The principles of science

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

But in reality the combinations are implied to be

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:‍—

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—

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:‍—

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.

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

Classification in the Biological Sciences.

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

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

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

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

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

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

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

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

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

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

Classification by Types.

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

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

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

Natural Genera and Species.

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

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

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

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

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

With the progress of botany intermediate and additional groups have gradually been introduced. According to the Laws of Botanical Nomenclature adopted by the International Botanical Congress, held at Paris‍600 in August 1867, no less than twenty-one names of classes are recognised—namely, Kingdom, Division, Sub-division, Class, Sub-class, Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe, Genus, Sub-genus, Section, Sub-section, Species, Sub-species, Variety, Sub-variety, Variation, Sub-variation. It is allowed by the authors of this scheme, that the rank or degree of importance to be attributed to any of these divisions may vary in a certain degree according to individual opinion. The only point on which botanists are not allowed discretion is as to the order of the successive sub-divisions; any inversion of the arrangement, such as division of a genus into tribes, or of a tribe into orders, is quite inadmissible. There is no reason to suppose that even the above list is complete and inextensible. The Botanical Congress itself recognised the distinction between variations according as they are Seedlings, Half-breeds, or Lusus Naturæ. The complication of the inferior classes is increased again by the existence of hybrids, arising from the fertilisation of one species by another deemed a distinct species, nor can we place any limit to the minuteness of discrimination of degrees of breeding short of an actual pedigree of individuals.