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.”

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.

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.”

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.”

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 remarks‍564 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:

“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

“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:‍—

Name of Planet. Period of Axial 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.

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

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 Faraday‍571 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.

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, AbC, aBc, abc. The order of arrangement will now be of importance; for if we place them in the order

ABC AbC

aBc 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 aBc

AbC 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:‍—

ABCdE aBCdE

ABcde aBcde

AbCDE abCDE

AbcDe abcDe

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:‍—

ABCdE AbCDE

ABcde AbcDe

aBCdE abCDE

aBcde abcDe

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:‍—

ABCdE ABcde

aBCdE aBcde

AbCDE AbcDe

abCDE abcDe,

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 = Ac. Our classification is then identical with the following bifurcate one:‍—

A

AB Ab

ABC = 0 ABc AbC Abc = 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 ABcd, AbCd, and AbcD.

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 mammæ = Mammifera. deprived of mammæ

with wings = Birds. without wings = 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 ABcD ABcd Ab.

But in reality the combinations are implied to be

ABCd = Mammifera, ABcD = Birds, ABcd = Reptiles, Abcd = Fish,

and we imply at the same time that the other four conceivable combinations containing B, C, or D, namely ABCD, AbCD, AbCd, and AbcD, do not exist in nature.

Mr. Bentham points out‍573 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.

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 arrangement‍574 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.

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, Ab 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 AbCdE

ABCDe AbCde.

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.

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 words‍576 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.

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 Bentham‍578 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.

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.

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 ABc. The term A is wider still, for it includes the four species ABC, ABc, AbC, Abc. 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.

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 kind‍581 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.

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.

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.”

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 remarks‍585 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.”

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.