or does not come under the term.

My own experience entirely bears out the opinion of De Morgan, that classification according to the name of the author is the only one practicable in a large library, and this method has been admirably carried out in the great catalogue of the British Museum. The name of the author is the most precise circumstance concerning a book, which usually dwells in the memory. It is a better characteristic of the book than anything else. In an alphabetical arrangement we have an exhaustive classification, including a place for every name. The following remarks[585] of De Morgan seem therefore to be entirely correct. “From much, almost daily use, of catalogues for many years, I am perfectly satisfied that a classed catalogue is more difficult to use than to make. It is one man’s theory of the subdivision of knowledge, and the chances are against its suiting any other man. Even if all doubtful works were entered under several different heads, the frontier of the dubious region would itself be a mere matter of doubt. I never turn from a classed catalogue to an alphabetical one without a feeling of relief and security. With the latter I can always, by taking proper pains, make a library yield its utmost; with the former I can never be satisfied that I have taken proper pains, until I have made it, in fact, as many different catalogues as there are different headings, with separate trouble for each. Those to whom bibliographical research is familiar, know that they have much more frequently to hunt an author than a subject: they know also that in searching for a subject, it is never safe to take another person’s view, however good, of the limits of that subject with reference to their own particular purposes.”

[585] *Philosophical Magazine*, 3rd Series (1845), vol. xxvi. p. 522. See also De Morgan’s evidence before the Royal Commission on the British Museum in 1849, Report (1850), Questions, 5704*-5815*, 6481–6513. This evidence should be studied by every person who wishes to understand the elements of Bibliography.

It is often desirable, however, that a name catalogue should be accompanied by a subordinate subject catalogue, but in this case no attempt should be made to devise a theoretically complete classification. Every principal subject treated in a book should be entered separately in an alphabetical list, under the name most likely to occur to the searcher, or under several names. This method was partially carried out in Watts’ *Bibliotheca Britannica*, but it was excellently applied in the admirable subject index to the *British Catalogue of Books*, and equally well in the *Catalogue of the Manchester Free Library* at Campfield, drawn up under the direction of Mr. Crestadoro, this latter being the most perfect model of a printed catalogue with which I am acquainted. The Catalogue of the London Library is also in the right form, and has a useful index of subjects, though it is too much condensed and abbreviated. The public catalogue of the British Museum is arranged as far as possible according to the alphabetical order of the authors’ names, but in writing the titles for this catalogue several copies are simultaneously produced by a manifold writer, so that a catalogue according to the order of the books on the shelves, and another according to the first words of the title-page, are created by a mere rearrangement of the spare copies. In the *English Cyclopædia* it is suggested that twenty copies of the book titles might readily have been utilised in forming additional catalogues, arranged according to the place of publication, the language of the book, the general nature of the subject, and so forth.[586] An excellent suggestion has also been made to the effect that each book when published should have a fly-leaf containing half a dozen printed copies of the title, drawn up in a form suitable for insertion in catalogues. Every owner of a library could then easily make accurate printed catalogues to suit his own purposes, by merely cutting out these titles and pasting them in books in any desirable order.

[586] *English Cyclopædia, Arts and Sciences*, vol. v. p. 233.

It will hardly be a digression to point out the enormous saving of labour, or, what comes to the same thing, the enormous increase in our available knowledge, both literary and scientific, which arises from the formation of extensive indices. The “State Papers,” containing the whole history of the nation, were practically sealed to literary inquirers until the Government undertook the task of calendaring and indexing them. The British Museum Catalogue is another national work, of which the importance in advancing knowledge cannot be overrated. The Royal Society is doing great service in publishing a complete catalogue of memoirs upon physical science. The time will perhaps come when our views upon this subject will be extended, and either Government or some public society will undertake the systematic cataloguing and indexing of masses of historical and scientific information which are now almost closed against inquiry.

*Classification in the Biological Sciences.*

The great generalisations established in the works of Herbert Spencer and Charles Darwin have thrown much light upon other sciences, and have removed several difficulties out of the way of the logician. The subject of classification has long been studied in almost exclusive reference to the arrangement of animals and plants. Systematic botany and zoology have been commonly known as the Classificatory Sciences, and scientific men seemed to suppose that the methods of arrangement, which were suitable for living creatures, must be the best for all other classes of objects. Several mineralogists, especially Mohs, have attempted to arrange minerals in genera and species, just as if they had been animals capable of reproducing their kind with variations. This confusion of ideas between the relationship of living forms and the logical relationship of things in general prevailed from the earliest times, as manifested in the etymology of words. We familiarly speak of a *kind* of things meaning a class of things, and the kind consists of those things which are *akin*, or come of the same race. When Socrates and his followers wanted a name for a class regarded in a philosophical light, they adopted the analogy in question, and called it a γένος, or race, the root γεν- being connected with the notion of generation.

So long as species of plants and animals were believed to proceed from distinct acts of Creation, there was no apparent reason why methods of classification suitable to them should not be treated as a guide to the classification of other objects generally. But when once we regard these resemblances as hereditary in their origin, we see that the sciences of systematic botany and zoology have a special character of their own. There is no reason to suppose that the same kind of natural classification which is best in biology will apply also in mineralogy, in chemistry, or in astronomy. The logical principles which underlie all classification are of course the same in natural history as in the sciences of lifeless matter, but the special resemblances which arise from the relation of parent and offspring will not be found to prevail between different kinds of crystals or mineral bodies.

The genealogical view of the relations of animals and plants leads us to discard all notions of a regular progression of living forms, or any theory as to their symmetrical relations. It was at one time a question whether the ultimate scheme of natural classification would lead to arrangement in a simple line, or a circle, or a combination of circles. Macleay’s once celebrated system was a circular one, and each class-circle was composed of five order-circles, each of which was composed again of five tribe-circles, and so on, the subdivision being at each step into five minor circles. Macleay held that in the animal kingdom there are five sub-kingdoms--the Vertebrata, Annulosa, Radiata, Acrita, and Mollusca. Each of these was again divided into five--the Vertebrata, consisting of Mammalia, Reptilia, Pisces, Amphibia, and Aves.[587] It is evident that in such a symmetrical system the animals were made to suit themselves to the classes instead of the classes being suited to the animals.

[587] Swainson, “Treatise on the Geography and Classification of Animals,” *Cabinet Cyclopædia*, p. 201.

We now perceive that the ultimate system will have the form of an immensely extended genealogical tree, which will be capable of representation by lines on a plane surface of sufficient extent. Strictly speaking, this genealogical tree ought to represent the descent of each individual living form now existing or which has existed. It should be as personal and minute in its detail of relations, as the Stemma of the Kings of England. We must not assume that any two forms are exactly alike, and in any case they are numerically distinct. Every parent then must be represented at the apex of a series of divergent lines, representing the generation of so many children. Any complete system of classification must regard individuals as the infimæ species. But as in the lower races of animals and plants the differences between individuals are slight and apparently unimportant, while the numbers of such individuals are immensely great, beyond all possibility of separate treatment, scientific men have always stopped at some convenient but arbitrary point, and have assumed that forms so closely resembling each other as to present no constant difference were all of one kind. They have, in short, fixed their attention entirely upon the main features of family difference. In the genealogical tree which they have been unconsciously aiming to construct, diverging lines meant races diverging in character, and the purpose of all efforts at so-called natural classification was to trace out the descents between existing groups of plants or animals.

Now it is evident that hereditary descent may have in different cases produced very different results as regards the problem of classification. In some cases the differentiation of characters may have been very frequent, and specimens of all the characters produced may have been transmitted to the present time. A living form will then have, as it were, an almost infinite number of cousins of various degrees, and there will be an immense number of forms finely graduated in their resemblances. Exact and distinct classification will then be almost impossible, and the wisest course will be not to attempt arbitrarily to distinguish forms closely related in nature, but to allow that there exist transitional forms of every degree, to mark out if possible the extreme limits of the family relationship, and perhaps to select the most generalised form, or that which presents the greatest number of close resemblances to others of the family, as the *type* of the whole.

Mr. Darwin, in his most interesting work upon Orchids, points out that the tribe of Malaxeæ are distinguished from Epidendreæ by the absence of a caudicle to the pollinia; but as some of the Malaxeæ have a minute caudicle, the division really breaks down in the most essential point. “This is a misfortune,” he remarks,[588] “which every naturalist encounters in attempting to classify a largely developed or so-called natural group, in which, relatively to other groups, there has been little extinction. In order that the naturalist may be enabled to give precise and clear definitions of his divisions, whole ranks of intermediate or gradational forms must have been utterly swept away: if here and there a member of the intermediate ranks has escaped annihilation, it puts an effectual bar to any absolutely distinct definition.”

[588] Darwin, *Fertilisation of Orchids*, p. 159.

In other cases a particular plant or animal may perhaps have transmitted its form from generation to generation almost unchanged, or, what comes to the same result, those forms which diverged in character from the parent stock may have proved unsuitable to their circumstances, and perished. We shall then find a particular form standing apart from all others, and marked by many distinct characters. Occasionally we may meet with specimens of a race which was formerly far more common but is now undergoing extinction, and is nearly the last of its kind. Thus we explain the occurrence of exceptional forms such as are found in the Amphioxus. The Equisetaceæ perplex botanists by their want of affinity to other orders of Acrogenous plants. This doubtless indicates that their genealogical connection with other plants must be sought for in the most distant ages of geological development.

Constancy of character, as Mr. Darwin has said,[589] is what is chiefly valued and sought after by naturalists; that is to say, naturalists wish to find some distinct family mark, or group of characters, by which they may clearly recognise the relationship of descent between a large group of living forms. It is accordingly a great relief to the mind of the naturalist when he comes upon a definitely marked group, such as the Diatomaceæ, which are clearly separated from their nearest neighbours the Desmidiaceæ by their siliceous framework and the absence of chlorophyll. But we must no longer think that because we fail in detecting constancy of character the fault is in our classificatory sciences. Where gradation of character really exists, we must devote ourselves to defining and registering the degrees and limits of that gradation. The ultimate natural arrangement will often be devoid of strong lines of demarcation.

[589] *Descent of Man*, vol. i. p. 214.

Let naturalists, too, form their systems of natural classification with all care they can, yet it will certainly happen from time to time that new and exceptional forms of animals or vegetables will be discovered and will require the modification of the system. A natural system is directed, as we have seen, to the discovery of empirical laws of correlation, but these laws being purely empirical will frequently be falsified by more extensive investigation. From time to time the notions of naturalists have been greatly widened, especially in the case of Australian animals and plants, by the discovery of unexpected combinations of organs, and such events must often happen in the future. If indeed the time shall come when all the forms of plants are discovered and accurately described, the science of Systematic Botany will then be placed in a new and more favourable position, as remarked by Alphonse Decandolle.[590]

[590] *Laws of Botanical Nomenclature*, p. 16.

It ought to be remembered that though the genealogical classification of plants or animals is doubtless the most instructive of all, it is not necessarily the best for all purposes. There may be correlations of properties important for medicinal, or other practical purposes, which do not correspond to the correlations of descent. We must regard the bamboo as a tree rather than a grass, although it is botanically a grass. For legal purposes we may continue with advantage to treat the whale, seal, and other cetaceæ, as fish. We must also class plants according as they belong to arctic, alpine, temperate, sub-tropical or tropical regions. There are causes of likeness apart from hereditary relationship, and *we must not attribute exclusive excellence to any one method of classification*.

*Classification by Types.*

Perplexed by the difficulties arising in natural history from the discovery of intermediate forms, naturalists have resorted to what they call classification by types. Instead of forming one distinct class defined by the invariable possession of certain assigned properties, and rigidly including or excluding objects according as they do or do not possess all these properties, naturalists select a typical specimen, and they group around it all other specimens which resemble this type more than any other selected type. “The type of each genus,” we are told,[591] “should be that species in which the characters of its group are best exhibited and most evenly balanced.” It would usually consist of those descendants of a form which had undergone little alteration, while other descendants had suffered slight differentiation in various directions.

[591] Waterhouse, quoted by Woodward in his *Rudimentary Treatise of Recent and Fossil Shells*, p. 61.

It would be a great mistake to suppose that this classification by types is a logically distinct method. It is either not a real method of classification at all, or it is merely an abbreviated mode of representing a complicated system of arrangement. A class must be defined by the invariable presence of certain common properties. If, then, we include an individual in which one of these properties does not appear, we either fall into logical contradiction, or else we form a new class with a new definition. Even a single exception constitutes a new class by itself, and by calling it an exception we merely imply that this new class closely resembles that from which it diverges in one or two points only. Thus in the definition of the natural order of Rosaceæ, we find that the seeds are one or two in each carpel, but that in the genus Spiræa there are three or four; this must mean either that the number of seeds is not a part of the fixed definition of the class, or else that Spiræa does not belong to that class, though it may closely approximate to it. Naturalists continually find themselves between two horns of a dilemma; if they restrict the number of marks specified in a definition so that every form intended to come within the class shall possess all those marks, it will then be usually found to include too many forms; if the definition be made more particular, the result is to produce so-called anomalous genera, which, while they are held to belong to the class, do not in all respects conform to its definition. The practice has hence arisen of allowing considerable latitude in the definition of natural orders. The family of Cruciferæ, for instance, forms an exceedingly well-marked natural order, and among its characters we find it specified that the fruit is a pod, divided into two cells by a thin partition, from which the valves generally separate at maturity; but we are also informed that, in a few genera, the pod is one-celled, or indehiscent, or separates transversely into several joints.[592] Now this must either mean that the formation of the pod is not an essential point in the definition of the family, or that there are several closely associated families.

[592] Bentham’s *Handbook of the British Flora* (1866), p. 25.

The same holds true of typical classification. The type itself is an individual, not a class, and no other object can be exactly like the type. But as soon as we abstract the individual peculiarities of the type and thus specify a finite number of qualities in which other objects may resemble the type, we immediately constitute a class. If some objects resemble the type in some points, and others in other points, then each definite collection of points of resemblance constitutes intensively a separate class. The very notion of classification by types is in fact erroneous in a logical point of view. The naturalist is constantly occupied in endeavouring to mark out definite groups of living forms, where the forms themselves do not in many cases admit of such rigorous lines of demarcation. A certain laxity of logical method is thus apt to creep in, the only remedy for which will be the frank recognition of the fact, that, according to the theory of hereditary descent, gradation of characters is probably the rule, and precise demarcation between groups the exception.

*Natural Genera and Species.*

One important result of the establishment of the theory of evolution is to explode all notions about natural groups constituting separate creations. Naturalists long held that every plant belongs to some species, marked out by invariable characters, which do not change by difference of soil, climate, cross-breeding, or other circumstances. They were unable to deny the existence of such things as sub-species, varieties, and hybrids, so that a species of plants was often subdivided and classified within itself. But then the differences upon which this sub-classification depended were supposed to be variable, and thus distinguished from the invariable characters imposed upon the whole species at its creation. Similarly a natural genus was a group of species, and was marked out from other genera by eternal differences of still greater importance.

We now, however, perceive that the existence of any such groups as genera and species is an arbitrary creation of the naturalist’s mind. All resemblances of plants are natural so far as they express hereditary affinities; but this applies as well to the variations within the species as to the species itself, or to the larger groups. All is a matter of degree. The deeper differences between plants have been produced by the differentiating action of circumstances during millions of years, so that it would naturally require millions of years to undo this result, and prove experimentally that the forms can be approximated again. Sub-species may sometimes have arisen within historical times, and varieties approaching to sub-species may often be produced by the horticulturist in a few years. Such varieties can easily be brought back to their original forms, or, if placed in the original circumstances, will themselves revert to those forms; but according to Darwin’s views all forms are capable of unlimited change, and it might possibly be, unlimited reversion if suitable circumstances and sufficient time be granted.

Many fruitless attempts have been made to establish a rigorous criterion of specific and generic difference, so that these classes might have a definite value and rank in all branches of biology. Linnæus adopted the view that the species was to be defined as a distinct creation, saying,[593] “Species tot numeramus, quot diversæ formæ in principio sunt creatæ;” or again, “Species tot sunt, quot diversas formas ab initio produxit Infinitum Ens; quæ formæ, secundum generationis inditas leges, produxere plures, at sibi semper similes.” Of genera he also says,[594] “Genus omne est naturale, in primordio tale creatum.” It was a common doctrine added to and essential to that of distinct creation that these species could not produce intermediate and variable forms, so that we find Linnæus obliged by the ascertained existence of hybrids to take a different view in another work; he says,[595] “Novas species immo et genera ex copula diversarum specierum in regno vegetabilium oriri primo intuitu paradoxum videtur; interim observationes sic fieri non ita dissuadent.” Even supposing in the present day that we could assent to the notion of a certain number of distinct creational acts, this notion would not help us in the theory of classification. Naturalists have never pointed out any method of deciding what are the results of distinct creations, and what are not. As Darwin says,[596] “the definition must not include an element which cannot possibly be ascertained, such as an act of creation.” It is, in fact, by investigation of forms and classification that we should ascertain what were distinct creations and what were not; this information would be a result and not a means of classification.

[593] *Philosophia Botanica* (1770), § 157, p. 99.

[594] *Ibid.* § 159, p. 100.

[595] *Amœnitates Academicæ* (1744), vol. i. p. 70. Quoted in *Edinburgh Review*, October 1868, vol. cxxviii. pp. 416, 417.

[596] *Descent of Man*, vol. i. p. 228.

Agassiz seemed to consider that he had discovered an important principle, to the effect that general plan or structure is the true ground for the discrimination of the great classes of animals, which may be called branches of the animal kingdom.[597] He also thought that genera are definite and natural groups. “Genera,” he says,[598] “are most closely allied groups of animals, differing neither in form, nor in complication of structure, but simply in the ultimate structural peculiarities of some of their parts; and this is, I believe, the best definition which can be given of genera.” But it is surely apparent that there are endless degrees both of structural peculiarity and of complication of structure. It is impossible to define the amount of structural peculiarity which constitutes the genus as distinguished from the species.

[597] Agassiz, *Essay on Classification*, p. 219.

[598] *Ibid.* p. 249.

The form which any classification of plants or animals tends to take is that of an unlimited series of subaltern classes. Originally botanists confined themselves for the most part to a small number of such classes. Linnæus adopted Class, Order, Genus, Species, and Variety, and even seemed to think that there was something essentially natural in a five-fold arrangement of groups.[599]

[599] *Philosophia Botanica*, § 155, p. 98.

With the progress of botany intermediate and additional groups have gradually been introduced. According to the Laws of Botanical Nomenclature adopted by the International Botanical Congress, held at Paris[600] in August 1867, no less than twenty-one names of classes are recognised--namely, Kingdom, Division, Sub-division, Class, Sub-class, Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe, Genus, Sub-genus, Section, Sub-section, Species, Sub-species, Variety, Sub-variety, Variation, Sub-variation. It is allowed by the authors of this scheme, that the rank or degree of importance to be attributed to any of these divisions may vary in a certain degree according to individual opinion. The only point on which botanists are not allowed discretion is as to the order of the successive sub-divisions; any inversion of the arrangement, such as division of a genus into tribes, or of a tribe into orders, is quite inadmissible. There is no reason to suppose that even the above list is complete and inextensible. The Botanical Congress itself recognised the distinction between variations according as they are Seedlings, Half-breeds, or *Lusus Naturæ*. The complication of the inferior classes is increased again by the existence of *hybrids*, arising from the fertilisation of one species by another deemed a distinct species, nor can we place any limit to the minuteness of discrimination of degrees of breeding short of an actual pedigree of individuals.

[600] *Laws of Botanical Nomenclature*, by Alphonse Decandolle, translated from the French, 1868, p. 19.

It will be evident to the reader that in the remarks upon classification as applied to the Natural Sciences, given in this and the preceding sections, I have not in the least attempted to treat the subject in a manner adequate to its extent and importance. A volume would be insufficient for tracing out the principles of scientific method specially applicable to these branches of science. What more I may be able to say upon the subject will be better said, if ever, when I am able to take up the closely-connected subjects of Scientific Nomenclature, Terminology, and Descriptive Representation. In the meantime, I have wished to show, in a negative point of view, that natural classification in the animal and vegetable kingdoms is a special problem, and that the particular methods and difficulties to which it gives rise are not those common to all cases of classification, as so many physicists have supposed. Genealogical resemblances are only a special case of resemblances in general.

*Unique or Exceptional Objects.*

In framing a system of classification in almost any branch of science, we must expect to meet with unique or peculiar objects, which stand alone, having comparatively few analogies with other objects. They may also be said to be *sui generis*, each unique object forming, as it were, a genus by itself; or they are called *nondescript*, because from thus standing apart it is difficult to find terms in which to describe their properties. The rings of Saturn, for instance, form a unique object among the celestial bodies. We have indeed considered this and many other instances of unique objects in the preceding chapter on Exceptional Phenomena. Apparent, Singular, and Divergent Exceptions especially, are analogous to unique objects.

In the classification of the elements, Carbon stands apart as a substance entirely unique in its powers of producing compounds. It is considered to be a quadrivalent element, and it obeys all the ordinary laws of chemical combination. Yet it manifests powers of affinity in such an exalted degree that the substances in which it appears are more numerous than all the other compounds known to chemists. Almost the whole of the substances which have been called organic contain carbon, and are probably held together by the carbon atoms, so that many chemists are now inclined to abandon the name Organic Chemistry, and substitute the name Chemistry of the Carbon Compounds. It used to be believed that the production of organic compounds could be effected only by the action of vital force, or of some inexplicable cause involved in the phenomena of life; but it is now found that chemists are able to commence with the elementary materials, pure carbon, hydrogen, and oxygen, and by strictly chemical operations to combine these so as to form complicated organic compounds. So many substances have already been formed that we might be inclined to generalise and infer that all organic compounds might ultimately be produced without the agency of living beings. Thus the distinction between the organic and the inorganic kingdoms seems to be breaking down, but our wonder at the peculiar powers of carbon must increase at the same time.

In considering generalisation, the law of continuity was applied chiefly to physical properties capable of mathematical treatment. But in the classificatory sciences, also, the same important principle is often beautifully exemplified. Many objects or events seem to be entirely exceptional and abnormal, and in regard to degree or magnitude they may be so termed; but it is often easy to show that they are connected by intermediate links with ordinary cases. In the organic kingdoms there is a common groundwork of similarity running through all classes, but particular actions and processes present themselves conspicuously in particular families and classes. Tenacity of life is most marked in the Rotifera, and some other kinds of microscopic organisms, which can be dried and boiled without loss of life. Reptiles are distinguished by torpidity, and the length of time they can live without food. Birds, on the contrary, exhibit ceaseless activity and high muscular power. The ant is as conspicuous for intelligence and size of brain among insects as the quadrumana and man among vertebrata. Among plants the Leguminosæ are distinguished by a tendency to sleep, folding their leaves at the approach of night. In the genus Mimosa, especially the Mimosa pudica, commonly called the sensitive plant, the same tendency is magnified into an extreme irritability, almost resembling voluntary motion. More or less of the same irritability probably belongs to vegetable forms of every kind, but it is of course to be investigated with special ease in such an extreme case. In the Gymnotus and Torpedo, we find that organic structures can act like galvanic batteries. Are we to suppose that such animals are entirely anomalous exceptions; or may we not justly expect to find less intense manifestations of electric action in all animals?

Some extraordinary differences between the modes of reproduction of animals have been shown to be far less than was at first sight apparent. The lower animals seem to differ entirely from the higher ones in the power of reproducing lost limbs. A kind of crab has the habit of casting portions of its claws when much frightened, but they soon grow again. There are multitudes of smaller animals which, like the Hydra, may be cut in two and yet live and develop into new complete individuals. No mammalian animal can reproduce a limb, and in appearance there is no analogy. But it was suggested by Blumenbach that the healing of a wound in the higher animals really represents in a lower degree the power of reproducing a limb. That this is true may be shown by adducing a multitude of intermediate cases, each adjoining pair of which are clearly analogous, so that we pass gradually from one extreme to the other. Darwin holds, moreover, that any such restoration of parts is closely connected with that perpetual replacement of the particles which causes every organised body to be after a time entirely new as regards its constituent substance. In short, we approach to a great generalisation under which all the phenomena of growth, restoration, and maintenance of organs are effects of one and the same power.[601] It is perhaps still more surprising to find that the complicated process of reproduction in the higher animals may be gradually traced down to a simpler and simpler form, which at last becomes undistinguishable from the budding out of one plant from the stem of another. By a great generalisation we may regard all the modes of reproduction of organic life as alike in their nature, and varying only in complexity of development.[602]

[601] Darwin, *The Variation of Animals and Plants*, vol. ii. pp. 293, 359, &c.; quoting Paget, *Lectures on Pathology*, 1853, pp. 152, 164.

[602] *Ibid.* vol. ii. p. 372.

*Limits of Classification.*

Science can extend only so far as the power of accurate classification extends. If we cannot detect resemblances, and assign their exact character and amount, we cannot have that generalised knowledge which constitutes science; we cannot infer from case to case. Classification is the opposite process to discrimination. If we feel that two tastes differ, the tastes of two kinds of wine for instance, the mere fact of difference existing prevents inference. The detection of the difference saves us, indeed, from false inference, because so far as difference exists, inference is impossible. But classification consists in detecting resemblances of all degrees of generality, and ascertaining exactly how far such resemblances extend, while assigning precisely the points at which difference begins. It enables us, then, to generalise, and make inferences where it is possible, and it saves us at the same time from going too far. A full classification constitutes a complete record of all our knowledge of the objects or events classified, and the limits of exact knowledge are identical with the limits of classification.

It must by no means be supposed that every group of natural objects will be found capable of rigorous classification. There may be substances which vary by insensible degrees, consisting, for instance, in varying mixtures of simpler substances. Granite is a mixture of quartz, felspar, and mica, but there are hardly two specimens in which the proportions of these three constituents are alike, and it would be impossible to lay down definitions of distinct species of granite without finding an infinite variety of intermediate species. The only true classification of granites, then, would be founded on the proportions of the constituents present, and a chemical or microscopic analysis would be requisite, in order that we might assign a specimen to its true position in the series. Granites vary, again, by insensible degrees, as regards the magnitude of the crystals of felspar and mica. Precisely similar remarks might be made concerning the classification of other plutonic rocks, such as syenite, basalt, pumice-stone, lava.

The nature of a ray of homogeneous light is strictly defined, either by its place in the spectrum or by the corresponding wave-length, but a ray of mixed light admits of no simple classification; any of the infinitely numerous rays of the continuous spectrum may be present or absent, or present in various intensities, so that we can only class and define a mixed colour by defining the intensity and wave-length of each ray of homogeneous light which is present in it. Complete spectroscopic analysis and the determination of the intensity of every part of the spectrum yielded by a mixed ray is requisite for its accurate classification. Nearly the same may be said of complex sounds. A simple sound undulation, if we could meet with such a sound, would admit of precise and exhaustive classification as regards pitch, the length of wave, or the number of waves reaching the ear per second being a sufficient criterion. But almost all ordinary sounds, even those of musical instruments, consist of complex aggregates of undulations of different pitches, and in order to classify the sound we should have to measure the intensities of each of the constituent sounds, a work which has been partially accomplished by Helmholtz, as regards the vowel sounds. The different tones of voice distinctive of different individuals must also be due to the intermixture of minute waves of various pitch, which are yet quite beyond the range of experimental investigation. We cannot, then, at present attempt to classify the different kinds or *timbres* of sound.

The difficulties of classification are still greater when a varying phenomenon cannot be shown to be a mixture of simpler phenomena. If we attempt to classify tastes, we may rudely group them according as they are sweet, bitter, saline, alkaline, acid, astringent or fiery; but it is evident that these groups are bounded by no sharp lines of definition. Tastes of mixed or intermediate character may exist almost *ad infinitum*, and what is still more troublesome, the tastes clearly united within one class may differ more or less from each other, without our being able to arrange them in subordinate genera and species. The same remarks may be made concerning the classification of odours, which may be roughly grouped according to the arrangement of Linnæus as, aromatic, fragrant, ambrosiac, alliaceous, fetid, virulent, nauseous. Within each of these vague classes, however, there would be infinite shades of variety, and each class would graduate into other classes. The odours which can be discriminated by an acute nose are infinite; every rock, stone, plant, or animal has some slight smell, and it is well known that dogs, or even blind men, can discriminate persons by a slight distinctive odour which usually passes unnoticed.

Similar remarks may be made concerning the feelings of the human mind, called emotions. We know what is anger, grief, fear, hatred, love; and many systems for classifying these feelings have been proposed. They may be roughly distinguished according as they are pleasurable or painful, prospective or retrospective, selfish or sympathetic, active or passive, and possibly in many other ways; but each mode of arrangement will be indefinite and unsatisfactory when followed into details. As a general rule, the emotional state of the mind at any moment will be neither pure anger nor pure fear, nor any one pure feeling, but an indefinite and complex aggregate of feelings. It may be that the state of mind is really a sum of several distinct modes of agitation, just as a mixed colour is the sum of the several rays of the spectrum. In this case there may be more hope of some method of analysis being successfully applied at a future time. But it may be found that states of mind really graduate into each other so that rigorous classification would be hopeless.

A little reflection will show that there are whole worlds of existences which in like manner are incapable of logical analysis and classification. One friend may be able to single out and identify another friend by his countenance among a million other countenances. Faces are capable of infinite discrimination, but who shall classify and define them, or say by what particular shades of feature he does judge? There are of course certain distinct types of face, but each type is connected with each other type by infinite intermediate specimens. We may classify melodies according to the major or minor key, the character of the time, and some other distinct points; but every melody has, independently of such circumstances, its own distinctive character and effect upon the mind. We can detect differences between the styles of literary, musical, or artistic compositions. We can even in some cases assign a picture to its painter, or a symphony to its composer, by a subtle feeling of resemblances or differences which may be felt, but cannot be described.

Finally, it is apparent that in human character there is unfathomable and inexhaustible diversity. Every mind is more or less like every other mind; there is always a basis of similarity, but there is a superstructure of feelings, impulses, and motives which is distinctive for each person. We can sometimes predict the general character of the feelings and actions which will be produced by a given external event in an individual well known to us; but we also know that we are often inexplicably at fault in our inferences. No one can safely generalise upon the subtle variations of temper and emotion which may arise even in a person of ordinary character. As human knowledge and civilisation progress, these characteristic differences tend to develop and multiply themselves, rather than decrease. Character grows more many-sided. Two well educated Englishmen are far better distinguished from each other than two common labourers, and these are better distinguished than two Australian aborigines. The complexities of existing phenomena probably develop themselves more rapidly than scientific method can overtake them. In spite of all the boasted powers of science, we cannot really apply scientific method to our own minds and characters, which are more important to us than all the stars and nebulæ.

BOOK VI.